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Semi.gurroa.czObservational evidence favors a static universe David F. CrawfordSydney Institute for Astronomy,School of Physics, University of Sydney.
Correspondence: 44 Market St, Naremburn, 2065,NSW, Australiaemail: email@example.com The common attribute of all Big Bang cosmologies is that they are based on the assumption that the universe is expanding. However exam-ination of the evidence for this expansion clearly favors a static universe.
The major topics considered are: Tolman surface brightness, angular size,type 1a supernovae, gamma ray bursts, galaxy distributions, quasar dis-tributions, X-ray background radiation, cosmic microwave background ra-diation, radio source counts, quasar variability and the Butcher–Oemlereffect. An analysis of the best raw data for these topics shows that theyare consistent with expansion only if there is evolution that cancels theeffects of expansion. An alternate cosmology, curvature cosmology, is atired-light cosmology that predicts a well defined static and stable uni-verse and is fully described. It not only predicts accurate values for theHubble constant and the temperature of cosmic microwave backgroundradiation but shows good agreement with most of the topics considered.
Curvature cosmology also predicts the deficiency in solar neutrino pro-duction rate and can explain the anomalous acceleration of Pioneer 10.
Note:The editor of the Journal of Cosmology required this paper to be dividedinto three parts. Except for the abstracts, introductions and conclusionsthe text of the three papers is identical to this version. For convenience atable of contents is included in this version. The correspondence betweenthe sections in this version and those in the Journal of Cosmology are: cosmology: observations, large-scale structure of universe, theory arXiv:1009.0953v4 [physics.gen-ph] 13 Apr 2011 Cosmographic Parameters Big Bang cosmology (BB) . . . . . . . . . . .
Curvature cosmology (CC) Numerical comparison of BB and CC . . . . . . . .
Expansion and Evolution Tolman surface brightness . . . . . . . . . . .
Surface brightness in BB . . . . . . . . . .
Surface brightness in CC Conclusion for surface brightness . . . . . . .
Type 1a supernovae Supernova conclusion Galaxy distribution . . . . . . . . . . . . .
Quasar distribution . . . . . . . . . . . . .
Quasar variability in time . . . . . . . . . . .
The Butcher–Oemler effect X-ray background radiation . . . . . . . . . . .
Cosmic microwave background radiation . . . . . . .
CMBR conclusions . . . . . . . . . . .
CMBR at large redshifts . . . . . . . . . . . .
Fluctuations in the CMBR The Sunyaev–Zel'dovich effect . . . . . . . . . .
Gravitational lensing . . . . . . . . . . . . .
5.10 The Gunn–Peterson trough in high redshift quasars . . . .
5.11 Nuclear abundances 5.12 Galactic rotation curves . . . . . . . . . . . .
5.13 Redshifts in our Galaxy . . . . . . . . . . . .
5.14 Anomalous redshifts . . . . . . . . . . . . .
Curvature Cosmology Theory Derivation of curvature redshift . . . . . . . . . .
Photons in Curved Spacetime . . . . . . . .
Curvature redshift secondary photons . . . . . .
Inhibition of curvature redshift . . . . . . . .
Possible laboratory tests . . . . . . . . . .
Interactions for other particles Derivation of curvature pressure Gravitation is not a force . . . . . . . . .
A Newtonian model . . . . . . . . . . .
General relativistic model . . . . . . . . .
Local curvature pressure . . . . . . . . . .
The curvature cosmological model The Friedmann equations . . . . . . . . .
Temperature of the cosmic plasma . . . . . . .
Hubble constant: theory . . . . . . . . . .
Luminosities and magnitudes . . . . . . . .
Application of Curvature Cosmology Large number coincidences Solar neutrino production . . . . . . . . . . .
Heating of the solar corona . . . . . . . . . . .
Pioneer 10 acceleration . . . . . . . . . . . .
A Analytic methods The common attribute of all Big Bang cosmologies (BB) is that they are basedon the assumption that the universe is expanding An early al-ternative was the steady-state theory of Hoyle, Bondi and Gold (described withlater extensions by that required continuous creation ofmatter. However steady-state theories have serious difficulties in explaining thecosmic microwave background radiation. This left BB as the dominant cosmol-ogy but still subject to criticism. Recently and havecontinued major earlier criticisms of Big Bang cosmologies Whereas most of theses criticismshave been of a theoretical nature this paper concentrates on whether observa-tional data supports BB or a static cosmological model, curvature cosmology(CC), described below.
Expansion produces two distinct effects. The first effect of expansion is the increasing redshift with distance as described by Hubble's law. This could bedue to either a genuine expansion or resulting from a tired-light phenomenon.
The second effect of expansion is time dilation resulting from the slowing downof the arrival times of the photons as the source gets further away. Section concentrates on the evidence for expansion as shown by time dilation and showsthat the evidence is only consistent with BB if there is evolution in either lu-minosity or in angular size that closely cancels the effects of time dilation. Toillustrate that a static cosmology can explain the data, a particular model, cur-vature cosmology (CC), is used. Curvature cosmology is based on the hypothesisof curvature redshift and the hypothesis of curvature pressure. Curvature red-shift arises from the principle that any localized wave travelling in curved spacetime will follow geodesics and be subject to geodesics focussing. Since this willalter the transverse properties of the wave some of its properties such as angu-lar momentum will be altered which is contrary to quantum mechanics. For aphoton the result is an interaction that results in three new photons. One withalmost identical energy and momentum as the original and two extremely lowenergy secondary photons. In effect the photon loses energy via an interactionwith curved space-time. The concept of curvature pressure arises from the ideathat the density of particles produce curved space-time. Then as a function oftheir velocity there will be a reaction pressure that acts to decrease the localspace-time curvature.
It is the evaluation of evidence for this expansion that forms the major ba- sis of this paper. Consequently minor differences between different expansioncosmologies are not particularly important here; it is the broad brush approachthat is relevant. Nevertheless to provide appropriate numerical quantities theevaluation is based on a particular BB cosmology, the (ΛCDM) model, whichis defined in Section by the equations for angular size, volume and distancemodulus. Since the major difference between BB and CC arises from the expan-sion in BB, the major results of the comparison are applicable to other staticcosmologies and do not depend on the validity of CC. A problem in evaluatinga well established cosmology like Big Bang cosmology is that all of the obser-vations have been analyzed within the BB paradigm. Thus there can be subtleeffects that may lead to a possible bias. In order to avoid this bias and whereverpossible comparisons are made using original observations.
Section discusses the theoretical justification for the basic and additional hypotheses that have been incorporated into the cosmologies. For BB theseincludes inflation, dark matter and dark energy. For CC the hypotheses arecurvature redshift and curvature pressure.
The evaluation of BB and its comparison with CC using observational evi- dence is divided into four major parts. The first in Section concentrates onthose observations that have measurements that are strongly dependent on ex-pansion. It is found that in all cases where there is direct evidence for evolutionthis evolution is close to what is required to cancel the expansion term in theBB equations. This is an extraordinary coincidence. The simplest conclusion isthat the universe is not expanding. Section looks at observations that havedifferent explanations in CC from what they have in BB. It is found that notonly does CC gave better agreement with most of the observations but it doesso without requiring extra ad hoc parameters or hypotheses. The next sectionis a description of CC and possible experimental tests of its validity. FinallySection examines additional important topics that are relevant to CC.
The test for Tolman surface brightness in Section is through the expected variation of apparent surface brightness with redshift. The results strongly favor a static universe but could be consistent with BB provided there is luminosityevolution.
The relationship between angular size and linear size in BB includes a aber- ration factor of (1 + z) that does not occur in static cosmologies. However theavailable data does not support the inclusion of this factor and is more consistentwith a static universe.
Next it is argued that the apparent time dilation of the supernova light curves is not due to expansion. The analysis is complex and is based on thepremise that the most constant characteristic of the supernova explosion is itstotal energy and not its peak magnitude. If this is correct, then selection effectscan account for the apparent time dilation. The CC analysis is in completeagreement with the known correlation between peak luminosity and light curveduration. Furthermore the analysis overcomes a serious problem with the cur-rent redshift distribution of supernovae. Finally using CC the distribution ofthe total energy for each supernova as a function of (1 + z) has an exponent of0.047 ± 0.089. This shows that there is no redshift dependence that occurs inthe BB analysis. Thus there is no need for dark energy.
The raw data of various time measures taken from the light curves for gamma ray bursts (GRB) show no evidence of the time dilation that is expected inBB. Since it can be argued that evolutionary and other effects that may havecancelled the expected time dilation in BB are unlikely a reasonable conclusionis that there is no time dilation in GRB.
It is shown for galaxies with types E–Sa that have a well a defined peak in their luminosity distribution the magnitude of this peak is independent ofredshift when the analysis was done using a static cosmology.
Analysis of quasar distributions in BB shows that luminosity evolution is required to explain the observations. A novel method is used to analyst thequasar distribution. Because the quasar distribution is close to an exponentialdistribution in absolute magnitude (power law in luminosity) then for a smallredshift range it is also an exponential distribution in apparent magnitude.
Then for a small redshift range it is possible to use statistical averages to get anestimate of the distance modulus directly from the raw data. The only inputrequired from the cosmological model is the variation of volume with redshift.
The results are shown in Fig. They show much better agreement with CCthan with BB.
An analysis of the distribution of radio sources is included in this section not because of explicit evidence of time dilation but because it has been generallyaccepted that the distribution can only be explained by having strong evolution.
It is shown that a distribution of radio sources in a static universe can have agood fit to the observations.
has been monitoring quasar variability using a Fourier method since about 1975 and finds no variation in their time scales withredshift. Although it is generally accepted that the variations are intrinsic tothe quasar there is a possibility that they may be due to micro-lensing whichcould place their origin to modulation effects in our own galaxy.
The Butcher–Oemler effect of the increasing proportion of blue galaxies in clusters at higher redshift is shown to be non-existent or at least greatly exag-gerated.
Apart from its lack of expansion, curvature cosmology makes further specific predictions that can be compared with BB. These are considered in Section Whereas the conclusions about expansion and evolution in Section would beapplicable for any reasonable static cosmology this section considers topics thatare specific to CC and BB.
The topic of X-ray background radiation is very important for CC. Not only can CC explain the radiation from 10–300 keV but the results enable estimatesfor the temperature and density of the cosmic gas (the gas external to clustersof galaxies). For the measured density of N = 1.55 ± 0.01 m−3 the calculatedvalue of the Hubble constant is H = 64.4 ± 0.2 km s−1 Mpc−1 (c.f. equation whereas the value estimated from the type 1a supernova data (Section is63.8 ± 0.5 kms−1 Mpc−1 and the result from the Coma cluster (Section is 65.7 kms−1 Mpc−1. In CC the theoretical temperature for the cosmic gasis 2.56 × 109 K and the temperature estimated from fitting the X-ray data is(2.62 ± 0.04) × 109 K.
In CC the CMBR is produced by very high energy electrons via curvature- redshift radiation in the cosmic plasma.
The predicted temperature of the CMBR is 3.18 K to be compared with an observed value of 2.725 K The prediction does depend on the nuclei mix in the cosmic gas andcould vary from this value by several tenths of a degree. It is argued that in CCthe apparent larger CMBR temperature at large redshifts could be explainedby the effects of curvature redshift on the width of spectral lines. Evidence forcorrelations between CMBR intensity and galaxy density is consistent with CC.
Regarding dark matter not only does CC have a quite different explanation for the velocity dispersion in clusters of galaxies but it can make a good estimate,without any free parameters, of its value for the Coma cluster. In BB it isassumed that the redshift dispersion is a genuine velocity dispersion and themass of a cluster of galaxies is determined by using the virial theorem. In CCthe redshift dispersion is due to curvature redshift produced by the intra-clustergas.
The Sunyaev–Zel'dovich effect, gravitational lens, the Lyman-α forest and the Gunn–Peterson trough can be explained by or are fully consistent with CC.
BB offers a good explanation for the primordial abundances of the light nuclei,albeit with some uncertainty of the density of the early universe at their timeof formation. In CC the distribution of light elements is determined by nuclearreactions in the very high temperature cosmic gas. This explanation needs aquantitative analysis.
Galactic rotation curves are a problem for both cosmologies. BB requires an extensive halo of dark matter around the galaxy while CC requires a reasonablehalo of normal matter to produce the apparent rotation via curvature redshift.
Its problem is getting the required asymmetry in the halo distribution.
Anomalous redshifts are the controversial association of high redshift quasars with much lower redshift galaxies. Although they are inexplicable in BB, CCcould offer a partial explanation for some observations.
Finally voids and other large scale structures in the redshift distribution of quasars and galaxies is easily explained in CC by the extra redshift due tocurvature redshift in higher density gas clouds. In BB it is a complicated resultof the evolution of these objects.
Section provides a complete description of CC and its two major hypothe- ses: curvature redshift and curvature pressure. Although it is not a new idea itis argued that gravitation is an acceleration and not a force. This idea is usedto justify the averaging of accelerations rather than forces in deriving curvature COSMOGRAPHIC PARAMETERS The next Section includes the topics of entropy, Olber's paradox, black holes, astrophysical jets and large-number coincidences that are particularlyrelevant for CC but are not important for choosing between the two cosmologies.
Although the explanation for the deficiency in observed neutrinos from the sun can be explained by neutrino oscillations it is include here because curvaturepressure makes excellent estimates of the expected numbers without any freeparameters. The heating of the solar corona is a very old problem and still notfully explained. It is treated here simply to show that curvature redshift offersno help.
Finally it is shown the Pioneer 10 anomalous acceleration can be explained by the effects of curvature redshift that is produced by interplanetary dustprovided the density of the dust is a little higher than current estimates.
Except for cross-references the sub-sections on observational topics are self contained and can be read independently. Many of the topics use statisticalestimation methods and in particular linear regression. A brief summary of thegeneral linear regression and the treatment of uncertainties is provided in theappendix.
Cosmographic Parameters Just like the Doppler shift the cosmological redshift z is independent of thewavelength of the spectral line. In terms of wavelength, the redshift z is z =(λ0/λ−1) where λ0 is the observed wavelength and λ is the emitted wavelength.
In terms of frequency, ν, and photon energy, E, the redshift is z = ν/ν0 − 1 =E/E0 − 1. The basic cosmological equations needed to analyst observationsprovide the conversion from apparent magnitude to absolute magnitude, therelationship between actual lateral measurement and angle and the volume asa function of redshift.
The conversion from apparent magnitude, m, to absolute magnitude, M , is given by the equation M = m − µ(h) − Kz(λ0), where µ(h) is the distance modulus that strongly depends on the assumed cos-mology and Kz(λ0) is the K-correction that allows for the difference in thespectrum between the emitted wavelength and the observed wavelength and is independent of the assumed cosmol-ogy. For a small bandwidths and luminosity L(λ) it is Kz(λ0) = −2.5 log Note that the bandwidth ratio is included in the definition of the K-correction.
The Hubble constant H0 is the constant of proportionality between the appar-ent recession speed v and distance d. That is v = Hd. It is usually writtenH0 = 100h km s−1 Mpc−1 where h is a dimensionless number. Unless otherwisespecified it is assumed to have the value h = 0.7.
COSMOGRAPHIC PARAMETERS Big Bang cosmology (BB) The fundamental premise of Big Bang cosmology is that the universe is expand-ing with a scale factor proportional to (1 + z). A more detailed account canbe found in The analysis is simplified by usingcomoving coordinates that describe the non-Euclidean geometry without expan-sion. Note that in BB the Hubble constant is a function of redshift hence theuse of a zero subscript to denote the current value. A problem with BB is thatit is only the distances between large objects that are subject to the expansion.
It is generally accepted that any objects smaller than clusters of galaxies whichare gravitationally bound do not follow the Hubble flow.
The current version of Big-Bang cosmology is the cold dark matter (ΛCDM) or concordance cosmology that has a complex expression for its parameters thatdepends on the cosmological energy density ΩΛ. Regardless of the name, theBB model used here is defined by the following equations. Following (with corrections from the function f (x) is defined by (p((1 + z)3 − 1)ΩM + 1 where ΩM is the cosmological energy-density parameter. For observations onthe transverse size of objects, such as galactic diameters that do not follow theHubble flow, the linear size S 2.998 × 109θf (z) where θ is its angular size in radians.
For θ in arcseconds the constant is 1.453 × 104. The total comoving volume out to a redshift z is 4π 2.998f (z) 3 Note that the actual volume, which would be relevant for the cosmic gas density,is the comoving volume divided by (1 + z)3, which shows that the density of thecosmic gas (i.e. inter-galactic gas outside clusters of galaxies) increases rapidlywith increasing redshift. The distance modulus is Curvature cosmology (CC) Curvature cosmology (Section is a complete cosmol-ogy that shows excellent agreement with all major cosmological observationswithout needing dark matter or dark energy and is fully described in section It is compatible with both (slightly modified) general relativity and quantummechanics and obeys the perfect cosmological principle that the universe is sta-tistically the same at all places and times. This new theory is based on twomajor hypotheses. The first hypothesis is that the Hubble redshift is due to aninteraction of photons with curved spacetime where they lose energy to other COSMOGRAPHIC PARAMETERS very low energy photons. Thus it is a tired-light model. It assumes a simple uni-versal model of a uniform high temperature plasma (cosmic gas) at a constantdensity. The important result of curvature redshift is that the rate of energyloss by a photon (to extremely low energy secondary photons) as a function ofdistance, ds, is given by where MH is the mass of a hydrogen atom and the density in hydrogen atomsper cubic meter is N = ρ/MH. Eq. shows that the energy loss is proportionalto the integral of the square root of the density along the photon's path. Thisequation can be integrated to get The Hubble constant is predicted to be 51.69N 1/2 kms−1 Mpc−1 64.4 ± 0.2 kms−1 Mpc−1 (N = 1.55 ± 0.01 m−3), where the density N comes from the background X-ray analysis.
The second hypothesis is that there is a pressure, curvature pressure, that acts to stabilize expansion and provides a static stable universe. This hypothesisleads to modified Friedmann equations which have a simple solution for a unformcosmic gas. In this model the distance travelled by a photon from a redshift, z,to the present is r = Rχ, where χ = ln(1 + z)/ 3 and R is the radius of the universe. Since the velocity of light is a universalconstant the time taken is Rχ/c. There is a close analogy to motion on thesurface of the earth with radius R. Light travels along great circles and χ is theangle subtended along the great circle between two points. The geometry of thiscurvature cosmology is that of a three-dimensional surface of a four-dimensionalhypersphere. It is identical to that for Einstein's static universe. For a staticuniverse, there is no ambiguity in the definition of distances and times. Onecan use a universal cosmic time and define distances in light travel times or anyother convenient measure.
The linear size S , at a redshift z with an angular size of θ is For this geometry the area of a three dimensional sphere with radius, r = Rχ,is given by A(r) = 4πR2 sin2(χ).
COSMOGRAPHIC PARAMETERS Table 1: Comparison angular size and volume The surface is finite and χ can vary from 0 to π. The total volume V , is givenby sin(2χ) kpc3.
The distance modulus is obtained by combining the energy loss rate with the area equation to get + 2.5 log(1 + z) + 42.384.
Numerical comparison of BB and CC It turns out that the two cosmologies can be simply related as a function of(1 + z). The approximation equations are where the parameters were determined by averaging them from z = 0 to thelisted value. To avoid any bias the redshifts used were for 15,339 quasars fromthe Sloan Digital Sky Survey Quasar catalogue).
Table shows these the relevant parameters for angular size and total volumeand Table shows them for the distance modulus. The uncertainties in theparameters were all less that 0.003 in the exponents and less than 0.002 in thecoefficients. The first column (labelled Point) is the actual value of the param-eter at that redshift. Examination of Table shows that the major differencein angular size comes from the (1 + z) aberration factor in the denominator ofthe BB equation. Except for a constant scale factor the CC volume is almostthe same as the BB comoving volume. Again note that the actual BB volumehas an additional factor of (1 + z)−3.
Table shows that for the same apparent magnitude CC predicts a fainter absolute magnitude. For example for z = 5 we get M Table 2: Comparison of distance modulus 2.376 which shows that for the same apparent magnitude, in CC the absoluteluminosities are fainter than they are in BB by a factor of 0.112.
Examination of these tables and in particular bS and bM shows that the major difference between BB and CC is the inclusion of the expansion factor(1 + z) in the equations for BB.
This section briefly examines the ad hoc hypotheses for BB and the basic hy-potheses for CC. Although the basis of BB in general relativity is sound ithas acquired some additional hypotheses that are questionable. has listed some of the basic theoretical problems with BB as being the horizonproblem, the flatness problem, the antimatter problem, the structure problemand the expansion problem. Although it does not overcome all these problemsthe solution that was suggested by and developed by was the concept of inflation. The idea is that very early inthe evolution of the universe there is a brief but an extremely rapid accelerationin the expansion. There is no support for this concept outside cosmology andit must therefore be treated warily. Nevertheless with the adjustment of severalparameters it can explain most of the listed problems.
In 1937 found in an analysis of the Coma cluster of galaxies that the ratio of total mass obtained by using the virial theorem to the totalluminosity was 500 whereas the expected ratio was 3. This huge discrepancy wasthe start of the concept of dark matter. It is surprising that in more than sevendecades since that time there is no direct evidence for dark matter. Similarlythe concept of dark energy (some prefer quintessence) has been introduced toexplain discrepancies in the observations of type 1a supernovae. The importantpoint is that these three concepts have been introduced in an ad hoc manner tomake BB fit the observations. None has any theoretical or experimental supportoutside the field of cosmology.
As already stated CC is based on the hypotheses of curvature redshift and that of curvature pressure. Both are described in section and are supported bystrong physical arguments. Curvature redshift is testable in the laboratory andsome support for curvature pressure may come from solar neutrino observations.
Nevertheless they are new hypotheses and must be subject to strong scrutiny.
EXPANSION AND EVOLUTION Expansion and Evolution The original models for BB had all galaxies being formed shortly after the be-ginning of the universe. Then as stars aged the characteristics of the galaxieschanged. Consequently these characteristics are expected to show an evolutionthat is a function of redshift. Since nearly all cosmological objects are associ-ated with galaxies we would also expect their characteristics to show evolution.
Clearly in CC there is no evolution in the average characteristics of any ob-ject. The individual objects will evolve but the average characteristics of thepopulation remain constant. Any strong evidence of evolution of the averagecharacteristics would constitute serious evidence against CC.
The simple BB evolution concept is complicated by galactic collisions and mergers In many cases these canproduce new large star-forming regions. Clearly the influx of a large numberof new stars will alter the characteristics of the galaxy. It is likely that a sig-nificant number of galaxies have been reformed by collisions and mergers andconsequently the average evolution may be very little or at least somewhat re-duced over the simple model.
Tolman surface brightness This test, suggested by relies on the observation that the sur-face brightness of objects does not depend on the geometry of the universe.
Although it is obviously true for Euclidean geometry it is also true for non-Euclidean geometries. For a uniform source, the quantity of light received perunit angular area is independent of distance. However, the quantity of light isalso sensitive to non-geometric effects, which make it an excellent test to dis-tinguish between cosmologies. For expanding universe cosmologies the surfacebrightness is predicted to vary as (1 + z)−4, where one factor of (1 + z) comesfrom the decrease in energy of each photon due to the redshift, another factorcomes from the decrease in rate of their arrival and two factors come from theapparent decrease in area due to aberration. This aberration is simply the rateof change of area for a fixed solid angle with redshift. In a static, tired-light,cosmology (such as CC) only the first factor is present. Thus an appropriatetest for Tolman surface brightness is the value of this exponent.
Surface brightness in BB The obvious candidates for surface brightness tests are elliptic and S0 galaxieswhich have minimal projection effects compared to spiral galaxies . The majorproblem is that surface brightness measurements are intrinsically difficult due tothe strong intensity gradients across their images. In a series of papers ; (hereafter SL01) have investigatedthe Tolman surface brightness test for elliptical and S0 galaxies. More recentlyhas done a more comprehensive analysis but since he came tothe same conclusion as the earlier papers and since the earlier papers are betterknown this analysis will concentrate on them. The observational difficulties arethoroughly discussed by with the conclusion that theuse of Petrosian metric radii helps solve many of the problems. showed that if the EXPANSION AND EVOLUTION Table 3: Galactic properties for Petrosian radius η = 2.0 ratio of the average surface brightness within a radius is equal to η times thesurface brightness at that radius then that defines the Petrosian metric radius,η. The procedure is to examine an image and to vary the angular radius untilthe specified Petrosian radius is achieved.
Thus, the aim is to measure the mean surface brightness for each galaxy at the same value of η. The choice of Petrosian radii greatly diminishes thedifferences in surface brightness due to the luminosity distribution across thegalaxies. However, there still is a dependence of the surface brightness on thesize of the galaxy which is the Kormendy relationship Thepurpose of the preliminary analysis done by SL01 is not only to determine thelow redshift absolute luminosity but also to determine the surface brightnessverses linear size relationship that can be used to correct for effects of sizevariation in distant galaxies. The data on the nearby galaxies used by SL01was taken from and consists of extensive data on thebrightest cluster galaxies (BCG) from 119 nearby Abell clusters. All magnitudesfor these galaxies are in the R (Cape/Landolt) system. Since the results for different Petrosian radii are highly correlated the analysis repeated here usingsimilar procedures will use only the Petrosian η = 2 radius. Although the actualvalue used for h does not alter any significant results here it is set to h = 0.5 fornumerical consistency. A minor difference is that the angular radius used hereis provided by Eq. whereas they used the older Mattig equation.
The higher z data also comes from SL01. They made Hubble Space Telescope observations of galaxies in three clusters and measured their surface brightnessand radii. The names and redshifts of these clusters are given in Table whichalso shows the number of galaxies in each cluster, N , the logarithm of theaverage metric radius in kpc, log(S ), and the average absolute magnitude.
Note that the original magnitudes for Cl 1324+3011 and Cl 1604+4304 wereobserved in the I band.
The bracketed numbers are the root-mean-square (rms) values for each vari- able. In order to get a reference surface brightness at z = 0 all the surfacebrightness values, SB, of the nearby galaxies were reduced to absolute surfacebrightnesses by using Eq. Since all the redshifts are small, this reductionis essentially identical for all cosmological models. However the calculation ofthe metric radii for the distant galaxies is very dependent on the cosmologicalmodel. This procedure of using the BB in analyzing a test of BB is discussedin SL01. Their conclusion is that it reduces the significance of a positive resultfrom being strongly supportive to being consistent with the model. Of interestis that Table shows that on average the distant galaxies are smaller than thenearby galaxies.
Then a linear least squares fit of the absolute surface brightness as a function EXPANSION AND EVOLUTION Table 4: Fitted exponents for distant clusters (η = 2.0) ), the Kormendy relationship, for the nearby galaxies results in the SB = 9.29 ± 0.50 + (2.83 ± 0.11) log(S whereas SL01 found a slightly different equation SB = 8.69 ± 0.06 + (2.97 ± 0.05) log(S Although a small part of the discrepancy is due to slightly different proce- dures the main reason for the discrepancy is unknown. Of the 74 galaxies usedthere were 19 that had extrapolated estimates for either the radius or the surfacebrightness or both. In addition there were only three galaxies that differed fromthe straight line by more than 2σ. They were A147 (2.9σ), A1016 (2.0σ)andA3565 (-2.4σ). omission of all or some of these galaxies did not improve theagreement. The importance of this preliminary analysis is that Eq. containsall the information that is needed from the nearby galaxies in order to calibratethe distant cluster galaxies.
Next we use the galaxies' radius and Eq. to correct the apparent sur- face brightness of the distant galaxies for the Kormendy relation and then doleast squares fit to the difference between the corrected surface brightness andits absolute surface brightness as a function of 2.5 log(1 + z) to estimate theexponent, n, where SB ∝ (1 + z)n. If needed the non-linear corrections givenby were applied to the nearby surface brightness values. Forthe I band galaxies the absolute surface brightness included the color correction< R − I >= 0.62 The results for the exponent, n, foreach cluster are shown in Table together with the values from SL01 (column 5)where the second column is the band (color) in which the cluster was observed.
Because the definition of magnitude contains a negative sign the expected value for n in BB is four. Nearly all of the difference between these resultsand those from SL01 arises from the use of a different Kormendy relationship.
If the Kormendy relationship used by SL01 Eq. is used instead of Eq. the agreement is excellent. If it is assumed that there is no evolutionary orother differences between the three clusters and all the data are combined theresulting exponent is n = 2.16 ± 0.13.
Clearly there is a highly significant disagreement between the observed ex- ponents and the expected exponent of four. Both SL01 and claim that the difference is due to the effects of luminosity evolution. Based ona range of theoretical models SL01 show that the amount of luminosity evolu-tion expressed as the exponent, p = 4 − n , varies between p =0.85–2.36 in the R band and p =0.76–2.07 in the I band. In conclusion to their analysis theyassert that they have either (1) detected the evolutionary brightening directlyfrom the SB observations on the assumption that the Tolman effect exists or EXPANSION AND EVOLUTION Table 5: Radii and fitted exponents for distant clusters (η = 2.0) (2) confirmed that the Tolman test for the reality of the expansion is positive,provided that the theoretical luminosity correction for evolution is real.
SL01 also claim that their results are completely inconsistent with a tired light cosmology. Although this is explored for CC in the next sub-section it isinteresting to consider a very simple model. The essential property of a tiredlight model is that it does not include the time dilation factor of (1 + z) in itsangular radius equation. Thus assuming BB but without the (1 + z) term allvalues of log(S ) will be increased by log(1 + z). Hence the predicted absolute surface brightness will be (numerically) increased by (2.83/2.5)log(1 + z). Forexample, the exponent for all clusters will be changed to ntired light = 2.16 ± 0.16 − This is clearly close to the expected value of unity predicted by a tired-lightcosmology and thus disagrees with the conclusion of SL01 that the data areincompatible with a tired light cosmology.
There are two major criticisms of this work. The first is that relying on theoretical models to cover a large gap between the expected index and themeasured index makes the argument very weak. Although SL01 indirectly con-sider the effects of relatively common galaxy interactions and mergers in thevery wide estimates they provide for the evolution, the fact that there is such awide spread makes the argument that Tolman surface brightness for this data isconsistent with BB possible but weak. Ideally there would be an independent es-timate of p based on other observations. The second criticism is that the nearbygalaxies are not the same as the distant cluster galaxies. The nearby galaxiesare all brightest cluster galaxies (BCG) whereas the distant cluster galaxies arenormal cluster galaxies. It is well known that BCG are in general much brighter and larger than would be expected for thelargest member of a normal cluster of galaxies. Whether or not this amountsto a significant variation is unknown but it does violate the basic rule that likeshould be compared with like.
Surface brightness in CC Unsurprisingly it is found that using CC the relationship between absolute sur-face brightness and radius is identical to that shown in Table What is differentis the average radii, the absolute magnitudes and the observed exponent n These are shown in Table where as before the bracketed terms are the rmsspreads.
EXPANSION AND EVOLUTION The result for all clusters is n = 1.38 ± 0.13 which is in agreement with unity. Note that the critical difference from BB is in the size of the radii. Theyare not only much closer to the nearby galaxy radii but because they are largerthey do not require the non-linear corrections for the Kormendy relation. Asbefore we note that the nearby galaxies are BCG which may have a brighterSB than the normal field galaxies. If this is true it would bias the exponentto a larger value. If we assume that CC is correct then this data shows thaton average the BCG galaxies are −0.64 ± 0.08 mag (which is a factor of 1.8 inluminosity) brighter than the general cluster galaxies.
Conclusion for surface brightness The SL01 data for the surface brightness of elliptic galaxies is consistent withBB but only if a large unknown effect of luminosity evolution is included. Thedata do not support expansion and are in complete agreement with CC.
Closely related to surface brightness is relationship between the observed angu-lar size of a distant object and its actual linear transverse size. The variationof angular size as a function of redshift is one of the tests that should clearlydistinguish between BB and CC. The major distinction is that CC like all tired-light cosmologies does not include the (1 + z) aberration factor. Its relationship(Eq. between the observed angular size and the linear size is very close (forsmall redshifts) to the Euclidean equation. provide a comprehensive history of studies for a wide range of objects that gen-erally show a 1/z or Euclidean dependence. Most observers suggest that theprobable cause is some form of size evolution. Recently used 393 galaxies with redshift range of 0.2 < z < 3.2 in order to test manycosmologies. Briefly his conclusions are The average angular size of galaxies is approximately proportional to z−α with α between 0.7 and 1.2.
Any model of an expanding universe without evolution is totally unable to fit the angular size data . .
Static Euclidean models with a linear Hubble law or simple tired-light fit the shape of the angular size vs z dependence very well: there is a differencein amplitude of 20%–30%, which is within the possible systematic errors.
It is also remarkable that the explanation of the test results with an expanding model require four coincidences: 1. The combination of expansion and (very strong evolution) size evolu- tion gives nearly the same result as a static Euclidean universe witha linear Hubble law: θ ∝ z−1.
2. This hypothetical evolution in size for galaxies is the same in normal galaxies as in quasars, as in radio galaxies, as in first ranked clustergalaxies, as the separation among bright galaxies in cluster EXPANSION AND EVOLUTION 3. The concordance (ΛCDM) model gives approximately the same (dif- ferences of less than 0.2 mag within z < 4.5) distance modulus in aHubble diagram as the static Euclidean universe with a linear law.
4. The combination of expansion, (very strong) size evolution, and dark matter ratio variation gives the same result for the velocity dispersionin elliptical galaxies (the result is that it is nearly constant with z)as for a simple static model with no evolution in size and no darkmatter ratio variation.
With a redshift range of z < 3 the value of S is approximately proportional to z0.68 which shows that it is consistent with these results. A full analysis re-quires a fairly complicated procedure to correct the observed sizes for variationsin the absolute luminosity.
A simple example of the angular size test can be done using double-lobed Using quasar catalogues, carefully selected 103 edge-brightened, double-lobed sources from the VLA FIRST survey andmeasured their angular sizes directly from the FIRST radio maps. These areFaranoff-Riley type II objects and exhibit radio-brighthot-spots near the outer edges of the lobes. Since claimthat three different Friedmann (BB) models fit the data well but that a Eu-clidean model had a relatively poor fit a re-analysis is warranted. The angularsizes were converted to linear sizes for each cosmology and were divided intosix bins so that there were 17 quasars in each bin. Because these double-lobedsources are essentially one dimensional a major part of their variation in size isdue to projection effects. For the moment assume that in each bin they havethe same size, ˆ S, and the only variation is due to projection then the observed S sin(θ) where θ is the projection angle. Clearly we do not know the projection angle but we can assume that all angles are equally likely so that ifthe N sources, in each bin, are sorted into increasing size the i'th source in thislist should have, on average, an angle θi = π(2i − 1)/4N . Thus the maximumlikelihood estimate of ˆ Note that the sum in the denominator is a constant and that the commonprocedure of using median values is the same as using only the central term inthe sum. Next a regression (Appendix was done between logarithm of theestimated linear size in each bin and log(1 + z) where z is the mean redshift.
Then the significance of the test was how close was the exponent, b, to zero.
For BB the exponent was b = −0.79 ± 0.44 and for CC it was b = 0.16 ± 0.44.
Although the large uncertainties show that this is not a decisive discriminationbetween the two cosmologies the slope for BB suggests that no expansion islikely. The overall conclusion is strongly in favor of no expansion.
Type 1a supernovae Type 1a supernovae make ideal cosmological probes. Nearby observations showthat they have an essentially constant peak absolute magnitude and the widthsof the light curves provide an ideal probe to investigate the dependence of time EXPANSION AND EVOLUTION delay as a function of redshift.
The current model for type 1a supernovae is that of a white dwarf steadily acquiring mat-ter from a close companion until the mass exceeds the Chandrasekhar limitat which point it explodes. The light curve has a rise time of about 20 daysfollowed by a fall of about 20 days and then a long tail that is most likelydue to the decay of 56Ni. The widths are measured in the light coming fromthe expanding shells before the radioactive decay dominates. Thus the widthsare a function of the structure and opacity of the initial explosion and havelittle dependence on the radioactive decay. The type 1a supernovae are distin-guished from other types of supernovae by the absence of hydrogen lines andthe occurrence of strong silicon lines in their spectra near the time of maximumluminosity. Although the theoretical modeling is poor, there is much empiricalevidence, from nearby supernovae, that they all have remarkably similar lightcurves, both in absolute magnitude and in their time scales. This has led to aconsiderable effort to use them as cosmological probes. Since they have beenobserved out to redshifts with z greater than one they have been used to testthe cosmological time dilation that is predicted by expanding cosmologies.
Several major projects have used both the Hubble space telescope and large earth-bound telescopes to obtain a large number of type 1a supernova obser-vations, especially to large redshifts. They include the Supernova CosmologyProject theSupernova Legacy Survey the Hubble Higher z supernovaeSearch and the ESSENCE supernova survey Recently have provideda re-analysis of these survey data and all other relevant supernovae in the liter-ature and have provided new observations of some nearby supernovae. Becausethese Union data are comprehensive, uniformly analyzed and include nearly allprevious observations, the following analysis will be confined to this data. Thedata provided (Bessel) B-band magnitudes, stretch factors, and B − V colorsfor supernovae in the range 0.015 6 z 6 1.551. Since there is a very smallbut significant dependence of the absolute magnitude on the B − V color , ineffect the K-correction, following the magnitudes werereduced by a term β(B − V ) where β was determined by minimizing the χ2 ofthe residuals after fitting M verses 2.5 log(1 + z). This gave β = 1.54 which can be compared with β = 2.28 given by The widths (relative to the standard width), w, of the supernova light curves are derived from the stretch factors, s, provided by by theequation w = (1 + z)s. The uncertainty in each width was taken to be (1 + z)times the quoted uncertainty in the stretch value. For convenience in determin-ing power law exponents a new variable W is defined by W = 2.5 log(w). Sincethe width is relative to a standard template the reference value for W is W0 = 0.
Fig. /refsnf1 shows a plot of these widths as a function of redshift. An prelim-inary fit for W as a function of 2.5 log(1 + z) showed an offset −0.095 ± 0.014.
This offset was removed from the supernova widths, W , before further analysiswas done. The same color correction and width offset will be used for bothcosmologies.
What is relevant for both cosmologies is the selection procedure. The cur- rent technique for the supernova observations is a two-stage process The first stage is toconduct repeated observations of many target areas to look for the occurrence EXPANSION AND EVOLUTION of supernovae. Having found a possible candidate the second stage is to con-duct extensive observations of magnitude and spectra to identify the type ofsupernova and to measure its light curve. This second stage is extremely ex-pensive of resources and it is essential to be able to determine quickly the typeof the supernova so that the maximum yield of type 1a supernovae is achieved.
Since current investigators assume that the type 1a supernovae have essentiallya fixed absolute BB magnitude (with possible corrections for the stretch factor),one of the criteria they used is to reject any candidate whose predicted abso-lute peak magnitude is outside a rather narrow range. The essential point isthat the absolute magnitudes are calculated using BB and hence the selectionof candidates is dependent on the BB luminosity-distance modulus. In a com-prehensive description of the selection procedure for a major survey state: Best fits required consistency in the light curve shape and peakcolor (to within magnitude limits) and in peak luminosity in that the derivedabsolute magnitude in the rest-frame B band had to be consistent with observeddistribution of absolute B-band magnitudes shown in .
Fundamental to any cosmology that explains the Hubble redshift as being dueto an expanding universe is the requirement that exactly the same dependencemust apply to time dilation. The raw data of the widths of the type 1a su-pernovae light curves as a function of redshift is shown in Fig. /refsnf1 for theUnion data provided by The fitted straight line showsthat the exponent of (1 + z) is 1.097 ± 0.032 which is in good agreement with theexpected value of unity. These results, which show excellent quantitative agree-ment with the predicted time dilation, have been hailed as one of the strongestpieces of evidence for an expanding cosmology. However the regression of MBBon 2.5 log(1 + z) shows that the luminosity is proportional to (1 + z)0.230±0.070which shows that the absolute luminosity is slowly decreasing as the universeevolves. The standard explanation for this change is the ad hoc introduction ofdark energy or quintessence A further problem with BB is a shortfall in the number of high redshift supernovae that are found. Since to the first order the discovery of a supernovadepends only on apparent magnitude this search procedure (at high z) shouldfind all the candidates out to a redshift where the apparent magnitude is too faintfor the telescope. Then the expected distribution of supernovae as a function ofredshift should be proportional to the comoving volume. To check the redshiftdistribution the 300 acceptable supernovae were put in six bins in ascendingorder of redshift so that there were 50 supernovae in each bin. Then the cutoffapparent magnitude (24.95 mag) was chosen so that all supernovae in the firstfive bins would be included. The results are shown in Table where the columnsare: the bin number, the redshift range, the included number in the bin, theratio of BB volume in that bin to the BB volume in bin 2 and an estimatebased on CC (see below). The results for bin one are not unexpected. It simplyshows that there have been many more searches done for supernovae at nearbyredshifts. The results for bin six show that 33 out of 50 supernovae had anapparent magnitude brighter than the cut-off. To compensate the volume forbin 6 was computed for a redshift limit of z = 1.139 which was the highestredshift for an included supernova. The problem with the BB results is that
EXPANSION AND EVOLUTION Figure 1: Width of supernovae type 1a verses redshift for Union data. Thedashed line (black) is expected time dilation in an expanding cosmology. Thesolid line (red) is the function µ there is a dramatic shortage of supernovae in the high redshift bins. The usualexplanation is that this shortage is due to evolutionary effects. However thisexplanation must be able to show why there is a dramatic decrease in the rateof occurrence of supernovae at redshifts for z near one when there is no obviouschange in the stellar contents of galaxies with these redshifts.
Since the redshift in CC arises from an interaction with the intervening gas, itis not always a good measurement of distance. In particular the halo aroundour galaxy and that around any target galaxy will produce an extra redshiftthat results in an overall redshift that is larger than would be expected fromthe distance and a constant inter-galactic gas density. Since this is an additiveeffect it is important only for nearby objects. In fact the nearby supernovae(defined as those with z < 0.15) show an average absolute magnitude that isbrighter than the extrapolated magnitude from more distant supernovae. Inorder to make a partial correction for this bias all redshifts were reduced bysubtracting 0.006 from each redshift, z. This correction brought the near andmore distant magnitudes into agreement. A plot of absolute magnitude, M verses width, W , is shown in Fig. /refsnf2. For later analysis the data are dividedinto the same 6 redshift bins used in Table The best-fit straight line to all thesupernova has a (global) slope of 0.695 ± 0.072 (for M it is −0.391 ± 0.056).
EXPANSION AND EVOLUTION Table 6: Density distribution aThe ratio of volumes in BB: V (n)/V (2) bThe ratio of selection probability times CC volumecRedshift range used columns 4 and 5 is 0.821–1.139dThe number brighter than the cutoff of 24.95 mag Table 7: rms and local dependence of M verses width, W .
This implies that supernovae that are brighter have narrower widths, or theweaker are wider. Table gives the rms of the reduced magnitudes and the slopeof the reduced magnitudes verses the width, W , for each bin. However argue for a local dependence ofmagnitude on stretch that has the opposite sign to the fitted straight line.
In the first bin the stretch and magnitude are essentially identical so that we can compare the local result of −1.19 ± 0.21 with −1.56 ± 0.25 reported byto show good agreement. The slope for the other bins showsthat although this local slope is not so well defined it is clearly present at higherredshifts. The challenge is to explain why the slope in a small redshift range isopposite to that for the global redshift range. If ∆M is a variation in magnitudeand ∆W is a variation in width then the variations can be summarized: (1). Local: ∆M ∝ −(1.56 ± 0.25)∆W (2). Global: ∆M ∝ (0.70 ± 0.07)∆W .
(3). Constant energy: ∆M ∝ ∆W .
where the last item (constant energy) assumes that the shape of the supernovalight curve is the same for all supernova with a height proportional to thepeak luminosity. Hence the total energy is proportional to the peak luminositymultiplied by the width.
EXPANSION AND EVOLUTION Figure 2: Absolute magnitudes, M , (Bessel B-band) of type 1a supernovae verses widths of their light curves for Union data. The solid line (red) is theline of best fit. The data are divided into 6 redshift bins identical to the binsshown in Table Note that the brightest supernovae are at the top.
For type 1a supernovae it will be shown that the choice of BB magnitudes does have an important effect on the selection of supernovae and the use ofBB leads to a biased sample. Because the Chandrasekhar limit places a well-defined limit on the supernova mass and hence its energy, it is expected thateach supernova has the same total energy output. Since the total energy of asupernova is proportional to the area under its light curve it is proportional tothe product of the maximum luminosity and the width of the light curve. Theproposed model is based on the principle that the most unchanging characteristicof type 1a supernovae is their total energy and not their peak magnitude. Dueto local effects the total energy will have small variations about a constant value.
Since the distance modulus for BB is always larger than that for CC a selec- tion based on absolute magnitude that uses BB will select at greater redshiftssupernovae that are fainter and hence supernovae with wider light curves. Thusfinding that a selection that has width increasing with redshift could mimic timedilation.
Define the magnitude of the total energy, E, in the same way as the magni- tude of the luminosity, that is E = −2.5 log(E /E0) where E0 is a reference energy.
Then the first assumption is that although individual supernovae will show vari-ation in total energy, magnitude and width we expect that the averages overmany supernovae will satisfy the equation E = M − W . Since, by definition,the reference width, W0, is zero the expected value of the energy is E0 = M0where M0 is the reference magnitude. Since its expected value is constant E EXPANSION AND EVOLUTION will be a better standard candle than M .
To summarize, by relying on BB distance modulus the (distant) supernovae search method consistently selects supernova that are weaker than expected.
This pushes the selection towards the limits of the natural variation and alsoselects supernovae with wider light curves. If the error in the BB magnitudesis due to the inclusion of the time dilation term the correct absolute luminosityis smaller than the BB luminosity by a factor of (1 + z). Thus with constantenergy, the width is larger by a factor of (1 + z) which agrees with the resultsshown in Fig. Using CC and a constant energy model the dependence is ∆M = M which is shown as the solid (red) line in Fig. /refsnf1. Considering that the selected supernovae are a biasedsample the agreement with the widths is reasonable. There is still a problemof explaining the local slope. In BB the average of the local slope over the sixbins is −1.24 ± 0.15 compared with the global slope of −0.391 ± 0.056 and forCC the average local slope is −1.07 ± 0.20. Consider a supernovae with aboveaverage energy. If the only change in the shape of the light curve is a largerscaling factor the peak luminosity will be proportional the width which agreeswith the average local slope.
Finally this model can be used to get an approximate estimate of the ex- pected number of supernovae that would be selected. The nearby supernovaecome from a wide range of heterogenous surveys and serendipitous observa-tions. Consequently their selection probability is essentially unknown. Howeverthe more recent, distant supernovae come from deliberate surveys that scan asmall area of the sky looking for sudden outbursts. The crucial point is thatprovided the apparent magnitude of the supernova is within the observationalconstraints the probability of detection is independent of redshift. Thus as afirst approximation all the surveys may be combined in order to determine theexpected number of supernovae that should be detected.
Assuming that the intrinsic distribution in magnitude is normal (with a stan- dard deviation of σ) and that a supernova is selected if M falls with a narrow range about the reference value, then the probability of selection is proportionalto exp(−(µ (z))2/2σ2). The test is whether the number of expected supernovae as a function of redshift is similar to the observed number. The ex-pected numbers (probability times volume, V ), are shown for σ = 0.37 in the last column of Table Note that the redshift ranges for each bin are determinedby the observed supernovae which are heavily biased in bin 1 and probably alsoin bin 2 due to the inclusion of supernovae from many local surveys that donot fit the selection model. Consequently the value of σ = 0.35 was chosen toprovide roughly equal values for bins 3, 4 and 5. The average of the rms of MCCshown in Table is 0.32 mag which is reasonably consistent with σ = 0.37.
It has been argued that the total energy of type 1a supernovae makes a good standard candle. Now we investigate whether the energies of type 1a supernovaeare independent of redshift. Fig. /refsnf3 shows the expected energy, definedhere for each supernova as E − W , of the Union supernovae as a function of redshift.
The slope of the fitted line is 0.047 ± 0.089 which is in excellent agreement with zero. Thus showing that CC can provide an good fit to the supernova datawithout the fitting of any free parameters and that E is an good standard candle.
Since there is no deviation from the straight line at large redshifts there is no
EXPANSION AND EVOLUTION Figure 3: Absolute energy, M − W , verses redshift, 2.5 log(1 + z), for Union data. Solid line (red) is the line of best fit with the equation (−19.070 ± 0.042) +(0.047 ± 0.089)2.5 log(1 + z) need for dark energy or quintessence (both of which are meaningless in CC). The estimate of the intercept,that is M0, is −19.070 ± 0.042. have measured accuratedistances to two galaxies containing nearby supernovae.
Together with two earlier measurements, they derive an absolute magnitude of type 1a supernovaeof −19.17 ± 0.07. Hence the reduced Hubble constant, h, can be estimated from−19.070 − 5 log(0.7) = −19.17 − 5 log(h) to get h = 0.638 ± 0.05. Thus themeasured Hubble constant is 63.8 ± 0.5 kms−1 Mpc−1.
Supernova conclusion It has been shown that there is very strong support for the proposition thatthe most invariant property of type 1a supernovae is their total energy andnot their peak magnitude. Given an essentially constant energy there is aninverse relationship between the peak luminosity and the width of the lightcurve.
Since the prime characteristic used for selecting these supernovae is the peak magnitude which is computed using BB there is a strong bias thatresults in intrinsically weaker supernovae being selected at higher redshifts. Forconstant energy these weaker supernovae must have wider light curves. Using asimple model for the selection process it was shown that it predicts the observeddependence for the light curve widths on redshift (Fig. It is also consistentwith the observed local variation of magnitudes on widths. When the observedmagnitudes are corrected for the supernova width, they are independent ofredshift (Fig. /refsnf3). The conclusion is that with a simple selection model EXPANSION AND EVOLUTION these supernovae observations are fully compatible with CC and there is noneed for dark energy or quintessence. This is strong support for the premisethat there is no time dilation and hence no universal expansion.
Gamma ray bursts (GRB) Gamma-Ray bursts (GRB) are transient events with time scales of the orderof seconds and with energies in the X-ray or gamma-ray region. provides (a mainly theoretical) review and givea review of observations and analysis. Although the reviews by and cover more recent research and provide extensive referencesthey are mainly concerned with GRB models. This paper considers only thedirect GRB observations and makes no assumptions about GRB models.
The search for the time dilation signature in data from GRB has a long his- tory and before redshifts were available claimed evidence for a time dilation effect by compar-ing dim and bright bursts. However found no evidencefor time dilation. found rather inconclusive re-sults from a comparison between brightness measures and timescale measures.
They also provide a brief summary of earlier results. Since redshifts have becomeavailable and using a Fourier energyspectrum method and using an autocorrelation method claimevidence of time dilation. The standard understanding, starting with and is that time dilation is present but because ofan inverse relationship between luminosity and time measures it cannot be seenin the raw data. Their argument is that because a strong luminosity-dependentselection produces an average luminosity that increases with redshift there willbe a simultaneous selection for time measures that decrease with redshift whichcan cancel the effects of time dilation.
has argued that within the paradigm of BB that there is no evidence for strong luminosity selection or luminosity evolution. Consequentlythose time measures that show a strong relationship with luminosity must haveevolved in a similar manner. Although it is possible that a combination ofluminosity selection, selection of GRB by other characteristics and evolutionmay be sufficient to cancel time dilation it does require a fortuitous coincidenceof these effects to completely cancel the effects of time dilation in the raw data.
Another explanation is that the universe is not expanding and thus there is notime dilation.
Galaxy distribution Recently, large telescopes with wide fields and the use of many filters haveenabled a new type of galactic survey. The light-collecting capability of the largetelescopes enables deep surveys to apparent magnitudes of 24 mag or better andthe wide field provides a fast survey over large areas. A major innovation is theuse of many filters whose response can be used to classify the objects withgreat accuracy. Thus, galaxies can be separated from quasars without needingmorphological analysis. This photometric method of analysis works becausephotometric templates are available for a wide range of types of galaxies andother types of objects. In addition, accurate redshifts are obtained from fitting EXPANSION AND EVOLUTION Table 8: M ∗ for SED Type 1 galaxy luminosity distributions.
aAbsolute magnitude for the SDSS r-band the templates without the tedious procedure of measuring the spectrum of eachobject.
A typical example of this photometric method is the COMBO-17 survey (Classifying Objects by Medium-Band Observations in 17 filters) provided byThe goal of this survey was to provide a sample of 50,000galaxies and 1000 quasars with rather precise photometric redshifts based on17 colors. In practice, such a filter set provides a redshift accuracy of 0.03 forgalaxies and 0.1 for quasars. The central wavelength of the 17 filters variedfrom 364 nm to 914 nm and consisted of 5 broadband filters (U, B, V, R, andI)and 12 narrower-band filters. have analyzed this data andclaim that there is strong evolution for 0.2 < z < 1.2. Instead of using genericK-corrections, the restframe luminosity of all galaxies are individually measuredfrom their 17-filter spectrum. For each galaxy, three restframe passbands areconsidered, (i) the SDSS r-band, (ii) the Johnston B-band and (iii) a syn-thetic UV continuum band centered at λrest= 280 nm with 40 nm FWHM andrectangular transmission function. A spectral energy distribution, SED, wasdetermined for each galaxy by template matching. For the evolution analysisthey were assigned to one of four types. The only type that showed a well de-fined peak in their luminosity distribution was Type 1 which covers the E-Sagalactic types. The characteristics of the luminosity distribution were obtainedby fitting a Schechter function which is where the luminosity L∗ (and its magnitude M ∗) is a measure of location andα is a measure of shape. They found that a fixed value for α works quite wellfor the luminosity functions of individual SED types. Examination of theirestimate of M ∗ for Type 1 galaxies showed that if they were converted to CCmagnitudes they were independent of redshift. This is shown in Table wherethe data are taken from the appendix to The second column isthe difference, µ , between BB and CC distance moduli. The remaining columns show the absolute magnitude for the three restframe passbands. Thelast row shows the χ2 for the five magnitudes relative to their mean using thegiven uncertainties (all in the range 0.14-0.23).
With four degrees of freedom the first two bands show excellent agreement with a constant value. The values for M ∗ have less than a 2.5% chance of being constant. However since most of the discrepancy comes from the z = 0.3 EXPANSION AND EVOLUTION value of -17.38 mag and most of this band at small redshifts is outside the rangeof the 17 filters this discrepancy can be ignored. If this value is ignored the χ2 isreduced from 12.81 to 6.12 (with 3 D0F) which is consistent with being constant.
Since α is independent of redshift the result is that if the data had been analyzedusing CC the magnitude for these Type 1 galaxies does not vary with redshift.
Thus we have the surprising result that using BB a class of galaxies has a welldefined luminosity evolution that is predicted by CC. In other words there is noexpansion.
Quasar distribution A major difference between BB and CC is that at a redshift just greater thanz = 5 the absolute luminosity of a quasar is a factor of ten smaller for CC thanfor BB. have made a comprehensive study of quasarsfrom the Sloan Digital Sky Survey (SDSS) and provide tables of absolute (BB)magnitudes and selection probabilities for 15,343 quasars. The sample extendsfrom i = 15 to 19.1 at z 6 3 and i = 20.2 for z > 3. There was an additionalrequirement that the absolute magnitude was M < −22 mag. Only some low redshift quasars failed this test. The final selection criterion was that each hada full width at half-maximum of lines from the broad-line region greater than1000 km s−1. provided the redshift, apparent magnitude,selection probability and the K-correction for each quasar. The K-correctionhad two parts. The first part was a function only of the redshift and thereforeit was independent of the nature of each quasar. However the second part wasvery important since it depended on the color difference g − i. They computedthe quasar luminosity function in eleven redshift bins and in each case it wasclose to a power law in luminosity or an exponential function in magnitude.
Effectively this meant that distributions were scale free and that there was noway the magnitudes could be directly used to compare cosmologies.
Let us assume that the magnitude distribution is exponential and can be φ(M ) dM = V ρβ exp(βM ) dM where β is the basic parameter of the exponential distribution, V is the accessiblevolume and ρ is the quasar density. Then using Eq. we get φ(M ) dM = V ρβ exp β(m − µ(z) − K(z)) dM Now consider a small range of redshifts centered on zk, then because m will alsohas an exponential distribution the expected number in this redshift range is βk exp(βk(m − m b k is the cutoff for the apparent magnitude and K z is the average K- correction for this z range. This is necessary because there are color-dependentcorrections that are a property of the individual quasars. Another change is tochange the increment in the independent variable from dM to dm. The reasonfor the separation of the two exponents in Eq. is the first line is the samefor all quasars in the range and the second line contains all the details of thequasar distribution. All variables that are common to the all the quasars in the EXPANSION AND EVOLUTION redshift range have a suffix of k. The result of integrating with respect to M onthe left and with respect to m on the right is the expected number of quasarsin this redshift range, Nk. Thus rearranging Eq. we get The essence of this method is that because the luminosity distribution is a powerlaw we can easily change the independent variable from absolute magnitude toapparent magnitude. Thus Eq. provides an estimate of the distance moduluswhere cosmology enters only through the volume, Vk. The overall density ρ iscommon to all redshift ranges and can be estimated by a least squares fit to allthe ranges.
The next step is to estimate the exponential parameter βk. A small com- plication is that the apparent magnitudes have a measurement uncertainty sothat assuming a Gaussian error distribution the expected distribution is theconvolution of a Gaussian with the exponential distribution to get p(m) dm = β exp(−β(m − m) + where σ is the standard deviation of the magnitude uncertainty. Note thatthe second term shows that there is an excess of quasars moved from faintermagnitudes compared to those moved to fainter magnitudes. The maximumlikelihood estimate for β is the solution to the quadratic equation β2σ2 − (m − m)β + 1 where m is the mean magnitude and its variance is var(β) = N(1 − β2σ2) This analysis has been done for the SDSS data and the results are shown in Fig. ?? where filled circles (red) are for CC volumes andthe filled diamonds (blue) are for BB volumes. Both sets of points have beennormalized to be the same for z = 0.9. The selection probability for each quasaris allowed for by including only those quasars that have a selection probabilitygreater than 0.3 and by dividing each quasars contribution to the distributionby the selection probability. As expected the different volumes between thecosmologies have only a small effect. The full (blue) line shows the BB distancemodulus and the dashed (red) line is for the CC modulus. The uncertaintieswere estimated from the deviations from a smooth curve: in this case µ The plotted lines are the theoretical distance moduli normalized in the sameway as the data. The apparently lower values for observations near z=2.8 areprobably due to confusion between the spectra of stars and that for quasars inthis region which not only produces lower selection probabilities but also makestheir estimates more uncertain. The very clear result is that the quasars areconsistent with CC but they are not consistent with BB without evolution.
The conclusion is that CC clearly fits the data whereas BB would require evolution that cancels the expansion term in its distance modulus.
Radio Source Counts The count of the number of radio sources as a function of their flux densityis one of the earliest cosmological tests that arose from the development of
EXPANSION AND EVOLUTION Figure 4: The distance modulus of SDSS quasars as a function of redshift. Thediamonds (blue) are for BB and the circles (red) are for CC. The full (blue) lineis the theoretical distance modulus for BB and the dotted (red) line is for CC.
All have been normalized to be the same at z = 0.9.
radio astronomy after World War II. Indeed, this test played a pivotal rolein the rejection of the steady state cosmology of Bondi, Gold, and Hoyle infavor of the Big-Bang evolutionary model. In recent years, the study of radiosource counts has declined for several reasons both theoretical and experimental.
An important experimental problem is that many radio sources are double orcomplex in structure. Whether or not they are resolved depends on their angularsize and the resolution of the telescope. Since their distance is unknown, thecounts are distorted in a way that cannot be readily determined. The maintheoretical problem in Big-Bang cosmology is that the counts are of a collectionof quite different objects such as galaxies and quasars that can have differenttypes of evolution. Thus, the radio source counts are not very useful in the studyof these objects. However, in CC, the source number density must be the sameat all places and at all times. Curvature cosmology demands that radio sourcecounts are consistent with a reasonable luminosity number density distributionthat is independent of redshift. Thus it provides a critical test of CC.
In order to clarify the nature of the radio source count distribution let us start with a simple Euclidean model. Let the observed flux density of a source atan observed frequency of ν0 be S(ν0) in units of W m−2 Hz−1 and its luminosityat the emitted frequency ν be L(ν) in units of W Hz−1. For simplicity, let usassume that all the sources have the same luminosity and that they have avolume density of N sources per unit volume. Then using the inverse square EXPANSION AND EVOLUTION law, the observed number of sources with a flux density greater than S is Thus the number density of observed sources is dn/ds = S−5/2. The importanceof this result is that it is customary to multiply the observed densities by S5/2so that if the universe had Euclidean geometry the distribution as a function ofS would be constant. It has the further advantage in that it greatly reduces therange of numbers involved. However this practice is not used in this analysis.
For CC the area at a distance r is A(r) = 4πR2 sin2(χ) where r = sin(χ) and χ = ln(1 + z)/ 3. Note that the actual light travel distance is Rχ . Thus 4πR2 sin2(χ)(1 + z) where ν = (1 + z)ν0 and the (1 + z) in the divisor allows for the energy loss dueto curvature redshift. Since the ratio of the differentials (the bandwidth factor)contributes a factor that cancels the energy loss, the result is It is convenient to replace the distance variable by the redshift parameter z.
Then the differential volume is If the luminosity number density is N (L, (1 + z)ν0) the expected radio- source-count distribution (allowing for the dS/dL term needed to match thedifferentials) is 16π2R5 Z zm sin4 (χ) N (L, (1 + z)ν0) dz, where zm is the limiting redshift.
A major problem with the observations is the difficulty in knowing the se- lection criteria. Typically, all sources greater than a chosen flux density arecounted in a defined area. Since the flux density measurements are uncertainand the number of sources is a strong function of the flux density, it is difficultto assess a statistically valid cut-off for the survey. In a static cosmology, thechange in the distribution due to the change in emitting frequency as a functionof z is an added complication. Thus, an essential test of CC is to show thatthere is an intrinsic distribution of radio sources that is identical at all redshifts.
Unfortunately, it is not feasible to obtain a definitive distribution. What will bedone is to show that the observations are consistent with a possible distribution.
Thus the aim of this section is to show that there is a distribution N (L, νe) thatprovides a reasonable fit to the observations at all frequencies.
A simple distribution has been found that provides a good first approxima- tion to the intrinsic distribution. Define the variable x by EXPANSION AND EVOLUTION Table 9: Origin of radio source-count data.
Table 10: Parameters for radio source distribution.
where L∗ and β are constants. The first term is the ratio of the absolute fluxdensity to a reference value, L∗, and the absolute flux density is obtained fromthe measured flux density by using Eq. The second term in Eq. containsthe frequency of emission, ν0(1 + z), and is the only frequency contribution inthis simple model. Then the model for the intrinsic radio-source distribution is N (L, ν) = Ax−α exp(−γx).
where α, β, γ, and A are constants that are found by fitting the model to thedata listed in Table In order to provide realistic values a value of h=0.7 hasbeen adopted. The results of fitting this distribution to the data are shown inTable The χ2 goodness of fit was 4360 for 252 DoF. Because of the poor fit,the estimate of statistical uncertainties has been omitted.
A plot of the data (references in Table with the flux densities in Jy, and the results of this model is shown in Fig. For clarity the source density (numberper Gpc3) for each set of points for a given frequency has been multiplied by afactor of 10 relative to the adjacent group. The density scale is correct for the1400 MHz data.
EXPANSION AND EVOLUTION Figure 5: Logarithm of the radio source volume density (Gpc−3) distribution asa function of flux density. The legend shows the frequency (in Mhz) for eachgroup. Each group is displaced vertically from the adjacent group by a factorof 10. The vertical scale is exact for the 1400 MHz group.
Clearly, the model satisfies the basic structure of the distributions but there is a poor fit at low flux densities, which is probably due to the limitations ofthe simple distribution (Eq. It should be emphasized that the only freeparameters are those shown in Table Since the observed densities at lowflux densities are larger than those predicted by the model it suggests that theremay be a second population that is intrinsically fainter that the flux densitiesfitted by the model. This analysis shows that a simple distribution of radiosource flux densities can be found that is consistent with the observations. InBig-Bang cosmology even with the inclusion of evolution there is no model ofradio sources that is as simple as the one described here. Thus, the observationsof the number distribution of radio sources as a function of flux density andfrequency shows that they have no need for luminosity evolution.
Quasar variability in time One of the major differences between a tired-light cosmology and an expandinguniverse cosmology is that any expanding universe cosmology predicts that timevariations and clocks have the same dependence on redshift as does the frequencyof radiation. has analyzed the variability of 800quasars covering time scales from 50 days to 28 years.
His data permitted the straightforward use of Fourier analysis to measure the time scale of thevariability. He showed that there was no significant change in the time scaleof the variability with increasing redshift. He considered and rejected various EXPANSION AND EVOLUTION explanations including that the time scales of variations were shorter in bluerpass bands or that the variations were not intrinsic but were due to interveningprocesses such as gravitational micro-lensing. His conclusion was either thatthe quasars are not at cosmological distances or that the expanding universecosmologies are incorrect in this prediction. Curvature cosmology on the otherhand would predict just such results. To summarize, the sparse data on quasarvariability strongly supports no time dilation.
The Butcher–Oemler effect If there were evidence of significant change in the universe as a function of red-shift, it would be a detrimental to any static cosmology. Probably the mostimportant evidence for this cosmic evolution that appears to be independentof any cosmological model is the effect. Althoughthey had discussed the effect in earlier papers, the definitive paper is They observed that the fraction of blue galaxies in galacticclusters appears to increase with redshift. Clusters allow the study of large num-bers of galaxies at a common distance and out to large redshifts, which makesthem ideal for studies in evolution. The core regions in a cluster are dominatedby early-type (elliptical and lenticular) galaxies, which have a tight correla-tion between their colors and magnitudes. We can calculate R30, the projectedcluster-centric radius that contains 30% of the total galaxy population. Theblue fraction, fB, is defined to be the fraction of galaxies within R30 which arebluer than the color-magnitude relationship for that cluster. At first sight, thismay appear to be a simple test that could be done with apparent magnitudes.
However to compare the ratio for distant clusters with that for nearby ones thecolors must be measured in the rest frame of each cluster, hence the need to useK-corrections. The major advantage of the Butcher–Oemler effect is that it isindependent of the luminosity-distance relationship that is used. Therefore, tobe more precise fB is the fraction that has an absolute magnitude MV , whoserest frame (B-V) color is at least 0.2 magnitudes bluer than expected. A reviewby summaries the important observations.
In its original form the Butcher–Oemler effect is dependent on the apparent magnitude cut-off limits. It is essential that selection effects are the same inthe rest frame for each cluster. There are further complications in that thepercentage of blue galaxies may or may not depend on the richness of the clusterand the effect of contamination from background galaxies. Although concluded there was a definite effect, his Fig. 1 shows that this conclusionis open to debate. Since then there have been several attempts to measurean unambiguous effect. Even though they attempted to duplicate the originalmethodology of Butcher & Oemler, found essentiallyno effect for K-selected galaxies. examinedthree clusters around z=0.7 and did not find clear-cut evidence for the effect.
To quote one of their conclusions: Twenty years after the original intuitionby Butcher & Oemler, we are still in the process of ascertaining the reality ofthe Butcher–Oemler effect. The Butcher–Oemler effect remains uncertain, andtherefore does not provide evidence to refute a static cosmology.
The Hubble redshift The increase in the redshift of an object with distance is well documented andboth cosmologies provide a solution. In BB it is due to the expansion of theuniverse which is predicted by general relativity. The basic instability of theEinstein model is well known Although the inclusionof a cosmological constant provides a static solution it is still unstable andafter a small perturbation will either expand or contract. Thus the BB is infull agreement with general relativity. On the other hand in CC the Hubbleredshift is due to a gravitational interaction between the photons and the cosmicgas whose density produces a curved space-time.
Both cosmologies predict that for nearby distances the Hubble redshift is a linear function of distance.
However in CC it depends on the integral of the square root of the densityalong the path length and can because of density variations in a particularsituation it may vary from the average value. Another important difference isthat CC predicts the actual value for the Hubble constant. For the measureddensity of N = 1.55 ± 0.01 m−3 the calculated value of the Hubble constant isH = 64.4 ± 0.2 km s−1 Mpc−1 whereas the value estimated from the type 1asupernova data is 63.8 ± 0.5 kms−1 Mpc−1 and the result from the Coma cluster(Section is 65.7 kms−1 Mpc−1.
X-ray background radiation Since observed the X-ray background there have beenmany suggestions made to explain its characteristics. Although much of theunresolved X-ray emission comes from active galaxies, there is a part of thespectrum between about 10 keV and 1 MeV that is not adequately explained byemission from discrete sources. The very high energy range is most likely dueto external point sources. It is the intermediate range that is examined here.
In Big-Bang cosmology for the intermediate X-ray range of about 10–300 keV,the production of X-rays in hot cosmic plasma through the process of bremsstrahlunghas been suggested by In a review of the spectrum of theX-ray background radiation concluded that the measured spectra ofdiscrete sources are not consistent with the observations in the intermediate en-ergy range but there is a remarkable fit to a 40 keV (4.6×108 K) bremsstrahlungspectrum from a diffuse hot gas. However, in an expanding universe most ofthe X-rays are produced at redshifts of z > 1 where the density is large enoughto scatter the CMBR. This scattering known as the Sunyaev–Zel'dovich effect(see Section makes a distinctive change in the spectrum of the CMBR. Thispredicted change in the spectrum has not been observed and this is the majorreason why the bremsstrahlung model is rejected in Big-Bang cosmology.
In CC, the basic component of the universe is plasma with a very high tem-perature, and with low enough density to avoid the Sunyaev–Zel'dovich effect(Section The background X-ray emission is produced in this plasma bythe process of free-free emission (bremsstrahlung).
The observations of the background X-ray emission are analyzed in order to measure the density andtemperature of the plasma. In CC, this density is the major free parameter andit determines the size of the universe and the value of the Hubble constant. Inaddition, the temperature of the plasma determined from the X-ray measure-ments can be compared with the predicted value from CC of 2.56 × 109 K.
The first step is to calculate the expected X-ray emission from high tem- perature plasma in thermal equilibrium.
Here the dominant mechanism is bremsstrahlung radiation from electron-ion and electron-electron collisions. Witha temperature T and emission into the frequency range ν to ν + dν the volumeemissivity per steradian can be written as ×g(ν, T ) exp − where g(ν, T ) is the Gaunt factor, Ne is the electron number density, Ni isthe ion number density and r0 is the classical electron radius and the othersymbols have their usual significance InSI jν(ν) has the units of W m−3 Hz−1. As it stands this equation does notinclude the electron-electron contribution. and have done accurate calculations for many light elements. Based ontheir calculations Professor Naoki Itoh (http://www.ph.sophia.ac.jp/) providesa subroutine to calculate the Gaunt factor that is accurate for temperaturesgreater than 3 × 108 K. It is used here. Let the average density be expressed asthe number of hydrogen atoms per unit volume (N = ρ/MH m−3). Then it isconvenient to define ne = Ne/N and where the sum is over all species present. Because of the very high temperature,we can assume that all atoms are completely ionized. Thus, Eq. includingthe Gaunt factor provides the production rate of X-ray photons as a function ofthe plasma temperature and density. The next step is to compute the expectedintensity at an X-ray detector. Consider an X-ray photon that is producedat a distance Rχ from the detector. During its travel to the detector, it willhave many curvature-redshift interactions. Although the photon is destroyedin each interaction, there is a secondary photon produced that has the samedirection but with a slightly reduced energy. It is convenient to consider thissequence of photons as a single particle and to refer to it as a primary photon.
The important result is that the number of these primary photons is conserved.
Therefore, we need the production distribution of the number of photons perunit energy interval. The number of photons emitted per unit volume per unit Table 11: List of background X-ray data used.
time in the energy interval ε to ε + dε is given by where ε = hν, h is Plank's constant and jν(ν) is the energy distribution perunit frequency interval. Now consider the contribution to the number of X-raysobserved by a detector with unit area. Because the universe is static, the areaat a distance R from the source is the same as the area at a distance R from thedetector. Since there is conservation of these photons, the number coming froma shell at radius R per unit time and per steradian within the energy interval εto ε + dε is dn(r) dε = jn(ε)dεR dχ.
Next, we integrate the photon rate per unit area and per steradian from eachshell where the emission energy is ε and the received energy is ε0 to get where ε = (1 + z)ε0 and it is assumed that the flux is uniform over the 4πsteradians. Furthermore, it is useful to change the independent coordinate tothe redshift parameter z. Then using Eq. we get where H is the Hubble constant and the change in bandwidth factor dν/dνo,cancels the (1 + z) factor that comes from the change in variable from dχ todz but there is another divisor of (1 + z) that accounts for the energy lost byeach photon. Thus the energy flux per unit area, per unit energy interval, perunit frequency and per solid angle is given by Eq. where Plank's constant isincluded to change the differential from frequency to energy. The zm limit of8.2 comes from the limit of χ 6 π.
Table 12: Background X-ray data: rejected points.
keV/(keV cm2 s sr) Table 13: Abundances for four models.
1.9094 × 103 keV mc2 1/2 The density N is obtained by fitting Eq. to the observed data as a function of the temperature T , and then extracting N from the normalization constant.
The X-ray data used is tabulated in Table It consists of the backgroundX-ray data cited in the literature and assessed as being the latest or moreaccurate results. Preliminary analysis showed that there were some discrepantdata points that are listed in Table in order of exclusion.
Very hard X-rays cannot be produced even by this hot plasma and are pre- sumably due to discrete sources Since bremsstrahlung is very sen-sitive to the presence of heavy elements, results are presented for four differentabundances of hydrogen, helium, and ‘metals'. The ‘metals', which is a descrip-tor for all the other elements, were simulated by an element with < Z >=8.4,< Z2 >=75.3 and < A >=17.25.
These values were derived from the abundances given by The details of the four different abundances are shown in Table where the per-centages are by number and the last two columns show the relative number ofelectrons and average value of Z2 per unit hydrogen mass. Thus the models areA: pure hydrogen, B: hydrogen with 8.5% helium, C: normal abundance and D:similar to C but with enhanced ‘metals'.
The results of the fit of the data to these models is given in Table where the errors are the fitted uncertainties (1σ). Fig. shows the flux density for the
Table 14: Fitted parameters for four abundance models.
aTemperature in units of 109Kball for 74 DoFcthe electron number density (m−3) fitted curve for model B and for the observations as a function of energy.
Figure 6: Background X-ray spectrum. See Table for list of observations.
The dashed (black) line is best fit from 10 keV to 300 keV.
Most of the X-ray flux below 10 keV and part of the flux just above 10 keV is emission from discrete sources. The deviation from the curve at energiesabove about 300 keV arises from X-rays coming from discrete sources. In theintermediate region where bremsstrahlung should dominate, there are clear signsof some minor systematic errors. In addition, there appears to be some variationbetween the data sets. It is not clear whether the discrepancy between theobserved points and the predicted flux densities is due to an inadequate theory,inadequate X-ray emission model, or systematic errors in the observations. Afterall the measurements are very difficult and come from a wide range of rocket, Table 15: Fitted parameters for model B.
1.55 ± 0.01 m−3 1.35 ± 0.01 m−3 Electron temperature (2.62 ± 0.04) × 109 K Power law value at 1 Mev balloon and satellite experiments. In particular, the recent HEAO results differ from earlier results reported by It is apparent from Table that although the measured temperature is relatively insensitive to the assumed abundance the density estimate is stronglydependent. This is because bremsstrahlung depends mainly on the number andto a lesser extent the type of charged particles whereas the density also dependson the number of neutrons in each nucleus.
The power law fit parameters are the same for all the models and are shown in Table for model B. This model was chosen because it uses the standardabundances that might be expected in the plasma and it has a relatively goodfit to the observations. The quoted errors are the formal uncertainties of the fit.
There are certainly larger, unknown systematic errors.
For the measured density of N = 1.55 ± 0.01 m−3 the calculated value of the Hubble constant is H = 64.4 ± 0.2 km s−1 Mpc−1. Further properties of the uni-verse based on this density are shown Table That said, this bremsstrahlungmodel for the background X-ray emission within CC provides a good fit to therelevant observations. A crucial test of CC is that it predicts a temperatureof 2.56 × 109 K for the cosmic plasma. The temperature estimated from fit-ting the X-ray data is (2.62 ± 0.04) × 109 K. There is remarkable agreementbetween these values. It should be emphasized that the predicted temperatureis a pure prediction from the theory without any dependence on observations.
This agreement and the good fit to the observations gives strong support to CC.
In CC the argument against bremsstrahlung based on the Sunyaev–Zel'dovich effect is not valid because the density of the gas is much less and the CMBRhas a different source. It has been shown that the X-ray data in the range fromabout 10 Kev to about 300 kev can be explained by bremsstrahlung from thecosmic gas. The fitted temperature was 2.62±0.04×109 K and the fitted densitywas N = 1.55 ± 0.01 hydrogen atoms per cubic meter (2.57×10−27 kg m−3).
At present BB does not have a good explanation for background X-ray ra- diation in the intermediate range of energies. Curvature cosmology can com-pletely explain these observations as coming from bremsstrahlung in a hot cos-mic plasma.
Cosmic microwave background radiation The cosmic microwave background radiation (CMBR) is one of the major suc-cess stories for BB. The observed radiation has a spectrum that is extremely (bynormal cosmological standards) close to a black body spectrum which meansthat it can be described by a single parameter, its temperature. Observations of the CMBR spectrum were obtained from the FIRAS instrument on the Cobesatellite by They measured the temperature of the CMBRto be 2.725 K. This temperature is in agreement with the observations of who measured a temperature of 2.729(+0.023, −0.031) K usingcyanogen excitation in diffuse interstellar clouds. It must be remembered thatthere is nothing special about a black body spectrum. If the radiation is quan-tized and all energy levels are freely available the black body (Plank function)is the thermal equilibrium spectrum. It is the maximum entropy solution. Theblack body is only required to permit the number of photons in each energylevel come into equilibrium. Thus in a general sense the black body spectrumis the default spectrum.
In BB the CMBR is the relic radiation that has been redshifted from the hightemperature radiation that was in equilibrium with matter at the time when theions combined with electrons to produce neutral atoms which are transparent.
This decoupling of the radiation from matter occurred at a redshift of aboutz = 1000. The exact temperature of the subsequent CMBR depends on thebaryonic density parameter. Over time the redshift of the photons results in adecrease of their energy corresponding to an identical decrease in temperaturewithout changing the shape of the spectrum. Thus BB predicts a black bodyspectrum with only a poor estimate of its temperature.
In CC, the CMBR comes from the curvature-redshift process acting on thehigh-energy electrons and ions in the cosmic plasma. Examination of Eq. shows that even for very high temperature plasma the emission from electronswill dominate that from other ions. The energy loss occurs when an electronthat has been excited by the passage through curved spacetime interacts with aphoton or charged particle and loses its excitation energy. Ideally, the theoreticalmodel would provide the number distribution of secondary photons as a functionof their energy. This distribution would then be combined with the distributionof electron energies to obtain the production spectrum of low-energy secondaryphotons from the plasma. The final step would be to integrate this productionspectrum over all distances allowing for the geometry and curvature redshift.
The result would be the spectrum of photons that would be observed at anypoint in space.
We assume that the production spectrum for the photons is peaked at much larger energies than the cosmic microwave background photons. Then the cos-mic microwave photons are seen to have had many curvature-redshift interac-tions. At each of these interactions the photons lose a small fraction of energy tovery low energy photons that have frequencies less than the plasma frequency.
Thus these low energy secondaries only exist as evanescent waves with their en-ergy heating the plasma. Now since the radiation field is quantized the choice ofthe precise frequency of the large secondary photon is controlled by the numberof available modes of oscillation. Being evanescent the very low energy pho-tons are not relevant and the number of modes of oscillation is determined bythe wavelength of the large secondary photon. Thus although the average en- ergy loss rate is determined by the average density the selection of the numberand occurrence of individual interactions depends on the quantization of theradiation field. All modes are available and in equilibrium the rate of photonsentering a mode will equal the rate of photons leaving a mode. Because of thevery large number of curvature-redshift interactions that have occurred sincethe original photon was produced the distribution of number of photons in eachmode is essentially determined by the curvature-redshift interactions and notby the original spectrum. Thus the observed spectrum will be the maximumentropy spectrum determined by the allowed modes of the radiation field. Inequilibrium there is a constant energy density for these photons and as originallyshown by Einstein the maximum entropy solution is that fora black body with a well defined temperature. The assumption of equilibriumenables us to equate the energy loss by the electrons to the energy loss by theCMBR photons and then to use Stefan's law to determine the temperature ofthe CMBR.
This brings up the problem of how the excited electrons produce the CMBR photons. Since conservation of energy and momentum prevents an excited freeelectron from emitting a photon, there must be an interaction with a third par-ticle. A quick calculation shows that Thompson (Compton) scattering with theexisting CMBR photons is too infrequent. The only other suitable interactionis Rutherford scattering off other electrons and ions. Since its last gravitationalinteraction, the electron will have become excited and have an excess of energydue to its passage through curved spacetime. At the Rutherford scattering thisexcitation energy is transferred to secondary photons which become the CMBRafter many curvature-redshift interactions.
The balance between the energy loss by the X-ray electrons and the energy loss by the CMBR photons implies that there is an overall conservation of energywith the photon energy loss being returned to the electrons. Since the secondaryphotons produced by curvature-redshift interaction of the CMBR photons havefrequencies well below the plasma frequency (of about 10 Hz), their energy mustgo into plasma waves which are dissipated with their energy going to heat theplasma. These processes are not driven by temperature differences so that thereis no change in entropy.
For equilibrium, the energy gained by these secondary photons must equal the energy lost by the electrons. Since the dominant energy loss by photonsin the cosmic space is via curvature redshift, we can equate the two loss ratesto determine the average energy of these photons. For electrons, or indeed anynon-zero rest mass particle, the energy loss rate is given by Eq. Thus theenergy loss rate for an electron is )1/2(γ − 1) mec2, where to prevent confusion with the symbol for temperature the electron's ki-netic energy is denoted by ε = (γ − 1)mec2 and the an extra factor of β comesfrom conversion of distance rate to time rate. The next step is to average thisenergy loss over the distribution of electron energies. Since the electrons arerelativistic, the appropriate distribution is J¨ uttner distribution, which is Table 16: Some values for function f (T ).
With a change of variable to γ it becomes n(γ) dγ ∝ γ(γ2 − 1)1/2 exp Then integrating Eq. over all the electron energies we get dε = HNemec2f(Te), where Ne is the density of the electrons and f (Te) is average of the γ terms.
Where we have γ2 − 1 1/2 β3(γ − 1) n(γ) dγ uttner distribution can be integrated analytically in terms of modified Bessel functions, it is just as easy to evaluate both integrals numeri-cally. Table shows some values for the function f (Te) as a function of theelectron temperature Te.
Consider the CMBR photons at one point in space. All of these photons will have been produced in one of the shells surrounding that point. In a staticcosmology the contribution from each shell depends only on the thickness of theshell and is independent of the radius of the shell. However in CC there is anenergy loss due to curvature redshift which means that the average energy thatcomes from a shell at redshift z is reduced by the factor (1 + z)−1. Thus theaverage energy at the select point is less than the average energy of productionby the integration of this factor with respect to distance. By using Eq. toconvert from distance to redshift the ratio is Thus the electron energy loss rate must be 3 times larger than the energy loss rate by the CMBR photons. This is because the CMBR photons have already lost a major part of their energy since production during which timetheir spectrum is transformed in that for a black body.
The next step is to calculate the energy loss rate for the CMBR photons.
If the CMBR photons are the result of curvature redshift acting on the cosmicelectrons and the system is in equilibrium these two loss rates should be equal.
For a black body spectrum then the energy density of the CMBR photons nearus must be the same as that for a uniform black body radiation with the sametemperature. However, because the universe is homogeneous, the energy densitymust be the same everywhere. Then using Eq. and Stefan's equation we get where σ is the Stefan-Boltzmann constant and, not surprisingly, the Hubbleconstant cancels. Then from Table we get Ne = 1.35 and for a temperatureof (2.62±0.04)×109 K the calculated value of the function f (Te) is 1.215. Thesenumbers result in a predicted temperature for the CMBR of 3.18 K. Probablythe largest error in these temperature estimates comes from the uncertaintyin the nuclear abundances. For the four abundance models (section thepredicted temperatures of the CMBR are 3.48 K, 3.18 K, 2.95 K and 2.74 Kfor the models 1, 2, 3 and 4 respectively. The main dependence is due to thedifferences in the electron density. Another important factor is the assumptionthat the universe has uniform density when it is apparent that it has largedensity variations.
Both cosmologies offer argument to support the black body spectrum. Those forBB are well founded those for CC are less well founded. Against that BB doesnot have a good prediction for the temperature while CC predicts a narrow rangeof temperatures that is in excellent agreement with the observed temperature.
CMBR at large redshifts The temperature of the CMBR has been measured at large redshifts using twodifferent methods. The first method measures the column density ratio of thefine structure absorption lines originating from the fundamental and first ex-cited states of carbon These lines are seen in theLyman-α forest that is observed in the spectra from a bright quasar. The tem-perature estimate is based on the relative strengths of these spectral lines. Forthese measurements to be valid, it is essential that the line widths and columndensities are well understood. In CC the width of a spectral line is increased bythe differential redshift as the photons traverse the absorbing gas. This changein the widths of spectral lines makes the very complex interpretation of the spec-tra required to estimate the temperature of the radiation suspect. Thus, untilthis interpretation is fully understood in the context of CC, CMBR temperatureresults from this method cannot be trusted.
The second method uses the Sunyaev–Zel'dovich effect acting on the CMBR by the gas in clusters of galaxies By using multiple frequencies, it is possible to minimize the effects due to properties of the clusterson the result. However the method is flawed in CC because the CMBR has adifferent cause from that in Big-Bang cosmology. Thus, these results cannot betaken as showing a dependence of the temperature of the CMBR on redshiftuntil the complete mechanism is understood in the context of CC.
Fluctuations in the CMBR One of the arguments for the interpretation of the CMBR in BB is that thereare extensive models that can explain the density and polarization of spatialfluctuations in the observed radiation. In the model proposed for curvatureradiation these fluctuations will also occur but in this case they are due tovariations in the density of the cosmic plasma. The CMBR seen through thedenser gas within a galactic cluster will have lower than average temperature.
show some support for this model in that they have correlateddata from the Wilkinson Microwave Anisotropy Probe (WMAP) with galaxysamples from the SDSS DR4 galaxy survey and found a significant correlationfor the intensity fluctuations with galaxy density.
The theoretical problems with dark matter have already been canvassed. Herewe concentrate on the observational evidence. All observational evidence fordark matter comes from the application of Newtonian gravitational physics toeither clusters of objects or the rotation of galaxies. Galaxy rotation will bedealt with in Section The original concept for dark matter comes fromapplying the virial theorem to the Coma cluster of galaxies Thevirial theorem is a statistical theorem that states that for aninverse square law the average kinetic energy of a bound system is equal to halfthe potential energy (i.e. 2T + V = 0). Then with knowing the linear size of thecluster and measuring the mean square spread of velocities we can estimate thetotal mass of the cluster. There is no doubt that applying the virial theoremto the Coma and other clusters of galaxies provides mass estimates that canbe several hundred times the mass expected from the total luminosity. Eventhe mass of inter-galactic gas is not enough to overcome this imbalance. In BBcosmology dark matter has been introduced to make up for the shortfall of mass.
However if CC is valid then it is possible that the observed redshifts are not due to kinematic velocities but are curvature redshifts produced by the inter-galactic gas. The purpose of this section is to show that curvature redshift canexplain the galactic velocities without requiring dark matter. For simplicity, wewill use the Coma cluster as a test bed. Not only is it very well studied, butit also has a high degree of symmetry and the presence of an inter-galactic gascloud is known from X-ray observations. and have fitted the density of the gas cloud to an isothermal-model with the form with a center at 12h59m10s, 27◦5905600 (J2000) and with re = 8.80 ± 0.70, α =1.37 ± 0.09, ρ0 = (2.67 ± 0.22) × 103h2 m H m−3. The central density is obtained Table 17: Coma velocity dispersions for some distances.
Dispersion /km s−1 from the X-ray luminosity and has a strong dependence on the distance. assumed a Hubble constant of 50 km s−1 Mpc−1. With a meanvelocity of 6,853 km s−1 and with this Hubble constant,the distance to the Coma cluster is 137 Mpc. Recently using theTully–Fisher relation to measure the distance modulus to the galaxies in theComa cluster, to observe a value of 34.4±0.2 mag whereas using infrared surface brightness fluctuations get 34.99±0.21 mag. The averageis 34.7 mag that corresponds to a distance of 87.1 Mpc. This is consistent withthe distance of 85.6 Mpc given by Thus putting h = 0.7gives a corrected central gas density of ρ0 = (6.61 ± 0.54) × 10−3 mH m−3.
The galactic velocity data are taken from who provide information for 583 galaxies. The velocity centroid of theComa cluster is 12h59m19s, 27◦520200 (J2000). They find that early-type galaxies(E+S0+E/S0) have a mean velocity of 9,926 km s−1 and a rms (root-mean-square) velocity, dispersion velocity, of 893 km s−1. Let us assume that all thegalactic velocities are due to curvature redshift. That is we assume that theactual velocities, the peculiar velocities, are negligible. Then the redshifts forthe galaxies are calculated (in velocity units) by N (Z) dZ km s−1, where Z is the distance from the central plane of the Coma cluster to the galaxymeasured in Mpc, N (Z) is the density of the inter-galactic gas cloud and v0 isthe average velocity of the galaxies in the cluster. The problem here is thatwe do not know Z distances. Nevertheless, we can still get a good estimateby assuming that the distribution in Z is statistically identical to that in Xand in Y . In a Monte Carlo simulation, each galaxy was given a Z distancethat was the same as the X (or Y ) distance of one of the other galaxies in thesample chosen at random. For 50 trials, the computed dispersion was 554 km s−1which can be compared with the measured dispersion of 893 km s−1. Curvaturecosmology has predicted the observed dispersion of galactic velocities in theComa cluster to within a factor of two. Considering that this is a predictionof the cosmological model without fitting any parameters and ignoring all thecomplications of the structure both in the gas and galactic distributions theagreement is remarkable.
Since the distance to the Coma cluster is an important variable, the com- puted velocity dispersion from the Monte Carlo simulation for some differentdistances (all the other parameters are the same) is shown in Table Thus,the redshift dispersion (in velocity units) is approximately a linear function ofthe Coma distance. This is not surprising since in this context the distance ismainly a scale factor.
note that a better fit to the velocity distribution is provided by the sum of two Gaussian curves.
parameters for these two Gaussians are v1 = 7, 501 ± 187 km s−1, with σ1 =650 ± 216 km s−1 and v2 = 6641 ± 470 km s−1, with σ2 = 1, 004 ± 120 km s−1.
This double structure is supported by who argue for anongoing merger between two sub clusters centered in projection on the dominantgalaxies NGC 4874 and NGC 4889. In addition, found evidence for substructure in the X-ray emission and and have measured the X-ray luminosityof individual galaxies in the Coma cluster showing that the model for the gasused above is too simple. The net effect of this substructure is that the observedvelocity dispersion would be different from that predicted by a simple symmetricmodel. Thus, it appears that substructure makes it very difficult to achieve amore accurate test of CC using the Coma cluster.
There is an important difference between curvature redshift and models that assume that the redshifts of the galaxies within a cluster are due to their veloc-ities. Since the laws of celestial mechanics are symmetric in time, any galaxycould equally likely be going in the opposite direction. Thus a galaxy with ahigh relative (Z) velocity could be in the near side of the cluster or equally likelyon the far side of the cluster. However, if the redshifts are determined by curva-ture redshift then there will be a strong correlation in that the higher redshiftswill come from galaxies on the far side of the cluster. A possible test is to seeif the apparent magnitudes are a function of relative redshift. With a distanceof 87.1 Mpc the required change in magnitude is about 0.025 mag Mpc−1. Asimple regression between magnitude of Coma galaxies (each relative to its typeaverage) and velocity did not show any significant dependence. Although thiswas disappointing, several factors can explain the null result. The first is thepresence of substructure; the second is that the magnitudes for a given galac-tic type have a standard deviation of about one magnitude, which in itself issufficient to wash out the predicted effect; and thirdly mistyping will produceerroneous magnitudes due to the different average velocities of different types.
In support of the second factor we note that for 335 galaxies with known typesand magnitudes, the standard deviation of the magnitude is 1.08 mag and if weassume that the variance of the Z distribution is equal to the average of thevariances for the X and Y distributions then the expected standard deviationof the slope is 0.076 mag Mpc−1. Clearly, this is such larger than the expectedresult of 0.025 mag Mpc−1. It is expected that better measurements or newtechniques of measuring differential distances will in the future make this a veryimportant cosmological test.
In BB observations of the velocity dispersion of clusters of galaxies cannot be explained without invoking an ad hoc premise such as dark matter. HoweverCC not only explains the observations but also makes a good prediction, withoutany free parameters, of its numerical value.
The Sunyaev–Zel'dovich effect The Sunyaev–Zel'dovich effect isthe effect of Thompson scattering of background radiation by free electrons inthe intervening medium. The technique depends on knowing the spectrum ofthe background source and then measuring the changes in the spectrum dueto the intervening plasma. In particular, it is the scattering in both angle andfrequency of the cosmic microwave background radiation (CMBR) by electrons in the cosmic plasma. Because of the rapidly increasing density (like (1 + z)3)with redshift this is an important effect in BB.
The effect is often characterized by the dimensionless Compton y-parameter, which for a distance x through non-relativistic thermal plasma with an electrondensity of Ne has the value σT Nex = 3.46 × 10−16NeTex Mpc, where σT is the Thompson cross-section. An object at redshift z is at the distance x = Rχ = 5.80 × 103Ne log(1 + z) Mpc. Hence, using Te = 2.62 × 109 K, Ne = 1.35 m−3 we get y = 9.2 × 10−6 log(1 + z).
Using the CMBR as a source the Sunyaev–Zel'dovich effect has been ob- served and report an observed upper limit of y = 0.001,and more recently report y = 1.5 × 10−5. Using this limitwith Eq. shows that there is no effect in CC if z < 4.1. Although in CCthe CMBR has a more local origin it is of interest to note that this analysisassumes that each photon has many Compton interactions. For this electrondensity the Compton mean free path is 575 Gpc whereas the distance to z = 4.1is about 3.7 Gpc which means that a negligible number of the photons will havean interaction with the high temperature electrons. Furthermore the photonenergy distribution for a single interaction has a different spectrum for that forthe normal Sunyaev–Zel'dovich effect extend the results of to show thatnot only was the Sunyaev–Zel'dovich effect less than what was expected butthat it tendered to disappear as the redshift went from 0.1 to 0.3. The conclu-sion is that CC is completely consistent with the experimental observations ofthe Sunyaev–Zel'dovich effect on the CMBR. The conclusion is that althoughthe Sunyaev–Zel'dovich effect is important in BB it is not important in CC.
Gravitational lensing There are more than 50 known gravitational lens where a quasar or distantgalaxy has one or more images produced by a nearer lensing galaxy or clusterof galaxies. A set of these lensing systems has been examined in the contextof CC to see if it offers a consistent and possibly simpler explanation. The twoimportant measures are the prediction of the mass of the lensing galaxy and thedetermination of the Hubble constant from the time delays between variationsin the luminosity of different images. Since the delay measurement is easily doneall that is needed is to measure the different path lengths. This path differenceinvolves both geometric and general relativistic corrections.
One of the remarkable properties of gravitational lenses is that the geometry is completely determined by a two-dimensional lensing potential which can beexpressed in terms off a surface density at the position of the lensing galaxy.
For thin lenses, any two systems with the same surface density distribution havethe same lens effect. Now the usual way to determine the surface density is tomeasure the widths of spectral lines, assume that the width is due to velocityand then use the virial theorem to obtain the surface density. However in CC thewidths of spectral lines are likely to have a large component due to the effectsof curvature redshift from dust and gas in the lensing object. Thus the widthsare not a reliable measure of area density and this method cannot be used.
Instead some double image gravitational lens were investigated using a verysimple point sources lens in order to see if the observations could be consistentwith CC. However because of the paucity of examples and the wide range ofcharacteristics there was no test that showed a significant difference betweenBB and CC. The data was consistent with both cosmologies. Currently themodeling and the data are not sufficient to choose between the cosmologies.
However a more thorough analysis within the paradigm of CC may be moredefinite.
Lyman alpha forest The Lyman-α (Lyα) forest is the large number of absorption lines seen in thespectra of quasars. Most of the lines are due to absorption by clouds of neutralhydrogen in the line of sight to the quasar. Some of the lines are due to otherelements or due to Lyman-β absorption. Because of the redshift between theabsorbing cloud and us, the lines are spread out over a range of wavelengths.
Usually the analysis is confined to lines between the Lyα (at a wavelength of121.6 nm) and Lyβ (at 102.5 nm). Thus, each quasar provides a relativelynarrow spectrum of Ly-α lines at a redshift just less than that for the quasar.
Since the advent of spacecraft telescopes, in which can observe the ultravioletlines, and by using many quasars the complete redshift range up to the mostdistant quasar has been covered. The large redshift range makes the Lyman αspectra potentially a powerful cosmological tool.
The obvious cosmological observation is the density of lines as a function of redshift but as discussed by in an excellent review, there are manyimportant observational problems. The first, which has now been overcome, isthat the spectra must have sufficient resolution to resolve every line. The secondis that most lines are very weak and the number of resolved lines can dependgreatly on the signal to noise ratio.
This is accentuated because the steep spectrum for the density of lines as a function of their strength means thata small decrease in the acceptance level can drastically increase the number ofobserved lines. The third problem is that each quasar only provides a set of linesin a narrow range of redshift and there are considerable difficulties in gettinguniform cross-calibrations. In addition to these problems, it will be shown thatcurvature redshift can have a profound effect on the interpretation of the linewidths and column densities.
Since in CC the distribution of clouds is independent of time or distance the expected density of lines as a function of redshift is where N0 is the volume density and A is the average area of a cloud. Mostobservers have fitted a power law with the form (1 + z)γ to the observed linedensities with a wide range of results. They vary from γ = 1.89 to γ = 5.5All of which are inconsistent with the CC prediction of γ = −1.
In CC there is the additional effect that much of the line broadening may be dueto curvature redshift. Curvature redshift will be operating within the clouds sothat the observed line width will be a combination of the usual Voigt profileand the change in the effective central frequency as the photons pass throughthe cloud. If the cloud has a density ρ(x) at the point x, measured along the photon trajectory then the change in frequency from the entering frequency dueto curvature redshift is In units of N (x) = ρ(x)/mH this is (with N in m−3 and dx in kpc) 1.724 × 10−7pN (x)dx.
Then the final profile will be the combination of the natural line width, theDoppler width due to temperature, any width due to bulk motions and thecurvature redshift width. Now assuming pure hydrogen, the hydrogen columndensity is given by NH = R N (x)dx. Although it is unlikely that the line ofsight goes through the center of the cloud, it is reasonable to expect a roughlysymmetric distribution of gas with a shape similar to a Gaussian. We can definean effective density width by (x − x)2 N (x)dx / Also define N0 = NH /xw and an effective velocity width ∆v = 51.68ηxw N0and where η is a small numeric constant that depends on the exact shape of thedensity distribution. Eliminating the central density, we get (with xw in kpc) ∆v2 = 8.656 × 10−17η2NH xw.
For values NH = 1019 m−2, xw=1 kpc and with η=1 we get ∆v=29 km s−1.
Since there is a wide variation in column densities and the effective widthsare poorly known it is clear that curvature redshift could completely dominatemany of the Lyman-α line widths and the others would require a convolutionof the Doppler profile with the curvature redshift density effect. What is alsoapparent is that the very broad absorption lines may be due to curvature redshiftacting in very dense clouds. Although there is uncertainty about the observedrelationship between the line width and the column density, we note that for afixed effective density width, Eq. predicts a square relationship that may becompared with the exponent of 2.1 ± 0.3 found by Clearly,there needs to be a complete re-evaluation of profile shapes, column-densities,and cloud statistics that allows for the effects of CC. We must await this analysisto see whether the Lyman-α forest can provide a critical test of CC.
The Gunn–Peterson trough in high redshift quasars The Gunn–Peterson trough is a feature of the spectra of quasars probably dueto the presence of neutral hydrogen in the intergalactic medium. The troughis characterized by suppression of electromagnetic emission from the quasar atwavelengths less than that of the Lyman-α line at the redshift of the emittedlight. This effect was originally predicted by Althoughthe Gunn–Peterson-trough has now been seen in several high redshift quasarsit is notseen in all quasars and appears to be strongly dependent on redshift. In BB the explanation is that it is only seen in very high redshift quasars where theintergalactic medium is still neutral but it is not seen in closer quasars becausethe medium has been re-ionized. In CC it has a more prosaic explanation.
We assume that the quasar is surrounded by a large halo or it lies within a cluster of galaxies which has, like many clusters, an internal gas cloud. Thehypothesis is that the halo or gas cloud is cool enough to have a small butimportant density of neutral hydrogen that absorbs much of the quasar emissionat shorter wavelengths than the Lyman-α emission. The important point is thatas the radiation traverses the cloud it is redshifted by curvature redshift due tothe density of the cloud. The Lyman-α radiation from the quasar is redshiftedby the curvature redshift due to the density of the whole cloud. The absorptionlines are only shifted by part of the cloud and appear at a shorter wavelengththan the quasar Lyman-α emission. Consider the scattering probability of aphoton with wavelength λ by the neutral hydrogen. The differential probabilityis N (r)g (ν − να) dr, where r0 is the classical electron radius, fH is the fraction of neutral hydrogenatoms, f is the oscillator strength (here f = 0.416), N (r) is the gas density andg is the profile function that is strongly peaked about ν = να. At a distance rinside the cloud ν = νq exp (−ar) where νq) is its frequency at the beginning ofthe cloud and = 5.588 × 10−27pn (r) m−1.
A change of variable from r to ν and noting that the width of g(ν) is smallcompared to the value of να and assuming a constant density N = N (r) resultsin the approximation = 8.01 × 104fH N .
Clearly all the photons are scattered for quite low densities and a very smalldensity of hydrogen atoms.
Having shown that total scattering is feasible the next step is to see why the effect is more pronounced for high redshift quasars. The basic equation forcurvature redshift is 1 + z = exp(cr/H) where r is the distance, H is the Hubbleconstant and dz = a(1 + z)dr. Thus the wavelength range that is observedover which the Gunn–Peterson effect is seen should scale as (1 + z). Thus theprobability of detection which depends on the wavelength range will be lowerat smaller redshifts. Naturally the effect may not be seen or could be partiallyobscured depending on the individual characteristics of the cloud around eachquasar.
report details of the quasar SDSS 1030+0524 which has an emission redshift of 6.29 and a Gunn–Peterson depression that has a widthdz = 0.21. With this value we find that it can be explained by a cloud thathas the constraint N 1/2dr = 167 where dr is in Mpc. With dr = 2 Mpc therequired density is 7 × 103 m−3 which is a feasible size and density for a largecluster cloud. Note that if the gas has high enough temperature fH will be toosmall and no effect will be seen.
Nuclear abundances One of the successes of BB is in its explanation of the primordial abundancesof the light elements. Since the proposed CC is static, there must be anothermethod of getting the ‘primordial' abundances of light elements. In CC, theprimordial abundance refers to the abundance in the cosmic gas from whichthe galaxies are formed. The first point to note is that in CC the predictedtemperature of the cosmic gas is 2.56 × 109 K at which temperature nuclearreactions can proceed. The major difference with the production of helium anddeuterium in the BB early universe is that the densities were incredibly higherin BB than they are in CC. It is postulated that in CC there is a continuousrecycling of material from the cosmic gas to galaxies and stars and then backto the gas. Because of the high temperature, nuclear reactions will take placewhereby the more complex nuclei are broken down to hydrogen, deuterium,and helium. Although this cycling can take many billions of years the verylow density of the gas means that the cycle time may not be long enough forthe nuclei densities to achieve statistical equilibrium. In addition, the majorreactions required are the breaking down of heavier nuclei to lighter ones andnot those that construct nuclei. It is through the interactions of cosmic gas inCC that the light nuclei abundances are produced.
Galactic rotation curves One of the most puzzling questions in astronomy is: why does the observedvelocity of rotation in spiral galaxies not go to zero towards the edge of thegalaxy. Simple Keplerian mechanics suggests that there should be a rapid riseto a maximum and then a decrease in velocity that is inversely proportional tothe square root of the radius once nearly all the mass has been passed. Althoughthe details vary between galaxies, the observations typically show a rapid riseand then an essentially constant tangential velocity as a function of radius outto distances where the velocity cannot be measured due to lack of material. TheBB explanation is that this is due to the gravitational attraction of a halo ofdark matter that extends well beyond the galaxy. We examine whether thisrotation curve can be explained by curvature redshift.
Observations show that our own Galaxy and other spiral galaxies have a gas halo that is larger than the main concentration of stars. It is clear that if theobserved redshifts are due to curvature redshift acting within this halo, the halomust be asymmetric; otherwise, it could not produce the asymmetric rotationcurve. Now the observed velocities in the flat part of the curves are typically100 to 200 km s−1. The first step is to see if curvature redshift provides theright magnitude for the velocity. For a gas with an average density of NH the predicted redshift (in velocity units) is 5.17 × 10−2d N km s−1 where d is thedistance in kpc. For realistic values of d = 10 kpc and N = 1.0 × 105 m−3 thevelocity is 163 km s−1. Thus, the magnitude is feasible. Although there could bea natural asymmetry in a particular galaxy, the fact that the flattened rotationcurve is seen for most spiral galaxies suggests that there is a common cause forthe asymmetry. One possibility is that the asymmetry could arise from rampressure. Since most galaxies are moving relative to the cosmic medium, it isexpected that there will be an enhanced density towards the leading point of thegalaxy. This asymmetric density could produce an apparent velocity gradient across the galaxy that could explain the apparent rotation curve. Naturally,there would be range of orientations and the apparent velocity gradient mustbe added to any intrinsic rotation curve to produce a wide diversity of results.
Thus, curvature redshift could explain the galactic rotation curves if there is anasymmetric distribution of material in the galactic halo. Both cosmologies haveproblems with galactic rotation curves. BB not only requires dark matter butdoes not have any definite models for its distribution. Curvature cosmology hasthe problem of achieving sufficient asymmetry to mimic a rotation curve.
Redshifts in our Galaxy In our Galaxy, the Milky Way, there is an interesting prediction. The density ofthe inter-stellar ionized gas is high enough to inhibit curvature redshift for radiofrequencies. From Eq. it was shown that for wavelengths longer than about20.6N −1/2 m the effect of refractive index in fully ionized plasma will inhibitcurvature redshift. The refractive index of neutral hydrogen is too low to inhibitcurvature redshift. However, any fully ionized plasma with N > 104m−3 willinhibit curvature redshift for the 21 cm hydrogen line. Since the local interstellarmedium has an electron density of about 105 m−3 curvatureredshift will be inhibited for the 21 cm hydrogen in regions of the galaxy nearthe sun. Thus for sight lines close to the Galactic plane we can assume a similardensity and thus a similar inhibition with the result that the observed radioredshifts can be correctly interpreted as velocities. Thus, there is little changeneeded to the current picture of Galactic structure and rotation derived from 21cm redshifts. However, there may be some curvature redshift present in sightlines away from the plane and especially in the Galactic halo.
Since optical redshifts have the full effects of curvature redshift, it should be possible to find objects with discrepant redshifts where the optical redshift isgreater than the radio redshift. The difficulty is that the two types of radiationare produced in radically different environments: the optical in compact hightemperature objects, such as stars, and the radio in very low-density cold clouds.
In addition, there is the complication that within the galactic plane, opticalextinction due to dust limits the optical range to about one kpc.
Curvature redshift may help to explain an old stellar mystery. There is a long history provided by of observations of anomalous redshifts inbright hot stars, which is called the K-term or K-effect. states thatB0 stars typically show an excess redshift of 5.1 m s−1, A0 have 1.4 km s−1 andF0 have 0.3 km s−1. This can be explained if these stars have a large coronathat produces a curvature redshift. It is probably no coincidence that such starshave large stellar winds and mass outflows. In order to see if it is feasible letus consider a simple model for the outflow in which the material has a constantvelocity v0, and conservation of matter (Gauss's Law) then requires that thedensity has inverse square law dependence. Although this is incorrect at smallstellar radii, it is a reasonable approximation further from the star. Then if ρ1is the density at some inner radius r1, then integration of Eq. out to a radiusr2, the expected redshift in velocity units is M is the observed stellar mass-loss-rate. Then with ˙ M in solar masses per year, with v and v0 in km s−1, the redshift is M = 10−5M yr−1 v0 = 1 km s−1 and r2/r1 = 103 the predicted redshift (in velocity units) is 2 km s−1 which is in reasonable agreementwith the observed K-effects mentioned above.
Anomalous redshifts have argued that there is strong observational evi-dence for anomalous redshifts between quasars and galaxies. Typically there is aquasar very close to a galaxy with a material bridge or other evidence that sug-gests that they are associated. report on five X-ray emittingblue stellar objects located less than 120 from the X-ray Seyfert galaxy NGC3516. In this case the association is that the objects lie close to a straight lineon either side of the galaxy and that their redshifts are proportional to log(θ)where θ is the angular distance from the central galaxy. Furthermore the line ofobjects is within a few degrees of the minor axis of NGC 3516. The measuredredshifts are 0.33, 0.69, 0.93, 1.4 and 2.1. NGC 3516 is a barred spiral galaxyand it has a redshift of 0.00884.
Can CC explain this redshift anomaly? If the objects are seen through a large dense cloud, such as a galactic halo, then curvature redshift will produce anextra redshift due to the photons passage through the cloud. the extra redshift,δ, is δ = 1.72 × 10−10 where N (x) is the number density and distances are measured in pc. If z is thecosmological redshift then the extra-observed redshift is ∆z = (1 + z)(eδ − 1).
In order to achieve an extra redshift δ ≈ 1 with a distance of 104 pc the gasnumber density must be about 3 × 1011 m−3. Now although cold interstellarmolecular clouds can have densities reaching this value it is still a very highdensity. But if the size is increased by a factor of two the required density isdecreased by a factor of four. Moreover the objects with the largest redshiftsare the furthermost away from the galaxy. These redshifts could be explainedby curvature redshift in a very large, very dense galactic halo with a hole in themiddle. Since NGC 3516 has a very low redshift and is seen nearly face on theimplication is that this gas cloud is probably shaped like a torus and it lies inthe galactic plane of NGC3516. A further test is to compare an estimate of themass of this torus with that for a typical galaxy. Since a torus formed by therotation of a circle with radius r about a axis in the plane of the circle where theradius of rotation is R, its volume is V = 2π2Rr2. With R and r in kpc and anaverage density of N its mass is M = 0.484Rr2N M. Then with R = 15 kpc,r = 10 kpc and N = 3 × 1011 the mass is 2 × 1014M which considerably largerthan a normal galaxy. Since these anomalous redshifts are completely outsideany BB model the only reason that these observations are not fatal to BB istheir the controversial nature.
Table 18: Velocity at, and average velocity within various projected radii in theComa cluster (distance = 87.1 Mpc).
aprojected radius If CC is valid then the redshift of the galaxies in the Coma cluster (Section will have been increased, on average, by the additional redshift due to the inter-galactic gas. Thus, they will have, on average, a larger redshift than an isolatedgalaxy at the same distance. Table shows the predicted (effective) velocity fora galaxy in the center plane of the Coma cluster as a function of the projectedradius. The second column is the velocity at that exact radius and the thirdcolumn shows the average velocity of galaxies (uniformly spread in area) withinthat radius. This simulation also showed that the average velocity offset for thegalaxies in the Coma cluster is 1206 kms−1 which means that the redshift of thecenter of the Coma cluster is 6926-1206=5720 kms−1. This offset is importantfor calculating the Hubble constant which from these figures is 5270/87.1=65.7kms−1 Mpc−1.
In addition, the redshift of objects seen through a cluster will be increased by curvature-redshift from the inter-galactic gas. claim to have seen this effect. They examined radio galaxies and classified theminto region A if their light does not pass through a cluster and region B iftheir light passes through a cluster. They found no significant differences inmagnitudes between the two regions but they did find a significant differencein the average redshifts that was consistent over the complete range. Theirresult is that radio galaxies seen through a cluster had an average extra redshift(in velocity units) of 2412±1327 km s−1. Overall the difference in the distancemodulus was µ = 0.16 ± 0.04, which is just significant. Since the density anddistribution of the gas in the clusters is unknown and the limiting radius of thecluster is not stated it is impossible to get an accurate prediction. Nevertheless,we note that for the Coma cluster with a radius of 2 Mpc the average extraredshift (from Table with a factor of two) corresponds to 1866 km s−1 showingthat curvature-cosmology could explain the effect. In a different study, and compared the magnitude of the brightest CURVATURE COSMOLOGY THEORY galaxy in a cluster with that in another cluster with similar redshift. They foundthat there was no significant difference in magnitudes between clusters but thatthe clusters with the largest number of galaxies had the higher redshift differencebetween the pairs. On average the redshift difference (in velocity units) was292±85 km s−1. This can be explained by the expected correlation betweennumber of galaxies and size and density of the inter-galactic gas. However itshould be noted that these observations have been disputed by In his review of voids in the distribution of galaxies, quotes who observed a large void in the distribution of galaxies in frontof the Coma cluster. This void has a magnitude of about 3000 kms−1, whichalthough somewhat larger, is not inconsistent with the expected value of about1200 km s−1. In other words, the Coma cluster galaxies have an extra curvature-redshift due to the inter-galactic gas. However, the galaxies just outside thecluster nearer to us do not have this extra redshift and would appear to becloser to us. Hence, we see an apparent void in the redshift distribution in frontof the Coma cluster.
A consequence of gas clouds and curvature-redshift is that the distribution of redshifts is similar to but not identical to the distribution of z distances.
Galaxies that are behind a cloud will have a higher redshift than would beexpected from a simple redshift distance relationship. Thus, we would expectto see anomalous voids and enhancements in the redshift distribution. This willbe accentuated if the gas clouds have a higher than average density of galaxies.
show a redshift plot for a region of the skythat includes the Coma cluster. Their data are from the Center for Astrophysicsredshift survey and their plot clearly shows several voids. They suggest thatthe galaxies are distributed on the surfaces of shells. However, this distributioncould also arise from the effects of curvature-redshift in clouds of gas.
Curvature Cosmology Theory Curvature cosmology (CC) is a static tired-light cosmology where the Hubbleredshift (and many other redshifts) is produced by an interaction of photonswith curved spacetime called curvature redshift. It is a static solution to theequation of general relativity that is described by the Friedmann equationswith an additional term that stabilizes the solution. This term called curvaturepressure is a reaction of high speed particles back on the material producing thecurved spacetime. This sense of this reaction is to try and reduce the curvature.
The basic cosmological model is one in which the cosmic gas dominates the massdistribution and hence the curvature of spacetime. In this first order model, thegravitational effects of galaxies are neglected. The geometry of this CC is thatof a three-dimensional surface of a four-dimensional hypersphere. It is almostidentical to that for Einstein's static universe. For a static universe, there isno ambiguity in the definition of distances and times. One can use a universalcosmic time and define distances in light travel times or any other convenientmeasure. In a statistical sense CC obeys the perfect cosmological principle ofbeing the same at all places and at all times.
CC makes quite specific predictions that can be refuted. Thus, any obser- vations that unambiguously show changes in the universe with redshift would CURVATURE COSMOLOGY THEORY Table 19: Published papers First mention of photon extent and gravity Photon decay near the sun: limb effecta Aust. J. Phys.40, 440 First mention of curvature redshiftb Aust. J. Phys., 40, 459 Application to background X-rays Astrophysical Journal, 377, 1 More on curvature redshift and applications Astrophysical Journal, 410, 488 A static stable universe: Newtonian cosmology Astrophysical Journal, 440, 466 Angular size of radio sources Astrophysical Journal, 441, 488 Quasar distribution Aust. J. Phys., 52, 753 Curvature pressure and many other topics "Curvature Cosmology" Major updated of the book aNot only is the theory discredited but also the observations have not stoodthe test of time.
bThis gives the equation for photons but not for non-zero rest mass particles.
chttp://www.davidcrawford.bigpondhosting.comdSuperseded by this paper.
invalidate CC. In CC, there is a continuous process in which some of the cos-mic gas will aggregate to form galaxies and then stars. The galaxies and starswill evolve and eventually all their material will be returned to the cosmic gas.
Thus, a characteristic of CC is that although individual galaxies will be born,live and die, the overall population will be statistically the same for any observ-able characteristic.
This paper is the culmination of many years of work and is a complete re-synthesis of many approaches that I have already published. Because hy-potheses and notations have changed and evolved, direct references to theseearlier versions of the theory would be misleading. Table (all with authorD. F. Crawford) is provided briefly stating each reference and the major topicin each paper. In nearly all cases, the data analyzed in the papers has beensuperseded by the more recent data that are analyzed in this paper.
Derivation of curvature redshift The derivation of curvature redshift is based on the fundamental hypothesis ofEinstein's general theory of relativity that spacetime is curved. As a conse-quence, the trajectories of initially-parallel point particles, geodesics, will movecloser to each other as time increases. Consequently in space with a positive cur-vature, the cross sectional area of a bundle of geodesics will slowly decrease. Inapplying this idea to photons, we assume that a photon is described in quantummechanics as a localized wave where the geodesics correspond to the rays of thewave. Note that this wave is quite separate from an electromagnetic wave thatcorresponds to the effects of many photons. It is fundamental to the hypothesisthat we can consider the motion in spacetime of individual photons. Becausethe curvature of spacetime causes the focussing of a bundle of geodesics, this CURVATURE COSMOLOGY THEORY focussing also applies to a wave. As the photon progresses, the cross sectionalarea of the wave associated with it will decrease. However, in quantum mechan-ics properties such as angular momentum are computed by an integration of aradial coordinate over the volume of the wave. If the cross sectional area of thewave decreases, then the angular momentum will also decrease. However, an-gular momentum is a quantized parameter that has a fixed value. The solutionto this dilemma is that the photon splits into two very low-energy photons anda third that has the same direction as the original photon and nearly all theenergy. It is convenient to consider the interaction as a primary photon losing asmall amount of energy into two secondary photons. Averaged over many pho-tons this energy loss will be perceived as a small decrease in frequency. Sincein quantum mechanics electrons and other particles are considered as waves,a similar process will also apply. It is argued that electrons will interact withcurved spacetime to lose energy by the emission of very low-energy photons.
Photons in Curved Spacetime Einstein's general theory of relativity requires that the metric of spacetime bedetermined by the distribution of mass (and energy). In general this spacetimewill be curved such that in a space of positive curvature nearby geodesics thatare initially parallel will come closer together as the reference position movesalong them. This is directly analogous to the fact that on the earth lines oflongitude come closer together as they go from the equator to either pole. Inflat spacetime, the separation remains constant. For simplicity, let us considergeodesics in a plane. Then the equation for geodesic deviation can be writtenp 30 as where ξ is normal to the trajectory and s is measured along the trajectory. Thequantity 1/a2 is the Gaussian curvature at the point of consideration. For asurface with constant curvature, that is the surface of a sphere, the equationis easily integrated to get (ignoring a linear term) ξ = ξ0 cos(s/a). Note thatthis equation also describes the separation of lines of longitude as we movefrom the equator to either pole.
Now geodesics describe the trajectories of point particles. Null-geodesics are associated with mass-less particles. However,photons are not point particles. The experiment of using single photons in atwo-slit interferometer shows that individual photons must have a finite size.
Quantum mechanics requires that all particles are described by wave functionsand therefore we must consider the propagation of a wave in spacetime. Becausephotons are bosons, the usual quantum mechanical approach is to describe theproperties of photons by creation and destruction operators.
of this approach is on the production and absorption of photons with littleregard to their properties as free particles. Indeed because photons travel atthe speed of light, their lifetime in their own reference frame between creationand destruction is zero. However, in any other reference frames they behavelike normal particles with definite trajectories and lifetimes. haspointed out that the concept of a single photon is rather tenuous. There is noway we can tell the difference between a single photon and a bundle of photons CURVATURE COSMOLOGY THEORY with the same energy, momentum, and spin. However, it is an essential part ofthis derivation that a single photon has an actual existence.
Assume that a photon can be described by a localized wave packet that has finite extent both along and normal to its trajectory. This economic descriptionis sufficient for the following derivation. We define the frequency of a photon asν = E/h and its wavelength as λ = hc/E where E is its energy. These definitionsare for convenience and do not imply that we can ascribe a frequency or awavelength to an individual photon; they are properties of groups of photons.
The derivation requires that the wavelength is short compared to the size ofthe wave packet and that this is short compared to variations in the curvatureof spacetime. Furthermore, we assume that the rays of any wave follow nullgeodesics and therefore any deviations from flat spacetime produce change inshape of the wave packet. In other words, since the scale length of deviationsfrom flat space are large compared to the size of the wave packet they act as avery small perturbation to the propagation of the wave packet.
Consider a wave packet moving through a spacetime of constant positive curvature. Because of geodesic deviation, the rays come closer together as thewave packet moves forward. They are focussed. In particular the direction θ, ofa ray (geodesic) with initial separation ξ0 after a distance s is (assuming smallangles) where a is the local radius of curvature. Since the central geodesic is the direc-tion of energy flow, we can integrate the wave-energy-function times the com-ponent of θ normal to the trajectory, over the dimensions of the wave packetin order to calculate the amount of energy that is now travelling normal to thetrajectory. The result is a finite energy that depends on the average lateralextension of the wave packet, the local radius of curvature, and the originalphoton energy. The actual value is not important but rather the fact that thereis a finite fraction of the energy that is moving away from the trajectory of theoriginal wave packet. This suggests a photon interaction in which the photoninteracts with curved spacetime with the hypothesis that the energy flow nor-mal to the trajectory goes into the emission of secondary photons normal to itstrajectory. From a quantum-mechanical point of view, there is a strong argu-ment that some interaction must take place. If the spin of the photon is directlyrelated to the angular momentum of the wave packet about its trajectory thenthe computation of the angular momentum is a similar integral. Then becauseof focussing the angular momentum clearly changes along the trajectory, whichdisagrees with the quantum requirement that the angular momentum, that isthe spin, of the photon is constant. The Heisenberg uncertainty principle re-quires that an incorrect value of spin can only be tolerated for a finite timebefore something happens to restore the correct value. We now consider theconsequences.
Consider motion on the surface of a three dimensional sphere with radius r. As described above, two adjacent geodesics will move closer together due tofocussing. Simple kinematics tells us that a body with velocity v associated withthese geodesics has acceleration v2/r, where r is the radius of curvature. Thisacceleration is directly experienced by the body. In addition, it experiences atidal acceleration within itself. This tidal acceleration is equivalent to the fo-cussing of the geodesics. Although the focussing and acceleration are closely CURVATURE COSMOLOGY THEORY linked, we need to consider whether the occurrence of one implies the occur-rence of the other. Does the observation of focussing (tidal acceleration) implyacceleration in the orthogonal direction? It is true in two and three dimensions,but it needs to be demonstrated for four dimensions.
The geometry of a three dimensional surface with curvature in the fourth dimension is essentially the same as motion in three dimensions except that thefocussing now applies to the cross-sectional area and not to the separation. Doesthis acceleration have the same physical significance? Assuming it does, a wavepacket that is subject to focussing has acceleration in an orthogonal dimension.
For instance if we could constrain a wave packet (with velocity c) to travel onthe surface of a sphere in three dimensions it would not only show a focussingeffect but also experience an acceleration of c2/r normal to the surface of thesphere. Then a wave packet (and hence a photon) that has its cross-sectionalarea focussed by curvature in the fourth dimension with radius r would have anenergy loss rate proportional to this acceleration. The essence of the curvature-redshift hypothesis is that the tidal distortion causes the photon to interact andthat the energy loss rate is proportional to c2/r. For a photon with energy Ethe loss rate per unit time is cE/r, and per unit distance it is E/r.
In general relativity the crucial equation for the focussing of a bundle of geodesics was derived by also see and and for the current context we can assume that the bundle haszero shear and zero vorticity. Since any change in geodesic deviation along thetrajectory will not alter the direction of the geodesics we need consider only thecross-sectional area A of the geodesic bundle to get the equation = −RαβUαUβ = − where R is the Ricci tensor (it is the contraction of the Riemann-Christoffeltensor), U is the 4-velocity of the reference geodesic and a is the local radius ofcurvature. This focussing can be interpreted as the second order rate of changeof cross-sectional area of a geodesic bundle that is on the three-dimensionalsurface in four-dimensional space. Then if we consider that a photon is a wavepacket we find that the rate at which the photon loses energy per unit distanceis E/a or more explicitly = − RαβUαUβ1/2 , What is interesting about this equation is that, for the Schwarzschild (and Kerr)solutions for the external field for a mass, the Ricci tensor is zero; hence, thereis no focussing and no energy loss. A geodesic bundle passing a mass such asthe sun experiences a distortion but the wave packet has not changed in area.
Hence, this model predicts that photons passing near the limb of the sun willnot suffer any energy loss due to curvature redshift.
The field equation for Einstein's general theory of gravitation is Rαβ = 8πG Tαβ − T gαβ where T is the contracted form of Tαβ the stress-energy-momentum tensor, gis the metric tensor, G is the Newtonian gravitational constant and Λ is the CURVATURE COSMOLOGY THEORY cosmological constant. It states that the Ricci tensor describing the curvatureof spacetime is determined by the distribution of mass (and energy). Directapplication of the field equations (without the cosmological constant) in termsof the stress-energy-momentum tensor Tαβ, the metric tensor g and with thematerial having a 4-velocity V gives TαβUαUβ − T gαβVαVβ For null geodesics gαβVαVβ is zero which leaves only the first term. For a perfect fluid the stress-energy-momentum tensor is where p is the proper pressure and ρ is the density. Combining Eq. withEq. gives for null geodesics For cases where the proper pressure is negligible compared to the density we can ignore the pressure and get −1.366 × 10−13 ρ m−1.
For many astrophysical types of plasma, it is useful to measure density by the equivalent number of hydrogen atoms per cubic metrae: that is we can putρ = N mH and get = −5.588 × 10−27 N m−1.
The rate of energy loss per distance travelled depends only on the square root ofthe density of the material, which may consist of gas, plasma, or gas and dust.
This equation can be integrated to get Curvature redshift secondary photons The above derivation does not define the form of energy loss. The most realisticmodel is that the photon decays into three secondary photons, one of whichtakes nearly all the energy and momentum and two very low-energy secondaryphotons. It is convenient (although not strictly correct) to think of the high-energy secondary as a continuation of the primary but with slightly reducedenergy. Two secondary photons are required to preserve spin and, by symmetry,they are emitted in opposite directions with the same energy This assumptionthat the two secondary photons have the same energy is made without properjustification.
What can be said is that if they are not, they will still have CURVATURE COSMOLOGY THEORY nearly equal energies because the probability of having one with a much longerrelative wavelength is very low. From symmetry they are ejected at right anglesto the original trajectory.
Thus, the primary photon is not deflected.
can get an estimate of how often these interactions occur and hence what thesecondary energies are by using the Heisenberg uncertainty principle appliedto the primary. For linear momentum and distance it is ∆p∆x ∼ putting X = ∆x we get ∆E = hc/4πX. Now after the photon with energy E0has travelled a distance X the energy-loss is ∆E = E0X/a, and hence If each secondary photon takes half the energy-loss, we find Therefore the secondary photons have a wavelength of For example consider a visible photon with wavelength 600 nm travelling ingas with density N , then X = 2.93 × 109N −1/4 m and the wavelength is λ =7.36×1010N −1/4 m which corresponds to a frequency of ν = 4.07N 1/4 mHz Nowfor fully ionized plasma the plasma frequency is ν = 4.55 × 10−4N−1/4.
Thus, for optical photons and all plasmas with densities greater than N =0.14 m−3 the secondary photons have frequencies well below the plasma fre-quency and therefore cannot propagate but will be quickly absorbed by theplasma. The energy lost by the primary photon is dissipated into heating theplasma.
Inhibition of curvature redshift From the discussion above it is clear that the process of curvature redshiftrequires a gradual focussing to a critical limit, followed by the emission of sec-ondary photons. It is as if the photon gets slowly excited by the focussinguntil the probability of secondary emission becomes large enough for it to oc-cur. If there is any other interaction the excitation due to focussing will benullified. That is, roughly speaking, curvature-redshift interaction requires anundisturbed path length of at least X (Eq. for significant energy loss to oc-cur. A suitable criterion for inhibition to occur is that the competing interactionhas an interaction length less than X. Although Compton or Thompson scat-tering are possible inhibitors there is another interaction that has a much largercross-section. This is the coherent multiple scattering that produces refractiveindex.
CURVATURE COSMOLOGY THEORY In classical electro-magnetic theory, the refractive index of a medium is the ratio of the velocity of light in vacuum to the group velocity in the medium.
However, in quantum mechanics photons always travel at the velocity of lightin vacuum. In a medium, a group of photons appears to have a slower velocitybecause the individual photons interact with the electrons in the medium andeach interaction produces a time delay. Because the interaction is with manyelectrons spread over a finite volume, the only possible result of each interactionis the emission of another photon with the same energy and momentum. Nowconsider the absorption of a wave. In order to cancel the incoming wave a newwave with the same frequency and amplitude but with opposite phase must beproduced. Thus, the outgoing wave will be delayed by half a period with respectto the incoming wave. For example if the phase difference was not exactly halfa period for an electro-magnetic wave incident on many electrons, the principleof conservation of energy would be violated. This simple observation enables usto compute the interaction length for refractive index n. If L is this interactionlength then it is where n is the refractive index and the modulus allows for plasma and othermaterials where the refractive index is less than unity. Note that L is closelyrelated to the extinction length derived by Ewald and Oseen (see or which is a measure of the distance needed foran incident electromagnetic wave with velocity c to be replaced by a new wave.
For plasmas the refractive index is where Ne is the electron density and r0 is the classical electron radius. We cancombine these two equations to get (for a plasma) L = (Ner0λ0)−1.
Thus, we would expect the energy loss to be inhibited if the average curvature-redshift interaction distance is greater than that for refractive-index interactions,i.e. if X > L. Therefore, we can compute the ratio (assuming a plasma withN ∼ = Ne) and using Eq. to get X/L = 0.0106N 3/4λ This result shows that curvature redshift will be inhibited if this ratio is greaterthan one, which is equivalent to λ0 > 20.7N −1/2 m. For example, curvatureredshift for the 21 cm hydrogen line will be inhibited if the electron density isgreater than about 104 m−3.
Possible laboratory tests It is apparent from the above analysis that to observe the redshift in the labora-tory we need to have sufficient density of gas (or plasma) to achieve a measurableeffect but not enough for there to be inhibition by the refractive index. Theobvious experiment is to use the M¨ ossbauer effect for γ-rays that enables very CURVATURE COSMOLOGY THEORY precise measurement of their frequency. Simply put, the rays are emitted by nu-clei in solids where there is minimal recoil or thermal broadening of the emittedray. Since the recoil-momentum of the nucleus is large compared to the atomicthermal energies and since the nucleus is locked into the solid so that the recoilmomentum is precisely defined, then the γ-ray energy is also precisely defined.
The absorption process is similar and has a very narrow line width. Such anexperiment has already been done by They measuredgravitational effects on 14.4 keV γ-rays from 57Fe being sent up and down avertical path of 22.5 m in helium near room pressure. They found agreementto about 1% with the predicted fractional redshift of 1.5 × 10−15, whereas frac-tional curvature redshift predicted by Eq. for this density is 1.25 × 10−12.
Clearly, this is much larger. At γ-ray frequencies, the electrons in the heliumgas are effectively free and we can use Eq. to compute the refractive indexinteraction length. For helium at STP, it is L = 0.077 m, which is much lessthan curvature-redshift interaction length which for these conditions is X=11m. Hence, we do not expect to see any significant curvature redshift in theirresults. Pound and Snyder did observe one-way frequency shifts but they weremuch smaller than curvature redshift and could be explained by other aspects ofthe experiment. However, the Pound and Snyder experiment provides a guideto a possible test for the existence of curvature redshift. Because curvatureredshift has a different density variation to that for the inhibiting refractive in-dex it is possible to find a density for which curvature redshift is not inhibited.
Although there is a slight advantage in using heavier gases than helium due totheir higher atomic number to atomic weight ratio, their increased absorptionto γ-rays rules them out. Hence, we stay with helium and from Eq. we cancompute curvature-redshift interaction length to be where p is the pressure and p0 is the pressure at STP. For the same gas therefractive index interaction length is It follows that the curvature redshift will not be inhibited if X < L or in thiscase, the pressure is less that 0.0014p0 which is about 1 mm of Hg. For thispressure, we find that X = 57 m which requires that the apparatus must bemuch longer than 57 m. For argument let us take the length to be 100 m thenthe fractional redshift expected is 2.1 × 10−13 which is detectable. The experi-mental method would use a horizontal (to eliminate gravitational redshifts) tubefilled with helium and with accurately controlled temperature. Then we wouldmeasure the redshift as a function of pressure. The above theory predicts thatif it is free of inhibition then the redshift should be proportional to the squareroot of the pressure.
Alternatively, it may be possible to detect the secondary photons. For he- lium with a pressure of 1 mm Hg the expected frequency of the secondaryradiation is about 100 kHz. The expected power from a 1 Cu source is about5 × 10−22 W. Unfortunately, the secondary radiation could be spread over afairly wide frequency band which makes its detection somewhat difficult but itmay be possible to detect the radiation with modulation techniques.
CURVATURE COSMOLOGY THEORY Table 20: Curvature redshift in air.
aGravitational interaction length in metrosbRefractive index interaction length in metroscDistance to halve beam intensity in metrosdSecondary frequency in Mhz Another possibility is to use γ-rays of much shorter wavelength where it may be possible to detect the secondary radiation in an experiment that didnot try to measure the redshift. For example consider the passage of keV toMev gamma rays from radioactive elements or synchrotron sources in air. For airat a density of 1.20 kg m−3 and with the γ-ray energy E0 in keV the frequencyof the secondaries is derived from Eq. to be and the gravitational interaction length is Now for there to be no inhibition the gravitational interaction length must beless than the refractive index interaction length (L) which from Eq. and forair has the equation In addition the gamma rays must have a path length greater than X.
appropriate measure of this path length is the distance over which the numberof γ-rays have been attenuated to half the original number. Table showsthese quantities for a range of primary energies.
Note that the curvature redshift will be inhibited by the attenuation length until the γ-rays have an energy a bit less than 20 keV. There is no inhibition fromeither cause for energies larger than 20 keV. The expected power per gammaray per meter of path length is given by ∆P = 7.24 × 10−21 Clearly a powerful γ-ray source with energies greater than about 50 keV isrequired.
CURVATURE COSMOLOGY THEORY Yet another possible test is to measure the frequency from a spacecraft at two receivers as a function of the differential distance between the receiversand the spacecraft. For example in the analysis of the Pioneer 10 accelerationanomaly (section it was shown that the interplanetary dust density couldcontribute a measurable frequency shift. Comparison of this frequency shift atthe same time at two receivers at different distances would remove most othercauses of frequency shifts. One advantage of this test is that it does not requirevery accurate frequency generation on the satellite. Typically the two receiverswould be two ground stations. The major problem is the uncertainty and indeedlarge variation in the density of the exosphere and any other frequency shifts dueto earth rotation that cannot be accurately modelled. Note that at the typicalX-band frequencies inhibition will prevent the neutral atmosphere showing anycurvature-redshift effects.
Interactions for other particles Since the focussing due to spacetime curvature applies to the quantum wave, it isexpected that electrons and other particles would interact with curved spacetimein a manner similar to photons. The argument is the same up to Eq. butnow we have to allow for nonzero mass. The problem (not solved here) is tofind a covariant expression that properly describes the energy-momentum lossto secondary particles and yet preserves the correct normalization of the energy-momentum 4-vector. An alternate approach is to consider the motion in a localMinkowskian reference frame. In this case the loss equations (with P0 denotingthe energy component) are where β is the usual velocity ratio, ae is the local radius of curvature for electronsand as required by normalization and the conservation of proper mass, we havefrom Eq. dP α Pα = 0.
Noting that for a nonzero rest mass particle VαVα = −1. The radius of curva-ture ae can be evaluated for the simple case of a uniform gas (or plasma) usingEq. and Eq. to get 1 − β2. Then with the further simplification of negligible pres- sure and with the material at rest and where T = (γ − 1) mc2 is the kineticenergy, the energy loss rate is 8πGρ γ2 − 1 CURVATURE COSMOLOGY THEORY It shows that for nonzero rest mass particles, the energy loss rate has a strongdependence on velocity, and for extreme relativistic velocities, the fractionalenergy-loss rate is proportional to γ. Because of the strong velocity dependence,the energy loss rate for electrons will be much higher than that for nuclei in anyplasma near thermal equilibrium. In addition, Eq. shows that the energy lossrate has the same square root dependence on density as the energy loss rate forphotons.
Since an electron interacts without being absorbed and re-emitted, we do not expect the same type of inhibition that applies to photons. Instead theelectron slowly gets excited with the addition of energy which it releases as low-energy photons when it interacts with some other particle. The need to preservespin and momentum prevents it from emitting photons without the presenceof another particle. In the cosmic medium, the most likely interactions areelectro-magnetic scattering off other charged particles and the inverse-Comptoneffect off 3K background radiation photons. In high temperature plasma theelectromagnetic (Rutherford) scattering is probably dominant since there willbe many small angle deflections with large impact parameters. Thus the modelfor curvature redshift of non-photon particles is one in which an excited electronemits most of its excitation energy as a low-energy photon during the scatteringoff another photon, electron or nucleus.
Derivation of curvature pressure The hypothesis of curvature pressure is that for moving particles there is a pres-sure generated that acts back on the matter that causes the curved spacetime.
In this case, curvature pressure acts on the matter (plasma) that is producingcurved spacetime in such a way as to try to decrease the curvature. In otherwords, the plasma produces curved spacetime through its density entering thestress-energy tensor in Einstein's field equations. The magnitude of the curva-ture is an increasing function of the plasma density.
Gravitation is not a force The phrase gravitational force is not only a popular expression but is endemicthroughout physics. In particular, gravitation is classified as one of the fourfundamental forces with its heritage going back to Newton's law of gravitation.
I argue that the formulation of gravitation as a force is a misconception. Inboth Newtonian theory and general relativity, gravitation is acceleration. Tobegin let us examine the original Newtonian gravitation equation where (following we identify m as the inertial mass of the test object, M as the active gravitational mass of the second object and m passive gravitational mass of the test object. The vector a is its acceleration andr is its displacement from the second object. This equation is usually derivedin two steps: first, the derivation of a gravitational field and second, the forceproduced by that field on the test mass. By analogy with Coulombs law, thepassive gravitational mass has a similar role to the electric charge.
CURVATURE COSMOLOGY THEORY However many experiments by and have shown that the passive gravitational massis equal to the inertial mass to about one part in 1012. The usual interpretationof the agreement is that they are fundamentally the same thing. However, analternative viewpoint is that the basic equation is wrong and that the passivegravitational mass and the inertial mass should not appear in the equation. Inthis case the correct equation is Thus, the effect of gravitation is to produce accelerations directly; there is noforce involved. Some might argue that since the two masses cancel the distinc-tion is unimportant. On the other hand, I would argue that the application ofOckham's razor dictates the use of Eq. instead of Eq. The agreement of the inertial mass with the passive gravitational mass is the basis of the weak equivalence principle in that it applies regardless of thecomposition of the matter used. Shows that it applies to boththe potential and the kinetic energy in the body. The theory of general rela-tivity is based on the principle of equivalence as stated by Einstein: All local,freely falling, non-rotating laboratories are fully equivalent for the performanceof physical experiments. The relevance here is that it is impossible to distinguishbetween acceleration and a uniform gravitational field. Thus when gravitationis considered as acceleration and not a force the passive gravitational mass is aspurious quantity that is not required by either theory.
A Newtonian model A simple cosmological model using Newtonian physics in four-dimensional spaceillustrates some of the basic physics subsequently used to derive the features ofcurvature pressure. The model assumes that the universe is composed of gasconfined to the three-dimensional surface of a four-dimensional hypersphere.
Since the visualization of four dimensions is difficult let us suppress one of thenormal dimensions and consider the gas to occupy the two-dimensional surfaceof a normal sphere. From Gauss's law (i.e. the gravitational effect of a sphericaldistribution of particles with radial symmetry is identical to that of a pointmass equal in value to the total mass situated at the center of symmetry) thegravitational acceleration at the radius r of the surface is normal to the surface,directed inward and it has the magnitude where M is the total mass of the particles and the dots denote a time derivative.
For equilibrium, and assuming all the particles have the same mass and velocitywe can equate the radial acceleration to the gravitational acceleration and getthe simple equation from celestial mechanics of If there is conservation of energy, this stable situation is directly analogous tothe motion of a planet about the sun. When there is a mixture of particles with CURVATURE COSMOLOGY THEORY different masses, there is an apparent problem. In general, particles will have adistribution of velocities and the heavier ones can be expected to have, on aver-age, lower velocities. Thus, equilibrium radii will vary with the velocity of theparticles. However, the basis of this model is that all particles are constrainedto have the same radius regardless of their mass or velocity with the value ofthe radius set by the average radial acceleration. Thus for identical particleswith a distribution of velocities we average over the squared velocities to get If there is more than one type of particle with different masses then we invokethe precepts of Section and average over the accelerations to get the sameresult as Eq. The effect of this balancing of the accelerations against thegravitational potential is seen within the shell as a curvature pressure that isa direct consequence of the geometric constraint of confining the particles to ashell. If the radius r decreases then there is an increase in this curvature pressurethat attempts to increase the surface area by increasing the radius. For a smallchange in radius in a quasi-equilibrium process where the particle velocities donot change the work done by this curvature pressure (two-dimensions) with anincremental increase of area dA is pcdA and this must equal the gravitationalforce times the change in distance to give where M = P mi with the sum going over all the particles. Therefore, usingEq. we can rewrite the previous equation in terms of the velocities as Now dA/dr = 2A/r, hence the two-dimensional curvature pressure is Thus in this two-dimensional model the curvature pressure is like the average kinetic energy per unit area. This simple Newtonian model provides a guide as towhat the curvature pressure would be in the full general relativistic model. Theessential result is that there is a curvature pressure that is due to the constraintof requiring all the particles to stay within the two-dimensional surface.
General relativistic model In deriving a more general model in analogy to the Newtonian one, we firstchange dA/dr = 2A/r to dV /dr = 3V /r and secondly we include the correctionγ2 needed for relativistic velocities. The result is In this case the constraint arises from the confinement of all the particles within athree-dimensional hyper-surface. Now we expect to be dealing with fully ionized CURVATURE COSMOLOGY THEORY high temperature plasma with a mixture of electrons, protons, and heavier ionswhere the averaging is done over the accelerations. Define the average densityby ρ = M/V then the cosmological curvature pressure is γ2 − 1 ρc2.
In effect, my hypothesis is that the cosmological model must include this cur- vature pressure as well as thermodynamic pressure. Note that although this hasa similar form to thermodynamic pressure it is quite different. In particular, it isproportional to an average over the squared velocities and the thermodynamicpressure is proportional to an average over the kinetic energies. This meansthat, for plasma with free electrons and approximate thermodynamic equilib-rium, the electrons will dominate the average due to their much larger velocities.
From a Newtonian point of view, curvature pressure is opposed to gravitationalmutual acceleration. In general relativity, the plasma produces curved space-time through its density entering the stress-energy tensor in Einstein's fieldequations. Then the constraint of confining the particles to a three-dimensionalshell produces a pressure whose reaction is the curvature pressure acting to de-crease the magnitude of the curvature and hence decrease the density of theplasma.
For high temperature plasma in equilibrium, the J¨ uttner distribution can be used to evaluate the curvature pressure. For a gas with temperature T and formolecules with mass m, showed that γ2 (α) = 3αK3(1/α)/K2(1/α), where α = kT /mc2 and Kn(1/α) are the modified Bessel functions of the secondkind For small, α this has the approximation γ2(α) = 1 + 3α + 152α2 + 458α3 + . .
For a Maxwellian (non-relativistic) distribution, the first two terms are exactand the α2 term is the first term in the correction for the J¨ Local curvature pressure For the universe, the calculation of curvature pressure is simple because of theconstant curvature and homogeneous medium. However, for a localized regionsuch as a star with inhomogeneous medium and curvature the calculation ismuch more difficult. We start with the premise that it is the motion of particlesthat reacts back on the material producing the curvature by producing a pres-sure that tends to reduce the curvature. The problem is that the calculationof the curvature at any point requires the integration of Einstein's equationsof general relativity. Then if the particles' motion produces a reaction force,the problem is to determine how that reaction force is apportioned amongstthe matter that produces the curvature. One approach that is valid for mostastrophysical applications where the spacetime curvature is small is to use theNewtonian approximation. Let a, be the effective radius of curvature of thefour dimensional space where the particles' are constrained. Then the premiseis that this constraint produces an acceleration due to curvature (assuming for CURVATURE COSMOLOGY THEORY the moment that there is only one type of particle) of where the angular brackets denote an averaging over all the velocities. Nowconsider a spherically symmetric distribution of gas. If the distribution is static,the central gravitational attraction is balanced by some pressure pg, so that dpg = −ρ(r)g(r), where ρ (r) is the density at radius r and g(r) is the gravitational accelerationat r. Similarly, we define a curvature pressure by dpc = −ρ(r)gc(r).
However, if there is a mixture of particles there is an important difference.
Because electrons have a much lighter mass than ions the velocity average formixed particles (provided the gas is ionized) will be dominated by the electronsand the appropriate density to use in Eq. is that for the electrons. Now thecurvature radius a, is given by Eq. and for a gas with relativistic particleswe put γ2 − 1 c2.
We need to include a factor of one third because only the velocity componentorthogonal to the direction of the acceleration is relevant. Then the curvaturepressure acceleration is γ2 − 1 pρ(r) c2 Since the hypothesis is that this curvature pressure is a reaction to the accel-erations produced by the gas at radius r, the averaging over velocities must beover all the gas that is being accelerated. By Gauss's law and symmetry this isthe gas with radii greater than r thus we get r2(γ2 − 1)pρ(ˆ γ2 − 1 pρ(r) where N (r) is the particle number density. Now for plasmas where the temper-atures less than about 108 K we can use Eq. to get Hence the working equation for local curvature pressure is CURVATURE COSMOLOGY THEORY where the function in angular brackets is and Ne(r) is the electron number density.
A theory of curvature pressure in a very dense medium where quantum me- chanics dominates and where general relativity may be required is needed todevelop this model. Nevertheless, without such a theory, we expect the pres-sure to be proportional to the local gravitational acceleration and an increasingfunction of the temperature of the particles. Thus, we might expect a curvaturepressure that would resist a hot compact object from collapsing to a black hole.
Because of the energy released during collapse, it is unlikely for a cold objectto stay cold enough to overcome the curvature pressure and collapse to a blackhole.
The curvature cosmological model Curvature cosmology can now be derived by including curvature redshift andcurvature pressure into the equations of general relativity. This is done by usinghomogeneous isotropic plasma as a model for the real universe. The generaltheory of relativity enters through the Friedmann equations for a homogeneousisotropic gas. Although such a model is simple compared to the real universe,the important characteristics of CC can be derived by using this model. Thefirst step is to obtain the basic relationship between the density of the gasand the radius of the universe. The inclusion of curvature pressure is not onlyimportant in determining the basic equations but it also provides the necessarymeans of making the solution static and stable. Then it is shown that the effectof curvature redshift is to produce a redshift that is a function of distance, andthe slope of this relationship is (in the linear limit of small distances) the Hubbleconstant.
The first-order model considers the universe to be a gas with uniform density and complications such as density fluctuations, galaxies, and stars are ignored.
In addition, we assume (to be verified later) that the gas is at high temperatureand is fully ionized plasma. Because of the high symmetry, the appropriatemetric is the one that satisfies the equations of general relativity for a homo-geneous, isotropic gas. This metric was first discovered by A. Friedmann andfully investigated by H. P. Robertson and A. G. Walker. The Robertson-Walkermetric for a space with positive curvature can be written as ds2 = c2dt2 − [R(t)]2 + r2 dθ2 + sin2(θ)dϕ2 where ds is the interval between events, dt is time, R(t)dr is the comovingincrement in radial distance, R(t) is the radius of curvature and R0 is the valueof R(t) at the present epoch.
The Friedmann equations Based on the Robertson-Walker metric, the Friedmann equations for the ho-mogeneous isotropic model with constant density and pressure are CURVATURE COSMOLOGY THEORY where R is the radius, ρ is the proper density, p is the thermodynamic pressure,G is the Newtonian gravitational constant, Λ is the cosmological constant, c isthe velocity of light and the superscript dots denote time derivatives. Workingto order of me/mp thermodynamic pressure may be neglected but not curvaturepressure. How to include curvature pressure is not immediately obvious. Thethermodynamic pressure appears only as a relativistic correction to the inertialmass density whereas curvature pressure is closer in spirit to the cosmologicalconstant. My solution is to include curvature pressure (with a negative sign)with the thermodynamic pressure and to set the cosmological constant to zero.
This is an ad hoc variation to general relativity and its only justification is thatit provides sensible equations and show good agreement with observations. In-cluding curvature pressure from Eq. and from Eq. the modified Friedmannequations are Clearly there is a static solution if < γ2 − 1 >= 1, in which case ¨ second equation, with ˙ R = 0 provides the radius of the universe which is given Thus, the model is a static cosmology with positive curvature. Although thegeometry is similar to the original Einstein static model, this cosmology differsin that it is stable. The basic instability of the static Einstein model is wellknown On the other hand, the stability of CC isshown by considering a perturbation ∆R, about the equilibrium position. Thenthe perturbation equation is 3c2 dhγ2 − 1i For any realistic equation of state for the cosmic plasma, the average velocitywill decrease as R increases. Thus the right hand side is negative, showing thatthe result of a small perturbation is for the universe return to its equilibriumposition. Thus, CC is intrinsically stable. Of theoretical interest is that Eq. predicts that oscillations could occur about the equilibrium position.
Temperature of the cosmic plasma One of the most remarkable results of CC is that it predicts the temperature ofthe cosmic plasma from fundamental constants. That is the predicted temper-ature is independent of the density and independent of any other characteristic CURVATURE COSMOLOGY THEORY of the universe. For a stable solution to Eq. we need that < γ2 − 1 >= 1, (i.e.
< γ2 >= 2) where the average is taken over the electron and nucleon numberdensities, that is for equal numbers of electrons and protons where the terms on the right are for electrons and protons. Provided the tem-peratures are small enough for the proton's kinetic energy to be much less thanits rest mass energy, we can put γ2 = 1 and thus for pure hydrogen, the result γ2 = 3. Using a more realistic composition that has 8.5% by number of helium we find that γ2 = 2.927. Hence using Eq. the predicted electron temperature is 2.56 × 109 K. For this temperature γ2 = 1.0007. This shows that the temperature is low enough to justify the assumption made earlier,that the proton's kinetic energy is much smaller than its rest mass energy.
To recapitulate the stability of CC requires that ¨ R = 0. This requires that the plasma has the precise temperature that makes < γ2 − 1 >= 1. The basisfor this result is that curvature pressure exists and critical to its derivation isthe averaging over accelerations and not over forces. This is where the assertionthat gravitation is acceleration and is not a force is important.
Hubble constant: theory The Hubble constant is proportional to the local energy loss rate given by Equa-which gives 1.671 × 10−18N 1/2 m−1 51.69N 1/2 kms−1 Mpc−1.
The usual redshift parameter z is defined in terms of the wavelengths, frequen-cies and energies as If the plasma density is constant then we can integrate the energy loss alongthe path to get where r is the distance travelled.
The Robertson-Walker metric shown in Eq. is not in the simplest form thatexplicitly shows the geometry. Following we can introduce anew variable χ, where r = R sin χ and the new metric is ds2 = c2dt2 − R2 dχ2 + sin2 χ dθ2 + sin2 θdφ2 .
In this metric the distance travelled by a photon is Rχ , and since the velocity oflight is a universal constant the time taken is Rχ/c. There is a close analogy to CURVATURE COSMOLOGY THEORY motion on the surface of the earth with radius R. Light travels along great circlesand χ is the angle subtended along the great circle between two points. Thegeometry of this CC is that of a three-dimensional surface of a four-dimensionalhypersphere. For this geometry the area of a sphere with radius R is given by A(r) = 4πR2 sin2(χ).
The surface is finite and χ can vary from 0 to π. Integration of this equationwith respect to χ gives the volume V , namely, V (r) = 2πR3 χ − Clearly the maximum volume is 2π2R3 and we can using Eq. to get R wehave 3.100 × 1026N −1/2 m 10.05N −1/2 Gpc.
Examination of Equations and shows that there is a simple rela- tionship between R and H, namely The next step is to replace r in Eq. with r = Rχ to get z = exp( 3χ) − 1, This is the fundamental relationship between z and χ. Since the geometry ofCC does not involve a time coordinate, it is much simpler than that for BB.
The key equations define the CC geometry are Eq. which defines the radiusof the universe in terms of the Hubble constant and Eq. which defines thedistance variable χ in terms of the redshift parameter z. We now examine sometopics that are relevant practical applications.
Luminosities and magnitudes Let a source have a luminosity L(ν) (W Hz−1) at the emission frequency ν. Thenif energy is conserved, the observed flux density (W m−2 Hz−1) at a distanceparameter χ is the luminosity divided by the area, which is APPLICATION OF CURVATURE COSMOLOGY However, because of curvature redshift there is an energy loss such that thereceived frequency ν0 is related to the emitted frequency νe by Eq. Includingthis effect the result is 4π (R sin(χ))2 (1 + z) The apparent magnitude is defined as m = −2.5 log(S) where the base of thelogarithm is 10 and the constant 2.5 is exact. Since the absolute magnitude is theapparent magnitude when the object is at a distance of 10 pc (3.0857 × 1017 m),the flux density at 10 pc is where because 10 pc is negligible compared to R, approximations have beenmade. The flux density ratio is Defining M as the absolute magnitude and putting νe = (1 + z)ν0 we get for(m − M ) + Kz(ν0) + 2.5 log(1 + z), where the K-correction is described in section Furthermore, we can use Eq. to replace R byH since where h is the reduced Hubble constant. Hence, we get the distance modulus + 2.5 log(1 + z) + 42.384.
Application of Curvature Cosmology These topics are relevant to CC but are not part of the comparison of CC withBB. However they are either very important for any cosmology or offer furtherobservational support for CC.
Consider a stellar cluster or an isolated cloud of gas in which collisions arenegligible or elastic. In either case the virial theorem states that the averagekinetic energy K, is related to the average potential energy V , by the equationV = V0 − 2K where V0 is the potential energy when there is zero kinetic energy.
APPLICATION OF CURVATURE COSMOLOGY Let U be the total energy then U = K + V = V0 − K . Thus, we get thesomewhat paradoxical situation that since V0 is constant; an increase in totalenergy can cause a decrease in kinetic energy. This happens because the aver-age potential energy has increased by approximately twice as much as the lossin kinetic energy. Since the temperature is proportional to (or at the least amonotonic increasing function of) the average kinetic energy it is apparent thatan increase in total energy leads to a decrease in temperature. This explains theoften-quoted remark that a self-gravitationally bound gas cloud has a negativespecific heat capacity. Thus, when gravity is involved the whole construct ofthermodynamics and entropy needs to be reconsidered. One of the commonstatements of the second law of thermodynamics is that Theenergy of the universe is constant: the entropy of the Universe tends to a max-imum, the entropy of the universe is always increasingor from Wikipedia the second law of thermodynamics is an expression of theuniversal law of increasing entropy, stating that the entropy of an isolated sys-tem which is not in equilibrium will tend to increase over time, approaching amaximum value at equilibrium.
Now the normal proof of the second law considers the operation of reversible and non-reversible heat engines working between two or more heat reservoirs.
If we use a self-gravitating gas cloud as a heat reservoir then we will get quitedifferent results since the extraction of energy from it will lead to an increasein its temperature. Thus if the universe is dominated by gravity the second lawof thermodynamics needs reconsideration. In addition, it should be noted thatwe cannot have a shield that hides gravity. To put it another way there is noadiabatic container that is beyond the influence of external gravitational fields.
Thus we cannot have an isolated system.
This discussion shows that in a static finite universe dominated by gravity simple discussions of the second law of thermodynamics can be misleading. Thepresence of gravity means that it is impossible to have an isolated system. Tobe convincing any proof of the second law of thermodynamics should includethe universe and its gravitational interactions in the proof.
For CC, Olber's Paradox is not a problem. Curvature redshift is sufficient tomove distant starlight out of the visible band. Visible light from distant galaxiesis shifted into the infrared where it is no longer seen. Of course, with a finiteuniverse, there is the problem of conservation of energy and why we are not sat-urated with very low frequency radiation produced by curvature redshift. Theselow-energy photons are eventually absorbed by the cosmic plasma. Everythingis recycled. The plasma radiates energy into the microwave background radi-ation and into X-rays. The galaxies develop from the cosmic plasma and passthrough their normal evolution. Eventually all their material is returned to thecosmic plasma. Note that very little, if any, is locked up into black holes. Cur-vature pressure causes most of the material from highly compact objects to bereturned to the surrounding region as high-velocity jets.
APPLICATION OF CURVATURE COSMOLOGY Black holes and Jets The existence of curvature pressure provides a mechanism that could prevent thecollapse of a compact object into a black hole. A theory of curvature pressure ina very dense medium where quantum mechanics dominates is needed to developthis model. Nevertheless, without a full theory we can assume that curvaturepressure will depend on the local gravitational acceleration and it will be anincreasing function of the temperature of the particles. Thus, we might expecta curvature pressure that would resist a hot compact object from collapsing toa black hole. Because of the energy released during collapse it is unlikely fora cold object to stay cold long enough to overcome the curvature pressure andcollapse to a black hole.
What is expected is that the final stage of gravitational collapse is a very dense object, larger than a black hole but smaller than a neutron star. Thiscompact object would appear very much like a black hole and would have mostof the characteristics of black holes. Such objects could have large masses andbe surrounded by accretion discs. Thus, many of the observations that arethought to show the presence of black hole could equally show the presence ofthese compact objects. However, there is one observational difference in thatmany of the mass estimates of black holes come from observations of redshiftsfrom nearby stars. Since part or most of these redshifts may be due to curvatureredshift in the surrounding gas, these mass estimates may need to be revised.
If the compact object is rotating there is the tantalizing idea that curvature pressure may produce the emission of material in two jets along the spin axis.
This could be the ‘jet engine' that produces the astrophysical jets seen in stellar-like objects and in many huge radio sources. Currently there are no acceptedmodels for the origin of these jets.
The postulate here is that the jets are a property of the compact object and do not come from the accretion disk.
The spinning object provides the symmetry necessary to generate two jets andcurvature pressure provides the force that drives the jets. This mechanism isapplicable to both stellar and galactic size structures.
Large number coincidences It is appropriate to have a brief discussion of famous numerical coincidencesin cosmology First, however we need the results for the sizeparameters for the CC universe which are shown in Table where the NH isthe density divided by the mass of a hydrogen atom. The first large numbercoincidence is the ratio of the radius of the universe to the classical electronradius (R/r0). The result is 9.49 × 1040 which is to be compared with the ratioof the electrostatic force to that of the gravitation force between and electronand a proton. This is 4.3 × 1038 which being about 200 times smaller than R/r0shows that it is hardly a coincidence and although interesting probably has littlephysical significance.
investigated the use of Mach's principle and the role of inertia in general relativity. By direct analogy to Maxwell's equations, hederived for rectilinear motion a combination of Newton's laws of motion and ofgravitation, with the inertial frame determined by Mach's principle (his italics).
In effect, there is an acceleration term added to Newton's gravitational equation.
The consequence is that the total energy (inertial plus gravitational) of a particle APPLICATION OF CURVATURE COSMOLOGY Table 21: Size of CC universe.
2.58 × 10−27 kg m−3 at rest in the universe is zero. He further assumed that matter receding witha velocity greater than that of light makes no contribution. The equivalentdistance in CC is the radius, R. The implication of his theory is that 2πGρR2 ≈ 1.
Now using Eq. we get the actual value for the left hand side to be 3/4 andthis value does not depend on the size of the universe. The closeness of thisvalue to unity suggests that Sciama's ideas are worthy of further investigation.
Solar neutrino production Since the Homestead mine neutrino detector started operation in the late 1960's,its observations have shown a deficiency in the observed intensity of solar neutri-nos compared to accurate theoretical calculations. This has led to an enormousactivity in the development and testing of solar models. Currently the standardexplanation for the deficiency in the arrival rate of solar neutrinos is that it isdue to neutrino oscillations. Basically the electron neutrinos produced near thecenter of the sun are converted into a mix of muon and tau neutrinos by thetime they reach the earth. Because of the high densities the matter oscillationas well as vacuum oscillations are important. Although there are several freeparameters that must be estimated the most convincing evidence comes fromthe Sudbury Neutrino Observatory, where the solar neutrino problem was finallysolved. There it was shown that only 34% of the electron neutrinos (measuredwith one charged current reaction of the electron neutrinos) reach the detector,whereas the sum of rates for all three neutrinos (measured with one neutral cur-rent reaction) agrees well with the expectations. The only reason that I includethe following, alternative explanation is that I was surprised at how accuratewere the results predicted by curvature pressure with no additional parameters.
Since this is the only place where local curvature pressure is used it is feasiblethat its derivation in section is flawed. Nevertheless Because it provides accu-rate predictions and the neutrino oscillation model has to fit several parametersit is worth examination.
The solar model used here is based on that described by For a local context, curvature pressure is given by Eq. What was doneis to use the tables (for solution BS05) generously provided by Bahcall in hisweb site and used them to calculate curvature pressure. It was then assumedthat the thermodynamic pressure was reduced by the value of the curvaturepressure and then we used the thermodynamic pressure as an index into thesame tables to get the temperature. This largely avoids all the complications of APPLICATION OF CURVATURE COSMOLOGY Table 22: Computed production rates for solar neutrinos for the standard modelincluding curvature pressure.
equations of state and changing compositions. Naturally, this will only work ifthe corrections, as they are here, are small. Then this temperature was used asan index into the neutrino production table to get the production rate for eachof the eight listed reactions. As a calibration and a check, the same programwas used to compute the rates with no curvature pressure. In this test, themaximum discrepancy from the expected rates was 1.3%.
At a radius of 0.1 solar radii, the reduction in thermodynamic pressure was 12.5% and the reduction in temperature was 4.1%. The computed rates withcurvature pressure included in the solar model are shown in Table Thestandard rates are from The solar neutrino unit (SNU) is a product of the production rate times theabsorption cross section and has the units of events per target atom per secondand one SNU is defined to be 10−36 s−1. For example for each 71Ga target atomin the detector the expected event rate due to solar neutrinos for the pp reactionwould be 57.7 × 10−36 s−1. The last row shows the expected event rates for37Cl and 71Ga target atoms where the uncertainties are proportional to thoseprovided by Another type of detector uses Cherenkovlight from the recoiling electron that is scattered by the neutrino. Because thiselectron requires high-energy neutrinos to give it enough energy to produce theCherenkov light this type of experiment is essentially sensitive only to the 8Bneutrinos.
provides a list of recent observational results and they are compared with the predictions in Table The columns show the name of theexperiment, the type of detector, the unit, the predicted rate (with curvaturepressure), the observed rate, and the χ2 of the difference from the predictedvalue. The statistical and systematic uncertainties have been added in quadra-ture to get the observed uncertainty. The result in the last row from SNO isfrom the charged current reaction ( νe+d→p+p+e) that is the expected rate ifthere are no neutrino oscillations. The agreement is excellent. However, theremay be some biases that could be either theoretical or experimental in origin.
The crucial test requires computation with a solar model that includes curva-ture pressure so that the more subtle effects are properly handled. The benefitof this agreement is that it gives very strong support for curvature pressure in APPLICATION OF CURVATURE COSMOLOGY Table 23: Comparison of predicted and observed solar neutrino production rates.
a106 cm−2 s−1 a non-cosmological context.
Heating of the solar corona For over fifty years, astrophysicists have been puzzled by what mechanism isheating the solar corona. Since the corona has a temperature of about 2 × 106 Kand lies above the chromosphere that has a temperature of about 6000K, theproblem is where the energy comes from to give the corona this high tempera-ture. Let us consider whether curvature redshift due to the gas in the coronacan heat the corona via the energy loss from the solar radiation. quotes the number distribution of electrons in the corona to be Ne = 2.99 × 1014r−16 + 1.55 × 1014r−6 + 3.6 × 1012r−1.5 m−3, where r is the distance from the solar center in units of solar radii. If we assumespherical symmetry then all the radiation leaving the sun must pass througha shell centrad on the sun and we can use Eq. and Eq. to compute thefractional energy loss in that shell. To the accuracy required, we can also assumethat the hydrogen number density is the same as the electron density and thenthe integration of Eq. from the solar surface to 4 solar radii above the surfacegives a total fractional energy loss of 1.32 × 10−11. Thus with a solar poweroutput of 3.83 × 1026 W the total energy loss to the solar corona by curvatureredshift is 5.1 × 1015 W which is equivalent to 8.3 × 10−4 W m−2 at the surfaceof the sun. This may be compared with the energy losses from the coronato conduction, solar wind and radiation. The total loss rates are quoted byto be 8×102 W m−2 for coronal holes, 3×103 W m−2 for thequiet corona and 104 W m−2 for an active corona. Since these are about sevenmagnitudes larger than the predicted loss, curvature redshift is not importantin the inner corona. Although it is not pursued here, there is a similar problemin that the Milky Way has a corona with a high temperature. It is intriguingto speculate that curvature redshift may explain the high temperature of thegalactic halo.
Pioneer 10 acceleration Precise tracking of the Pioneer 10/11, Galileo and Ulysses spacecraft have shown an anomalous constant acceleration for Pioneer10 with a magnitude (8.74 ± 1.55) × 10−10 m s−2 directed towards the sun. The APPLICATION OF CURVATURE COSMOLOGY major method for monitoring Pioneer 10 is to measure the frequency shift ofthe signal returned by an active phase-locked transponder. These frequencymeasurements are then processed using celestial mechanics in order to get thespacecraft trajectory. The simplicity of this acceleration and its magnitude sug-gests that Pioneer 10 could be a suitable candidate for investigating the effectsof curvature redshift. There is a major problem in that the direction of theacceleration corresponds to a blue shift whereas curvature redshift predicts aredshift. Nevertheless, we will proceed, guided by the counter-intuitive obser-vation that a drag force on a satellite actually causes it to speed up. This isbecause the decrease in total energy makes the satellite change orbit with aredistribution of kinetic and potential energy.
The crucial point of this analysis is that the only information available that can be used to get the Pioneer 10 trajectory is Doppler shift radar. There isno direct measurement of distance. Thus the trajectory is obtained by applyingcelestial mechanics and requiring that the velocity matches the observed fre-quency shift. Since the sun produces the dominant acceleration we can considerthat all the other planetary perturbations and know drag effects have been ap-plied to the observations and the required celestial mechanics is to be simpletwo body motion. If the observed velocity (away from the sun) is increased(in magnitude) by an additional apparent velocity due to curvature redshift theorbit determination program will compensate by assuming that the spacecraftis closer to the sun than its true distance. It will be shown that this distancediscrepancy produces an extra apparent acceleration that is directed towardsthe sun. The test of this model is whether the densities required by curvatureredshift agree with the observed densities.
Let the actual velocity of Pioneer 10 at a distance r, be denoted by v(r), then since the effect of curvature redshift is seen as an additional velocity, ∆v(r)where from Eq. it is given by where the factor of 2 allows for the two-way trip and the density at the distancer from the sun is ρ(r). Since Pioneer 10 has a velocity away from the sun thisredshift shows an increase in the magnitude of its velocity. We will assume thatall the perturbations and any other accelerations that may influence the Pioneer10 velocity have been removed as corrections to the observed velocity and theremaining velocity, v(r), is due to the gravitational attraction of the sun. Inthis case the energy equation is where µ = GM is the gravitational constant times the mass of the sun (µ =1.327 × 1020 m3 s−2) and v∞ is the velocity at infinity. The essence of thisargument is that the tracking program is written to keep energy conserved sothat an anomalous change in velocity, ∆v(r), will be interpreted as a change inradial distance which is Thus an increase in magnitude of the velocity will be treated as a decreasein radial distance which, in order to keep the total energy constant, implies APPLICATION OF CURVATURE COSMOLOGY an increase in the magnitude of the acceleration. Either by using Newton'sgravitational equation or by differentiating Eq. the acceleration a(r) is givenby Hence with v∞ = 0 and therefore v(r) = and then to the first order an increase in velocity of ∆v(r) will produce anapparent decrease in acceleration of ∆a(r), and πµG r−1/2 < where for the last equations we measure the distance in AU so that r = 1.496 ×1011R and the angle brackets show an average value. Now fig. 7 from shows that after about 20 AU the anomalous acceleration is essen-tially constant. The first step is to get an estimate of the required density andsee if is feasible. Using the observed acceleration of aP = 8.74 × 10−10 m s−2the required average density for the two-way path is 1.60 × 10−20R kg m−3 andfor R=20 it is 3.21 × 10−19 kg m−3.
The only constituent of the interplanetary medium that approaches this density is dust. One estimate by ofthe interplanetary dust density at 1 AU is 1.3 × 10−19 kg m−3 and more recently, suggests a value of 10−19 kg m−3 which is consistent with their earlier estimate of 9.6 × 10−20 kg m−3 the authors do not provide uncertainties it is clear that their densities couldbe in error by a factor of two or more. The main difficulties are the paucityof information and that the observations do not span the complete range ofgrain sizes. The meteoroid experiment on board Pioneer 10 measures the fluxof grains with masses larger than 10−10 g. The results show that after it left theinfluence of Jupiter the flux was essentially constant(in fact there may be a slight rise) out to a distance of 18 AU. It is thoughtthat most of the grains are being continuously produced in the Kuiper belt. Asthe dust orbits evolve inwards due to Poynting-Robertson drag and planetaryperturbations, they achieve a roughly constant spatial density. The conclusionis that interplanetary dust could provide the required density to explain theanomalous acceleration by a frequency shift due to curvature redshift.
also reports a annual velocity variation of (1.053 ± 0.107)×10−4 m s−1 with a phase angle relative to conjunction of 5◦.7±1◦.7. Thecause of this variation is the changing path length through the dust at about1 AU as the earth cycles the sun. However if this annual variation is due tocurvature redshift it cannot be easily distinguished from a position displacementin the plane of the ecliptic: for example this anomalous velocity correspondsto a position shift of about 5 × 10−3 arcsec. From Eq. and a density of10−19 kg m−3 the predicted curvature-redshift velocity is 3.9 × 10−3 m s−1 which is an order of magnitude larger than the reported anomalous diurnal velocity.
Clearly most of the predicted velocity could have been interpreted by the orbitdetermination program as a very small angular displacement. This could alsoexplain the phase angle. The predicted phase angle is 90◦ from conjunction,whereas the observed phase angle is very close to the line of conjunction.
Finally reports a diurnal component. Reading from their fig. 18 the diurnal velocity amplitude is about 1.4 × 10−4 m s−1. Notethat due to inhibition there is no curvature redshift to be expected from theatmosphere. The major redshift will come from the inter-planetary dust. Thenusing the earths radius and a density of 10−19 kg m−3 the expected diurnalvelocity amplitude due to curvature redshift is 1.7 × 10−7 m s−1 which is threeorders of magnitude too small. The average density that is needed is about7.2 × 10−14 kg m−3. Unless there is such a density it is unlikely that curvatureredshift could explain the diurnal velocity effect. Note a critical test would beto compare the simultaneous observation of the Pioneer 10 velocity from twotracking stations as a function of their different distances from Pioneer 10.
Overall, this analysis has shown that it is possible to explain the acceleration anomaly of Pioneer 10 but that a more definitive result requires curvature red-shift to be included in the fitting program and more accurate estimates of thedust density are certainly needed. Subject to the caveat about the dust density,curvature redshift could explain the anomaly in the acceleration of Pioneer 10(and by inference other spacecraft).
The major conclusion from this evaluation of Big Bang cosmologies is that allof the topics covered in Section namely Tolman surface brightness, type 1asupernova, angular size, gamma ray bursts, galaxy distribution, quasar distri-bution, radio source counts, quasar variability in time and the Butcher–Oemlereffect are in excellent agreement with a static universe. Furthermore quantita-tive estimates of evolution derived within the BB paradigm are very close towhat would be predicted by BB with a time dilation term of (1 + z) removedfrom the equations.
Results for the topics of the Hubble redshift, X-ray background radiation, the cosmic background radiation and dark matter show strong support forcurvature cosmology. In particular CC predicts that the Hubble constant is64.4 ± 0.2 km s−1 Mpc−1 whereas the value estimated from the type 1a super-nova data is 63.8 ± 0.5 kms−1 Mpc−1 and the result from the Coma cluster(Section is 65.7 kms−1 Mpc−1. In CC the theoretical cosmic temperatureis 2.56×109 K for the cosmic gas and the temperature estimated from fitting theX-ray data is (2.62 ± 0.04) × 109 K. The predicted temperature for the CMBRis 3.18 K. whereas measured the temperature to be 2.725K. This prediction does depend on the nuclei mix in the cosmic gas and couldvary from this value by several tenths of a degree. Curvature cosmology doesnot need dark matter to explain the velocity dispersion in clusters of galaxies orthe shape of galactic rotation curves. Nor does it need dark energy to explaintype 1a supernovae observations.
Other topics in Section namely the Sunyaev–Zel'dovich effect, gravita- tional lens, the Gunn–Peterson trough, redshifts in our Galaxy and voids can be explained by CC or are fully compatible with CC. Currently CC providesa qualitative explanation for the abundances of light elements (the 'primor-dial' abundances) but not a quantitative predictions. The remaining topics inSection namely Lyman-α forest, galaxy rotation and anomalous redshift arecompatible but with problems.
Curvature pressure can explain the non-cosmological topic of solar neutrino production but since this already explained by neutrino oscillations it must re-main a curiosity. The explanation of the Pioneer 10 anomalous acceleration isfeasible if the inter-planetary dust density is a little larger than current esti-mates.
This research has made use of the NASA/IPAC Extragalactic Database (NED)that is operated by the Jet Propulsion Laboratory, California Institute of Tech-nology, under contract with the National Aeronautics and Space Administra-tion. The calculations have used Ubuntu Linux and the graphics have used theDISLIN plotting library provided by the Max-Plank-Institute in Lindau.
Traditional least squares regression provides two lines, the regression of y onx and the regression of x on y. In general, these regression lines do not havereciprocal slopes. If the data includes values for the uncertainties in both coordi-nates, the traditional minimization of χ2 methods would ignore the uncertaintyin the independent variable. A better solution is obtained by the subroutine fitexy provided by in their book Numerical Recipes, whichalso gives the basic references. This solution uses both sets of uncertainties ina completely symmetric manner that has a single solution regardless of whichvariable is deemed to be the independent. The method for estimating a and bin y = a + bx is to minimize the χ2 function defined by X (yi − a − bxi)2 σ2 + b2ε2 + η2 where σi is the uncertainty in yi, εi is the uncertainty in xi and η is describedbelow. It is apparent that this equation has the basic symmetry in that if η iszero then interchanging x and y will give the reciprocal of the slope b. If all theε are zero it has an analytic solution otherwise the minimization can be donenumerically.
A problem arises when the χ2 value is significantly different from the number of degrees of freedom. The case where the value of χ2 is much less than thenumber of degrees of freedom is handled by multiplying the uncertainties bythe square root of the ratio of χ2 to the degrees of freedom. If the χ2 is largerthan the number of degrees of freedom the approach used here is to assumethat there is an additional, unknown contribution which is uncorrelated withthe existing uncertainties. This is a common occurrence when the uncertaintiesare measurement errors and may be precise but some other factor makes asignificant contribution the scatter of values. the solution is to add a quadratic term, η2, to the denominator and estimate its value by making the χ2 equal tothe degrees of freedom. The prime advantage of this technique is that it providesa smooth transition from the situation where the given uncertainties dominateto the alternative where they are negligible. The main effect is a change in therelative weights.
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Advanced Diagnostic Approaches and Current Managemnet of Proventricular Dilatation Disease Ady Y. Gancz, DVM, MSc, DVSc, DABVP (avian practice),a Susan Clubb, DVM, DABVP (avian practice), b and H. L. Shivaprasad, BVSc, PhD, DACPV c aThe Exotic Clinic, Herzliya, Israel. bRainforest Clinic for Birds and Exotics, Loxahatchee, Florida. cUniversity of California, Davis, California.