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Summary of Curvature Cosmology
INTRODUCTION
This is a summary of the major aspects of curvature cosmology (book: Curvature Cosmology BrownWalker Press). For brevity most of the arguments and references are omitted. Figure references are to the book and they are also available on my web site. http://www.davidcrawford.bigpondhosting.com For historic reasons many redshifts are expressed in velocity units. Curvature cosmology (CC) describes a new cosmological theory that is an alternative to the current Big-Bang cosmology. It rests on two hypotheses: curvature-redshift and curvature-pressure and is fully compatible with General Relativity and quantum mechanics. CURVATURE REDSHIFT In space with a positive curvature a bundle of geodesics will decrease in cross-sectional area just like lines of longitude get closer as they go from the equator to the poles. This decrease is applicable to all wave motion and if we consider a photon to be a localised wave packet its cross-sectional dimensions will also decrease as it propagates. The implication is that its angular momentum should also decrease which contrary to quantum mechanics. The hypothesis of curvature redshift is that this impasse is overcome by the emission of two very low energy secondary photons. In effect there is a gravitational interaction with curved space time. Because the photon interacts with a very large effective mass and because of symmetries there is no angular scattering and the usual argument against tired-light redshifts is not applicable. The ‘focussing theorem’ shows that the rate of loss of energy is proportional to the product square root of the local gas (plasma) density and the photon energy. There is not curvature redshift for a photon passing a large mass like the sun. Since the process is applicable to any wave a similar interaction occurs for electrons and other particles. Because the basic interaction is one of a slow build up any other interaction that occurs before the interaction takes place can inhibit curvature redshift. Refractive index is a very important inhibiting interaction. It makes it very difficult to see the effect in a laboratory. Nevertheless a possible laboratory test is suggested. It is similar to the famous Pond and Snyder Harvard tower experiment. CURVATURE PRESSURE Consider a point at the bottom of the ocean. It is clear that the large pressure is due to the opposition by the elastic forces of the ocean floor opposing the gravitation acceleration of the water above it. There is no pressure if the water was in free fall. Hence the simple principle that follows is that if gravitational accelerations are partly or fully opposed by some other forces a pressure will be produced. The hypothesis of curvature pressure is that in plasma that is producing the local curvature of space the non-geodesic motion of electrons and nuclei due to electromagnetic forces will produce a pressure that reacts back upon the plasma. In this case it is a self interacting pressure that will act against changes in density. Since the curvature depends on density an increase in density will produce an increase in curvature and hence an increase in pressure that will act to try and decrease the density. Although the curvature pressure is negligible in most circumstances it is very important in cosmology and may be important in solar neutrino production. THE COSMOLOGICAL MODEL The simplest cosmological model is for a universe that consists only of uniform plasma. In the Friedmann equations the curvature pressure replaces the cosmologic constant. Except for this modification there is no change needed in the equations for general relativity. The result is a set of equations that are easily solved to get a static cosmology. Of most importance is that the model is stable and thus it obeys the perfect cosmological principle. There is only one free parameter, the plasma density. The Hubble constant and the radius of the universe are directly determined by the density. What is interesting is that this model predicts that the temperature of the plasma is independent of the density and is 2.56E9 K. The real universe clearly has large density variations. Unfortunately these variations are not in equilibrium and it is difficult to determined appropriate equations of state. While they may not greatly change the broad picture they will make second order corrections to many predictions. OLBER’S PARADOX There is no problem with Olber’s paradox in this model. Any radiation will be redshifted until all of its energy is lost. This energy goes into heating the cosmic plasma. In turn galaxies and stars are being continuously created from the cosmic plasma so that in a statistical sense the universe is the same everywhere and for all time. CREATION OF THE ELEMENTS A major problem in any cosmology is the creation of deuterium and helium from hydrogen. In this model the cosmic plasma is hot enough to sustain nuclear interactions. Furthermore the major interactions will be the destruction of heavier nuclei that were produced in stars and supernovae. BLACK HOLES The collapse of a body to something smaller than a neutron star will be invariably accompanied by the production of energy so that such an object will be very hot. In such an object there will be curvature pressure due to the strong nuclear forces preventing the particles from following geodesics. It is argued that this will prevent the final collapse to a black hole. Moreover if the body is rotating the combination of curvature pressure and instabilities may provide a mechanism for the production of two jets of material. Could this explain the occurrence of astrophysics jets on both stellar and galactic dimensions? The result is an object smaller than a neutron star with all the external characteristics of a black hole. COSMOLOGICAL TESTS Because it has very few free parameters CC is easily refutable. Probably it most important prediction that can be tested is that there is no evolution. To be more precise if it is shown that there is any characteristic of any object that can be shown to change with time this would be a major blow to CC. The tests to be discussed have two objectives. The first is to show that they have strong support for the model and the second is to critically examine the evidence for evolution. BACKGROUND X-RAY OBSERVATIONS These are very important in that they can provide an estimate of the cosmic plasma density and an estimate of its temperature. Although the energy range below about 10 kev and that above about 300 kev are well explained by the presence of discrete sources; the explanation for the intermediate range is uncertain. Although in big bang (BB) cosmology the explanation of X-rays being due to bremsstrahlung provides a good fit the large plasma densities required are ruled out by the Sunyaev-Zel’dovitch effect. In CC the required densities are lower and Figs. 4 & 5 show an excellent fit. For the plasma composition of hydrogen with 8.5% helium the density is 1.34 H atoms/cubic metre. The fitted temperature is (2.62 +/- 0.04)E9 K to be compared with the predicted temperature of 2.56E9 K. The temperature agreement is excellent. COSMIC MICROWAVE BACKGROUND RADIATION The prediction and black body spectrum of the CMBR is claimed as a major victory for BB cosmology. First we note that a black body spectrum is the maximum entropy distribution for photons with a fixed energy density provided that all energy levels are available and there is a mechanism for transferring photons between levels. Thus the black body spectrum is a default spectrum. Without belittling the claim I would be more impressed if BB cosmology had predicted a non-black body spectrum which was then found. Note that in BB cosmology the prediction of an exact temperature is difficult and the best a priori prediction was a lower limit of 5K. In CC the CMBR arises from the secondary photons of electrons undergoing the curvature redshift interaction. What we observe is the local production of the CMBR photons (say z1). They are continuously produced and redshifted in a two stage process that will populate all levels and therefore a black body spectrum is predicted. We can estimate the temperature of the CMBR by requiring that the energy loss rate by the electrons must equal the energy loss rate by the CMBR photons. That is we ignore other radiation such as the background X-ray radiation. This gives a temperature of 3.48 K to be compared with the observation of 2.729 K. Note that the prediction is sensitive to the second order effects of density variations. SUNYAEV-ZEL’DOVICH EFFECT The major criticism of tired-light redshifts is that we do not observe the Sunyaev-Zel’dovich effect in the CMBR. In this model the mean free path for a density of 1.34 is 435 Gpc whereas the production distance is less than 4.9 Gpc (z=1) which means that only about 1% of the photons have a Compton interaction. Thus the effect is negligible. TYPE 1A SUPERNOVAE The observations of these supernovae are critical since they appear to show a well defined time-dilation of their light curves which is contrary to CC. Fig. 6 shows the light curve widths relative to a template of supernovae from the Supernovae Cosmology Project. The straight line shows the fitted straight line which shows a power law slope of 0.956+/- 0.022 which is in full agreement with BB cosmology. The only worry is that the Chi^2 fit has a value of 509.4 for 48 (DoF). The width uncertainties are accurate so why is the Chi^2 a factor of about 10 too large. In CC this width verses z dependence is explained as a selection effect. Fig. 7 shows the absolute magnitudes verses width factor and shows that the power law between luminosity and width has a value of -1.07+/-0.18 which is opposite in sign to that suggested by Phillips (1993) & Hamuy et al(1996). My analysis of their data (local supernovae in Fig. 8) gave an exponent of 1.16+/-0.47. Using BB cosmology there is no variation of absolute magnitude with width. The particular choice of BB parameters used produces a luminosity-distance equation that differs from the CC one by a factor close to (1+z). Thus in BB cosmology the luminosity-width relation is hidden by the BB luminosity-distance equation. The interesting physics is that a slope of -1.07 shows that the total energy output of a type 1a supernova is constant, that is in full agreement with the Chandrasekar limit. The BB result shows that the total energy varies with width exponent of 2.31! Thus the CC result shows that the supernovae maximum luminosity is inversely proportional to the width of its light curve. Now Richardson et al (2002) have shown that for local supernovae the standard deviation of their absolute magnitudes is 0.56 mag. Thus if there is a selection effect there is sufficient variation in magnitudes and hence widths to cover the observed distribution. However this is not sufficient since we still must explain why there is a strong dependence of width on z and not a scatter plot. The process getting these supernovae results is a two stage process. The first stage is to monitor a large number of galaxies to wait for possible supernovae. The second stage is to make repeated observations of it to determine its parameters. Now this second stage is very expensive and as a result there is considerable pressure to eliminate supernovae of other types as soon as possible. After all the major objective was to get magnitudes and the width verses redshift relationship was deemed to be well understood. Using the BB cosmological model the rms value for the absolute magnitudes was 0.22 mag which is a surprising small value (c.f.0.56 mag cited above). Thus a simple selection model is that after initial observations any candidate that had an absolute magnitude (BB) greater than a chosen value was rejected. Then because of the absolute magnitude verses width relationship a curve (dashed line in Fig. 6) is expected. Given the absolute magnitude and width for a type 1a supernova we can use the width to correct the absolute magnitude to a “standard candle”. This is shown for SCP data in Fig. 9 and for data from Riess et al (2004) in Fig. 10. This latter set goes to a larger redshift than the SCP data. A minor problem is that in CC the redshift is a good measure of distance for distant objects but it is distorted by galactic halos and gas clouds for nearby objects. In Fig 10 the fitted straight line for all the supernova with z0.1 has an exponent of -0.061+\-0.088 with Chi^2=57.0 (62 DoF) which is consistent with no variation of magnitude with distance. Thus there is no indication of any dark energy. The simple CC equation to convert from apparent to absolute magnitudes has very strong support from the supernovae data provided the magnitudes are corrected for width. QUASAR VARIABILITY WITH TIME Hawkins (2001,2003) has measured the variability of quasars over an eighteen year period and has found no dependence on redshift. Exactly what is predicted by CC. LINEAR SIZE OF RADIO SOURCES Gurvits, Kellerman & Frey (1999) have measured the angular size of extended radio sources from the central peak to the most distant part that had a brightness of at least 2% of the peak brightness. Fig. 11 shows a plot of the linear size of there sources out to a redshift of about 3.9. The sources were placed in bins and the median value was taken for each bin. The straight line fitted to the medians has a slope of 47.3+/-10.9 kpc/z. Because of the omission of large nearby sources and distant small objects (lack of resolution) we would expect a small positive slope. Therefore the data is consistent with no change in linear size with redshift. Buchalter et al (19980) have looked at 103 double-lobed sources from the VLA First survey. These sources are about one order of magnitude large than the previous ones. The result was a slope of 0.39+/-0.15 Mpc/z, which is not significant. Both these data sets show good support for the geometry of CC. TOLMAN SURFACE BRIGHTNESS This well known test is often quoted to show that tired-light redshifts are invalid. I have used the data from Sandage & Lubin (in four papers in 2001) of the surface brightness of elliptical and S0 galaxies. I have followed their analysis except that I have used CC geometry. Results are shown in Figs. 13 & 14. Of interest is that the surface brightness as a function of the logarithm of linear radius Has a slope of -0.401+/-0.053 which shows that the surface luminosity is proportional to the radius. At the end of the analysis what we have is the surface brightness for a large number of nearby galaxies and that for three distant clusters of galaxies. Now the expected dependence is a power law with a slope of -1 for tired-light redshifts and a slope of -4 for an expansion cosmology. Fig 15 shows a plot of the corrected surface brightness verses redshift. The fitted slope was -1.81+/-0.11 which is inconsistent with both models. However a critical point is that all the nearby galaxies were the brightest members of their clusters! If we fit to all the nearby galaxies and only the three brightest members of each of the three distant clusters the slope is -0.58+/-0.24. Although it is not conclusive we could argue that it rules out BB cosmology. CLUSTERS OF GALAXIES One of the major reasons for introducing dark matter was to explain the apparent fact that the virial theorem when applied to the velocities of galaxies in a cluster gave one or more orders of magnitude mass for the cluster than that obtained from luminosities. The Coma cluster is fairly symmetric and has a gas cloud in its centre. Curvature redshift predicts that galaxies seen through this gas cloud will have increased redshifts. For 583 galaxies the rms velocity is 893 km/s. Unfortunately we do not know the Z distance of each galaxy. However if we assume that our position is not privileged we can use a Monte Carlo method and give each galaxy a Z position that is the same as an X or Y position for another arbitrary galaxy. This was done many times and using the gas density and distribution obtained from X-ray observations the average rms was estimated to be 554 km/s. Although this differs from the observed value it must be emphasised that there were no free parameters, it is very sensitive to the distance to the Coma cluster and it should be noted that both the galaxy and the gas cloud density distributions are quite complex. Although the Virgo cluster is fairly irregular a plot (Fig. 16) of the redshifts of inner galaxies verses their (non Hubble) distance shows a strong dependence and a straight line has a slope of 436+/-152 km/s/Mpc which is consistent with the Coma results. An important consequence of redshift due to cluster gas is that galaxies within the cluster and those seen though it will have increased redshifts. For the Coma cluster the average increase is about 1200 km/s. This is evidence for this from Karaji et al (1976). Furthermore there is a well known void in front of the Coma cluster with a width of about 3000 km/s. Although the numerical agreement is not perfect there is no doubt that many of the voids and structure in the redshift distribution of galaxies could be due to the presence of gas clouds. THE HUBBLE CONSTANT The basic theory says that the Hubble constant (parameter?) is given by H=42.21sqrt(N) km/s/Mpc, and with N=1.34 we get H=48.9 km/s/Mpc. For the Coma cluster with an average velocity of 6926 km/s and a distance of 87.1 Mpc we get H=(6926-1205)/87.1=65.7 km/s/Mpc. The supernovae 1a results provide a value of 69 km/s/Mpc. Given the unknown corrections needed for inhomogeneous cosmic plasma in the theoretical value and uncertainties in the observational values the agreement is reasonable. LYMAN-ALPHA FOREST The obvious cosmological test is to determine the density of Lyman-alpha absorption lines as a function of redshift. There are two big problems in that each quasar only provides a narrow range of redshifts and the number of lines is critically dependent on measuring the correct base levels. It is now thought that the problem of resolution is now largely overcome. The result is that there does not appear to be a consistent set of observations that can be used for definitive cosmological tests. In addition the occurrence of curvature redshift in an absorbing cloud will broaden any spectral line. This may explain the occurrence of lines with very large widths. EVOLUTION There are four topics that will be covered in this section. They are the distribution and evolution of galaxies and quasars, the Butcher-Oemler effect and the radio source counts as a function of flux density. The analysis of galaxies and quasars follow essentially the same procedure. The absolute magnitude is computed for each object and then given the characteristics of the data set the total volume in which that object could be have been observed and still accepted into the data set is calculated. The reciprocal of this volume is the density for that object. This is down for each object and the densities are added into bins in absolute magnitude. For both galaxies and quasars a distribution was observed that was very close to a Gaussian with a narrow width. For galaxies the curves are shown in Fig. 17 and the distribution had a peak (B magnitude of -19.14 mag and a HWHM of 1.16 mag. The quasar results are shown in Fig.23 with a peak of -22.82 mag and HWHM of 0.854 mag. To look for evolution it is essential to only use those objects that can be observed at all redshifts within the range. A suitable measure is the average absolute magnitude as a function of redshift. For galaxies the selection criteria are shown in Fig. 18 and the magnitude verses redshift dependence is shown in Fig. 19. For quasars the figures are Fig. 22 and Fig. 24. In both cases the spread in magnitude as a function of z is small and there is no consistent trend. Thus are no obvious signs of evolution. The Butcher-Oemler effect is the apparent increase in the fraction of blue galaxies in a cluster as the redshift of the cluster increases. There appears to be great difficulty in achieving consistent results over a wide range of wavelengths. Often one cannot tell whether K-corrections (which are essential) have been applied. Adreon et al (2004) state that “Twenty years after the original intuition by Butcher & Oemler, we are still in the process of ascertaining the reality of the Butcher-Oemler effect”. I have been able to devise a simple distribution for radio sources that shows excellent agreement with observed results. The results are shown in Fig. 25 where the counts have been corrected to a Euclidean distribution and the points for each frequency group have been increased by a factor of 10 from the set immediately below it. Except for the early low frequency results the model shows all of the basic structure and a good fit to the data. These results are meant to be a proof of concept rather than definitive results. In conclusion after examining four of the major sets of observations which show strong evolution in bag bang cosmology I have shown that the observations are consistent with CC without any sign of evolution. MISCELLANEOUS TOPICS One topic for which CC does not have a clear cut answer is that of galactic rotation curves. The typical spiral galactic rotation curve shows a rapid rise in (linear) velocity and then an almost constant value to points beyond the visible galaxy. This is usually ascribed to the presence of a huge halo of dark matter. In CC the magnitude of the velocity is readily achieved by curvature redshift in the normal galactic halo. The problem is that a symmetric halo would produce a symmetric velocity curve unlike the observed asymmetric ones. One possibility is that the halos are in fact asymmetric because of ram pressure due to the galaxy moving through the external medium. In the Milky Way galaxy the density of interstellar gas is sufficient to inhibit curvature redshift for radio frequency photons but not for optical photons. Consequently any object that can be seen at both frequencies should show curvature redshift for the optical lines but not for 21 cm and other radio frequency lines. Although the deficiency in rate of solar neutrinos can be explained by a non-zero neutrino mass and neutrino mixing it is interesting to investigate the application of curvature pressure in the sun. Using the standard solar model the presence of curvature pressure decreased the pressure at 0.1 solar radii by 12.5% and the temperature by 4.1%. The resulting predictions for the solar neutrino experiments a Table 23: Predicted and observed solar neutrino production rates. Experiment, Unit, Predicted, Observed, Chi^2 Homes SNU, 2.66±0.42, 2.56±0.23, 0.04 GALLEX+GNO, SNU, 85.6±5.4, 70.8±5.9, 3.42 SAGE, SNU, 85.6±5.4, 70.9±6.4, 3.08 Kamiokande, 10E6cm-2/s, 1.45±0.26, 2.8±0.38, 8.60 Super-Kamiokande,10E6cm-2/s, 1.45±0.26, 2.35±0.08, 10.95 SNO, 10E6cm-2/s, 1.45±0.26, 1.76±0.10, 0.25 The anomalous acceleration of Pioneer 10 can be explained by the occurrence of curvature redshift due to the inter-planetary dust. Although the apparent acceleration from curvature redshift is away from the sun it must be remembered that the only navigation information comes from Doppler radar. In solving the celestial mechanics equations the navigation program in effect reverses the sign of the apparent acceleration. The required dust density is about 8.8E-19 kg/m^3. Although this is higher than measurements made with a limited range of grain sizes it is feasible. -- David F. Crawford (Please remove the bird) http://www.davidcrawford.bigpondhosting.com |
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