The complicated regression approach, used with the standard 𝛬 cold dark matter model, can be changed to a single parameter regression approach for a finite and symmetrical spacetime manifold. Rather than using Euclidean geometry, as for the current standard approach, it is now proposed that it is more appropriate to use spherical geometry when the Universe has positive curvature. The calculations are analogous to determining the change in latitude on the surface of the Earth for a known distance over the surface of the Earth and a known ratio of the of latitude and longitude changes.
Using the assumption that the spacetime manifold has a single radius, R which is equal for all three space dimensions and for the time dimension (when the space dimensions are measured in light years and the time dimension is measured in years) leads to a single parameter model to describe the expansion of space with time from the Big Bang event. This expansion of space causes the wavelength of photons to increase in direct proportion with the expansion (when ignoring any potential gravitational redshift or relative motion effects). The spherical geometry is used to locate the time at which the measured photon is emitted.
The space dimension will reach a maximum at time T. If the time elapsed from the Big Bang, t, is expressed as the ratio t/T then this ratio can be multiplied by π/2 to get a change in angle, θ in radians. This is analogous to a change in latitude except that the proposed convention is to measure the angle from a time zero starting point which is at a position analogous to a pole on a globe. The determination of time zero, at a pole, may be a purely theoretical extrapolation since there is no way to observe what happens prior to the release of the first free photons. The measured redshift corresponding to a photon emitted at time t1 and detected at time t2 is then determined by the equation:
1+z = sinθ2/sinθ1
Assuming that our photon detector near Earth is located at about 13.8 billion light years after the Big Bang then the general formula is:
where t is the time after the Big Bang time zero when the photon was emitted.
As an illustration, let’s assume that a cosmic background radiation (CMB) photon was emitted at about 7.15 million years after time zero, then the redshift for this photon will be given by:
When T=24, then 1+z = 1089 which agrees with our observations. If the estimate for the current age of the Universe or the time at which the first photons are produced is changed then a new value for T can be found to match the observed redshift.
The highest-known redshift radio galaxy (TGSS J1530+1049)5 is at 1+z = 6.72. For this galaxy:
sin(13.8/24)π/2 = 0.785389, so sin(t/24)π/2 = 0.785389/6.72 = 0.116873 which gives t = 1.79 billion years after the Big Bang which is about 12 billion years ago.
So far, the contribution from gravitational redshift has been ignored. This may be reasonable when the spacing between the source and detector is very large. It is assumed, in the following discussion, that the reported cosmological scale factor data has been appropriately interpreted to account for any gravitational redshift effects.
Relative motion for our point of observation has also been ignored. Comprehensive surveys of type 1a supernovae redshift data have recently allowed for correction of this effect4.
The light path distance through space for a photon, measured in light years, is always equivalent to the time elapsed, measured in years, since the emission of the photon. This is known because special relativity works. Localized adjustments, using general relativity, may be necessary where spacetime is distorted by matter.
The radius of the spacetime manifold, R is related to T by the formula:
R = 2T/π
If T is set to 2.4 × 1010 years then 2πR = (360/90) × 2.4 × 1010 so R = (4.8 × 1010)/π
What has been found is that photons emitted from galaxies which are dated using the standard 𝛬 cold dark matter model approach have red shift values which are very similar to the values calculated as described above for a Universe with positive curvature. This is most likely because the additional fitting parameters have transformed the data from the true spherical basis to a flat basis in much the same way that maps are transformed to flat paper representations of the curved surface of the Earth.
Ringermacher and Mead6 have provided a useful plot of the cosmological scale factor, a(t) = 1/(1+z) as a function of lookback time.
For a(t) = 0.5 which corresponds to 1+z = 2.0, they report a value of t = 0.432 x 13.8 = 5.96 billion years.
Using this reported value for t, the single fitted parameter model prediction for 1+z is as follows:
Although this is not an exact match, it is well within the data scatter in Figure 2 from Ringermacher and Mead6.
For a lookback time of 0.6 x 13.8 = 8.28 billion years the calculation gives:
so a(t) = 0.657 and this is also well within the reported data scatter.
In general, for the data range provided by type 1a supernovae, the fits for the new single fit parameter model and the standard 𝛬 cold dark matter model are both within the data scatter.