Single parameter model for cosmic scale photon redshift in a closed Universe

A successful single parameter model has been formulated to match the observations of photons from type 1a supernovae which were previously used to corroborate the standard 𝛬 cold dark matter model. The new single parameter model extrapolates all the way back to the cosmic background radiation (CMB) without requiring a separate model to describe in�ation of the space dimensions after the Big Bang. The model for the redshift progression of a photon is:


Introduction
An interpretation of the Planck Legacy 2018 release (by Di Valentino, Melchiorri and Silk 1 ) concluded that spacetime has a positive curvature at more than the 99% con dence level.Positive curvature implies that the spacetime manifold is nite.
Prior to this discovery, the calculation of the expansion of space assumed that spacetime is at but expanding with time.
When photons are detected from very distant sources, a calculation is necessary which considers the expansion of space with time.As a result, it is necessary to assume the rate of expansion of space in order to calculate the distance travelled by the photon from the source to the detector.The intensity of the detected photon is related to the light path of the photon and it is the light path distance that is measured.This measurement, using photon intensity, must be corrected to account for the density of matter, referred to as cosmic dust, along the light path.
To date, the usual approach to locating the distance for a far-away galaxy assumes that the expansion of space is related to the increase of the wavelength of the photon.A complicated technique is required to calculate a distance travelled through the expanding space dimension.The calculated progression of the space dimensions is then tted to a modi ed Friedmann equation into which at least three parameters are added, and the values of these parameters are regressed to get a good t.The three added

Results And Discussion
The complicated regression approach, used with the standard  cold dark matter model, can be changed to a single parameter regression approach for a nite 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 rst free photons.The measured redshift corresponding to a photon emitted at time t 1 and detected at time t 2 is then determined by the equation: 1+z = sinθ 2 /sinθ 1 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: Loading [MathJax]/jax/output/CommonHTML/fonts/TeX/fontdata.jsWhen 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 rst photons are produced is changed then a new value for T can be found to match the observed redshift.
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 effect 4 .
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 × 10 10 years then 2πR = (360/90) × 2.4 × 10 10 so R = (4.8× 10 10 )/π What has been found is that photons emitted from galaxies which are dated using the standard  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 tting parameters have transformed the data from the true spherical basis to a at basis in much the same way that maps are transformed to at paper representations of the curved surface of the Earth.
Ringermacher and Mead 6 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 tted parameter model prediction for 1+z is as follows: Loading [MathJax]/jax/output/CommonHTML/fonts/TeX/fontdata.js