A Tired Light/Contracting Universe Model from the Union2.1 Supernovae Data

A tired light/contracting universe (TLCU) model is shown to be an excellent fit to the redshift/distance modulus data for the 580 supernovae 1a in the Union2.1 compilation. The data reveal that the Milky Way is in a static region with a radius of about 450 Mpc. Beyond the static region the universe is contracting with a space velocity which is linearly proportional to distance over the whole range of the data (K=−7.6±2.3km⋅s−1⋅Mpc−1). The other constant of the model is the Hubble constant for which a value of H=69.51±0.86 km⋅s−1⋅Mpc−1 is obtained. The fit of the TLCU model to the Union2.1 data is at least as good as the fit of the two constant ΛCDM model to the same data. A formula for photon travel distance is derived and an experiment for the possible detection of the tired light process is proposed.


Introduction
Theory [1] predicts that the universe is either expanding or contracting with a space velocity which is linearly proportional to distance. An expansion may continue for ever or it may halt and then contract giving rise to the possibility of a "periodic world".
When it was observed that the redshift and distance for galaxies beyond the Local Group had a linear relationship [2] the expanding universe theory became established. Observations of supernovae 1a show that at higher redshifts [3], [4] the distances measured are greater than expected by the expanding universe theory and this is interpreted as an accelerating expansion for which a new force called "dark energy" is proposed.
The tired light theory [5] is an alternative explanation for the systematic How to cite this paper: Glover, J. (2017) A Tired Light/Contracting Universe Model redshift. The viewpoint on the supernovae 1a observations from the tired light theory is that for a given distance the observed redshift is less than expected as a result of the blueshift of a contracting universe. Hence it is suggested that the new physics required by the supernovae 1a observations may possibly be the old idea of tired light instead of the new idea of dark energy. A tired light/contracting universe (TLCU) model is developed here using the Union2.1 supernovae 1a data [6] and compared to the ΛCDM model. The TLCU model uses photon travel distance for which a formula is derived. A possible mechanism for the tired light effect is discussed and an experiment to test this mechanism is proposed.

The TLCU Model
The TLCU model is built on the idea [7] that the observed systematic redshift (z) has two components tl c z z z = + (1) where tl z is the result of an energy loss process and c z is the result of space contraction. Assuming that photons lose energy by a first order rate process [8] the tired light component is given by where d is the photon travel distance, H is the Hubble constant 1 and c is the speed of light. The distance between the emitter and observer at the moment the photon is emitted ( 0 d ) is here called the "initial distance" and 0 d d ≈ for low values of 0 d . The exact relationship between 0 d and d is considered later.
Assuming a flat (i.e. Euclidean) universe the initial distance is related [9] to the luminosity distance (D) by The luminosity distance (D) is obtained from the distance modulus (dm) by the standard relationship.
The distance modulus (dm) is defined as dm m M = − , where m is the observed apparent magnitude of an object and M is it's absolute magnitude. For supernovae 1a "m" is the peak observed apparent magnitude (with appropriate corrections).

Preliminary Calculations
Initially z c was calculated from the redshift and distance modulus data for 1 The Hubble constant used in the TLCU model is a constant of nature and assumed to be indepen-  each of the 580 SN 1a in the Union2.1 compilation using Equations (1) to (4) and with the assumptions that ). Since c z is essentially negative the initial assumption of a contracting universe is confirmed.
Although the overall picture is of a contracting universe the local situation is different. There are 176 supernovae with 0.015 0.101 z < < for which the c z values show 78 redshifts and 98 blueshifts which is consistent ( 0.13 p = ) with an even split. So the next conclusion is that the Milky Way is situated in a static region of about 450 Mpc radius.

The Final Model
In a gravitationally bound region of space the force of Newtonian gravity is greater than the cosmic force of expansion/contraction and although the region as a whole will take part in the universal cosmic expansion/contraction the effect of cosmic expansion/contraction cannot be measured within the region. It is now assumed that the static region extending to about 450 Mpc around the Milky Way is gravitationally bound. For the purpose of the model it is assumed that the Milky Way is located at the center of a static sphere with a radius of 450 Mpc. In order to be consistent with the observations it is also assumed that the cosmic contraction starts at the edge of the static sphere. For a cosmic contraction the velocity of contraction is proportional to distance, so that where k (km·s −1 ·Mpc −1 ) is the constant for cosmic contraction. Within the static region 0 d d = , but, beyond the static region it is necessary to allow for the contraction that occurs during the photon travel time. The relationship between 0 d , k and d is derived in Appendix 1, from which, Equation (11) re-arranged as is more convenient for finding d from o d and k by repeated substitution.
Equation (6)  while the rms un-weighted residual 0.226 mag dm = . Figure 1 shows a large scatter in c z so the data were binned in order to make the relationship between velocity of contraction and distance more precise. The 580 SN 1a were sorted into z order and divided into 29 bins each containing 20 SN 1a. The weighted average redshift and distance modulus were calculated for each bin (see Table 1). The procedure for finding H and k was repeated using The values of the contraction velocity calculated from the binned data using

Comparison of TLCU and ΛCDM Models
In the two constant ΛCDM model the initial distance (from Equation (13) of ref.
[10]) is In order to compare the models on the same basis the constants for the ΛCDM model were found by fitting Equation (8) to the Union2.1 data using the best fit criterium (Equation (6)). This gave      Figure 4. The hint of periodicity shown in Figure 2 is repeated in Figure 4 and not only for the TLCU model but also for the ΛCDM model.
Another method of comparing the models is to fix the minor constants and then to calculate the value of the Hubble constant for each bin of the binned data. This calculation (with  Table 1. The values of the uncertainty in the Hubble constant show that the TLCU model is a better fit to the binned data than is the ΛCDM model.

Discussion
Although the TLCU model only shows a contracting universe it is reasonable to assume that there was a prior expansion which would be consistent with the "periodic world" predicted by Friedman [1]. In this case the linear contraction revealed by the model can be expected to reverse at higher redshifts and eventually show the expanding phase. The periodicity hinted at in Figure 2 and Figure 4 may possibly be harmonics of the fundamental period. More accurate observations at higher redshifts are needed to reveal the truth.
The reality of the contracting universe depends, of course, on the reality of the tired light effect and although the TLCU model is an excellent fit to the observed data such a fit is no guarantee of the reality of the assumptions on which the model is based. It is also claimed [11] that time dilation falsifies the tired light theory although the assumption that the thirteen high redshift supernovae used are not subject to a Malmquist type bias may not possibly be the case. Nevertheless independent evidence for the tired light effect is essential. A possible Figure 4. dm residuals for ΛCDM and TLCU models using binned data (see text). mechanism for the tired light process and a terrestrial experiment to test this mechanism are discussed in Appendix 2.

Conclusion
It is concluded that further experimental work on a possible photon energy loss process would be justified.