_{1}

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.

Theory [

When it was observed that the redshift and distance for galaxies beyond the Local Group had a linear relationship [

The tired light theory [

A tired light/contracting universe (TLCU) model is developed here using the Union2.1 supernovae 1a data [

The TLCU model is built on the idea [

where

^{1}The Hubble constant used in the TLCU model is a constant of nature and assumed to be independent of time and space. The usual units used for the Hubble constant are ^{−1}, which is characteristic of a first order rate process. (

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 (

The luminosity distance (D) is obtained from the distance modulus (dm) by the standard relationship.

The distance modulus (dm) is defined as

Initially

The 580 values of

Although the overall picture is of a contracting universe the local situation is different. There are 176 supernovae with

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 con- traction 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

is more convenient for finding d from

where the weighting factor (w) is proportional to the inverse square of the estimated error in dm. For the Union2.1 data (

In the two constant ΛCDM model the initial distance (from Equation (13) of ref. [

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

z | dm | d_{o}_{ } Mpc | d Mpc | z_{c} | z_{tl} | TLCU | ΛCDM | ||
---|---|---|---|---|---|---|---|---|---|

H | resid. | H_{o} | resid. | ||||||

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) |

0.0157 | 34.158 | 67 | 67 | 0 | 0.0157 | 69.99 | −0.79 | 70.31 | −0.31 |

0.0192 | 34.589 | 81 | 81 | 0 | 0.0192 | 70.33 | −1.13 | 70.71 | −0.71 |

0.0231 | 35.033 | 99 | 99 | 0 | 0.0231 | 69.05 | 0.15 | 69.51 | 0.49 |

0.0260 | 35.294 | 112 | 112 | 0 | 0.0260 | 68.99 | 0.21 | 69.51 | 0.49 |

0.0304 | 35.659 | 131 | 131 | 0 | 0.0304 | 68.34 | 0.86 | 68.93 | 1.07 |

0.0338 | 35.862 | 144 | 144 | 0 | 0.0338 | 69.21 | −0.01 | 69.88 | 0.12 |

0.0437 | 36.409 | 183 | 183 | 0 | 0.0437 | 69.90 | −0.70 | 70.76 | −0.76 |

0.0597 | 37.109 | 249 | 249 | 0 | 0.0597 | 69.72 | −0.52 | 70.87 | −0.87 |

0.0894 | 38.033 | 371 | 371 | 0 | 0.0894 | 69.15 | 0.05 | 70.82 | −0.82 |

0.1287 | 38.894 | 532 | 532 | −0.0020 | 0.1307 | 69.18 | 0.02 | 70.49 | −0.49 |

0.1632 | 39.444 | 665 | 664 | −0.0051 | 0.1683 | 70.20 | −1.00 | 70.94 | −0.94 |

0.1976 | 39.935 | 810 | 807 | −0.0085 | 0.2061 | 69.58 | −0.38 | 69.96 | 0.04 |

0.2352 | 40.378 | 964 | 957 | −0.0122 | 0.2474 | 69.22 | −0.02 | 69.44 | 0.56 |

0.2641 | 40.646 | 1065 | 1056 | −0.0146 | 0.2786 | 69.76 | −0.56 | 70.04 | −0.04 |

0.2914 | 40.899 | 1171 | 1159 | −0.0171 | 0.3085 | 69.52 | −0.32 | 69.80 | 0.20 |

0.3243 | 41.135 | 1274 | 1258 | −0.0195 | 0.3438 | 70.42 | −1.22 | 70.88 | −0.88 |

0.3554 | 41.416 | 1416 | 1395 | −0.0229 | 0.3782 | 68.96 | 0.24 | 69.33 | 0.67 |

0.3897 | 41.671 | 1553 | 1526 | −0.0261 | 0.4159 | 68.33 | 0.87 | 68.73 | 1.27 |

0.4221 | 41.862 | 1658 | 1625 | −0.0286 | 0.4507 | 68.66 | 0.54 | 69.19 | 0.81 |

0.4538 | 42.076 | 1789 | 1749 | −0.0317 | 0.4855 | 67.85 | 1.35 | 68.37 | 1.63 |

0.5017 | 42.279 | 1902 | 1855 | −0.0344 | 0.5361 | 69.38 | −0.18 | 70.24 | −0.24 |

0.5427 | 42.440 | 1994 | 1941 | −0.0366 | 0.5793 | 70.60 | −1.40 | 71.73 | −1.73 |

0.5895 | 42.712 | 2194 | 2126 | −0.0413 | 0.6308 | 68.97 | 0.23 | 69.98 | 0.02 |

0.6302 | 42.860 | 2289 | 2214 | −0.0436 | 0.6737 | 69.74 | −0.54 | 70.94 | −0.94 |

0.7080 | 43.180 | 2533 | 2437 | −0.0493 | 0.7573 | 69.36 | −0.16 | 70.62 | −0.62 |

0.8011 | 43.624 | 2946 | 2810 | −0.0591 | 0.8602 | 66.21 | 2.99 | 67.06 | 2.94 |

0.8799 | 43.722 | 2953 | 2817 | −0.0593 | 0.9391 | 70.49 | −1.29 | 72.01 | −2.01 |

1.0060 | 44.130 | 3340 | 3160 | −0.0684 | 1.0745 | 69.22 | −0.02 | 70.49 | −0.49 |

1.2559 | 44.763 | 3974 | 3712 | −0.0835 | 1.3394 | 68.64 | 0.56 | 69.45 | 0.55 |

(1) weighted bin average redshift; (2) weighted bin average distance modulus; (3) initial proper distance-Equations (3) & (4); (4) photon travel distance Equation (6); (5) cosmic blueshift If d_{o} < 450 then z_{c} = 0 else z_{c} = (−7.6/c) ´ (d_{o} − 450); (6) tired light redshift z_{tl} = z − z_{c}; (7) H = (c/d) ´ ln(1 + z_{tl}); (8) resid = 69.2 − H; (9) Equation (8) Ω_{M} = 0.278; (10) resid = 70.0 − H_{o}.

weighted dm residual = 0.266. These residuals are identical to the residuals from the TLCU model. There is also a close correlation between the un-weightd dm residuals for the ΛCDM model and those for the TLCU model as seen in

Fitting the ΛCDM model to the binned data (n = 29) gives

The hint of periodicity shown in

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

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 [

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 [

mechanism for the tired light process and a terrestrial experiment to test this mechanism are discussed in Appendix 2.

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

I am grateful to Graham P. Gerrard for help with computing and many stimulating discussions. I used TurboBasic for the calculations which were recoded in Python and the results checked by GPG using SageMathCloud (SMC). I am also grateful to the late Roger H. Beresford for a copy of the Nelder & Mead minimisation routine coded in Basic. I used SMC to create the figures and SageTex to prepare the document.

Glover, J. (2017) A Tired Light/Contracting Universe Model from the Union2.1 Supernovae Data. Journal of Modern Physics, 8, 147-155. https://doi.org/10.4236/jmp.2017.82013

The initial distance (

where x is distance and k is a constant. Integrating Equation (9) between

gives

Equation (11) is re-arranged as Equation (6) for use in Section 2.2.

Mechanisms for the tired-light effect which involve photon/photon interactions or photon/baryon interactions would involve deflection and blurring of images which is not observed. However a possible mechanism which avoids the blurring problem is spontaneous photon decay [

Spontaneous decay of the primary radiation from the sun would produce secondary photons amounting to about

It is suggested here that the tired light radio emission which would be produced from a pulsed femtosecond optical laser could be detected in a terrestrial experiment. It would be necessary to conduct the laser beam through an evacuated tube in order to prevent the radio emission which would otherwise result [