index $ \def \hieruit {\quad \Longrightarrow \quad} \def \slechts {\quad \Longleftrightarrow \quad} \def \SP {\quad ; \quad} \def \MET {\quad \mbox{with} \quad} \def \half {\frac{1}{2}} \def \artanh {\operatorname{artanh}} \def \EN {\quad \mbox{and} \quad} $

Cosmology Models

Our major source of information meanwhile has become Observations at tension with cosmological models by Louis Marmet. UAC theory is not deviant from standard trouble in one respect: Hubble tension is our main problem as well. But there is an essential difference: we can have more Hubble parameters than just one. The reason is that the (intrinsic) Hubble parameters in UAC may be related to Age, according to formulas in Hubble Parameter ($T_0=$ nowadays timestamp, mind that $A$ is negative most of the time): \begin{matrix} T_0-A = 2(1/H_i) & & 1/H_i = (T_0-A)/2 \\ \mbox{Creation time} = 2 \times \mbox{Hubble time} & & \mbox{Hubble time} = \mbox{Creation time}\;/\;2 \end{matrix}

The $R_h=ct$ Universe

Even at first glance, it is clear that The $R_h=ct$ Universe too, is part of the mainstream GR cosmology, though it is flat as much as it can be. Almost every paper about this cosmological model has the signature of one and the same author: Fulvio Melia. Wikipedia says: Melia and his students have developed the so-called $R_h=ct$ Universe, a cosmological theory that, they argue, has accounted for the observational data better than all other models proposed thus far, In this cosmology, the Universe has no horizon problem, and therefore evolved without inflation. In short: everybody has a pet universe and this happens to be the one of Melia. Probably We do not live in the $R_h=ct$ universe at all. It is nevertheless provided with some interesting features. One of these is suggested in the critical paper as mentioned; the authors are calling it the '$Ht=1$ model'. Indeed, in The Apparent (Gravitational) Horizon in Cosmology near formula (38) we find $\,H(t)=1/t\,$ and (39) as sort of a Milne's formula look alike. Of course the main formula $\,R_h=ct\,$, for the cosmic horizon, has no meaning in the framework of UAC. Summarizing:

$R_h=ct$	UAC
$$ H(t)=\frac{1}{t} \\ R_\gamma(t) = ct\ln\left(\frac{t_0}{t}\right) $$	$$ H_i = \frac{2}{T_0-A} \\ t-t_0 = (T_0-A)\ln\left(\frac{T-A}{T_0-A}\right) $$

It is seen that the cosmic time $t$ in Melia's universe corresponds with the orbital time of creation $(T_0-A)$ in the UAC universe, which is a quite reasonable correspondence. It's interesting to investigate when the right hand side of Milne's formula becomes zero: $$ t-t_0 = (T_0-A)\ln\left(\frac{T-A}{T_0-A}\right) = 0 \slechts t=t_0 \; \wedge \; (T=T_0 \; \vee \; T_0 = A) $$ The first case is when $\,t=t_0\,$ and $\,T=T_0\,$, which is the trivial case because orbital and atomic clocks are always synchronized here and now. The second case is when $\,t=t_0\,$ and $\,T_0=A\,$. We haven't seen this case before. The physical meaning of it is that, when an observer were present at the orbital timestamp of creation, atomic time doesn't run, its clock stands still for ever and ever. Perhaps we have just unravelled the physical meaning of Eternity.

Tired-Light Redshift Models

The notion that the cosmological redshift is a non-Doppler phenomenon in which photons continuously undergo an energy depletion or "aging" effect is not new. [ .. ]
The photon energy-depletion interpretation of the cosmological redshift, or "tired light cosmology" as it is now commonly referred to, implies the following energy loss relation: $$ E(r) = E_0\,e^{-\beta r} \; , \qquad \mbox{(1)} $$ where $\,E_0\,$ is the initial photon energy, $\,E(r)\,$ is the photon's energy after it travels a distance $\,r\,$, and $\,\beta\,$ is the energy attenuation coefficient. Alternatively, relation (1) may be expressed as the following redshift-distance relation: $$ z(r) = \Delta\lambda/\lambda_0 = e^{\beta r}-1 \; , \qquad \mbox{(2)} $$ where $\,z\,$ is the redshift of the photon's initial wavelength $\,\lambda_0\,$ after the photon has travelled a distance $\,r\,$. For cosmologically short propagation distances where $r \ll \beta^{-1}$ these exponential formulae become approximated by the following linear relations: $$ E_0-E=-\beta\,E_0\,r \; , \qquad \mbox{(3)} $$ or $$ z(r) = \beta\,r \; , \qquad \mbox{(4)} $$ Expressed in Doppler terminology, $\,\beta = H_0/c\,$, where $\,H_0\,$ is the Hubble constant and $\,c\,$ is the velocity of light. Thus relation (4) is equivalent to Hubble's linear redshift-distance relation.

• Larger Intrinsic redshift corresponds with younger matter
• Larger TiredLight redshift corresponds with older photons

But things shall become even more confusing. In our Relativizing Newton section, the following expression for gravitational redshift is mentioned, with $\lambda=$ wavelength, $c_0=$ speed of light in vacuum, $r=$ distance, $g=$ (gravitational) acceleration. $$ \frac{d\lambda}{\lambda} = \frac{g\,dr}{c_0^2} $$ Assume that $\,g\,$ is a constant and integrate (for the meaning of $\Gamma$ see here). $$ \int\frac{d\lambda}{\lambda} = \ln\left(\frac{\lambda}{\lambda_0}\right) = \frac{g\,r}{c_0^2} \hieruit \frac{\lambda}{\lambda_0} = e^{g\,r/c_0^2} \SP \frac{\lambda_0}{\lambda_0}+\frac{\lambda-\lambda_0}{\lambda_0} = 1+z = e^{\Gamma r} $$ Let $\,g\,$ be the acceleration near the boundary of a heavy object with mass $M$ and radius $R$, for example our milky way, or half the universe, then: $$ e^{g\,r/c_0^2}=e^{\Gamma r} \hieruit g\,r/c_0^2 = \Gamma r \MET g = \frac{GM}{R^2} \EN \Gamma = \frac{H}{c_0} \\ \large \boxed{g = H c_0} \normalsize = \frac{GM}{R^2} = a_0 $$ The latter quantity is precisely Milgrom's constant, as it appears in MOND = UAC.

Conclusion. It seems that Tired Light can hardly be distinguished from Variable Mass.
Question. Maybe TL and VM are equivalent representations of the same phenomenon?