Tired light as a mechanism producing cosmological redshift is suggested as a possible explanation of the findings. As described by Anton Petrov, this mechanism leads to many long-known contradictions with cosmological observations, and in my opinion it can be safely forgotten. However, the effect may be real, even though it has been reported by only one researcher hitherto.

Redshift is real and in general relativity it would most naturally be interpreted as a direct evidence that energy is not conserved. In TGD, where spacetimes are surfaces, the explanation for the cosmological redshift is much simpler and consistent with conservation of energy. The 4-D tangent spaces of the 4-D surfaces related to the 3-surfaces corresponding to the detector and the source differ from each other by the Lorentz transformation and this produces an analogy of the Doppler effect. The energy of the photons is preserved, but one could say that they are perceived as if from systems in different states of motion. The projections of the three-surface tangent spaces M^{4} to the sender and the receiver differ by the Lorentz transformation and this results in a redshift.

A possible TGD based explanation for the observed effect relies on many-sheeted spacetime. The galaxies rotating in opposite directions could correspond to space-time sheets for which Hubble constants are slightly different at the moment of the emission of the radiation. In the GRT framework this would mean that the density of matter is slightly different for these space-time regions.

I have proposed that the fluctuations of h_{eff} at quantum criticality induce fluctuations of density and temperature. If the regions of many-sheeted space-time tend to contain galaxies with the same direction of rotation, one can imagine that the h_{eff} depends on the direction of rotation. The CMB temperature behaves as T(a)=T_{0}(a_{0}/a) and a naive dimensional guess for the dependence of h_{eff} is T_{0}(h_{eff})= (h_{eff}/h)T_{0}. This would scale the energy density of radiation by a factor (h_{eff}/h)^{4} and the following
little calculations show that the value of H increases.

Using Einstein's equations, Hubble constant can be expressed as

H^{2}== [(da/dt)/a]^{2}=(8πG/3)ρ -k/a^{2}+Λ/3 ,

The expression for Hubble constant reads as

H(a)=H_{0}X^{1/2} ,

X=Ωka^{-2}+Ω_{m} a^{-3} + Ω_{r}a^{-4} +Ω_{DE}a^{-3(1+w)} .

Here parameter w depends on the model of dark energy and w=1 is a possible value. From this formula one sees that if the temperature of CMB background is proportional to h_{eff}, regions of larger h_{eff} have a large Hubble constant.

The critical density and density parameter are defined

ρ_{c}=3H^{2}/8πG, Ω =ρ/ρ _{c} .

The parameters Ω_{k} (k∈{0,-1,1}, Ω_{m}, Ω_{r}, and Ω_{DE} refer to various contributions to the density corresponding to the curvature of 3-space (k=0 corresponds to flat space), matter, radiation and dark energy. If dark energy corresponds to the cosmological constant, one obtains

ρ_{c}= 3H_{0}^{2}/8πG ,

Ω_{m}== ρ_{m0}/ρ_{c} = (8π G/3H_{0}^{2})ρ_{m0} ,
Ω_{k}== -k/a_{0}^{2}H_{0}^{2},
Ω_{Λ}== Λ/3H_{0}^{2} .

The question is whether the measured two different values of H could reflect slightly different temperatures for the Hubble constant in some space-time regions induced by different values of h_{eff} and whether these regions could correspond to regions containing preferentially galaxies, which rotate in the same or opposite direction as the Milky Way. Some kind of parity violation in cosmic scales is suggestive.

This mechanism could also provide insights to two other cosmological problems.

- The proposal might explain the observed two values of the Hubble constant. The two Hubble constants could correspond to stars of galaxies rotating in different directions as compared to the Milky Way.
Note that TGD suggests the formula for G in terms of the fundamental length scale as G= kR

^{2}/h_{eff}. This would induce factor 1/h_{eff}to Ω_{m}and Ω_{r}but the conclusions would not be changed in the radiation dominated phase. - Could the accelerated expansion of the Universe could relate to the increase of h
_{eff}suggested by the number theoretic evolution possibly explaining the apparent disappearance of the baryonic matter. One expects that the average value of h_{eff}increases and that this corresponds to the gradual transformation of the baryonic matter to dark matter in the TGD sense.From the formula for the Hubble constant one can calculate the dH/dt as

dH/dt= -H

^{2}(1+q) , q== -[d^{2}a/dt^{2}]a/ (da/dt)^{2}.From this one can estimate the change of the parameter q caused by the time evolution of h

_{eff}. The additional term Δ q in q due to T_{0}∝ h_{eff}dependence would beΔ q=H

_{0}^{2}/H^{2}T_{0}× 4Ω_{r}a^{-4}(dh_{eff}/dt)/h_{eff}.If h

_{eff}increases, the sign of Δ q= -a(d^{2}a/dt^{2})/(da/dt)^{2}is positive so that the acceleration is positive.

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