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A recent arXiv paper by Capozziello et al (27 Feb 2023) claims:

the H0 tension is not due to neither systematic errors, nor to some new physics beyond the $\Lambda$CDM model, but simply to the fact that, in measuring the same quantity at different redshift, different results arise thanks to the look-back time evaluated at different epochs.

Now, maybe the solution is really this "simple", and I might be missing something, but I thought the age of the universe/lookback times depend on input cosmological parameters, including the Hubble parameter, i.e., just the redshift isn’t enough. I am noting this paper claims the age of the universe by Planck collaboration was derived without using $H_0$ via six independent parameters.

How can the Capozziello et al claim stack up?

It would seem to me to be a reheated version of this claim by the same author.

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    $\begingroup$ I am actually baffled by the notion of $H_0^{(z)}$ in the paper. Should $H_0$ be $z$ independent? $H$ is $z$ dependent, but not $H_0$, right? $\endgroup$
    – MadMax
    Mar 3 at 17:15
  • $\begingroup$ @MadMax: $H_0$ is of course $z$ independent, however in order to evaluate it we need some data from some nonzero redshift. $\endgroup$
    – A.V.S.
    Mar 4 at 9:52
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    $\begingroup$ For me, a red flag is that the paper doesn't seem to explain why the existing publications are wrong. That is, presumably the two "sides" of the Hubble tension are actually measuring different things, but where does that show up in previous papers? $\endgroup$
    – Javier
    Mar 4 at 15:22
  • $\begingroup$ @A.V.S. if the evaluated $H_0$ from nonzero redshift data shows that $H_0$ is $z$ dependent, does it mean that there is something wrong with either the $\Lambda$CDM model or the measurement? But the paper claims that both are right, is it contradictory? $\endgroup$
    – MadMax
    Mar 8 at 16:02

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Their equation (5) says

$$T_{lt} = T_0 − T(z) = T_0 − a(t)T_0$$

where $T_{lt}$ is the light travel time and $T_0$ is the present age of the universe. That implies they're assuming $T=a\,T_0$. In a footnote they say

The parametrization $T(z) = a(t)T_0$ is assumed as a label for the age of the Universe at a given redshift.

Fine, but you can't do that and also equate $T$ with the cosmological time, which they do.

They could be equated in a Milne cosmology, where $a\propto t$. It may be true that Milne cosmology is a better fit to this data, but Milne cosmology requires either $Ω\approx 0$, which is strongly excluded by other data, or general relativity being wrong. The paper claims to resolve the tension within the standard ΛCDM model, so it's just wrong.

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  • $\begingroup$ @A.V.S. Equation (1) says that for any $z$, $T_{lt}(z)$ is the cosmological time interval from redshift $z$ to redshift $0$ (that's what the right hand side evaluates to), and (5) says $T_{lt}=T_0-T(z)$, and in the text they say $T_0$ is the age of the universe, so as far as I can tell, $T(z)$ must be the cosmological time at redshift $z$. $\endgroup$
    – benrg
    Mar 9 at 6:14
  • $\begingroup$ You are right, eqs. 1-3 are not compatible with eq. 5 … $\endgroup$
    – A.V.S.
    Mar 9 at 11:09
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While benrg would seem to be on the right track with the answer, just after this question was posted, there seems to have been professional interest as well, such that a Comment on: paper was put up on arXiv by Escamilla-Rivera et al (3rd March 2023). The comment papers two main conclusions, respectively:

The lookback approach...cannot address the $H_0$ tension completely. Recovering the values from current collaborations under this approach does not correspond to an answer on why such an issue could be associated with $z$ measurements; and

The circularity problem is not avoided entirely, since the age of the Universe is sensitive to both $\Omega_m$ and $H_0$ for a flat $\Lambda$CDM cosmology. However, the local determination of H0 is not sensitive to $\Omega_m$.

The comment on paper has now been withdrawn, so obviously something wrong. However, since most of the citations on Capozziello paper are self-citations by the authors, and nobody seems to be taking any notice of it, benrg is no doubt correct in any case.

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    $\begingroup$ The (v2) of that preprint from 6 Mar. has a comment field “Strong corrections on the calculations” and the body of the preprint is 0kb in size, so presumably the (v1) has some problems. $\endgroup$
    – A.V.S.
    Mar 8 at 17:56
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Look back time also known as light-travel distance isn't well defined, so it doesn't make sense to use it for astronomical calculations.

For more information please see Edward Wright's Article https://astro.ucla.edu/~wright/Dltt_is_Dumb.html I can't see these points discussed in the reference you show.

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