The Hubble Tension, the M Crisis of Late Time H(z) Deformation Models and the Reconstruction of Quintessence Lagrangians†
Round 1
Reviewer 1 Report
In the manuscript by A. Theodoropoulos and L. Perivolaropoulos ``The Hubble tension, the $M$ crisis of late time $H(z)$ deformation models and the reconstruction of quintessence Lagrangians''
the authors confront the recent observational SnIa, CMB, BAO data with the standard
cosmological models with different EoS for Dark Energy: $\Lambda$CDM, $w$CDM, CPL.
Their analysis of the mentioned observational SnIa, CMB, BAO data is deep and useful
from the pedagogical point of view. The authors interpret the well known Hubble constant
tension as a mismatch in the Pantheon SnIa absolute magnitudes $M$, this approach may be
promising.
However, the manuscript contains serious shortcomings and require the corresponding essential revision.
The major points of criticism are:
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1. The $1\sigma$--$3\sigma$ contour plots in figures 17, 18 and 19 show that the authors minimize
their $\chi^2$ function not over all free model parameters, but they fix some parameters and vary
only two of them in each panel. This approach simplifies calculations, but leads to the radical mismatch
between (for example) the error bounds for the parameter $w_a$ in 4 planes in Fig. 19 and in Table 3.
The authors should minimize $\chi^2$ over all free model parameters for each model and draw the corresponding
contours (with coinciding bounds for parameters), like other researchers in this field.
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2. The point, connected with the previous one. The authors calculate the marginal errors for model parameters
from Equation 62 and the Fisher matrix, determined via a 2-parameter distribution.
But in the authors' approach the results will depend on a choice of a 2-parameter distribution: $w_a-M$, $w_a-h$, etc.
The marginal errors should be independent on this choice.
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The minor points of criticism are:
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3. The authors can include into consideration the observations (estimations) of the
Hubble parameter $H(z)$ at different redshifts or motivate their approach.
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4. The authors considered only the spatially-flat cases ($\Omega_k=0$) for all scenarios.
This assumption needs some motivation.
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To my mind, publication will be possible only after the essential revision in the major points.
Comments for author File: 
Comments.pdf
Author Response
Please see atached pdf file.
Author Response File: Author Response.pdf
Reviewer 2 Report
Thanks for the opportunity to review this. As a pedagogical review, the scope of the work is pretty thorough. I was impressed by some of the observational details delved into by the authors, and what details are not covered in the text are further expanded in the appendices. It is a nice comprehensive piece of work.
I noticed a few typos. Line 495 "he"-> "the", line 510 "exra"->"extra" and line 544 "constrains"-> "constraints", but in a long text, these are expected. So, I would recommend another proofreading to catch these typos.
On line 373/374, it is claimed that LCDM has only a single parameter in the Hubble parameter. Do the authors mean two parameters? What about H0?
On the science, I think this work (https://arxiv.org/abs/1505.05781) could be relevant to section 5.2. The author reconstructs the Quintessence model corresponding to CPL.
Given the comments in section 5.3, it may be useful to explicitly state in section 5.1 that w \geq -1 for a Quintessence model with Lambda saturating the bound, w = -1. Thus, it is expected that any Quintessence model will break down trying to describe some phantom regime.
While the dark energy models introduced are the standard ones in the literature, I think that CPL is not a good dynamical dark energy model. It is true that one can fit CMB, but since (1-a) is a small parameter this leads to larger errors in wa. Thus, it is less likely that one will find evidence for dynamical (wa \neq 0) dark energy. There are a number of other (w0, wa) parametrizations and the Alcaniz-Barboza model overcomes this "sensitivity" problem (https://arxiv.org/pdf/0805.1713.pdf), but obviously is ad hoc in the sense that it is not a Taylor expansion. However, any truncation is also ad hoc, so I see no reason why CPL is the model of choice.
Lastly, one can argue pretty generically that Quintessence, being a w > -1 model (if one neglects Lambda) will only make Hubble tension worse. While no explicit proof exists, the closest argument to that effect has been given in (https://arxiv.org/pdf/2006.00244.pdf). This could be mentioned somewhere. I think these arguments (and others) are useful in helping to cut down on the space of dark energy models. Moreover, as the authors have touched upon with varying M, we appear to be in the process of moving away from late Universe resolutions to the cosmological tensions.
Author Response
Please see attached pdf file.
Author Response File: Author Response.pdf
Reviewer 3 Report
The manuscript contains original work which can be published in Universe journal.
Author Response
We thank the reviewer for taking the time to review our manuscript and for finding it suitable for publication at its current form.
Round 2
Reviewer 1 Report
The authors have not made corrections, connected with my 1st comment (about the fixed parameters in figures 17, 18 and 19).
In their response they explained: ``In the contour
plots shown, we fix each time the parameters not used (with the best fit values provided from the minimization
of the 2 function), in order to obtain a two-dimensional plot. Essentially, what we do is that we take two-
dimensional cuts from an ellipsoid in the 4-dimensional (or 5-dimensional in the CPL case) parameter space.''
I suggest that the cited (or similar) clarification should be included after Eq.~(127)
(where the results for the wCDM model are described) with the corresponding words for the CPL case.
Author Response
In order to address the remaining concerns of the referee we have added the following comments:
1. “In order to produce the contour plots for the $wCDM$ model we vary all four parameters in the $\chi^2$ function and then we show a two-dimensional projection of the ellipsoid produced in the four-dimensional parameter space.” after Equation 127
2. “In order to produce the contour plots for the $CPL$ model we vary all five parameters in the $\chi^2$ function and then we show a two-dimensional projection of the ellipsoid produced in the five-dimensional parameter space.” after Equation 132.
The changes in the resubmitted paper are written in red fonts.
