The Spectrum of Gravitational Waves, Their Overproduction in Quintessential Inflation and Its Influence in the Reheating Temperature

Round 1

Reviewer 1 Report

The manuscript considers various reheating scenarios in the quintessential inflation model, and provide constraints on the reheating temperature and mass parameters using the upper bound on gravitational waves obtained from Big Bang Nucleosynthesis. The authors also discuss the range of the model parameters explaining the present abundance of dark matter. The paper first provides an overview of quintessential inflation and reheating mechanisms in the model. In the last half of the paper, it gives detailed calculations on how to connect model parameters and observations (abundance of GWs and dark matter). Although the topic itself is not new, the paper is a comprehensive and good-quality review. Therefore I think the paper is worth publication, while there are minor issues that need to be addressed.

1. I do not see any difference between the upper and lower panels of Figure 2 (same for the three cases in Figure 3). I am guessing the difference in the models can be seen at much earlier epochs (N << -8) and I wonder why the authors only show evolutions of the late stage (-8 < N < 4).

2. Figure 4 is never mentioned in the text. Although I can guess that it is showing the discussion in Section 3.2, I think a more detailed explanation is necessary. For example, it is not clear at the first sight that the colored lines for different values of T_R are indicating the bounds shown in Table 1.

3. I wonder where the logarithmic term in (66) comes from, although it is eventually neglected.

4. The constraints on reheating in the quintessential inflation model by the BBN bound have been extensively discussed in the literature, while I see only a limited number of references, and some papers closely related to this paper are not mentioned.

For example, the constraint on instant preheating is discussed in

\bibitem{Sami:2004xk}
M.~Sami and V.~Sahni,
%``Quintessential inflation on the brane and the relic gravity wave background,''
Phys. Rev. D \textbf{70} (2004), 083513
doi:10.1103/PhysRevD.70.083513
[arXiv:hep-th/0402086 [hep-th]].

The constraint on gravitational particle production is recently discussed in detail by (although I think this is not the first paper discussing it)

\bibitem{Figueroa:2018twl}
D.~G.~Figueroa and E.~H.~Tanin,
%``Inconsistency of an inflationary sector coupled only to Einstein gravity,''
JCAP \textbf{10} (2019), 050
doi:10.1088/1475-7516/2019/10/050
[arXiv:1811.04093 [astro-ph.CO]].

I just gave two examples but there are more related works. I think the authors should go through previous works and make a proper list of citations.

Author Response

Our answer is in the pdf.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper studies in detail reheating in the scenario of quintessential inflation originally proposed by Peebles and Vilenkin. In these models the inflaton does not oscillate around the minimum of the potential decaying to SM particles but rolls down a potential till present epoch, leading to a final stage of quintessence. Reheating of the SM fields takes place through particle production during or right after inflation.

The paper is carefully written and the results seem compatible with previous literature. The main claim is that strong constraints on this scenario arise from the production of gravity waves that are constrained by BBN.

While I believe the results are correct the paper is rather difficult to follow. Several important relations (concerning in particular the all important heating efficient coefficient \Theta) are not explicit and some rely on other references making the paper not self-contained. I would urge the authors to streamline the presentation perhaps removing some details. Some technical parts could be moved to technical appendixes or some equations removed. It would help perhaps to organise the paper differently considering one scenario at the time. Some physical intuition on the results would also help.

Other comments about the draft:

- The potential considered is a modified version of the one considered in the original work by Peebles and Vilenkin because that is currently excluded by constraints on the scalar tensor ratio. I understand that this problem is solved with the new potential but I could not find the value of r in the paper.

- The paper does not take into account the backreaction of particle production on the evolution of the inflaton field. Some explanation of why this is justified would be needed.

- From an effective field theory point of view a non analytic potential is problematic. The authors argue that such effective behaviour can be obtained in theories with extra fields but it is not clear how their dynamics would affect the analysis here. I would suggest  to clarify this point, stating clearly how model dependent the results are.

- Scenarios where DM is produced through inflationary fluctuations often suffer from isocurvature constraints. Are there bounds in this case?

- The last statement at the end of remark 5.1 is confusing. Why the bound is different from the one on effective neutrino species during BBN?

- The precision with which numerical values are reported is certainly excessive.

In summary I think this paper contains some interesting material but it needs to be simplified to make it useful for the community.

Author Response

Our answer is in the pdf.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The new version of the paper has improved and is appropriate for publication on Universe.

The authors should check the sentence below eq. 55 which I find confusing:

As has been shown in [65], the main contribution of the integral (55)  comes from the modes that leave the Hubble radius before the inflationary epoch and finally re-enter during the inflation, that means, for kend ≤ k ≤ kkin, where kend = aendHend and kkin = akinHkin.

Author Response

The sentence was wrong, so we have corrected them.