Gravitational Waves as a Probe to the Early Universe
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsPlease find the attached PDF while containing the report.
Comments for author File: 
Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis is interesting work and should be published after the following points are addressed: (1) The general Running Vacuum Model of Refs. 17 and 18 has a vacuum energy density which depends on both H and its time derivative, and in which G also depends on H. For clarity it should be mentioned that the version used here is a reduced version with only a vacuum energy density that depends on H. (2) The third sentence of text should be changed to something like 'In particular, the “B-mode” pattern expected in the polarization of the Cosmic
Microwave Background (CMB) would be due to these stochastic GWs' -- since this pattern has not yet been seen. (3) In the abstract, the phrase 'the equality time when the vacuum energy density meets the radiation energy density' seems awkward and should be changed to something like 'the time at which there is a transition from vacuum energy dominance to radiation dominance'. And 'manipulating' might be changed to 'varying'. (4) 'Running Vacuum Model' is capitalized in the text and not in the abstract. (5) The authors may want to cite the recent data released by the Dark Energy Spectroscopic Instrument (DESI) collaboration as providing strong motivation for studies like this.
Author Response
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Reviewer 3 Report
Comments and Suggestions for AuthorsThe manuscript "Gravitational Waves as a Probe to the Early Universe" studies gravitational waves in a FLRW spacetime and in the context of the Running Vacuum Model (RVM). The later could serve as an alternative (or complementary framework) to re-heating and impacts the spectrum of gravitational waves, as shown by the authors. The paper is interesting and generally well-written. However, it lacks important aspects that would improve the quality of the presentation and the overall reliability of the claims in the manuscripts. Once these omissions are corrected (specially in the text of the manuscript), I will be happy to recommend this paper for publication in Universe. They are the following:
1. Eq. (2) is key to the RVM, yet it is not properly justified. It is true that the authors point the reader to the relevant references---e.g. Ref. [19]. However, it would serve best to include a paragraph explaining the underpinnings behind it.
2. Eq. (3) made use of the equation of state p = w \rho. However, there is no p in Eq. (3). The authors used some other equation or set of equations to derive (3). What are those? Please, specify the manipulations required.
3. Should we assume here that H = a'/a and derive (4) form (3)?
4. Eq. (5) does not seem to go to a = \eta^{-1} for \eta —> — \infty, as claimed by the authors. This is probably clearer only after Eq. (45) is presented; but the latter depend on some assumptions. Please, amend this issue with the presentation.
5. In the second line below Eq. (9), it is assumed that \nabla^j h_{ij} =0 as part of the transverse-traceless gauge conditions. Why is the covariant derivative used instead of the partial (ordinary) derivative?
6. Does Eq. (12) come from the wave equation in Fourier space? Please, say it explicitly in the manuscript.
7. In connection to Eq. (13): Why is (a''/a) interpreted as a bell-shaped potential barrier? Is it in hindsight with respect to Fig. 2? But Fig. 2 depend on the particular functional form of a(\eta). This should be stated in the paper.
8. Eq. (22) uses "t" as the dummy integration variable. Wouldn't it be better to use another choice of letter in orther to avoid confusion with the cosmic time?
9. In the second line of page 6 it is claimed that "the contribution from the backreaction source, which is expected to be negligible during the very early stages of cosmic evolution." Why is that so? Can this be demonstrated from your solution for \eta —> — infinity?
10. The definition of \eta_H should appear immediately after Eq. (26).
11. There is a typo in Eq. (28): It should be (a'/a), not (a'/a^2).
12. Eq. (37): A_0 does not appear in the definition of \mu_n. Why should we introduce it here and not there?
13. Page 8, line 1: "This is because the mode function in the remote past must be regarded as the adiabatic vacuum in de Sitter spacetime." I do not understand how this explains Eq. (39). Please explain.
14. Please specify which equations were use to get (40).
15. Please specify which equations were use to get (41).
16. Second line below Eq. (41): "Consequently, we derive the mode function at \eta_H, leading to the spectrum (...)". How? Lots os details could be given here, including the values of the integration constants.
17. Which equation is being used in deriving (44)?
18. Below Eq. (44): "Ultimately, it is possible to ascertain A_0, (...)". How?
19. The values for 1/H_{max}^2 in the plots shown in Figs. 1, 2 and 3 correspond to H_max of order 10^{26}. What is the meaning of such high values of the Hubble parameter? What was the criterium for choosing these values?
20. Line 247: "(...) correlate to e-folding numbers N = 26.02 and N = 25.91, (...)". How did the authors conclude this?
21. With respect to Fig. 3: Express the formula for h(n,eta) in terms of the frequency.
22. With respect to Fig. 4: What is the definition of \Omega_g? And what is its equation as a function of the frequency?
23. Line 277 onward: "By varying its parameters, we investigate the interplay between the gravitational wave spectrum and pivotal factors such as the deviation timescale, equality time, and the initial Hubble parameter." This claim is not backed by the content in the manuscript. The "pivotal factors" do appear in the equations, but were not studied in the face of the variation of the model parameters, as implied by the beginning of the sentence. Rewrite, please.
24. From line 277 on: "Conversely, observational constraints on these factors derived from the gravitational wave spectrum facilitate a more profound comprehension of cosmic history." Was this actually done in the paper? If not, rewrite please.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe authors have given an appropriate response to my comments and suggestions. The manuscript is accepted in its current form.
