Numerical simulations of inflationary dynamics: slow roll and beyond

Dear Editor,
Thanks a lot for sending us the reports by the two referees. The comments and suggestions made by the two referees were insightful and detailed. We have made a sincere attempt to address them in the revised manuscript, and all changes appear in violet colour text. The suggestions have greatly improved the quality of our paper, for which we are thankful to the referees . We hope the revised version will be suitable for publication at the Scipost Physics Codebases.

With Regards,
-Swagat

In reference to the comments and suggestions made by the two referees, we have made the following changes which appear in violet colour text in the revised manuscript.

1) On page 4, in the paragraph below Eq. (12), we have added a statement -

'While we have fixed N∗ = 60 for the most part of this work, it is important to note that the exact value of N∗ depends upon the post-inflationary reheating history [16].'

2) At the beginning of sec. 3 on page 5, we have included -

'At linear order in perturbation theory, one gauge-invariant scalar degree of freedom (which is approximately massless during slow-roll inflation), and two gauge-invariant (transverse and traceless) massless tensor degrees of freedom are guaranteed to exist in the single-field inflationary paradigm [53, 54].'

3) On page 6, in the paragraph below Eq. (27), we have included -

'where, H and $\epsilon_H$ appearing in the right-hand side of the above equation should be calculated at the time of Hubble-exit of the mode k, namely, when k = aH. '

4) We have added footnote 2 on page 7, which reads

'Note that in Eq. (32) we have assumed the tensor modes to be propagating along the z-direction, i.e. along (0,0,1); while Eq. (33) is valid in general, independent of the aforementioned assumption.'

5) On page 8, in the paragraph below Eq. (47), we have included

'for the slow-roll inflationary paradigm of a single scalar field with a canonical kinetic term' ;

along with footnote 4 on the same page, which reads -

'Note that the consistency relation does not hold for a non-canonical scalar field for which the speed of sound $c_s^2 ̸= 1$, as well as for multi-filed inflation, in general.'

6) On page 9, in the paragraph below Eq. (56), we have included the phrase -

'...for simple slow-roll potentials that do not possess any features on small scales outside the CMB window,'

7) In the last paragraph on page 9, we have noted -

'Furthermore, the current obser vational constraints [66] are consistent with predominantly Gaussian primordial fluctuations. Within the canonical single-field inflationary paradigm, this provides support... '

8) On page 13, in the paragraph below Eq. (77), we have included the comment -

'where the last three Eqs. are valid only under the slow-roll approximations, as discussed in Sec. (4.2).'

9) We have added footnote 11 on page 15, which reads -

'We again stress that the exact value of $N_e$ depends upon the reheating history in the post-inflationary universe. While we fix it to $N_e = 60$ for the purpose of illustration, in principle, our numerical framework allows for incorporating a different value of $N_e$ without any trouble.'

10) On page 19, in the paragraph below Eq. (82), we have added the phrase -

' for a relatively shorter duration of time';

along with footnote 14 on the same page, which reads -

'Keeping in mind the important caveat that the initial time should be sufficiently early enough to impose Bunch-Davies initial conditions, and the final time should be sufficiently late enough for the mode to be frozen outside the Hubble radius. As discussed below, for most of the potentials considered in this work, as well as for most single-field slow-roll violating models [68] relevant for PBH formation in the literature, imposing initial conditions for about 5-6 e-folds before the end of inflation is sufficient.

Nevetheless, in general, especially for slow-roll violating models, one might need to evolve the mode functions for longer duration and our numerical set-up easily allows the user to incorporate a longer evolution duration for the mode functions.'

11) On page 21, at the end of the paragraph within the point 3., we have included the caveat -

' (however, see the caveat given in footnote 14).'

12) We have removed an annotation from figure 18 and we have added the following to the caption of figure 18-

'the amplitude of the power spectrum roughly matches that of the base KKLT potential, although its spectral index is shifted due to a gain in the number of e-folds because of the presence of the bump, see Ref. [44] for a detailed discussion on this.'

13) In the discussion section, at the end of page 30 and at the beginning of page 31, we have included the following comment -

'...even more sophisticated, frameworks relevant to inflation have been suggested in the literature. For example, Ref. [140] discusses a python package called PyTransport, while Ref. [141] discusses a C++ based platform called CppTransport for the numerical computation of inflationary correlators. Similarly,...'

14) We have added a new paragraph on page 31 in the discussion section, which reads -

'Furthermore, it is important to note that our work uses cosmic time t (along with a suitable mass scale to generate a dimensionless variable) as the time variable. In the literature, most frameworks incorporate number of e-folds N as the time variable since it is arguably more intuitive in describing the evolution during inflation. On the other hand, the usage of cosmic time t is quite convenient under certain circumstances. Firstly, for quasi dS (near-exponential) inflation, since H is almost constant, N ≃ Ht, therefore both N and t are equally intuitive, especially for vanilla slow-roll models. Additionally, cosmic time is particularly more suitable for describing the post-inflationary oscillations, and hence it can also be used conveniently to simulate the dynamics scalar-field dark matter. Moreover, our framework also describes the study of initial conditions for inflation, where the initial phase-space trajectories may correspond to pre-inflationary (decelerating) epoch. Nevertheless, given a vast majority of numerical frameworks on inflation do utilize N , our work provides an alternative time variable. However, it is worth pointing out that computing higher- order correlators as well as quantities at a higher-order in perturbation theory (loop corrections to power spectra), the cosmic time variable might present challenges. Similarly, it may be important to investigate the efficiency of cosmic time as a time variable while comparing models against cosmological data. We plan to return to these issues in the near future.'

15) We have included the references [52,60,140,141] in the revised manuscript.

16) We have created an Python Notebook as a user guide in our GitHub folder (with link https://github.com/bhattsiddharth/NumDynInflation/blob/main/Num_Dyn_Inflation.ipynb)

17) We have added a scale factor 'a' in the final expression of Eq. (19).