SciPost Submission Page
Numerical simulations of inflationary dynamics: slow roll and beyond
by Siddharth S. Bhatt, Swagat S. Mishra, Soumen Basak, Surya N. Sahoo
Submission summary
Authors (as registered SciPost users):  Siddharth Bhatt · Swagat Saurav Mishra 
Submission information  

Preprint Link:  scipost_202403_00022v2 (pdf) 
Code repository:  https://github.com/bhattsiddharth/NumDynInflation 
Date accepted:  20241023 
Date submitted:  20240915 23:47 
Submitted by:  Mishra, Swagat Saurav 
Submitted to:  SciPost Physics Codebases 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Computational, Phenomenological 
Abstract
Numerical simulations of the inflationary dynamics are presented here for a single canonical scalar field minimally coupled to gravity. We spell out the basic equations governing the inflationary dynamics in terms of cosmic time $t$ and define a set of dimensionless variables convenient for numerical analysis. We then provide a link to our simple numerical \texttt{Python} code on \texttt{GitHub} that can be used to simulate the background dynamics as well as the evolution of linear perturbations during inflation. The code computes both scalar and tensor power spectra for a given inflaton potential $V(\phi)$. We discuss a concrete algorithm to use the code for various purposes, especially for computing the enhanced scalar power spectrum in the context of Primordial Black Holes and scalarinduced Gravitational Waves. We also compare the efficiency of different variables used in the literature to compute the scalar fluctuations. We intend to extend the framework to simulate the dynamics of a number of different quantities, including the computation of scalarinduced secondorder tensor power spectrum in the near future.
Author comments upon resubmission
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
List of changes
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 postinflationary reheating history [16].'
2) At the beginning of sec. 3 on page 5, we have included 
'At linear order in perturbation theory, one gaugeinvariant scalar degree of freedom (which is approximately massless during slowroll inflation), and two gaugeinvariant (transverse and traceless) massless tensor degrees of freedom are guaranteed to exist in the singlefield 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 righthand side of the above equation should be calculated at the time of Hubbleexit 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 zdirection, 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 slowroll 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 noncanonical scalar field for which the speed of sound $c_s^2 ̸= 1$, as well as for multifiled inflation, in general.'
6) On page 9, in the paragraph below Eq. (56), we have included the phrase 
'...for simple slowroll 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 singlefield 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 slowroll 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 postinflationary 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 BunchDavies 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 singlefield slowroll violating models [68] relevant for PBH formation in the literature, imposing initial conditions for about 56 efolds before the end of inflation is sufficient.
Nevetheless, in general, especially for slowroll violating models, one might need to evolve the mode functions for longer duration and our numerical setup 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 efolds 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 efolds 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 (nearexponential) inflation, since H is almost constant, N ≃ Ht, therefore both N and t are equally intuitive, especially for vanilla slowroll models. Additionally, cosmic time is particularly more suitable for describing the postinflationary oscillations, and hence it can also be used conveniently to simulate the dynamics scalarfield dark matter. Moreover, our framework also describes the study of initial conditions for inflation, where the initial phasespace trajectories may correspond to preinflationary (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 higherorder 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).
Current status:
Editorial decision:
For Journal SciPost Physics Codebases: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report #3 by L. Sriramkumar (Referee 1) on 20241012 (Invited Report)
Strengths
1. As I had said in my original report, the manuscript is written
clearly.
2. It deals with a topic that is important in the context of
primordial cosmology, viz. the dynamics of inflaton and
generation of perturbations during inflation.
3. The manuscript discusses models/scenarios that have
drawn attention in the recent literature.
Weaknesses
1. I had hoped that, in the revised version, the authors will use
efolds as the independent variable. But, the authors have
continued with cosmic time. I believe this limits the efficiency
of the code.
Report
Barring the weakness I have indicated, the authors have
addressed the other points I had raised. They have specifically
commented on my last point concerning the evaluation of the
power spectrum (in nontrivial scenarios) at suitably late
times. So, I recommend the manuscript for publication.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #2 by L. Sriramkumar (Referee 1) on 20241012 (Invited Report)
Strengths
1. As I had said in my original report, the manuscript is written clearly.
2. It deals with a topic that is important in the context of primordial cosmology, viz. the dynamics of inflaton and generation of perturbations during inflation.
3. The manuscript discusses models/scenarios that have drawn attention in the recent literature.
Weaknesses
1. I had hoped that, in the revised version, the authors will use efolds as the independent variable. But, the authors have continued with cosmic time. I believe this limits the efficiency of the code.
Report
Barring the weakness I have indicated, the authors have addressed the other points I had raised. They have specifically commented on my last point concerning the evaluation of the power spectrum (in nontrivial scenarios) at suitably late times. So, I recommend the manuscript for publication.
Recommendation
Publish (meets expectations and criteria for this Journal)
Report #1 by David Wands (Referee 2) on 2024920 (Invited Report)
Report
The authors have made numerous small changes and corrections in response to my earlier report. In particular they have added a python notebook as a user guide for their numerical code. They have addressed all my comments and I believe that their code meets the journals acceptance criteria. I am happy to recommend this work for publication.
Recommendation
Publish (meets expectations and criteria for this Journal)