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Numerical simulations of inflationary dynamics: slow roll and beyond
by Siddharth S. Bhatt, Swagat S. Mishra, Soumen Basak, Surya N. Sahoo
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Siddharth Bhatt · Swagat Saurav Mishra 
Submission information  

Preprint Link:  scipost_202403_00022v1 (pdf) 
Code repository:  https://github.com/bhattsiddharth/NumDynInflation 
Date submitted:  20240318 08:08 
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.
Current status:
Reports on this Submission
Report 2 by David Wands on 2024613 (Invited Report)
Strengths
1  clear pedagogical review of primordial power spectra generated by a period of inflation in the very early universe
2  calculations of both background (homogeneous and isotropic) solutions, and linear perturbations, in minimallycoupled, singlefield models of inflation
3  focusses on the topical subject of ultraslowroll models of inflation
4  some exploration of the efficiency of numerical schemes making different choices for the variables used to track the evolution of perturbations
Weaknesses
1  there are more sophisticated codes already publicly available, e.g., Py and Mtransport, able to calculate perturbation spectra in multifield inflation and beyond linear order
Report
Overall this paper represents a useful addition to the literature and with some improvements listed below it should be suitable for publication in SciPost
Requested changes
1 Early on (in the text after eq.12) the authors fix the CMB scale to 60 efolds before the end of inflation. In fact the CMB scale is dependent on the duration of reheating and this is a major uncertainty in observational predictions in particular models. This deserves more consideration by the authors. At the very least, as a limitation of the present analysis it should be noted. This also applies to points 4 and 5 on pages 15 and 16 (section 4).
2 Near the start of section 3, the authors comment “two gaugeindependent massless fields, one scalar, and one transverse and traceless tensor, are guaranteed to exist during inflation”. I found this a bit confusing. There are two massless, transverse and traceless degrees of freedom (the free gravitational field) but the scalar degree of freedom is not strictly massless; it is only approximately massless during slowroll inflation.
3 There is a factor of a (the scale factor) missing in the last equation in eq.(19).
4 The righthandside of Eq.(27) must be evaluated at Hubble exit, k=aH. In general it will evolve on superHubble scales, whereas the curvature perturbation remains constant on superHubble scales.
5 Eq.(32) presumably corresponds to the specific case of a gravitational wave propagating along the direction (0,0,1). This should be noted. Eq.(33) is a more general form.
6 The normalization of the polarization tensors, εij, in the text following eq.(33) seems unusual. Is the factor 2 correct? It is purely conventional, of course, if followed consistently, but usually one chooses orthonormal polarisation tensors (which would make the factor of the squareroot of 2 unnecessary in eq.(33)).
7 The consistency relation (47) is described as a “smokinggun test of inflation” but it only holds for the specific class of single, canonical field (with sound speed equal to the speed of light), slowroll inflation, and can be broken in models with multiple fields or noncanonical sound speed. These caveats should be given.
8 There are a few typos that should be corrected, such as the extra “)” in the text after eq.(20) and the extra “In fact, “ between eqs.(50) and (51).
9 After eq.(56), and again at the start of section 3.2, the authors state that observations favour an “asymptoticallyflat” potential, which is overstating what observations can tell us. Observations only probe a finite range of the potential, not the asymptotic behaviour.
10 The authors suggest at the start of section 3.2 that the low tensortoscalar ratio provides strong evidence for singlefield inflation. Why? For example, additional fields tend to suppress the tensortoscalar ration (by providing additional scalar sources). So a lowtensortoscalar ratio could be considered evidence for multiple fields or a low effective sound speed.
11 Eqs.(7577) apply only for perturbations evaluated at Hubbleexit during slowroll inflation, in which case they were previously given in Eqs.(4245) so they seem out of place in section 4 if the numerical results are not limited to slow roll.
12 In section 5, between Eqs.(82) and (83), the authors say that mode functions for perturbations only need to be evolved for a few efolds around Hubble exit. But in nonslowroll inflation models, especially models with a sudden, nonadiabatic transition, perturbation modes can be affected on scales far from the Hubble scale. See for example Jackson et al, arXiv:2311.03281. This also relates to point 5, page 22, where the authors evolve the mode function until 5 efolds after Hubble exit. This may be long enough for the models considered by the authors, but why not evolve until the end of inflation? Or leave the number of efolds after Hubble exit as a userspecified parameter.
13 Figure 18, in the caption, the authors claim that the power spectrum at CMB scales matches that of the base KKLT model, also shown. While the power at the CMB scale looks the same, the tilt appears to be different. I would certainly it expect it to be different because of the additional 15 efolds that occurs due to the presence of the bump, changing the relationship between the background field value and the number of efolds with respect to the base model (see point 1 above).
14 Note that Ananda et al, astroph/0612013, studied the induced GW background from primordial perturbations in the radiation era, after Matarrese et al [58] but shortly before Baumann et al [59].
15 There are a some limitations of the code which the authors should consider:
 The numerical code uses cosmic time as the time variable. This makes it harder to estimate the interval required to simulate the observable duration of inflation. The logarithmic expansion (“efolds”) is usually more useful for this purpose.
 It would be helpful to provide a user guide to the code with expected example outputs using jupyter notebooks for example.
 There is currently no way for the used to verify outputs.
Recommendation
Ask for minor revision
Strengths
1. 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.
Weaknesses
1. The numerical computation of the inflationary power spectrum is a well understood and often implemented technique and the manuscript does not provide any new insights.
2. There are other publicly available codes, which are likely to be more efficient.
3. As I have described in my report, I am not convinced that the code developed is efficient, and it will possibly work more efficiently if efolds is used as the independent variable.
4. Lastly, in models involving strong departures from slow roll inflation, it would be prudent to evaluate the inflationary spectra close to or at the end of inflation.
Report
In this manuscript, the authors construct a Python code to calculate the scalar and tensor power spectra generated during inflation. The authors consider the simplest of scenarios involving inflation driven by a single canonical scalar field. Apart from the effects that arise on scales relevant for the cosmic microwave background (say, 10^{4} < k < 1 Mpc^{1}), the aim is to also take into account effects that occur on small scales (say, k > 10^4 Mpc^{1}), such as those that lead to the production of significant number of primordial black holes.
The manuscript begins with a broad discussion on inflation (in Sec. 2) before it goes on to analytically derive the standard results for the scalar and tensor power spectra in slow roll inflation (in Sec. 3). Thereafter, it describes the typical scenario involving an epoch of ultra slow roll inflation that leads to enhanced power on small scales and to an increased production of primordial black holes (in Sec. 3.2). Then (in Secs. 4 and 5), the authors discuss the primary topic of the manuscript, viz. the numerical computation of the inflationary background and the scalar and tensor power spectra. If I understand it correctly (see point 4 on page 21), they use a dimensionless version of cosmic time as the independent variable to integrate the equations governing the background and the perturbations [as described in Secs. 4 and 5, see Eqs. (63)(72) and Eqs. (80)(87)]. They also explore (in Sec. 6) the evolution of different dependent variables (characterizing the perturbations) to understand if the evolution can be more accurate and/or efficient.
Requested changes
The manuscript is written clearly. But, I have the following comments about the manuscript:
1. At the outset, I should say that the numerical computation of the inflationary power spectrum is a well understood and often implemented technique and the manuscript does not provide any new insights.
2. Secondly, it is also well known that working with cosmic time as the independent variable is inefficient and it is more effective to work with efolds as the independent variable. If the authors wish to develop codes to calculate more involved quantities (such as loop corrections to power spectra, as they indicate in the concluding section) or compare with the cosmological data, then efficiency will become an important issue.
3. I am surprised that the authors need to choose by hand an initial value of A (see their comment 1 on page 14). I would have thought that the initial value of the scale factor (see Eq. (67) which relates A to the scale factor a) is determined by the condition that the pivot scale, say, k_\ast, leaves the Hubble radius at N_\ast number of efolds before the end of inflation.
4. In point 3 on page 21, the authors say that they impose the initial conditions on the perturbations 5 efolds before the mode leaves the Hubble radius. And, in point 4, they say that they evaluate the spectrum at 5 efolds after the mode leaves the Hubble radius. While it is largely fine to impose the initial conditions 5 efolds before Hubble exit, there can be models/situations wherein this may not be adequate (the axion monodromy model comes to mind) and one may have to impose the initial conditions at earlier times. More importantly, if there arise deviations from slow roll  specifically, strong departures involving a phase of ultra slow roll inflation  to avoid the transient effects, the power spectrum needs to be evaluated either close to or, preferably, at the end of inflation. It is not clear if the authors do so in the model with a bump that they consider in Sec. 5.2.
I would urge the authors to attend to the above points.
Recommendation
Ask for major revision
Author: Swagat Saurav Mishra on 20240915 [id 4784]
(in reply to Report 1 on 20240511)
Dear Referee, We are thankful to you for your critical and insightful comments. We have attempted to address them in the revised version of our manuscript. Your suggestions have greatly improved the quality of our paper. Specifically, we have addressed the following 
1) We agree with this comment, which is also similar to a comment made by the second referee (Prof. David Wands). Our response is:
We completely agree with the referee(s) that there exist more sophisticated numerical codes in the literature. We have added a discussion on this in the discussion section of our revised manuscript and included two new references to PyTransport and CppTransport frameworks.
However, we would like to stress that our framework is very simple and userfriendly. It is quite fast and works very well for computing twopoint functions. We are keen on extending the framework to compute higherpoint functions to study nonGaussianity and we will be happy to report on the chanellenges and results from our analysis in the future.
2) We have added a long paragraph in the discussion section commenting on the usage of cosmic time 't' as the time variable and the potential advantages and disadvantages over the usage of number of efolds 'N'.
3) We would like to stress that, on page 14, the comment 1 regarding the initial value of A concerns only the background evolution. This does not affect any observable quantity. When we solve the MukhanovSasaki equation, we then read off the scalefactor corresponding to the Hubbleexit of a given mode for the same background.
It is also possible to fix the initial value of A by fixing the presentday scalefactor to be equal to 1, and by fixing the reheating history. Then one can go backward in time to find the corresponding value of the initial scalefactor. However, this is unnecessary, since we are interested in the evolution until the end of inflation, and the inflationary observables are insensitive to A_i once we fix the fact that the pivot scale k_* made its Hubbleexit at a given number of efolds before the end of inflation (e.g. 60 efolds in our paper).
4) This is an important point, which is also made by the second referee (Prof. David Wands). Our response is:
We agree with the referee(s). We have added footnote 14, discussing this point. We should also stress that our numerical framework is flexible enough and the user can simulate a mode for much longer duration, rather than the 56 efolds prescription we had mentioned in the paper. We have highlighted this in footnote 14 and mentioned it again on page 21.
Please note that for the specific case of the bump model, 5 efolds of expansion before and after the Hubbleexit is enough, since it is wellknown that the ultra slowroll phase lasts for less than 3 efolds (in order to avoid overproduction of PBHs)
Author: Swagat Saurav Mishra on 20240915 [id 4783]
(in reply to Report 2 by David Wands on 20240613)Dear Referee, We are thankful for your detailed and insightful comments and suggestions on the earlier version of our manuscript. We have made an honest attempt to take all of your suggestions into account in the revised version of the manuscript. In particular, the following is our response to your comments and suggestions.
1) Regarding the number of efolds before the end of inflation, corresponding to the Hubbleexit of the CMB pivot scale, we agree and accordingly we have added notes at the respective places in the revised manuscript.
2) We agree, that the phrase involving 'two gaugeindependent massless fields' was confusing, therefore, we have edited the text accordingly.
3) Indeed, the scale factor 'a' was missing in Eq. (19), this has been corrected.
4)Thank you for this important comment. We have now explicitly mentioned below Eq. (27) that H and $\epsilon_H$ must be computed at the Hubbleexit time of k.
5) We have mentioned the convention (assumption) used in Eq. (32) in the revised manuscript as a footnote. We have also noted that Eq. (33) is more general. Thanks for this suggestion which brings more clarity to the text of our paper.
6) The factor of 2, and therefore our field redefinitions, are indeed less conventional. However, we rechecked the formulae and calculations and we found that what we had written was correct. Therefore, we have kept it unchanged in the revised manuscript.
7) We agree with the referee's suggestion here. therefore we have changed the text below Eq. (47) and we have also added a footnote highlighting the point made by the referee.
8) We apologize for the typos and thank the referee for pointing them out. They have been corrected in the revised manuscript.
9) Following the referee's suggestion, we have changed the text below Eq. (56), highlighting this point.
10) We agree with the referee's point here, the statement is valid only for canonical single field models. Therefore we have modified the text at the start of sec. 3.2 taking the referee's suggestion into account.
11) We have now made it explicit that the formulae given in Eqs. (75), (76) and (77) are valid under the slowroll approximations and hence they are applicable only for the potentials discussed in sec. 4.2.
12) We agree with the referee. We have added footnote 14, discussing this point. We should also stress that our numerical framework is flexible enough and the user can simulate a mode for much longer duration, rather than the 56 efolds prescription we had mentioned in the paper. We have highlighted this in footnote 14 and mentioned it again on page 21.
13) This is an important point and we completely agree with the referee. Therefore, we have removed the extra annotation from fig. 18 and highlighted this point in the caption of that figure. We have further referred the reader to Mishra and Sahni 2019 paper (https://arxiv.org/abs/1911.00057) where this point was discussed more explicitly.
14) We apologize for missing out such an important early paper on SIGWs. We have included it in the revised manuscript and we thank the referee for pointing this out.
15) Regarding the limitations of our numerical framework
a) We have added a long paragraph in the discussion section commenting on the usage of cosmic time 't' as the time variable and the potential advantages and disadvantages over the usage of number of efolds 'N'.
b) We have now included an IPython notebook as a user guide to implement our numerical framework and we have provided a link to the notebook (in the same GitHub account) in the revised manuscript. We thank the referee for this important suggestion.
In response to the weakness of our framework : We completely agree with the referee that there exist more sophisticated numerical codes in the literature. We have added a discussion on this in the discussion section of our revised manuscript and included two new references to PyTransport and CppTransport frameworks.
However, we would like to stress that our framework is very simple and userfriendly. It is quite fast and works very well for computing twopoint functions. We are keen on extending the framework to compute higherpoint functions to study nonGaussianity and we will be happy to report on the chanllenges and results from our analysis in the future.