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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

Submission summary

Authors (as registered SciPost users): Siddharth Bhatt · Swagat Saurav Mishra
Submission information
Preprint Link: scipost_202403_00022v1  (pdf)
Code repository:
Date submitted: 2024-03-18 08:08
Submitted by: Mishra, Swagat Saurav
Submitted to: SciPost Physics Codebases
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approaches: Computational, Phenomenological


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 scalar-induced 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 scalar-induced second-order tensor power spectrum in the near future.

Current status:
In refereeing

Reports on this Submission

Anonymous Report 1 on 2024-5-11 (Invited Report)


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.


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 e-folds 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.


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 e-folds 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 e-folds 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 e-folds before the mode leaves the Hubble radius. And, in point 4, they say that they evaluate the spectrum at 5 e-folds after the mode leaves the Hubble radius. While it is largely fine to impose the initial conditions 5 e-folds 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.


Ask for major revision

  • validity: high
  • significance: ok
  • originality: low
  • clarity: good
  • formatting: good
  • grammar: good

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