Numerical simulations of inflationary dynamics: slow roll and beyond

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 minimally-coupled, single-field models of inflation
3 - focusses on the topical subject of ultra-slow-roll 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

1 - there are more sophisticated codes already publicly available, e.g., Py- and M-transport, able to calculate perturbation spectra in multi-field inflation and beyond linear order

Overall this paper represents a useful addition to the literature and with some improvements listed below it should be suitable for publication in SciPost

1- Early on (in the text after eq.12) the authors fix the CMB scale to 60 e-folds 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 gauge-independent 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 slow-roll inflation.

3- There is a factor of a (the scale factor) missing in the last equation in eq.(19).

4- The right-hand-side of Eq.(27) must be evaluated at Hubble exit, k=aH. In general it will evolve on super-Hubble scales, whereas the curvature perturbation remains constant on super-Hubble 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 square-root of 2 unnecessary in eq.(33)).

7- The consistency relation (47) is described as a “smoking-gun test of inflation” but it only holds for the specific class of single, canonical field (with sound speed equal to the speed of light), slow-roll inflation, and can be broken in models with multiple fields or non-canonical 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 “asymptotically-flat” 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 tensor-to-scalar ratio provides strong evidence for single-field inflation. Why? For example, additional fields tend to suppress the tensor-to-scalar ration (by providing additional scalar sources). So a low-tensor-to-scalar ratio could be considered evidence for multiple fields or a low effective sound speed.

11- Eqs.(75-77) apply only for perturbations evaluated at Hubble-exit during slow-roll inflation, in which case they were previously given in Eqs.(42-45) 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 e-folds around Hubble exit. But in non-slow-roll inflation models, especially models with a sudden, non-adiabatic 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 e-folds 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 e-folds after Hubble exit as a user-specified 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 e-folds that occurs due to the presence of the bump, changing the relationship between the background field value and the number of e-folds with respect to the base model (see point 1 above).

14- Note that Ananda et al, astro-ph/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 (“e-folds”) 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.

Ask for minor revision

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.

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