Bridging perturbation theory and simulations: initial conditions and fast integrators for cosmological simulations

In this lecture note, the author gave a detailed overview on generating the required initial conditions for cosmological N-body simulations. The manuscript started off with an introduction on Gaussian random fields (GRF), which corresponds to the key theoretical concept for cosmological observations. Then, the author moves on with the Vlasov-Poisson system, which can be solved using the characteristics method. The early-time approximation corresponds to a vanishing scale factor, i.e. taking the limit $a(t) \rightarrow 0$, further leads to the Zel'dovich approximation for the system of equations. With a set of initial conditions, the author obtains a perturbative expansion of the $n$th-order solution to the characteristic equations using Lagrangian perturbation theory (LPT). The $n$th Lagrangian perturbation theory is further generalized to N-body simulations to obtain solutions that go beyond shell-crossing.

In short, the lecture note is well-structured, including the necessary theoretical concepts and code technicalities required for the simulation. It gives detailed instructions to readers who want to simulate their first $N-$body cosmological simulation.

I recommend the manuscript to be published in SciPost Physics Lecture Notes after minor revision.

There are a few grammar mistakes or parts that required more clarification in this manuscript:

1- Typo in line 67: since we since 2-line 149: explain what does the notation $\asymp$ mean 3-Typo in line 190: implementatio 4-Figure 2: Explain what do the color lines indicate in the caption of the right figure 5-Grammar mistake in line 195: Ever since the some of the first cosmological N-body simulations in the 1980s 6-Typo in line 201: hiher 7-Typo in line 235: intertial 8-Grammar mistake in line 278: Error on small scales appear that are larger,

Ask for minor revision

These lecture notes provide a clear, self-contained introduction to the generation of initial conditions for cosmological N-body simulations, as well as fast simulation techniques. The notes successfully bridge the gap between perturbation theory and practical simulation techniques by presenting a logical flow from the statistical properties of Gaussian random fields, to the generation of initial conditions (ICs) and the dynamical evolution of cold dark matter particles using Lagrangian Perturbation Theory (LPT). The discussion of LPT is effectively presented in the context of IC generation, and some state of the art PT-informed time integration schemes (Fast-PM and Bullfrog) are discussed. The notes end with some critical insights into discreteness errors, which limit the precision of currently feasible cosmological simulations. The reference to a demonstrative Jupyter notebook also adds educational value to the notes.

Some typos, figure-to-text scalings, and sentence structures need to be revised (see comments).

The content is up to date, of good quality, and effectively references other lectures from the same school for related topics beyond the scope of this lecture. The lectures notes are also accompanied with a Jupyter notebook, which helpfully demonstrates some of the discussed concepts. I recommend publication in this SciPost lecture note series, after addressing the minor comments below.

Some minor comments to ensure that the notes remain clear and self-contained for the non-expert reader:

1. Line 56: It could be worthwhile to mention (e.g. in a footnote) that the change of variables works that way because it has a unit Jacobian. Elaboration on other mathematical assumptions to bring the rest of the discussion to a similar level could also be helpful to the non-expert reader.

2. Eq. (7) implicitly assumes a discrete Fourier basis on a periodic domain, while the discussion has been so far in terms of a continuous variable k. A short elaboration on this may be helpful.

3. Figures 1, 3: For readability, it could be better to re-scale the figures such that the font sizes of the labels are comparable to that of the main text (similarly to e.g. Fig. 2).

4. The text needs to be checked again for typos and sentence structures. Some I could spot so far:

5. Section 3.4 title: “Discreteness”.
6. Line 67: The sentence is structured in a confusing way. I believe the author intended “since we took the real part without additional…” or similar.
7. Line 190: “implementaio” --> “implementation”.
8. Line 123: “hierachy” --> “hierarchy”.
9. Line 187: “conergence” --> “convergence”.

Publish (meets expectations and criteria for this Journal)