Late-Time Correlators and Complex Geodesics in de Sitter Space

The authors approximate two-point functions of massive scalars in de Sitter using geodesics.

While the existence of this approximation is not particularly surprising, the authors apply an interesting method, and highlight the role played by complex geodesics. I think the paper should be published for these reasons, meeting the criteria of SciPost Physics.

However there are certain aspects of the work that should be addressed as below.

1. It is not clear to what extent the authors method is a self-contained derivation of the propagator. i.e. it appears from the discussion surrounding (3.19) that the known answer is used as an input to the calculation, informing which geodesics to include. I would like the authors to clarify if their method can be used to compute the two-point function without using the known answer.

2. The demonstration of the level of numerical agreement between the approximation and the exact answer needs significant improvement, particularly in light of statements like "we show that a complex geodesic correctly reproduces the propagator in this case.". The main evidence supporting such statements appears to be figures 3,4,5. e.g. for figure 5 the authors state that the agreement is excellent at large T/l, but it looks to me like both curves are getting smaller and it is not obvious that the relative error is decreasing. Similar comments apply to figures 3,4. A more suitable approach would be to indicate the relative error and the convergence towards zero as the approximation is refined.

1 - it is clearly presented, with ample background material.

1 - there is only one novel computation, and it is very similar to what has appeared in a special case before.

This paper studies the geodesic approximation to the two-point correlation function of a massive (free) scalar in de Sitter space. The exact correlator is known, so the main purpose is to show how, in an expansion of the heat kernel, complex geodesics are required to give the right answer. While the paper is mostly review, the point made is an important one so I recommend it be published.

The framing of the paper does not give due credit to the authors' reference [36]. That paper showed the contribution of complex geodesics in the case of dS$_3$, and the purpose of this paper is to generalize that result to higher dimensions. While even before [36] the result of complex geodesics dominating the answer was known, I don't know of a paper where it appears explicitly, so [36] should be cited prominently as being the first appearance of this fact.

1) Modify the referencing of [36].
2) The last sentence in the first paragraph is inaccurate, in particular the line ``indicating that these effects are a universal property of event horizons [7-13]". This makes it seem like there are such corrections to the entropy of Hawking radiation of the dS horizon. In the papers [7-13], one of the following things has been done: a dramatic change of the geometry or quantum state ([12-13]), an error in the calculation of fine-grained entropy ([10]), an added rule of subtracting points to obtain nontrivial islands ([9]), islands which are relevant only to the BH region of dS ([8]), and an ambiguity in choice of contour ([7]). I have no idea what [11] is doing. But in any case, the situation for dS is very much unlike the case for AdS black holes, so this line should be modified.
3) (2.31) needs elaboration. The words before it say $\sigma(x,y)$ is half the square of the geodesic distance, but (2.31) seems to be displaying something else (the integrand is squared instead of the total length and there is a $1/s$).
4) The words describing the expansion in (2.36) are a bit confusing. In particular, ``we now know that the leading term in the $s$-expansion should reduce to the flat space result...where the dots describe higher-curvature terms that are suppressed by powers of $s$." This makes it seem as if the leading piece is the flat space answer and then \emph{all} of the curvature dependence is encoded in the $\dots$. But $\sigma(x,y)$ and $\Delta(x,y)$ both depend on the curvatures, so this is not exactly an expansion in curvatures.
5) On pg. 10 it is stated ``Thus, for small geodesic distance $\sigma$ they should be subdominant." Isn't $\sigma$ the square of the geodesic distance? And the convention here is that the square of the geodesic distance can be negative, so does ``small" mean small in magnitude?
6) On pg. 11 it says ``As discussed, for large mass equation (3.1) should give a good approximation to the exact Green function as long as a single geodesic gives the dominant contribution." As far as I could see this isn't discussed/explained earlier in the text.
7) There is a choice made in (3.7) which neglects the second solution for the Wightman function which is a similar hypergeometric function with $Z \rightarrow -Z$ and has the same short-distance structure. (See e.g. https://arxiv.org/pdf/hep-th/0110007.pdf). Eliminating all non-coincident singularities can eliminate this solution but it should at least be mentioned.
8) At the beginning of section 3.2 it says ``It is natural to expect that the approximation (3.11) should involve a sum over geodesics in that case. This is relatively easy to see when we want to connect two antipodal points, for example located at $t=0$ in static coordinates." Doesn't this example have zero proper time separation and therefore break the large-proper-time assumption behind (3.11)?
9) Figure 7 is confusingly drawn. It seems that $t_E$ is the separation between $x$ and $y$ such that $t_E = 0$ puts them on top of each other. But according to (3.22) $t_E = 0$ means they are antipodal points. Further labeling the drawing would help.
10) It's not totally clear what problem is being addressed in the last paragraph of the paper. Are the authors working on a Schwarzschild-dS background, and then arguing that wormholes in the black hole context will allow for shortcuts for the geodesic, as in https://arxiv.org/pdf/1910.10311.pdf? What are the ``(higher-order) corrections" referred to in this paragraph -- higher order in what? Furthermore, the authors assume that there exist real geodesics that span the black hole interior, but at late times such geodesics do not exist.