The Asymptotic Structure of Cosmological Integrals

We thank the referees for recommending our paper for publication. In the following answer, we address the referee's comments/concerns.

We added a discussion after the one loop two point example, relating it to the existing literature on $\lambda \phi^4$ in $dS_4$, which hopefully addresses some of the concerns of referee 1.

Referee 2

We find that both the alleged weaknesses contradict the strengths. Precisely because the paper develops a mathematical technology (strength1) typically unfamiliar to cosmologists, it is unavoidable that it is an unusual paper for the 'cosmological correlator community'. However, we reviewed just the basic material needed to follow the paper. Indeed, as the review part has this aim (and thus is not a full-fledged review, which, in our opinion, would have been unnecessary and would have made the paper unnecessarily longer than what it already is) is also unavoidable that understanding more deeply some of the statements in it requires to go back to the literature. The style we chose for writing the paper aims to make the paper understandable, providing that certain statements already proved in the literature are accepted by the reader (and referring to the literature for a deeper comprehension). Secondly, as the language is novel in cosmology, it unavoidably requires an extra effort on the reader's side: the reader cannot expect to read the paper with the same fluidity as it was written in a more standard field theoretical language.

Secondly, precisely because our approach mirrors similar techniques in flat space (avoiding many of the pitfalls of cosmological loop calculations -- strength 2), the analysis concerns individual graphs. Indeed, we agree with the referee that a single graph is not a full-fledged physical process. However, in the case of scalars (which is the case we treat here), no particular simplification occurs over the sum of graphs (which is true even in flat space), and, no particular redundancy is associated with them (except field redefinition in our case, which, in any case, does not alter the singularities at finite location of the graphs and of the physical process). Hence, from the analysis of individual graphs, it is still possible to extract general conclusions for the full process. Furthermore, it is likely that, as it happens in flat space, processes involving spinning states can be decomposed in terms of scalar graphs. Hence, our findings have the potential to be extended beyond the realm of scalar theories. For this reason, we find the comment "there is very little physics in the paper" too bold (and a bit unfair). Also, for the asymptotic behaviour -- which is the subject of our paper -- it is clear that graphs contribute to singularities in different ways. This can also be seen field theoretically: analysing the asymptotic behaviour of a certain physical process is often phrased in how (and which) individual graph contributes to the divergences. For this reason, we find that weakness two does not hold.

In general, we have the impression that there is some uneasiness from the referee's side to deal with a novel, and hence, unusual language. However, the purpose of our paper is exactly to develop such a language that we think is going to help us understand both the IR and UV behaviour of actual physical processes.

Referee 1

We will start with the weakness found by the referee. It is not true that in this paper we aim to solve any universal questions on cosmological correlators. Those questions are indeed the motivation for us to embark on the work that originated this paper. However, the paper in and of itself is the first step towards such a solution, and it concerns the development of a novel technology based on combinatorics, and recognizing that the asymptotic structure of the integrals that contribute to physical processes is controlled by a combinatorial object that can be predicted for an arbitrary graph, and with it the degree of divergence and the divergence coefficients. We found such a proof important enough to propose a paper to be published, but we have never claimed to have solved the full problem. We do think that these findings will help us to solve the problem in a very transparent way, but this will be the subject of future publications.  We now move on to comment on the requested changes. 1. We find that this request is intimately tied to what we explained in the paragraph above: our paper does not aim to solve the full IR problem, which is indeed one of the motivations for writing it, but to develop a technology that can allow having a deeper understanding of it, and it identifies that the asymptotic behaviour can be generally predicted (and the coefficients of the divergences more generally computed) because it is governed by a special combinatorial structure for all graphs. Looking at problems like $\lambda \phi^4$ in $dS_4$ -- the case more extensively studied in the literature -- is beyond the scope of this paper, and it will be the subject of future publications. We would also like to point out that our findings already required a very long paper. Requiring to face the IR problem in it, will make the paper inevitably and unnecessarily longer, as the method+structure and the physical problem can be separated. And this was our declared intention from the start. The cosmological integrals depend on certain parameters -- which we indicate with α -- that encode a substantial amount of information, among which is how fast the universe expands. If the universe expands sufficiently fast, these power-law divergences can appear. Is this a physical scenario? From our point of view, one of our general goals (not of this paper but of our approach) is to understand whether there is a pattern (or more generally some structure) in the IR divergences in perturbation theory that tells us that they are bound to resum, cancel or make the theory pathological without having to do perform any other computation. For this reason, we need to consider any possible behaviour, irrespectively of the scenario. Said differently, we aim to find consistency conditions on the IR behaviour that tells us whether the theory is healthy or not. We only comment on the renormalization group in the outlook, as an interesting link to understand. Even in flat space, an understanding of the RG that is more suited to modern techniques (e.g. the on-shell approach which is the inspiration for the cosmological bootstrap) is still not available. We only argued that our combinatorial approach might lead to a different understanding of the RG in the long run. In the specific case of dS, the leading IR divergences are found to resum so that the probability distribution satisfies a Fokker-Planck equation (compatibly with stochastic inflation). However, a Fokker-Planck/diffusion equation is an RG equation (and any RG equation can be put in a diffusion equation form). In this sense, if our approach can shed light on the IR problem, it also has the potential to provide a novel understanding of the RG. Now, it is true that the RG concerns UV divergences, whose physical meaning is completely different from the IR. However, our combinatorial approach unifies them (putting them on the same mathematical footing), thus allowing a complete understanding of the RG. Furthermore, having a combinatorial/polytope description gives us novel mathematical rules whose exploitation might lead us beyond the realm in which such a description has been formulated. Again, all this is based on our intuition, and for this reason, this point has been just mentioned in the outlook as a future research direction to explore.