A Plausible Model of Inflation Driven by Strong Gravitational Wave Turbulence
Round 1
Reviewer 1 Report
see file attached
Comments for author File: 
Comments.pdf
Author Response
Point 1: First of all, it is worth remembering that in the very first theory of inflation proposed long ago by Starobinsky the rapid expansion of the universe was indeed caused by non-linear gravitational dynamics through R^2 terms in the action. It would be interesting to know if there is a relation with an effective description of gravitational turbulence.
Response 1: The Starobinsky’s theory is different because it introduces an extra (R^2) term into the action in addition to the Einstein-Hilbert term. The referee refers to this as a nonlinear term, which is certainly true. On the contrary, we consider the system with the pure (non-modified) classical Einstein-Hilbert action. The nonlinear (with respect to h, not R) effects that we consider are already present in the Ricci tensor (whose dependence on h is nonlinear). On the other hand, we treat the quantum effects (eg. primary black hole merger) that usually show up in the energy-stress tensor, as an external forcing of gravitational waves (GW) at the smallest scales. In the Starobinsky’s theory, the extra term mimics some quantum effects and can also lead to strong GW. In our paper we would like avoid making preferences to specific generation mechanisms of the initial small-scale GWs because they do not have much impact on the turbulent mechanisms such as the inverse cascade and the critical balance state. However, we have now added a discussion of the Starobinsky’s theory in the context of the possible framework for describing the GW forcing. A modification (in red) is made around lines 40 and 60.
Point 2: Although the paper pretends to sketch a plausible scenario, rather than a full quantitative model, some of the statements are rather vague or imprecise. For example, in sect. 3, after eq. (10), it states that turbulence ‘… needs to be already partly strong’ in the regime considered. What is partly strong? please specify.
Response 2: We have modified this part. In our scenario, the GW injected in our turbulent system are supposed to be relatively strong in the sense that the weak GW turbulence theory is borderline relevant for our scenario, and the strong GW turbulence scenario, called critical balance, is the main regime. We have modified our paper to better explain this point. The word ‘strong’ has been introduced (in red) to specify the type of turbulence regime (abstract and lines 191). We have also modified the figure caption, and the lines 127 and 167 to explain that the initial excitation is close to the wavenumber k_s where GW turbulence becomes strong.
Point 3: A few lines below it discusses non-zero curvature as a next stage after including metric perturbation $h_{\mu,\nu}$; however, any non-trivial perturbation generates curvature, even at the linear level. Maybe the authors have in mind some kind of explanation scheme in the gravitational coupling constant?
Response 3: We agree, any perturbation leads to curvature. Actually in weak or strong GW turbulence we do have a non-zero local curvature. By curvature we meant a large-scale curvature that could be included. We have included this comment line 128.
Point 4: A a side remark, if the authors have not already discovered this, on the first line after eq. (12) the statement about $h_{\mu 0}$ is repeated twice. Presumably this is a typing error.
Response 4: Actually, it is not a typing error because the indices are not the same. We have modified one word just after this equation.
Point 5: Finally, although it is mentioned in the introduction, the authors do not discuss any conditions related to the expansion by 60 e-folds which is required for viable theories of cosmic inflation, nor to what extent the recent acceleration of the cosmic expansion might still require a cosmological constant or a quintessence field; presumably the gravitational turbulence scenario is of no help here.
Response 5: As the referee mentioned, the turbulence scenario cannot give a prediction about the e-folds. For the recent acceleration it seems difficult to imagine that our scenario is relevant because the condition today are quite different. We made a comment about that in the conclusion (lines 197 and 201).
Reviewer 2 Report
Authors present mechanism for inflation from interacting gravitational waves rather than from inflaton field in traditional approaches. I find this work as very interesting and intriguing also supported by rigid mathematical theory of turbulent behaviour. I think that critical behaviour phenomenology and BE condensates are really attractive tools for understanding the early universe, especially after experimental evidencies for gravitational waves (GW). The paper is well written and the presentation is very good.
However, I have a conceptual problem with this work i.e. a kind of 'beginning' problem. Considering gravitational waves as strong excitations of spacetimes emerging from Planck era requires pre-spacetime nonclassical state (e.g. black holes in a sufficiently large amount and all they carry spacetime singularities) and spacetime would not be well-defined. We do not understand the emergence of classical spacetime from this quantum regime but there presumably emerge strongly nonlinear gravitational waves. The approach in the paper deals with weak GW on spacetime which evolve anyway to the strong GW. Authors chose Ricci flat spacetime with $\Lambda = 0$ on which they consider weak GW. However, the question about the origin of weak GW remains unclear in their presentation: one starts with strong GW (with singular spacetime) then obtain weak GW with well-formed classical spacetime, which next leads to strong GW again. Then the description as in the paper would apply. A kind of phase transition would be helpful in understanding of this problem. I would expect that Authors will comment on this issue. In the standard formulation of inflation by scalar inflaton field this kind of question does not appear since the source for inflation is just this quantum field on spacetime.
I certainly recommend this paper for publication in Universe but after addressing the above issue.
Author Response
Our scenario starts at a time close to 10**(-36)s, which is far enough from the Planck time to use Einstein’s equations (although inflaton models also use general relativity). We need a source of space-time excitation around this time. We propose primary black hole (PBH) merger (among others) as a possible source of GW excitation. General relativity cannot describe locally quantum effects associated with PBH, however, we only need to take into account the consequences of the merger, ie. GWs, in our statistical modeling. We implicitly assume a decoupling between the physics of PBH (structures) and that of space-time (fluctuations). The interaction between them is modeled as a GW excitation (‘a forcing’ in the language of turbulence). These waves are supposed to be relatively strong in the sense that the weak GW turbulence theory is borderline relevant for our scenario, and the strong GW turbulence scenario called critical balance is the main regime. By strong we mean that GWs cannot be used in a perturbation theory, however, there are not strong enough to produce black holes. Then, our inflation scenario is based on strong wave turbulence.
We have modified our paper to better explain this point. The word ‘strong’ has been introduced (in red) to specify the type of turbulence regime (abstract and lines 185). We have also modified the figure caption, and the lines 121 and 161 to explain that the initial excitation is close to the wavenumber k_s where GW turbulence becomes strong.
Reviewer 3 Report
Dear Editor,
we think this paper takes a different approach to inflation. Since inflation is such a broad scenario, we believe this paper is interesting by exploring how to get some sort of inflation out of a strong gravitational turbolence.
The paper is sound, the idea interesting, so that we would suggest the paper to be published as it is.
Author Response
We thank the referee for this very positive comment.
Round 2
Reviewer 1 Report
I agree with the author's modifications/improvements.
Reviewer 2 Report
The issues raised by me are now answered and I understand better the approach. It is not fully explaining the quantum origins of spacetime and inflation since it can not do this (QG issue). It is still valid and give insight how inflation could be generated within unmodified GR. The paper should be published in the present form.
