Cosmological Perturbations via Quantum Corrections in M-Theory
Round 1
Reviewer 1 Report
I attached my report as a PDF file.
Comments for author File: 
Comments.pdf
Author Response
Please see the attachment.
Author Response File: 
Author Response.docx
Reviewer 2 Report
The starting point of the paper is the theory of eleven-dimensional supergravity, deformed by quartic Weyl terms, at order-eight in the low-energy derivative expansion of the theory. This (Weyl)^4-deformation is part of a more general eight-derivative correction obtained by viewing
eleven-dimensional supergravity as a low-energy limit of M-theory.
The authors consider the equations of motion of the (Weyl)^4-corrected theory and show that they admit inflationary solutions of FRLW type. In the following they consider tensor perturbations around these inflationary solutions: they obtain the equations of motion for the tensor perturbations as well as an effective action from which these equations of motion can be derived consistently. The authors then derive an analytical solution for the tensor perturbations, to first order in the deformation of the theory.
The paper contains a substantial amount of work on an interesting subject, and is relatively clearly written. However I think the authors should address certain points before I can recommend its publication:
1. A mentioned above, the (Weyl)^4-deformation is only part of the full M-theory correction at the same order in derivatives. In particular the full correction to the action at this order contains a Chern-Simons term which sources the field equation of the four-form field G: d*G-1/2G^G=X_8, where X_8 is a quartic polynomial of the Riemann tensor. X_8 is crucial for the quantum consistency of the theory and can be used to determine the full eight-derivative correction by supersymmetry (see for example hep-th/0305129 [hep-th] for a discussion of these issues). As can be seen from the equation above, if X_8 is not zero neither can the four-form G be zero. The authors have neglected the eight-derivative corrections containing G. This is a priori inconsistent, unless X_8 vanishes in the space of the solutions they are considering.
2. In the perturbation expansion, it not always clear to me that the correction terms are small compared to the order-zero terms. For example in eqn.(10) the terms linear in Gamma on the right-hand sides are multiplied by large numbers c_h and c_g. Perhaps it would be useful to have a plot, as a function of tau, separately of the zeroth- and first-order terms on the right hand sides of (10).
3. I am confused about the integral (17): if the internal directions y_m are compact then instead of the l-integral there should be a discrete sum over Fourier modes.
4. In equation (6): I could not find the definition of the spaces described by the coordinates x_i and y_m. I believe the internal space is a flat 7-torus but, unless I have missed it, did not see this spelled out. Similarly for the internal space: I believe it is a flat R^3 (k=0 in cosmology), but I do not think this is mentioned explicitly.
Author Response
Please see the attachment.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Dear editor,
In the previous report, I pointed out that, because the boundary condition of perturbations is imposed at the future infinity, the model in this manuscript reduces the predictability. The authors described in their reply that it is inevitable for a consistent treatment in quantum gravity. Accordingly, I do not further criticize that point, and as the quantum gravity discussion of inflation itself is of physical importance, I recommend the manuscript for publication.
Best regards,
The referee.
Reviewer 2 Report
The authors have adequately responded to the points in my report. I therefore recommend the manuscript for publication.
