Graviton to Photon Conversion in Curved Space-Time and External Magnetic Field

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper deals with conversion of gravitational waves (GWs) into photons in an external magnetic field but in a curved spacetime, with particular regards of a LFRW metric. The authors apply the results to suppression of GWs in an expanding universe with a magnetic field O(~1nG) showing that low frequency GWs can be highly suppressed.

The derivation of GW-photon conversion equations (SoDE)  is rather intricate and the notation is not always transparent. I must assume that the derivation is correct since it is not easy to reproduce the SoDE. The results are interesting although are based on the simplified hypothesis that the background  magnetic field is constant (apart the a^-2 dependence). Very little is known on cosmological magnetic fields, but a more realistic field has a stochastic behaviour.   

The text can be improved. In particular, the conclusions and prospects can be enhanced (it looks that the prospects are just drafted). I have found few typos in the text. In Eq. (1.0.3) "M" in the trace should be M^-1. After Eq. (5.2.11) the authors state that Dirac matrices must commute (actually they anti-commute). A "flolows" (follows) after Eq. (6.1.13). I cannot check all the equations as they are too  complicated. The bibliography is messy and must be standardized. 

Author Response

Thanks for your valuable comments. We have highlighted text improvements and also corrected the bibliography. Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Referee report on the manuscript Universe-2729463
"Graviton to photon conversion in curved space-time and in external
magnetic field" by Alexander Dolgov, Lyubov Panasenko, and Vladimir Bochko.


In the manuscript under consideration the authors study graviton to
photon conversion in a manifold with curved metric. They derived
the coupled system of equations describing gravitational and electromagnetic wave propagation in an arbitrary curved space-time and in external magnetic field. The obtained system of equations is
solved numerically in the Friedman-LeMaitre-Robertson-Walker metric for the upper limit of the intergalactic magnetic field strength of 1nGs.
It was concluded that the gravitational wave conversion into photons
in the intergalactic magnetic field can significantly change the amplitude of the relic gravitational wave and their frequency spectrum.

The paper can be published after minor (optional) improvement of the text:

1. Definition of the acronym FLRW should be done where it appears
for the first time (2nd sentence of the 2nd paragraph of Introduction).

2. Introduction of full electromagnetic tensor and etc. (see,
text between Eqs.(1.0.1) and (1.0.3) can be improved.
E.g., one can say that we introduce the full electromagnetic
field \bar A_{\mu} the sum of an external classical
component electromagnetic field A_{\mu} and a small quantum
fluctuation f_{\mu}, which is considered as a small perturbation,

 \bar A_{\mu} =  A_{\mu} +  f_{\mu}

Then, the stress tensors of (\bar A, \bar A, and f) are introduced accordingly:

\bar F_{\mu,\nu} = \partial_\mu \bar A_\nu - \partial_\mu \bar A_\nu

     F_{\mu,\nu} = ...

     f_{\mu,\nu} = ...

The full metric tensor $\bar g_{\mu,\nu}$ is expanded around
the metric tensor of the background space-time  $g_{\mu,\nu}$
as $\bar g_{\mu,\nu} = g_{\mu,\nu} + h_{\mu,\nu}^c$ and
$\bar g^{\mu,\nu} = g^{\mu,\nu} -  h^{\mu,\nu}$ with
$h_{\mu,\nu}$ being a small/perturbative fluctuation of the metric.
The properties of the metric tensor $g_{\mu,\nu}$ are specified by:
(1) the orthogonality condition ...,
(2) rising and lowering of the indices of the tensors $h$ and $f$.
Note, that the indices of the full and classical stress tensors of
the electromagnetic fields are rised and lowered with full metric tensor $\bar g$.


2. There are in the text the question marks (??). Possibly,
they are the LaTeX-typos and should corrected accordingly.  


3. After Eq.(2.0.1) the gravitational constant G, the covariant
derivative $D_\mu$,
the Ricci and energy-momentum tensors should be specified.

4. The comma in the math-expression $B = 0 F = 0$
after Eq.(3.0.3) is missing.

5. After Eq.(5.2.11) the authors defined the commutator relations
for the gamma-matrices in the flat and curved space-time.
Must be anticommutators? Right?

6. The vielbein factor $a(\tau)$ in Eq.(5.2.10) relating
the curved and flat metric should be specified. I suppose that
$a(\tau) = a$. For nonspecialist a remark that the curved
metric reduces to the flat at $a \to 1$ would be useful.

7. Eq.(5.2.11) is written in non-fully convenient form:
some terms are expressed in terms of flat and curved
metric tensor. In this vein, Eq.(5.2.12) is fine.

8. Just before Eq.(5.2.13):

"value of $\alpha$."   ---> "value of the electromagnetic coupling $\alpha$."


9. After Eq.(5.2.13):

"e.g. for quarks  $q = ....$"  ---->
"e.g., for down or up quarks  $q = ....$"  

Author Response

Thank you for the valuable comments. We have highlighted text improvements, supplemented the conclusion, corrected typos and also corrected the bibliography. Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Author Response

Thank you for the valuable comments. We tried to answer your comments in the text of the article as fully as possible. Thank you for finding an error in the formula (it remained as a rudiment of old versions of the work). We tried to place more emphasis on our assumptions when formulating the problem, for example, regarding the negligence of gravity from the background magnetic field, and the rejection at the first stages of the stochastic nature of relict GW and magnetic field. Taking the latter into account will greatly influence the final result of further research. We focus on the fact that we solved a simplified problem to obtain an estimate (from above) of the order of magnitude of the suppression of GW by the conversion effect. The derivation and solution of a system of equations even for such a simplified problem is complex and cumbersome, the article turned out to be large, and an attempt to cover more will lead to difficulties for the reader to understand the presentation. So the current publication is the initial stage of further work. And, of course, there is still a lot of work for future publications.

We have highlighted text improvements. Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

Comments are attached in pdf file.

Comments for author File: Comments.pdf

Author Response

Thank you for your valuable comments.
Please see the attachment. Modified text is highlighted in red

(1) - we have corrected after (5.1.7)

(2) - we have corrected (8.2.6) and (8.2.14)

(3) - we have improved on p. 32-33

(5) In order to derive (5.1.10) from (5.1.9) we use h_{ij}=g_{il}h^l_j and divide both sides of the equation by (-a^2).
Using (8.2.10) we obtain eq. (9.0.11) from eq. (5.1.10)

(4)We have found two errors in equation (9.0.11) with the powers of the scale factor. The corrected results of the work at this point became opposite. We have rewritten sections 9.2, 10 and abstract. Thank you for the comment that helped us to avoid the error in the equation and consequent error in the solution and conclusions. As you requested we added an explanation of
the physical reason for the suppression on page 35.

Author Response File: Author Response.pdf

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

Review is attached.

Comments for author File: Comments.pdf

Author Response

Thank you very much for your comments. We will correct the paragraph explaining the reason for suppression and recheck the algebra.

We wish you a Happy New Year!