This paper studies the gravitational wave power spectra formed by
instantaneously initiated magnetohydrodynamic turbulence. Informed by
a combination of new simulations and previously established properties
of the shear stress tensor, the authors develop an analytical model of
the gravitational wave power spectrum and demonstrate agreement with
their simulation results. Their focus is the low-wavenumber behaviour
of the power spectrum, motivated by the possibility of an inverse
cascade transferring power to longer length scales.

The paper is interesting. The work addresses important issues,
representing real progress in this area of research. I will be able to
recommend publication after the following issues have been addressed:

(1) The final four paragraphs of the Introduction ("To address the
problem..." onward) mix elements of a literature review and the
motivation for the present work in a way which is not quite successful
at either. I suggest the authors try to make clearer how the existing
literature has informed the present work, and to make clearer what is
being done here for the first time.

(2) It is also not clear from the Introduction what the relationship is
with the simulations of Ref. 23. The authors state that "we also
provide the evolution of the stress spectrum for these simulations
[presumably those in Ref. 23]". This would seem to imply that the
simulations in the present work are the same or have the same
parameters as those in Ref. 23, or that the results of those
simulations are reanalyzed in the current work, but as far as I can
tell that is not the case.

Answer- A table in appendix with our parameter and those with Alberto Roper Pol

(3) Many of the plots lack clear legends, which is an impediment to
understanding the results of the paper. It is not at all clear what
the different colors mean in Figures 1 and 2 - presumably different
times. The colors for Figures 3 and 4 are given in the caption text,
but that may not be sufficient for color-blind readers. Even with
color vision and a color printer, it is hard to distinguish the curves
in Figure 5; in this case it would be good to explain in the caption
which curves exactly coincide with which other curves. The caption in
Figure 8, left, is not optimally placed while Figure 8, right, appears
to lack a k_p=1 curve despite it being present in the legend. Figure
12 has arrows identifying each curve, but the caption again relies on
colors that are difficult to interpret.

Answer- In figure 5, write t on the curves with arrow. Oragne-> Blue dashed

(4) The model of Eq. 5, while undoubtedly leading to good agreement (as
seen in Figure 12), does not seem to be supported by the behaviour of
\tilde{T} for the helical case (seen in Figure 3, top left) which does
not seem to have a clear power-law. Could the authors perhaps comment
on how the model nevertheless seems to work? They may also wish to
expand the discussion in the third paragraph of the left column on
page 3, left, where these results are introduced. That would allow a
better motivation of the model in Eq. 5. They could also extend their
results in Figures 3 and 4 to the k=3 mode shown in Fig. 11 which
appears to have a clearer increase for both cases.

Response- Investigate top left panel of Figure (3) and think about the comparison with the plot in figure  (11) left panel.

(5) The time at which the results shown in Figure 12 are plotted does
not appear to be stated everywhere. Is agreement consistently good at
all times?

Response- Make a plot with a particular wavenumber for GW spectrum at different time

I have some further minor points:

- In the Introduction, the authors write about "the various phase
transitions in the early Universe" but we have not established that
any such phase transitions took place (as far as we know, the QCD and
electroweak transitions were crossovers).

- The run labels 'hel' and 'nonhel' are sometimes hard to pick out of
the text, perhaps they could be typeset in a different typeface.
Furthermore, in some cases they are instead referred to as the
"helical and nonhelical runs" (e.g. in the text two paragraphs below
Table I) or as "... with magnetic helicity" and "... for the
nonhelical case". These should be consistently labeled throughout the
paper, and I would suggest erring on the side of verbosity (thus: "...
for the nonhelical (HEL) case").

Response- HEL and NHEL


- Table II has column headings 'p' but it is not clear that what is
actually in the table is the power p but rather the overall time
dependence of the shear stress.