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\begin{document}
\begin{center}
\textbf{Response to the Referee report-DS13341/Sharma}
(\today)
\end{center}
We would like to thank the referee for a constructive review and highly
appreciate the comments and suggestions, which helped us to improve
the manuscript. We also thank the referee for the positive comments
about our work. Please find below our detailed response to each of the
comments. \textcolor{blue}{In accounting for the comments, we have made
changes in the manuscript and marked them in blue.}

\vskip .2cm
\noindent
{\bf Referee's comment 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.

\noindent
{\bf Authors' response:} We have now modified
paragraphs 3 and 5 of the introduction to make a clear distinction between
the idea presented in the paper and a comparison with the earlier studies
on the subject.

\vskip .2cm
\noindent
{\bf Referee's comment 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.

\noindent
{\bf Authors' response:} Our simulations are similar to those of
Ref.~[26] (which was previously Ref.~[23]).
There are differences in the parameters, and their resolution is
lower, but the important point is that our analysis is different.
The important point is that we are using a more accurate representation
of the time dependence of the stress, which turned out to be important.
We also highlight a systematic difference between helical and nonhelical runs.
This is now stated in the penultimate paragraph of the introduction.

\vskip .2cm
\noindent
{\bf Referee's comment 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.

\noindent
{\bf Authors' response:} Yes, different colors in Figure 1 and 2 represent
different times.
We have now added a legend to clarify this.
We have now added different line styles in Figures 3 and 4 to make the
curves distinguishable.
We have now changed the colors in Figure 5, added a legend and also added
the text in the caption explaining which curves coincide with which
other curves.
The legend in the left panel of Figure 8 is now placed
more conveniently.
Actually, there was the $k_p=1$ line in the right panel of Figure 8.
However it was not visible since the red curve corresponding to the
$k_p=1$ line was on the top of this curve.
We have now changed the color's opacity and it is evident now.
We have now chosen different line styles for different curves in Figure~12.
We have also added a legend in Figure 12 to have a clear distinction
between different curves.
We hope that, with these changes, different curves in the Figures are
distinguishable now.

\vskip .2cm
\noindent
{\bf Referee's comment 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 behavior
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.

\noindent
{\bf Authors' response:}
The evolution at selected wave vectors is not representative of the average.
Therefore we are estimating the time evolution
of the $\tilde{T}$ that is used in the estimate of the GW spectrum from
the evolution of stress spectrum $\tilde{T}=\sqrt{\Sp(\tilde{T})/4\pi
k^2}$. This is shown in Figure~11. The reason for not having the same
behavior in Figure~3 is due to the fact that we only have here a few
wave vectors of the stress for particular values of $k$ in Figure 3 as
opposed to the case shown in Figure 11 where we have an average over all
wave vectors of nearly the same length when we calculate it from the
stress spectrum. To clarify this, we have now added some text in paragraph
3 of section IIB (lower left of page 3).
We are now showing the time evolution for $k=0.3$
and $0.5$ in Figure 11 instead of $k=3$ (as in the previous version)
to have a comparison with Figure 3 and 4. The behavior of $\tilde{T}$
for $k=0.3$ and $0.5$ is similar to the case $k=3$ and it is still fitted
with the same double power law.

We also note here that the results for other wave vectors of the same
length show a similar behavior of
$\arg(\tilde{T})$ and
$\arg(\dot{\tilde{h}})$ as is shown in
the figures of the Supplemental Material provided along with the data
in Ref.~[44].

\vskip .2cm
\noindent
{\bf Referee's comment 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?

\noindent
{\bf Authors' response:} The results in Figure 12 are in the saturated
phase (the value of $\Sp(\dot{\tilde{h}}$) for all wave numbers plotted
in the figures have reached to approximately a constant value). The actual
time is $t=175.5$ for HEL and $t=99.5$ for NHEL, our new labels for the
helical and nonhelical runs (see also Table~I). It is
now mentioned in the manuscript (see the text at the end of the first
paragraph of section IV). The mean value of $\Sp(\dot{\tilde{h}})$ is
in reasonably good agreement with the values obtained from the model at
all times.

\vskip .2cm
\noindent
{\bf Minor comments:} \\
{\bf Referee's comment 6:} 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).

\noindent
{\bf Authors' response:} We agree with the referee's comment.
We have now added a footnote in the manuscript to clarify this point
(left column of page 1).
Magnetic fields may still be generated during a cross-over, and this would then also produce turbulence.

\vskip .2cm
\noindent
{\bf Referee's comment 7:} 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").

\noindent
{\bf Authors' response:} We have now labeled the runs with capital
letters HEL and NHEL for helical and nonhelical runs respectively.
We have also made analogous changes at all relevant places in the manuscript.

\vskip .2cm
\noindent
{\bf Referee's comment 8:} Table II has column headings 'p' but it is not clear what is
actually in the table is the power p but rather the overall time
dependence of the shear stress.

\noindent
{\bf Authors' response:} Thank you for pointing this out. We have corrected this now.

In addition to the changes to incorporate the above comments, we have
also added a new table (Table~II on page~6) that provides a summary of
section~IIC regarding the time evolution of $\Sp(T)$ and $|\tilde{T}|$.

\vskip .5cm

We once again thank the referee for the comments and we hope that with the
clarifications and modifications, the paper will be found suitable for publication in Phys. Rev. D.\\

\end{document}
