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\textbf{Response to the Referee report - JCAP\_002P\_0923}
(\today)
\noindent\rule{16.5cm}{0.4pt}
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We would like to thank the referee for a constructive review.
We 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.
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:} During the phase transition, the fluid is put into motion from the interaction with the scalar field and then travels freely after percolation. In
2.4, the initial conditions in the simulation are chosen such that the energy
density is homogeneous in the beginning [mentioned below (2.25)]. I would
think that this (in the irrotational case) leads to a collection of standing
waves rather than waves moving at the speed of sound. This is probably also
the main reason for the ringing in the energy density mentioned at the end
of page 10. It is unclear to me why the authors chose these initial conditions,
while initializing the density according to traveling waves seems much more
natural.

\noindent
{\bf Authors' response:} 
We agree with the referee's remarks regarding our chosen initial conditions for the energy density.
To address this comment, we have conducted new simulations incorporating a non-uniform energy density such that
$\widetilde{\ln\rho}(k) = \boldsymbol{k} \cdot \tilde{\boldsymbol{u}}/(c_s^2 k)$.
Here $\widetilde{\ln\rho}(k)$ and $\tilde{\boldsymbol{u}}(\boldsymbol{k})$
represent the Fourier transformed logarithmic energy density and velocity field, respectively, and $c_s$ is the sound speed.
In the new appendix~D of the paper (see also \Fig{acoustic_vs_acoustic} below),
we provide a comparative analysis of simulations initialized with uniform
energy density (runs A1 and A3) and those initiated with non-uniform
energy density (runs A1$'$ and A3$'$).
The red and black curves represent the uniform case, while the blue and green curves correspond to the non-uniform case at simulation times $\eta=1.5$ and $\eta=41$, respectively.
From this figure, it is evident that the ringing effect is reduced in the non-uniform case.
Additionally, it is worth noting that the resulting GW spectra at $\eta=41$ exhibit negligible differences between the two cases.
Thus, we agree with the referee that initializing the energy
density in a manner that satisfies the continuity equation
is important for explaining the ringing.
However, its impact on the resulting GW spectrum is negligible.
We have now added some text after equation~(2.25) in section~2.5,
at the end of section~3.3, and in Appendix~D to clarify this point.

\begin{figure*}\begin{center}
\includegraphics[scale=1.3]{acoustic_vs_acoustic.eps}
\end{center}\caption[]{Kinetic and normalized GW spectra at different times. The red and black curves are for the non-uniform initial energy density case and the blue and green curves are for the non-uniform initial energy density case. The red and blue curves represent the spectra at $\eta=1.5$, while the black and green curves correspond to $\eta=41$. }\label{acoustic_vs_acoustic}\end{figure*}

\vskip .2cm
\noindent
{\bf Referee's comment 2:} The authors report a subdominant GW component that is stationary
but scales as $k^1$. Since it is stationary, it’s tempting to associate this effect to
initial conditions. Why would this effect appear in a real phase transition?
Along the same lines: One of the main assumptions mentioned on page 4 is
that the velocity is assumed to be Gaussian. This is probably not true right after the phase transition. Hence, the low $k$ behavior is probably affected
also by this assumption and could contain other stationary components.

\noindent
{\bf Authors' response:} We understand the referee's concern regarding the association of the low $k$ linear behavior of the GW spectrum with the initial condition. 
To illustrate this point, we present the time evolution of the normalized GW energy density parameter at early times for various $k$ values in \Fig{time_evolution_of_spectrum_wo_exp_kp30} here or figure 9 in the updated manuscript. As it is evident from this figure, the $\OmGW/\Omega_0$ increases with time initially and saturates to a constant value. Additionally, the duration required for this saturation is inversely proportional to the value of $k$. 
In this analysis, we assume a constant kinetic spectrum. This is because we assume that the kinetic energy develops within a time duration less than $1/\kp$.
We have now added some text in section 3.1 and in Appendix~B to clarify this point.
\begin{figure*}\begin{center}
\includegraphics[scale=0.6]{time_evolution_of_spectrum_wo_exp_kp30.eps}
\includegraphics[scale=0.6]{time_evolution2_of_spectrum_wo_exp_kp30.eps}
\end{center}\caption[]{In this figure, we show the evolution of $\OmGW/(k\Omega_0)$ with time for $k=2,4,6,$ and 8. In the right panel, we normalized the time with $\eta_k=2\pi/k$.
}\label{time_evolution_of_spectrum_wo_exp_kp30}
\end{figure*}

In our analysis, we have adopted the assumption that the velocity follows a Gaussian random field distribution.
We acknowledge that any non-Gaussianity may introduce an additional contribution, which we have neglected in our current study.
We did already mention this point in the first paragraph of section~2.3, where we have now made minor improvements highlighted in blue.

\vskip .2cm
\noindent
{\bf Referee's comment 3:} How does the calculation in the appendix relate to the results of [46].
There a $k^3$ behavior was reported for small $k$. Since both papers start from
the sound shell model, at what point do they deviate to come to different
conclusions?

\noindent
{\bf Authors' response:} Regarding the low $k$ behavior of $\OmGW$, our
results match with the results presented in [46], which is now ref.~[38].
As in that paper, we also have a $k^3$ scaling for the wave numbers,
$k<1/\eta$ as shown in figures 1 and 5 of our manuscript for the
non-expanding and expanding case respectively.
The main differences to their work are now highlighted in the penultimate
paragraph of the introduction.

We recall that in Appendix A, we present the argument to show the linear
scaling of $\OmGW$ for the intermediate wave numbers. This linear behavior
applies to the wave numbers below the peak wave number and satisfying
the condition $k \eta>1$.  We have now added a line in Appendix A to
clarify this.

\vskip .2cm
\noindent
{\bf Referee's comment 4:} Does the field $\tilde{\phi}(k)$ around (2.20) contain a random phase? How are the
amplitudes sampled from the kinetic spectrum? Some additional comments
would be helpful in this context.

\noindent
{\bf Authors' response:}
Yes, $\tilde{\phi}(k)$ has a random phase. The amplitude of $\tilde{\phi}(k)$ is chosen such that the resultant kinetic spectrum ($\EK$) has a broken power low and $\int \EK dq = \bra{\boldsymbol{u}^2}_V/2$, where $\bra{\,}_V$ represents a volume average.
We now clarify this in the paragraph below equation (2.20).
Regarding sampling of the amplitudes in the kinetic spectrum, we have now added a sentence after equation~(2.11).
\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 JCAP.\\

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