% In this section, we will calculate the GW spectrum for the scenario where sound waves play the dominant role. The stress tensor for for the fluid is given by,
%  \begin{equation}
% %     T_{ij}=(\rho+p)\gamma^2 u_i u_j,
%      \Pi_{ij}=(\rho+p)\gamma^2 u_i u_j,
%      %rs changed T_{ij} to \Pi_{ij} to be consistent with our notation
%  \end{equation}
%  where $\rho$ and $p$ represent the energy density and pressure of the fluid respectively. Here, $u_i$ and $\gamma=(1-\uu^2)^{-1/2}$ denote the fluid velocity and Lorentz factor, respectively.
%  To calculate $|\tilde{T}|^2(k,\eta_1,\eta_2)$ for sound waves, we have followed the Ref.~\cite{Hindmarsh_2019} and it is given by Equation~(3.34) in there.
%  \begin{equation}
%      |\tilde{T}|^2(k,\eta_1,\eta_2)=\frac{16 \pi^2\bar{\omega}^2}{ k} \int_0^{\infty} dq \int_{|q-k|}^{q+k} d\tilde{q}~\frac{q}{\tilde{q}^3} (1-\mu^2)^2 E_K(q) E_K(\tilde{q}) \cos(c_s q(\eta_1-\eta_2))  \cos(c_s \tilde{q}(\eta_1-\eta_2)).
%  \end{equation}
% %In the above expression, $E_k$ ($\int E_k dk =<v^2>/2$) represents
% %AB: use \bra{}. Should we use v or u?
% %rs We use u. I have corrected this now.
% In the above expression, $E_K$ ($\int E_K dk =\bra{u^2}/2$) represents
% the linear energy spectrum, $c_s$ denotes sound speed, $\bar{\omega}=\rho+p$ and
% $$\mu=\frac{q^2+k^2-\tilde{q}^2}{2kq}.$$


