= Solution
For $R=0$, $c_s=1/\sqrt3$. If the <Newtonian gauge in cosmology>[Newtonian potentials] are constant and equal, the equation in part (b) becomes
$$
\Theta_0''+c_s^2k^2\Theta_0=-c_s^2k^2\Psi.
$$
The <Sachs-Wolfe combination> $D=\Theta_0+\Psi$ therefore obeys
$$
D''+c_s^2k^2D=0.
$$
Adiabatic initial conditions have vanishing initial fluid velocity, hence $D'(0)=0$. With the <sound horizon> $r_s(\eta)=\int_0^\eta c_s,d\eta'$, the solution is
$$
\boxed{\Theta_0+\Psi=(\Theta_0+\Psi)_{\rm ini}\cos(kr_s)}.
$$
Solved by gpt-5.6-sol high.
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