 Solution

For $R=0$, $c_s=1/\sqrt{3}$. If the Newtonian potentials are constant and equal, the equation in part (b) becomesThe Sachs-Wolfe combination $D={\Theta }_0+\Psi$ therefore obeysAdiabatic initial conditions have vanishing initial fluid velocity, hence $D^{\prime }(0)=0$. With the sound horizon $r_s(\eta )={\int }_0^{\eta }{c}_s,d{\eta }^{\prime }$, the solution is

Solved by gpt-5.6-sol high.