= Solution
{c}
There are two first-order effects: an intrinsic <temperature> change from <adiabatic compression>, and the <gravitational redshift> of the emerging <photons>. Evaluate the excess <Newtonian gravitational potential> at the boundary of the spherical fluctuation, with the exterior reference potential subtracted:
$$
\delta\Phi=-\frac{G\delta M}{r_0}.
$$
In the <weak-field approximation>, <gravitational time dilation> gives $d\tau_{\rm local}/dt\simeq1+\delta\Phi/c^2$. A local emitter's successive wave crests therefore arrive farther apart at a reference observer outside the potential well. Their <frequency> changes by $\delta\nu/\nu=\delta\Phi/c^2<0$. Since a redshifted <black-body radiation> spectrum has its <temperature> scaled by the same factor as its <frequency>, the gravitational contribution is
$$
\left(\frac{\Delta T}{T}\right)_{\rm grav}
=\frac{\delta\Phi}{c^2}=-\frac{G\delta M}{r_0c^2}.
$$
For the intrinsic effect, compare the local <scale factor> $R+\Delta R$ with the unperturbed value $R$ at the reference epoch. An equilibrium <photon gas> has <energy density> $u=a_{\rm rad}T^4$ and <pressure> $p=u/3$. For <adiabatic compression>, the <first law of thermodynamics> applied to a comoving volume $V\propto R^3$ gives
$$
d(uV)=-p\,dV
\quad\Longrightarrow\quad
4\frac{dT}{T}=-\frac43\frac{dV}{V}
\quad\Longrightarrow\quad
\frac{dT}{T}=-\frac{dR}{R}.
$$
Thus $TR$ is constant along this <adiabatic process>, and the initial intrinsic contribution is $\Delta T/T=-\Delta R/R$. The smaller local expansion associated with the fluctuation leaves its radiation hotter; this is the compression or local expansion-history contribution, distinct from the clock-rate effect on the escaping radiation.
Multiplying the two first-order factors, and cancelling the common subsequent background redshift, gives
$$
\frac{T_{\rm obs}}{\bar T_{\rm obs}}
\simeq\left(1-\frac{\Delta R}{R}\right)
\left(1+\frac{\delta\Phi}{c^2}\right),
\qquad
\boxed{\frac{\Delta T}{T}=-\frac{G\delta M}{r_0c^2}-\frac{\Delta R}{R}.}
$$
For the overdense region produced by <adiabatic compression>, $\Delta R<0$, so \b[the final term is positive]. It competes with the negative <gravitational redshift> term. Their sum is the intrinsic-plus-endpoint part of the <Sachs-Wolfe combination>; computing a full observed <Cosmic microwave background anisotropy> can also require <Doppler CMB anisotropy> and the <Integrated Sachs-Wolfe effect>.<Cosmology by David Tong>[Sections 3.4 and 3.4.1, equations (3.210)–(3.212): adiabatic temperature perturbations and gravitational redshift contributions]{ref}
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