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:

δ Φ = - \frac{G δ M}{r _{0}} .

In the weak-field approximation, gravitational time dilation gives

d τ_{local} / d t ≃ 1 + δ Φ/ 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

δ ν / ν = δ Φ/ 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

(\frac{Δ T}{T})_{grav} = \frac{δ Φ}{c ^{2}} = - \frac{G δ M}{r _{0} c ^{2}} .

(2)

For the intrinsic effect, compare the local scale factor

R + Δ R

with the unperturbed value

R

at the reference epoch. An equilibrium photon gas has energy density

u = a_{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 (u V) = - p d V ⟹ 4 \frac{d T}{T} = - \frac{4}{3} \frac{d V}{V} ⟹ \frac{d T}{T} = - \frac{d R}{R} .

(3)

Thus

TR

is constant along this adiabatic process, and the initial intrinsic contribution is

Δ T / T = - Δ 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