Photon brightness perturbation ($\Delta =\int q^3{f}_{1}dq/\int q^3{f}_{0}dq=4\Theta$)

The photon brightness perturbation is the fractional frequency-integrated directional energy perturbation. For a small blackbody temperature shift $\Theta =\delta T/T$, $f_1=-q{f}_0^{\prime }(q)\Theta$. Integration by parts gives $\Delta =4\Theta$. Its direction average is the photon density contrast. This temperature identification assumes a perturbation of blackbody form; the integrated brightness definition itself is more general.

// Target: cosmology.bigb

For collisionless photons in synchronous gauge, their perturbed momentum obeys $q^{\prime }=-q{h}_{ij}^{\prime }{n}^i{n}^j/2$. Terms in $q^{\prime }{\partial }_q{f}_1$ and direction deflection times angular derivatives of $f_1$ are second order. Integrating the remaining linear Collisionless Boltzmann equation over $q^{3}dq$ gives the displayed equation, because $\int q^4{f}_0^{\prime }dq=-4\int q^3{f}_{0}dq$.

// Target: cosmology.bigb

For synchronous gauge metric $d{s}^2=a^2[-d{\tau }^2+({\delta }_{ij}-h_{ij})d{x}^{i}d{x}^j]$, conserved-background comoving momentum obeys $q^{\prime }=q{h}_{ij}^{\prime }{n}^i{n}^j/2$. In the linear Collisionless Boltzmann equation, insert $\Delta =-4f_1/(q{f}_0^{\prime })$ to obtain the displayed source. Using ${\delta }_{ij}+h_{ij}$ instead reverses both signs; it does not change the physics. The scalar contraction is $h_{ij}^{\prime }{n}^i{n}^j=(h^{\prime }-h_s^{\prime })/3+{\mu }^2{h}_s^{\prime }$.

For a photon in $d{s}^2=a^2[-d{\tau }^2+({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j]$, the null geodesic condition cancels the homogeneous $2\mathcal{H}{p}^0$ term in the derivative of $q=a^2{p}^0$. The remaining first-order geodesic equation gives the displayed momentum change. Its direction is constant at zeroth order, so first-order redshift and brightness sources can be evaluated on the unperturbed ray; direction deflection multiplied by an already first-order anisotropy is second order.