Line-of-sight solution for free-streaming photons

The line-of-sight solution integrates the gravitational source along the unperturbed photon trajectory. In Fourier space the integrating factor $e^{ik\mu \eta }$ accounts for free streaming between emission and observation.

In the negative-spatial-perturbation convention, a matter-era scalar solution $h=h_s+C=A{k}^{1/2}{\tau }^2/4$ gives the displayed exact photon brightness perturbation. Integration gives emission bracket ${\Delta }_{\ast }+iA{k}^{-1/2}\mu {\tau }_{\ast }-A{k}^{-3/2}$ times the propagation phase, plus observer monopole $A{k}^{-3/2}$ and dipole $-iA{k}^{-1/2}\mu {\tau }_0$. Under ordinary adiabatic growing-mode initial conditions and $k{\tau }_{\ast }\ll 1$, the emission gravitational term dominates higher angular multipoles. Arbitrary freely specified initial brightness need not obey this dominance.

With instantaneous photon decoupling and tight-coupling initial brightness, integrate the synchronous photon brightness equation along the unperturbed ray $\mathbf{x}(\tau )={\mathbf{x}}_0-\mathbf{n}({\tau }_0-\tau )$. The direction $\mathbf{n}$ here is photon propagation, opposite to the observer-to-source sky direction. Emission terms are evaluated at the emitter, not the observer. The synchronous metric integral combines endpoint gravitational redshifts and evolving-potential effects when expressed in Newtonian gauge; it is not solely the Integrated Sachs-Wolfe effect.

// Target: cosmology.bigb

Use $h_{ij}{n}^i{n}^j=h/3+({\mu }^2-1/3)h_s$. Two integrations by parts of the anisotropic source give the exact Cosmic microwave background anisotropy amplitude $W_{\ast }[{\delta }_{\gamma }/4+3i\mu {\delta }_{\gamma }^{\prime }/(4k)+i\mu (h^{\prime }-h_s^{\prime })/(2k)+h_s^{\prime \prime }/(2k^2){]}_{\ast }+[i\mu h_s^{\prime }/(2k)-h_s^{\prime \prime }/(2k^2){]}_0-{\int }_{\ast }^{0}W[(h^{\prime }-h_s^{\prime })/6-h_s^{\prime \prime \prime }/(2k^2)]d\tau$. The emission velocity was eliminated using ${\delta }_{\gamma }^{\prime }+4ik\cdot v/3+2h^{\prime }/3=0$. The local observer bracket is a monopole plus a dipole and can be discarded only after explicitly restricting to the measured higher angular multipoles. This identity fixes endpoint signs and integral factors without importing a different scalar-metric convention.