Cosmological visibility function ($g(\eta )$)

The visibility functionis the probability density for the conformal time of a CMB photon's last scattering.

This probability density describes last scattering during cosmological recombination and is built from the optical depth, rather than naming the recombination process itself.

The CMB line-of-sight solution separates sharply localized last-scattering sources, weighted by the cosmological visibility function, from integrated gravitational sources weighted by $e^{-\tau }$ along the photon trajectory.

Ignoring the last-scattering angular source and later scattering, the tensor-induced temperature anisotropy is $\Theta =-\frac{1}{2}{\int }_{{\eta }_{\ast }}^{{\eta }_0}{\dot{h}}_{ij}(\eta ,{\mathbf{x}}_0-({\eta }_0-\eta )\mathbf{e})e^i{e}^{j}d\eta$. An instantaneous cosmological visibility function alone leaves a scattering source at last scattering; leading tensor tight coupling supplies the additional reason to neglect it.