One-particle distribution function ($f(x,p)$)

A scalar function on the future mass shell describing the expected particle count in a phase-space distribution cell. In a local orthonormal tetrad, the number crossing a spacelike surface element is $f{p}^{\mu }d{\Sigma }_{\mu }{d}^{3}p/E$. On a constant-time slice it becomes $f{d}^{3}x{d}^{3}p$. Including degeneracy and phase-space normalization in $f$ gives the stress-energy tensor $T^{\mu \nu }=\int (d^{3}p/E)f{p}^{\mu }{p}^{\nu }$. Other conventions may put the degeneracy or $(2\pi {)}^{-3}$ explicitly in the measure.

The temporal-temporal component of the photon stress-energy tensor in a local orthonormal tetrad. For an isotropic one-particle distribution function expressed using comoving energy $ϵ=aE$, it is ${\bar{\rho }}_{\gamma }=4\pi a^{-4}\int dϵ{ϵ}^3\bar{f}(ϵ)$. A thermal photon temperature perturbation gives a first-order directional energy perturbation four times its fractional temperature change. Boundary conditions on the distribution justify this factor through integration by parts.