= Accumulated free-streaming distance in a radiation-dominated universe
{title2=$\chi=\frac1{aH}\frac{\operatorname{arsinh}(m/T_r)}{m/T_r}$}
Suppose a collisionless particle has <momentum> $p=T_r\propto a^{-1}$ during pure <radiation domination>. Since $a\propto t^{1/2}$, integrating $\chi=\int_0^t v(t')dt'/a(t')$ gives the displayed <comoving distance>. The physical distance at observation is $a\chi=H^{-1}\operatorname{arsinh}(y)/y$, $y=m/T_r$. The printed expression without $a^{-1}$ is consequently a physical distance, or a comoving distance in coordinates normalized to $a=1$ at observation. Its ratio tends to one in the relativistic limit and to $\log(2y)/y$ in the <nonrelativistic limit>. The accumulated scale differs from an instantaneous velocity-over-expansion-rate estimate.
Back to article page
