Conformal time

Conformal time satisfies $d\tau =dt/a(t)$ and factors the FLRW metric into $a^2(\tau )$ times a static conformal metric.

In a flat universe containing pressureless matter and a positive cosmological constant, normalize the scale factor to one today. The Friedmann equation and $d\tau =dt/a$ giveThe change of variable is $a=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}y$. Including radiation in cosmology makes this an approximation, with its omitted early contribution of order $H_0^{-1}\sqrt{{\Omega }_r}/{\Omega }_m$. When the upper limit is large, the integral approaches $I=\Gamma (1/6)\Gamma (1/3)/(3\sqrt{\pi })$; this follows by setting $z=y^3$ and using the beta function integral.

For positive curvature and zero cosmological constant, the single-fluid Friedmann solution has a bang-to-crunch conformal time interval $2\pi /(1+3w)$ for $w\ge 0$. A radial null geodesic in FLRW spacetime travels that same great-circle comoving distance. Relative to a three-sphere circumference $2\pi$, it is $1/(1+3w)$ circuits: one for dust and one half for radiation, with the final point reached only in the singular-endpoint limit.

For the de Sitter scale factor in flat slicingchoose the additive constant so thatAn eternal expanding patch has $-\infty <\tau <0$.

Comoving radial distance uses the coordinate $\chi =\int dr/\sqrt{1-k{r}^2}$ and removes the radial curvature factor.

This radial coordinate is the comoving member of the comoving and proper distances comparison; multiplying by the scale factor gives proper radial distance at fixed cosmic time.

For nearby redshifts in a homogeneous expanding universe, the comoving radial distance increment is $c\delta z/H(z)$. Multiplying by the scale factor at the absorber's epoch gives the displayed proper distance in cosmology. The conversion neglects peculiar velocities; observed redshift differences can also contain such velocities and errors in the quasar emission redshift.

In conformal coordinates a radial light ray obeys $d\chi /d\tau =\pm c$, so the cosmological light cone is at forty-five degrees after setting $c=1$.