Friedmann-Lemaître-Robertson-Walker metric

The FLRW metric is the homogeneous and isotropic spacetime metric with scale factor $a(t)$ and constant spatial curvature $k$.

A cosmological metric can measure the interval in units of time rather than length. With dimensionless comoving spatial coordinates, its scale factor $R$ then has units of length and a comoving clock has $ds=dt$. Proper spatial length is $c\sqrt{-d{s}^2}$ at constant cosmic time. This convention is equivalent to the more common length-valued interval $d{\ell }^2=c^{2}d{t}^2-R^{2}d{\Sigma }_k^2$, not a different cosmology.

Normalize the present scale factor to one. In a dust-only open cosmology with no cosmological constant, the Friedmann equation gives $-k{c}^2=H_0^2(1-{\Omega }_m)$. The present comoving and physical curvature radii are therefore $r_c=c/[H_0\sqrt{1-{\Omega }_m}]$. At another epoch the physical curvature radius is $a{r}_c$.

// Target: cosmic-microwave-background-anisotropy.bigb

A round three-sphere of radius $R$ embedded by $|\mathbf{x}{|}^2+u^2=R^2$ has metric $R^2\{d{r}^2/(1-r^2)+r^{2}d{\Omega }^2\}$ for $r=|\mathbf{x}|/R$. Multiplying by $b(t{)}^2$ gives a scale factor $a=Rb$ and normalized positive spatial curvature of an FLRW universe $K=1$. This radial chart covers a hemisphere; $r=\sin \chi$ gives the global polar description.

Cosmic time is the proper time measured by observers comoving with the homogeneous cosmological fluid. Constant-cosmic-time hypersurfaces are the homogeneous spatial slices of an FLRW spacetime.

The Milne universe is the empty $k=-1$, $\Lambda =0$ FLRW model with scale factor proportional to proper time. It is a hyperbolic-coordinate description of the interior of a future light cone in Minkowski spacetime.

In Cartesian comoving coordinates and signature $(-,+,+,+)$, a spatially flat FLRW universe has

For coordinates $(t,x,y,z)$ and line element $d{s}^2=c^{2}d{t}^2-a^2(t)d{\mathbf{x}}^2$, these are the nonzero Christoffel symbols. The prime means differentiation with respect to $t$, and coordinate zero is $t$, not $ct$. They follow either from the metric-derivative formula or the Euler-Lagrange equations for the affine geodesic Lagrangian. Both mixed lower-index orders must be counted to obtain the factor of two in the spatial geodesic equation.

The cases $k>0$, $k=0$, and $k<0$ give spherical, Euclidean, and hyperbolic constant-curvature spatial slices.

Proper distance is the spatial metric length measured along a fixed-cosmic-time slice.

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 $c=1$.