FLRW Spacetimes

FLRW spacetimes are the geometries whose spatial slices are homogeneous and isotropic. Those symmetries force each slice to have constant curvature, leaving one time-dependent scale factor. Einstein’s equation then reduces to the Friedmann system. The reduction is exact, but interpreting a particular scale factor as our universe additionally requires matter modeling and observational inference.

Required background. Einstein’s Equation supplies the gravitational field equation; Continuum Kinematics and Balance Laws supplies perfect-fluid density, pressure, and conservation.

Helpful background. The Riemann Curvature Tensor supplies constant curvature; Exact Solutions of Einstein’s Equations supplies the family context; Killing Fields and Isometry Groups supplies spatial homogeneity and isotropy.

Symmetry and the metric

Isotropy about every point means the isotropy group acts as all spatial rotations on each tangent space. No nonzero spatial vector or trace-free symmetric two-tensor can be invariant under that action. Consequently the intrinsic spatial Ricci tensor must obey

$${}^{(3)}R_{ij}=2K\,h_{ij},$$

and in three dimensions the Riemann tensor is determined by the Ricci tensor:

$${}^{(3)}R_{ijkl}=K(h_{ik}h_{jl}-h_{il}h_{jk}).$$

Homogeneity makes $K$ constant on each slice. Factoring out a scale $a(t)$ gives $K=k/a^2$ with dimensionless normalized $k=+1,0,-1$ when nonzero curvature is rescaled conventionally.

In cosmic time,

$$ds^2=-dt^2+a^2(t) \left[ \frac{dr^2}{1-kr^2}+r^2d\Omega_2^2 \right].$$

The coordinate range of $r$ depends on $k$ and this chart does not cover the entire three-sphere when $k=+1$. Conformal time $d\eta=dt/a(t)$ gives $ds^2=a^2(\eta)(-d\eta^2+d\Sigma_k^2)$, exhibiting a conformal relation to a static metric with constant-curvature spatial slices. For $k=0$, that static metric is explicitly Minkowski; for $k=\pm1$, further coordinates are needed to display a flat conformal representative. Ellis and van Elst derive the symmetry and covariant kinematics in Ellis and van Elst 1999, §§2–3, Open PDF.

For $k=+1$, set $r=\sin\chi$ to obtain

$$d\Sigma_{+1}^2=d\chi^2+\sin^2\chi\,d\Omega_2^2, \qquad 0\leq\chi\leq\pi.$$

The original $r$ chart turns around at the equator and needs a second patch. For $k=-1$, $r=\sinh\chi$ gives $d\chi^2+\sinh^2\chi\,d\Omega_2^2$ with $\chi\geq0$. These substitutions distinguish a coordinate endpoint from the topology and completeness of a spatial slice.

FLRW’s Weyl tensor vanishes. That is a useful independent check of isotropy, but the converse is not sufficient without the symmetry, matter alignment, and foliation hypotheses: conformal flatness alone does not define an FLRW cosmology.

Friedmann equations

For comoving four-velocity $u^a=(\partial_t)^a$ and perfect fluid

$$T_{ab}=(\rho+p)u_au_b+pg_{ab},$$

define $H=\dot a/a$. Einstein’s equation with cosmological constant yields

$$H^2+\frac{k}{a^2}=\frac{8\pi G}{3}\rho+\frac{\Lambda}{3},$$ $$\dot H-\frac{k}{a^2}=-4\pi G(\rho+p).$$

Stress-energy conservation is

$$\dot\rho+3H(\rho+p)=0.$$

Only two of these three equations are independent away from a degenerate branch: differentiating the first and using conservation gives the second. The acceleration equation follows as

$$\frac{\ddot a}{a} =-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}.$$

This redundancy is an important sign check. Reall develops the same reduction and representative solutions in Reall 2013, §13, pp. 149–173, Open PDF.

The comoving congruence is geodesic when pressure is spatially homogeneous, as FLRW symmetry requires. Its expansion is $\theta=3H$, while shear and vorticity vanish. Those kinematic statements are invariant and distinguish cosmological expansion from the coordinate growth of a single radial distance. For two comoving worldlines at fixed spatial separation, their instantaneous proper distance on a cosmic-time slice scales with $a(t)$; this does not define a relative velocity by parallel transport over a finite curved separation.

Spatial curvature and expansion should likewise remain distinct. The intrinsic scalar of a cosmic slice is ${}^{(3)}R=6k/a^2$, whereas four-dimensional curvature contains both $H$ and $\dot H$. A spatially flat model can therefore have strongly curved spacetime, and a curved spatial slicing can occur in flat spacetime, as the Milne example below demonstrates.

Exact families and invariant checks

For a constant equation of state $p=w\rho$, conservation gives $\rho\propto a^{-3(1+w)}$. With positive fluid density, $k=\Lambda=0$, and $w\gt-1$, the expanding branch, with its initial singularity placed at $t=0$, has

$$a(t)\propto t^{\frac{2}{3(1+w)}}.$$

Thus dust gives $a\propto t^{2/3}$ and radiation gives $a\propto t^{1/2}$. For a constant phantom equation of state $w\lt-1$, the expanding solution is instead $a(t)\propto(t_*-t)^{2/[3(1+w)]}$ for $t\lt t_*$: its negative exponent makes $a$ grow without bound as the finite future time $t_*$ is approached. Vacuum energy $p=-\rho$ is constant and produces exponential expansion on the flat slicing. Curvature or multiple components generally leave an integral rather than a single power law.

The four-dimensional Ricci scalar provides an independent check:

$$R=6\left(\dot H+2H^2+\frac{k}{a^2}\right) =8\pi G(\rho-3p)+4\Lambda.$$

The equality of geometric and matter expressions tests the signature and trace conventions. A vanishing scale factor is not classified by coordinates alone: one checks curvature scalars, tidal frames, and geodesic extendibility. Milne coordinates have $a=t$ and $k=-1$ but describe a wedge of flat spacetime, so $a=0$ there is a coordinate boundary rather than curvature blow-up.

At a proposed bounce, $H=0$ and $\ddot a>0$. The first Friedmann equation fixes which combinations of $k$, $\rho$, and $\Lambda$ can reach $H=0$, while the acceleration equation tests whether the extremum is a minimum. Drawing a smooth positive $a(t)$ is not enough; both equations and matter conservation must hold.

Exercise

For $k=\Lambda=0$ and constant $w>-1$, derive the power-law exponent.

Solution

Conservation gives $\rho=\rho_0a^{-3(1+w)}$. The first Friedmann equation gives $\dot a/a=C a^{-3(1+w)/2}$, so $a^{(1+3w)/2}\dot a=C$. Integration gives $a^{3(1+w)/2}\propto t-t_0$, hence $a\propto(t-t_0)^{2/[3(1+w)]}$.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models.” In Theoretical and Observational Cosmology, edited by Marc Lachièze-Rey, 1–116. Dordrecht: Springer, 1999. DOI; Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.