The Cosmological Principle and the FLRW Metric

General relativity applied to the whole universe needs an input that no local experiment supplies: the large-scale distribution of matter. The machinery of the preceding modules — the metric, geodesics, the Einstein equation — is local, valid in any patch. Extending it to a cosmological model requires a statement about how matter is arranged on scales far larger than any galaxy. That statement is the cosmological principle, and it is strong enough to fix the metric up to one function of time and one discrete choice of spatial curvature.

The cosmological principle

Two symmetry assumptions define the standard model of the universe.

• Homogeneity: at a fixed cosmic time the universe looks the same at every spatial point. No location is special. Formally, there is a symmetry that maps any point of a constant-time slice to any other.
• Isotropy: at a fixed cosmic time the universe looks the same in every direction from a given point. No direction is special.

The two are independent. A universe with a uniform magnetic field threading it would be homogeneous but not isotropic; a universe with a single density peak would be isotropic about that peak but not homogeneous. Isotropy about every point does imply homogeneity, and this is the version the data support: observers at widely separated galaxies would each see an isotropic sky.

These are statements about averages on scales of order hundreds of megaparsecs, not about the clumpy distribution of stars and galaxies. On smaller scales the universe is manifestly inhomogeneous. The cosmic microwave background is isotropic to about one part in $10^5$ after the dipole from our own motion is removed, and deep galaxy surveys show the same statistical distribution in every direction. The cosmological principle promotes these observations to an exact symmetry of the model, to be corrected perturbatively when structure is added.¹

Spatial slices of constant curvature

Homogeneity and isotropy are symmetries of space at each instant of cosmic time. A three-dimensional space that is homogeneous and isotropic is maximally symmetric: it has the largest possible symmetry group, six independent isometries (three translations, three rotations). A maximally symmetric space has constant curvature, the same value of the Ricci scalar everywhere, and there are exactly three cases distinguished by the sign of that curvature.

Write the spatial line element on a slice using a radial coordinate. The isotropic form is

where $k$ has been normalized to one of three values by rescaling $r$:

• $k=+1$: a closed space, the three-sphere $S^3$, with finite volume and no boundary. Angles of a large triangle sum to more than $\pi$.
• $k=0$: a flat space, ordinary Euclidean ${\mathbb{R}}^3$, infinite in extent. Triangle angles sum to exactly $\pi$.
• $k=-1$: an open space, the hyperbolic three-space $H^3$, also infinite. Triangle angles sum to less than $\pi$.

An equivalent and often cleaner form uses a dimensionless radial coordinate $\chi$ measuring geodesic distance in units of the curvature radius,

The function $S_k(\chi )$ is the area radius: the surface area of a sphere at geodesic distance $\chi$ is $4\pi S_k(\chi {)}^2$. In the flat case this is the familiar $4\pi {\chi }^2$. In the closed case the area grows, reaches a maximum at $\chi =\pi /2$, then shrinks back to zero at the antipode $\chi =\pi$, the geometry of a three-sphere. In the open case the area grows faster than Euclidean, $4\pi {\sinh }^2\chi$.²

The FLRW metric

The full spacetime adds time and allows the spatial slice to scale with cosmic time. Homogeneity forbids any position dependence in the scaling, so a single function $a(t)$ multiplies the entire spatial metric. Isotropy forbids any cross term $\mathbf{d}t\mathbf{d}{x}^i$, which would pick out a spatial direction, and lets the lapse between slices be set to unity by choosing proper time of comoving observers as the time coordinate. The result is the Friedmann–Lemaître–Robertson–Walker (FLRW) metric,

Its ingredients each carry a definite meaning.

• Cosmic time $t$ is the proper time read by an observer at rest in the comoving coordinates. All such observers share a common time because the homogeneous slices provide a preferred synchronization: the surfaces of constant $t$ are the surfaces of constant density.
• Comoving coordinates $(r,\theta ,\phi )$ label a galaxy by a fixed address that does not change as the universe expands. A galaxy with no peculiar motion keeps constant $(r,\theta ,\phi )$ for all time.
• The scale factor $a(t)$ carries all the time dependence. It is dimensionless when $r$ is given the dimension of length, or has the dimension of length when $r$ is an angle-like comoving coordinate; the convention here keeps $a$ dimensionless and normalized so that $a(t_0)=1$ today, with $t_0$ the present cosmic time.
• The curvature index $k\in \{-1,0,+1\}$ fixes the spatial geometry once and for all. Expansion rescales distances but cannot change the sign of the spatial curvature.

The metric is spatially curved through $k$ and dynamically evolving through $a(t)$, but at each instant it is perfectly uniform. The one function $a(t)$ and the one number $k$ are all the freedom the cosmological principle leaves; the Einstein equation, taken up in the next lesson, determines $a(t)$ from the matter content.

Comoving and proper distance

The distinction between the fixed comoving label and the growing physical separation is the central bookkeeping of cosmology. Consider two galaxies on a radial line, one at the origin and one at comoving coordinate $\chi$ (using the $S_k$ form). The proper distance between them at cosmic time $t$, measured along the constant-$t$ slice, is the spatial line element integrated at fixed time,

The comoving distance $\chi$ is fixed; the proper distance grows in proportion to $a(t)$. Differentiating gives the recession velocity of the distant galaxy,

This is the Hubble law: recession velocity is proportional to proper distance, with the proportionality constant the Hubble parameter $H(t)$. Its present value is the Hubble constant $H_0=H(t_0)$. The law is exact in the FLRW model, not an approximation, and it holds for every comoving observer, so it does not single out a center. The recession is not motion of galaxies through space but the growth of the space between them; galaxies at rest in comoving coordinates have no peculiar velocity.

At large enough $d_p$ the recession speed ${\dot{d}}_p=H{d}_p$ formally exceeds $c$. This is not a violation of special relativity, which limits relative speeds of objects passing each other at the same event. Comoving galaxies are not at the same event, and their separation speed is a rate of change of a distance defined on a curved spacetime, not a local relative velocity. The surface of constant recession speed $c$, at $d_p=c/H$, is the Hubble radius; it is a rate scale, not a horizon in the causal sense.

Cosmological redshift

Light propagating through the expanding universe stretches with it. Consider a radial light ray from a distant galaxy at comoving coordinate ${\chi }_e$ emitted at time $t_e$ and received at the origin at time $t_0$. A radial null geodesic has $\mathbf{d}{s}^2=0$ with $\mathbf{d}\theta =\mathbf{d}\phi =0$, so in the $S_k$ coordinates

for an incoming ray. Integrating from emission to reception,

The right side is the fixed comoving coordinate of the source. Now follow a second wave crest emitted one period later, at $t_e+\delta t_e$, and received at $t_0+\delta t_0$. Its comoving distance is the same ${\chi }_e$, so

Subtracting the two integrals leaves the small end intervals, and over one period the scale factor is essentially constant, so

The intervals $\delta t$ are the wave periods, $\delta t=\lambda /c$. The observed and emitted wavelengths therefore satisfy

Define the cosmological redshift $z$ by $1+z={\lambda }_0/{\lambda }_e$. Then

using the normalization $a(t_0)=1$. Redshift measures the factor by which the universe has expanded since the light was emitted. Light from $z=1$ left when the universe was half its present size; light from the surface of last scattering at $z\approx 1100$ left when the universe was about a thousand times smaller. The wavelength is stretched in step with the scale factor, not by a Doppler shift of the source through space nor by a gravitational potential difference, though it reduces to the Doppler formula for nearby sources where $z\approx H_0{d}_p/c$.³

The observed universe as an FLRW model

The FLRW metric organizes the primary cosmological observables into a single geometric frame.

• Redshift replaces emission time as the natural clock: $a(t_e)=1/(1+z)$ labels an epoch directly by what a spectrograph measures.
• Proper distance at the present epoch, $d_p(t_0)={\chi }_e$, is not directly observable; distances are inferred through the redshift–distance relation set by $a(t)$, which the next lesson computes from the dynamics.
• The spatial curvature $k$ enters observables only through $S_k(\chi )$ in areas and angles, so it is measured by comparing the apparent size or number density of distant standard objects against a flat-space expectation. Current data are consistent with $k=0$ to within a few percent.

Everything so far is kinematics: the metric follows from symmetry alone, and redshift and distance follow from the metric, but the time dependence of $a(t)$ is undetermined. Fixing $a(t)$ requires the Einstein equation with a matter source, which converts the geometric scale factor into a dynamical variable obeying the Friedmann equations.