---
title: The Cosmological Principle and the FLRW Metric
module: A Bridge to Cosmology
moduleNumber: 10
lessonNumber: 1
order: 1001
summary: >
  Homogeneity and isotropy restrict the spacetime of the universe to a single
  family of metrics: a flat cosmic-time slicing of spatial sections of constant
  curvature, scaled by a time-dependent factor a(t). This lesson builds the
  Friedmann–Lemaître–Robertson–Walker metric from those symmetries, separates
  comoving from proper distance, and derives cosmological redshift as the
  stretching of wavelengths with the scale factor.
topics: [A Bridge to Cosmology]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 18 — Cosmological Models"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §8 — Cosmology, arXiv:gr-qc/9712019"
---

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.[^hartle-cp]

$$
% caption: Homogeneity moves any point to any other; isotropy rotates any
% direction into any other. A single uniform field would break isotropy without
% breaking homogeneity, so the two conditions are logically independent.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % homogeneous isotropic panel
  \foreach \x in {0,0.7,1.4,2.1} {
    \foreach \y in {0,0.7,1.4,2.1} {
      \fill (\x,\y) circle (0.055);
    }
  }
  \node[black, anchor=north] at (1.05,-0.35) {homogeneous and isotropic};
  \draw[->, black] (1.05,1.05) -- (1.75,1.05);
  \draw[->, black] (1.05,1.05) -- (1.05,1.75);
  \draw[->, black] (1.05,1.05) -- (0.55,0.55);
  % anisotropic panel
  \begin{scope}[shift={(4.6,0)}]
    \foreach \x in {0,0.7,1.4,2.1} {
      \foreach \y in {0,0.7,1.4,2.1} {
        \fill[black] (\x,\y) circle (0.055);
        \draw[acc, thick, ->] (\x,\y) -- (\x,\y+0.42);
      }
    }
    \node[black, anchor=north] at (1.05,-0.35) {aligned arrows break isotropy};
  \end{scope}
\end{tikzpicture}
$$

## 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

$$
\d\sigma^2 = \frac{\d r^2}{1 - k r^2} + r^2\left(\d\theta^2 + \sin^2\theta\,\d\varphi^2\right),
$$

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,

$$
\d\sigma^2 = \d\chi^2 + S_k(\chi)^2\left(\d\theta^2 + \sin^2\theta\,\d\varphi^2\right),
\qquad
S_k(\chi) =
\begin{cases}
\sin\chi, & k = +1,\\[2pt]
\chi, & k = 0,\\[2pt]
\sinh\chi, & k = -1.
\end{cases}
$$

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, $4\pi\sinh^2\chi$.[^carroll-flrw]

$$
% caption: The three constant-curvature spatial geometries read off from the
% angle sum of a large triangle: positive curvature closes the space and swells
% the angle sum above pi, flat space keeps it at pi, negative curvature opens the
% space and drops it below pi.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % closed: bulging triangle
  \draw[thick] (0,0) to[bend left=22] (1.6,0) to[bend left=22] (0.8,1.5) to[bend left=22] (0,0);
  \node[black, anchor=north] at (0.8,-0.3) {closed: k = +1};
  \node[black, anchor=north] at (0.8,-0.72) {angles sum above 180};
  % flat: straight triangle
  \begin{scope}[shift={(3.4,0)}]
    \draw[thick] (0,0) -- (1.6,0) -- (0.8,1.5) -- cycle;
    \node[black, anchor=north] at (0.8,-0.3) {Euclidean: k = 0};
    \node[black, anchor=north] at (0.8,-0.72) {angles sum to 180};
  \end{scope}
  % open: pinched triangle
  \begin{scope}[shift={(6.8,0)}]
    \draw[thick] (0,0) to[bend right=22] (1.6,0) to[bend right=22] (0.8,1.5) to[bend right=22] (0,0);
    \node[black, anchor=north] at (0.8,-0.3) {open: k = -1};
    \node[black, anchor=north] at (0.8,-0.72) {angles sum below 180};
  \end{scope}
\end{tikzpicture}
$$

## 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 $\d t\,\d x^i$, 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**,

$$
\d s^2 = -c^2\,\d t^2 + a(t)^2\left[\frac{\d r^2}{1 - k r^2}
+ r^2\left(\d\theta^2 + \sin^2\theta\,\d\varphi^2\right)\right].
$$

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** $(r,\theta,\varphi)$ label a galaxy by a fixed
  address that does not change as the universe expands. A galaxy with no
  peculiar motion keeps constant $(r,\theta,\varphi)$ 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.

$$
% caption: Comoving galaxies keep fixed coordinate labels while the scale factor
% a(t) multiplies every spatial separation; the physical distance between any two
% galaxies is their fixed comoving separation times a(t).
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % early grid
  \foreach \x in {0,0.55,1.1,1.65} {
    \draw[black] (\x,0) -- (\x,1.65);
    \draw[black] (0,\x) -- (1.65,\x);
  }
  \fill[acc] (0.55,0.55) circle (0.06);
  \fill[acc] (1.1,1.1) circle (0.06);
  \fill[acc] (1.65,0.55) circle (0.06);
  \node[black, anchor=north] at (0.82,-0.3) {early: small a};
  \draw[->, black, thick] (2.1,0.82) -- (3.2,0.82);
  \node[black, anchor=south] at (2.65,0.85) {expand};
  % late grid
  \begin{scope}[shift={(3.7,-0.35)}]
    \foreach \x in {0,0.85,1.7,2.55} {
      \draw[black] (\x,0) -- (\x,2.55);
      \draw[black] (0,\x) -- (2.55,\x);
    }
    \fill[acc] (0.85,0.85) circle (0.06);
    \fill[acc] (1.7,1.7) circle (0.06);
    \fill[acc] (2.55,0.85) circle (0.06);
    \node[black, anchor=north] at (1.27,-0.3) {late: large a};
  \end{scope}
\end{tikzpicture}
$$

## 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,

$$
d_p(t) = \int_0^\chi a(t)\,\d\chi' = a(t)\,\chi.
$$

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

$$
\dot d_p = \dot a\,\chi = \frac{\dot a}{a}\,(a\chi) = H(t)\,d_p,
\qquad
H(t) \equiv \frac{\dot a}{a}.
$$

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
$\d s^2 = 0$ with $\d\theta = \d\varphi = 0$, so in the $S_k$ coordinates

$$
c^2\,\d t^2 = a(t)^2\,\d\chi^2
\quad\Longrightarrow\quad
\frac{c\,\d t}{a(t)} = -\,\d\chi
$$

for an incoming ray. Integrating from emission to reception,

$$
\int_{t_e}^{t_0} \frac{c\,\d t}{a(t)} = \int_0^{\chi_e} \d\chi = \chi_e.
$$

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

$$
\int_{t_e + \delta t_e}^{t_0 + \delta t_0} \frac{c\,\d t}{a(t)} = \chi_e
= \int_{t_e}^{t_0} \frac{c\,\d t}{a(t)}.
$$

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

$$
\frac{\delta t_0}{a(t_0)} = \frac{\delta t_e}{a(t_e)}.
$$

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

$$
\frac{\lambda_0}{\lambda_e} = \frac{a(t_0)}{a(t_e)}.
$$

Define the **cosmological redshift** $z$ by $1 + z = \lambda_0 / \lambda_e$.
Then

$$
1 + z = \frac{a(t_0)}{a(t_e)} = \frac{1}{a(t_e)},
$$

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
$z \approx H_0 d_p / c$.[^hartle-redshift]

$$
% caption: A wave crest and the next crest travel the same comoving distance, so
% the received period scales with a(t); the wavelength is stretched by the factor
% by which the universe expanded during the light's travel, giving 1 plus z equal
% to a today over a at emission.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % emitted short wavelength
  \draw[acc, thick] (0,1.4)
    sin (0.25,1.7) cos (0.5,1.4) sin (0.75,1.1) cos (1.0,1.4)
    sin (1.25,1.7) cos (1.5,1.4) sin (1.75,1.1) cos (2.0,1.4);
  \node[black, anchor=east] at (-0.1,1.4) {emitted};
  \draw[<->, black] (0,0.95) -- (0.5,0.95);
  \node[black, anchor=north] at (0.25,0.92) {short};
  \draw[->, black, thick] (2.5,1.4) -- (3.6,1.4);
  \node[black, anchor=south] at (3.05,1.45) {expansion};
  % received long wavelength
  \begin{scope}[shift={(4.1,0)}]
    \draw[acc, thick] (0,1.4)
      sin (0.5,1.75) cos (1.0,1.4) sin (1.5,1.05) cos (2.0,1.4)
      sin (2.5,1.75) cos (3.0,1.4);
    \node[black, anchor=west] at (3.1,1.4) {received};
    \draw[<->, black] (0,0.85) -- (1.0,0.85);
    \node[black, anchor=north] at (0.5,0.82) {stretched};
  \end{scope}
\end{tikzpicture}
$$

## 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.

[^hartle-cp]: **Hartle**, _Gravity_, Ch. 18 — the cosmological principle, homogeneity and isotropy of the observed universe, and the construction of the homogeneous isotropic cosmological models.

[^carroll-flrw]: **Carroll**, _Lecture Notes on General Relativity_, §8 — maximally symmetric spatial sections, the three constant-curvature geometries, and the Robertson–Walker metric. arXiv:gr-qc/9712019.

[^hartle-redshift]: **Hartle**, _Gravity_, Ch. 18 — comoving and proper distance, the Hubble law in the FLRW model, and the derivation of cosmological redshift from radial null geodesics.