---
title: The Expanding Universe and Hubble's Law
draft: false
module: Cosmic Expansion and Dynamics
moduleNumber: 11
lessonNumber: 1
order: 1101
summary: >
  The universe is homogeneous and isotropic on large scales, so its expansion is
  captured by a single function of time, the scale factor. Comoving coordinates
  stay fixed while proper distances grow in proportion to the scale factor,
  producing Hubble's law and a cosmological redshift that measures stretched space
  rather than a Doppler shift. A Newtonian energy argument reproduces the dynamics,
  and the same finite, expanding cosmos resolves Olbers' paradox.
topics: [Cosmology]
sources:
  - book: Ryden
    ref: "Ch. 2 — Fundamental Observations; Ch. 3 — Newtonian Cosmology"
  - book: Carroll & Ostlie
    ref: "Ch. 29 — Cosmology; §29.1 The Cosmological Principle and the Hubble Law"
  - book: Maoz
    ref: "Ch. 8 — Tests and Probes of Big Bang Cosmology"
---

Cosmology is the attempt to treat the universe as a single physical system whose
history and structure follow from the same laws that govern a laboratory. The
attempt only becomes tractable because the universe, viewed on scales larger than
a few hundred megaparsecs, reduces to something simple: it looks the same everywhere and
in every direction. That simplicity, elevated to a working assumption, reduces the
entire spatial geometry and its time evolution to a single function, the scale
factor $a(t)$. This lesson sets up the cosmological principle, defines comoving and
proper distance in terms of the scale factor, derives Hubble's law and the
cosmological redshift as direct consequences, recovers the expansion dynamics from a
Newtonian energy argument, and closes with Olbers' paradox, whose resolution was the
first hint that the universe is neither static nor infinitely old.

## The cosmological principle

The foundational assumption of modern cosmology is the **cosmological principle**:
on sufficiently large scales the universe is **homogeneous** (the same at every
place) and **isotropic** (the same in every direction). Homogeneity is invariance
under translation; isotropy about every point is the stronger statement, and
isotropy holding everywhere implies homogeneity. The two are logically distinct — a
universe with a uniform magnetic field is homogeneous but not isotropic — but the
observed universe appears to satisfy both.[^ry-cosmo]

The evidence is empirical, not a priori. Redshift surveys such as the Sloan Digital
Sky Survey show that galaxies cluster into filaments and voids up to scales of order
$100\ \text{Mpc}$, but that averaging over larger volumes washes the structure out:
the mean density in one $300\ \text{Mpc}$ box matches another to a fraction of a
percent. Isotropy is even sharper. The cosmic microwave background, the relic
radiation from the hot early universe, has the same temperature in every direction
to one part in $10^{5}$ once the dipole from our own motion is removed. A homogeneous
and isotropic distribution of matter with an isotropic radiation bath is what the
cosmological principle asserts.

> **Definition (Cosmological principle).** On scales large compared to the largest
> structures ($\gtrsim 100\ \text{Mpc}$), the universe is homogeneous and isotropic:
> no location and no direction is special. There is no center and no edge.

$$
% caption: On small scales the galaxy distribution is clumpy, but averaged over
% large volumes the mean density in any box matches any other: statistical
% homogeneity.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left panel: clumpy small scale
\draw[black] (0,0) rectangle (3.6,3.6);
\node[black, anchor=south] at (1.8,3.65) {small scale: clumpy};
\foreach \x/\y in {0.5/0.6, 0.7/0.9, 0.9/0.5, 1.2/2.7, 1.4/2.9, 2.9/1.1, 3.0/1.4, 2.7/1.2, 1.9/2.0, 2.2/0.4, 0.4/3.1, 3.2/3.0, 3.1/0.5, 0.6/2.4, 2.5/3.2}
  \fill[black] (\x,\y) circle (0.055);
% right panel: coarse-grained boxes uniform
\draw[black] (5.0,0) rectangle (8.6,3.6);
\node[black, anchor=south] at (6.8,3.65) {large scale: uniform};
\foreach \x in {5.0,5.9,6.8,7.7}
  \draw[black] (\x,0) -- (\x,3.6);
\foreach \y in {0,0.9,1.8,2.7}
  \draw[black] (5.0,\y) -- (8.6,\y);
\foreach \cx in {5.45,6.35,7.25,8.15}
  \foreach \cy in {0.45,1.35,2.25,3.15}
    \fill[black] (\cx,\cy) circle (0.16);
\node[black, anchor=north] at (6.8,-0.15) {equal mean density per cell};
\end{tikzpicture}
$$

## The scale factor and comoving coordinates

A homogeneous, isotropic universe can expand or contract, but it can do so only in
one way: uniformly, preserving the shape of every configuration while rescaling all
distances by a common time-dependent factor. Any position-dependent rescaling would
introduce a preferred location and violate homogeneity. This single degree of freedom
is the **scale factor** $a(t)$, a dimensionless function of cosmic time normalized so
that its present value is $a(t_0) = a_0 = 1$.

To separate the expansion from the fixed pattern of galaxies, introduce **comoving
coordinates**. Assign each galaxy a coordinate $\vec{x}$ that does not change as the
universe expands — the galaxies are at rest in the comoving frame, apart from small
peculiar velocities. The physical, or **proper**, distance between two comoving
points separated by comoving distance $x$ is

$$
d(t) = a(t)\, x .
$$

The comoving separation $x$ is a fixed label; all of the time dependence lives in
$a(t)$. A useful picture is a rubber sheet or a rising loaf of raisin bread: the
raisins keep their coordinates in the dough, but the dough stretches and the physical
distance between any two raisins grows in proportion to the expansion.

$$
% caption: Comoving coordinates label galaxies on a grid that stretches with the
% scale factor. The grid labels are fixed; proper distances grow as a(t).
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% early grid (small a)
\begin{scope}[shift={(0,0)}]
\foreach \i in {0,1,2,3}{
  \draw[black] (\i*0.55,0) -- (\i*0.55,1.65);
  \draw[black] (0,\i*0.55) -- (1.65,\i*0.55);
}
\foreach \i in {0,1,2,3}\foreach \j in {0,1,2,3}
  \fill[black] (\i*0.55,\j*0.55) circle (0.045);
\node[black, anchor=north] at (0.82,-0.2) {early: a small};
\end{scope}
% arrow
\draw[->, black, thick] (2.4,0.82) -- (3.4,0.82);
\node[black, anchor=south] at (2.9,0.9) {expand};
% late grid (large a)
\begin{scope}[shift={(4.0,-0.5)}]
\foreach \i in {0,1,2,3}{
  \draw[black] (\i*0.9,0) -- (\i*0.9,2.7);
  \draw[black] (0,\i*0.9) -- (2.7,\i*0.9);
}
\foreach \i in {0,1,2,3}\foreach \j in {0,1,2,3}
  \fill[black] (\i*0.9,\j*0.9) circle (0.045);
\node[black, anchor=north] at (1.35,-0.2) {late: a large};
\end{scope}
\end{tikzpicture}
$$

## Hubble's law

Differentiate the proper distance with respect to cosmic time. Because the comoving
separation $x$ is constant,

$$
\dot d = \dot a\, x = \frac{\dot a}{a}\,(a x) = \frac{\dot a}{a}\, d .
$$

The rate of change of proper distance — the **recession velocity** of one comoving
point relative to another — is proportional to the proper distance between them. The
proportionality constant is the **Hubble parameter**

$$
H(t) \equiv \frac{\dot a}{a},
$$

so that $v = H(t)\, d$. Evaluated at the present epoch this is **Hubble's law**,

$$
v = H_0\, d , \qquad H_0 = \left.\frac{\dot a}{a}\right|_{t_0},
$$

with $H_0$ the **Hubble constant**. Note that $H_0$ is a constant in space, not in
time: the subscript denotes its value now, and $H(t)$ generally changes over cosmic
history. The linear relation is exact and unavoidable for uniform expansion; any other
distance dependence would single out an origin. Every observer riding along with the
expansion sees exactly the same law with the same $H_0$, so the recession is not
evidence that we sit at a center. It is not motion through space but the growth of
space between comoving observers.[^ry-hubble]

Hubble measured the relation in 1929 from the redshifts and estimated distances of
nearby galaxies, finding the recession velocity proportional to distance. His slope
was some seven times too steep because his distance calibration was in error, but the
linearity was correct. The modern value, from the Cepheid–supernova distance ladder
and from the CMB, is

$$
H_0 \approx 67\text{–}73\ \text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1},
$$

the range reflecting a persistent tension between early-universe and late-universe
determinations that later lessons return to. The units make the meaning transparent: a
galaxy $1\ \text{Mpc}$ away recedes at about $70\ \text{km}\,\text{s}^{-1}$, one at
$100\ \text{Mpc}$ at about $7000\ \text{km}\,\text{s}^{-1}$.

$$
% caption: Recession velocity rises linearly with distance. Nearby galaxies define
% the slope H0; deep data curve away as H changes over lookback time.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {distance};
\draw[->, black] (0,0) -- (0,4.3) node[above, black!70] {recession velocity};
% linear Hubble law
\draw[acc, very thick] (0,0) -- (6.3,3.8);
\node[acc, anchor=west] at (5.0,3.4) {v = H0 d};
% scatter points near the line
\foreach \x/\y in {0.7/0.5, 1.3/0.7, 1.9/1.25, 2.6/1.4, 3.2/2.05, 3.9/2.2, 4.5/2.85, 5.1/2.9}
  \fill[black] (\x,\y) circle (0.06);
% deep-data deviation (dashed) curving below
\draw[black, densely dashed, thick] (4.2,2.5) .. controls (5.6,3.1) and (6.8,3.3) .. (7.8,3.4);
\node[black, anchor=south west] at (6.2,2.75) {deep: H(t) dif\/fers};
\end{tikzpicture}
$$

## Cosmological redshift

A photon travelling from a distant galaxy to us moves through space that is
stretching, and its wavelength stretches with it. Consider a light wave emitted at
time $t_e$ with wavelength $\lambda_e$ and observed now at $t_0$ with wavelength
$\lambda_0$. Successive wave crests are emitted a comoving distance apart; as they
travel, that comoving separation is fixed, but the proper wavelength scales with
$a(t)$. Carrying the argument through the null geodesic of the photon (done rigorously
from the metric in the next lesson) gives the exact result

$$
\frac{\lambda_0}{\lambda_e} = \frac{a(t_0)}{a(t_e)} = \frac{a_0}{a} .
$$

Defining the **redshift** $z$ through the fractional wavelength shift,

$$
1 + z \equiv \frac{\lambda_0}{\lambda_e} = \frac{a_0}{a(t_e)} = \frac{1}{a(t_e)} ,
$$

using $a_0 = 1$. This is the single most important relation in observational
cosmology: a measured redshift is a **direct readout of the scale factor** at the
time the light was emitted. A galaxy observed at $z = 1$ emitted its light when the
universe was half its present size; the CMB at $z \approx 1100$ shows us the universe
when it was about a thousandth of its current scale.

It is essential to distinguish this **cosmological redshift** from a Doppler shift.
The galaxy is essentially at rest in comoving coordinates; it is not moving through
space away from us in the ordinary kinematic sense. The wavelength grew because the
space the photon traversed grew. For small distances the two pictures agree
numerically — expanding $1 + z = a_0/a(t_e)$ to first order in lookback time recovers
$z \approx H_0 d / c$, the Doppler form $v = cz$ — but at large $z$ the naive Doppler
interpretation fails, and redshifts $z > 1$ do not imply superluminal motion.

> **Definition (Cosmological redshift).** The stretching of a photon's wavelength by
> the expansion of space, $1 + z = a_0/a(t_e)$. The redshift measures the scale factor
> at emission, not a velocity through space.

$$
% caption: As the wave propagates through expanding space, each wavelength stretches
% in proportion to the scale factor, so the observed wavelength exceeds the emitted.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% emitted short wave
\draw[very thick] plot[domain=0:3.4, samples=120] (\x, {1.9 + 0.35*sin(deg(\x*7))});
\node[anchor=south] at (1.7,2.45) {emitted: short wavelength};
% observed long wave
\draw[very thick] plot[domain=0:3.4, samples=120] (\x, {0.0 + 0.35*sin(deg(\x*3))});
\node[anchor=north] at (1.7,-0.45) {observed: long wavelength};
% arrow between
\draw[->, black, thick] (3.7,1.9) -- (3.7,0.2);
\node[black, anchor=west] at (3.85,1.05) {space stretches};
\end{tikzpicture}
$$

## A Newtonian derivation of expansion dynamics

The full dynamics of $a(t)$ follow from general relativity, but a Newtonian argument
reproduces the same equation for pressureless matter and makes the physics
transparent. This works because of a theorem due to Newton and Birkhoff: in a
spherically symmetric mass distribution the gravitational field at radius $r$ depends
only on the mass interior to $r$, and the exterior shells exert no net force. In a
homogeneous universe we may therefore isolate a sphere, analyze it with Newtonian
gravity, and appeal to homogeneity to argue the result holds everywhere.

Consider a sphere of comoving radius $x$ centered on an arbitrary point, with proper
radius $R(t) = a(t)\, x$ and enclosing a fixed mass $M = \tfrac{4}{3}\pi R^3 \rho$.
A test galaxy of mass $m$ on the surface feels only the interior mass. Its energy —
kinetic plus gravitational potential — is conserved:

$$
E = \frac{1}{2} m \dot R^{2} - \frac{G M m}{R}
  = \frac{1}{2} m \dot R^{2} - \frac{4\pi}{3} G \rho R^{2} m .
$$

Substitute $R = a x$ and $\dot R = \dot a\, x$, and divide by $\tfrac{1}{2} m x^{2}$:

$$
\left(\frac{\dot a}{a}\right)^{2}
  = \frac{8\pi G}{3}\,\rho - \frac{2E}{m x^{2}}\,\frac{1}{a^{2}} .
$$

The constant $-2E/(m x^{2})$ must, by homogeneity, be independent of which sphere we
chose; write it as $-k c^{2}$ with $k$ a constant. Then

$$
\left(\frac{\dot a}{a}\right)^{2} = \frac{8\pi G}{3}\,\rho - \frac{k c^{2}}{a^{2}} ,
$$

which is precisely the **Friedmann equation**, derived rigorously from general
relativity in a later lesson. The Newtonian energy constant $E$ reappears there as
the spatial curvature: a bound sphere ($E < 0$, $k > 0$) corresponds to a closed
universe that recollapses, an unbound one ($E > 0$, $k < 0$) to an open universe that
expands forever, and the marginal case ($E = 0$, $k = 0$) to a spatially flat universe
poised between them. The Newtonian picture correctly captures the matter-dominated
dynamics; it misses the pressure contribution to gravity and the behaviour of
radiation and vacuum energy, which require the relativistic treatment.[^ry-newton]

$$
% caption: A test galaxy on a comoving sphere feels only the interior mass. Its
% conserved energy fixes whether expansion halts and reverses or continues forever.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% sphere
\draw[black] (0,0) circle (2.2);
\fill[acc!10] (0,0) circle (2.2);
\node[black, anchor=center] at (0,0.0) {interior mass M};
% center
\fill[black] (0,0) circle (0.05);
% test galaxy on surface
\fill[acc] (2.2,0) circle (0.09);
\node[acc, anchor=west] at (2.35,0.15) {test galaxy m};
% radius
\draw[->, black] (0,0) -- (2.15,0);
\node[black, anchor=north] at (1.1,-0.1) {R = a x};
% velocity arrow outward
\draw[->, acc, very thick] (2.3,0.35) -- (3.15,0.35);
\node[acc, anchor=south] at (2.75,0.42) {recession};
% gravity arrow inward
\draw[->, black, thick] (2.3,-0.35) -- (1.55,-0.35);
\node[black, anchor=north] at (1.9,-0.4) {gravity};
\end{tikzpicture}
$$

## Olbers' paradox

Why is the sky dark at night? In a universe that is infinite, static, and eternally
old, every line of sight would eventually terminate on the surface of a star, and the
whole sky would blaze with the surface brightness of a stellar photosphere. The
argument is quantitative. In a homogeneous static universe of number density $n$ of
stars each of radius $R_\star$, the number of stars in a shell of radius $r$ and
thickness $\d r$ is $4\pi r^{2} n\,\d r$. Each subtends a solid angle $\propto
R_\star^{2}/r^{2}$, so the light received from the shell, or equivalently the sky area
it covers, is independent of $r$: the falloff of flux as $1/r^{2}$ exactly cancels the
growth of shell volume as $r^{2}$. Integrating over all shells to infinity gives a
divergent total — an infinitely bright sky. Even accounting for nearer stars blocking
farther ones, the whole sky should shine at stellar surface brightness.

The dark night sky therefore rules out the infinite–static–eternal universe. The
resolution has two parts, both supplied by an expanding universe of finite age. First,
the universe has a finite age $t_0$, so light can only have reached us from within a
finite horizon of radius $\sim c\, t_0$; more distant stars have not had time to send
us their light, truncating the integral. Second, the expansion redshifts distant
starlight, so photons from remote sources arrive with reduced energy and at a reduced
rate, dimming the far contributions further. The finite age is the dominant effect: it
is not that there is too little matter, but that we see only a finite, and only a
finitely old, portion of it. Olbers' paradox is thus the earliest and simplest
cosmological observation, and its resolution already contains the two ingredients —
finite age and expansion — that the rest of this module develops quantitatively.[^ry-olbers]

$$
% caption: In a static infinite universe every sightline ends on a star, so the sky
% should blaze. A finite horizon set by the age of the universe truncates the sum.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% observer
\fill[acc] (0,0) circle (0.09);
\node[acc, anchor=north] at (0,-0.15) {observer};
% horizon circle
\draw[acc!55, thick, densely dashed] (0,0) circle (3.2);
\node[acc, anchor=south] at (0,3.25) {horizon = c times age};
% stars inside (reachable)
\foreach \x/\y in {1.1/0.6, -1.4/0.9, 0.6/-1.5, -0.9/-1.2, 1.8/-0.7, -1.9/-0.3, 0.3/1.9, 2.2/1.1, -0.5/2.3, 1.5/1.8}
  \fill[black] (\x,\y) circle (0.06);
% stars outside (unreachable, faint)
\foreach \x/\y in {3.8/1.2, -3.9/0.6, 2.4/-3.0, -2.7/-2.6, 4.0/-0.8, -1.2/3.8}
  \fill[black] (\x,\y) circle (0.06);
\node[black, anchor=west] at (3.4,-1.6) {beyond horizon: unseen};
% a sightline that would end on a distant star but is cut off
\draw[black, densely dotted] (0,0) -- (3.1,0.35);
\end{tikzpicture}
$$

## Summary

The cosmological principle — large-scale homogeneity and isotropy — reduces the
expanding universe to a single function of time, the scale factor $a(t)$, normalized
to $a_0 = 1$ today. Proper distances scale as $d = a x$ with comoving separation $x$
fixed, and differentiating gives Hubble's law $v = H d$ with $H = \dot a/a$, a linear
relation that holds for every comoving observer and implies no center. The wavelength
of light stretches with the same factor, so the cosmological redshift measures the
scale factor at emission, $1 + z = 1/a(t_e)$, and is not a Doppler shift through space.
A Newtonian energy argument on a comoving sphere reproduces the Friedmann equation
$(\dot a/a)^2 = 8\pi G\rho/3 - kc^2/a^2$, with the energy constant reappearing as
spatial curvature. Olbers' paradox — the dark night sky — already excludes an
infinite, static, eternal universe and points to the finite age and expansion this
module now makes precise. The next lesson replaces the Newtonian sphere with the
[Robertson–Walker
metric](/astrophysics-cosmology/cosmology-expansion-and-dynamics/the-frw-metric-and-cosmological-redshift),
deriving the geometry, the distances, and the redshift rigorously.

[^ry-cosmo]: Ryden, *Introduction to Cosmology*, Ch. 2 — Fundamental Observations: the evidence for large-scale homogeneity and isotropy.
[^ry-hubble]: Ryden, Ch. 2 — the Hubble law $v = H_0 d$ as a consequence of uniform expansion; Carroll & Ostlie §29.1.
[^ry-newton]: Ryden, Ch. 3 — Newtonian Cosmology: the energy-conservation derivation of the Friedmann equation and the identification of the curvature constant.
[^ry-olbers]: Ryden, Ch. 2 — Olbers' paradox and its resolution by the finite age and expansion of the universe.