---
title: Galaxies, Cosmology, and the Evolving Universe
draft: false
module: Orientation
moduleNumber: 1
lessonNumber: 3
order: 103
summary: >
  Galaxies come in elliptical, spiral, and irregular forms, and their redshifts
  obey Hubble's law, evidence that space itself is expanding. The critical
  density and the density parameter decide whether the universe is open, flat, or
  closed; baryons, dark matter, and dark energy each contribute. The cosmic
  microwave background and primordial helium anchor the Big Bang, whose thermal
  history runs from inflation through nucleosynthesis to the atoms of today.
topics: [Galaxies, Hubble's law, The Big Bang]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 13 — Astrophysics and Cosmology; §13-6 Galaxies, §13-7 Cosmology and Gravitation, §13-8 Cosmology and the Evolution of the Universe"
---

Beyond the Milky Way, a barred spiral about $28{,}000\ \text{ly}$ across with the
Sun a third of the way out, lie roughly $10^{10}$ other galaxies. Their light
carries two facts that together define modern cosmology: galaxies fall into a few
structural types, and almost all of them are receding, with speed proportional to
distance. The second fact means the universe is expanding, and running that
expansion backward leads to the Big Bang.

## The interstellar medium

The space between the stars is not empty. The **interstellar medium (ISM)** holds
a thin distribution of dust, silicate and carbide grains a few hundred nanometers
across that absorb and scatter starlight; because blue scatters more than red,
distant starlight is dimmed and reddened. By mass the ISM is about 70% hydrogen,
mostly helium for the rest, with dust a few percent.

Cold hydrogen clouds, at $30$ to $150\ \text{K}$, sit in their ground state, yet
they can be mapped through the **21-cm line**. The hydrogen ground state is split
by the [hyperfine interaction](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm)
between the electron and proton magnetic moments: the antiparallel-spin state lies
below the parallel-spin state by only

$$
\Delta E = 5.9 \times 10^{-6}\ \text{eV}.
$$

A spin flip back to the ground state emits a photon of wavelength $21\ \text{cm}$.
The transition is extraordinarily slow for a single atom, but the clouds hold so
many atoms that radio telescopes see a steady 21-cm glow, tracing the gas.

## Classifying galaxies

Edwin Hubble resolved the "spiral nebulae" into separate galaxies in the 1920s
using Cepheid variables to measure their distances, showing them far outside the
Milky Way. He sorted them into a few classes.

$$
% caption: Hubble's galaxy classes split by shape; disk galaxies further divide
% into ordinary and barred spirals, with irregulars as the remainder.
\begin{tikzpicture}[scale=1.0, font=\footnotesize,
  bx/.style={draw, black, minimum height=7mm, inner sep=3pt, align=center, font=\scriptsize}]
\definecolor{acc}{HTML}{4A6FA5}
\node[bx, draw=acc, fill=acc!10] (g) at (0,0) {galaxies};
\node[bx] (e) at (3.2,1.6) {elliptical};
\node[bx] (d) at (3.2,0.0) {disk};
\node[bx] (i) at (3.2,-1.6) {irregular};
\draw[->, acc] (g) -- (e);
\draw[->, acc] (g) -- (d);
\draw[->, acc] (g) -- (i);
\node[bx] (s) at (6.6,0.6) {ordinary spiral};
\node[bx] (b) at (6.6,-0.6) {barred spiral};
\draw[->, black] (d) -- (s);
\draw[->, black] (d) -- (b);
\node[black, anchor=west, font=\scriptsize] at (7.9,-0.6) {Milky Way};
\end{tikzpicture}
$$

- **Ellipticals** are roundish, with little gas or dust, mostly old stars, and
  random rather than rotational stellar motion. Most are far smaller than spirals.
- **Spirals** (ordinary and barred) are rotating disks with substantial gas and
  dust and ongoing star formation. The Milky Way is a barred spiral.
- **Irregulars** lack regular shape and often hold much gas and dust.

A small fraction are **active galaxies**, where the core outshines all the stars
and varies fast enough to be less than a light-year across: Seyfert galaxies, radio
galaxies, and **quasars** (quasi-stellar objects). Quasars carry the largest
redshifts known, implying recession above $0.95c$ and power outputs near
$10^{40}\ \text{W}$, more than $10^{12}$ Suns from a region light-hours across.

## Hubble's law

Hubble found a linear relation between a galaxy's recession velocity $v$ and its
distance $r$:

$$
v = H_0 r, \qquad H_0 = 71 \pm 4\ \text{km/s per Mpc}.
$$

$$
% caption: Recession velocity rises linearly with distance; the slope of the
% Hubble line is the Hubble constant, and its reciprocal sets a timescale.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {distance};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {recession velocity};
\draw[acc, very thick] (0,0) -- (6.6,4.0);
% data points scattered along the line
\foreach \x/\y in {0.9/0.45, 1.8/1.2, 2.7/1.55, 3.6/2.3, 4.5/2.6, 5.4/3.35, 6.0/3.4}
  \fill[black] (\x,\y) circle (2pt);
\node[acc, anchor=east] at (4.2,3.6) {slope $= H_0$};
\end{tikzpicture}
$$

Because no location is special, an observer in any galaxy would measure the same
law: every galaxy recedes from every other, so space itself expands. Light
traveling through expanding space is stretched, giving the **cosmological
redshift**

$$
z = \frac{\lambda - \lambda_0}{\lambda_0} = \frac{\Delta\lambda}{\lambda_0}.
$$

This is not the [relativistic Doppler effect](/relativity/foundations/time-dilation-length-contraction);
it reflects the change in the scale of the universe between emission and
observation. Writing the scale factor as $R(t)$, the fractional wavelength change
equals the fractional change in scale, so

$$
\frac{R_{\text{observed}}}{R_{\text{emitted}}} = 1 + z.
$$

A galaxy at $z = 2$ is now three times farther than when its light left, and the
universe was then one-third its present size. Looking to higher $z$ is looking
further back in time; the reciprocal of the Hubble constant, the **Hubble time**
$1/H_0 \approx 1.3 \times 10^{10}\ \text{yr}$, estimates the age of the universe
when gravity is ignored.[^tl-gal]

## The critical density

Whether expansion continues forever depends on how strongly gravity pulls back,
which depends on the mass density. Consider a galaxy of mass $m$ at distance $R$,
enclosing total mass $M$. Its energy is

$$
E = \tfrac{1}{2} m v^2 - \frac{GMm}{R}.
$$

The galaxy escapes if $E \geq 0$ and falls back if $E < 0$, exactly as for a
projectile. Setting $E = 0$ and substituting $v = H_0 R$ and
$M = \tfrac{4}{3}\pi R^3 \rho$ gives the **critical density**:

$$
\rho_c = \frac{3 H_0^2}{8\pi G} \approx 10^{-26}\ \text{kg/m}^3,
$$

about five hydrogen atoms per cubic meter. The ratio to critical is the **density
parameter** $\Omega = \rho / \rho_c$, which fixes the fate and geometry of the
universe.

$$
% caption: The density parameter decides the outcome: below critical the
% universe expands forever, at critical it coasts, above critical it recollapses.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {time};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {scale of universe};
% open (accelerating up)
\draw[acc, very thick] (0,0.4) .. controls (2.5,1.6) and (4.5,3.2) .. (7.2,4.3);
\node[acc, anchor=south east, font=\scriptsize] at (6.7,4.15) {open};
% flat (coasting)
\draw[black, very thick] (0,0.4) .. controls (3.0,1.8) and (5.5,2.4) .. (7.2,2.7);
\node[black!70, anchor=west, font=\scriptsize] at (6.0,2.6) {critical};
% closed (recollapse)
\draw[black, very thick, dashed] (0,0.4) .. controls (2.2,2.0) and (3.6,2.1) .. (4.6,1.4)
  .. controls (5.2,1.0) and (5.6,0.5) .. (5.9,0.1);
\node[black!70, anchor=south, font=\scriptsize] at (3.4,2.15) {closed};
\end{tikzpicture}
$$

- $\Omega < 1$ (open): total energy positive, expansion continues forever.
- $\Omega = 1$ (flat): total energy zero, expansion coasts to a halt as
  $t \to \infty$.
- $\Omega > 1$ (closed): total energy negative, expansion reverses into a "Big
  Crunch".

Visible baryonic matter supplies only about 4% of $\rho_c$. Adding the
[dark matter](/particle-physics/beyond-standard-model/beyond-standard-model) inferred
from galactic and cluster dynamics raises the total to about 26%. The redshifts
of distant Type Ia supernovae are smaller than a decelerating universe predicts,
so expansion is accelerating; the repulsive **dark energy** driving it supplies
the remaining 74%.[^tl-cosmo]

## A Newtonian expansion model

The same energy accounting, applied to a thin spherical shell of comoving radius
$r$ in a homogeneous, isotropic universe, yields the equation governing
expansion. Only the enclosed mass $M_r = \tfrac{4}{3}\pi r^3 \rho$ acts on the
shell; the mass outside exerts no net force. Conservation of energy, written with
a constant $k$ of units of inverse length squared, gives

$$
v^2(t) = \frac{8\pi}{3} G \rho(t)\, r^2(t) - k c^2 r^2(t_0),
$$

and in terms of the **scale factor** $R(t)$ defined by $r(t) = R(t)\, r(t_0)$,

$$
H^2(t) - \frac{8\pi}{3} G \rho(t) = -\frac{k c^2}{R^2(t)}.
$$

The sign of $k$ sets the geometry: $k > 0$ closed, $k = 0$ flat, $k < 0$ open,
matching the density-parameter cases. The scale factor connects to redshift by
$1 + z = 1/R(t)$, with $R(t_0) = 1$ today.

General relativity adds the pressure through the **acceleration equation**,

$$
\frac{1}{R}\frac{\d^2 R}{\d t^2} = -\frac{4\pi G}{3 c^2} \left( \varepsilon c^2 + 3P \right),
$$

where $\varepsilon c^2$ is the energy density and $P$ the pressure. When a
negative-pressure component dominates, the acceleration is positive, the
mechanism behind both early inflation and today's dark-energy expansion. The best
current density parameters are $\Omega_\Lambda \approx 0.73$ and
$\Omega_m \approx 0.27$, close to flat.

## The Big Bang and the cosmic microwave background

Two 1960s discoveries fixed the **Big Bang** as the standard model, dating the
hot dense origin to $(13.7 \pm 0.2) \times 10^9$ years ago. First, distant radio
galaxies are more numerous than nearby ones, so the universe looked different at
earlier times and has evolved. Second, stellar fusion cannot account for the
cosmic abundance of helium; that helium must have formed in the hot early
universe, which implies a thermal radiation field that has since cooled.

That relic field is the **cosmic microwave background (CMB)**, predicted to cool
to a few kelvin and discovered by Penzias and Wilson in 1965. COBE and WMAP fixed
its temperature at

$$
T = 2.725 \pm 0.001\ \text{K},
$$

with a spectrum matching a [Planck blackbody](/quantum-mechanics/old-quantum-theory/blackbody-radiation-and-the-planck-quantum)
more precisely than any source made in a laboratory. Its near-perfect isotropy
confirms the **cosmological principle**, that the universe is homogeneous and
isotropic at large scale. The tiny WMAP temperature fluctuations, about
$30\ \mu\text{K}$, mark the density seeds that grew into galaxies. The finite age
also resolves **Olbers' paradox**: the night sky is dark because lines of sight
reach back to a time before stars formed.

## Thermal history

Running the expansion forward from $t = 0$, the universe cooled through a
sequence of eras, each ending when the temperature dropped below a
characteristic energy.

$$
% caption: The universe cools as it expands, passing through unification,
% hadron, lepton, and nucleosynthesis eras before matter and radiation decouple.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (10.4,0) node[right, black!70] {time (cooling)};
\foreach \x/\lab/\t in {
  0.7/inflation/{10^{-36} s},
  2.7/{quark-hadron}/{10^{-6} s},
  4.7/lepton/{1 s},
  6.7/nucleosynthesis/{3 min},
  8.9/{atoms form}/{380 kyr}} {
  \fill[black] (\x,0) circle (2.3pt);
  \draw[black] (\x,0) -- (\x,0.28);
}
\node[black, anchor=south, font=\scriptsize] at (0.7,0.32) {rapid expansion};
\node[black, anchor=south, font=\scriptsize] at (2.7,0.32) {quarks bind};
\node[black, anchor=south, font=\scriptsize] at (4.7,0.32) {lepton era};
\node[black, anchor=south, font=\scriptsize] at (6.7,0.32) {nuclei form};
\node[black, anchor=south, font=\scriptsize] at (8.9,0.32) {atoms form};
\node[black, anchor=north, font=\scriptsize] at (0.7,-0.1) {hot};
\node[black, anchor=north, font=\scriptsize] at (8.9,-0.1) {cool};
\end{tikzpicture}
$$

- **Inflation.** An early phase of negative-pressure energy drove an
  exponentially accelerated expansion, explaining the CMB's uniformity, the
  near-flat geometry, and the absence of monopoles.
- **Grand-unification era.** The strong, weak, and electromagnetic forces were
  unified; a slight excess of quarks over antiquarks, about one part in $10^9$,
  set the matter content of the universe.
- **Quark-hadron transition.** As the strong force separated, quarks bound into
  [hadrons](/particle-physics/electroweak-higgs/standard-model); at $kT \sim 1\
  \text{GeV}$ nucleon-antinucleon pairs annihilated, leaving the small matter
  excess.
- **Lepton and radiation eras.** At $kT \sim 1\ \text{MeV}$ electron-positron
  pairs annihilated, leaving photons and neutrinos dominant.
- **Nucleosynthesis.** Within minutes, protons and neutrons fused into
  deuterium, helium, and a little lithium; further expansion halted fusion, and
  heavier elements waited for stars.
- **Recombination.** At about $3000\ \text{K}$ (scale factor near
  $2.8 \times 10^{-4}$), electrons and nuclei combined into atoms and radiation
  decoupled from matter, releasing the CMB.[^tl-evolve]

Matter then dominated for most of cosmic history, until the recent dark-energy
phase resumed acceleration. The present energy budget is about 4% ordinary
matter, 22% dark matter, and 74% dark energy.

$$
% caption: Present mass-energy budget of the universe: ordinary atoms are a small
% fraction, with dark matter and dark energy dominating.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% pie approximated by wedges
\draw[black] (0,0) -- (90:2.4) arc (90:75.6:2.4) -- cycle;
\draw[black] (0,0) -- (75.6:2.4) arc (75.6:0:2.4) -- cycle;
\filldraw[acc, fill=acc!12] (0,0) -- (0:2.4) arc (0:-266.4:2.4) -- cycle;
\draw[thick] (0,0) circle (2.4);
\node[black!70, anchor=west] at (2.9,2.0) {atoms: 4 percent};
\node[black!70, anchor=west] at (2.9,1.2) {dark matter: 22 percent};
\node[black!70, anchor=west] at (2.9,0.4) {dark energy: 74 percent};
\draw[black] (2.4,1.6) -- (2.85,2.0);
\draw[black] (2.4,0.6) -- (2.85,1.2);
\draw[black] (-1.2,-1.0) -- (2.85,0.4);
\end{tikzpicture}
$$

Whether the universe expands forever or recollapses turns on whether the present
density exceeds the critical value. Measurement places it close to critical, and
the accelerating expansion driven by dark energy points toward indefinite
expansion, though the precise fate remains at the edge of what current data can
decide.

[^tl-gal]: Tipler & Llewellyn, §13-6 — Galaxies: the interstellar medium and the 21-cm line, Hubble's galaxy classes, active galaxies and quasars, and Hubble's law with the cosmological redshift.
[^tl-cosmo]: Tipler & Llewellyn, §13-7 — Cosmology and Gravitation: the cosmological principle, the critical density, and the density parameter.
[^tl-evolve]: Tipler & Llewellyn, §13-8 — Cosmology and the Evolution of the Universe: the Newtonian shell model, the Big Bang and the cosmic microwave background, and the thermal history from inflation to recombination.