Galaxies, Cosmology, and the Evolving Universe

Beyond the Milky Way, a barred spiral about $28, 000 ly$ across with the Sun a third of the way out, lie roughly $1 0^{10}$ 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 $150 K$ , 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 between the electron and proton magnetic moments: the antiparallel-spin state lies below the parallel-spin state by only

Δ E = 5.9 \times 1 0^{- 6} eV .

A spin flip back to the ground state emits a photon of wavelength $21 cm$ . 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.

Hubble's galaxy classes split by shape; disk galaxies further divide into ordinary and barred spirals, with irregulars as the remainder.

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 $0.95 c$ and power outputs near $1 0^{40} W$ , more than $1 0^{12}$ 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, H_{0} = 71 \pm 4 km/s per Mpc .

Recession velocity rises linearly with distance; the slope of the Hubble line is the Hubble constant, and its reciprocal sets a timescale.

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{λ - λ _{0}}{λ _{0}} = \frac{Δ λ}{λ _{0}} .

This is not the relativistic Doppler effect; 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 _{observed}}{R _{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 $1/ H_{0} \approx 1.3 \times 1 0^{10} yr$ , estimates the age of the universe when gravity is ignored.¹

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 = \frac{1}{2} m v^{2} - \frac{GM m}{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 $M = \frac{4}{3} π R^{3} ρ$ gives the critical density:

ρ_{c} = \frac{3 H _{0}^{2}}{8 π G} \approx 1 0^{- 26} kg/m^{3},

about five hydrogen atoms per cubic meter. The ratio to critical is the density parameter $Ω = ρ / ρ_{c}$ , which fixes the fate and geometry of the universe.

The density parameter decides the outcome: below critical the universe expands forever, at critical it coasts, above critical it recollapses.

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

Visible baryonic matter supplies only about 4% of $ρ_{c}$ . Adding the dark matter 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%.²

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 $M_{r} = \frac{4}{3} π r^{3} ρ$ 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 π}{3} Gρ (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 π}{3} Gρ (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 π G}{3 c ^{2}} (ε c^{2} + 3 P),

where $ε c^{2}$ 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 $Ω_{Λ} \approx 0.73$ and $Ω_{m} \approx 0.27$ , 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 $(13.7 \pm 0.2) \times 1 0^{9}$ 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 K,

with a spectrum matching a Planck blackbody 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 $30 μ K$ , 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.

The universe cools as it expands, passing through unification, hadron, lepton, and nucleosynthesis eras before matter and radiation decouple.

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 $1 0^{9}$ , set the matter content of the universe.
Quark-hadron transition. As the strong force separated, quarks bound into hadrons; at $k T \sim 1 GeV$ nucleon-antinucleon pairs annihilated, leaving the small matter excess.
Lepton and radiation eras. At $k T \sim 1 MeV$ 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 $3000 K$ (scale factor near $2.8 \times 1 0^{- 4}$ ), electrons and nuclei combined into atoms and radiation decoupled from matter, releasing the CMB.³

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.

Present mass-energy budget of the universe: ordinary atoms are a small fraction, with dark matter and dark energy dominating.

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.

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. ↩
Tipler & Llewellyn, §13-7 — Cosmology and Gravitation: the cosmological principle, the critical density, and the density parameter. ↩
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. ↩