Cosmological Models and Distances

The Friedmann equation is a differential equation for the scale factor whose solution depends on the mixture of components filling the universe. Choosing simple mixtures gives the benchmark models — idealized universes dominated by a single component or by two — that bracket the possibilities and build intuition, before assembling the realistic concordance model. From each model's expansion history follow the age of the universe, the lookback time to a given redshift, and the several distinct distance measures. Because we can measure only redshift directly, these distance measures diverge at high redshift, and one of them behaves so counterintuitively that distant galaxies appear to grow with distance. This lesson develops the models and the distance ladder of cosmology, and closes with the horizon problem that motivates inflation.

The Friedmann equation in density parameters

To solve for $a(t)$, write the Friedmann equation with each component's known scaling. Using ${\epsilon }_i\propto a^{-3(1+w_i)}$ and normalizing by the present critical density, the Friedmann equation becomes

where the curvature term is written as an effective density parameter ${\Omega }_{k,0}=1-{\Omega }_{r,0}-{\Omega }_{m,0}-{\Omega }_{\Lambda ,0}$ scaling as $a^{-2}$. This single equation, sometimes written with $E(z)\equiv H(z)/H_0$ and $a=1/(1+z)$, is the master equation of cosmological modelling. The four terms fall off at different rates, so as $a$ grows the dominant term changes from radiation to matter to curvature or vacuum. Each benchmark model switches off all but one or two of the $\Omega$'s.¹

Benchmark models

Radiation-dominated (${\Omega }_r=1$, all else zero). Early universe. The Friedmann equation reads $\dot{a}/a=H_0{a}^{-2}$, integrating to

The expansion decelerates and the scale factor rises as the square root of time.

Einstein-de Sitter (${\Omega }_m=1$, flat, matter only). The historical default. Here $\dot{a}/a=H_0{a}^{-3/2}$, giving

The age is exactly two-thirds of the Hubble time $H_0^{-1}$. The expansion decelerates forever but never halts; the density asymptotically approaches zero. This model is the simplest flat universe, and it was the assumed cosmology for most of the twentieth century.

Open, matter-dominated (${\Omega }_m<1$, ${\Omega }_{\Lambda }=0$, $k<0$). The curvature term ${\Omega }_k{a}^{-2}$ eventually dominates over matter's $a^{-3}$, and the expansion coasts toward $a\propto t$ (free expansion, gravity negligible). The universe expands forever, and its age approaches the full Hubble time $H_0^{-1}$ as ${\Omega }_m\to 0$.

Lambda-dominated (de Sitter) (${\Omega }_{\Lambda }=1$). The far future. With constant vacuum energy the Friedmann equation is $\dot{a}/a=H_0=\text{const}$, integrating to exponential growth,

The expansion accelerates without bound; this de Sitter phase is where a vacuum-dominated universe is heading and where inflation put the very early universe.

Concordance Lambda-CDM (${\Omega }_{m,0}\approx 0.31$, ${\Omega }_{\Lambda ,0}\approx 0.69$, flat, radiation negligible today). The real universe. Early on matter dominates and the expansion decelerates as in Einstein-de Sitter; as matter dilutes the vacuum term takes over and the expansion transitions to acceleration. The scale-factor history interpolates between the $t^{2/3}$ deceleration at early times and the exponential $e^{H_{0}t}$ growth in the future, with the inflection (where $\ddot{a}=0$) at redshift $z\approx 0.6$.²

The age of the universe

The age follows from integrating $\mathbf{d}t=\mathbf{d}a/\dot{a}$. From $\dot{a}=aH(a)$,

The prefactor $H_0^{-1}$ is the Hubble time, $H_0^{-1}\approx 14\text{Gyr}$ for $H_0=70\text{km}{\text{s}}^{-1}{\text{Mpc}}^{-1}$, and the dimensionless integral is an order-unity correction set by the contents. For Einstein-de Sitter the integral gives exactly $2/3$, an age of about $9\text{Gyr}$ — uncomfortably young, less than the ages of the oldest globular clusters, which was one of the classic problems with the matter-only model. For the concordance model the integral evaluates to almost exactly $1$ (the deceleration and acceleration phases nearly cancel), giving

in excellent agreement with the Planck determination and comfortably older than the oldest stars. The lookback time to redshift $z$ — how long ago the light we now see at that redshift was emitted — is the same integral with the lower limit changed from $0$ to $a=1/(1+z)$:

Distance measures

Because only redshift is directly observed, distance splits into several inequivalent operational definitions, each tied to a different observable. All are built from the comoving distance

with $E(z)=H(z)/H_0$. The comoving distance is the coordinate separation and is the same for all observers today.

The luminosity distance $d_L$ is defined so that the inverse-square law holds in its usual form: a source of known luminosity $L$ observed to have flux $F$ is assigned $d_L=\sqrt{L/(4\pi F)}$. Two effects of expansion dim the flux beyond the geometric $1/d_C^2$ — each photon is redshifted in energy by $(1+z)$ and photons arrive less frequently by another $(1+z)$ — so

in a flat universe (with $S_k$ replacing $d_C$ if curved). This is the distance that enters the supernova Hubble diagram.

The angular-diameter distance $d_A$ is defined so that an object of known physical size $\ell$ subtending angle $\theta$ is assigned $d_A=\ell /\theta$. The object's light was emitted when the universe was smaller by $(1+z)$, so the same physical size subtends a larger angle, and

The three distances therefore stand in the fixed ratio

known as the Etherington reciprocity relation. At low redshift all three converge to $cz/H_0$, the naive Hubble distance; at high redshift they diverge by powers of $(1+z)$, and the distinction becomes essential for interpreting any high-$z$ observation.³

The turnover of angular size

The angular-diameter distance has a strange consequence. Since $d_A=d_C/(1+z)$ and $d_C$ grows only logarithmically slowly at high redshift while $(1+z)$ grows linearly, $d_A$ rises, reaches a maximum, and then decreases with redshift. The maximum occurs near $z\approx 1.6$ in the concordance model. Beyond it, the angular size $\theta =\ell /d_A$ of a fixed physical object increases with redshift: the most distant galaxies of a given size appear larger on the sky than nearer ones.

The physical reason is that when the light left a very high-redshift object, the universe was so small that the object was actually close to us in proper distance at the time of emission; the expansion then carried it far away, but the light preserves the wide angle it subtended when it was near. This counterintuitive turnover is a genuine, measurable prediction — the apparent sizes of galaxies and the acoustic scale imprinted on the CMB both sit beyond the turnover — and it is a clean illustration that distance in an expanding universe is not a single number but a family of them.

The horizon problem

A finite age implies a finite reach for causal contact. The particle horizon is the maximum comoving distance light could have travelled since the Big Bang,

the comoving radius of the region with which a given point could have exchanged signals. In a decelerating universe this integral converges to a finite value: only a finite patch has ever been in causal contact. At the epoch of last scattering, when the CMB was released, the particle horizon subtended an angle of only about $1^{\circ }$ to $2^{\circ }$ on today's sky. Regions of the CMB separated by more than a couple of degrees were, according to the standard Friedmann expansion, never in causal contact — yet they share the same temperature to one part in $10^5$.

This is the horizon problem: how did causally disconnected patches reach the same temperature, if they never had time to exchange heat? Nothing in the Friedmann dynamics of a radiation- or matter-dominated universe explains it, because the expansion decelerates and the horizon grows faster than comoving scales, exposing ever more previously disconnected regions. The resolution is an early phase of accelerated expansion — inflation — which stretches a tiny, causally connected patch to encompass the entire observable universe, so that the uniformity is inherited from a pre-inflationary thermal equilibrium. Inflation is developed in the hot-Big-Bang module; here the horizon problem stands as the sharpest way in which the benchmark models are incomplete.⁴

Summary

Solving the Friedmann equation for a chosen mixture yields the benchmark models: radiation-dominated $a\propto t^{1/2}$, matter-dominated Einstein-de Sitter $a\propto t^{2/3}$ with age $\frac{2}{3}{H}_0^{-1}$, open coasting toward $a\propto t$, vacuum-dominated de Sitter $a\propto e^{H_{0}t}$, and the concordance Lambda-CDM that decelerates then accelerates with age $\approx H_0^{-1}\approx 13.8\text{Gyr}$. The age and lookback time are integrals of $1/(aH)$, and the several distance measures — comoving, luminosity $d_L=(1+z)d_C$, angular-diameter $d_A=d_C/(1+z)$ — obey the Etherington relation $d_L=(1+z{)}^2{d}_A$ and diverge at high redshift, with $d_A$ turning over near $z\approx 1.6$ so that distant objects can appear larger. The finite particle horizon at last scattering exposes the horizon problem, the failure of the decelerating models to explain the CMB's uniformity. The next lesson turns to the observation that first revealed the accelerating term: dark energy and the accelerating universe.