---
title: Cosmological Models and Distances
draft: false
module: Cosmic Expansion and Dynamics
moduleNumber: 11
lessonNumber: 4
order: 1104
summary: >
  Integrating the Friedmann equation for particular mixtures gives the benchmark
  models, from the matter-only Einstein-de Sitter universe to the concordance
  Lambda-CDM, each with its own scale-factor history and age. Because the redshift is
  the only direct observable, several distance measures diverge at high redshift, and
  the angular-diameter distance even turns over so that the most distant objects look
  larger. The horizon and lookback time set what is causally and observationally
  reachable.
topics: [Cosmology]
sources:
  - book: Ryden
    ref: "Ch. 5 — Model Universes; Ch. 6 — Measuring Cosmological Parameters"
  - book: Carroll & Ostlie
    ref: "Ch. 29 — Cosmology; §29.4 The Age and Fate of the Universe"
  - book: Maoz
    ref: "Ch. 9 — The Big Bang and the Expansion of the Universe"
---

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 $\varepsilon_i \propto a^{-3(1+w_i)}$ and normalizing by the present critical
density, the Friedmann equation becomes

$$
H^{2}(a) = H_0^{2}\left[\Omega_{r,0}\,a^{-4} + \Omega_{m,0}\,a^{-3}
  + \Omega_{k,0}\,a^{-2} + \Omega_{\Lambda,0}\right] ,
$$

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.[^ry-models]

## 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

$$
a(t) \propto t^{1/2} , \qquad \varepsilon \propto a^{-4} \propto t^{-2} .
$$

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

$$
a(t) = \left(\frac{t}{t_0}\right)^{2/3} , \qquad t_0 = \frac{2}{3 H_0} .
$$

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,

$$
a(t) \propto e^{H_0 t} .
$$

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
$z \approx 0.6$.[^co-models]

$$
% caption: Scale-factor histories for the benchmark models. Einstein-de Sitter and open
% models decelerate; the concordance model turns up as vacuum energy takes over.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {time};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {scale factor};
% now marker
\draw[black, densely dotted] (4.4,0) -- (4.4,3.0);
\node[black, anchor=north] at (4.4,-0.1) {now};
\fill[black] (4.4,2.0) circle (0.06);
% Einstein-de Sitter t^2/3: passes through now, decelerating
\draw[black, very thick, densely dashed] (0.15,0.35) .. controls (1.6,1.15) and (3.0,1.7) .. (8.2,2.95);
\node[black, anchor=south] at (6.6,2.55) {Einstein-de Sitter};
% open ~ toward linear: higher at late times
\draw[black, very thick, densely dotted] (0.15,0.3) .. controls (2.0,1.35) and (4.0,2.2) .. (8.2,3.6);
\node[black, anchor=south east] at (8.1,3.4) {open};
% concordance: decelerate then accelerate up, acc solid
\draw[acc, very thick] (0.15,0.3) .. controls (2.4,1.4) and (4.4,2.0) .. (8.2,4.15);
\node[acc, anchor=east] at (7.6,3.75) {Lambda-CDM};
\end{tikzpicture}
$$

## The age of the universe

The age follows from integrating $\d t = \d a/\dot a$. From $\dot a = a H(a)$,

$$
t_0 = \int_0^{t_0} \d t = \int_0^{1} \frac{\d a}{a\,H(a)}
= \frac{1}{H_0}\int_0^{1}
\frac{\d a}{a\sqrt{\Omega_{r,0}a^{-4} + \Omega_{m,0}a^{-3} + \Omega_{k,0}a^{-2} + \Omega_{\Lambda,0}}} .
$$

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

$$
t_0 \approx H_0^{-1} \approx 13.8\ \text{Gyr} ,
$$

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
$a = 1/(1+z)$:

$$
t_L(z) = \frac{1}{H_0}\int_{1/(1+z)}^{1}
\frac{\d a}{a\sqrt{\Omega_{r,0}a^{-4} + \Omega_{m,0}a^{-3} + \Omega_{k,0}a^{-2} + \Omega_{\Lambda,0}}} .
$$

$$
% caption: The age integral accumulates cosmic time as one over a-dot from the Big Bang
% at a = 0 to the present at a = 1; the area under the curve is the age.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {scale factor};
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {one over a-dot};
% integrand: large at small a, decreasing then flattening
\draw[acc, very thick] (0.35,3.9) .. controls (1.2,1.9) and (2.6,1.1) .. (7.6,1.5);
\node[acc, anchor=south west] at (2.6,1.6) {integrand};
% shaded area under curve
\fill[acc!12] (0.35,0) -- (0.35,3.9) .. controls (1.2,1.9) and (2.6,1.1) .. (7.6,1.5) -- (7.6,0) -- cycle;
\node[black, anchor=center] at (4.0,0.7) {area = age};
% a=1 marker
\draw[black, densely dotted] (7.6,0) -- (7.6,1.5);
\node[black, anchor=north] at (7.6,-0.1) {a = 1};
\node[black, anchor=north] at (0.35,-0.1) {a = 0};
\end{tikzpicture}
$$

## 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

$$
d_C(z) = c\int_0^{z} \frac{\d z'}{H(z')} = \frac{c}{H_0}\int_0^{z} \frac{\d z'}{E(z')} ,
$$

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

$$
d_L = (1 + z)\, d_C
$$

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

$$
d_A = \frac{d_C}{1 + z} .
$$

The three distances therefore stand in the fixed ratio

$$
d_L = (1 + z)^{2}\, d_A , \qquad d_A = \frac{d_C}{1+z} , \qquad d_L = (1+z)\,d_C ,
$$

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.[^ry-dist]

$$
% caption: The distance measures coincide at low redshift but diverge at high z: the
% luminosity distance grows fastest, the angular-diameter distance turns over.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {redshift};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {distance};
% luminosity distance: rises fastest, acc solid
\draw[acc, very thick] (0.2,0.2) .. controls (2.5,1.5) and (4.5,3.2) .. (8.0,4.15);
\node[acc, anchor=south east] at (7.3,3.9) {luminosity};
% comoving: middle, dashed
\draw[black, very thick, densely dashed] (0.2,0.2) .. controls (2.5,1.4) and (5.0,2.3) .. (8.0,2.9);
\node[black, anchor=south] at (6.0,2.55) {comoving};
% angular-diameter: rises then turns over, dotted
\draw[black, very thick, densely dotted] (0.2,0.2) .. controls (1.6,1.3) and (2.6,1.75) .. (3.6,1.75)
  .. controls (5.2,1.75) and (6.8,1.35) .. (8.0,1.0);
\node[black, anchor=south] at (2.4,1.85) {angular-diameter};
\end{tikzpicture}
$$

## 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.

$$
% caption: Because the angular-diameter distance turns over, a fixed physical size
% subtends a shrinking then growing angle: distant objects can look larger than nearer ones.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% observer
\fill[black] (0,0) circle (0.08);
\node[black, anchor=north] at (0,-0.15) {observer};
% near object: small angle
\draw[black] (0,0) -- (3.2,0.55);
\draw[black] (0,0) -- (3.2,-0.05);
\draw[very thick] (3.2,0.55) -- (3.2,-0.05);
\node[black, anchor=west] at (3.35,0.25) {near: narrow angle};
% far object beyond turnover: wide angle
\draw[black] (0,0) -- (6.4,1.9);
\draw[black] (0,0) -- (6.4,0.7);
\draw[acc, very thick] (6.4,1.9) -- (6.4,0.7);
\node[black, anchor=west] at (6.55,1.3) {far: wider angle};
\end{tikzpicture}
$$

## 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,

$$
d_{\text{hor}}(t) = c\int_0^{t} \frac{\d t'}{a(t')} ,
$$

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.[^ry-horizon]

## 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 $\tfrac{2}{3}H_0^{-1}$, 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 $1/(a H)$, 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](/astrophysics-cosmology/cosmology-expansion-and-dynamics/dark-energy-and-the-accelerating-universe).

[^ry-models]: Ryden, *Introduction to Cosmology*, Ch. 5 — Model Universes: the Friedmann equation in density parameters and the single-component solutions.
[^co-models]: Carroll & Ostlie §29.4 — the concordance model and the transition from deceleration to acceleration.
[^ry-dist]: Ryden, Ch. 6 — Measuring Cosmological Parameters: the luminosity and angular-diameter distances and their relation.
[^ry-horizon]: Ryden, Ch. 5 — the particle horizon and the horizon problem; the inflationary resolution in Ch. 10.