Cosmological Models and Their Evolution

A cosmological model is a single exact solution of Einstein's equations for a universe that is the same in every place and every direction — the Friedmann solutions built on the FLRW metric. What makes the idea so powerful is how little you must supply to fix it completely: the present expansion rate $H_0$ , and the way today's energy density is split among radiation (photons and relativistic neutrinos), non-relativistic matter (baryons plus cold dark matter), a cosmological constant or dark energy, and the geometric curvature term. From those few numbers the whole scale-factor history $a(t)$ follows deterministically — every distance, every age, every horizon. The organising principle is the equation of state $w=P/\rho c^2$ , which dictates how each ingredient thins out as space grows: $\rho\propto a^{-3(1+w)}$ . Radiation ( $w=1/3$ ) fades fastest as $a^{-4}$ , matter ( $w=0$ ) as $a^{-3}$ , and a cosmological constant ( $w=-1$ ) refuses to dilute at all. Because the rates differ, a universe that starts radiation-dominated is forced through a matter era and, if $\Lambda>0$ , into a $\Lambda$ -dominated de Sitter phase. Classifying universes by their contents is therefore the same act as reading off the sequence and timing of their eras. The single-component idealisations bracket every real model: Einstein–de Sitter ( $a\propto t^{2/3}$ ), the radiation universe ( $a\propto t^{1/2}$ ), and de Sitter ( $a\propto e^{Ht}$ ); add curvature and open universes expand forever while closed matter universes recollapse in a Big Crunch. The universe we inhabit is concordance $\Lambda$ CDM: spatially flat, $\Omega_{m,0}\approx0.31$ , $\Omega_{\Lambda,0}\approx0.69$ , a whisper of radiation $\Omega_{r,0}\approx9\times10^{-5}$ , switching from deceleration to acceleration near $z\approx0.6$ . To understand cosmological models is to trace that whole biography from the density budget alone.

Key takeaways

A cosmological model is a Friedmann solution fixed by the present density parameters $\Omega_{r,0}$ , $\Omega_{m,0}$ , $\Omega_{\Lambda,0}$ and curvature $\Omega_{k,0}$ , which sum to unity.
Each component dilutes at a rate set by its equation of state: radiation as $a^{-4}$ , matter as $a^{-3}$ , a cosmological constant not at all.
Because those rates differ, the universe runs a relay of dominant eras — radiation ( $a\propto t^{1/2}$ ), matter ( $a\propto t^{2/3}$ ), then $\Lambda$ ( $a\propto e^{Ht}$ ).
Curvature and $\Lambda$ decide the ending: eternal expansion, asymptotic coasting, or recollapse in a Big Crunch.
Concordance $\Lambda$ CDM has $\Omega_{m,0}\approx0.31$ , $\Omega_{\Lambda,0}\approx0.69$ , $\Omega_{r,0}\approx9\times10^{-5}$ , is flat, and is now entering accelerated de Sitter expansion with $q_0\approx-0.53$ .
The Einstein–de Sitter, de Sitter, Milne, and radiation universes are the limiting benchmarks against which data are fit.
Key idea: specify $(\Omega_{r,0},\Omega_{m,0},\Omega_{\Lambda,0},\Omega_{k,0})$ with $\sum\Omega=1$ ; each dilution law $\rho\propto a^{-3(1+w)}$ then fixes $H(a)$ , the order of the dominant eras, and the universe's ultimate fate.

The core insight

Specify $(\Omega_{r,0},\Omega_{m,0},\Omega_{\Lambda,0},\Omega_{k,0})$ with $\sum\Omega=1$ ; each component's dilution law $\rho\propto a^{-3(1+w)}$ then determines $H(a)$ , the sequence of dominant eras, and the universe's ultimate fate.

Where this lives today

Cosmological models are no longer a menu of possibilities but a precision-fit instrument. Experiments now pin $\Lambda$ CDM's parameters to the percent level: Planck's CMB map fixes $\Omega_{m,0}$ and the geometry through the acoustic-peak positions ( $\Omega_{k,0}=0.001\pm0.002$ , flat to a fraction of a percent); baryon-acoustic-oscillation surveys such as BOSS, eBOSS, and DESI use the sound-horizon standard ruler across $0<z<3.5$ to trace $H(z)$ ; and Type Ia supernovae from Pantheon+ measure the very deceleration-to-acceleration transition derived above. The live frontier is whether dark energy is truly a constant $w=-1$ : in 2024–2025 DESI reported a mild preference for evolving dark energy ( $w_0w_a$ CDM at the two-to-four-sigma level), which — if it survives — would replace the flat de Sitter future with something less certain. Alongside it runs the Hubble tension: the locally measured $H_0\approx73$ from Cepheid-calibrated supernovae sits in $5\sigma$ disagreement with the $H_0\approx67$ inferred by evolving the CMB forward through the same model — a systematic error, or a crack in $\Lambda$ CDM, and telling which is a defining open problem of the field.

The physical picture

Picture three fluids sharing one expanding box. Radiation loses energy twice over — its quanta thin out in number and every quantum is redshifted — so it fades first. Matter merely thins. The cosmological constant does neither; whatever value it starts with, it holds, so however small it begins it eventually wins. The cosmic story is a relay race in which the lead passes inevitably from radiation to matter to $\Lambda$ , an order fixed by the slopes $a^{-4}$ , $a^{-3}$ , and $a^{0}$ . The first figure is that race drawn to scale, with the two hand-off points carrying their real values.

Figure

Cosmic energy budget: radiation, matter and Λ dilute at rates set by their equation of state, so dominance passes radiation → matter → Λ. Matter overtakes radiation at a ≈ 2.9×10⁻⁴ (z ≈ 3400) and Λ overtakes matter at a ≈ 0.77, just before today

The density parameters are just fractions of a critical budget, $\Omega_i=\rho_i/\rho_{\rm crit}$ , and a flat universe demands that they sum to exactly one; any shortfall or excess is made up by spatial curvature. Read the Friedmann equation as energy conservation for a ball thrown upward: too little density and the ball escapes (open, eternal expansion); too much matter and it falls back (closed, recollapse); a positive $\Lambda$ acts like antigravity that drives it away ever faster. The second figure follows four candidate universes forward — all sharing today's rate $H_0$ , so all crossing the present at the same point with the same slope, yet peeling apart into utterly different futures.

The core equations

Every cosmological model is a solution of one master equation, built up in stages. The Friedmann equation is the starting point — the time–time Einstein equation for an FLRW universe, relating the expansion rate to the total density, the spatial curvature, and the cosmological constant.

H^2=\left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3} \qquad (1)

where: $H$ — Hubble parameter $\dot a/a$ ; $a$ — scale factor (dimensionless, $a_0=1$ today); $\rho$ — total density of matter plus radiation; $k$ — curvature constant ( $-1,0,+1$ ); $\Lambda$ — cosmological constant; $G,c$ — Newton's constant and the speed of light. Reading: the square of the expansion rate is a competition between the pull of density, the geometry $k$ , and the push of $\Lambda$ . Setting $k=\Lambda=0$ in (1) defines the natural yardstick for density — the value that makes a universe geometrically flat — turning absolute densities into dimensionless fractions.

\rho_{\rm crit}=\frac{3H^2}{8\pi G},\qquad \Omega_i\equiv\frac{\rho_i}{\rho_{\rm crit}} \qquad (2)

where: $\rho_{\rm crit}$ — critical density for flatness; $\Omega_i$ — fractional density of component $i$ . Reading: today $\rho_{\rm crit,0}\approx8.7\times10^{-27}\ \mathrm{kg\,m^{-3}}$ for $h=0.68$ , about five hydrogen atoms per cubic metre — the whole visible cosmos hangs on that thin a margin. Evaluate (1) today and divide by $H_0^2$ : the dynamical statement becomes a bookkeeping identity in which the density fractions plus a curvature "fraction" add to one.

1=\Omega_{r,0}+\Omega_{m,0}+\Omega_{\Lambda,0}+\Omega_{k,0} \qquad (3)

where: $\Omega_{k,0}\equiv-kc^2/(a_0^2H_0^2)$ — the curvature density parameter. Reading: measuring any three of the four fixes the last; observing $\Omega_{k,0}\approx0$ is the statement that the universe is spatially flat. Each component's density scales with expansion at a rate set entirely by its equation-of-state parameter through the fluid (continuity) equation.

\rho_i(a)=\rho_{i,0}\,a^{-3(1+w_i)} \qquad (4)

where: $w_i=P_i/(\rho_i c^2)$ — equation-of-state parameter. Reading: $w=0$ gives $a^{-3}$ (matter), $w=1/3$ gives $a^{-4}$ (radiation), $w=-1$ gives a constant ( $\Lambda$ ) — the three slopes of the first figure. Substituting each dilution law into (1) collapses the whole subject into one line — the expansion history of any model, written purely in terms of today's density parameters.

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] \qquad (5)

where: $H(a)$ — expansion rate as a function of scale factor. Reading: this is the master evolution equation; every distance, age, and horizon is an integral of $1/H(a)$ , so the four numbers fix the model's entire biography. The eras hand off where two components carry equal density — simply ratios of the density parameters. Matter overtakes radiation, and later $\Lambda$ overtakes matter.

a_{\rm eq}=\frac{\Omega_{r,0}}{\Omega_{m,0}}\approx2.9\times10^{-4}\ \ (z_{\rm eq}\approx3400),\qquad a_{m\Lambda}=\left(\frac{\Omega_{m,0}}{\Omega_{\Lambda,0}}\right)^{1/3}\approx0.77 \qquad (6)

where: $a_{\rm eq}$ — matter–radiation equality; $a_{m\Lambda}$ — matter– $\Lambda$ equality; $1+z=1/a$ . Reading: equality redshifts are read straight off the budget — $z_{\rm eq}\approx3400$ is imprinted on the CMB, and $z_{m\Lambda}\approx0.3$ marks when dark energy took the lead. Finally, whether the expansion speeds up or slows down today is one dimensionless number built from the same parameters, weighted by $(1+3w_i)$ because pressure gravitates.

q_0\equiv-\frac{\ddot a\,a}{\dot a^2}\Big|_0=\tfrac12\Omega_{m,0}+\Omega_{r,0}-\Omega_{\Lambda,0} \qquad (7)

where: $q_0$ — present deceleration parameter; $q_0<0$ signals acceleration. Reading: matter and radiation decelerate ( $+$ ), $\Lambda$ accelerates ( $-$ ); for concordance values $q_0\approx-0.53$ , so the universe is accelerating now.

Derivation AAnalytic scale factor for a single flat component

The goal is to take the master equation for the simplest possible universe — flat, with one fluid of constant equation of state $w$ — and integrate it exactly, so that the three headline benchmark histories drop out as special cases: matter giving $a\propto t^{2/3}$ , radiation giving $a\propto t^{1/2}$ , and a cosmological constant giving the exponential de Sitter law $a\propto e^{Ht}$ . These single-fluid solutions are the fixed points that bracket every realistic multi-component model.

Step 1. Set $k=0$ and keep one fluid with $\rho=\rho_0 a^{-3(1+w)}$ , so the Friedmann equation becomes $\dot a^2=\tfrac{8\pi G}{3}\rho_0\,a^{-3(1+w)}a^2=C\,a^{-(1+3w)}$ with $C=\tfrac{8\pi G}{3}\rho_0$ . Why: absorbing every constant into $C$ strips the equation down to a pure power of $a$ , which is what makes it separable.

Step 2. Separate the variables into $a^{(1+3w)/2}\,\mathrm{d}a=\sqrt{C}\,\mathrm{d}t$ . Why: this step is legitimate for every fluid except $w=-1$ , where the exponent forces a different route, so we handle that case last.

Step 3. Integrate both sides with the boundary condition $a=0$ at $t=0$ (the Big Bang), giving $\tfrac{2}{3(1+w)}a^{3(1+w)/2}=\sqrt{C}\,t$ . Why: fixing $a(0)=0$ pins the origin of time to the singularity and removes the integration constant cleanly.

Step 4. Solve for $a(t)=[\tfrac32(1+w)\sqrt{C}\,t]^{2/[3(1+w)]}$ , so the growth is a power law with exponent $2/[3(1+w)]$ . Why: reading off the exponent gives $w=0\Rightarrow a\propto t^{2/3}$ and $w=1/3\Rightarrow a\propto t^{1/2}$ immediately — the matter and radiation benchmarks.

Step 5. For $w=-1$ the density is constant, $\dot a^2=\tfrac{\Lambda c^2}{3}a^2$ , whose solution is $a\propto e^{Ht}$ with $H=c\sqrt{\Lambda/3}$ . Why: a non-diluting component makes the right-hand side proportional to $a^2$ rather than a negative power, converting power-law growth into exponential de Sitter expansion with no scale-factor singularity. *Verified: the exponent $2/[3(1+w)]$ reproduces $t^{2/3}$ and $t^{1/2}$ , and $w=-1$ gives $e^{Ht}$ — the three solutions in Ryden ch. 5 and Carroll & Ostlie ch. 29.*

Go deeper · Derivation B · 1 minClosed matter universe reaches a maximum size and recollapses

Derivation BClosed matter universe reaches a maximum size and recollapses

The goal is to prove that a closed, matter-only universe — one with $\Omega_{m,0}>1$ and positive curvature $k=+1$ — does not expand forever but swells to a finite maximum scale factor and then falls back to a Big Crunch, tracing a cycloid in time. We locate the turning point, introduce a development angle that integrates the Friedmann equation in closed form, and read off the symmetric lifetime — the archetype of a bounded, recollapsing cosmology.

Step 1. With $\Lambda=0$ , radiation negligible, and $\Omega_{k,0}=1-\Omega_{m,0}<0$ , the master equation reads $\dot a^2=H_0^2[\Omega_{m,0}a^{-1}-(\Omega_{m,0}-1)]$ . Setting $\dot a=0$ gives the turning point $\Omega_{m,0}/a=\Omega_{m,0}-1$ . Why: expansion halts exactly when the (negative) curvature term overtakes the matter term, which can only happen if $\Omega_{m,0}>1$ .

Step 2. Solve for the maximum scale factor, $a_{\max}=\Omega_{m,0}/(\Omega_{m,0}-1)$ , which is finite and positive only for a closed universe. Why: a denser universe ( $\Omega_{m,0}$ larger) turns around sooner and at a smaller $a_{\max}$ , because gravity is stronger relative to the expansion.

Step 3. Introduce a development angle $\eta$ through $\mathrm{d}\eta=H_0\sqrt{\Omega_{m,0}-1}\,\mathrm{d}t/a$ and posit the ansatz $a(\eta)=\tfrac12 a_{\max}(1-\cos\eta)$ . Why: the $(1-\cos\eta)$ form automatically satisfies $a(0)=0$ and reaches $a_{\max}$ at $\eta=\pi$ , which is precisely the turning-point structure Step 2 demands.

Step 4. Carrying the ansatz through $\mathrm{d}t=a\,\mathrm{d}\eta/[H_0\sqrt{\Omega_{m,0}-1}]$ and integrating gives the time as $t(\eta)=\dfrac{\Omega_{m,0}}{2H_0(\Omega_{m,0}-1)^{3/2}}(\eta-\sin\eta)$ . Why: the pair $a(\eta)$ , $t(\eta)$ is a cycloid — the curve traced by a point on a rolling wheel — the exact solution for a closed dust universe.

Step 5. Let $\eta$ run from $0$ to $2\pi$ : $a$ climbs to $a_{\max}$ at $\eta=\pi$ and returns to zero at $\eta=2\pi$ , giving a total lifetime $t_{\rm crunch}=\dfrac{\pi\,\Omega_{m,0}}{H_0(\Omega_{m,0}-1)^{3/2}}$ . Why: because $\eta-\sin\eta$ is symmetric about $\eta=\pi$ , the collapse is a perfect time-reversed image of the expansion — the Big Crunch mirrors the Big Bang. *Verified: the cycloid $a(\eta)=\tfrac12 a_{\max}(1-\cos\eta)$ with $t(\eta)\propto(\eta-\sin\eta)$ and prefactor $\Omega_{m,0}/[2H_0(\Omega_{m,0}-1)^{3/2}]$ is the standard closed-dust result (Ryden ch. 6, eq. 6.35); it requires $\Omega_{m,0}>1$ .*

Go deeper · Derivation C · 1 minThe redshift at which cosmic expansion switches to acceleration

Derivation CThe redshift at which cosmic expansion switches to acceleration

The goal is to find the moment a flat matter-plus- $\Lambda$ universe stops decelerating and begins to accelerate — the instant $\ddot a=0$ — and to express it as a redshift set only by the ratio $\Omega_{\Lambda,0}/\Omega_{m,0}$ . This transition is what distant Type Ia supernovae directly detect, and reproducing its measured value $z_{\rm acc}\approx0.6$ from the density budget is among the cleanest confirmations that dark energy exists and behaves like a cosmological constant.

Step 1. Start from the acceleration equation, which for matter ( $w=0$ ) and $\Lambda$ ( $w=-1$ ) reads $\ddot a/a=-\tfrac{4\pi G}{3}(\rho_m-2\rho_\Lambda)$ . Why: the combination $\rho+3P/c^2$ that sources acceleration is $+\rho_m$ for matter but $-2\rho_\Lambda$ for $\Lambda$ , because $\Lambda$ 's negative pressure over-compensates its energy density.

Step 2. Set $\ddot a=0$ , the boundary between slowing down and speeding up, which requires $\rho_m=2\rho_\Lambda$ . Why: acceleration turns on the instant matter's decelerating pull is exactly balanced by twice the dark-energy density — beyond that point $\Lambda$ dominates the source term.

Step 3. Insert the dilution laws with $\rho_\Lambda$ constant and $\rho_m\propto a^{-3}$ to get $\Omega_{m,0}a^{-3}=2\Omega_{\Lambda,0}$ , hence $a_{\rm acc}=(\Omega_{m,0}/2\Omega_{\Lambda,0})^{1/3}$ . Why: dividing out $\rho_{\rm crit,0}$ turns a statement about densities into one about scale factor, ready to convert to redshift.

Step 4. Convert to redshift through $1+z_{\rm acc}=1/a_{\rm acc}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}$ ; with $\Omega_{m,0}=0.31$ , $\Omega_{\Lambda,0}=0.69$ this gives $1+z_{\rm acc}\approx1.6$ , so $z_{\rm acc}\approx0.6$ . Why: this predicted turning redshift matches the supernova data almost exactly, closing the loop between the density budget and the observed onset of acceleration. *Verified: $1+z_{\rm acc}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}\approx1.64$ for concordance parameters, giving $z_{\rm acc}\approx0.6$ (Ryden ch. 6, Dodelson ch. 2).*

Limiting cases

The single-component flat models are the canonical scaling laws, and together they bracket every realistic universe between eternal coasting and recollapse.

Case	Result	Meaning
Radiation-dominated	$a\propto t^{1/2},\ \ \rho\propto a^{-4}$	The early universe before equality; $w=1/3$ , strongly decelerating.
Matter-dominated (Einstein–de Sitter)	$a\propto t^{2/3},\ \ \rho\propto a^{-3}$	Flat, $\Omega_m=1$ ; the classic benchmark; decelerating with $q_0=1/2$ .
$\Lambda$ -dominated (de Sitter)	$a\propto e^{Ht},\ \ \rho=\text{const}$	$w=-1$ ; exponential accelerating expansion; the asymptotic future of $\Lambda$ CDM.
Empty / curvature-dominated (Milne)	$a\propto t,\ \ \Omega_k=1$	Open, negatively curved, coasting at constant velocity; the divide between eternal expansion and recollapse for matter.
Closed matter	$a_{\max}=\dfrac{\Omega_{m,0}}{\Omega_{m,0}-1}$ , cycloid	$\Omega_{m,0}>1$ , $k=+1$ ; reaches maximum size, then recollapses in a Big Crunch.

Common trapTreating the density parameters as fixed constants of the universe

The single most common slip is to speak of "the" $\Omega_m$ or "the" $\Omega_\Lambda$ as though they were permanent properties. They are not: each $\Omega_i(a)=\rho_i(a)/\rho_{\rm crit}(a)$ evolves, because the numerators dilute at different rates $a^{-3(1+w)}$ and the denominator $\rho_{\rm crit}\propto H^2$ is itself changing. The universe that is matter-dominated today was radiation-dominated at $a<a_{\rm eq}$ and is becoming $\Lambda$ -dominated for $a>a_{m\Lambda}$ — the same universe, three balances. The subscript zero on $\Omega_{i,0}$ is a warning: these are pinned to the present epoch, and only the present. Plugging today's $\Omega_{\Lambda,0}=0.69$ into an equation meant for recombination gives nonsense, because at $z\approx1100$ dark energy was a part in $10^{9}$ of the budget, not two-thirds of it.

Worked examples

1 · Reading the cosmic timeline off the density budget (fully worked)

Take concordance $\Lambda$ CDM: $\Omega_{r,0}=9\times10^{-5}$ , $\Omega_{m,0}=0.31$ , $\Omega_{\Lambda,0}=0.69$ . Find the redshifts of matter–radiation equality, matter– $\Lambda$ equality, and the onset of acceleration, and confirm the universe is accelerating today.

Step 1. Matter–radiation equality is a ratio of parameters: $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}=9\times10^{-5}/0.31=2.9\times10^{-4}$ , so $1+z_{\rm eq}=1/a_{\rm eq}=3448$ , giving $z_{\rm eq}\approx3400$ — the epoch stamped into the height of the CMB acoustic peaks.

Step 2. Matter– $\Lambda$ equality from $\Omega_{m,0}a^{-3}=\Omega_{\Lambda,0}$ : $a_{m\Lambda}=(0.31/0.69)^{1/3}=0.766$ , so $1+z=1.306$ and $z_{m\Lambda}\approx0.31$ — dark energy took the lead only in the recent past.

Step 3. Acceleration begins where $\rho_m=2\rho_\Lambda$ : $1+z_{\rm acc}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}=(4.45)^{1/3}=1.645$ , giving $z_{\rm acc}\approx0.65$ — higher than $z_{m\Lambda}$ , so acceleration began before dark energy even reached half the budget, because it is pressure, not just density, that gravitates.

Step 4. Today's deceleration parameter: $q_0=\tfrac12\Omega_{m,0}+\Omega_{r,0}-\Omega_{\Lambda,0}=0.155+0.00009-0.69=\mathbf{-0.53}$ . Because $q_0<0$ , the expansion is accelerating now, exactly as the supernova Hubble diagram shows.

2 · The age of the universe: Einstein–de Sitter versus ΛCDM (fully worked)

With $h=0.68$ , compute the Hubble time, then the age of the universe in a flat matter-only (Einstein–de Sitter) model and in concordance $\Lambda$ CDM, and explain the gap.

Step 1. The Hubble time is $1/H_0$ . With $H_0=68\ \mathrm{km\,s^{-1}\,Mpc^{-1}}=2.20\times10^{-18}\ \mathrm{s^{-1}}$ , we get $1/H_0=4.54\times10^{17}\ \mathrm{s}=\mathbf{14.4\ Gyr}$ . This is the age the universe would have if it had always coasted at today's rate (the Milne limit).

Step 2. Einstein–de Sitter: pure matter decelerates the whole time, and the exact integral gives $t_0=\tfrac{2}{3H_0}$ . Numerically $t_0=\tfrac23\times14.4=\mathbf{9.6\ Gyr}$ . Because matter always slowed a faster past expansion, the model crams its history into a shorter span.

Step 3. $\Lambda$ CDM: the age integral $t_0=\int_0^1 \mathrm{d}a/[aH(a)]$ evaluates to $t_0=\tfrac{2}{3H_0\sqrt{\Omega_{\Lambda,0}}}\,\mathrm{arcsinh}\!\sqrt{\Omega_{\Lambda,0}/\Omega_{m,0}}=0.955/H_0$ . Numerically $t_0=0.955\times14.4=\mathbf{13.8\ Gyr}$ .

Step 4. The gap is the fingerprint of dark energy. The Einstein–de Sitter age of $9.6\ \mathrm{Gyr}$ is younger than the oldest globular clusters ( $\sim12\text{–}13\ \mathrm{Gyr}$ ) — a genuine historical crisis that a positive $\Lambda$ resolves, stretching the age to $13.8\ \mathrm{Gyr}$ and comfortably older than any star.

3 · The lifetime of a marginally closed universe

A closed, matter-only universe has $\Omega_{m,0}=1.1$ . Find its maximum scale factor and its lifetime from Big Bang to Big Crunch, using $1/H_0=14.4\ \mathrm{Gyr}$ .

Step 1. Maximum size: $a_{\max}=\Omega_{m,0}/(\Omega_{m,0}-1)=1.1/0.1=\mathbf{11}$ — it swells to eleven times its present size before turning around.

Step 2. Lifetime from the cycloid (Derivation 2): $t_{\rm crunch}=\dfrac{\pi\,\Omega_{m,0}}{H_0(\Omega_{m,0}-1)^{3/2}}=\dfrac{\pi\times1.1}{(0.1)^{3/2}}\,\dfrac{1}{H_0}=109\,\dfrac{1}{H_0}=109\times14.4\ \mathrm{Gyr}\approx\mathbf{1.6\times10^{3}\ Gyr}$ . A universe only ten percent above critical density is very long-lived: as $\Omega_{m,0}\to1^{+}$ the cycloid flattens and the lifetime diverges.

Physics of the portrait

The modern census of cosmological models — flat, open, closed, expanding forever or recollapsing — was not read off from data but written down from pure theory in 1922, by a meteorologist who took Einstein's equations more seriously than Einstein did.

Origins · 1 minAlexander Friedmann and how this was discovered

Alexander Friedmann

1888–1925 · dynamical solutions of general relativity that founded modern cosmology

When Einstein applied general relativity to the cosmos in 1917 he assumed the universe must be static, inserting the cosmological constant $\Lambda$ by hand to force it. Alexander Friedmann, trained in fluid dynamics and meteorology, refused the assumption. In his 1922 paper "On the Curvature of Space" he solved the field equations for a homogeneous, isotropic universe without demanding it stand still, and found an entire family of dynamical solutions — universes that expand, that expand and recollapse, that carry positive, zero, or negative spatial curvature. He wrote down, decades before the observations, the very taxonomy this article is built on: the density decides the geometry, and the geometry decides the fate. Einstein at first published a note claiming Friedmann had erred, then retracted it in 1923, though he still thought the expanding solutions physically irrelevant. Friedmann never saw himself vindicated — he died of typhoid in 1925 at thirty-seven, four years before Hubble's velocity–distance relation turned his abstract expanding universe into the measured one. Lemaître rediscovered the expanding solution in 1927 and connected it to galaxy recession; but the equations that classify cosmological models still bear Friedmann's name, and $\Omega_m$ , $\Omega_\Lambda$ , $\Omega_k$ are modern labels on the branches of his 1922 tree.

History

1917

— Einstein introduces the cosmological constant to enforce a static universe

1922

— Friedmann derives the expanding and recollapsing model solutions

1927

— Lemaître links the expanding solutions to observed galaxy recession

1929

— Hubble measures the velocity–distance relation, evidence for expansion

1998

— Type Ia supernova surveys reveal cosmic acceleration and revive Λ

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 5–6 (the definitive undergraduate treatment of single- and multi-component models) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29 · Dodelson, Modern Cosmology, 2nd ed., ch. 1–2 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Peacock, Cosmological Physics, ch. 3 · Planck 2018 results VI, Cosmological Parameters, A&A 641, A6 (2020).

Provenance. The Friedmann, fluid, and acceleration equations and the dilution law $\rho\propto a^{-3(1+w)}$ are recalled by name from the prerequisite articles and re-grounded here from Ryden ch. 4–6 and Carroll & Ostlie ch. 29. A verifier should confirm: the single-fluid solutions $a\propto t^{2/[3(1+w)]}$ and de Sitter $e^{Ht}$ ; $z_{\rm eq}\approx3400$ , $a_{\rm eq}\approx2.9\times10^{-4}$ , and $t_0\approx13.8\ \mathrm{Gyr}$ for Planck 2018 parameters ( $\Omega_{m,0}\approx0.31$ , $\Omega_{\Lambda,0}\approx0.69$ , $\Omega_{r,0}\approx9\times10^{-5}$ , $h\approx0.68$ ); the corrected closed-dust cycloid prefactor $t(\eta)=\Omega_{m,0}(\eta-\sin\eta)/[2H_0(\Omega_{m,0}-1)^{3/2}]$ ; $q_0=\tfrac12\Omega_{m,0}+\Omega_{r,0}-\Omega_{\Lambda,0}\approx-0.53$ ; and $\rho_{\rm crit,0}\approx8.7\times10^{-27}\ \mathrm{kg\,m^{-3}}$ for $h=0.68$ . Numerical parameter values are from Planck 2018 (A&A 641, A6).

Prerequisites. The Friedmann Equations · The Fluid and Acceleration Equations · Equation of State in Cosmology · The FLRW Metric · The Expansion of the Universe and the Scale Factor. Leads to. Spatial Curvature of the Universe · Cosmological Redshift · The Hubble Constant.

Ask the tutor: ask me to trace any universe's whole life from its density budget — hand me $\Omega_m$ , $\Omega_\Lambda$ , and $\Omega_k$ and I will tell you its eras, its age, and whether it ends in fire, ice, or a Crunch.

What makes something "a cosmological model" as opposed to just a Friedmann solution?

A cosmological model is a fully specified member of the Friedmann family — you have chosen $H_0$ and the density parameters $(\Omega_{r,0},\Omega_{m,0},\Omega_{\Lambda,0},\Omega_{k,0})$ , and with them the entire history $a(t)$ is locked in. The bare Friedmann equation is the framework; a model is the framework with the numbers filled in. "Einstein–de Sitter" is a model (flat, matter only, $\Omega_m=1$ ); "de Sitter" is a model ( $\Lambda$ only); " $\Lambda$ CDM with Planck parameters" is the one we believe describes reality. So the phrase implies a definite point in parameter space, from which distances, ages, and the ultimate fate follow deterministically.

If the density parameters change with time, why tabulate today's values?

Because they are enough. The present-day $\Omega_{i,0}$ , together with $H_0$ , are a complete set of initial data: substitute them into the master equation and you can compute $\Omega_i$ at any other epoch you like. We anchor at "today" simply because that is where we take our measurements — the CMB, supernovae, and galaxy surveys all report their findings as constraints on $\Omega_{i,0}$ . The evolving fractions $\Omega_i(a)$ are outputs of the model, not independent inputs.

Why does adding a cosmological constant make the universe older?

Age is the integral $t_0=\int_0^1 \mathrm{d}a/[aH(a)]$ , and $\Lambda$ lowers $H(a)$ over much of cosmic history. A matter-only universe decelerates the whole way, so it expanded faster in the past and reached its present size quickly — a short age, $2/(3H_0)$ . Adding $\Lambda$ inserts an early epoch of comparatively slow expansion followed by late acceleration; that slow middle stretches the time to reach $a=1$ , lengthening the age from $9.6$ to $13.8\ \mathrm{Gyr}$ — exactly what rescued cosmology from the embarrassment of stars older than the universe.

How can two universes with the same H₀ today have different ages and futures?

$H_0$ fixes only the current slope $\dot a/a$ , not the curvature of the $a(t)$ curve — that is set by $q_0$ , which depends on the density mix. In the second figure all four models cross the present with identical tangents, but a closed universe is bending downward (headed for recollapse), Einstein–de Sitter bends down gently, Milne is a straight coast, and $\Lambda$ CDM has already turned to bend upward. Same instantaneous velocity, different accelerations: run them backward and they hit $a=0$ at different times (different ages); run them forward and they end in a Crunch, an eternal coast, or an exponential runaway.

Is dark energy the same thing as the cosmological constant Λ?

Not necessarily — $\Lambda$ is the simplest candidate, but not the only one. A cosmological constant has an exactly fixed equation of state $w=-1$ and a density that never changes. "Dark energy" is the broader label for whatever drives acceleration, and it might have $w\neq-1$ or a $w$ that evolves with time (quintessence and similar models). The two are tested by measuring $w$ : so far it is consistent with $-1$ to within about five percent, which is why $\Lambda$ CDM remains the default. The 2024–2025 DESI hints of evolving dark energy, if confirmed, would be the first evidence that the accelerator is something richer than a bare constant.

Where does the deceleration parameter q₀ come from, and why the factor 1 + 3w?

$q_0\equiv-\ddot a a/\dot a^2$ is a dimensionless measure of how the expansion is changing, defined so positive means slowing and negative means speeding up. It comes straight from the acceleration equation $\ddot a/a=-\tfrac{4\pi G}{3}\sum_i(\rho_i+3P_i/c^2)$ . The combination $\rho+3P/c^2=\rho(1+3w)$ is what gravitates in general relativity — pressure has weight — so each component enters $q_0$ with weight $(1+3w_i)$ : matter $+1$ , radiation $+2$ , $\Lambda$ gives $-2$ . Collecting and normalising yields $q_0=\tfrac12\Omega_{m,0}+\Omega_{r,0}-\Omega_{\Lambda,0}$ , so the sign of $q_0$ is a direct verdict on whether gravity or dark energy is currently winning.