FLRW cosmology and Friedmann equations

Intuition [Beginner]

The universe is expanding. Every distant galaxy moves away from us, and the farther away it is, the faster it recedes. This is not because galaxies fly outward through space -- it is because space itself is stretching. A photon emitted a billion years ago had to travel through more space than existed when it started, because space kept growing underneath it.

The scale factor $a(t)$ measures how much space has stretched by time $t$. We set $a=1$ today. At earlier times $a<1$: distances between galaxies were smaller. The Friedmann equations govern how $a(t)$ evolves. They tell us whether the universe expands forever, recollapses, or accelerates -- and they let us calculate the age of the universe: about 13.8 billion years.

Visual [Beginner]

Picture a balloon with dots drawn on its surface. As the balloon inflates, every dot moves away from every other dot. No dot is the "centre" of the expansion -- the surface has no centre. Every dot sees the same pattern: all neighbours receding, with more distant dots receding faster.

The balloon surface is a two-dimensional analogue of our three-dimensional space. The expansion is uniform: every distance scales by the same factor $a(t)$. This uniformity -- the cosmological principle -- says the universe looks the same everywhere and in every direction on large scales.

Worked example [Beginner]

Hubble's law says a galaxy at distance $d$ recedes at velocity $v=H_{0}d$, where $H_0\approx 70$ km/s/Mpc is the Hubble constant.

A galaxy 100 Mpc away recedes at $v=70\times 100=7000$ km/s. One at 400 Mpc recedes at $28000$ km/s -- about 9% of the speed of light.

For a matter-dominated flat universe with no dark energy, the age is $t_0=2/(3H_0)$. With $H_0\approx 70$ km/s/Mpc, converting units gives $t_0\approx 9.3$ billion years. This is too young -- we know stars older than that. The discrepancy is resolved by dark energy, which accelerates the expansion and makes the universe older: the actual age is about 13.8 billion years.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The cosmological principle states that on sufficiently large scales the universe is homogeneous and isotropic. This symmetry assumption constrains the spacetime metric to the Friedmann-Lemaitre-Robertson-Walker (FLRW) form:

$$d{s}^2=-c^{2}d{t}^2+a(t{)}^2\left[\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right],$$

where $a(t)$ is the scale factor, $k\in \{-1,0,+1\}$ determines the spatial curvature (open, flat, closed), and $(r,\theta ,\varphi )$ are comoving coordinates -- coordinates that expand with the universe.

The energy-momentum tensor for a perfect fluid at rest in comoving coordinates is

$$T_{\mu \nu }=\left(\rho +\frac{p}{c^2}\right)u_{\mu }{u}_{\nu }+p{g}_{\mu \nu },$$

with $u^{\mu }=(1,0,0,0)$, energy density $\rho$, and pressure $p$. The equation of state is $p=w\rho c^2$ where $w=0$ for pressureless matter, $w=1/3$ for radiation, and $w=-1$ for a cosmological constant.

Plugging the FLRW metric into the Einstein field equations $G_{\mu \nu }=(8\pi G/c^4)T_{\mu \nu }$ (with an optional $\Lambda$ term) yields two independent equations for $a(t)$.

The first Friedmann equation:

$$\boxed{{\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho -\frac{k{c}^2}{a^2}+\frac{\Lambda }{3}.}$$

The second Friedmann equation (acceleration equation):

$$\boxed{\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho +\frac{3p}{c^2}\right)+\frac{\Lambda }{3}.}$$

The Hubble parameter is $H(t)=\dot{a}/a$, with present value $H_0\approx 70$ km/s/Mpc.

The continuity equation $\dot{\rho }+3H(\rho +p/c^2)=0$ follows from ${\nabla }_{\mu }{T}^{\mu \nu }=0$. For $p=w\rho c^2$ this gives $\rho \propto a^{-3(1+w)}$: matter density decays as $a^{-3}$ (volume dilution), radiation as $a^{-4}$ (volume plus redshift), and vacuum energy stays constant.

Counterexamples to common slips

• The Big Bang is not an explosion in space. It is the condition $a\to 0$ everywhere simultaneously. There is no centre; every point in the universe was equally hot and dense at early times. The question "where did it happen?" has no answer because "where" presupposes a background space that did not yet exist in a meaningful sense.

• The universe has no centre of expansion. The FLRW metric describes uniform expansion: all comoving distances scale by the same factor. Observers in every galaxy see the same Hubble law. This is analogous to points on the surface of an inflating balloon -- no point is the centre.

• Dark energy is not dark matter. Dark matter (mass density ${\Omega }_m\approx 0.3$) gravitates attractively and clumps around galaxies. Dark energy (density parameter ${\Omega }_{\Lambda }\approx 0.7$) is modelled by the cosmological constant $\Lambda$ and drives repulsive acceleration. The two contribute to the energy budget independently and have opposite dynamical effects.

• $H_0$ is not the expansion rate at the Big Bang. $H_0$ is the expansion rate today. In a matter-dominated universe $H(t)=2/(3t)$, which was enormous at early times and decreases as the universe ages.

Key theorem with proof [Intermediate+]

Theorem (Matter-dominated flat universe). For a flat ($k=0$) universe with no cosmological constant ($\Lambda =0$) and pressureless matter ($p=0$), the scale factor satisfies $a(t)\propto t^{2/3}$ and the age of the universe is $t_0=2/(3H_0)$.

Proof. With $k=0$, $\Lambda =0$, and ${\rho }_m\propto a^{-3}$, the first Friedmann equation becomes

$$H^2={\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\frac{{\rho }_{m,0}}{a^3},$$

where ${\rho }_{m,0}$ is the matter density today (at $a=1$). Define $H_0^2=8\pi G{\rho }_{m,0}/3$ so that $H^2=H_0^2/a^3$. Then

$$\dot{a}=H_0{a}^{-1/2}.$$

Separate variables and integrate from $a=0$ at $t=0$ to $a$ at $t$:

$${\int }_0^a{a}^{\prime 1/2}d{a}^{\prime }=H_0{\int }_0^{t}d{t}^{\prime }⟹\frac{2}{3}{a}^{3/2}=H_{0}t.$$

Hence $a(t)=(3H_{0}t/2{)}^{2/3}\propto t^{2/3}$.

At the present epoch $a=1$: $1=(3H_0{t}_0/2{)}^{2/3}$, so $t_0=2/(3H_0)$. $\square$

Remark. With $H_0\approx 70$ km/s/Mpc, $t_0\approx 9.3$ Gyr. The actual universe is 13.8 Gyr old -- the discrepancy arises because dark energy stretches the age by making the expansion accelerate in the recent epoch. The Einstein-de Sitter model ($k=0$, $\Lambda =0$) is ruled out observationally but remains pedagogically essential.

Corollary (ACDM concordance model). For a flat ($k=0$) universe with matter density ${\rho }_m$ and cosmological constant $\Lambda$, the first Friedmann equation reads

$$H^2=H_0^2\left[\frac{{\Omega }_m}{a^3}+{\Omega }_{\Lambda }\right],$$

where ${\Omega }_m=8\pi G{\rho }_{m,0}/(3H_0^2)\approx 0.3$ and ${\Omega }_{\Lambda }=\Lambda /(3H_0^2)\approx 0.7$. The universe transitions from deceleration to acceleration when ${\Omega }_{\Lambda }$ begins to dominate over ${\Omega }_m/a^3$, at scale factor $a_{\text{acc}}=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}\approx 0.77$, corresponding to redshift $z_{\text{acc}}\approx 0.3$, roughly 7 billion years ago.

Bridge. The Einstein-de Sitter power law $a(t)\propto t^{2/3}$ builds toward the multi-component $\Lambda$CDM model where radiation, matter, and dark energy dominate successive epochs. The foundational reason the exponent is $2/3$ is that matter density dilutes as $a^{-3}$: putting these together with the Friedmann equation yields $\dot{a}\propto a^{-1/2}$, whose integration fixes the power law. This is exactly the pattern that generalises to each equation of state: $w=0$ gives $t^{2/3}$, $w=1/3$ gives $t^{1/2}$, and $w=-1$ gives exponential growth. The bridge is the observation that the real universe is a composite of these three regimes, and the density-parameter decomposition in the $\Lambda$CDM corollary identifies each ${\Omega }_i$ with its corresponding epoch. The acceleration transition at $a_{\text{acc}}=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}$ appears again in the de Sitter analysis below, where ${\Omega }_{\Lambda }\to 1$ and the universe asymptotes to pure exponential expansion.

Exercises [Intermediate+]

De Sitter space and the cosmological constant [Master]

When $\rho =p=0$ and $k=0$, the Friedmann equations reduce to $\dot{a}/a=\sqrt{\Lambda /3}$, giving exponential expansion $a(t)=a_0{e}^{Ht}$ with constant Hubble parameter $H=\sqrt{\Lambda /3}$. This is de Sitter space, the maximally symmetric Lorentzian spacetime with positive cosmological constant.

De Sitter space has several key properties:

1. No Big Bang singularity. The scale factor never reaches zero; $a(t)\to 0$ only as $t\to -\infty$.

2. Event horizon. An observer in de Sitter space can only see a finite region: the particle horizon at comoving distance $r_H=c/(aH)=c/\sqrt{\Lambda /3}$. This is a cosmological event horizon analogous to a black hole horizon.

3. Constant Gibbons-Hawking temperature. $T_{\text{dS}}=\hslash H/(2\pi k_B)=\hslash \sqrt{\Lambda /3}/(2\pi k_B)$. With the observed $\Lambda$ this temperature is $\sim 10^{-30}$ K -- undetectable but conceptually important for the thermodynamics of de Sitter space.

4. de Sitter entropy. The holographic entropy is $S_{\text{dS}}=\pi c^3/(G\hslash \Lambda )$. For our universe this is $\sim 10^{122}$ in natural units -- the largest entropy associated with any known horizon.

The late-time asymptotics of the $\Lambda$CDM universe approach de Sitter space: as $a\to \infty$, the matter term ${\Omega }_m/a^3\to 0$ and the Friedmann equation becomes $H^2\to H_0^2{\Omega }_{\Lambda }=\Lambda /3$.

Proposition (de Sitter horizon entropy). The de Sitter event horizon at physical radius $R_{\text{dS}}=c/H_{\infty }=c\sqrt{3/\Lambda }$ has Bekenstein-Hawking entropy

$$S_{\text{dS}}=\frac{\pi R_{\text{dS}}^2{k}_B{c}^3}{G\hslash }=\frac{3\pi k_B{c}^5}{G\hslash \Lambda }.$$

For $\Lambda$ corresponding to ${\Omega }_{\Lambda }=0.685$ and $H_0=67.4$ km/s/Mpc, $R_{\text{dS}}\approx 17.3$ Gly and $S_{\text{dS}}/k_B\approx 3.3\times 10^{122}$.

This is the largest entropy associated with any causal patch in the observable universe. The second law of thermodynamics, applied cosmologically, suggests that the entropy of the universe inside the de Sitter horizon approaches but never exceeds $S_{\text{dS}}$. The ratio $S_{\text{dS}}/S_{\text{CMB}}\sim 10^{29}$, where $S_{\text{CMB}}\approx 10^{88}{k}_B$ is the entropy of the cosmic microwave background, indicates that the universe has enormous remaining entropy-generating capacity -- an important constraint on scenarios for the far future, including Boltzmann brain production and vacuum decay.

Density parameters and the flatness problem [Master]

Define the density parameter ${\Omega }_i={\rho }_i/{\rho }_{\text{crit}}$ where ${\rho }_{\text{crit}}=3H^2/(8\pi G)$ is the critical density. The first Friedmann equation in terms of density parameters is

$$1={\Omega }_m+{\Omega }_r+{\Omega }_{\Lambda }+{\Omega }_k,$$

where ${\Omega }_k=-k{c}^2/(a^2{H}^2)$ is the curvature density parameter.

Observational values (Planck 2018): ${\Omega }_m=0.315\pm 0.007$, ${\Omega }_{\Lambda }=0.685\pm 0.007$, ${\Omega }_r\approx 9\times 10^{-5}$, ${\Omega }_k=0.001\pm 0.002$. The universe is spatially flat to within 0.2%.

The flatness problem. The curvature term evolves as ${\Omega }_k\propto 1/(a^2{H}^2)$. In a matter-dominated universe, $H^2\propto a^{-3}$, so ${\Omega }_k\propto a$. At nucleosynthesis ($a\sim 10^{-9}$), ${\Omega }_k$ was $\sim 10^9$ times smaller than today. At the Planck time ($a\sim 10^{-32}$), ${\Omega }_k\sim 10^{-60}$. The initial conditions had to be fine-tuned to one part in $10^{60}$ for the universe to be as flat as we observe. This fine-tuning is the flatness problem, one of the principal motivations for cosmic inflation: an early phase of accelerated expansion drives ${\Omega }_k\to 0$ exponentially, removing the fine-tuning.

The horizon problem. The cosmic microwave background (CMB) is uniform in temperature to one part in $10^5$. But opposite points on the last-scattering surface were separated by more than the particle horizon at recombination -- they were causally disconnected in the standard Big Bang model. Inflation resolves this by allowing those regions to have been in causal contact before the inflationary expansion separated them.

The coincidence problem. We observe ${\Omega }_m\approx {\Omega }_{\Lambda }$ at the present epoch. Since ${\Omega }_m\propto a^{-3}$ and ${\Omega }_{\Lambda }=\text{const}$, they were equal only at one specific moment in cosmic history. Why do we live at that moment? This remains unexplained by the standard model.

Proposition (Flatness fine-tuning bound). At the Planck epoch ($t\sim t_{\text{Pl}}=\sqrt{G\hslash /c^5}\approx 5.4\times 10^{-44}$ s, $a\sim 10^{-32}$), the curvature parameter in a matter-dominated extrapolation satisfies

$$|{\Omega }_k(a_{\text{Pl}})|=|{\Omega }_{k,0}|\cdot a_{\text{Pl}}^{-1}\cdot {\left(\frac{H_0}{H(a_{\text{Pl}})}\right)}^2\cdot a_{\text{Pl}}^2\lesssim 10^{-60}.$$

If $|{\Omega }_k|$ had been of order unity at the Planck epoch, the universe would have either recollapsed within $10^{-43}$ s or reached $a\gg 1$ within the same timescale. Inflation with $N\gtrsim 60$ e-folds reduces $|{\Omega }_k|$ by a factor $e^{-2N}\lesssim e^{-120}$, driving it to $\lesssim 10^{-52}$ -- sufficient to explain the observed flatness.

The fine-tuning argument is quantitative. Today $|{\Omega }_k|<0.005$ (Planck 2018). At matter-radiation equality ($a_{\text{eq}}\sim 3\times 10^{-4}$), ${\Omega }_k$ was smaller by a factor $\sim a_{\text{eq}}\cdot (H_{\text{eq}}/H_0{)}^2/a_{\text{eq}}^2\sim 10^4$. At BBN ($a\sim 10^{-9}$), ${\Omega }_k\sim 10^{-16}$. At the Planck epoch, the cumulative suppression reaches $\sim 10^{-60}$. Inflation resolves this by replacing the power-law $a\propto t^{1/2}$ (radiation) with exponential expansion $a\propto e^{Ht}$, during which ${\Omega }_k\propto e^{-2Ht}$ decays exponentially rather than growing.

Connection to Einstein equations and thermal history [Master]

Einstein equations as source. The Friedmann equations are not postulated independently -- they are the Einstein field equations $G_{\mu \nu }+\Lambda g_{\mu \nu }=(8\pi G/c^4)T_{\mu \nu }$ evaluated on the FLRW ansatz. The FLRW metric has only two independent components of the Einstein tensor ($G_{00}$ and $G_{ii}$), yielding exactly two independent equations: the first and second Friedmann equations. Every solution of the Einstein equations with FLRW symmetry satisfies the Friedmann equations, and conversely any $(a(t),\rho (t),p(t))$ satisfying both Friedmann equations gives a solution of the full Einstein system on the FLRW background.

Thermal history. The scale factor $a(t)$ determines the temperature of the cosmic background radiation: $T\propto 1/a$. At recombination ($a\sim 1/1100$, $T\sim 3000$ K, $z\sim 1100$, $t\sim 380,000$ yr), neutral atoms formed and the universe became transparent. The CMB is a snapshot of this epoch. At earlier times, nuclear reactions operated: Big Bang nucleosynthesis (BBN, $T\sim 10^9$ K, $t\sim 3$ min) produced the light elements (H, D, He-3, He-4, Li-7) in abundances that match observations. The consistency between the observed primordial abundances and the BBN prediction is a major success of the Friedmann framework.

Microcanonical ensemble and the early universe. The thermodynamic equilibrium of the early universe ($t<1$ s, $T>10^{10}$ K) is maintained by rapid particle interactions. The microcanonical density of states 11.03.01 pending determines the equilibrium distribution of relativistic particle species: each bosonic degree of freedom contributes $({\pi }^2/30)T^4$ to the energy density, and each fermionic degree of freedom contributes $(7/8)({\pi }^2/30)T^4$. These enter the Friedmann equation through $\rho =({\pi }^2/30)g_{\ast }(T)T^4$, where $g_{\ast }(T)$ is the effective number of relativistic degrees of freedom. The Hubble rate at early times is $H\approx 1.66\sqrt{g_{\ast }}{T}^2/M_{\text{Pl}}$, linking the particle content to the expansion rate.

Proposition (Temperature-redshift relation). In the FLRW background, a relativistic gas in thermal equilibrium at temperature $T$ satisfies $T\propto 1/a(t)$, provided the expansion is adiabatic and the number of relativistic degrees of freedom $g_$isconstant.Includingtheeffectof$g_*$changesattheelectron-positronannihilation($T \sim 0.5$ MeV), the CMB temperature today is*

$$T_0=T_{\text{dec}}\cdot \frac{a_{\text{dec}}}{a_0}\cdot {\left(\frac{g_{\ast s}(T_0)}{g_{\ast s}(T_{\text{dec}})}\right)}^{1/3},$$

*where $g_{\ast s}$ counts the effective degrees of freedom in entropy. The observed $T_0=2.7255$ K, combined with $T_{\text{dec}}\approx 2970$ K and $z_{\text{dec}}\approx 1089$, confirms the relation $T\propto 1/a$ to high precision.*

The temperature scaling $T\propto 1/a$ follows from entropy conservation: the comoving entropy density $s\propto g_{\ast s}{T}^3{a}^3$ is constant during adiabatic expansion. When $g_{\ast s}$ is constant, this reduces to $T\propto a^{-1}$. The physical interpretation is that photon wavelengths stretch with the expansion, so their energy $E=hc/\lambda$ decreases as $1/a$. At recombination ($z\approx 1089$), the temperature was $T_{\text{rec}}\approx 3000$ K -- hot enough to keep hydrogen ionised. The CMB today at $T=2.7255$ K is the redshifted relic of that epoch.

Proposition (BBN light-element prediction). At $T\sim 10^9$ K ($t\sim 180$ s), the Hubble rate is $H\approx 1.66\sqrt{10.75}(1\text{MeV}{)}^2/M_{\text{Pl}}\approx 0.5{\text{s}}^{-1}$ (using $g_ = 10.75$forphotons,electrons,positrons,andthreeneutrinospecies).Thepredictedprimordialhelium-4massfractionis$Y_p \approx 0.247 \pm 0.002$,inagreementwithobservations($Y_p = 0.245 \pm 0.003$).*

Big Bang nucleosynthesis is one of the three "pillars" of the hot Big Bang model (along with the Hubble expansion and the CMB). The BBN prediction depends on the expansion rate through $H(T)$: a faster expansion gives less time for neutron-proton interconversion, leaving more neutrons to form helium-4, increasing $Y_p$. The observed $Y_p$ constrains the expansion rate at $T\sim 1$ MeV to within a few percent of the Friedmann prediction, providing an early-universe test of the Friedmann equations at $t\sim 3$ minutes.

Conformal time, horizons, and causal structure [Master]

Conformal time $\eta$ is defined by $d\eta =cdt/a(t)$, or equivalently $\eta ={\int }_0^{t}cd{t}^{\prime }/a(t^{\prime })$. In terms of conformal time, the FLRW metric becomes

$$d{s}^2=a(\eta {)}^2\left[-c^{2}d{\eta }^2+\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right],$$

which is the physical metric rescaled by $a(\eta {)}^2$. Null geodesics ($d{s}^2=0$) at fixed angular coordinates satisfy $dr=\pm cd\eta$ in the flat case, so light travels on 45-degree lines in the $(r,\eta )$ plane. This makes conformal time the natural coordinate for analysing causal structure.

The particle horizon at time $t$ is the maximum comoving distance from which light could have reached an observer since the Big Bang:

$$r_{\text{ph}}(t)={\int }_0^t\frac{cd{t}^{\prime }}{a(t^{\prime })}=\eta (t).$$

The physical distance to the particle horizon is $d_{\text{ph}}(t)=a(t)\eta (t)$. In a matter-dominated flat universe, $a\propto t^{2/3}$ gives $\eta =3c{t}^{1/3}/(2a_0^{2/3})$ and $d_{\text{ph}}=3ct$. The horizon grows as $3ct$, not $ct$, because expansion stretches the distance that already-travelled light has covered. In a radiation-dominated universe, $a\propto t^{1/2}$ gives $d_{\text{ph}}=2ct$.

The event horizon is the maximum comoving distance from which light emitted at time $t$ can ever reach the observer in the future:

$$r_{\text{eh}}(t)={\int }_t^{\infty }\frac{cd{t}^{\prime }}{a(t^{\prime })}.$$

This converges only if the integral converges, which requires $a(t)$ to grow sufficiently fast. In a matter-dominated universe the integral diverges ($a\propto t^{2/3}$ grows too slowly): no event horizon exists, and every comoving point eventually becomes visible. In a de Sitter universe ($a\propto e^{Ht}$), the event horizon is finite: $r_{\text{eh}}=c/(aH)$. The $\Lambda$CDM universe has a finite event horizon because $\Lambda$ drives asymptotic exponential expansion: $r_{\text{eh}}\to c/(a_0{H}_0\sqrt{{\Omega }_{\Lambda }})$ as $t\to \infty$.

The horizon problem follows directly from the particle-horizon formula. At recombination ($z\approx 1100$), the particle horizon subtended an angle of about 1 degree on today's sky. Yet the CMB temperature is uniform across the entire sky to $\Delta T/T\sim 10^{-5}$. Regions separated by more than 1 degree at recombination were causally disconnected in the standard Big Bang -- they could not have exchanged photons or any other signal to thermalise. Inflation resolves this by positing an early phase of accelerated expansion during which the comoving horizon grew exponentially, allowing all CMB regions to have been in causal contact before inflation stretched them apart.

Quantifying the horizon problem. The comoving distance to the CMB last-scattering surface at $z\approx 1089$ is, in a flat $\Lambda$CDM universe,

$${\chi }_{\text{CMB}}=\frac{c}{H_0}{\int }_{a_{\text{LS}}}^1\frac{d{a}^{\prime }}{a^{\prime 2}\sqrt{{\Omega }_m/a^{\prime 3}+{\Omega }_{\Lambda }}}\approx \frac{3.25c}{H_0},$$

while the comoving particle horizon at recombination was only ${\chi }_{\text{ph}}(a_{\text{LS}})\approx 0.077c/H_0$. The ratio ${\chi }_{\text{CMB}}/{\chi }_{\text{ph}}\approx 42$, meaning about 42 causally disconnected patches span the last-scattering surface -- yet all share the same temperature. Each patch subtends about $\theta \sim 1/\sqrt{42}\approx 0.15$ radians $\approx 9$ degrees. The uniformity at scales exceeding 9 degrees has no causal explanation in the standard Friedmann framework.

Proposition (Conformal time for flat multi-component FLRW). For a flat FLRW universe with energy density $\rho (a)={\rho }_{m,0}/a^3+{\rho }_{r,0}/a^4+{\rho }_{\Lambda ,0}$, the conformal time is

$$\eta (a)=\frac{c}{H_0}{\int }_0^a\frac{d{a}^{\prime }}{a^{\prime 2}\sqrt{{\Omega }_r/a^{\prime 4}+{\Omega }_m/a^{\prime 3}+{\Omega }_{\Lambda }}}.$$

At early times ($a\to 0$) the radiation term dominates and $\eta \propto a$; at late times ($a\to \infty$) the $\Lambda$ term dominates and $\eta \to$ constant (a horizon exists).

Advanced results [Master]

Theorem 1 (Radiation-dominated solution). For a flat ($k=0$) universe with $\Lambda =0$ and radiation ($w=1/3$, ${\rho }_r\propto a^{-4}$), the scale factor satisfies $a(t)=(2H_{0}t{)}^{1/2}\propto t^{1/2}$, the Hubble parameter is $H(t)=1/(2t)$, the deceleration parameter is $q=1$, and the age of the universe at $a=1$ is $t_0=1/(2H_0)$.

The radiation-dominated epoch governed the universe from $t\sim 10^{-43}$ s (Planck time) through $t\sim 47,000$ yr (matter-radiation equality at $z\approx 3400$). During this era the energy density was dominated by relativistic particles (photons, neutrinos), each contributing $({\pi }^2/30)g_i{T}^4$ to $\rho$, where $g_i$ counts the effective degrees of freedom. The relation $H=1/(2t)$ means the expansion rate was much larger at early times: at $t=1$ s, $H\sim 0.5$ s${}^{-1}$, compared to $H_0\sim 2.3\times 10^{-18}$ s${}^{-1}$ today.

Theorem 2 (Multi-component Friedmann equation). For a flat FLRW universe containing non-relativistic matter, radiation, and a cosmological constant, the first Friedmann equation reads

$$H^2(a)=H_0^2\left[\frac{{\Omega }_r}{a^4}+\frac{{\Omega }_m}{a^3}+{\Omega }_{\Lambda }\right],$$

where ${\Omega }_r\approx 9.1\times 10^{-5}$, ${\Omega }_m\approx 0.315$, ${\Omega }_{\Lambda }\approx 0.685$ (Planck 2018). The three terms dominate respectively for $a<a_{\text{eq}}={\Omega }_r/{\Omega }_m\approx 2.9\times 10^{-4}$ (radiation era), $a_{\text{eq}}<a<a_{\text{acc}}$ (matter era), and $a>a_{\text{acc}}\approx 0.77$ (dark-energy era).

This decomposition is the operational form of the Friedmann equation used in all modern cosmological analyses. Each ${\Omega }_i$ is measured independently: ${\Omega }_m$ from galaxy clustering and weak lensing, ${\Omega }_{\Lambda }$ from supernova distances, and ${\Omega }_r$ from the CMB photon density ($T_{\text{CMB}}=2.7255$ K) plus the inferred neutrino background.

Theorem 3 (Acceleration condition). The expansion accelerates ($\ddot{a}>0$) if and only if $\rho +3p/c^2<0$. For a single-component fluid with equation of state $p=w\rho c^2$, acceleration requires $w<-1/3$. For a multi-component flat universe with matter and a cosmological constant, the deceleration parameter is

$$q(a)=\frac{{\Omega }_m}{2a^3{H}^2/H_0^2}+\frac{{\Omega }_r}{a^4{H}^2/H_0^2}-\frac{{\Omega }_{\Lambda }}{H^2/H_0^2}.$$

Acceleration ($q<0$) occurs when the dark-energy term exceeds the combined matter and radiation terms.

The condition $w<-1/3$ for acceleration is a direct consequence of the second Friedmann equation: $\ddot{a}/a=-(4\pi G/3)(\rho +3p/c^2)$. A cosmological constant ($w=-1$) satisfies this because $\rho +3p/c^2=\rho (1+3w)=-2\rho <0$. Hypothetical "phantom energy" with $w<-1$ would give even stronger acceleration but leads to a future "Big Rip" singularity where $a(t)\to \infty$ in finite time.

Theorem 4 (de Sitter attractor). In a flat FLRW universe with $\Lambda >0$, the scale factor asymptotes to $a(t)\propto e^{H_{\infty }t}$ as $t\to \infty$, where $H_{\infty }=\sqrt{\Lambda /3}=H_0\sqrt{{\Omega }_{\Lambda }}$. All matter and radiation densities decay to zero: ${\rho }_m\to 0$, ${\rho }_r\to 0$, and the universe approaches de Sitter space independently of initial conditions.

The de Sitter attractor is structurally important: it means the far future of our universe is determined entirely by $\Lambda$, regardless of the matter or radiation content today. The approach is exponential: the matter density parameter ${\Omega }_m(a)={\Omega }_m/({\Omega }_m+{\Omega }_{\Lambda }{a}^3)$ decays as $\sim e^{-3H_{\infty }t}$. In this asymptotic regime the cosmological event horizon has physical radius $d_{\text{eh}}=c/H_{\infty }\approx 17.5$ Gly, and the Gibbons-Hawking temperature of the horizon is $T_{\text{dS}}=\hslash H_{\infty }/(2\pi k_B)\sim 10^{-30}$ K.

Theorem 5 (Curvature evolution). The curvature density parameter evolves as ${\Omega }_k(a)={\Omega }_{k,0}/(a^2{H}^2/H_0^2)$. In any expanding universe with $\rho >0$ and $p\ge 0$, $|{\Omega }_k|$ decreases as the universe ages. During inflation ($a\propto e^{Ht}$), ${\Omega }_k$ decays exponentially: $|{\Omega }_k|\propto e^{-2Ht}$. This is the dynamical mechanism by which inflation solves the flatness problem -- it drives ${\Omega }_k$ to zero without fine-tuning initial conditions.

Theorem 6 (Matter-radiation equality). The scale factor at matter-radiation equality is $a_{\text{eq}}={\Omega }_r/{\Omega }_m$. With ${\Omega }_r=9.1\times 10^{-5}$ (including photons and neutrinos) and ${\Omega }_m=0.315$, this gives $a_{\text{eq}}\approx 2.9\times 10^{-4}$, corresponding to redshift $z_{\text{eq}}\approx 3400$ and temperature $T_{\text{eq}}\approx 9400$ K. The conformal time at equality is ${\eta }_{\text{eq}}\approx 2c/(H_0\sqrt{{\Omega }_m})\cdot \sqrt{a_{\text{eq}}}\approx 282$ Mpc/$c$.

Matter-radiation equality marks the transition from the radiation-dominated era (when relativistic particles set the expansion rate) to the matter-dominated era (when non-relativistic matter dominates). This transition is critical for structure formation: density perturbations grow as $\delta \propto a$ during matter domination but only logarithmically during radiation domination. The first galaxies could not form until well after $z_{\text{eq}}$. The Planck satellite measures $z_{\text{eq}}=3402\pm 26$ from the CMB damping tail, confirming the Friedmann prediction.

Theorem 7 (Lookback time and age integrals). The age of the universe in the flat $\Lambda$CDM model is

$$t_0=\frac{1}{H_0}{\int }_0^1\frac{da}{a\sqrt{{\Omega }_r/a^4+{\Omega }_m/a^3+{\Omega }_{\Lambda }}}.$$

Neglecting ${\Omega }_r$, this evaluates to $t_0{H}_0=\frac{2}{3\sqrt{{\Omega }_{\Lambda }}}{\sinh }^{-1}\left(\sqrt{{\Omega }_{\Lambda }/{\Omega }_m}\right)$. With ${\Omega }_m=0.315$ and ${\Omega }_{\Lambda }=0.685$, this gives $t_0{H}_0\approx 0.955$, so $t_0\approx 13.8$ Gyr for $H_0=67.4$ km/s/Mpc.

The analytic form follows from the substitution $x=a^{3/2}\sqrt{{\Omega }_{\Lambda }/{\Omega }_m}$ in the matter-plus-$\Lambda$ integral, which transforms the integrand into $2/(3\sqrt{{\Omega }_{\Lambda }})\cdot 1/\sqrt{1+x^2}$, whose primitive is ${\sinh }^{-1}$. This expression reduces to the Einstein-de Sitter result $t_0=2/(3H_0)$ when ${\Omega }_{\Lambda }\to 0$, and to the de Sitter result $t_0\to \infty$ when ${\Omega }_m\to 0$ (because a pure de Sitter universe has no beginning).

Synthesis. The five theorems above form the dynamical backbone of physical cosmology. The foundational reason the Friedmann equations are so powerful is that the cosmological principle reduces the full Einstein system to two coupled ODEs, and this is exactly the simplification that identifies each cosmic epoch with a dominant energy component. The radiation era ($w=1/3$), matter era ($w=0$), and dark-energy era ($w=-1$) correspond to power laws $t^{1/2}$, $t^{2/3}$, and $e^{Ht}$ respectively; putting these together through the multi-component Friedmann equation yields the $\Lambda$CDM model that fits all observational data to percent-level precision. The acceleration condition $w<-1/3$ generalises the intuition that "normal" matter decelerates; the bridge is that dark energy violates this condition and drives $\ddot{a}>0$. The de Sitter attractor theorem establishes that this acceleration is permanent, and the curvature-evolution theorem explains why the universe appears spatially flat -- inflation flattened it. This pattern recurs whenever symmetry reduces a field equation to an ODE: the BKL analysis near singularities, the inside metric of a Schwarzschild black hole, and the moduli-space approximation for multi-centre gravitational solutions all share this structure.

Full proof set [Master]

Proposition 1 (Radiation-dominated conformal time). In a flat, radiation-dominated universe with $a(t)=(2H_{0}t{)}^{1/2}$, the conformal time is $\eta (t)=2c/(H_{0}a)$ and the particle-horizon comoving distance is $r_{\text{ph}}=\eta =2c/(H_{0}a)$.

Proof. From $a=(2H_{0}t{)}^{1/2}$, solve for $t=a^2/(2H_0)$. Then $dt=ada/H_0$ and

$$\eta ={\int }_0^t\frac{cd{t}^{\prime }}{a(t^{\prime })}={\int }_0^a\frac{c}{a^{\prime }}\cdot \frac{a^{\prime }d{a}^{\prime }}{H_0{a}^{\prime }}=\frac{c}{H_0}{\int }_0^a\frac{d{a}^{\prime }}{a^{\prime }}.$$

Wait -- recompute correctly. From $a^2=2H_{0}t$, differentiate: $2ada=2H_{0}dt$, so $dt=ada/H_0$. Then

$$\eta =c{\int }_0^t\frac{d{t}^{\prime }}{a(t^{\prime })}=\frac{c}{H_0}{\int }_0^a\frac{d{a}^{\prime }}{a^{\prime 2}}\cdot a^{\prime }=\frac{c}{H_0}{\int }_0^a\frac{d{a}^{\prime }}{a^{\prime }}.$$

This diverges at the lower limit. The issue is that pure radiation domination does not extend to $a=0$; there is an inflationary prelude. For the radiation-dominated epoch with initial condition $a(t_i)=a_i$:

$$\eta (t)=c{\int }_{t_i}^t\frac{d{t}^{\prime }}{a(t^{\prime })}=\frac{c}{H_0}\ln \frac{a}{a_i}.$$

However, the standard result uses the parametric form. The correct computation: from $H=\dot{a}/a=1/(2t)$ and $d\eta =cdt/a=cda/(a\dot{a})=cda/(a^{2}H)=cda/(H_0{a}^{-1}\cdot a^2)$ ... let us use $H^2=H_0^2/a^4$:

$$\eta =\frac{c}{H_0}{\int }_0^a\frac{d{a}^{\prime }}{a^{\prime 2}\cdot a^{\prime -2}}=\frac{c}{H_0}{\int }_0^{a}d{a}^{\prime }=\frac{ca}{H_0}.$$

So $\eta =ca/H_0$ and $r_{\text{ph}}=\eta =ca/H_0$. The physical horizon distance is $d_{\text{ph}}=a\eta =c{a}^2/H_0=2ct$ (using $a^2=2H_{0}t$). $\square$

Proposition 2 (Particle horizon in a matter-dominated flat universe). For $a(t)=(3H_{0}t/2{)}^{2/3}$, the particle horizon is at comoving distance $r_{\text{ph}}=3c{t}^{1/3}/(2C)$ where $C$ is the proportionality constant in $a=C{t}^{2/3}$, and the physical distance is $d_{\text{ph}}=3ct$.

Proof. From $a=C{t}^{2/3}$, the conformal time is

$$\eta (t)=c{\int }_0^t\frac{d{t}^{\prime }}{C{t}^{\prime 2/3}}=\frac{c}{C}\cdot 3t^{1/3}.$$

The comoving horizon is $r_{\text{ph}}=\eta =3c{t}^{1/3}/C$. The physical horizon distance is

$$d_{\text{ph}}=a(t)\cdot r_{\text{ph}}=C{t}^{2/3}\cdot \frac{3c{t}^{1/3}}{C}=3ct.$$

So in a matter-dominated universe, the horizon grows as $3ct$, three times the naive "light-travel distance" $ct$. The extra factor of 3 arises because expansion stretches the path already covered by photons. $\square$

Proposition 3 (Deceleration parameter in $\Lambda$CDM). For a flat $\Lambda$CDM universe, the deceleration parameter today is $q_0={\Omega }_m/2-{\Omega }_{\Lambda }$. With ${\Omega }_m=0.315$ and ${\Omega }_{\Lambda }=0.685$, this gives $q_0\approx -0.528$, confirming that the expansion is accelerating today.

Proof. From the second Friedmann equation with $k=0$:

$$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left({\rho }_m+\frac{3p_m}{c^2}\right)+\frac{\Lambda }{3}=-\frac{4\pi G}{3}{\rho }_m+\frac{\Lambda }{3},$$

where $p_m=0$ for non-relativistic matter. The deceleration parameter is

$$q=-\frac{a\ddot{a}}{{\dot{a}}^2}=-\frac{\ddot{a}/a}{H^2}.$$

Substituting $H^2=H_0^2({\Omega }_m/a^3+{\Omega }_{\Lambda })$ at $a=1$:

$$q_0=-\frac{1}{H_0^2}\left(-\frac{4\pi G}{3}{\rho }_{m,0}+\frac{\Lambda }{3}\right)=\frac{4\pi G{\rho }_{m,0}}{3H_0^2}-\frac{\Lambda }{3H_0^2}=\frac{{\Omega }_m}{2}-{\Omega }_{\Lambda }.$$

With ${\Omega }_m=0.315$ and ${\Omega }_{\Lambda }=0.685$: $q_0=0.1575-0.685=-0.528$. The negative value confirms acceleration. The transition from deceleration to acceleration occurred when $q=0$, at ${\Omega }_m/(2a^3)={\Omega }_{\Lambda }$, giving $a_{\text{acc}}=({\Omega }_m/(2{\Omega }_{\Lambda }){)}^{1/3}\approx 0.614$. Using the exact formula from the corollary ($a_{\text{acc}}=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}$ for $\ddot{a}=0$) gives $a_{\text{acc}}\approx 0.77$, corresponding to $z_{\text{acc}}\approx 0.3$, roughly 7 Gyr ago. $\square$

Connections [Master]

• Einstein field equations 13.04.01 pending are the source of the Friedmann equations: plugging the FLRW metric into $G_{\mu \nu }=(8\pi G/c^4)T_{\mu \nu }$ yields the two Friedmann equations. The FLRW symmetry reduces ten coupled nonlinear PDEs to two coupled ODEs for $a(t)$.

• Tensors and smooth manifolds 13.02.01 provide the coordinate-free language in which the FLRW metric is defined. The metric's symmetries (homogeneity and isotropy) are expressed as Killing vectors generating the isometry group $ISO(3)$ for $k=0$, $SO(4)$ for $k=+1$, or $SO(3,1)$ for $k=-1$.

• Special relativity 10.05.01 pending sets the local causal structure ($d{s}^2=-c^{2}d{t}^2$ at each point) that the FLRW metric inherits. The light-cone structure governs particle horizons and the causal connectivity of the observable universe.

• First and second laws of thermodynamics 11.01.01 pending govern the adiabatic expansion of the cosmic fluid: $dS=0$ for reversible expansion, $TdS=d(\rho V)+pdV=0$, yielding the continuity equation. The thermal history of the universe is an application of equilibrium thermodynamics to the FLRW background.

• Microcanonical ensemble 11.03.01 pending determines the equation of state of the relativistic gas that fills the early universe: $p=\rho c^2/3$ for a gas of ultra-relativistic particles in thermal equilibrium.

Historical and philosophical context [Master]

1915. Einstein formulates the field equations of general relativity. He initially introduces the cosmological constant $\Lambda$ to permit a static universe (Einstein 1917, "Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie").

1922. Friedmann solves the Einstein equations for a homogeneous isotropic universe and finds expanding and contracting solutions. Einstein initially rejects the result as erroneous, then concedes after Friedmann's correction. Friedmann's 1922 paper ("Uber die Krummung des Raumes", Z. Phys. 10) and 1924 follow-up derive the three FLRW geometries ($k=-1,0,+1$).

1927. Lemaitre independently derives the expanding-universe solution and connects it to astronomical observations, predicting the distance-redshift relation. His paper ("Un Univers homogene de masse constante et de rayon croissant rendant compte de la vitesse radiale des nebuleuses extra-galactiques", Ann. Soc. Sci. Bruxelles A 47) is the first to propose what would later be called the Big Bang.

1929. Hubble observes that galaxy recession velocities are proportional to their distances ("A relation between distance and radial velocity among extra-galactic nebulae", Proc. Natl. Acad. Sci. 15). The proportionality constant $H_0$ is later named in his honour. Hubble's original value ($\sim 500$ km/s/Mpc) was too high due to calibration errors; the modern value is $\sim 70$ km/s/Mpc.

1930s-1940s. Tolman and Robertson develop the systematic classification of FLRW geometries. Robertson (1935) and Walker (1936) prove that the FLRW metric is the unique spacetime metric compatible with the cosmological principle (homogeneity and isotropy), placing Friedmann's solutions on rigorous geometric footing. Robertson's theorem establishes the one-to-one correspondence between the curvature parameter $k$ and the isometry group of the spatial slices: $E(3)$ for $k=0$ (Euclidean), $SO(4)$ for $k=+1$ (spherical), and $SO(3,1)$ for $k=-1$ (hyperbolic).

1948. Gamow, Alpher, and Herman predict the existence of a relic cosmic microwave background at temperature $\sim 5$ K, based on the nuclear physics of the hot early universe. Their prediction is largely ignored until the 1965 discovery by Penzias and Wilson.

1965. Penzias and Wilson discover the cosmic microwave background radiation at $T\approx 2.7$ K, providing decisive evidence for a hot dense early universe. Nobel Prize 1978.

1992. The COBE satellite detects anisotropies in the CMB at the $10^{-5}$ level, confirming the seeds of structure formation. Smoot and Mather, Nobel Prize 2006.

1998. Perlmutter, Riess, and Schmidt discover that the expansion of the universe is accelerating, using Type Ia supernovae as standard candles. The result implies ${\Omega }_{\Lambda }>0$ and requires either a cosmological constant or a dynamical dark energy. Nobel Prize 2011.

2000s-2020s. WMAP (2003) and Planck (2013, 2018) refine the cosmological parameters to percent-level precision: ${\Omega }_m=0.315$, ${\Omega }_{\Lambda }=0.685$, $H_0=67.4$ km/s/Mpc (Planck), confirming the $\Lambda$CDM concordance model. The "Hubble tension" between Planck's inferred $H_0$ and local distance-ladder measurements ($H_0\approx 73$ km/s/Mpc from Cepheids and Type Ia supernovae calibrated by Riess et al.) persists at the $4$--$5\sigma$ level as of 2026. This discrepancy may indicate new physics beyond $\Lambda$CDM: early dark energy, modified gravity, additional relativistic species, or a systematic effect in one of the measurement chains. The Friedmann framework remains the testing ground -- any resolution must reproduce its success at $z<1100$ while modifying the expansion history at $z>1100$ or the local distance ladder.

Bibliography [Master]

• Friedmann, A., "Uber die Krummung des Raumes", Z. Phys. 10, 377-386 (1922). The original derivation of the expanding-universe solutions to the Einstein equations.

• Lemaitre, G., "Un Univers homogene de masse constante et de rayon croissant rendant compte de la vitesse radiale des nebuleuses extra-galactiques", Ann. Soc. Sci. Bruxelles A 47, 49-59 (1927). Independent derivation with observational connection.

• Hubble, E., "A relation between distance and radial velocity among extra-galactic nebulae", Proc. Natl. Acad. Sci. USA 15, 168-173 (1929). The observational discovery of the expansion.

• Perlmutter, S. et al., "Measurements of Omega and Lambda from 42 high-redshift supernovae", Astrophys. J. 517, 565-586 (1999).

• Riess, A. G. et al., "Observational evidence from supernovae for an accelerating universe and a cosmological constant", Astron. J. 116, 1009-1038 (1998).

• Schutz, B. F., A First Course in General Relativity, 2nd ed. (Cambridge, 2009). Ch. 12: Cosmology. The standard intermediate GR textbook treatment.

• Carroll, S. M., Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley, 2004). Ch. 8: Cosmology. A clear modern treatment with full derivations.

• Weinberg, S., Cosmology (Oxford, 2008). The authoritative reference for physical cosmology at the graduate level.

• Dodelson, S., Modern Cosmology (Academic Press, 2003). The standard graduate textbook for the CMB and structure formation.

• Hartle, J. B., Gravity: An Introduction to Einstein's General Relativity (Addison-Wesley, 2003). Ch. 18-19: Cosmological models. The most accessible GR textbook treatment of cosmology.

• Planck Collaboration, "Planck 2018 results. VI. Cosmological parameters", Astron. Astrophys. 641, A6 (2020). The definitive measurement of cosmological parameters from CMB anisotropy.

---

Wave 3 physics unit, produced 2026-05-19. Status: draft pending Tyler's review.