Friedmann Equations

From ten field equations to two ODEs

Einstein's 1915 general relativity is a system of ten coupled nonlinear partial differential equations relating the geometry of spacetime to its matter content: G_μν + Λg_μν = 8πG T_μν / c⁴. Solving them in full generality is hopeless. But cosmology imposes an enormous simplification — the so-called cosmological principle: on scales above a hundred megaparsecs, the universe is statistically homogeneous (no preferred location) and isotropic (no preferred direction). Modern surveys confirm this beautifully — the cosmic microwave background is the same temperature to one part in 10⁵ in every direction, and galaxy counts agree across the sky at the per-cent level.

Homogeneity and isotropy force the metric into a unique form, the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, parameterised by a single function a(t) (the scale factor, conventionally set to 1 today) and a discrete spatial curvature k ∈ {−1, 0, +1}:

ds² = −c² dt² + a(t)² [ dr² / (1 − kr²) + r² (dθ² + sin²θ dφ²) ]

The matter content must, by the same symmetries, behave like a perfect fluid characterised by a uniform energy density ρ(t) and pressure p(t). Substituting this metric and stress–energy tensor into the Einstein equations collapses the system to two independent ordinary differential equations for a(t), plus one continuity equation that follows from energy conservation.

The first Friedmann equation

The 00-component of the Einstein equations gives the celebrated first Friedmann equation:

(ȧ/a)² = (8πG/3) ρ − kc²/a² + Λc²/3

The left-hand side is the square of the Hubble parameter H(t) ≡ ȧ/a — the fractional rate at which the universe is expanding. The three contributions on the right are, in order, ordinary energy density, spatial curvature, and the cosmological constant. Read it as an energy-conservation identity: the kinetic term of a co-moving observer is balanced by a gravitational potential proportional to ρ, a curvature contribution, and a vacuum-energy contribution.

Setting k = Λ = 0 in this equation yields a special density called the critical density:

ρ_crit = 3H²/(8πG)
ρ_crit,0 ≈ 9.5 × 10⁻²⁷ kg/m³   (about 5 hydrogen atoms per cubic metre)

Divide every density by ρ_crit and we get the dimensionless density parameters Ω_i = ρ_i / ρ_crit. The first Friedmann equation evaluated today then reads

1 = Ω_m + Ω_r + Ω_Λ + Ω_k

— a single constraint linking all the ingredients of the universe. Observation puts Ω_m ≈ 0.31, Ω_Λ ≈ 0.69, Ω_r ≈ 9 × 10⁻⁵, and Ω_k consistent with zero. The geometry is essentially flat.

The acceleration equation

The spatial components of the Einstein equations, combined with the first equation, give the second Friedmann equation — usually called the acceleration equation:

ä/a = − (4πG/3) (ρ + 3p/c²) + Λc²/3

This is where pressure enters. A surprise of GR is that positive pressure also gravitates: an ordinary fluid with p > 0 contributes positively to (ρ + 3p/c²) and thus accelerates the expansion backwards. The only way to get ä > 0 — accelerating expansion — is to find an ingredient with sufficiently negative pressure that ρ + 3p/c² becomes negative. For a fluid with equation of state p = wρc², this requires w < −1/3.

The 1998 discovery, by the Supernova Cosmology Project and the High-Z Supernova Search Team, that distant Type Ia supernovae are dimmer than expected was a measurement of ä > 0. Translated into the acceleration equation, it demanded that the dominant component of the present universe has w ≈ −1 — a cosmological constant, or something close to one. The Nobel Prize in 2011 was awarded for that direct measurement of cosmic acceleration.

The continuity equation

The two Friedmann equations are not independent. Their compatibility — equivalently, the local conservation of stress-energy ∇_μ T^μν = 0 — gives the continuity equation:

ρ̇ + 3 (ȧ/a) (ρ + p/c²) = 0

Combined with an equation of state p = wρc² (with w constant), this integrates immediately:

ρ(a) ∝ a^(−3(1+w))

This single relation tells you how every cosmic fluid dilutes with expansion. The standard ingredients are:

Component	w	ρ scaling	Why this w
Cold matter (CDM + baryons)	0	ρ ∝ a⁻³	Pressureless; density just dilutes with volume
Radiation (photons + relativistic neutrinos)	1/3	ρ ∝ a⁻⁴	Volume dilution plus cosmological redshift
Spatial curvature	−1/3	ρ_k ∝ a⁻²	Bookkeeping equivalent for the kc²/a² term
Cosmological constant Λ	−1	ρ_Λ = const	Vacuum energy does not dilute
Quintessence / dynamical DE	−1 < w < −1/3	ρ ∝ a^(−3(1+w))	Slowly rolling scalar field
Phantom dark energy	w < −1	ρ grows with a	Speculative; leads to a Big Rip in finite time

Inserting these scalings into the first Friedmann equation gives the standard form of H(z) used in every cosmology fitting code:

H(z) = H₀ √( Ω_m (1+z)³ + Ω_r (1+z)⁴ + Ω_k (1+z)² + Ω_Λ )

This is the master formula for translating redshifts into expansion histories. Plug in the measured Ω_i and you can compute the look-back time to any z, the angular-diameter distance, the volume per redshift slice — every quantity that links observation to model.

Three eras of cosmic history

Because different components dilute at different rates, the universe has been dominated by different fluids in different eras. The transitions are set by equality of densities. The H(z) formula and the continuity equation together imply three canonical analytic solutions.

Era	Dominant fluid	a(t)	Redshift range	Approx age
Radiation-dominated	w = 1/3	a ∝ t^(1/2)	z > 3400	0 → 50 kyr
Matter-dominated (Einstein–de Sitter)	w = 0	a ∝ t^(2/3)	3400 > z > 0.3	50 kyr → 9 Gyr
Λ-dominated (de Sitter)	w = −1	a ∝ exp(H_Λ t)	z < 0.3	9 Gyr → ∞

Radiation-matter equality occurred at z_eq ≈ 3400 — about 50,000 years after the Big Bang. Matter-Λ equality is more recent, at z ≈ 0.3, only about 4 billion years ago. We live in the transition zone — close enough to the Λ era that expansion is now accelerating, but still close enough to the matter era that structure formed within the past few billion years. This timing is the so-called coincidence problem: why are Ω_m and Ω_Λ comparable today, when they have such different histories?

Canonical analytic solutions

Specialise the first Friedmann equation to a single component and integrate.

Radiation-dominated (w = 1/3, flat): ρ ∝ a⁻⁴, so (ȧ/a)² ∝ a⁻⁴ and ȧ ∝ a⁻¹. Integrating gives a(t) ∝ t^(1/2). The early universe — nucleosynthesis at three minutes, recombination at 380,000 years — is set by this scaling.

Matter-dominated (w = 0, flat — the Einstein–de Sitter solution): ρ ∝ a⁻³, so (ȧ/a)² ∝ a⁻³ and ȧ ∝ a⁻^(1/2). Integrating gives a(t) ∝ t^(2/3). Most of cosmic structure formed in this era.

Lambda-dominated (w = −1, flat — the de Sitter solution): ρ_Λ is constant, so (ȧ/a)² = Λc²/3 = constant. The unique solution is exponential: a(t) = a₀ exp(H_Λ t), where H_Λ = c √(Λ/3) is the asymptotic Hubble rate. The far future is de Sitter — exponential expansion forever, structure freezing out, every distant galaxy redshifting to infinity.

Inflation in the very early universe is a fourth de Sitter phase, driven by a much higher effective Λ from the inflaton field — and it left its imprint in the near-flatness of the universe and the scale-invariant primordial fluctuation spectrum that seeded structure.

Worked example: age of the universe

For the standard flat ΛCDM cosmology, the age of the universe is

t₀ = ∫₀^1 da / [a H(a)]
   = (1/H₀) ∫₀^1 da / √(Ω_m/a + Ω_Λ a²)

Adopt Planck values H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, Ω_Λ = 0.685. Numerically the integral evaluates to 0.951, so

t₀ = 0.951 / H₀
   = 0.951 × (9.78 Gyr × 100 / 67.4)
   = 13.79 Gyr

which agrees with the Planck quoted value 13.787 ± 0.020 Gyr. (The 9.78 factor is 1/H₀ in Gyr when H₀ is in km/s/Mpc divided by 100.) Note how the integral folds in the entire expansion history — a different cosmology gives a different age. An Einstein–de Sitter universe (Ω_m = 1) gives t₀ = (2/3)/H₀ ≈ 9.7 Gyr, too young for the oldest known stars; the addition of Λ is what reconciles the model with stellar ages.

The Hubble tension

The value of H₀ is now the most-debated single number in cosmology. Two independent measurement chains, both rigorous, disagree by more than 5σ.

Method	H₀ (km/s/Mpc)	What is measured	What is assumed
Planck CMB (early)	67.4 ± 0.5	CMB power spectrum at z ≈ 1100	ΛCDM + Friedmann eqs to today
SH0ES distance ladder (late)	73.0 ± 1.0	Parallax → Cepheids → SN Ia	Only local physics; no cosmology
TRGB (Carnegie)	69.8 ± 1.7	Tip of red giant branch instead of Cepheids	Local; partial overlap with both
BAO + BBN	67.5 ± 1.0	Baryon acoustic oscillations + Big-Bang nucleosynthesis prior	Early-universe physics
Strong lensing (H0LiCOW)	73.3 +1.7/−1.8	Time-delay between lensed quasar images	Local; lens models
Gravitational-wave standard sirens	~70 ± 5	NS–NS merger luminosity + redshift	Local; few events so far

The pattern is striking: every method that measures H₀ directly in the local universe gives 72–74. Every method that measures the early universe and propagates forward through the Friedmann equation gives 67–68. If both are right, then the propagation step is wrong — and that requires new physics modifying the expansion history between recombination and today. Leading proposals include early dark energy (a brief boost to ρ_Λ around z ≈ 3000), a non-standard neutrino sector that changes Ω_r, or modifications to gravity that alter Friedmann's right-hand side.

Why the universe is flat

The curvature term in the Friedmann equation goes as kc²/a². As the universe expands, this term dilutes more slowly than radiation (a⁻⁴) or matter (a⁻³), so curvature should dominate at late times if it has any non-zero starting value. Yet observation puts |Ω_k| < 0.005 today. To be that flat now, the universe must have been flat to at least one part in 10⁶² at the Planck epoch — a colossal fine-tuning known as the flatness problem.

Inflation solves it. A brief de Sitter phase in the very early universe stretches any initial curvature by a factor of ~e⁶⁰ ≈ 10²⁶, driving Ω_k toward zero exponentially. The same mechanism explains why the CMB is uniform on scales never causally connected by ordinary expansion — the horizon problem.

Variants and extensions

• ΛCDM (the standard model). Flat (Ω_k = 0) FLRW with cold dark matter, baryons, photons, three neutrino species, and a cosmological constant. Six free parameters, fits the CMB plus large-scale structure plus supernovae plus BAO to remarkable precision — but with the H₀ tension and the σ₈ tension as open puzzles.
• wCDM. Replaces Λ with a constant-w dark-energy fluid. Current data give w = −1.03 ± 0.03, consistent with Λ. Best constraints come from DES and DESI BAO measurements.
• w₀-w_a parameterisation. Allows w(a) = w₀ + w_a (1 − a) — a leading-order Taylor expansion of an evolving equation of state. DESI Year-1 results in 2024 hinted at non-zero w_a.
• Early dark energy (EDE). A scalar field that briefly behaves like dark energy around recombination, then decays away. Designed specifically to relieve the H₀ tension by raising H(z) at z ~ 3000 without spoiling the CMB fit.
• Modified gravity (f(R), DGP, MOND-like). Replace the left-hand side of the Einstein equations rather than the right. These typically modify the Friedmann equation by adding extra terms — often constrained by combining background expansion with structure-growth observations.
• Bianchi anisotropic cosmologies. Drop the isotropy assumption while keeping homogeneity. Useful for studying CMB anomalies but constrained tightly by observation.
• Brans–Dicke and scalar–tensor. Promote Newton's G to a dynamical field; modifies the first Friedmann equation in ways constrained by primordial nucleosynthesis abundances.

Why these two equations are foundational

• Predicted the Big Bang. Friedmann's 1922 paper showed — five years before Hubble's data — that the universe must be expanding or contracting. Extrapolated backward, a(t) → 0 in finite time. That mathematical singularity, refined by Lemaître's 1927 'primeval atom', is the Big Bang.
• Predicted the CMB. Run the equations into the past and the universe gets hotter; at z ≈ 1100 (T ≈ 3000 K), hydrogen recombines and photons decouple. Gamow, Alpher and Herman predicted this relic radiation in 1948; Penzias and Wilson discovered it in 1965.
• Predicted Big-Bang nucleosynthesis. The radiation-dominated solution a ∝ t^(1/2) sets the temperature–time relation T ≈ 1 MeV (t/s)^(−1/2). Combined with nuclear cross sections, this predicts the primordial abundances of ²H, ³He, ⁴He and ⁷Li to within per-cent accuracy — confirmed.
• Quantified dark matter. Galaxy rotation curves and CMB acoustic peaks each require Ω_m ≈ 0.3 — but Big-Bang nucleosynthesis caps baryons at Ω_b ≈ 0.05. The Friedmann framework forces the difference to be non-baryonic dark matter.
• Quantified dark energy. 1998 supernovae require ä > 0, which the acceleration equation translates into Ω_Λ ≈ 0.69. The Friedmann equations turned a single observation into the discovery of 69 % of the cosmic budget.
• Established the Hubble tension. The very precision of the equations is what makes the H₀ tension a tension at all. Comparing 67 with 73 only matters because the inference chain is so tight — a 1 % effect would be invisible in any pre-Friedmann cosmology.

Common pitfalls

• Reading H = ȧ/a as a velocity. H has units of inverse time, not velocity. Recession velocity is v = Hd; v > c is allowed at large d because cosmic expansion is not motion through space, it is the stretching of space itself.
• Confusing curvature k with the curvature of spacetime. k ∈ {−1, 0, +1} is the spatial curvature of the universe at a fixed time — a slicing-dependent concept. Spacetime is always curved by virtue of having a(t) at all.
• Assuming Λ is "dark energy". Λ is the simplest model of dark energy, with w = −1. "Dark energy" allows for w ≠ −1 and w(t). Current data are consistent with Λ but do not require it.
• Forgetting that "the universe is expanding" is a coordinate statement. What is invariant is that distances measured by ruler at different times grow — but this only applies on scales above galaxy clusters. The Milky Way is bound by gravity and does not participate in cosmic expansion.
• Misusing Hubble's law at z > 1. The linear law v = H₀d is only valid at small z. At higher redshift you must use the full integral over H(z) to get distances correctly.
• Adding Ω_Λ to ρ but not p. A cosmological constant contributes equally with sign (ρ_Λ = +Λc²/8πG, p_Λ = −Λc²/8πG), so ρ_Λ + 3p_Λ/c² = −2Λc²/8πG ≠ +Λ. The acceleration equation makes this difference explicit; the first Friedmann equation tucks it inside a sign convention.