The Fluid and Acceleration Equations

The first Friedmann equation is only half a cosmology. It fixes the expansion rate $H=\dot a/a$ from the total energy density at one instant, but says nothing about what happens next — not how that density changes as space grows, nor whether the expansion is winding down or running away. The two equations in this article supply the missing dynamics, and between them they organize every era from the radiation fireball to the dark-energy future. The fluid equation is energy conservation for an expanding universe. As a comoving patch grows, the matter inside does pressure work on its surroundings, and because every neighbour pushes back symmetrically, there is nowhere for energy to flow — yet the total energy in a comoving volume still changes whenever the pressure is nonzero. That is not a paradox but the exact general-relativistic bookkeeping in $\nabla_\mu T^{\mu\nu}=0$ , and it is what makes radiation dilute faster than dust and vacuum energy not dilute at all. The acceleration equation — the second Friedmann equation — carries one of the deepest surprises in physics: pressure gravitates. Newton let only mass attract; Einstein adds a $3P/c^2$ term, so the source of cosmic gravity is the combination $\varepsilon+3P$ . For matter and radiation this is positive and the expansion decelerates, as intuition demands. But a component with sufficiently negative pressure flips the sign, turning gravity repulsive — the single mechanism behind both primordial inflation and the accelerating expansion measured today. The three equations are not independent: any one follows from the other two, a redundancy that is the cosmological face of the Bianchi identity.

Key takeaways

The fluid (continuity) equation, $\dot\varepsilon+3\tfrac{\dot a}{a}(\varepsilon+P)=0$ , is local energy conservation: it couples energy density $\varepsilon$ to pressure $P$ through the expansion rate.
It follows either from the first law $\mathrm dE=-P\,\mathrm dV$ applied to a comoving volume, or from the vanishing covariant divergence of the stress-energy tensor, $\nabla_\mu T^{\mu\nu}=0$ .
The acceleration equation, $\tfrac{\ddot a}{a}=-\tfrac{4\pi G}{3c^2}(\varepsilon+3P)$ , shows that pressure as well as energy density gravitates, so matter and radiation slow the expansion.
Any component with $w<-1/3$ (so $P<-\varepsilon/3$ ) makes $\varepsilon+3P<0$ and drives acceleration; a cosmological constant, $w=-1$ , gives exponential de Sitter growth.
Feeding a constant equation of state $P=w\varepsilon$ into the fluid equation gives $\varepsilon\propto a^{-3(1+w)}$ : matter as $a^{-3}$ , radiation as $a^{-4}$ , vacuum as constant.
Only two of the three equations (Friedmann, fluid, acceleration) are independent — a consistency check rooted in the contracted Bianchi identity.
Key idea: pressure does two jobs in cosmology — it does $P\,\mathrm dV$ work in the fluid equation (setting how density dilutes) and it sources gravity in the acceleration equation (with $\varepsilon+3P$ deciding accelerate versus decelerate).

The core insight

Pressure does two jobs in cosmology: through the fluid equation it does work as the universe expands, altering how energy density dilutes, and through the acceleration equation it acts as a source of gravity, with the combination $\varepsilon+3P$ determining whether the expansion accelerates or decelerates.

Where this lives today

These equations are the working core of every modern cosmological analysis. The Planck cosmic-microwave-background measurement, combined with Type Ia supernova compilations (from the 1998 High-z and Supernova Cosmology Project teams to today's Pantheon+ sample) and baryon acoustic oscillation surveys like DESI, is in essence a precision fit of the Friedmann, fluid and acceleration equations to data, extracting $\Omega_m$ , $\Omega_\Lambda$ and the dark-energy equation of state $w$ . The headline result — $w=-1.03\pm0.03$ , consistent with a pure cosmological constant — is a direct verdict from the acceleration equation, since it is $\varepsilon+3P$ that the accelerating expansion measures. The live frontier is whether $w$ is exactly $-1$ or drifts with time: DESI's 2024–2025 data hint at evolving dark energy, which would force the fluid equation to be integrated with a time-varying $w$ rather than the tidy power law of equation (5). At the other end of cosmic history, the same acceleration equation with $P\approx-\varepsilon$ underwrites inflation, and the hunt for its imprint — primordial gravitational waves in CMB polarization (BICEP/Keck, Simons Observatory, the future LiteBIRD) — is a search for the era when $\varepsilon+3P$ was hugely negative. Every large code, from CAMB and CLASS to N-body suites, integrates the fluid equation per species and audits with the acceleration equation.

The physical picture

Picture a comoving box of proper volume $V\propto a^3$ filled with cosmic fluid. As the universe expands the box grows and the pressure inside pushes outward, doing work on everything around it. Every neighbouring box does exactly the same in return, so the boundary work is perfectly symmetric and no net energy crosses any surface — yet the fluid still loses internal energy, precisely as if it had expanded adiabatically against a piston. That is the entire physical content of the fluid equation, $\mathrm dE=-P\,\mathrm dV$ : there is no heat flow in a homogeneous universe, only $P\,\mathrm dV$ work. The first figure makes this bookkeeping and its gravitational consequence one picture.

Figure

Two jobs of pressure: a comoving box of proper volume V ∝ a³ does P dV work as it expands, draining internal energy through the fluid equation, while the sign of ε + 3P fixes whether the expansion decelerates for w above minus one third or accelerates for w below it, with the coasting threshold at w = minus one third

Why should pressure weigh in on gravity at all? In relativity, energy and the flux of momentum both curve spacetime, and pressure is momentum flux — a gas at high pressure streams large momentum across every surface, sourcing gravity in its own right. The factor of three in $\varepsilon+3P$ counts the three spatial directions of that flux (the trace of the pressure tensor). For matter and radiation the sum is positive and gravity halts the expansion; for a cosmological constant, $P=-\varepsilon$ makes $\varepsilon+3P=-2\varepsilon<0$ , and its "anti-gravity" pushes space apart forever. Which component wins is decided across cosmic history, because each dilutes at its own rate — the punchline of the second figure.

Figure

How the cosmic energy budget scales with the scale factor a: radiation ε ∝ a⁻⁴ in red, matter ε ∝ a⁻³ in blue and vacuum ε constant in amber, crossing at matter–radiation equality z ≈ 3400 and matter–Λ equality z ≈ 0.3, drawn with Planck values Ω_m ≈ 0.315, Ω_r ≈ 9×10⁻⁵ and Ω_Λ ≈ 0.685

The redundancy among the three equations is a feature: general relativity conserves energy and momentum automatically, so writing down Friedmann plus the fluid equation guarantees the acceleration equation for free. Every numerical cosmology code exploits this, integrating two equations and watching the third as a running consistency check.

The core equations

The fluid equation is the continuity equation for the cosmic energy density. It states that the rate of change of $\varepsilon$ is set by the expansion rate acting on the enthalpy density $\varepsilon+P$ — density and pressure enter together because both the volume change and the pressure work matter. It is the first law of thermodynamics written for an expanding, homogeneous universe.

\dot{\varepsilon} + 3\,\frac{\dot{a}}{a}\,(\varepsilon + P) = 0 \qquad (1)

where: $\varepsilon$ — total energy density (energy per unit proper volume); $P$ — isotropic pressure of the cosmic fluid; $a(t)$ — scale factor of the FLRW metric; $\dot a/a=H$ — Hubble parameter, the fractional expansion rate; $\dot\varepsilon$ — time rate of change of energy density. Reading it: with $P=0$ (dust) the equation says $\dot\varepsilon=-3H\varepsilon$ , pure dilution by volume; a positive pressure drains $\varepsilon$ faster, a negative one slower. The acceleration equation — the second Friedmann equation — governs the second derivative of the scale factor. Its content is that the "active gravitational mass" of a relativistic fluid is not $\varepsilon/c^2$ but $(\varepsilon+3P)/c^2$ . The minus sign means that ordinary, positive-pressure matter pulls $\ddot a$ downward: gravity decelerates cosmic expansion.

\frac{\ddot{a}}{a} = -\frac{4\pi G}{3 c^2}\,(\varepsilon + 3P) \qquad (2)

where: $\ddot a$ — second time derivative of the scale factor (cosmic acceleration); $G$ — Newton's gravitational constant; $c$ — speed of light; $\varepsilon+3P$ — the active gravitational combination, positive for matter and radiation. Reading it: acceleration ( $\ddot a>0$ ) requires $\varepsilon+3P<0$ , that is a large enough negative pressure — nothing in Newtonian gravity can produce it. For orientation, the first Friedmann equation is included here as the equation that fixes $H$ ; the fluid and acceleration equations are its dynamical companions, and any two of the three imply the third.

H^2 = \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3c^2}\,\varepsilon - \frac{k c^2}{a^2} \qquad (3)

where: $H$ — Hubble parameter; $k$ — curvature constant ( $+1,0,-1$ , or carrying units of inverse length squared); $a$ — scale factor. Reading it: curvature appears here but is absent from equation (2), because it enters as a constant that differentiates away. To close the system one specifies an equation of state relating pressure to density. For the cosmic fluids the simplest and most useful choice is barotropic and linear: a single dimensionless number $w$ per component.

P = w\,\varepsilon \qquad (4)

where: $w$ — dimensionless equation-of-state parameter: $0$ for pressureless matter (dust), $1/3$ for radiation, $-1$ for vacuum energy. Reading it: $w$ is the one input the dynamics needs from microphysics; everything else is geometry and conservation. Substituting the equation of state into the fluid equation and integrating gives the master scaling law of cosmology — how each component's density falls as the universe grows.

\varepsilon(a) = \varepsilon_0 \left(\frac{a}{a_0}\right)^{-3(1+w)} \qquad (5)

where: $\varepsilon_0$ — present-day density of the component; $a_0$ — present-day scale factor (usually set to 1); $w$ — equation-of-state parameter (assumed constant). Reading it: $w=0\Rightarrow a^{-3}$ (matter), $w=1/3\Rightarrow a^{-4}$ (radiation), $w=-1\Rightarrow a^0$ (vacuum) — exactly the three slopes of the second figure. To quantify whether the expansion speeds up or slows, one defines the dimensionless deceleration parameter, evaluated today.

q_0 \equiv -\frac{\ddot a\, a}{\dot a^2}\bigg|_0 = \frac{1}{2}\sum_i \Omega_{i,0}\,(1+3w_i) \qquad (6)

where: $q_0$ — present deceleration parameter; $q_0>0$ means the expansion is slowing, $q_0<0$ means accelerating; $\Omega_{i,0}$ — present density parameter of component $i$ ; $w_i$ — its equation-of-state parameter. Reading it: the sign is a weighted vote across the components, each weighted by its density and its $(1+3w_i)$ . Finally, for a flat universe dominated by a single component the Friedmann and scaling equations integrate to closed-form growth histories — power laws for ordinary fluids and an exponential for vacuum.

a(t) \propto t^{\frac{2}{3(1+w)}}\ \ (w\neq -1), \qquad a(t)\propto e^{Ht}\ \ (w=-1) \qquad (7)

where: $t$ — cosmic time; $w$ — equation-of-state parameter; $H$ — the (constant) Hubble rate in the vacuum-dominated case. Reading it: matter gives $a\propto t^{2/3}$ , radiation $a\propto t^{1/2}$ , and vacuum the de Sitter exponential that never stops.

Derivation AThe fluid equation from the first law of thermodynamics

The goal of this derivation is to show that the fluid equation is nothing more exotic than energy conservation for an adiabatically expanding gas, dressed in cosmological notation. We take a comoving sphere whose proper volume scales as $a^3$ , insist that no heat crosses its boundary because the universe is homogeneous, and apply the first law $\mathrm dE=-P\,\mathrm dV$ . Carrying the derivatives through delivers equation (1) and makes transparent why a nonzero pressure changes how density dilutes.

Step 1. Write the internal energy of the comoving sphere as $E=\varepsilon V$ with proper volume $V=\tfrac43\pi(a\,r_c)^3\propto a^3$ , where $r_c$ is the fixed comoving radius. Why: separating the constant comoving radius from the time-dependent scale factor isolates all the time dependence of the volume into $a^3$ .

Step 2. Impose the first law in adiabatic form, $\mathrm dE=-P\,\mathrm dV$ (that is, $\mathrm dQ=0$ ), so $\dot E=-P\dot V$ . Why: in a homogeneous universe every point is equivalent, so there is no temperature gradient and nowhere for heat to flow — the only way to change $E$ is the pressure work done as the boundary moves.

Step 3. Differentiate $E=\varepsilon V$ with the product rule to get $\dot\varepsilon\,V+\varepsilon\dot V=-P\dot V$ . Why: both the density and the volume change in time, and both contributions to $\dot E$ must be kept before the pressure term is balanced against them.

Step 4. Use $\dot V/V=3\dot a/a$ (from $V\propto a^3$ ) and divide through by $V$ to reach $\dot\varepsilon+3\tfrac{\dot a}{a}(\varepsilon+P)=0$ . Why: dividing by $V$ removes the arbitrary comoving size, leaving a purely local statement about $\varepsilon$ , $P$ and the expansion rate — the fluid equation, valid patch by patch everywhere.

Go deeper · Derivation B · 1 minThe acceleration equation from the first two

Derivation BThe acceleration equation from the first two

The aim here is to demonstrate the celebrated redundancy of the cosmological equations by deriving the acceleration equation without any new physics — using only the first Friedmann equation and the fluid equation. We differentiate Friedmann in time, notice the curvature term drops out, and eliminate $\dot\varepsilon$ with the fluid equation. What survives is equation (2), pressure factor of three and all, confirming that only two of the three equations carry independent content.

Step 1. Write the Friedmann equation as $\dot a^2=\tfrac{8\pi G}{3c^2}\varepsilon a^2-kc^2$ and differentiate with respect to time: $2\dot a\ddot a=\tfrac{8\pi G}{3c^2}(\dot\varepsilon a^2+2\varepsilon a\dot a)$ . Why: the constant curvature term $-kc^2$ has zero derivative and disappears, which is exactly why spatial curvature never appears in the acceleration equation.

Step 2. Divide through by $2\dot a\,a$ to isolate the acceleration ratio: $\tfrac{\ddot a}{a}=\tfrac{4\pi G}{3c^2}\big(\tfrac{\dot\varepsilon\,a}{\dot a}+2\varepsilon\big)$ . Why: this puts the equation into a form containing a single awkward term $\dot\varepsilon a/\dot a$ , which the fluid equation is tailor-made to replace.

Step 3. Insert the fluid equation in the form $\dot\varepsilon=-3\tfrac{\dot a}{a}(\varepsilon+P)$ , so that $\dot\varepsilon a/\dot a=-3(\varepsilon+P)$ . Why: this is the only place new information enters, and it is information we already trust from the thermodynamic derivation above.

Step 4. Combine the terms: $\tfrac{\ddot a}{a}=\tfrac{4\pi G}{3c^2}(-3\varepsilon-3P+2\varepsilon)=-\tfrac{4\pi G}{3c^2}(\varepsilon+3P)$ . Why: the $2\varepsilon$ and $-3\varepsilon$ collapse to $-\varepsilon$ , leaving $-(\varepsilon+3P)$ — the pressure survives tripled, which is the entire physical message that pressure gravitates.

Go deeper · Derivation C · 1 minHow density dilutes for a constant equation of state

Derivation CHow density dilutes for a constant equation of state

This derivation extracts the scaling law $\varepsilon\propto a^{-3(1+w)}$ that assigns each cosmic component its characteristic fall-off — the engine behind the whole cosmic-budget figure. We assume a constant $w$ , substitute $P=w\varepsilon$ into the fluid equation, separate variables and integrate. The one-line result explains why radiation redshifts away faster than matter and why vacuum energy refuses to dilute at all.

Step 1. Put $P=w\varepsilon$ into the fluid equation to get $\dot\varepsilon+3\tfrac{\dot a}{a}\varepsilon(1+w)=0$ . Why: a single constant $w$ collapses density and pressure into one term, turning a coupled statement into an ordinary differential equation for $\varepsilon$ alone.

Step 2. Separate variables using $\dot\varepsilon/\varepsilon=\mathrm d\ln\varepsilon/\mathrm dt$ , giving $\tfrac{\mathrm d\varepsilon}{\varepsilon}=-3(1+w)\tfrac{\mathrm da}{a}$ . Why: writing both sides as logarithmic differentials removes time entirely and relates the fractional change in density directly to the fractional change in scale factor.

Step 3. Integrate for constant $w$ to obtain $\ln\varepsilon=-3(1+w)\ln a+\text{const}$ . Why: constancy of $w$ is what lets the coefficient come outside the integral; a time-varying $w$ (as in some dark-energy models) would leave an integral $\int w\,\mathrm d\ln a$ instead.

Step 4. Exponentiate to reach $\varepsilon\propto a^{-3(1+w)}$ , i.e. $a^{-3}$ for matter, $a^{-4}$ for radiation, $a^{0}$ for vacuum. Why: the extra factor of $a^{-1}$ for radiation over matter is the photon redshift — each photon's energy falls as $1/a$ on top of the $a^{-3}$ number-density dilution.

Limiting cases

The acceleration equation and the scaling law specialize sharply once an equation of state is chosen. The table collects the four cases that anchor cosmic history.

Case	Result	Meaning
Pressureless matter, $w=0$	$\varepsilon_m\propto a^{-3}$ , $\ \ddot a/a=-\tfrac{4\pi G}{3c^2}\varepsilon_m<0$	Dust dilutes purely by volume; total energy $\varepsilon a^3$ is conserved; expansion always decelerates.
Radiation, $w=1/3$	$\varepsilon_r\propto a^{-4}$ , $\ \ddot a/a=-\tfrac{8\pi G}{3c^2}\varepsilon_r<0$	An extra $a^{-1}$ from photon redshift; pressure adds to gravity, so deceleration is stronger than matter at equal density.
Cosmological constant, $w=-1$	$\varepsilon_\Lambda=\text{const}$ , $\ \ddot a/a=+\tfrac{8\pi G}{3c^2}\varepsilon_\Lambda>0$	Density never dilutes; drives de Sitter exponential growth $a\propto e^{Ht}$ and eventually dominates everything.
Acceleration threshold, $w=-1/3$	$\varepsilon+3P=0\ \Rightarrow\ \ddot a=0$	The coasting boundary; any $w<-1/3$ yields acceleration. This is the operational definition of dark energy.

Common trapThe total energy in a comoving box is not conserved

A tempting but wrong shortcut is to assume the total energy in a comoving volume stays fixed, $\mathrm d(\varepsilon a^3)/\mathrm dt=0$ . This holds only for pressureless matter. The fluid equation actually says $\mathrm d(\varepsilon a^3)=-P\,\mathrm d(a^3)$ : whenever $P\neq0$ , pressure does work on the expanding boundary and the comoving energy changes. For radiation, $P=\varepsilon/3>0$ drains energy, so $\varepsilon_r a^3\propto a^{-1}$ falls and the density scales as $a^{-4}$ , not $a^{-3}$ . Assuming conservation would give you the matter scaling for radiation and wreck every prediction from nucleosynthesis to the cosmic microwave background. Energy is not globally conserved in an expanding universe — there is no time-translation symmetry to enforce it — and the fluid equation is the correct local bookkeeping in its place.

Worked examples

1 · Matter–radiation equality (confidence builder)

Radiation dominated the young universe even though matter rules today. Using the Planck values $\Omega_{m,0}=0.315$ and $\Omega_{r,0}=9.0\times10^{-5}$ , find the redshift $z_\mathrm{eq}$ at which their densities were equal.

Step 1. Use the scalings from equation (5): $\varepsilon_m\propto a^{-3}$ and $\varepsilon_r\propto a^{-4}$ . Their ratio is $\varepsilon_m/\varepsilon_r\propto a$ , so radiation, falling one power of $a$ faster, must have dominated at small $a$ and matter takes over as $a$ grows.

Step 2. Equality means $\Omega_{m,0}\,a^{-3}=\Omega_{r,0}\,a^{-4}$ , which rearranges to $a_\mathrm{eq}=\Omega_{r,0}/\Omega_{m,0}$ . Numerically $a_\mathrm{eq}=(9.0\times10^{-5})/0.315=2.86\times10^{-4}$ .

Step 3. Convert to redshift with $1+z=1/a$ : $\ 1+z_\mathrm{eq}=\Omega_{m,0}/\Omega_{r,0}=0.315/(9.0\times10^{-5})=3.5\times10^{3}$ . So $z_\mathrm{eq}\approx 3500$ , i.e. equality about $50\,000$ years after the Big Bang. This crossover — a direct consequence of the two different fluid-equation scalings — sets the size of the largest structures and prints a turnover into the matter power spectrum. $z_\mathrm{eq}\approx 3500$ (matter and radiation equal), from $1+z_\mathrm{eq}=\Omega_m/\Omega_r$ .

2 · When did cosmic acceleration begin, and how strong is it now?

In the flat $\Lambda$ CDM model with $\Omega_{m,0}=0.315$ and $\Omega_{\Lambda,0}=0.685$ , find (a) the redshift at which the expansion switched from decelerating to accelerating, and (b) today's deceleration parameter $q_0$ .

Step 1. Acceleration turns on when $\varepsilon+3P=0$ . Matter contributes $\varepsilon_m$ with $P_m=0$ ; the cosmological constant contributes $\varepsilon_\Lambda$ with $P_\Lambda=-\varepsilon_\Lambda$ . So $\varepsilon+3P=\varepsilon_m+\varepsilon_\Lambda+3(-\varepsilon_\Lambda)=\varepsilon_m-2\varepsilon_\Lambda$ , which vanishes when $\varepsilon_m=2\varepsilon_\Lambda$ .

Step 2. Insert the scalings $\varepsilon_m=\Omega_{m,0}\,\varepsilon_{c,0}(1+z)^3$ and $\varepsilon_\Lambda=\Omega_{\Lambda,0}\,\varepsilon_{c,0}$ (constant). The condition becomes $\Omega_{m,0}(1+z)^3=2\Omega_{\Lambda,0}$ , so $(1+z)^3=2(0.685)/0.315=4.35$ .

Step 3. Take the cube root: $1+z=4.35^{1/3}=1.63$ , so $z_\mathrm{acc}\approx0.63$ — the universe began accelerating roughly six billion years ago. Note the onset ( $z\approx0.63$ ) precedes matter– $\Lambda$ density equality ( $z\approx0.30$ ): acceleration begins while matter is still the larger density, because it takes only $\varepsilon_m<2\varepsilon_\Lambda$ , not $\varepsilon_m<\varepsilon_\Lambda$ .

Step 4. For $q_0$ use equation (6), $q_0=\tfrac12\sum_i\Omega_{i,0}(1+3w_i)$ . Matter has $w=0$ , so it contributes $\tfrac12\Omega_{m,0}(1)$ ; the constant has $w=-1$ , contributing $\tfrac12\Omega_{\Lambda,0}(1-3)=-\Omega_{\Lambda,0}$ . Thus $q_0=\tfrac12\Omega_{m,0}-\Omega_{\Lambda,0}=\tfrac12(0.315)-0.685=0.158-0.685$ .

Step 5. Evaluate: $q_0\approx-0.53$ . The negative value is the quantitative statement that the expansion is accelerating today, matching what Type Ia supernova surveys measured. Had there been no dark energy ( $\Omega_m=1$ ), the same formula would give $q_0=+\tfrac12$ — and, via $\varepsilon_m\propto a^{-3}$ in the flat Friedmann equation, a decelerating matter universe growing as $a\propto t^{2/3}$ . $z_\mathrm{acc}\approx0.63$ and $q_0\approx-0.53$ : acceleration is real.

Physics of the portrait

The fluid and acceleration equations are usually just called the Friedmann equations, and with good reason: they were first written down by a young Russian mathematician and meteorologist who took Einstein's brand-new field equations more seriously than Einstein himself did.

Origins · 1 minAlexander Friedmann and how this was discovered

Alexander Friedmann

1888–1925 · relativistic cosmology and the expanding universe

Alexander Alexandrovich Friedmann was a St. Petersburg mathematician and geophysicist — he flew reconnaissance and computed bombing tables in the First World War, and set a balloon-altitude record — who in 1922 did what no one else had dared: he solved Einstein's field equations without assuming the universe was static. Where Einstein had inserted a cosmological constant in 1917 specifically to hold the cosmos still, Friedmann treated the scale factor $a(t)$ as a genuine dynamical variable and derived the equations that now bear his name, showing that a homogeneous, isotropic universe must in general expand or contract. Einstein's first reaction, in a 1922 note to the Zeitschrift für Physik, was that Friedmann had erred; a year later he retracted the objection and conceded the solutions were correct, though still physically uninteresting to him. Friedmann's 1922 and 1924 papers laid down the full framework — expanding, static and oscillating models — five years before Hubble measured galaxy recession and seven before Lemaître independently tied the mathematics to observation. Friedmann saw none of it: he died of typhoid in 1925 at thirty-seven, having glimpsed the expanding universe purely at the end of a pen. The second of his relations, the acceleration equation, would wait until 1998 to reveal its strangest solution — a universe whose expansion speeds up.

History

1915

— Einstein completes the general theory of relativity

1917

— Einstein adds the cosmological constant to force a static universe

1922

— Friedmann derives the expanding-universe equations from the field equations

1927

— Lemaître independently links cosmic expansion to galaxy redshifts

1998

— supernova surveys reveal that the expansion is accelerating

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 4 (fluid and acceleration equations) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29 · Dodelson, Modern Cosmology, 2nd ed., Ch. 2–3 (tensor treatment) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 · Weinberg, Cosmology (2008), Ch. 1 · Binney & Tremaine, Galactic Dynamics, 2nd ed., cosmology appendix.

Provenance. The thermodynamic derivation follows Ryden Ch. 4 and Carroll & Ostlie Ch. 29; the covariant $\nabla_\mu T^{\mu\nu}=0$ route follows Dodelson Ch. 3. A verifier should confirm: (1) the fluid equation from $\mathrm dE=-P\,\mathrm dV$ with $V\propto a^3$ ; (2) the acceleration equation by differentiating $\dot a^2=\tfrac{8\pi G}{3c^2}\varepsilon a^2-kc^2$ and substituting the fluid equation; (3) the scaling $\varepsilon\propto a^{-3(1+w)}$ by integrating with constant $w$ ; (4) $z_\mathrm{eq}\approx3500$ from $\Omega_{m,0}/\Omega_{r,0}$ ; (5) $z_\mathrm{acc}\approx0.63$ from $(1+z)^3=2\Omega_\Lambda/\Omega_m$ ; (6) $q_0\approx-0.53$ from $\tfrac12\Omega_m-\Omega_\Lambda$ with $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ . All densities are standard Planck 2018 values.

Builds on: The Friedmann Equations · The FLRW Metric · The Expansion of the Universe and the Scale Factor · Equation of State in Cosmology · The Cosmological Principle. Leads to: Critical Density and Density Parameters · Cosmological Redshift · Spatial Curvature of the Universe.

Ask the tutor: Want to watch $\varepsilon+3P$ flip sign as dark energy overtakes matter, and trace when a universe starts to accelerate?

Is the fluid equation really independent of the acceleration equation?

No — and that is the point. Given the first Friedmann equation and the fluid equation, the acceleration equation follows by pure differentiation, as the boxed derivation shows; equivalently, Friedmann plus acceleration gives the fluid equation. Only two of the three ever carry independent information. This redundancy is the cosmological expression of the contracted Bianchi identity, $\nabla_\mu G^{\mu\nu}=0$ , which through Einstein's equations forces $\nabla_\mu T^{\mu\nu}=0$ . In practice cosmologists integrate the two most convenient equations and keep the third as a running numerical check.

How can pressure gravitate? In Newton only mass attracts.

In general relativity the source of gravity is the full stress-energy tensor, not just its energy component. Energy density curves spacetime, but so does the flux of momentum — and pressure is precisely momentum flux across a surface. The acceleration equation packages this as $\varepsilon+3P$ , the factor of three counting the three spatial directions of that flux. For everyday matter $P\ll\varepsilon$ , the pressure term is negligible and you recover Newton. It becomes decisive only for relativistic fluids (radiation, $P=\varepsilon/3$ ) and exotic negative-pressure components (vacuum, $P=-\varepsilon$ ), where it can flip the sign of gravity.

Why does radiation dilute as a⁻⁴ but matter only as a⁻³?

Both dilute by $a^{-3}$ because the volume grows as $a^3$ — spread the same stuff over more space and the number density falls. Radiation suffers one extra effect: each photon redshifts, its wavelength stretching with $a$ and its energy falling as $a^{-1}$ . Multiply the $a^{-3}$ number dilution by the $a^{-1}$ per-photon loss to get $a^{-4}$ . The fluid equation encodes this automatically: with $P=\varepsilon/3$ the pressure does positive work on the expanding boundary, and integrating gives exactly $\varepsilon_r\propto a^{-4}$ .

What exactly is the deceleration parameter q₀, and why the minus sign?

It is a dimensionless measure of how the expansion rate is changing, $q_0\equiv-\ddot a\,a/\dot a^2$ evaluated today. The minus sign is a historical convention from an era when everyone expected the universe to be slowing under its own gravity: it was defined so that a decelerating universe would have a positive $q_0$ . Nature embarrassed the convention — the measured value is negative, $q_0\approx-0.53$ , meaning the expansion is speeding up. Through equation (6), $q_0$ is a density-weighted average of $(1+3w_i)$ over all components, so it is really the acceleration equation in disguise: negative $q_0$ is the same statement as $\varepsilon+3P<0$ .

Does the accelerating universe violate energy conservation?

Not any law that applies. Global energy conservation follows, via Noether's theorem, from time-translation symmetry — and an expanding universe has none, since the state of space genuinely differs from epoch to epoch. What holds instead is the local statement $\nabla_\mu T^{\mu\nu}=0$ , the fluid equation: the energy in a comoving volume changes by exactly the pressure work $-P\,\mathrm d(a^3)$ . For dark energy this is striking — as space grows, more vacuum energy appears (constant density, rising volume), paid for by the negative-pressure work term. Nothing is violated; there is simply no conserved total energy in cosmology.

When did the universe switch from decelerating to accelerating?

Around redshift $z\approx0.63$ , roughly six billion years ago. The switch happens when $\varepsilon+3P$ first turns negative, which for $\Lambda$ CDM means the matter density drops to twice the dark-energy density, $\varepsilon_m=2\varepsilon_\Lambda$ . This is earlier than the moment the two densities become simply equal ( $z\approx0.30$ ): acceleration needs only $\varepsilon_m<2\varepsilon_\Lambda$ , not $\varepsilon_m<\varepsilon_\Lambda$ , thanks to the factor of three on the dark-energy pressure. Before then matter's positive $\varepsilon+3P$ won; after, dark energy's negative pressure took over.