Equation of State in Cosmology

The dark-energy equation of state is a headline number of observational cosmology. Type Ia supernovae calibrate the expansion history, baryon acoustic oscillations lay a "standard ruler" across redshift, and the cosmic microwave background anchors the early-time densities; together they currently place $w$ within a few percent of $-1$, consistent with a pure cosmological constant. The live question is whether $w$ is exactly $-1$ and constant, or evolves. Surveys fit the Chevallier–Polarski–Linder form $w(a)=w_0+w_a(1-a)$ and hunt for any $w_a\neq0$; in 2024–2025 the DESI spectroscopic survey, combined with supernovae and the CMB, reported a mild preference for evolving dark energy that has not yet reached decisive significance. A confirmed departure from $w=-1$ would rule out the cosmological constant and point to a dynamical field — quintessence, whose $w$ ranges from $-1$ to $+1$ as the field rolls. Euclid, the Vera Rubin Observatory's LSST, and the Nancy Grace Roman telescope are built in large part to measure $w(a)$ to the sub-percent precision that would settle whether the vacuum is static.

Picture a comoving box that stretches along with the Universe. When the scale factor doubles, every edge doubles and the volume grows eightfold. Any conserved particles — massive or massless — simply spread over that larger volume, so their number density falls as $a^{-3}$. That $a^{-3}$ is the common floor shared by every species and is pure geometry. What separates one component from another is what happens to the energy each particle carries as space stretches. A cold, massive particle keeps its rest energy, so matter fades at exactly the volume rate, $\varepsilon_m\propto a^{-3}$. A photon does not: its wavelength is stretched by the expansion, $\lambda\propto a$, so each photon loses energy as $a^{-1}$, and radiation fades one power faster, $\varepsilon_r\propto a^{-4}$.

The parameter $w$ is really a bookkeeping of how expensive pressure is for the fluid. A positive pressure ($w>0$) pushes outward on the receding boundary and does work, draining energy faster than volume alone — which is exactly why radiation fades faster than matter. A negative pressure ($w<0$) is a tension: the fluid resists being stretched, and the expansion does work on it. For $w=-1$ that inflow of work precisely balances the volume growth, so the density never changes — the vacuum gains energy at just the rate needed to fill the new space, which is why a cosmological constant behaves like a fixed property of spacetime rather than a diluting substance. The same $w$ decides the fate of the expansion. In general relativity pressure gravitates, entering the source of the acceleration equation with a factor of three: the active gravitational density is $\varepsilon+3p=(1+3w)\varepsilon$. Ordinary matter and radiation have $1+3w>0$ and decelerate the expansion, while a fluid with $w<-1/3$ makes that combination negative — its gravity repels — and the expansion accelerates. Track all three species at once and their crossings lay out cosmic history: the steep radiation line dominates first, then matter, then the flat dark-energy line takes over near the present day.

The whole subject rests on one modelling choice: that a comoving cosmic fluid is barotropic, its pressure a function of its energy density alone. For the dominant components that function is linear, and the definition below is the entire content of the equation of state.

where: $p$ — isotropic pressure of the fluid; $\varepsilon$ — energy density, equal to $\rho c^2$; $\rho$ — mass-equivalent density $\varepsilon/c^2$; $w$ — dimensionless equation-of-state parameter; $c$ — speed of light. Reading: $w$ is just the pressure measured in units of the energy density — dimensionless, positive or negative, and everything else in the article follows from its value. Local energy conservation in an expanding spacetime, $\nabla_\mu T^{\mu\nu}=0$, gives the fluid or continuity equation. It says the energy density falls both because the volume grows and because pressure does work on the expanding boundary.

where: $\dot\varepsilon$ — time rate of change of the energy density; $\dot a/a=H$ — the Hubble parameter; $\varepsilon+p$ — the enthalpy density, which is the combination that sources dilution beyond the pure $a^{-3}$ of volume. Reading: if pressure were zero the equation would give simple volume dilution; the pressure term is precisely the "extra" physics that distinguishes species. Combining (1) and (2) with $w$ held constant integrates to a single power law — the most-used result in the field. It is the master formula from which every scaling is read.

where: $\varepsilon_0$ — present-day energy density, evaluated at $a_0=1$; $a$ — scale factor normalised to unity today; $w$ — assumed constant. Reading: the exponent $-3(1+w)$ carries all the information. Set $w=0$ and get $a^{-3}$ (matter); set $w=1/3$ and get $a^{-4}$ (radiation); set $w=-1$ and the exponent vanishes, so $\varepsilon$ is constant. The second Friedmann or acceleration equation shows how the same $w$ controls dynamics. Because pressure gravitates in general relativity, the source term is $\varepsilon+3p$, not $\varepsilon$ alone.

where: $\ddot a$ — second time derivative of the scale factor; $G$ — Newton's constant; $1+3w$ — the sign-determining factor. Reading: since $\varepsilon>0$ and $G>0$, the sign of $\ddot a$ is set entirely by $-(1+3w)$. Ordinary fluids decelerate; a fluid with $1+3w<0$ makes gravity push outward. Setting $\ddot a=0$ in (4) isolates the single value of $w$ that marks the boundary between a slowing and a speeding-up universe.

where: $\ddot a>0$ — accelerating expansion; $-1/3$ — the critical equation of state at which gravity neither attracts nor repels on the largest scales. Reading: this is the operational definition of dark energy — any component negative enough in pressure to cross this line. For a barotropic fluid the adiabatic sound speed comes straight from the slope of $p(\varepsilon)$, which is another face of $w$.

where: $c_s$ — sound speed in the fluid; $\partial p/\partial\varepsilon|_S$ — the derivative at fixed entropy. Reading: radiation ($w=1/3$) has $c_s=c/\sqrt{3}$, the speed of sound in a photon gas; matter has $c_s\approx0$. A negative $w$ gives a formally imaginary sound speed — a signal that dark energy is not a simple barotropic fluid but usually a field with its own effective sound speed. Because $\Lambda$CDM contains several species at once, the total is a sum of terms, each scaling with its own $w$, and this is what actually enters the Friedmann equation.

where: $\rho_{r,0}$ — present radiation density ($w=1/3$); $\rho_{m,0}$ — present matter density ($w=0$); $\rho_{\Lambda,0}$ — present dark-energy density ($w=-1$). Reading: the dominant term changes with $a$ because the exponents differ — the ordering of exponents is the ordering of the cosmic eras.

The power of the linear equation of state is that a short table of $w$ values captures every fluid the standard model needs, plus the effective fluids that mimic curvature and stiff matter.

1 · When did radiation give way to matter? (fully worked)

Take the present density parameters $\Omega_{m,0}=0.31$ (matter, $w=0$) and $\Omega_{r,0}=9.1\times10^{-5}$ (radiation, $w=1/3$, photons plus relativistic neutrinos). Find the scale factor and redshift of matter–radiation equality, and the photon temperature there, given $T_0=2.725$ K.

Step 1. Equality is where the two densities cross. Using the dilution law for each, $\Omega_{m,0}\,a^{-3}=\Omega_{r,0}\,a^{-4}$. The $a^{-3}$ cancels on both sides, leaving $\Omega_{m,0}=\Omega_{r,0}\,a^{-1}$, so

Step 2. Convert to redshift with $1+z=1/a$: $1+z_{\rm eq}=1/(2.9\times10^{-4})=3407$, so $\mathbf{z_{\rm eq}\approx3.4\times10^{3}}$. Before this redshift radiation dominated the expansion; after it, matter did — the transition that lets structure begin to grow.

Step 3. The photon temperature scales as $T\propto a^{-1}=1+z$, so $T_{\rm eq}=T_0(1+z_{\rm eq})=2.725\times3407=\mathbf{9.3\times10^{3}\ K}$, corresponding to a thermal energy $k_BT_{\rm eq}\approx0.80$ eV. Equality falls at $a_{\rm eq}\approx2.9\times10^{-4}$, $z_{\rm eq}\approx3400$, $T_{\rm eq}\approx9300$ K — comfortably before recombination at $z\approx1100$, so the Universe was already matter-dominated when the CMB was released.

2 · When did the expansion start to accelerate? (fully worked)

Take a flat $\Lambda$CDM model with $\Omega_{m,0}=0.31$ and $\Omega_{\Lambda,0}=0.69$, neglecting radiation. Find the present deceleration parameter $q_0$, and the redshift at which acceleration began.

Step 1. The deceleration parameter is $q\equiv-\ddot a\,a/\dot a^{2}$, and for a mix of fluids the acceleration equation gives $q=\tfrac12\sum_i\Omega_i(1+3w_i)$. Today, with matter ($w=0$) and $\Lambda$ ($w=-1$),

The negative value is the direct statement that the Universe is accelerating now.

Step 2. Acceleration begins when $\ddot a=0$, i.e. when the total $\varepsilon+3p$ changes sign: $\rho_m-2\rho_\Lambda=0$. Using the scalings, $\Omega_{m,0}\,a^{-3}=2\,\Omega_{\Lambda,0}$, so

Step 3. In redshift, $1+z_{\rm acc}=1/0.61=1.65$, so $\mathbf{z_{\rm acc}\approx0.6}$. The cosmos was decelerating until $z\approx0.6$ and has been accelerating since, with $q_0\approx-0.54$ today. Note that matter and dark energy are only equal somewhat later, at $z\approx0.3$ (from $\Omega_{m,0}a^{-3}=\Omega_{\Lambda,0}$) — acceleration starts before $\Lambda$ formally dominates, because pressure enters with a factor of three.

3 · Reading $w$ off a scaling law (quick)

A hypothetical component dilutes as $\varepsilon\propto a^{-5}$. What is its equation of state? Match exponents in the dilution law: $-3(1+w)=-5$, so $1+w=5/3$ and $\mathbf{w=2/3}$. Its sound speed is $c_s=\sqrt{2/3}\,c\approx0.82\,c<c$, so it is causal, lying between radiation ($w=1/3$) and the stiff limit ($w=1$); it decelerates the expansion since $w>-1/3$.

The equation of state is where thermodynamics meets gravitation, and no one saw that junction more clearly, or earlier, than the Belgian priest-physicist who first read Einstein's cosmological constant as a pressure.

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 5 (fluid and acceleration equations) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29 · Dodelson, Modern Cosmology, 2nd ed., ch. 2 · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 · Peebles, Principles of Physical Cosmology, ch. 5. The key results of this article are: the dilution law $\varepsilon\propto a^{-3(1+w)}$ from integrating the fluid equation with constant $w$; the three canonical scalings $a^{-3}$, $a^{-4}$, and constant for $w=0,1/3,-1$; the acceleration threshold $w=-1/3$ from equation (4); and the sound speed $c_s^2=wc^2$, giving $c/\sqrt{3}$ for radiation. The worked numbers use standard $\Lambda$CDM parameters $\Omega_{m,0}=0.31$, $\Omega_{\Lambda,0}=0.69$, $\Omega_{r,0}=9.1\times10^{-5}$, yielding matter–radiation equality $z_{\rm eq}\approx3400$, acceleration onset $z_{\rm acc}\approx0.6$, and present deceleration parameter $q_0\approx-0.54$. The identification of vacuum pressure as $p=-\rho c^2$ traces to Lemaître (1934).

Prerequisites. The Fluid and Acceleration Equations · The Scale Factor and Cosmic Expansion · The FLRW Metric · Homogeneity and Isotropy in Cosmology. Leads to. The Friedmann Equations · Critical Density and the Density Parameters · Cosmological Redshift.

Ask the tutor: Ask me to slide $w$ from $+1$ down to $-2$ and watch the dilution law, the sound speed, and the fate of the expansion change at each step.