Equations of State and Scaling Laws

An equation of state relates a cosmic component’s pressure to its energy density. At background level the ratio $w=p/\rho$, together with separate stress-energy conservation, determines how that density scales as the universe expands. Dust, radiation, vacuum energy, and a stiff fluid therefore dilute at different rates. The same exponent can also arise from geometric bookkeeping, so a term proportional to $a^{-2}$ is not automatically a material fluid. Moreover, background $w$ is not in general the speed at which perturbations propagate.

Required background. The Einstein Equation supplies the Einstein Equations and Bianchi consistency; Flat-Background Hilbert Stress-Energy from Metric Variation supplies the Stress–Energy and Conservation meaning of density and pressure.

Helpful background. Relativistic Matter and Compact Objects supplies equations of state and thermodynamic closure beyond a constant parameter.

What the equation-of-state ratio controls

For a homogeneous component with $\rho\ne0$, define

$$w(a)\equiv\frac{p(a)}{\rho(a)}.$$

If that component is separately conserved, the continuity equation is

$$\frac{d\rho}{d\ln a}=-3(1+w)\rho$$

on a monotonic branch of $a(t)$. Integrating from a reference scale $a_\star$ gives the general background solution

$$\rho(a) =\rho_\star \exp\left[ -3\int_{a_\star}^{a} \left(1+w(\tilde a)\right)\frac{d\tilde a}{\tilde a} \right].$$

For constant $w$, this reduces to

$$\boxed{ \rho(a)=\rho_\star \left(\frac{a}{a_\star}\right)^{-3(1+w)} }.$$

The factor $a^{-3}$ is volume dilution. The additional $a^{-3w}$ records pressure work. These results and their standard components are derived in Tong 2019, §§1.2.1–1.2.2, pp. 24–29, eqs. (1.35) and (1.38)–(1.41) (PDF).

Benchmark scaling laws

The comparison assumes separate conservation and a constant background $w$.

Component	$w$	Density scaling	Physical reading
Nonrelativistic matter	$0$	$\rho_m\propto a^{-3}$	Particle energy is approximately fixed; number density dilutes with volume
Radiation	$1/3$	$\rho_r\propto a^{-4}$	Volume dilution contributes $a^{-3}$ and redshift contributes another $a^{-1}$
Vacuum energy	$-1$	$\rho_\Lambda=\text{constant}$	Negative pressure exactly offsets dilution
Stiff fluid	$1$	$\rho_s\propto a^{-6}$	Positive pressure work makes the density decay especially rapidly
Curvature-like exponent	$-1/3$	$\rho_{\rm eff}\propto a^{-2}$	Matches curvature algebraically, but need not represent matter

For a comoving volume $V\propto a^3$, the corresponding energy scales as $E=\rho V\propto a^{-3w}$. This gives a second check: dust has constant comoving energy, radiation has $E\propto a^{-1}$, and vacuum energy has $E\propto a^3$. The Particle Data Group summarizes the same dust, radiation, and vacuum limits in Particle Data Group 2026, §§22.1.5.1–22.1.5.4, pp. 4–5, especially eq. (22.16) (PDF).

The exponent alone does not identify the microphysics. The $a^{-6}$ shear contribution in source-free Bianchi I scales like a stiff component in the Hamiltonian constraint, but it is geometric anisotropy. Likewise, coherent scalar fields, particle mixtures, and effective descriptions can share an average $w$ while responding differently to perturbations.

Curvature is not a material fluid

The Friedmann constraint can be written

$$H^2 =\frac{8\pi G}{3}\rho+\frac{\Lambda}{3}-\frac{K}{a^2}.$$

Here $\rho$ excludes the geometrically displayed $\Lambda$. In the equivalent source-side convention one sets $\rho_\Lambda=\Lambda/(8\pi G)$, includes it in $\rho$, and removes the explicit $\Lambda/3$. The two conventions must not be added together.

One may define an algebraic effective density

$$\rho_K\equiv-\frac{3K}{8\pi G a^2},$$

whose exponent corresponds formally to $w_K=-1/3$. This shorthand is useful when comparing powers of $a$, but spatial curvature belongs to the metric. It has no independent particle current, rest frame, equation of state, or perturbation sound speed. In particular, applying fluid energy conditions to $\rho_K$ or interpreting its sign as a local negative-energy material source would confuse geometry with stress-energy.

The distinction also fixes the sign. With $\Omega_K=-K/(a^2H^2)$ where $H\ne0$, positive spatial curvature has $K\gt0$ and $\Omega_K\lt0$ in this bookkeeping convention. No physical energy density has become negative; the curvature term was moved across the equation.

Background pressure is not sound speed

The ratio $w=p/\rho$ controls homogeneous dilution. Perturbations require a closure for $\delta p$, and several inequivalent derivatives occur. The adiabatic background derivative is

$$c_a^2\equiv\frac{\dot p}{\dot\rho}.$$

This definition and its role in the pressure perturbation are stated in Malik and Wands 2009, §8.2.2, p. 48, eq. (8.34) (PDF).

For a separately conserved component with $p=w(t)\rho$,

$$c_a^2 =w-\frac{\dot w}{3H(1+w)}$$

where the denominator is nonzero. Only a strictly barotropic constant-$w$ fluid gives $c_a^2=w$. Even then, the physical rest-frame propagation speed must come from the component’s microphysics. A canonical scalar field can have negative background $w$ while its field perturbations propagate with unit sound speed in its well-defined rest frame; see Baumann 2012, §23.3.1, pp. 95–96, eqs. (332)–(334) (PDF). Thus the slogan “negative pressure means imaginary sound speed” is not generally valid.

For a true cosmological constant, $\dot p=\dot\rho=0$, so $c_a^2$ is the indeterminate ratio $0/0$ and there is no independent fluid perturbation. A time-dependent effective component reaching $w=-1$ with $\dot w\ne0$ instead has $\dot\rho=0$ but $\dot p=\dot w\,\rho$, so this single-fluid $c_a^2$ develops a pole. The pole diagnoses failure of that parameterization; it does not by itself prove a physical instability. Regular variables and a complete microphysical model are needed to assess a crossing.

Time dependence and interactions

If $w$ varies, use the integral solution rather than inserting an instantaneous value into the constant-$w$ power law. If components exchange energy,

$$\dot\rho_i+3H(1+w_i)\rho_i=Q_i,$$

then $Q_i$ also enters the solution, and only the summed source vanishes. Phase transitions, decays, particle thresholds, and oscillating fields can all invalidate a single power law over the full history.

An equation of state is also not always the single relation $p=p(\rho)$. Temperature, composition, chemical potentials, elastic strain, magnetic fields, or nonequilibrium variables may be needed to close the stress tensor. The constant-$w$ laws are exact within their assumptions and useful as limiting regimes, not universal material models.

Exercises

Let

$$w(a)=w_0+w_a(1-a), \qquad a_0=1.$$

Use the general integral to find $\rho(a)/\rho_0$ for a separately conserved component, and recover the constant-$w$ result when $w_a=0$.

Solution

Substitute $1+w(a)=1+w_0+w_a-w_a a$ into the integral:

$$\begin{aligned} \ln\frac{\rho(a)}{\rho_0} &=-3\int_1^a \left(1+w_0+w_a-w_a\tilde a\right) \frac{d\tilde a}{\tilde a}\\ &=-3(1+w_0+w_a)\ln a+3w_a(a-1). \end{aligned}$$

Therefore

$$\frac{\rho(a)}{\rho_0} =a^{-3(1+w_0+w_a)} \exp\left[3w_a(a-1)\right].$$

Setting $w_a=0$ gives $\rho/\rho_0=a^{-3(1+w_0)}$, as required. The power-law exponent contains the early-time value $w_0+w_a$, while the exponential factor records the linear change toward $w_0$ at $a=1$. Both pieces are required; replacing $w(a)$ by its instantaneous value inside a constant-$w$ power law would miss the integrated expansion history.

Explain why observing an $a^{-2}$ term in $H^2(a)$ is insufficient to prove the existence of a material component with $w=-1/3$.

Solution

Spatial curvature already contributes $-K/a^2$ to the Friedmann equation. Moving this term to the source side produces an effective density with the same exponent as a separately conserved $w=-1/3$ fluid. The expansion history alone therefore sees an algebraic degeneracy. A genuine material component would also require a covariant stress tensor and could carry independent perturbations, with a closure that determines whether they cluster or are pressure supported. Curvature has no independent fluid perturbation or material closure.

References

• Baumann, Daniel. “TASI Lectures on Inflation.” arXiv:0907.5424v2, revised 30 November 2012. Open PDF.
• Malik, Karim A., and David Wands. “Cosmological Perturbations.” Physics Reports 475 (2009): 1–51. DOI. Open PDF.
• Particle Data Group. “Big-Bang Cosmology.” In Review of Particle Physics 2026, revised August 2025. Official PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019. Official PDF.