Equation of state (cosmology)

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In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number w, equal to the ratio of its pressure p to its energy density ρ: w = p / ρ. It is closely related to the thermodynamic equation of state and ideal gas law.[1] The equation of state may be used in Friedmann-Lemaître-Robertson-Walker equations to describe the evolution of an isotropic universe filled with a perfect fluid. If a is the scale factor then

\rho\propto a^{-3(1+w)}.

If the fluid is the dominant form of matter in a flat universe, then

a\propto t^{\frac{2}{3(1+w)}},

where t is the proper time.

In general the Friedmann acceleration equation is

3\frac{\ddot{a}}{a} = \Lambda - 4 \pi G (\rho + 3p)

where Λ is the cosmological constant and G is Newton's constant, and \ddot{a} is the second proper time derivative of the scale factor.

If we define (what might be called "effective") energy density and pressure as

\rho^\prime \equiv \rho + \frac{\Lambda}{8 \pi G}
p^\prime \equiv p - \frac{\Lambda}{8 \pi G}

and

p^\prime = w^\prime\rho^\prime

the acceleration equation may be written as

\frac{\ddot a}{a}=-\frac{4}{3}\pi G\left(\rho^\prime + 3p^\prime\right) = -\frac{4}{3}\pi G(1+3w^\prime)\rho^\prime

The equation of state of ordinary non-relativistic matter (e.g. cold dust) is w = 0, which means that it is diluted as \rho\propto a^{-3}=V^{-1}, where V is the volume. This means that the energy density red-shifts as the volume, which is natural for ordinary non-relativistic matter. The equation of state of ultra-relativistic matter (e.g. radiation, but also matter in the very early universe) is w = 1 / 3 which means that it is diluted as a − 4. In an expanding universe, the energy density decreases more quickly than the volume expansion, because radiation has momentum and, by the de Broglie hypothesis a wavelength, which is red-shifted.

Cosmic inflation and the accelerated expansion of the universe can be characterized by the equation of state of dark energy. In the simplest case, the equation of state of the cosmological constant is w = − 1. In this case, the above expression for the scale factor is not valid and a\propto e^{Ht}, where the constant H is the Hubble parameter. More generally, the expansion of the universe is accelerating for any equation of state w < − 1 / 3. Phantom energy has equation of state w < − 1, and causes a Big Rip.

In an expanding universe, fluids with larger equations of state disappear more quickly than those with smaller equations of state. This is the origin of the flatness and monopole problems of the big bang: curvature has w = − 1 / 3 and monopoles have w = 0, so if they were around at the time of the early big bang, they should still be visible today. These problems are solved by cosmic inflation which has w\approx -1. Measuring the equation of state of dark energy is one of the largest efforts of observational cosmology. By accurately measuring w, it is hoped that the cosmological constant could be distinguished from quintessence which has w\ne -1.

A scalar field φ can be viewed as a sort of perfect fluid with equation of state

{w=\frac{\frac{1}{2}\dot{\phi}^2-V(\phi)}{\frac{1}{2}\dot{\phi}^2+V(\phi)},}

where \dot{\phi} is the time-derivative of φ and V(φ) is the potential energy. A free (V = 0) scalar field has w = 1, and one with vanishing kinetic energy is equivalent to a cosmological constant: w = − 1. Any equation of state in between is achievable, which makes scalar fields useful models for many phenomena in cosmology.

  1. ^ The perfect gas equation of state may be written as
    p = ρmRT = ρmC2
    where ρm is the mass density, R is the particular gas constant, T is the temperature and C = (RT)1/2 is a characteristic thermal speed of the molecules. Thus
    w = \frac{p}{\rho} = \frac{\rho_mC^2}{\rho_mc^2} = \frac{C^2}{c^2}\approx 0
    where ρ = ρmc2 and C < < c for a "cold" gas.
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