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. The equation of state is used to describe the evolution of a perfect fluid in a homogeneous isotropic universe, the Friedmann-Lemaître-Robertson-Walker universe. If a is the scale factor and t is the proper time, then they are related to the equation of state by \rho\propto a^{-3(1+w)} and, if the fluid is the dominant form of matter in a flat universe, a\propto t^{\frac{2}{3(1+w)}}.

In general the Friedmann acceleration equation is

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

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

We can define away Λ by redefining:

\rho \rightarrow \rho - \frac{\Lambda}{8 \pi G}

p \rightarrow p + \frac{\Lambda}{8 \pi G}

So that we finally we get the equation of state related to the acceleration or deceleration of the universe by

\frac{\ddot a}{a}=-\frac{4}{3}\pi G(1+3w)\rho

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\le-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.

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