# Equation of state (cosmology)

In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number ${\displaystyle w}$, equal to the ratio of its pressure ${\displaystyle p}$ to its energy density ${\displaystyle \rho }$ :

${\displaystyle w={\frac {p}{\rho }}}$.

It is closely related to the thermodynamic equation of state and ideal gas law.

## The equation

The perfect gas equation of state may be written as

${\displaystyle p=\rho _{m}RT=\rho _{m}C^{2}}$

where ${\displaystyle \rho _{m}}$ is the mass density, ${\displaystyle R}$ is the particular gas constant, ${\displaystyle T}$ is the temperature and ${\displaystyle C={\sqrt {RT}}}$ is a characteristic thermal speed of the molecules. Thus

${\displaystyle w={\frac {p}{\rho }}={\frac {\rho _{m}C^{2}}{\rho _{m}c^{2}}}={\frac {C^{2}}{c^{2}}}\approx 0}$

where ${\displaystyle c}$ is the speed of light, ${\displaystyle \rho =\rho _{m}c^{2}}$ and ${\displaystyle C\ll c}$ for a "cold" gas.

### FLRW equations and the equation of state

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 ${\displaystyle a}$ is the scale factor then

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

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

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

where ${\displaystyle t}$ is the proper time.

In general the Friedmann acceleration equation is

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

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

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

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

and

${\displaystyle p^{\prime }=w^{\prime }\rho ^{\prime }}$

the acceleration equation may be written as

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

### Non-relativistic matter

The equation of state of ordinary non-relativistic matter (e.g. cold dust) is ${\displaystyle w=0}$, which means that it is diluted as ${\displaystyle \rho \propto a^{-3}=V^{-1}}$, where ${\displaystyle V}$ is the volume. This means that the energy density red-shifts as the volume, which is natural for ordinary non-relativistic matter.

### Ultra-relativistic matter

The equation of state of ultra-relativistic matter (e.g. radiation, but also matter in the very early universe) is ${\displaystyle w=1/3}$ which means that it is diluted as ${\displaystyle \rho \propto a^{-4}}$. 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.

### Acceleration of cosmic inflation

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 ${\displaystyle w=-1}$. In this case, the above expression for the scale factor is not valid and ${\displaystyle a\propto e^{Ht}}$, where the constant H is the Hubble parameter. More generally, the expansion of the universe is accelerating for any equation of state ${\displaystyle w<-1/3}$. The accelerated expansion of the Universe was indeed observed.[1] According to observations, the value of equation of state of cosmological constant is near -1.

Hypothetical phantom energy would have an equation of state ${\displaystyle w<-1}$, and would cause a Big Rip. Using the existing data, it is still impossible to distinguish between phantom ${\displaystyle w<-1}$ and non-phantom ${\displaystyle w\geq -1}$.

### Fluids

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 ${\displaystyle w=-1/3}$ and monopoles have ${\displaystyle w=0}$, 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 ${\displaystyle w\approx -1}$. Measuring the equation of state of dark energy is one of the largest efforts of observational cosmology. By accurately measuring ${\displaystyle w}$, it is hoped that the cosmological constant could be distinguished from quintessence which has ${\displaystyle w\neq -1}$.

### Scalar modeling

A scalar field ${\displaystyle \phi }$ can be viewed as a sort of perfect fluid with equation of state

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

where ${\displaystyle {\dot {\phi }}}$ is the time-derivative of ${\displaystyle \phi }$ and ${\displaystyle V(\phi )}$ is the potential energy. A free ${\displaystyle (V=0)}$ scalar field has ${\displaystyle w=1}$, and one with vanishing kinetic energy is equivalent to a cosmological constant: ${\displaystyle w=-1}$. Any equation of state in between, but not crossing the ${\displaystyle w=-1}$ barrier known as the Phantom Divide Line (PDL),[2] is achievable, which makes scalar fields useful models for many phenomena in cosmology.

## Notes

1. ^ Hogan, Jenny. "Welcome to the Dark Side." Nature 448.7151 (2007): 240-245. http://www.nature.com/nature/journal/v448/n7151/full/448240a.html
2. ^ Vikman, Alexander (2005). "Can dark energy evolve to the Phantom?". Phys. Rev. D. 71 (2): 023515. arXiv:astro-ph/0407107. Bibcode:2005PhRvD..71b3515V. doi:10.1103/PhysRevD.71.023515.
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Relativistic Euler equations

In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity.

The equations of motion are contained in the continuity equation of the stress–energy tensor ${\displaystyle T^{\mu \nu }}$:

${\displaystyle \nabla _{\mu }T^{\mu \nu }=0.}$

For a perfect fluid,

${\displaystyle T^{\mu \nu }\,=(e+p)u^{\mu }u^{\nu }+p\eta ^{\mu \nu }.}$

Here ${\displaystyle e}$ is the relativistic rest energy density of the fluid, ${\displaystyle p}$ is the fluid pressure, ${\displaystyle u^{\mu }}$ is the four-velocity of the fluid, and ${\displaystyle \eta ^{\mu \nu }}$ is the Minkowski metric tensor with signature (−,+,+,+).

To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If ${\displaystyle n}$ is the number density of baryons this may be stated

${\displaystyle \nabla _{\mu }(nu^{\mu })=0.}$

These equations reduce to the classical Euler equations if the fluid three-velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density.

The relativistic Euler equations may be applied to calculate the speed of sound in a fluid with a relativistic equation of state (that is, one in which the pressure is comparable with the internal energy density ${\displaystyle e}$, including the rest energy; ${\displaystyle e=\rho c^{2}+\rho e^{C}}$ where ${\displaystyle e^{C}}$ is the classical internal energy per unit mass).

Under these circumstances, the speed of sound ${\displaystyle S}$ is given by

${\displaystyle S^{2}=c^{2}\left.{\frac {\partial p}{\partial e}}\right|_{\rm {adiabatic}}.}$

(note that

${\displaystyle e=\rho (c^{2}+e^{C})}$

is the relativistic internal energy density). This formula differs from the classical case in that ${\displaystyle \rho }$ has been replaced by ${\displaystyle e/c^{2}}$.

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