Equation of state (cosmology)
The equation
The equation
------------

The [perfect gas](/wiki/Perfect_gas "Perfect gas") equation of state may be written as 
p \= \\rho\_m RT \= \\rho\_m C^2
where  \\rho\_m is the mass density, R is the particular gas constant, T is the temperature and C\=\\sqrt{RT} is a characteristic thermal speed of the molecules. Thus
w \\equiv \\frac{p}{\\rho} \= \\frac{\\rho\_mC^2}{\\rho\_mc^2} \= \\frac{C^2}{c^2}\\approx 0
where c is the speed of light, \\rho \= \\rho\_mc^2 and 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](/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker "Friedmann–Lemaître–Robertson–Walker") (FLRW) equations to describe the evolution of an isotropic universe filled with a perfect fluid. If a is the [scale factor](/wiki/Scale_factor_%28cosmology%29 "Scale factor (cosmology)") then 
\\rho \\propto a^{\-3(1\+w)}.
If the fluid is the dominant form of matter in a [flat universe](/wiki/Shape_of_the_universe%23Flat_universe "Shape of the universe#Flat universe"), then
a \\propto t^{\\frac{2}{3(1\+w),
where t is the proper time.

In general the [Friedmann acceleration equation](/wiki/Friedmann_equations "Friedmann equations") is
3\\frac{\\ddot{a{a} \= \\Lambda \- 4 \\pi G (\\rho \+ 3p)
where  \\Lambda is the [cosmological constant](/wiki/Cosmological_constant "Cosmological constant") and G is [Newton's constant](/wiki/Newton%27s_constant "Newton's constant"), and \\ddot{a} is the second [proper time](/wiki/Proper_time "Proper time") derivative of the scale factor.

If we define (what might be called "effective") energy density and pressure as
\\rho' \\equiv \\rho \+ \\frac{\\Lambda}{8 \\pi G}
p' \\equiv p \- \\frac{\\Lambda}{8 \\pi G}
and  p' \= w'\\rho'
the acceleration equation may be written as
\\frac{\\ddot a}{a}\=\-\\frac{4}{3}\\pi G\\left(\\rho' \+ 3p'\\right) \= \-\\frac{4}{3}\\pi G(1\+3w')\\rho'

### Non\-relativistic particles

The equation of state for ordinary non\-[relativistic](/wiki/Theory_of_relativity "Theory of relativity") 'matter' (e.g. cold dust) is w \= 0, which means that its energy density decreases as \\rho \\propto a^{\-3} \= V^{\-1}, where V is a volume. In an expanding universe, the total energy of non\-relativistic matter remains constant, with its density decreasing as the volume increases.

### Ultra\-relativistic particles

The equation of state for ultra\-relativistic 'radiation' (including [neutrinos](/wiki/Neutrino "Neutrino"), and in the very early universe other particles that later became non\-relativistic) is w \= 1/3 which means that its energy density decreases as \\rho \\propto a^{\-4}. In an expanding universe, the energy density of radiation decreases more quickly than the volume expansion, because its wavelength is [red\-shifted](/wiki/Red-shift "Red-shift").

### Acceleration of cosmic inflation

[Cosmic inflation](/wiki/Cosmic_inflation "Cosmic inflation") and the [accelerated expansion](/wiki/Accelerating_universe "Accelerating universe") of the universe can be characterized by the equation of state of [dark energy](/wiki/Dark_energy "Dark energy"). In the simplest case, the equation of state of the [cosmological constant](/wiki/Cosmological_constant "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  is the [Hubble parameter](/wiki/Hubble_parameter "Hubble parameter"). More generally, the expansion of the universe is accelerating for any equation of state w \< \-1/3. The accelerated expansion of the Universe was indeed observed.Hogan, Jenny. "Welcome to the Dark Side." Nature 448\.7151 (2007\): 240\-245\. <http://www.nature.com/nature/journal/v448/n7151/full/448240a.html> According to observations, the value of equation of state of cosmological constant is near \-1\.

Hypothetical [phantom energy](/wiki/Phantom_energy "Phantom energy") would have an equation of state w \< \-1, and would cause a [Big Rip](/wiki/Big_Rip "Big Rip"). Using the existing data, it is still impossible to distinguish between phantom w \< \-1  and non\-phantom w \\ge \-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](/wiki/Flatness_problem "Flatness problem") and [monopole](/wiki/Monopole_problem "Monopole problem") problems of the [Big Bang](/wiki/Big_Bang "Big Bang"): [curvature](/wiki/Curvature "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](/wiki/Observational_cosmology "Observational cosmology"). By accurately measuring w, it is hoped that the cosmological constant could be distinguished from [quintessence](/wiki/Quintessence_%28physics%29 "Quintessence (physics)") which has w \\ne \-1.

### Scalar modeling

A [scalar field](/wiki/Scalar_field "Scalar field")  \\phi 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  \\phi and V(\\phi) 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, but not crossing the w \= \-1 barrier known as the Phantom Divide Line (PDL), is achievable, which makes scalar fields useful models for many phenomena in cosmology.


What is the equation of state for a free scalar field?