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Concept
Equation of state (cosmology)
Résumé
In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number , equal to the ratio of its pressure to its energy density :
It is closely related to the thermodynamic equation of state and ideal gas law.
The perfect gas equation of state may be written as
where is the mass density, is the particular gas constant, is the temperature and is a characteristic thermal speed of the molecules. Thus
where is the speed of light, and for a "cold" gas.
The equation of state may be used in Friedmann–Lemaître–Robertson–Walker (FLRW) equations to describe the evolution of an isotropic universe filled with a perfect fluid. If is the scale factor then
If the fluid is the dominant form of matter in a flat universe, then
where is the proper time.
In general the Friedmann acceleration equation is
where is the cosmological constant and is Newton's constant, and is the second proper time derivative of the scale factor.
If we define (what might be called "effective") energy density and pressure as
and
the acceleration equation may be written as
The equation of state for ordinary non-relativistic 'matter' (e.g. cold dust) is , which means that its energy density decreases as , where is a volume. In an expanding universe, the total energy of non-relativistic matter remains constant, with its density decreasing as the volume increases.
The equation of state for ultra-relativistic 'radiation' (including neutrinos, and in the very early universe other particles that later became non-relativistic) is which means that its energy density decreases as . In an expanding universe, the energy density of radiation decreases more quickly than the volume expansion, because its wavelength 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 . In this case, the above expression for the scale factor is not valid and , where the constant H is the Hubble parameter.
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