In
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
, the equation of state of a
perfect fluid
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure ''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in whi ...
is characterized by a
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
number
, equal to the ratio of its
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
to its
energy density :
It is closely related to the thermodynamic
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
and
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
.
The equation
The
perfect gas
In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. In all perfect gas models, intermolecular forces are neglected. This means that one ...
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.
FLRW equations and the equation of state
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
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
then
If the fluid is the dominant form of matter in a
flat universe
The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes gen ...
, then
where
is the proper time.
In general the
Friedmann acceleration equation is
where
is the
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
and
is
Newton's constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, and
is the second
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
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
Non-relativistic particles
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.
Ultra-relativistic particles
The equation of state for ultra-relativistic 'radiation' (including
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s, 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-shift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in fr ...
ed.
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 physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univ ...
. In the simplest case, the equation of state of the
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
is
. In this case, the above expression for the scale factor is not valid and
, where the constant is the
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
. More generally, the expansion of the universe is accelerating for any equation of state
. The accelerated expansion of the Universe was indeed observed. According to observations, the value of equation of state of cosmological constant is near -1.
Hypothetical
would have an equation of state
, and would cause a
Big Rip
In physical cosmology, the Big Rip is a hypothetical cosmological model concerning the ultimate fate of the universe, in which the matter of the universe, from stars and galaxies to atoms and subatomic particles, and even spacetime itself, is ...
. Using the existing data, it is still impossible to distinguish between phantom
and non-phantom
.
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
and monopoles have
, 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
. Measuring the equation of state of dark energy is one of the largest efforts of
observational cosmology
Observational cosmology is the study of the structure, the evolution and the origin of the universe through observation, using instruments such as telescopes and cosmic ray detectors.
Early observations
The science of physical cosmology as it is ...
. By accurately measuring
, it is hoped that the cosmological constant could be distinguished from
quintessence which has
.
Scalar modeling
A
scalar field can be viewed as a sort of perfect fluid with equation of state
where
is the time-derivative of
and
is the potential energy. A free (
) scalar field has
, and one with vanishing kinetic energy is equivalent to a cosmological constant:
. Any equation of state in between, but not crossing the
barrier known as the Phantom Divide Line (PDL),
is achievable, which makes scalar fields useful models for many phenomena in cosmology.
Notes
{{DEFAULTSORT:Equation Of State (Cosmology)
Physical cosmology
Equations of state