Equation of state (cosmology)

Equation of state (cosmology)

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In cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

, the equation of state of a perfect fluid
Perfect fluid
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p....

 is characterized by a dimensionless number , equal to the ratio of its pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

  to its energy density
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

 : . It is closely related to the thermodynamic equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

 and ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...

.

The equation


The perfect gas
Perfect gas
In physics, a perfect gas is a theoretical gas that differs from real gases in a way that makes certain calculations easier to handle. Its behavior is more simplified compared to an ideal 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 and for a "cold" gas, = speed of light.

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 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
Friedmann equations
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity...

 is
where is the cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

 and is Newton's constant, and is the second proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

 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 matter


The equation of state of ordinary non-relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

 matter (e.g. cold dust) is , which means that it is diluted as , where is the volume. This means that the energy density red-shifts as the volume, which is natural for ordinary non-relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

 matter.

Ultra-relativistic matter


The equation of state of ultra-relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

 matter (e.g. radiation, but also matter in the very early universe) is which means that it is diluted as . In an expanding universe, the energy density decreases more quickly than the volume expansion, because radiation has momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 and, by the de Broglie hypothesis a wavelength, which is red-shifted.

Acceleration of cosmic inflation


Cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...

 and the accelerated expansion
Accelerating universe
The accelerating universe is the observation that the universe appears to be expanding at an increasing rate, which in formal terms means that the cosmic scale factor a has a positive second derivative, implying that the velocity at which a given galaxy is receding from us should be continually...

 of the universe can be characterized by the equation of state of dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...

. In the simplest case, the equation of state of the cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

 is . In this case, the above expression for the scale factor is not valid and , where the constant H is the Hubble parameter. 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 phantom energy
Phantom energy
Phantom energy is a hypothetical form of dark energy with equation of state \! w Phantom energy is a hypothetical form of dark energy with equation of state...

 would have an equation of state , and would cause a Big Rip
Big Rip
The Big Rip is a cosmological hypothesis first published in 2003, about the ultimate fate of the universe, in which the matter of the universe, from stars and galaxies to atoms and subatomic particles, is progressively torn apart by the expansion of the universe at a certain time in the future...

. 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
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

: curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

 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
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...

 which has . Measuring the equation of state of dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...

 is one of the largest efforts of observational cosmology
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:...

. By accurately measuring , it is hoped that the cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

 could be distinguished from quintessence
Quintessence (physics)
In physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...

 which has .

Scalar modeling


A scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

  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 field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

s useful models for many phenomena in cosmology.