- Equation of state (cosmology)
In cosmology, the

**equation of state**of aperfect fluid is characterized by adimensionless number "w", equal to the ratio of itspressure "p" to itsenergy density ρ: $w=p/\; ho$. It is closely related to the thermodynamicequation of state andideal gas law .**The equation**The perfect gas equation of state may be written as :$p\; =\; ho\_m\; RT\; =\; ho\_m\; C^2$where $ho\_m$ is the mass density, "R" is the particular gas constant, "T" is the temperature and "C" = ("RT")

^{1/2}is a characteristic thermal speed of the molecules. Thus:$w\; =\; frac\{p\}\{\; ho\}\; =\; frac\{\; ho\_mC^2\}\{\; ho\_mc^2\}\; =\; frac\{C^2\}\{c^2\}approx\; 0$where $ho\; =\; ho\_mc^2$ and $C<math>\; for\; a\; "cold"\; gas.$**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 "a" is the scale factor then :$hopropto\; a^\{-3(1+w)\}.$If the fluid is the dominant form of matter in a flat universe, then:$apropto\; t^\{frac\{2\}\{3(1+w),$where $t$ is the proper time.In general the Friedmann acceleration equation is:$3frac\{ddot\{a\{a\}\; =\; Lambda\; -\; 4\; pi\; G\; (\; ho\; +\; 3p)$where $Lambda$ is the

cosmological constant and $G$ isNewton's constant , and $ddot\{a\}$ is the secondproper time derivative of the scale factor.If we define (what might be called "effective") energy density and pressure as:$ho^prime\; equiv\; ho\; +\; frac\{Lambda\}\{8\; pi\; G\}$:$p^prime\; equiv\; p\; -\; frac\{Lambda\}\{8\; pi\; G\}$and:$p^prime\; =\; w^prime\; ho^prime$the acceleration equation may be written as:$frac\{ddot\; a\}\{a\}=-frac\{4\}\{3\}pi\; Gleft(\; ho^prime\; +\; 3p^prime\; ight)\; =\; -frac\{4\}\{3\}pi\; G(1+3w^prime)\; ho^prime$

**Non-relativistic matter**The equation of state of ordinary non-relativistic matter (e.g. cold dust) is $w=0$, which means that it is diluted as $hopropto\; a^\{-3\}=V^\{-1\}$, where $V$ is the volume. This means that the energy density

red-shift s 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 $w=1/3$ which means that it is diluted as $a^\{-4\}$. In an expanding universe, the energy density decreases more quickly than the volume expansion, because radiation has

momentum and, by thede 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 ofdark energy . In the simplest case, the equation of state of thecosmological constant is $w=-1$. In this case, the above expression for the scale factor is not valid and $apropto\; e^\{Ht\}$, where the constant "H" is theHubble parameter . More generally, the expansion of the universe is accelerating for any equation of state $w<-1/3$.Phantom energy has equation of state $w<-1$, and causes aBig Rip .**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 $w=-1/3$ 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 bycosmic inflation which has $wapprox\; -1$. Measuring the equation of state ofdark energy is one of the largest efforts ofobservational cosmology . By accurately measuring "w", it is hoped that thecosmological constant could be distinguished from quintessence which has $w\; e\; -1$.**calar modeling**A

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) [*A. Vikman,``Can dark energy evolve to the phantom?,"Phys. Rev. D*] , is achievable, which makes**71**, 023515 (2005), http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=bb+astro-ph%2F0407107&FORMAT=WWW&SEQUENCE=scalar field s useful models for many phenomena in cosmology.**Value**According to a 2007 article by the journal Nature, scientists have discovered by using evidence from observation of

supernova , galaxy clusters, and thecosmic microwave background that the value of the equation of state parameter w is in a small range around -1. [*Hogan, Jenny. "Welcome to the Dark Side." Nature 448.7151 (2007): 240-245. http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=25801949&site=ehost-live*]**Notes**

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