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In cosmology we usually assume that the matter follows an equation of state given by

$$ P=w \rho. $$ Given that we have a lot of non-standard theoretical proposals for field theories and exotic matter, is this equation always linear? Are there examples of theories where it is not? and how is this motivated?

marRrR
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1 Answers1

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I'd put this as a comment, but don't have enough rep...anyway, as this answer and the comments within state, the equation of state isn't necessarily linear. One thing I'd add is that one can define $w$ to be the ratio $\frac{P}{\rho}$ (as it's dimensionless), and since in general both pressure and density depend on time (no $\vec x$ dependence is allowed in a perfect FRW universe), then $w$ will be time dependent as well, hence the equation of state would be non-linear. An example of this would be a scalar field $\phi(t)$ with $w=\frac{\frac{1}{2}\dot\phi^2-V(\phi)}{\frac{1}{2}\dot\phi^2+V(\phi)}$, which is derivable from the energy-momentum tensor, and you can see that the ratio is time dependant as the scalar field evolves with the equations of motion. I suppose you could imagine some other kind of field (spinor, vector etc.), but I'm not sure if it would be compatible with the symmetries of the FRW metric.

blueshift
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  • Thank you for your answer. Besides the fact that the relation (or w as you say) can be time dependent, my question is more related towards the fact that $\rho\propto P$ and not $\rho\propto P^2 $ for example. Given the fact that the ratio is dimensionless, the relation might include another dimensionfull quantity and become non-linear. I wonder if there is a model that considers such case and if so how it is motivated. – marRrR Jun 21 '16 at 12:04
  • Also, I don't mind considering cosmologies that don't assume FRW, maybe some cosmological model assuming a different cosmology could allow for that behavior for the equation of state. – marRrR Jun 21 '16 at 12:04
  • For an example of a nonlinear equation, you could for instance take the equation for a white dwarf, which is of the form $P\propto \rho^{\frac{5}{3}}$, source: http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec20.pdf. As for the second question, I can't say I'm aware of any cosmological (i.e. large scale) model that would use such an equation of state, mostly since there's plenty of evidence in favor of an FRW-like metric, and we're usually dealing with either matter, radiation, or a cosmological constant-like substance. – blueshift Jun 25 '16 at 22:25
  • In theory, if you have a collection of particles, I think you should be able to calculate the pressure and density directly from the following formulas: $P=\frac{g_x}{(2\pi)^3}\int d^3p; d^3x; f(x,p)\frac{p^2}{3E}$ and $\rho=\frac{g_x}{(2\pi)^3}\int d^3p; d^3x; f(x,p);E$, where $f(x,p)$ is the phase space density which depends on the physical coordinates and momenta, $g_x$ is the DOF of the particle, and $E$ is of course the energy of the particle. In kinetic equilibrium (the calculation performed usually), $f$ is just the Fermi or Bose distribution. – blueshift Jun 25 '16 at 22:58