# Is there any use of a quadratic equation of state in FLRW cosmology?

Consider standard FLRW cosmology. Usually, the relation between energy density $$\rho$$ and pressure $$p$$ of a cosmological fluid component is linear: $$$$\tag{1} p = w \, \rho,$$$$ where $$w$$ is just a constant ($$w = 0$$ for the dust model, $$w = \frac{1}{3}$$ for ultra-relativistic radiation, $$w = - 1$$ for the cosmological constant, etc). I'm interested in applications of the following non-linear equation of state (quadratic in this case): $$$$\tag{2} p = -\, \kappa \, \rho^2,$$$$ where $$\kappa$$ is a positive constant. I never saw any textbook or paper discussing such an equation of state. Is there a reason for that? I would like to find some references on it.

Notice that there is no sound speed from (2) and that the four main energy conditions may be violated, depending on the value of the energy density $$\rho$$ relative to the constant $$\kappa$$: \begin{align} &\textsf{NEC:} \qquad \rho + p \ge 0. \tag{3} \\[12pt] &\textsf{WEC:} \qquad \rho \ge 0, \quad \text{and} \quad \rho + p \ge 0. \tag{4} \\[12pt] &\textsf{SEC:} \qquad \rho + 3 p \ge 0, \quad \text{and} \quad \rho + p \ge 0. \tag{5} \\[12pt] &\textsf{DEC:} \qquad \rho \ge |\, p \,|. \tag{6} \end{align} These energy conditions are a subject of much controversy today, and nothing in physics imposes that they need to be satisfied in all cases.

The state equation (2) describes phantom energy when $$\rho > 1/\kappa$$ and may lead to a Big Rip scenario.

Local conservation of energy-momentum imposes the following constraint: $$$$\tag{7} \nabla_{\mu} \, T^{\mu \nu} = 0 \qquad \Rightarrow \qquad \dot{\rho} \, a + 3 \, (\rho + p) \, \dot{a} = 0.$$$$ Substituting (2) into (7) gives the energy density as a function of the scale factor $$a(t)$$: $$$$\tag{8} \rho = \frac{\mu}{\mu \kappa + a^3} = \frac{\rho_0}{\kappa \, \rho_0 + (1 - \kappa \, \rho_0)(a / a_0)^3},$$$$ where $$\mu$$ is an integration constant and $$\rho_0$$ is the energy density today (at time $$t_0$$ for which $$a(t_0) \equiv a_0$$). Notice that (8) gives $$\rho \approx 1 / \kappa = \text{cste}$$ (the fluid is behaving like a cosmological constant) when $$a / a_0 \ll 1$$, and $$\rho \propto a^{-3}$$ (like dust) when $$a / a_0 \gg 1$$ (but only if $$\kappa \, \rho_0 < 1$$, so that energy density stay positive: $$\rho > 0$$). So (2) may be a model for inflation before a matter (i.e. dust) era.

So was the equation of state (2) already discussed before (surely!)? Any reference on it?

EDIT: Here's a cool graph made with Mathematica, of the evolution of the scale factor $$a(t)/a_0$$ for a model containing a cosmological constant ($$\Lambda > 0$$) and a fluid described by equation of state (2) (with $$\kappa$$ very small). For the parameters selected, the universe is spatially closed ($$k = 1$$), and there's a Big Bounce without any singularity (no Big Bang and no Big Rip). There's a timid inflation era, a matter domination era and a Dark energy era (second inflation):