Another static solution of the Friedmann equations - interpretation of $p=-\rho c^2$

Question

Looking for solutions of the Friedmann equations

$$(\frac{\dot a}{a})^2+\frac{kc^2}{a^2} = \frac{8 \pi G \rho+\Lambda c^2}{3}, \tag{1}$$

$$\frac{\ddot a}{a} = \frac{-4 \pi G}{3} (\rho + \frac{3p}{c^2}) +\frac{\Lambda c^2}{3}. \tag{2}$$

There seems to be this possibility

with $\Lambda = 0$, $k=0$, constant $H$, and $a=e^{Ht}$, the equations reduce to

$$3H^2 = 8 \pi G \rho, \tag{3}$$

$$3H^2 = -4 \pi G(\rho + \frac{3p}{c^2}), \tag{4}$$

leading to the solution

$$\rho = \frac{3H^2}{8 \pi G}, \tag{5}$$

$$p = - \rho c^2. \tag{6}$$

A nice, simple solution with scaling symmetry and time symmetry. If the expansion happens to all length scales including the observer as in Cosmology - an expansion of all length scales then the solution is an apparently static universe (but with a redshift as described in the link) and a universe always at critical density.

But the question is, what is the best interpretation of $p = - \rho c^2$ in a universe with $\Lambda = 0$? One idea is that explosive events in the universe e.g. from the nuclei of galaxies provide the negative pressure - is there any other way to interpret this?

Answer 1

The solution you consider is well-known de Sitter space. What you've got is the cosmological constant interpreted not as a separate $\Lambda$-term but as a contribution to $T_{\mu\nu}$ that looks like an ideal fluid with $w=\frac{p}{\rho}=-1$ (I'll use $c=1$). Note that the stress-energy tensor for the ideal fluid looks like, \begin{equation} T_{\mu\nu}=(p+\rho)u_\mu u_\nu -p g_{\mu\nu}\underset{w=-1}{=} \rho g_{\mu\nu} \end{equation} This means that there's no velocity field for such a fluid and if no other matter is present $\nabla_\mu T^{\mu\nu}=0$ enforces $\rho=const$. If you compare such a $T_{\mu\nu}$ with the $\Lambda$-term in the Einstein equation then you will see that they are completely equivalent.

The cosmological constant is the best current explanation for the Dark Energy and I will refer you to the alternative Dark Energy models for the stuff that can approximately imitate this (e.g. "quintessence" which is simply a scalar field in the slow roll regime).