Is the equation of state of the cosmological vacuum $P=-(1/3)\rho$?

Question

I understand that the equation of state of the vacuum is assumed to be $P = -\rho$ due to the Lorentz invariance of its stress-energy tensor. But this argument assumes flat spacetime. We know at cosmological scales that spacetime is not flat.

I was wondering whether the true cosmological vacuum equation of state is $P_{\rm vac} = - (1/3) \rho_{\rm vac}$.

Consider two parallel co-moving conducting plates in space whose separation distance scales with the scale factor $a(t)$.

The Casimir effect states that the pressure between the plates, $P_{\rm inside}$, is:

$$P_{\rm inside} \propto - 1 / a^4.$$

Therefore the pressure outside the plates $P_{\rm outside}$ is given by

$$P_{\rm outside} \propto 1 / a^4$$

If the equation of state of the fluid outside the plates is $P_{\rm outside} = w \rho_{\rm outside}$ then

$$\rho_{\rm outside} \propto 1 / a^4$$

Using the relationship

$$\rho \propto a^{-3[1+w]}$$

We find that the equation of state of the fluid outside the plates is

$$P_{\rm outside} = (1/3) \rho_{\rm outside}$$

Therefore the equation of state of the vacuum inside the plates is

$$P_{\rm inside} = - (1/3) \rho_{\rm inside}$$

Therefore

$$P_{\rm vac} = - (1/3) \rho_{\rm vac}$$

Does this argument make sense?

Answer 1

I think this arguments kinda holds. However it is always a bit risky to associate $\Lambda$ with the Casimir effect, as the cosmological constant arise in GR which is a classical field theory without any quantum considerations. It is yet unclear how $\Lambda$ can be treated consistently in a quantum way (and if it is the quantum field's vacuum energy), as there is no consensual quantum theory of gravity. The equation of state for $\Lambda$ arises solely from GR and the considerations about the stress-energy tensor you are referring to. This however does not require to assume that the space-time is flat. It solely comes from the fact that 1) space-time is locally Minkowskian (always true in GR) 2) Dark energy behaves as a perfect fluid (assumption comings from the cosmological principle) i.e. space must be homogeneous and isotropic on large scales.

The derivation goes roughly like this: (for more detail have a look at classical cosmology textbooks as Dodelson or Peter & Uzan)

A) In Einstein equation $G = 8\pi G T - \Lambda g$, you can consider that the term -Lambda g is the stress energy tensor associated to the cosmological constant $T_\Lambda = - \Lambda/(8\pi G) g$. B) The cosmological principle assumes that all components of the Universes behaves as perfect fluids on large scales to preserve homogeneity and isotropy. As such, the stress-energy tensors of these components should be perfect fluid stress-energy tensors. C) Now you should always be able to find locally a frame in which the metric is diagonal and in which $T$ can be expressed as diag($-\rho,p,p,p$). From there you can derive the equation of state for dark energy without assuming anything about a flat space-time. The only thing you have to assume is homogeneity and isotropy of space, which is a fundamental building block of cosmology, from which almost all of the considerations about dark energy are derived.