Scalar field energy density and pressure in hot universe

[karlzr]

Kolb&Turner in "the early universe" mentioned that for a scalar field $\phi$ at finite temperature, $p=-V_T(\phi)$ and $\rho=-p+T\frac{d p(T)}{d T}$ where $V_T$ is potential energy including temperature correction. My question is: when we consider the evolution of the universe using Friedman's equation, should we use this kind of definition of $\rho$ and $p$ or we stick to those obtained from stress energy tensor? I would say the latter since that's what we can get from Einstein's equation. But then I don't understand what are Kolb&Turner's $\rho$ and $p$. Why doesn't kinetic term contribute to them?

 

[fzero]

Perhaps the notation is unfortunate, but $V_T(\phi)$ there is not the scalar potential, but rather the free-energy computed at finite temperature in perturbation theory. So one puts the particles in an appropriate box with finite temperature boundary conditions (periodicity) and computes the partition function

$$ Z = \int \mathcal{D}\phi \exp \left\{ -\int_0^\beta d\tau \int d^3x \, \mathcal{L}(\phi) \right\}.$$

The effective action is $-\ln Z$ and is the sum of 1-particle irreducible diagrams. This quantity is called the "effective potential" because the perturbative calculation is done over diagrams where the external field legs are set to the vacuum expectation value of the scalar field. There are no physical external particles in the process.

The results quoted there are the results once the sum over momentum modes has already been done. There is a reference to [3] for the result which is the classic paper of Dolan and Jackiw, but a random web-accessible source is this master's thesis that seems to include the relevant calculations.

For most calculations of late-term cosmology, the pedestrian free-particle equations of state are appropriate. However, for early times, when phase transitions are possible, the finite temperature, interacting theory results are necessary.