In the textbook introduction to cosmology by Barbara Ryden, the author say that the energy density of the inflaton field is:
$$\epsilon_{\phi} = \frac{\dot{\phi^2}}{2 \hbar c^3} + V(\phi)$$
However, she doesn't really explained how she get that expression. It is obvious? What am I missing?
Thanks
I wouldn't say obvious, especially to someone who isn't familiar with quantum field theory, but it's the type of expression that theorists write because it's the only possible thing it could be by analogy to other fields.
The first term is half a time derivative squared, so something analogous to $\frac{1}{2} (\dot{x})^2 = \frac{1}{2} v^2$. Hence, it's the kinetic energy, normalised by some universal constants to make the dimensions work. The second term is a generic potential, with no constraints apart from the usual restriction that it's a function of only the field. The total energy is thus just the sum of kinetic and potential energy, as it always is. Because the inflaton field is conjectured to be a scalar field, there's virtually nothing else the energy density could be.
