Definition of Energy Scale of Inflation

Question

It seems natural to all people, when talking about the energy scale of inflation to take simply $$E_{\text{inf}} = A\cdot r\cdot V^{1/4}$$ Where, $A$ is some factor, $r$ the tensor to scalar ratio and $V$ is the potential.

I know that $V$ is the potential of a Lagrangian density with dimension of 4 in energy, $M^4$ (natural units), and $V$ is an energy density in the space.

For what I understood one takes the DEFINITION of $E_{\text{inf}}$ by dimensional analysis.

Buy I thought that for the energy we can also take something like $\int V d^3x$ on some volume. Isn't this less arbitrary?

Something else that I don't catch now is in what time is $V$ calculated when we define the energy of the inflation?

Answer 1

I would actually define the scale of inflation as the Hubble scale, defined by $H = \kappa \sqrt{V_*/3}$, where $V_*$ is the value of the inflaton potential at Horizon Exit, so on top of the plateau of the inflaton potential.