What is the energy density of the inflaton field?

Question

I am trying to compare the theoretically calculated vacuum energy density according to quantum field theory with the energy density of the inflaton field, in joules per cubic meter (or Pascal). I found that the first is about $10^{113}$ joules per cubic meter according to Wikipedia, and that the second has an energy scale of about 10$^{16}$ GeV according to this paper (p. 215). However, I am unsure about how to convert these two different units. I also cannot find the energy density of the inflaton field in joules per cubic meter (maybe because the universe is much smaller than a cubic meter during inflation?).

I read that the energy density of the inflaton field is more or less constant while inflation is going on.

Answer 1

The energy density of inflation is dominated by its potential $V$, i.e. $\rho\simeq V$. Your linked paper defined inflation's energy scale to be $V^{1/4}\simeq 10^{16}~\mathrm{GeV}$, which implies that the associated energy density is

$$\rho = (10^{16}~\mathrm{GeV})^4 = 10^{64}~\mathrm{GeV}^4$$

Note that we don't really know the energy scale of inflation, but $10^{16}$ GeV is a typical value for inflation models. It can't be much higher than that.

How does this relate to conventional units like $\mathrm{J}/\mathrm{m}^3$? Just set $\hbar=c=1$, so that $1~\mathrm{GeV}\simeq 5.0677\times 10^{15}~\mathrm{m}^{-1}$. Meanwhile $1~\mathrm{GeV}\simeq 1.6022\times 10^{-10}~\mathrm{J}$, so $\mathrm{GeV}^4\simeq 2.085\times 10^{37}~\mathrm{J}/\mathrm{m}^3$, and hence

$$\rho \simeq 2\times 10^{101}~\mathrm{J}/\mathrm{m}^3.$$

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Of course, that's a very roundabout way to compare things. The Wikipedia value for the vacuum energy, $10^{113}~\mathrm{J}/\mathrm{m}^3$, is just $m_\mathrm{p}^4$, where $m_\mathrm{p}\simeq 1.22\times 10^{19}~\mathrm{GeV}$ is the Planck mass. So you are really just comparing the $(10^{19}~\mathrm{GeV})^4$ vacuum energy density to the $(10^{16}~\mathrm{GeV})^4$ inflation energy density.