Evolution of Vacuum Fluctuations of an Ultra-Light Massive Scalar Field generated during and before Inflation

Abstract

We consider an ultra-light scalar field with a mass comparable to (or lighter than) the Hubble parameter of the present universe, and calculate the time evolution of the energy-momentum tensor of the vacuum fluctuations generated during and before inflation until the late-time radiation-dominated and matter-dominated universe. The equation of state changes from $w=1/3$ in the early universe to $w=-1$ at present, and it can give a candidate for the dark energy that we observe today. It then oscillates between $w=-1$ and $1$ with the amplitude of the energy density decaying as $a^{-3}$. If the fluctuations are generated during ordinary inflation with the Hubble parameter $H_I\lesssim 10^{-5}{M}_{\mathrm{Pl}}$, where $M_{\mathrm{Pl}}$ is the reduced Planck scale, we need a very large e-folding number $N\gtrsim 10^{12}$ to explain the present dark energy of the order of $10^{-3}\mathrm{eV}$. If a Planckian universe with a large Hubble parameter $H_P\sim M_{\mathrm{Pl}}$ existed before the ordinary inflation, an e-folding number $N\sim 240$ of the Planckian inflation is sufficient.

Comments: 36 pages, 6 figures; v2:version publisehd in PTEP