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 $H_I \lesssim 10^{-5} M_{\rm Pl}$, where $M_{\rm Pl}$ 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 $10^{-3} {\rm eV}$. If a Planckian universe with a large Hubble parameter $H_P \sim M_{\rm Pl}$ existed before the ordinary inflation, an e-folding number $N \sim 240$ of the Planckian inflation is sufficient.