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