Evolution of vacuum fluctuations generated during and before inflation
  We calculate the time evolution of the expectation value of the
energy-momentum tensor for a minimally-coupled massless scalar field in
cosmological spacetimes, with an application to dark energy in mind. We first
study the evolution from inflation until the present, fixing the Bunch-Davies
initial condition. The energy density of a quantum field evolves as $\rho \sim
3(H_I H)^2 /32 \pi^2 $ in the matter-dominated (MD) period, where $H_I$ and $H$
are the Hubble parameters during inflation and at each moment. Its equation of
state, $w=\rho/p$, changes from a negative value to $w=1/3$ in the
radiation-dominated period, and from $1/3$ to $w=0$ in the MD period. We then
consider possible effects of a Planckian universe, which may have existed
before inflation, by assuming there was another inflation with the Hubble
parameter $H_P (> H_I)$. In this case, modes with wavelengths longer than the
current horizon radius are mainly amplified, and the energy density of a
quantum field grows with time as $\rho \sim (a/a_0)(H_P H)^2/32$ in the MD
period, where $a$ and $a_0$ are the scale factors at each time and at present.
Hence, if $H_P$ is of the order of the Planck scale $M_P$, $\rho$ becomes
comparable to the critical density $3(M_P H)^2$ at the present time. The
contribution to $\rho$ from the long wavelength fluctuations generated before
the ordinary inflation has $w=-1/3$ in the free field approximation. We mention
a possibility that interactions further amplify the energy density and change
the equation of state.
