  We reconsider and analyze in detail the problem of particle production in the
time dependent background of $c=1$ matrix model where the Fermi sea drains away
at late time. In addition to the moving mirror method, which has already been
discussed in hep-th/0403169 and hep-th/0403275, we describe yet another method
of computing the Bogolubov coefficients which gives the same result. We
emphasize that these Bogolubov coefficients are approximately correct for small
value of the deformation parameter.
  We also study the time evolution of the collective field theory stress-tensor
with a special point-splitting regularization. Our computations go beyond the
approximation of the previous treatments and are valid at large coordinate
distances from the boundary at a finite time and up-to a finite coordinate
distance from the boundary at late time. In this region of validity our
regularization produces a certain singular term that is precisely canceled by
the collective field theory counter term in the present background. The energy
and momentum densities fall off exponentially at large distance from the
boundary to the values corresponding to the static background. This clearly
shows that the radiated energy reaches the asymptotic region signaling the
space-time decay.
