Hawking-like radiation as tunneling from the apparent horizon in a FRW
  Universe
  We study Hawking-like radiation in a Friedmann-Robertson-Walker (FRW)
universe using the quasi-classical WKB/tunneling method which pictures this
process as a "tunneling" of particles from behind the apparent horizon. The
correct temperature of the Hawking-like radiation from the FRW spacetime is
obtained using a canonical invariant tunneling amplitude. In contrast to the
usual quantum mechanical WKB/tunneling problem where the tunneling amplitude
has only a spatial contribution, we find that the tunneling amplitude for FRW
spacetime (i.e. the imaginary part of the action) has both spatial and temporal
contributions. In addition we study back reaction and energy conservation of
the radiated particles and find that the tunneling probability and change in
entropy, ${\cal S}$ are related by the relationship: $\Gamma\propto\exp[-\Delta
{\cal S}]$ which differs from the standard result $\Gamma\propto\exp[\Delta
{\cal S}]$. By regarding the whole FRW universe as an isolated adiabatic system
the change in the total entropy is zero. Then splitting the entropy between
interior and exterior parts of the horizon ($\Delta {\cal S}_{total}=\Delta
{\cal S}_{int} + \Delta {\cal S}_{ext}=0$), we can explain the origin of the
minus sign difference with the usual result: our $\Delta {\cal S}$ is for the
interior region while the standard result from black hole physics is for the
exterior region.
