  The Hawking radiation forms the essential basis of the black hole
thermodynamics. The black hole thermodynamics denotes a nice correspondence
between black hole kinematics and the laws of ordinary thermodynamics, but has
been so far considered only in an asymptotically flat case. Does such the
correspondence rely strongly on the feature of the gravity vanishing at the
infinity? In order to resolve this question, it should be considered for the
first to extend the Hawking radiation to a case with a dynamical boundary
condition like an expanding universe. Therefore the Hawking radiation in an
expanding universe is discussed in this paper. As a concrete model of a black
hole in an expanding universe, we use the swiss cheese universe which is the
spacetime including a Schwarzschild black hole in the
Friedmann-Robertson-Walker universe. Further for simplicity, our calculation is
performed in two dimension. The resultant spectrum of the Hawking radiation
measured by a comoving observer is generally different from a thermal one. We
find that the qualitative behavior of the non-thermal spectrum is of dumping
oscillation as a function of the frequency measured by the observer, and that
the intensity of the Hawking radiation is enhanced by the presence of a
cosmological expansion. It is appropriate to say that it a black hole with an
asymptotically flat boundary condition stays in a lowest energy thermal
equilibrium state, and that, once a black hole is put into an expanding
universe, it is excited to a non-equilibrium state and emits its mass energy
with stronger intensity than a thermal one.

General Relativity and Quantum Cosmology