  We study the evolution of time-dependent fluctuations and particle production
in an expanding dS and contracting AdS universe. Using the functional
Schrodinger formalism we are able to probe the time dependent regime which is
out of the reach of the standard approximations like the Bogolyubov method. In
both cases, the evolution of fluctuations is governed by the harmonic
oscillator equation with time dependent frequency. In the case of an expanding
dS universe we explicitly show that the frequency of fluctuations produced at a
certain moment diminish in time, while the distribution of the created
particles quickly approaches the thermal radiation of the dS space. In the case
of a contracting AdS universe we show that the frequency of fluctuations
produced at a certain moment grow in time. Nominally, the temperature of
radiation diverges as the Big Crunch is approaching, however, increasing
oscillations of the spectrum make the temperature poorly defined, which is in
agreement with the fact that AdS space does not have an event horizon which
would cause thermal radiation. Unlimited growth of fluctuations indicates that
an eventual tunneling into AdS vacuum would have catastrophic consequences for
our universe.
