Noncommutative Double Scalar Fields in FRW Cosmology as Cosmical
  Oscillators
  We investigate effects of noncommutativity of phase space generated by two
scalar fields conformally coupled to curvature in FRW cosmology. We restrict
deformation of minisuperspace to noncommutativity between scalar fields and
between their canonical conjugate momenta. The investigation is carried out by
means of comparative analysis of mathematical properties of time evolution of
variables in classical model and wave function of universe in quantum level. We
find that impose of noncommutativity causes more ability in tuning time
solutions of scalar fields and hence, has important implications in evolution
of universe. We get that noncommutative parameter in momenta sector is the only
responsible parameter for noncommutative effects in flat universes. A
distinguishing feature of noncommutative solutions of scalar fields is that
they can be simulated with well known harmonic oscillators, depend on values of
spatial curvature. Namely free, forced and damped harmonic oscillators
corresponding to flat, closed and open universes. In this respect, we call them
cosmical oscillators. In closed universes, when noncommutative parameters are
small, cosmical oscillators have analogous effect with familiar beating effect
in sound phenomenon. The existence of non-zero constant potential does not
change solutions of scalar fields, but modifies scale factor. An interesting
feature of well behaved solutions of wave functions is that functional form of
its radial part is the same as commutative ones provided that given replacement
of constants, caused by noncommutative parameters, is performed. Further,
Noether theorem has been employed to explore effects of noncommutativity on
underlying symmetries in commutative frame. Two of six Noether symmetries of
flat universes, in general, are retained in noncommutative case, and one out of
three ones in non flat universes.
