One can translate this into a phenomenological Friedmann equation involving the scale factor and its
time derivative. The Hamiltonian equation of motion for implies
and thus
when this is inserted into Equation (34 A similar form of effective equations can be used for a closed model, which, due to its classical recollapse
now being combined with a quantum bounce, provides cyclic-universe models. These are no longer
precise effective equations, but one can show that additional quantum corrections during the
classical collapse phase are small. Thus, deviations from the phenomenological trajectory solving
Equation (35) build up only slowly in time and require several cycles to become noticeable, as
first analyzed and verified numerically in [28
]. If one pretends that the same kind of quadratic
corrections can be used for massive or self-interacting scalars, there is a rich phenomenology of cyclic
universes, some of which has been analyzed in [281, 270, 271, 314, 307, 311, 287, 313, 151].
However, for such long evolution times it is crucial to consider all quantum effects, which for
massive or self-interacting matter is not captured fully in the simple quadratic energy-density
corrections.
Qualitatively, the appearance of bounces agrees with what we saw for effective densities, although the
precise realization is different and occurs for different models. Moreover, if we bring together higher-power
corrections, as well as effective densities, we see that, although individually they both have
similar effects, together they can counteract each other. In the specific model considered here,
using effective densities in the matter term and possibly the gravitational term results in an
inequality of the form where
replaces the classical
used
above and has an upper bound. Thus, one cannot immediately conclude that
is bounded
away from zero and rather has to analyze the precise numerical constants appearing in this
relation. A conclusion about a bounce can then no longer be generic but will depend on initial
conditions. This shows the importance of bringing all possible quantum corrections together in a
consistent manner. It is also important to realize that, while Equation (34
) is a precise effective
equation for the dynamical behavior of a quantum state as described later, the inclusion of
effective densities would imply further quantum corrections, which we have not described so
far and which result from backreaction effects of a spreading state on its expectation values.
These corrections would also have to be included for a complete analysis, which is still not
finished.
Before all quantum corrections have been determined, one can often estimate their relative
magnitudes. In the model considered here, this is possible when one uses the condition that a
realistic universe must have a large matter content. Thus, must be large, which affects
the constants in the above relations to the extent that the maximum of
will not be
reached before
reaches the value one. In this case there is thus a bounce independent of
effective-density suppressions. Moreover, if one starts in a sufficiently semiclassical state at large
volume, quantum backreaction effects will not change the evolution too much before the bounce is
reached. Thus, higher power corrections are indeed dominant and can be used reliably. There are,
however, several caveats: First, while higher-power corrections are relevant only briefly near
the bounce, quantum backreaction is present at all times and can easily add up. Especially
for systems with a different matter content such as a scalar potential, a systematic analysis
has yet to be performed. Anisotropies and inhomogeneities can have a similar effect, but for
inhomogeneities an additional complication arises: not all the matter is lumped into one single
isotropic patch, but rather distributed over the discrete building blocks of a universe. Local
matter contents are then much smaller than the total one, and the above argument for the
dominance of higher-power corrections no longer applies. Finally, the magnitude of corrections in
effective densities has been underestimated in most homogeneous studies so far because effects of
lattice states were overlooked (see Appendix in [75
]). They can thus become more important at
small scales and possibly counteract a bounce, even if the geometry can safely be assumed
to be nearly isotropic. Whether or not there is a bounce in such cases remains unknown at
present.
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