From the linear Hamiltonian of Section 6.3 one obtains equations of motion
For such a solvable model one can determine many more properties in detail, such as the behavior of fluctuations through the bounce and the role of coherent states. But the solvability of the model also reveals its very special nature, and, as well, that of all related numerical results, which are available so far. Initially, it came as a surprise that those numerical investigations were showing a nearly smooth bounce, with hardly any quantum effects deforming the wave packet or even leading to significant spreading. As quantum systems go, this is certainly not the expected behavior. Relating those models to a solvable one, where, as in the harmonic oscillator, moments only couple weakly or not at all to expectation values, demonstrates that the results are, after all, consistent with general expectations.
On the other hand, these models by themselves cannot be taken as an indication of the general behavior. If a matter potential is included, or anisotropies or inhomogeneities are taken care of, there will be quantum backreaction, as in typical quantum systems. States spread and deform during evolution, and it is no longer guaranteed that a state starting out semiclassically at large volume will remain so when it approaches the classical singularity. While the underlying difference equation is non-singular independent of quantum backreaction, the geometrical picture of non-singular behavior may well deviate strongly from a smooth bounce when the state is no longer semiclassical on small scales. (Note that phenomenological equations including higher powers of the connection, which are sometimes taken as indications for bounces, are themselves also subject to severe additional corrections in models with quantum backreaction.)
Moreover, even in these models one makes use of the large size of matter in the form of a large
. For smaller values even semiclassical states can enter the small-volume regime much
more deeply, which then would require other corrections, such as those from effective densities.
Assuming a large
for the dynamics of local geometrical variables would no longer be
justified in an inhomogeneous context, in addition to the then unavoidable quantum backreaction.
Currently, robust demonstrations of bounces in loop quantum cosmology only refer to cases
where quantum backreaction can safely be ignored, which are hardly realistic ones. Work on
extending those results and on understanding the more general picture of non-singular quantum
evolution is currently in progress; see also [110] for a discussion of the generality of present bounce
results.
The model of Section 6.3 for loop quantum cosmology allows precise results about the behavior of dynamical coherent states when they evolve through the point of the classical singularity. Not only can expectation values for volume and curvature describing the bounce be computed, but also fluctuations and higher moments of the state. Thus, one can see how a state evolves and whether the regime around the bounce has any implications. One can also analyze the full range of parameters determining such states in general, and thus address questions about how general certain properties, such as the approach to semiclassicality, are.
Of particular interest in the context of coherent states are fluctuations, which for the harmonic oscillator
would remain constant. The system relevant for cosmology, however, is different and here fluctuations
cannot be constant. Nevertheless, dynamical coherent states for the Wheeler–DeWitt model or the loop
quantized model demonstrate that the ratios and
remain constant for any part of the
universe before or after the bounce. This implies that fluctuations can be huge because the solvable model
has an unbounded
, but fluctuations relative to
stay small if they are small in a semiclassical initial
state.
This is not true, however, if we consider the transition through the bounce. The bounce connects a
contracting and an expanding phase, each of which is described well by Wheeler–DeWitt evolution. In
these phases, fluctuations relative to expectation values remain nearly constant. During the
transition through the bounce, however, the magnitude of fluctuations can change dramatically by
factors, which do not need to be near because fluctuations before and after the bounce are
determined by independent free parameters of dynamical coherent states, as illustrated in Figure 10
.
The ratio of these parameters is related to the squeezing of the state of the universe [69
]. For
unsqueezed states, fluctuations before and after the bounce are symmetric, but this is an additional
assumption for which no observational basis exists. While the uncertainty relation (74
) restricts
squeezing, and thus the asymmetry, for given fluctuations through a bound on the covariance, a
controlled application would require tight control on fluctuations of isotropic variables of the
universe.
If one were to ask whether the state before the Big Bang was as classical as the state after the Big Bang,
this question could not be answered based solely on observational information available after the Big Bang;
fluctuations before the bounce simply have such a weak influence that they could not be discerned from observations
afterwards [72].3
Moreover, the ratio of fluctuations before and after the bounce depends very sensitively on state parameters
such that a state with symmetric fluctuations requires extreme fine tuning [74
].
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As mentioned in Section 6.3, after solving effective equations one has to impose reality conditions to ensure
that expectation values and fluctuations of observables are real. This is the same condition one requires to
determine the physical inner product, which at the representation level can be very complicated. For
solutions to effective equations, on the other hand, reality conditions are as straightforward to implement as
in the classical case. In this way, physical inner product issues are under much better control in the effective
treatment. This has been demonstrated not only for the exactly effective solvable system of a free scalar in a
flat isotropic universe, but also in the presence of a perturbative potential [87], where quantum
backreaction occurs. This issue is also important for the effective treatment of constrained
systems.
In addition to the physical inner product, potential anomalies are one major issue in quantum gravity. Here
also, though direct calculations for operators are hard to perform in general, effective constraints provide a
much more practical route. One can first derive effective equations for a given set of constraint operators,
not worrying about anomalies. Inconsistencies can only arise when one tries to solve the resulting effective
constraints, so before doing so one must analyze the anomaly issue at the effective level. This is feasible
because the effective constraints are of the classical type, although amended by quantum corrections.
Calculations thus require only the use of Poisson relations rather than commutators of operators.
Moreover, after effective constraints have been computed one can often incorporate systematic
approximation schemes such as perturbation theory, and then make sure that anomalies are
absent order by order. This is a further simplification, which has been used in several cases of
quantum cosmological perturbation theory. As one of the results, the possibility of anomaly
freedom in perturbative loop quantizations in the presence of non-trivial quantum corrections was
demonstrated [92, 90, 91], and is studied for non-perturbative spherical symmetry in [102]. There are
then standard techniques to compute evolution equations for gauge invariant observables from
the anomaly-free effective constraints, which can immediately be employed in cosmological
phenomenology.
These conclusions are also conceptually important for the general framework: the loop quantization does not necessarily remove covariance, as it is sometimes said. Non-trivial quantum corrections of the characteristic forms of loop quantum gravity are allowed while respecting covariance in effective equations. This issue is related to the question of local Lorentz invariance, although it has not been fully evaluated yet. Naive corrections in Hamiltonians or equations of motion could imply superluminal propagation, e.g., of gravitational waves. This would certainly violate causality, but superluminal motion disappears when anomaly freedom is properly implemented [91]. And yet, non-trivial quantum corrections due to the loop quantization remain.
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