When the universe has collapsed to a sufficiently small size, repulsion becomes noticeable and bouncing
solutions become possible as illustrated in Figure 1. Requirements for a bounce are that the conditions
and
can be fulfilled at the same time, where the first one can be evaluated with the
Friedmann equation, and the second one with the Raychaudhuri equation. The first condition can only be
fulfilled if there is a negative contribution to the matter energy, which can come from a positive curvature
term
or a negative matter potential
. In those cases, there are classical solutions with
, but they generically have
corresponding to a recollapse. This can easily be seen in the flat
case with a negative potential where (30
) is strictly negative with
at large
scales.
The repulsive nature at small scales now implies a second point where from (29
) at
smaller
since the matter energy now decreases also for
. Moreover, the modified
Raychaudhuri equation (30
) has an additional positive term at small scales such that
becomes
possible.
Matter also behaves differently through the modified Klein-Gordon equation (32). Classically, with
the scalar experiences antifriction and
diverges close to the classical singularity. With the
modification, antifriction turns into friction at small scales, damping the motion of
such that it
remains finite. In the case of a negative potential [68
] this allows the kinetic term to cancel the
potential term in the Friedmann equation. With a positive potential and positive curvature, on the
other hand, the scalar is frozen and the potential is canceled by the curvature term. Since the
scalar is almost constant, the behavior around the turning point is similar to a de Sitter bounce
[187, 203]. Further, more generic possibilities for bounces arise from other correction terms
[100, 97].
Repulsion can not only prevent collapse but also accelerates an expanding phase. Indeed, using the behavior
(26) at small scales in the effective Raychaudhuri equation (30
) shows that
is generically positive since
the inner bracket is smaller than
for the allowed values
. Thus, as illustrated by the
numerical solution in the upper left panel of Figure 2
, inflation is realized by quantum gravity effects for
any matter field irrespective of its form, potential or initial values [45
]. The kind of expansion at early
stages is generically super-inflationary, i.e., with equation of state parameter
. For
free massless matter fields,
usually starts very small, depending on the value of
, but
with a non-zero potential such as a mass term for matter inflation
is generically close to
exponential:
. This can be shown by a simple and elegant argument independently of the
precise matter dynamics [101]: The equation of state parameter is defined as
where
is the pressure, i.e., the negative change of energy with respect to volume,
and
energy density. Using the matter Hamiltonian for
and
, we
obtain
and thus in the classical case
as usually. In the modified case, however, we have
|
Since for small the numerator in the fraction approaches zero faster than the second part of the
denominator,
approaches minus one at small volume except for the special case
, which is
realized for
. Note that the argument does not apply to the case of vanishing potential since then
and
presents a unique solution to the linear equations for
and
. In
fact, this case leads in general to a much smaller
[45
].
One can also see from the above formula that , though close to minus one, is a little smaller than
minus one generically. This is in contrast to single field inflaton models where the equation of state
parameter is a little larger than minus one. As we will discuss in Section 4.15, this opens the door to
characteristic signatures distinguishing different models.
Again, also the matter behavior changes, now with classical friction being replaced by antifriction [77].
Matter fields thus move away from their minima and become excited even if they start close to a
minimum (Figure 2
). Since this does not only apply to the homogeneous mode, it can provide a
mechanism of structure formation as discussed in Section 4.15. But also in combination with
chaotic inflation as the mechanism to generate structure does the modified matter behavior
lead to improvements: If we now view the scalar
as an inflaton field, it will be driven to
large values in order to start a second phase of slow-roll inflation which is long enough. This is
satisfied for a large range of the ambiguity parameters
and
[67] and can even leave
signatures [197] in the cosmic microwave spectrum [134]: The earliest moments when the inflaton
starts to roll down its potential are not slow roll, as can also be seen in Figures 2
and 3
where
the initial decrease is steeper. Provided the resulting structure can be seen today, i.e., there
are not too many e-foldings from the second phase, this can lead to visible effects such as a
suppression of power. Whether or not those effects are to be expected, i.e., which magnitude of the
inflaton is generically reached by the mechanism generating initial conditions, is currently being
investigated at the basic level of loop quantum cosmology [27]. They should be regarded as
first suggestions, indicating the potential of quantum cosmological phenomenology, which have
to be substantiated by detailed calculations including inhomogeneities or at least anisotropic
geometries. In particular the suppression of power can be obtained by a multitude of other
mechanisms.
It is already clear that there are different inflationary scenarios using effects from loop cosmology. A scenario without inflaton is more attractive since it requires less choices and provides a fundamental explanation of inflation directly from quantum gravity. However, it is also more difficult to analyze structure formation in this context while there are already well-developed techniques in slow role scenarios.
In these cases where one couples loop cosmology to an inflaton model one still requires the same conditions for the potential, but generically gets the required large initial values for the scalar by antifriction. On the other hand, finer details of the results now depend on the ambiguity parameters, which describe aspects of the quantization that also arise in the full theory.
It is also possible to combine collapsing and expanding phases in cyclic or oscillatory models [148]. One
then has a history of many cycles separated by bounces, whose duration depends on details of the model
such as the potential. There can then be many brief cycles until eventually, if the potential is right, one
obtains an inflationary phase if the scalar has grown high enough. In this way, one can develop
ideas for the pre-history of our universe before the Big Bang. There are also possibilities to
use a bounce to describe the structure in the universe. So far, this has only been described
in effective models [137] using brane scenarios [151] where the classical singularity has been
assumed to be absent by yet to be determined quantum effects. As it turns out, the explicit
mechanism removing singularities in loop cosmology is not compatible with the assumptions made
in those effective pictures. In particular, the scalar was supposed to turn around during the
bounce, which is impossible in loop scenarios unless it encounters a range of positive potential
during its evolution [68]. Then, however, generically an inflationary phase commences as in [148],
which is then the relevant regime for structure formation. This shows how model building in
loop cosmology can distinguish scenarios that are more likely to occur from quantum gravity
effects.
Cyclic models can be argued to shift the initial moment of a universe in the infinite past, but they do not explain how the universe started. An attempt to explain this is the emergent universe model [110, 112] where one starts close to a static solution. This is difficult to achieve classically, however, since the available fixed points of the equations of motion are not stable and thus a universe departs too rapidly. Loop cosmology, on the other hand, implies an additional fixed point of the effective equations which is stable and allows to start the universe in an initial phase of oscillations before an inflationary phase is entered [160, 53]. This presents a natural realization of the scenario where the initial scale factor at the fixed point is automatically small so as to start the universe close to the Planck phase.
Cosmological equations displaying super-inflation or antifriction are often unstable in the sense that matter
can propagate faster than light. This has been voiced as a potential danger for loop cosmology, too [94, 95
].
An analysis requires inhomogeneous techniques at least at an effective level, such as those described
in Section 4.12. It has been shown that loop cosmology is free of this problem, because the
modified behavior for the homogeneous mode of the metric and matter is not relevant for matter
propagation [129
]. The whole cosmological picture that follows from the effective equations is thus
consistent.
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