that is restricted to be zero. If we assume a constant scalar potential , there is no
-dependence
and the scalar equations of motion show that
is constant. Thus, the potential for the motion of
is
essentially determined by the function
.
In the classical case, and the potential is negative and increasing, with a divergence at
. The scale factor
is thus driven toward
, which it will always reach in finite time where
the system breaks down. With the effective density
, however, the potential is bounded from below,
and is decreasing from zero for
to the minimum around
. Thus, the scale factor is now slowed
down before it reaches
, which, depending on the matter content, could avoid the classical
singularity altogether.
The behavior of matter is also different as shown by the Klein–Gordon equation (33). Most importantly,
the derivative in the
-term changes sign at small
since the effective density is increasing there. Thus,
the qualitative behavior of all the equations changes at small scales, which, as we will see, gives rise to many
characteristic effects. Nevertheless, for the analysis of the equations, as well as for conceptual
considerations, it is interesting that solutions at small and large scales are connected by a duality
transformation [214, 132], which even exists between effective solutions for loop cosmology and braneworld
cosmology [133].
We have seen that the equations of motion following from a phenomenological Hamiltonian incorporating effective densities are expected to display qualitatively different behavior at small scales. Before discussing specific models in detail, it is helpful to observe what physical meaning the resulting modifications have.
Classical gravity is always attractive, which implies that there is nothing to prevent collapse in
black holes or the whole universe. In the Friedmann equation this is expressed by the fact that
the potential, as used before, is always decreasing toward , where it diverges. With
the effective density, on the other hand, we have seen that the decrease stops and instead the
potential starts to increase at a certain scale before it reaches zero at
. This means
that at small scales, where quantum gravity becomes important, the gravitational attraction
turns into repulsion. In contrast to classical gravity, thus, quantum gravity has a repulsive
component, which can potentially prevent collapse. So far, this has only been demonstrated in
homogeneous models, but it relies on a general mechanism which is also present in the full
theory.
Not only the attractive nature of gravity changes at small scales, but also the behavior of matter in a
gravitational background. Classically, matter fields in an expanding universe are slowed down by a friction
term in the Klein–Gordon equation (33), where
is negative. Conversely, in a
contracting universe matter fields are excited and even diverge when the classical singularity is reached.
This behavior turns around at small scales, where the derivative
becomes positive. Friction
in an expanding universe then turns into antifriction such that matter fields are driven away from
their potential minima before classical behavior sets in. In a contracting universe, on the other
hand, matter fields are not excited by antifriction but freeze once the universe becomes small
enough.
These effects do not only have implications for the avoidance of singularities at
but also for the behavior at small but non-zero scales. Gravitational repulsion can not only
prevent the collapse of a contracting universe [279
] but also, in an expanding universe, enhance
its expansion. The universe then accelerates in an inflationary manner from quantum gravity
effects alone [53
]. Similarly, the new behavior of matter fields has implications for inflationary
models [111
].
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