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 effective Klein-Gordon equation (32). 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 conceptual considerations it is interesting that solutions at small and large scales are connected by a duality transformation , which even exists between effective solutions for loop cosmology and braneworld cosmology .
We have seen that the equations of motion following from an effective Hamiltonian 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 (32) 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 collapse of a contracting universe  but also, in an expanding universe, enhance its expansion. The universe then accelerates in an inflationary manner from quantum gravity effects alone . Similarly, the modified behavior of matter fields has implications for inflationary models .
© Max Planck Society and the author(s)