Despite what one might expect, much of cosmology is based on classical physics. Certainly the standard model of particle physics is constructed from classical gravity, namely an FRW model solution to Einstein's equations, which serves as a fixed “background” over which one considers in a semiclassical approximation quantum fields, say to model the cosmic microwave background radiation.
On the one hand it is remarkable how well the standard model of cosmology built this way compares to available experimental data. On the other hand it is clear that a more precise model will need to take quantum field theory into account more properly. In particular non-classical effects of quantum gravity will play a role at least close to the initial singularity, if not also for the explanation of the cosmological constant and other aspects of the standard model of cosmology which currently don’t have a conceptual explanation in the model.
Such a more comprehensively quantum mechanical discussion of cosmology goes by the term quantum cosmology.
However, despite the existence of this term, quantum cosmology is to a large extent only a hypothetical field of research. This is due to two major open conceptual problems of present day fundamental physics:
the nature of quantum gravity remains largely unsettled;
the very formulations of quantum mechanics/quantum field theory available all require, more or less explicitly and to lesser or greater extent, an external classical observer to make sense of the predictions of the theory. This makes sense, in principle, for every subsystem of the observable universe. But a comprehensive theory of “quantum cosmology” would seem to need a notion of observables of quantum phenomena by “quantum observers” internal to the quantum system which they observe, instead of by “classical observers” external to that quantum system.
It has been suggested (Doering-Isham, section 1) that the second of the conceptual problems above might have a natural solution in terms of the internal logic of a topos-incarnation of the space of states of a quantum mechanical system. One proposed formalization of this is the notion of a Bohr topos. While these constructions may have something to them, implications, if any, for actual quantum cosmology are not clear yet.
Calcagni, G.; Papantonopoulos, L.; Siopsis, G.; Tsamis, N. (Eds.), Quantum Gravity and Quantum Cosmology Lecture Notes in Physics, Vol. 863
Claus Kiefer, Barbara Sandhoefer, Quantum cosmology (arXiv:0804.0672)
J.J.Halliwell, Introductory Lectures on Quantum Cosmology (1990) (arXiv:0909.2566)
Edward Anderson, On the Semiclassical Approach to Quantum Cosmology Class. Quant. Grav. 28 185008 (2011) (arXiv:1101.4916)
On structure formation during cosmic inflation in quantum inhomogeneous cosmology:
Quantum cosmology for supergravity and its relation to the E10 U-duality group is discussed in
A survey of conceptual problems in quantum cosmology, or issues regarded as such, is in
Discussion of the problem of internal quantum observers with an eye towards the internal logic of a topos is in the introduction of
For more on this see at Bohr topos.