Studies in Loop quantum Cosmology

Abstract

The standard model of cosmology is based on Einstein's relativistic theory of gravity, together with an implementation of the cosmological principle. It is formulated using the Friedmann-Robertson Walker(FRW) solution of general relativity. The structure of FRW solution is dictated by the cosmological principle. The FRW solution could be classified in to three classes, viz., Spatially flat, Close and Open. This thesis mainly deals with spatially flat and close models. The standard model of cosmology is also reffered as standard big bang cosmology. The horizon problem is directly related with the fact that the standard model of classical cosmology contains an initial singularity. The most popular approach to solve the 'Horizon Problem of Big Bang Theory' is to postulate a phase of inflation. In a quest for a quantum theory of gravity, String theory and Loop quantum gravity, two promising candidate theories of quantum gravity have emerged. In Loop Quantum Gravity, one attempts to formulate a background independent, non-perturbative theory of quantum gravity. The thesis consists of studies of implications of a particular quantum theory, addressing the issues of cosmological singularities. This quantum theory is a detailed adaptation of methods used in loop quantum gravity to the cosmological context and known to be Loop Quantum Cosmology. The spirit of minisuperspace approach is followed in the Loop Quantum Cosmology, with the uses of the quantisation methods of LQG. The imposition of the Hamiltonian constraint (Wheeler-DeWitt equation) in loop quantum cosmology leads to a difference equation with eigenvalues of the densitized triad variable serving as labels. The effective Hamiltonian differs from the classical Hamiltonian due to the modifications in the differential equation derived from the difference equation. The two sources of modifications, the Matter sector and the gravity sector, have been obtained by exploiting non-separable nature of the kinematical Hilbert space of loop quantum cosmology. One of the questions addressed in this thesis is whether the modifications in the matter sector imply violation of strong energy condition. The non-perturbative modification coming from loop-quantum cosmology to the scalar matter sector is known to imply inflation. It is further proved that loop quantum cosmology modified scalar field generates near exponential inflation in the small scale factor regime, for all positive definite potentials, independent of initial conditions and independent of ambiguity parameters. Most of the implications of loop quantum cosmology can be obtained rather easily from the effective Hamiltonian. Further the effective description is free from potential pathologies such as stability, a-causality. The implications of loop quantization in the context of Bianchi-IX model is briefly discussed. Further the current open issues in loop quantum cosmology and recent attempts made in those directions are discussed.