Abstract. This is an introduction to loop quantum cosmology (LQC) reviewing mini-and midisuperspace models as well as homogeneous and inhomogeneous effective dynamics.

One of the qualitatively distinct and robust implication of Loop Quantum Gravity (LQG) is the underlying discrete structure. In the cosmological context elucidated by Loop Quantum Cosmology (LQC), this is manifested by the Hamiltonian constraint equation being a (partial) difference equation. One obtains an effective Hamiltonian framework by making the continuum approximation followed by a WKB approximation. In the large volume regime, these lead to the usual classical Einstein equation which is independent of both the Barbero-Immirzi parameter γ as well as . In this work we present an alternative derivation of the effective Hamiltonian by-passing the continuum approximation step. As a result, the effective Hamiltonian is obtained as a close form expression in γ. These corrections to the Einstein equation can be thought of as corrections due to the underlying discrete (spatial) geometry with γ controlling the size of these corrections. These corrections imply a bound on the rate of change of the volume of the isotropic universe. In most cases these are perturbative in nature but for cosmological constant dominated isotropic universe, there are significant deviations.

We consider a one-dimensional network in which the nodes at Euclidean distance l can have long range connections with a probability P(l) approximately l(-delta) in addition to nearest neighbor connections. This system has been shown to exhibit small-world behavior for delta<2, above which its behavior is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at delta=1. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicates that a continuous phase transition occurs at this point. Using these results, we find that the two transitions occurring in this network can be detected in any dimension by the behavior of a single quantity, the average bond length. The phase transitions in all dimensions are nontrivial in nature.

Models of neutron stars (NSs) with hyperon cores are constructed with an effective chiral model in mean-field approximation. The hyperon couplings are fixed by reproducing their experimentally determined binding energies. The impact of these couplings on population of different particles and the equation of state (EoS) are studied in this work. The global properties of NSs like gravitational mass, radius, baryonic mass and central density are calculated using parameterized Tolman-Oppenheimer-Volkoff equations (PTOV) with special emphasis on two effects of pressure -one contributing to total mass density and the other to self gravity of the star. We find that with PTOV solutions in static conditions, a softer EoS (including hyperons) can also lead to massive stellar configurations of NSs, which are in well agreement with the observed maximum mass bound of ≈ 2M (PSR J0348-0432). Estimates of R 1.4 and R 1.6 , obtained with the PTOV equations are consistent with the recent findings of the same from the data analysis of gravitational waves (GW170817) observation.