Analytic solutions of scalar field cosmology with minimal and nonminimal coupling and deformed discrete and finite quantum systems

Abstract

In this thesis first, we study analytic solutions of cosmology. We investigate the most general cosmological model with real scalar field which is minimally coupled to gravity and Brans- Dicke cosmology. Field equations consist of three differential equations. We switch independent variable from time to scale factor by change of variable ˙a/a = H(a). Thus a new set of differential equations are analytically solvable with known methods. a(t) can be explicitly found as long as methods of integration techniques are available. We investigate the dynamics of the universe at early times as well as at late times in light of these formulas. We find mathematical machinery which turns on and turns off early accelerated expansion. On the other hand late time accelerated expansion is explained by cosmic domain walls. φ 4 potential is studied in Brans-Dicke Cosmology. In this thesis we also study discrete and finite quantum systems. We define a deformed kinetic energy operator for a discrete position space with a finite number of points. The structure may be either periodic or nonperiodic with well-defined end points. It is shown that for the nonperiodic case the translation operator becomes nonunitary due to the end points. This uniquely defines an algebra which has the desired unique representation. Energy eigenvalues and energy wave functions for both cases are found. In addition, we uncover the mathematical structure of the Schwinger algebra and introduce almost unitary Schwinger operators which are derived by considering translation operators on a finite lattice.