Problems in cosmology and numerical relativity

Doctoral Thesis


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University of Cape Town

A generic feature of most inflationary scenarios is the generation of primordial perturbations. Ordinarily, such perturbations can interact with a weak magnetic field in a plasma, resulting in a wide range of phenomena, such as the parametric excitation of plasma waves by gravitational waves. This mechanism has been studied in different contexts in the literature, such as the possibility of indirect detection of gravitational waves through electromagnetic signatures of the interaction. In this work, we consider this concept in the particular case of magnetic field amplification. Specifically, we use non-linear gauge-in variant perturbation theory to study the interaction of a primordial seed magnetic field with density and gravitational wave perturbations in an almost Friedmann-Lemaıtre-Robertson- Walker (FLRW) spacetime with zero spatial curvature. We compare the effects of this coupling under the assumptions of poor conductivity, perfect conductivity and the case where the electric field is sourced via the coupling of velocity perturbations to the seed field in the ideal magnetohydrodynamic (MHD) regime, thus generalizing, improving on and correcting previous results. We solve our equations for long wavelength limits and numerically integrate the resulting equations to generate power spectra for the electromagnetic field variables, showing where the modes cross the horizon. We find that the interaction can seed Electric fields with non-zero curl and that the curl of the electric field dominates the power spectrum on small scales, in agreement with previous arguments. The second focus area of the thesis is the development a stable high order mesh refinement scheme for the solution of hyperbolic partial differential equations. It has now become customary in the field of numerical relativity to couple high order finite difference schemes to mesh refinement algorithms. This approach combines the efficiency of local mesh refinement with the robustness and accuracy of higher order methods. To this end, different modifications of the standard Berger-Oliger adaptive mesh refinement a logarithm have been proposed. In this work we present a new fourth order convergent mesh refinement scheme with sub- cycling in time for numerical relativity applications. One of the distinctive features of our algorithm is that we do not use buffer zones to deal with refinement boundaries, as is currently done in the literature, but explicitly specify boundary data for refined grids instead. We argue that the incompatibility of the standard mesh refinement algorithm with higher order Runge Kutta methods is a manifestation of order reduction phenomena which is caused by inconsistent application of boundary data in the refined grids. Indeed, a peculiar feature of high order explicit Runge Kutta schemes is that they behave like low order schemes when applied to hyperbolic problems with time dependent Dirichlet boundary conditions. We present a new algorithm to deal with this phenomenon and through a series of examples demonstrate fourth order convergence. Our scheme also addresses the problem of spurious reflections that are generated when propagating waves cross mesh refinement boundaries. We introduce a transition zone on refined levels within which the phase velocity of propagating modes is allowed to decelerate in order to smoothly match the phase velocity of coarser grids. We apply the method to test problems involving propagating waves and show a significant reduction in spurious reflections.

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