3+1 formulations of the Einstein field equations have become an invaluable tool in Numerical relativity, having been used successfully in modeling spacetimes of black hole collisions, stellar collapse and other complex systems. It is plausible that similar considerations could prove fruitful for modified gravity theories. In this article, we pursue from a numerical relativistic viewpoint the 3 + 1 formulation of metric f (R) gravity as it arises from the fourth order equations of motion, without invoking the dynamical equivalence with Brans-Dicke theories. We present the resulting system of evolution and constraint equations for a generic function f (R), subject to the usual viability conditions. We confirm that the time propagation of the f (R) Hamiltonian and Momentum constraints take the same Mathematical form as in general relativity, irrespective of the f (R) model. We further recast the 3+1 system in a form akin to the BSSNOK formulation of numerical relativity. Without assuming any specific model, we show that the ADM version of f (R) is weakly hyperbolic and is plagued by similar zero speed modes as in the general relativity case. On the other hand the BSSNOK version is strongly hyperbolic and hence a promising formulation for numerical simulations in metric f (R) theories.

We use non-linear gauge-invariant perturbation theory to study the interaction of an inflation produced seed magnetic field with density and gravitational wave perturbations in an almost FriedmannLemaître-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.

It has now become customary in the field of numerical relativity to couple high order finite difference schemes to mesh refinement algorithms. To this end, different modifications to the standard Berger-Oliger adaptive mesh refinement algorithm have been proposed. In this work we present a fourth order stable mesh refinement scheme with sub-cycling in time for numerical relativity. We do not use buffer zones to deal with refinement boundaries but explicitly specify boundary data for refined grids. We argue that the incompatibility of the standard mesh refinement algorithm with higher order Runge Kutta methods is a manifestation of order reduction phenomena, caused by inconsistent application of boundary data in the refined grids. 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.

The beyond-Horndeski gravity has recently been reformulated in the dark energy paradigm—which has been dubbed, Unified Dark Energy (UDE). The evolution equations for the given UDE appear to correspond to a non-conservative dark energy scenario, in which the total energy-momentum tensor is not conserved. We investigate both the background cosmology and, the large-scale imprint of the UDE by probing the angular power spectrum of galaxy number counts, on ultra-large scales; taking care to include the full relativistic corrections in the observed overdensity. The background evolution shows that only an effective mass smaller than the Planck mass is needed in the early universe in order for predictions in the given theory to match current observational constraints. We found that the effective mass-evolution-rate parameter, which drives the evolution of the UDE, acts to enhance the observed power spectrum and, hence, relativistic effects (on ultra-large scales) by enlarging the UDE sound horizon. Conversely, both the (beyond) Horndeski parameter and the kineticity act to diminish the observed power spectrum, by decreasing the UDE sound horizon. Our results show that, in a universe with UDE, a multi-tracer analysis will be needed to detect the relativistic effects in the large-scale structure. In the light of a multi-tracer analysis, the various relativistic effects hold the potential to distinguish different gravity models. Moreover, while the Doppler effect will remain significant at all epochs and, thus can not be ignored, the integrated Sachs-Wolfe, the time-delay and the potential (difference) effects, respectively, will only become significant at epochs near z = 3 and beyond, and may be neglected at late epochs. In the same vein, the Doppler effect alone can serve as an effective cosmological probe for the large-scale structure or gravity models, in the angular power spectrum—at all z.

In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial value problem for f (R) gravity. We consider, in particular, the vacuum field equations of Metric f (R) gravity in the Jordan frame, without utilising the dynamical equivalence with scalar-tensor theories. We present the full hierarchy of non-linear hypersurface and evolution equations necessary for numerical implementation in both tensorial and eth forms. Furthermore, we specialise the resulting equations to situations where the spacetime is almost Minkowski and almost Schwarszchild using standard linearization techniques. We obtain analytic solutions for the dominant ℓ = 2 mode and show that they satisfy the concomitant constraints. These results are ideally suited as testbed solutions for numerical codes. Finally, we point out that the Characteristic formulation can be used as a complementary analytic tool to the 1 + 1 + 2 semi-tetrad formulation. * Electronic address: bishop.mongwane@uct.ac.za Within the numerical relativity community, there are now a number of Characteristic codes being used, with differing levels of sophistication. For instance, some codes employ second order finite difference schemes [38], others use higher order schemes [68] while others have adopted Spectral methods [41]. Another point of distinction among different codes is the coordinate system used to cover the sphere labelling the null directions of the light cones. Common choices range from stereographic coordinate system [15] to multi-patch coordinate systems [38,67]. There has also been efforts to introduce Adaptive Mesh Refinement schemes to Characteristic evolution codes [64,78]. Overall, these codes have made it possible to demonstrate the versatility of Characteristic methods in numerical relativity and have found extensive applications in, for example, binary black hole mergers [4,15,42,65], stellar core collapse [61,66,73], Einstein-Klein-Gordon systems [6,39,62], Observational Cosmology [9, 79, 80] etc. These systems represent potential astrophysical laboratories for testing general relativity in the non-linear regime.Over the years, the theory of general relativity has been subjected to a wide range of experimental tests and has no doubt emerged as one of the most successful theories in Physics. However, there has been considerable interest in the literature to study gravity theories whose Lagrangians contain higher order curvature invariants such as 29,71,74]. The motivation for these alternative theories of gravity stems from a variety of grounds, most notably from within the dark sector in Cosmology [28]. Moreover, the inflationary paradigm arises naturally in alternative theories of gravity without postulating additional inflaton fields [29,74,75]. These higher order corrections also arise in the effective action of quantum gravity. For example, in the low energy limit of string theory o...