arXiv subjects

Joshua Ritchie

Publications and source records attributed to Joshua Ritchie.

8 recordsLinked to original sources

arXiv2026

Non-variational scalar field cosmology: Exact Bianchi I solutions for near-minimal scalar fields

The purpose of this work is to investigate spatially homogeneous and flat cosmological solutions of the Einstein equations coupled to a non-variational ``near-minimal'' scalar field. This coupling model represents a minimal departure from standard theory by decoupling the scalar field's self-interaction term from the derivative of its potential. By assuming a quadratic potential and a self-interaction term that is proportional to the potential, we derive four new exact Bianchi I solutions. We demonstrate that these solutions produce a diverse range of cosmological phenomena, including Big Bang, Big Crunch, and Big Rip singularities, as well as oscillatory (``cyclic'') behaviour. For our exact solutions, these singularities occur in infinite proper time and hence are never truly reachable by an observer. To assess the stability of these cosmologies, we perform a numerical stability analysis against spatially inhomogeneous perturbations of the mean curvature. We find that the oscillatory solution is unstable to perturbations of this type, as are solutions in possession of a crushing singularity. Conversely, solutions with a Big Rip singularity (at infinity) are stable to spatially inhomogeneous perturbations of the mean curvature.

gr-qc↗

arXiv2026

A general formalism for coupling scalar fields to the Einstein equations without a variational principle

The purpose of this work is to discuss how matter fields are coupled to gravity within the framework of General Relativity. Our particular focus here is on the coupling of scalar field models. In a first step, we suggest a new method for coupling scalar fields to the Einstein equations \emph{without} the use of a variational principle or Lagrangian. We show that, under the appropriate assumptions, this new method (for coupling scalar fields to gravity) reproduces the minimally and $k$-essence scalar field couplings with a non-zero potential. We therefore interpret this formalism as describing a \emph{generic} method for coupling scalar fields to gravity. The approach described here allows for a number of free fields which we interpret as constitutive freedoms. In a second step, we choose these free fields in such a way that the resulting system is somehow ``near minimal''. In this setting we investigate Bianchi I type solutions. We establish conditions under which the solutions are asymptotically Kasner, near the initial singularity, and investigate their stability properties.

gr-qc↗

arXiv2022

Turing instability and pattern formation on directed networks

Pattern formation, arising from systems of autonomous reaction-diffusion equations, on networks has become a common topic of study in the scientific literature. In this work we focus primarily on directed networks. Although some work prior has been done to understand how patterns arise on directed networks, these works have restricted their attentions to networks for whom the Laplacian matrix (corresponding to the network) is diagonalizable. Here, we address the question "how does one detect pattern formation if the Laplacian matrix is not diagonalizable?" To this end, we find it is useful to also address the related problem of pattern formation arising from systems of reaction-diffusion equations with non-local (global) reaction kinetics. These results are then generalized to include non-autonomous systems as well as temporal networks, i.e., networks whose topology is allowed to change in time.

nlin.PS↗

arXiv2022

Bianchi I "asymptotically Kasner" solutions of the Einstein scalar field equations

In this work we investigate the asymptotic behaviour of solutions to the Einstein equations with a minimally coupled scalar field. The primary focus of the present paper here establishing under what conditions a solution becomes "asymptotically Kasner" sufficiently close to the initial singularity. To address this question we restrict our attention to Bianchi I space-times. By restricting our attention to a strictly monotonic scalar field we are able to provide necessary conditions on a potential so that the resulting solution is asymptotically Kasner. Moreover, we provide both explicit and numerical examples of asymptotically Kasner space-times.

gr-qc↗

arXiv2022

Turing and wave instabilities in hyperbolic reaction-diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Hyperbolic reaction-diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction-diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction-diffusion systems which are distinct from the well studied Turing case.

nlin.PS↗

arXiv2021

Asymptotically hyperboloidal initial data sets from a parabolic-hyperbolic formulation of the Einstein vacuum constraints

In this paper, we continue our investigations of R\'acz's parabolic-hyperbolic formulation of the Einstein vacuum constraints. Our previous studies of the asymptotically flat setting provided strong evidence for unstable asymptotics which we were able to resolve by introducing a certain modification of R\'acz's parabolic-hyperbolic formulation. The primary focus of the present paper here is the asymptotically hyperboloidal setting. We provide evidence through a mixture of numerical and analytical methods that the asymptotics of the solutions of R\'acz's parabolic-hyperbolic formulation is stable, and, in particular, no modifications are necessary to obtain solutions that are asymptotically hyperboloidal.

gr-qc↗

arXiv2020

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

In this paper we continue earlier investigations of evolutionary formulations of the Einstein vacuum constraint equations originally introduced by Rácz. Motivated by the strong evidence from these works that the resulting vacuum initial data sets are generically not asymptotically flat we analyse the asymptotics of the solutions of a modified formulation by a combination of analytical and numerical techniques. We conclude that the vacuum initial data sets generated with this new formulation are generically asymptotically flat.

gr-qc↗

arXiv2019

Numerical construction of initial data sets of binary black hole type using a parabolic-hyperbolic formulation of the vacuum constraint equations

In this paper we investigate the parabolic-hyperbolic formulation of the vacuum constraint equations introduced by R{á}cz with a view to constructing multiple black hole initial data sets without spin. In order to respect the natural properties of this configuration, we foliate the spatial domain with 2-spheres. It is then a consequence of these equations that they must be solved as an initial value problem evolving outwards towards spacelike infinity. Choosing the free data and the "strong field boundary conditions" for these equations in a way which mimics asymptotically flat and asymptotically spherical binary black hole initial data sets, our focus in this paper is on the analysis of the asymptotics of the solutions. In agreement with our earlier results, our combination of analytical and numerical tools reveals that these solutions are in general not asymptotically flat, but have a cone geometry instead. In order to remedy this and approximate asymptotically Euclidean data sets, we then propose and test an iterative numerical scheme.

gr-qc↗