An important open question in cosmology is the degree to which the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions of Einstein's equations are able to model the large-scale behavior of the locally inhomogeneous observable universe. We investigate this problem by considering a range of exact n-body solutions of Einstein's constraint equations. These solutions contain discrete masses, and so allow arbitrarily large density contrasts to be modeled. We restrict our study to regularly arranged distributions of masses in topological 3-spheres. This has the benefit of allowing straightforward comparisons to be made with FLRW solutions, as both spacetimes admit a discrete group of symmetries. It also provides a time-symmetric hypersurface at the moment of maximum expansion that allows the constraint equations to be solved exactly. We find that when all the mass in the universe is condensed into a small number of objects ($\ensuremath{\lesssim}10$) then the amount of back-reaction in dust models can be large, with $O(1)$ deviations from the predictions of the corresponding FLRW solutions. When the number of masses is large ($\ensuremath{\gtrsim}100$), however, then our measures of back-reaction become small ($\ensuremath{\lesssim}1%$). This result does not rely on any averaging procedures, which are notoriously hard to define uniquely in general relativity, and so provides (to the best of our knowledge) the first exact and unambiguous demonstration of back-reaction in general relativistic cosmological modelling. Discrete models such as these can therefore be used as laboratories to test ideas about back-reaction that could be applied in more complicated and realistic settings.

An Exact quantification of backreaction in relativistic cosmology

The transfer matrix: A geometrical perspective

We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed.

The transfer matrix: a geometrical perspective

We construct the first exact statistically homogeneous and isotropic cosmological solution in which inhomogeneity has a significant effect on the expansion rate. The universe is modelled as a Swiss Cheese, with dust FRW background and inhomogeneous holes. We show that if the holes are described by the quasispherical Szekeres solution, their average expansion rate is close to the background under certain rather general conditions. We specialise to spherically symmetric holes and violate one of these conditions. As a result, the average expansion rate at late times grows relative to the background, \ie backreaction is significant. The holes fit smoothly into the background, but are larger on the inside than a corresponding background domain: we call them Tardis regions. We study light propagation, find the effective equations of state and consider the relation of the spatially averaged expansion rate to the redshift and the angular diameter distance.

/pdf/average-expansion-rate-and-light-propagation-in-a-33kz1j0dkv.pdf

Average expansion rate and light propagation in a cosmological Tardis spacetime

We construct the first exact statistically homogeneous and isotropic cosmological solution in which inhomogeneity has a significant effect on the expansion rate. The universe is modelled as a Swiss Cheese, with dust FRW background and inhomogeneous holes. We show that if the holes are described by the quasispherical Szekeres solution, their average expansion rate is close to the background under certain rather general conditions. We specialise to spherically symmetric holes and violate one of these conditions. As a result, the average expansion rate at late times grows relative to the background, i.e. backreaction is significant. The holes fit smoothly into the background, but are larger on the inside than a corresponding background domain: we call them Tardis regions. We study light propagation, find the effective equations of state and consider the relation of the spatially averaged expansion rate to the redshift and the angular diameter distance.

/pdf/average-expansion-rate-and-light-propagation-in-a-3ptfpi2r1h.pdf

We consider the optical properties of Lindquist-Wheeler (LW) models of the Universe. These models consist of lattices constructed from regularly arranged discrete masses. They are akin to the Wigner-Seitz construction of solid state physics, and result in a dynamical description of the large-scale Universe in which the global expansion is given by a Friedmann-like equation. We show that if these models are constructed in a particular way then the redshifts of distant objects, as well as the dynamics of the global space-time, can be made to be in good agreement with the homogeneous and isotropic Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions of Einstein's equations, at the level of $\ensuremath{\lesssim}3%$ out to $z\ensuremath{\simeq}2$. Angular diameter and luminosity distances, on the other hand, differ from those found in the corresponding FLRW models, while being consistent with the ``empty beam'' approximation, together with the shearing effects due to the nearest masses. This can be compared with the large deviations found from the corresponding FLRW values obtained in a previous study that considered LW models constructed in a different way. We therefore advocate the improved LW models we consider here as useful constructions that appear to faithfully reproduce both the dynamical and observational properties of space-times containing discrete masses.

Improved treatment of optics in the Lindquist-Wheeler models

The purpose of this paper is to provide an elementary introduction to the qualitative and quantitative results of velocity combination in special relativity, including the Wigner rotation and Thomas precession. We utilize only the most familiar tools of special relativity, in arguments presented at three differing levels: (1) utterly elementary, which will suit a first course in relativity; (2) intermediate, to suit a second course; and (3) advanced, to suit higher level students. We then give a summary of useful results and suggest further reading in this often obscure field.

/pdf/elementary-analysis-of-the-special-relativistic-combination-9qfe6t3rj8.pdf

Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession

The purpose of this paper is to provide an elementary introduction to the qualitative and quantitative results of velocity combination in special relativity, including the Wigner rotation and Thomas precession. We utilize only the most familiar tools of special relativity, in arguments presented at three differing levels: (1) utterly elementary, which will suit a first course in relativity; (2) intermediate, to suit a second course; and (3) advanced, to suit higher level students. We then give a summary of useful results, and suggest further reading in this often obscure field.

Kane O'Donnell

Papers

Improved treatment of optics in the Lindquist-Wheeler models

Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession

Elementary analysis of the special relativistic combination of velocities, Wigner rotation, and Thomas precession

Reply to ‘Comment on ‘Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession”