This paper examines the behaviour of closed `lattice universes' wherein masses are distributed in a regular lattice on the Cauchy surfaces of closed vacuum universes. Such universes are approximated using a form of Regge calculus originally developed by Collins and Williams to model closed FLRW universes. We consider two types of lattice universes, one where all masses are identical to each other and another where one mass gets perturbed in magnitude. In the unperturbed universe, we consider the possible arrangements of the masses in the Regge Cauchy surfaces and demonstrate that the model will only be stable if each mass lies within some spherical region of convergence. We also briefly discuss the existence of Regge models that are dual to the ones we have considered. We then model a perturbed lattice universe and demonstrate that the model's evolution is well-behaved, with the expansion increasing in magnitude as the perturbation is increased.Comment: 34 pages, 9 figures. Expanded Introduction: elaborated on cosmological context for work and referenced recent work on studying lattice universes using a wide variety of approaches. Corrected an error in the LW graphs in Fig

This paper examines the properties of 'lattice universes' wherein point masses are arranged in a regular lattice on space-like hypersurfaces; open, flat, and closed universes are considered. The universes are modelled using the Lindquist-Wheeler (LW) approximation scheme, which approximates the space-time in each lattice cell by Schwarzschild geometry. Extending Lindquist and Wheeler's work, we derive cosmological scale factors describing the evolution of all three types of universes, and we use these scale factors to show that the universes' dynamics strongly resemble those of Friedmann-Lemaître-Robertson-Walker (FLRW) universes. In particular, we use the scale factors to make more salient the resemblance between Clifton and Ferreira's Friedmann-like equations for the LW models and the actual Friedmann equations of FLRW space-times. Cosmological redshifts for such universes are then determined numerically, using a modification of Clifton and Ferreira's approach; the redshifts are found to closely resemble their FLRW counterparts, though with certain differences attributable to the 'lumpiness' in the underlying matter content. Most notably, the LW redshifts can differ from their FLRW counterparts by as much as 30%, even though they increase linearly with FLRW redshifts, and they exhibit a non-zero integrated Sachs-Wolfe effect, something which would not be possible in matter-dominated FLRW universes without a cosmological constant. PACS numbers: 98.80.Jk, 98.80.-k, 04.25.D-

If the brain processes incoming data efficiently, information should degrade little between early and later neural processing stages, and so information in early stages should match behavioral performance. For instance, if there is enough information in a visual cortical area to determine the orientation of a grating to within 1 degree, and the code is simple enough to be read out by downstream circuits, then animals should be able to achieve that performance behaviourally. Despite over 30 years of research, it is still not known how efficient the brain is. For tasks involving a large number of neurons, the amount of information encoded by neural circuits is limited by differential correlations. Therefore, determining how much information is encoded requires quantifying the strength of differential correlations. Detecting them, however, is difficult. We report here a new method, which requires on the order of 100s of neurons and trials. This method relies on computing the alignment of the neural stimulus encoding direction, f′, with the eigenvectors of the noise covariance matrix, Σ. In the presence of strong differential correlations, f′ must be spanned by a small number of the eigenvectors with largest eigenvalues. Using simulations with a leaky-integrate-and-fire neuron model of the LGN-V1 circuit, we confirmed that this method can indeed detect differential correlations consistent with those that would limit orientation discrimination thresholds to 0.5-3 degrees. We applied this technique to V1 recordings in awake monkeys and found signatures of differential correlations, consistent with a discrimination threshold of 0.47-1.20 degrees, which is not far from typical discrimination thresholds (1-2 deg). These results suggest that, at least in macaque monkeys, V1 contains about as much information as is seen in behaviour, implying that downstream circuits are efficient at extracting the information available in V1.

The Collins-Williams Regge calculus models of FLRW space-times and Brewin's subdivided models are applied to closed vacuum Λ-FLRW universes. In each case, we embed the Regge Cauchy surfaces into 3-spheres in E 4 and consider possible measures of Cauchy surface radius that can be derived from the embedding. Regge equations are obtained from both global variation, where entire sets of identical edges get varied simultaneously, and local variation, where each edge gets varied individually. We explore the relationship between the two sets of solutions, the conditions under which the Regge Hamiltonian constraint would be a first integral of the evolution equation, the initial value equation for each model at its moment of time symmetry, and the performance of the various models. It is revealed that local variation does not generally lead to a viable Regge model. It is also demonstrated that the various models do satisfy their respective initial value equations. Finally, it is shown that the models reproduce the correct qualitative dynamics of the space-time. Furthermore, the approximation's accuracy is highest when the universe is small but improves overall as we increase the number of tetrahedra used to construct the Regge Cauchy surface. Eventually though, all models gradually fail to keep up with the continuum FLRW model's expansion, with the models with lower numbers of tetrahedra falling away more quickly. We believe this failure to keep up is due to the finite resolution of the Regge Cauchy surfaces trying to approximate an ever expanding continuum Cauchy surface; each Regge surface has a fixed number of tetrahedra and as the surface being approximated gets larger, the resolution would degrade. Finally, we note that all Regge models end abruptly at a point when the time-like struts of the skeleton become null, though this end-point appears to get delayed as the number of tetrahedra is increased.

The late universe's matter distribution obeys the Copernican principle at only the coarsest of scales. The relative importance of such inhomogeneity is still not well understood. Because of the Einstein field equations' non-linear nature, some argue a non-perturbative approach is necessary to correctly model inhomogeneities and may even obviate any need for dark energy. We shall discuss an approach based on Regge calculus, a discrete approximation to general relativity: we shall discuss the Collins-Williams formulation of Regge calculus and its application to two toy universes. The first is a universe for which the continuum solution is well-established, the Λ-FLRW universe. The second is an inhomogeneous universe, the 'lattice universe' wherein matter consists solely of a lattice of point masses with pure vacuum in between, a distribution more similar to that of the actual universe compared to FLRW universes. We shall discuss both regular lattices and one where one mass gets perturbed.