Marcello Seri

In recent years, there have been increasing challenges to the cosmological principle, based on new observations of e.g. supernovae and the cosmic bulk flow. As a result, the cosmological community is speaking their concern for the cosmological principle, and from which scales onwards it should apply. In this context, there is a desire to understand more fully the properties and signatures of cosmologies not obeying the cosmological principle. In this paper, we let go of the demand of cosmic isotropy, and instead assume only spatial homogeneity in our cosmological models. We follow the results of our previous works [see citations in the list of references], and here bring these together into one unified picture, with the goal of describing the signature(s) of anisotropy in anisotropic cosmological models. We first introduce the Bianchi models — a particular instance of spatially homogeneous cosmologies — and show that a metric can be constructed for them when an appropriate collection of desired Killing vector fields is supplied. Then, we give the perturbations of the Friedmann equations in such Bianchi models, in the Newtonian gauge, derived using much the same methodology as applicable to the FLRW models. We show these can be combined into one characteristic partial differential equation. Finally, we use this equation in order to simulate the CMB of a toy Bianchi V example and produce its power spectrum. We close with a discussion and suggestions for further research.

This paper explores the relationship between Cartan symmetries, dynamical similarities, and dynamical symmetries in contact Hamiltonian mechanics. By introducing an alternative decomposition of vector fields, we characterize these symmetries and present a novel description in terms of tensor densities. Furthermore, we demonstrate that this framework allows, under specific conditions, for the recovery of integrals of motion. We also establish new criteria to assess their independence.

This special issue of the Journal of Mathematical Physics contains invited contributions by a selection of speakers at the conference Learning from Insulators: This special issue of the Journal of Mathematical Physics contains invited contributions by a selection of speakers at the conference Learning from Insulators:

A bstract: Which relativistic field theories give rise to Kepler dynamics in the two-body problem? We consider a class of Hamiltonians that is the unique relativistic extension of the Kepler problem preserving its so(4) algebra, and have orbits related through time reparametrisation to orbits of the original Kepler problem. For three explicit examples, we give a natural interpretation in terms of spin-0,-1 and -2 interacting field theories in 5D. These are organically connected via the classical double copy, which therefore preserves maximal superintegrability.

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler.

Leveraging techniques from the literature on geometric numerical integration, we propose a new general method to compute exact expressions for the Baker-Campbell-Hausdorff (BCH) formula. In its utmost generality, the method consists in embedding the Lie algebra of interest into a subalgebra of the algebra of vector fields on some manifold by means of an isomorphism, so that the BCH formula for two elements of the original algebra can be recovered from the composition of the flows of the corresponding vector fields. For this reason we call our method the flow method. Clearly, this method has great advantage in cases where the flows can be computed analytically. We illustrate its usefulness on some benchmark examples where it can be applied directly, and discuss some possible extensions for cases where an exact expression cannot be obtained.

Leveraging techniques from the literature on geometric numerical integration, we propose a new general method to compute exact expressions for the BCH formula. In its utmost generality, the method consists in embedding the Lie algebra of interest into a subalgebra of the algebra of vector fields on some manifold by means of an isomorphism, so that the BCH formula for two elements of the original algebra can be recovered from the composition of the flows of the corresponding vector fields. For this reason we call our method the flow method. Clearly, this method has great advantage in cases where the flows can be computed analytically. We illustrate its usefulness on some benchmark examples where it can be applied directly, and discuss some possible extensions for cases where an exact expression cannot be obtained.

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime.