arXiv subjects

Marcello Seri

Publications and source records attributed to Marcello Seri.

At least 19 recordsLinked to original sources

arXiv2026

Exploring the CMB in Anisotropic Universes

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 article, 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.

gr-qc↗

arXiv2026

Fundamental Cosmic Anisotropy and its Ramifications II: Perturbations in Bianchi spacetimes, and fixed in the Newtonian gauge

The standard cosmological model is challenged by an ever-growing collection of observations, which invites (and stimulates) inquiry into possible additions and/or alterations. One such alteration comes from letting cosmic isotropy -- as demanded by the cosmological principle -- go, whilst maintaining only homogeneity. This study concerns Bianchi models, a class of anisotropic, homogeneous spacetimes, and in particular their perturbations. Knowledge of their properties under perturbations (such as allowed wavemodes) aids in understanding cosmological signatures of such universes, e.g. CMBs, and thus allows for comparsion to observation and the theory of the standard model. This study develops linear perturbation theory of general Bianchi models, by working in a frame such that metric components depend solely on (cosmic) time. Perturbation equations in the Newtonian gauge, but for arbitrary metric, are derived for energy density $\rho$, (relativistic) pressure $p$, momentum density $q$, and anisotropic stress $\pi$, for the case of scalar and pure tensor perturbations. For the former, the equations for density and pressure are combined to yield the equivalent of the Mukhanov-Sasaki equation for Bianchi models. For a specific choice of metric and fluid flow $u$, the Friedmann equations for Bianchi models are also formulated, as this knowledge is necessary to fully formulate the perturbation equations. Finally, the obtained results are applied to the formulations of density contrasts in an Einstein-de Sitter universe and a Bianchi I universe.

gr-qc↗

arXiv2025

A geometric formulation of GENERIC stochastic differential equations

We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems.

math.DS↗

arXiv2024

Characterization of symmetries of contact Hamiltonian systems

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.

math-ph↗

arXiv2024

Fundamental Cosmic Anisotropy and its Ramifications I: Killing vector fields and constructing their metric

On the largest scales, the universe appears to be almost homogeneous and isotropic, adhering to the cosmological principle. In contrast, on smaller scales inhomogeneities and anisotropy become increasingly prominent, reflecting the origin, emergence, and formation of structure in the universe. Moreover, a range of tensions between various cosmological observations may suggest it necessary to explore the consequences of departure from the ideal, uniform universe on the fundamental level. Thus, in this work, the foundation of spatially homogeneous yet anisotropic universes is studied. Specifically, when given a 3D Lie algebra of \emph{desired} Killing vector fields (as would be the case for a homogeneous yet anisotropic universe), we provide an explicit construction for the metric that has exactly those as its Killing vector fields. This construction is presented accessibly, in a directly-usable, algorithmic fashion. Some examples demonstrating the construction are worked out, including a constructive method to separate out (cosmic) time dependence in spatially homogeneous, anisotropic cosmologies.

gr-qc↗

arXiv2024

Bertran\v{d}s Theorem and the Double Copy of Relativistic Field Theories

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.

hep-th↗

arXiv2023

Extremal Black Holes as Relativistic Systems with Kepler Dynamics

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 reparametrization to the Kepler problem, and that all Hamiltonians conserving a Laplace-Runge-Lenz-like vector are related in this way to Kepler.

gr-qc↗

arXiv2022

Chaotic light scattering around extremal black holes

We show that the scattering of light in the field of $N\geq 3$ static extremal black holes is chaotic in the planar case. The relativistic dynamics of such extremal objects reduce to that of a classical Hamiltonian system. Certain values of the dilaton coupling then allow one to apply techniques from symbolic dynamics and classical potential scattering. This results in a lower bound on the topological entropy of order $\log(N-1)$, thus proving the emergence of chaotic scattering for $N \geq 3$ black holes.

gr-qc↗

arXiv2022

The flow method for the Baker-Campbell-Hausdorff formula: exact results

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.

math-ph↗

arXiv2022

A Magnetic Pseudodifferential Calculus for Operator-Valued and Equivariant Operator-Valued Symbols

In this monograph we develop magnetic pseudodifferential theory for operator-valued and equivariant operator-valued functions and distributions from first principles. These have found plentiful applications in mathematical physics, including in rigorous perturbation theory for slow-fast systems and perturbed periodic operators. Yet, a systematic treatise was hitherto missing. While many of the results can be found piecemeal in appendices and as sketches in other articles, this article does contain new results. For instance, we have established Beals-type commutator criteria for both cases, which then imply the existence of Moyal resolvents for (equivariant) selfadjoint-operator-valued, elliptic Hörmander symbols and allows one to construct functional calculi. What is more, we give criteria on the function under which a magnetic pseudodifferential operator is (locally) trace class. Our aims for this article are three-fold: (1) Create a single, solid work that colleagues can refer to. (2) Be pedagogical and precise. And (3) give a straightforward strategy for extending results from the operator-valued to the equivariant case, pointing out some caveats and pitfalls that need to be kept in mind.

math-ph↗

arXiv2021

New directions for contact integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.

math.NA↗

arXiv2021

Geometric numerical integration of Liénard systems via a contact Hamiltonian approach

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.

math.NA↗

arXiv2019

Numerical integration in celestial mechanics: a case for contact geometry

Several dynamical systems of interest in celestial mechanics can be written in the form of a Newton equation with time-dependent damping, linear in the velocities. For instance, the modified Kepler problem, the spin-orbit model and the Lane-Emden equation all belong to such class. In this work we start an investigation of these models from the point of view of contact geometry. In particular we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators.

math.NA↗

arXiv2019

Contact variational integrators

We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz' variational principle. First we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle. Then we discuss the backward error analysis of our variational integrators, including the construction of a modified Lagrangian. Throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues.

math.NA↗

arXiv2017

Quantum confinement on non-complete Riemannian manifolds

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold $M$ equipped with a smooth measure $ω$, possibly degenerate or singular near the metric boundary of $M$, and in presence of a real-valued potential $V\in L^2_{\mathrm{loc}}(M)$. The main merit of this paper is the identification of an intrinsic quantity, the effective potential $V_{\mathrm{eff}}$, which allows to formulate simple criteria for quantum confinement. Let $δ$ be the distance from the possibly non-compact metric boundary of $M$. A simplified version of the main result guarantees quantum completeness if $V\ge -cδ^2$ far from the metric boundary and \[ V_{\mathrm{eff}}+V\ge \frac3{4δ^2}-\fracκδ, \qquad \text{close to the metric boundary}. \] These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of $M$; (ii) generalize the Kalf-Walter-Schmincke-Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace-Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace-Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [Boscain, Laurent - Ann. Inst. Fourier, 2013] .

math.DG↗

arXiv2017

A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators

In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g. CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a "reduction procedure" that allows to consider only a simple subset of sub-Riemannian geodesics. As an application, we derive isoperimetric-type and (p-)Hardy-type inequalities for a compact domain $M$ with piecewise $C^{1,1}$ boundary, and a universal lower bound for the first Dirichlet eigenvalue $λ_1(M)$ of the sub-Laplacian, \[ λ_1(M) \geq \frac{k π^2}{L^2}, \] in terms of the rank $k$ of the distribution and the length $L$ of the longest reduced sub-Riemannian geodesic contained in $M$. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations: \[ \mathbb{S}^1\hookrightarrow \mathbb{S}^{2d+1} \xrightarrow{p} \mathbb{CP}^d, \qquad \mathbb{S}^3\hookrightarrow \mathbb{S}^{4d+3} \xrightarrow{p} \mathbb{HP}^d, \qquad d \geq 1, \] where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, $L = π$ and $k=2d$ or $4d$, respectively.

math.DG↗

arXiv2016

Symbolic Dynamics of Magnetic Bumps

For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.

math.DS↗

arXiv2016

Resonances in the Two-Centers Coulomb Systems

We investigate the existence of resonances for two-centers Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalised complex eigenvalues of a non-selfadjoint deformation of the two-centers Schrödinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analysed by means of perturbation theory and numerical methods.

math-ph↗