arXiv subjects

Holger Waalkens

Publications and source records attributed to Holger Waalkens.

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 $ρ$, (relativistic) pressure $p$, momentum density $q$, and anisotropic stress $π$, 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

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↗

arXiv2025

Maslov $S^{1}$ Bundles and Maslov Data

We define Maslov $S^1$ bundles over a symplectic manifold $(M,ω)$. These are the determinant bundle $Γ_J$ of the unitary frame bundle defined by an almost complex structure compatible with $ω$, and the bundle $Γ_J^2 = Γ_J \big/ \{\pm1\}$. We analyze the properties of the Maslov $S^1$ bundles $Γ_J$ and $Γ_J^2$, focusing on the interplay between their geometry and the dynamics of a symplectic action of a compact Lie group $G$ on $M$ which induces lifted $G$ actions on $Γ_J$ and on $Γ_J^2$. We show that when $M$ is a homogeneous $G$-space and the first real Chern class $c_Γ$ is nonvanishing, $Γ_J$ and $Γ_J^2$ are also homogeneous $G$-spaces. Moreover, we give an alternative proof of the fact that when $[ω]=r\,c_Γ$ for some real number $r$, then the symplectic $G$ action on $(M,ω)$ is Hamiltonian. When the Maslov $S^1$ bundle $Γ_J^2$ is trivial, then an index generalizing the Maslov index can be defined. This is no longer true if $Γ_J^2$ is not trivial. However, if $G=S^1$ acts symplectically on $(M,ω)$ we define a quantity that we call Maslov data which serves as a non-integrable version of the notion of Maslov index in the case where $Γ_J^2$ is not trivial, and we associate the Maslov data at fixed points of the $G=S^1$ action to their resonance type. Finally, we consider three applications motivated by the study of integrable Hamiltonian systems. First, we discuss conditions under which an $S^1$ symmetry of a two degrees of freedom integrable Hamiltonian system can be extended to a $\mathbb T^2$ symmetry. Second, we show that the Maslov $S^1$ bundles over Lagrangian pinched tori are trivial. Third, we consider $S^2 \times S^2$ as a symplectic manifold with an $S^1$ action corresponding to simultaneous rotations of the two spheres, and we compute the corresponding Maslov data.

math.SG↗

arXiv2024

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↗

arXiv2024

Bertranďs 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↗

arXiv2022

A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

quant-ph↗

arXiv2021

Hill regions of charged three-body systems

For charged three-body systems, we discuss the configurations and orientations that are admissible for given values of the conserved total energy and angular momentum. The admissible configurations and orientations are discussed on a configuration space that is reduced by the translational, rotational and dilation symmetries of charged three-body systems. We consider the examples of the charged three-body systems given by the helium atom (two electrons and a nucleus) and the compound of two electrons and one positron. For comparison, the well known example of the Newtonian gravitational three-body system is discussed for the scheme presented in this paper first.

math.DS↗

arXiv2021

Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that the non-zero vector of Maslov indices is an eigenvector with eigenvalue 1 of the monodromy matrices of an integrable Hamiltonian system. We take a close look at the geometry behind this result and extend it to a more general context. We construct a bundle morphism defined on the lattice bundle of an (general) integrable system, which can be seen as a generalization of the vector of Maslov indices. The non-triviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1 of the monodromy matrices, and gives rise to a corank 1 toric foliation refining the original one induced by the integrable system. Furthermore, we show that in the case where the system has 2 degrees of freedom, this implies the global existence of a free S^{1} action.

math.DS↗

arXiv2021

Frames of group-sets and their application in bundle theory

We study fiber bundles where the fibers are not a group $G$, but a free $G$-space with disjoint orbits. These bundles closely resemble principal bundles, hence we call them semi-principal bundles. The study of such bundles is facilitated by defining the notion of a basis of a $G$-set, in analogy with a basis of a vector space. The symmetry group of these bases is a wreath product. Similar to vector bundles, using the notion of a basis induces a frame bundle construction, which in this case results in a principal bundle with the wreath product as structure group. This construction can be formalized in the language of a functor, which retracts the semi-principal bundles to the principal bundles. In addition, semi-principal bundles support parallel transport just like principal bundles, and this carries over to the frame bundle.

math.DG↗

arXiv2019

Phase space structures causing the reaction rate decrease in the collinear hydrogen exchange reaction

The collinear hydrogen exchange reaction is a paradigm system for understanding chemical reactions. It is the simplest imaginable atomic system with $2$ degrees of freedom modeling a chemical reaction, yet it exhibits behaviour that is still not well understood - the reaction rate decreases as a function of energy beyond a critical value. Using lobe dynamics we show how invariant manifolds of unstable periodic orbits guide trajectories in phase space. From the structure of the invariant manifolds we deduce that insufficient transfer of energy between the degrees of freedom causes a reaction rate decrease. In physical terms this corresponds to the free hydrogen atom repelling the whole molecule instead of only one atom from the molecule. We further derive upper and lower bounds of the reaction rate, which are desirable for practical reasons.

nlin.CD↗

arXiv2019

Comment on 'Winding around non-Hermitian singularities' by Zhong et al., Nat. Commun. 9, 4808 (2018)

In a recent paper entitled "Winding around non-Hermitian singularities" by Zhong et al., published in Nat. Commun. 9, 4808 (2018), a formalism is proposed for calculating the permutations of eigenstates that arise upon encircling (multiple) exceptional points (EPs) in the complex parameter plane of an analytic non-Hermitian Hamiltonian. The authors suggest that upon encircling EPs one should track the eigenvalue branch cuts that are traversed, and multiply the associated permutation matrices accordingly. In this comment we point out a serious shortcoming of this approach, illustrated by an explicit example that yields the wrong result for a specific loop. A more general method that has been published earlier by us and that does not suffer from this problem, is based on using fundamental loops. We briefly explain the method and list its various advantages. In addition, we argue that this method can be verified in a three wave-guide system, which then also unambiguously establishes the noncommutativity associated with encircling multiple EPs.

quant-ph↗

arXiv2018

Non-abelian nature of systems with multiple exceptional points

The defining characteristic of an exceptional point (EP) in the parameter space of a family of operators is that upon encircling the EP eigenstates are permuted. In case one encircles multiple EPs, the question arises how to properly compose the effects of the individual EPs. This was thought to be ambiguous. We show that one can solve this problem by considering based loops and their deformations. The theory of fundamental groups allows to generalize this technique to arbitrary degeneracy structures like exceptional lines in a three-dimensional parameter space. As permutations of three or more objects form a non-abelian group, the next question that arises is whether one can experimentally demonstrate this non-commutative behavior. This requires at least two EPs of a family of operators that have at least 3 eigenstates. A concrete implementation in a recently proposed $\mathcal{PT}$ symmetric waveguide system is suggested as an example of how to experimentally check the composition law and show the non-abelian nature of non-hermitian systems with multiple EPs.

math-ph↗

arXiv2018

Quantum Monodromy in the Isotropic 3-Dimensional Harmonic Oscillator

The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators, one of which is the Hamiltonian. We show that the joint spectrum of the Hamilton operator, the $z$ component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has quantum monodromy for sufficiently large energies. This means that one cannot globally assign quantum numbers to the joint spectrum. The effect can be classically explained by showing that the corresponding Liouville integrable system has a non-degenerate focus-focus point, and hence Hamiltonian monodromy.

math-ph↗

arXiv2018

Dynamical analysis of a chaos generator

Investigating the possibility of applying techniques from linear systems theory to the setting of nonlinear systems has been the focus of many papers. The pseudo linear form representation of nonlinear dynamical systems has led to the concept of nonlinear eigenvalues and nonlinear eigenvectors. When the nonlinear eigenvectors do not depend on the state vector of the system, then the nonlinear eigenvalues determine the global qualitative behaviour of a nonlinear system throughout the state space. The aim of this paper is to use this fact to construct a nonlinear dynamical system of which the trajectories of the system show continual stretching and folding. We first prove that the system is globally bounded. Next, we analyse the system numerically by studying bifurcations of equilibria and periodic orbits. Chaos arises due to a period doubling cascade of periodic attractors. Chaotic attractors are presumably of Hénon-like type, which means that they are the closure of the unstable manifold of a saddle periodic orbit. We also show how pseudo linear forms can be used to control the chaotic system and to synchronize two identical chaotic systems.

math.OC↗

arXiv2018

The phase space geometry underlying roaming reaction dynamics

Recent studies have found an unusual way of dissociation in formaldehyde. It can be characterized by a hydrogen atom that separates from the molecule, but instead of dissociating immediately it roams around the molecule for a considerable amount of time and extracts another hydrogen atom from the molecule prior to dissociation. This phenomenon has been coined roaming and has since been reported in the dissociation of a number of other molecules. In this paper we investigate roaming in Chesnavich's CH$_4^+$ model. During dissociation the free hydrogen must pass through three phase space bottleneck for the classical motion, that can be shown to exist due to unstable periodic orbits. None of these orbits is associated with saddle points of the potential energy surface and hence related to transition states in the usual sense. We explain how the intricate phase space geometry influences the shape and intersections of invariant manifolds that form separatrices, and establish the impact of these phase space structures on residence times and rotation numbers. Ultimately we use this knowledge to attribute the roaming phenomenon to particular heteroclinic intersections.

math.DS↗

arXiv2018

Scattering invariants in Euler's two-center problem

The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense.

math-ph↗

arXiv2016

Defect in the Joint Spectrum of Hydrogen due to Monodromy

In addition to the well known case of spherical coordinates the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, and the $z$-components of the angular momentum and quantum Laplace-Runge-Lenz vectors obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. This means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multi valued.

math-ph↗