arXiv subjects

Henk van Elst

Publications and source records attributed to Henk van Elst.

At least 19 recordsLinked to original sources

arXiv2022

The averaging problem on the past null cone in inhomogeneous dust cosmologies

Cosmological models typically neglect the complicated nature of the spacetime manifold at small scales in order to hypothesize idealized general relativistic solutions for describing the average dynamics of the Universe. Although these solutions are remarkably successful in accounting for data, they introduce a number of puzzles in cosmology, and their foundational assumptions are therefore important to test. In this paper, we go beyond the usual assumptions in cosmology and propose a formalism for averaging the local general relativistic spacetime on an observer's past null cone: we formulate average properties of light fronts as they propagate from a cosmological emitter to an observer. The energy-momentum tensor is composed of an irrotational dust source and a cosmological constant -- the same components as in the $Λ$CDM model for late cosmic times -- but the metric solution is not \emph{a priori}constrained to be locally homogeneous or isotropic. This generally makes the large-scale dynamics depart from that of a simple Friedmann--Lema\^ıtre--Robertson--Walker solution through \emph{backreaction} effects. Our formalism quantifies such departures through a fully covariant system of area-averaged equations on the light fronts propagating towards an observer, which can be directly applied to analytical and numerical investigations of cosmic observables. For this purpose, we formulate light front averages of observable quantities, including the effective angular diameter distance and the cosmological redshift drift and we also discuss the backreaction effects for these observables.

gr-qc↗

arXiv2022

An Introduction to Inductive Statistical Inference: from Parameter Estimation to Decision-Making

These lecture notes aim at a post-Bachelor audience with a background at an introductory level in Applied Mathematics and Applied Statistics. They discuss the logic and methodology of the Bayes-Laplace approach to inductive statistical inference that places common sense and the guiding lines of the scientific method at the heart of systematic analyses of quantitative-empirical data. Following an exposition of exactly solvable cases of single- and two-parameter estimation problems, the main focus is laid on Markov Chain Monte Carlo (MCMC) simulations on the basis of Hamiltonian Monte Carlo sampling of posterior joint probability distributions for regression parameters occurring in generalised linear models for a univariate outcome variable. The modelling of fixed effects as well as of correlated varying effects via multi-level models in non-centred parametrisation is considered. The simulation of posterior predictive distributions is outlined. The assessment of a model's relative out-of-sample posterior predictive accuracy with information entropy-based criteria WAIC and LOOIC and model comparison with Bayes factors are addressed. A brief discussion on the description of the generation of stationary time series data by means of autoregressive models is contained. Concluding, a conceptual link to the behavioural subjective expected utility representation of a single decision-maker's choice behaviour in static one-shot decision problems is established. Vectorised codes for MCMC simulations of multi-dimensional posterior joint probability distributions with the Stan probabilistic programming language implemented in the statistical software R are provided. The lecture notes are fully hyperlinked. They direct the reader to original scientific research papers, online resources on inductive statistical inference, and to pertinent biographical information.

stat.AP↗

arXiv2021

Tutorial on principal component analysis, with applications in R

This tutorial reviews the main steps of the principal component analysis of a multivariate data set and its subsequent dimensional reduction on the grounds of identified dominant principal components. The underlying computations are demonstrated and performed by means of a script written in the statistical software package R.

stat.OT↗

arXiv2019

Foundations of Descriptive and Inferential Statistics

These lecture notes were written with the aim to provide an accessible though technically solid introduction to the logic of systematical analyses of statistical data to both undergraduate and postgraduate students, in particular in the Social Sciences, Economics, and the Financial Services. They may also serve as a general reference for the application of quantitative--empirical research methods. In an attempt to encourage the adoption of an interdisciplinary perspective on quantitative problems arising in practice, the notes cover the four broad topics (i) descriptive statistical processing of raw data, (ii) elementary probability theory, (iii) the operationalisation of one-dimensional latent statistical variables according to Likert's widely used scaling approach, and (iv) null hypothesis significance testing within the frequentist approach to probability theory concerning (a) distributional differences of variables between subgroups of a target population, and (b) statistical associations between two variables. The relevance of effect sizes for making inferences is emphasised. These lecture notes are fully hyperlinked, thus providing a direct route to original scientific papers as well as to interesting biographical information. They also list many commands for running statistical functions and data analysis routines in the software packages R, SPSS, EXCEL and OpenOffice. The immediate involvement in actual data analysis practices is strongly recommended.

stat.AP↗

arXiv2015

An Introduction to Business Mathematics

These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief's stationary input-output matrix model, linear programming, elementary financial mathematics, as well as differential and integral calculus of real-valued functions of one real variable. A special focus is set on applications in quantitative economical modelling.

q-fin.GN↗

arXiv2014

Decision-theoretic approaches to non-knowledge in economics

We review two strands of conceptual approaches to the formal representation of a decision maker's non-knowledge at the initial stage of a static one-person, one-shot decision problem in economic theory. One focuses on representations of non-knowledge in terms of probability measures over sets of mutually exclusive and exhaustive consequence-relevant states of Nature, the other deals with unawareness of potentially important events by means of sets of states that are less complete than the full set of consequence-relevant states of Nature. We supplement our review with a brief discussion of unresolved matters in both approaches.

q-fin.GN↗

arXiv2012

How is non-knowledge represented in economic theory?

In this article, we address the question of how non-knowledge about future events that influence economic agents' decisions in choice settings has been formally represented in economic theory up to date. To position our discussion within the ongoing debate on uncertainty, we provide a brief review of historical developments in economic theory and decision theory on the description of economic agents' choice behaviour under conditions of uncertainty, understood as either (i) ambiguity, or (ii) unawareness. Accordingly, we identify and discuss two approaches to the formalisation of non-knowledge: one based on decision-making in the context of a state space representing the exogenous world, as in Savage's axiomatisation and some successor concepts (ambiguity as situations with unknown probabilities), and one based on decision-making over a set of menus of potential future opportunities, providing the possibility of derivation of agents' subjective state spaces (unawareness as situation with imperfect subjective knowledge of all future events possible). We also discuss impeding challenges of the formalisation of non-knowledge.

q-fin.GN↗

arXiv2011

On the Trace-Free Einstein Equations as a Viable Alternative to General Relativity

The quantum field theoretic prediction for the vacuum energy density leads to a value for the effective cosmological constant that is incorrect by between 60 to 120 orders of magnitude. We review an old proposal of replacing Einstein's Field Equations by their trace-free part (the Trace-Free Einstein Equations), together with an independent assumption of energy--momentum conservation by matter fields. While this does not solve the fundamental issue of why the cosmological constant has the value that is observed cosmologically, it is indeed a viable theory that resolves the problem of the discrepancy between the vacuum energy density and the observed value of the cosmological constant. However, one has to check that, as well as preserving the standard cosmological equations, this does not destroy other predictions, such as the junction conditions that underlie the use of standard stellar models. We confirm that no problems arise here: hence, the Trace-Free Einstein Equations are indeed viable for cosmological and astrophysical applications.

gr-qc↗

arXiv2009

Geometrical order-of-magnitude estimates for spatial curvature in realistic models of the Universe

The thoughts expressed in this article are based on remarks made by Jürgen Ehlers at the Albert-Einstein-Institut, Golm, Germany in July 2007. The main objective of this article is to demonstrate, in terms of plausible order-of-magnitude estimates for geometrical scalars, the relevance of spatial curvature in realistic models of the Universe that describe the dynamics of structure formation since the epoch of matter-radiation decoupling. We introduce these estimates with a commentary on the use of a quasi-Newtonian metric form in this context.

gr-qc↗

arXiv2008

Cosmological models (Cargèse lectures 1998)

The aim of this set of lectures is a systematic presentation of a 1+3 covariant approach to studying the geometry, dynamics, and observational properties of relativistic cosmological models. In giving (i) the basic 1+3 covariant relations for a cosmological fluid, the present lectures cover some of the same ground as a previous set of Cargèse lectures \cite{ell73}, but they then go on to give (ii) the full set of corresponding tetrad equations, (iii) a classification of cosmological models with exact symmetries, (iv) a brief discussion of some of the most useful exact models and their observational properties, and (v) an introduction to the gauge-invariant and 1+3 covariant perturbation theory of almost-Friedmann-Lema\^ıtre-Robertson-Walker universes, with a fluid description for the matter and a kinetic theory description of the radiation.

gr-qc↗

arXiv2005

Asymptotic silence of generic cosmological singularities

In this letter we investigate the nature of generic cosmological singularities using the framework developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum G2 cosmologies, models that are expected to capture the singular behavior of generic cosmologies with no symmetries at all. In particular, our results indicate that asymptotic silence holds, i.e., that particle horizons along all timelines shrink to zero for generic solutions. Moreover, we provide evidence that spatial derivatives become dynamically insignificant along generic timelines, and that the evolution into the past along such timelines is governed by an asymptotic dynamical system which is associated with an invariant set -- the silent boundary. We also identify an attracting subset on the silent boundary that organizes the oscillatory dynamics of generic timelines in the singular regime. In addition, we discuss the dynamics associated with recurring spike formation.

gr-qc↗

arXiv2003

Gowdy phenomenology in scale-invariant variables

The dynamics of Gowdy vacuum spacetimes is considered in terms of Hubble-normalized scale-invariant variables, using the timelike area temporal gauge. The resulting state space formulation provides for a simple mechanism for the formation of ``false'' and ``true spikes'' in the approach to the singularity, and a geometrical formulation for the local attractor.

gr-qc↗

arXiv2003

Asymptotic isotropization in inhomogeneous cosmology

In this paper we investigate asymptotic isotropization. We derive the asymptotic dynamics of spatially inhomogeneous cosmological models with a perfect fluid matter source and a positive cosmological constant near the de Sitter equilibrium state at late times, and near the flat FL equilibrium state at early times. Our results show that there exists an open set of solutions approaching the de Sitter state at late times, consistent with the cosmic no-hair conjecture. On the other hand, solutions that approach the flat FL state at early times are special and admit a so-called isotropic initial singularity. For both classes of models the asymptotic expansion of the line element contains an arbitrary spatial metric at leading order, indicating asymptotic spatial inhomogeneity. We show, however, that in the asymptotic regimes this spatial inhomogeneity is significant only at super-horizon scales.

gr-qc↗

arXiv2003

The past attractor in inhomogeneous cosmology

We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinski\vı, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models.

gr-qc↗

arXiv2002

BritGravII - The Second British Gravity Meeting

The second of the annual BritGrav meetings on current research in Gravitational Physics in Britain took place at the School of Mathematical Sciences of Queen Mary, University of London on June 10/11, 2002. 47 plenary talks of 12min duration were given. We make available the abstracts of the talks and the references to the electronic preprints at arXiv.org where they exist.

gr-qc↗

arXiv2001

Dynamical systems approach to G2 cosmology

In this paper we present a new approach for studying the dynamics of spatially inhomogeneous cosmological models with one spatial degree of freedom. By introducing suitable scale-invariant dependent variables we write the evolution equations of the Einstein field equations as a system of autonomous partial differential equations in first-order symmetric hyperbolic format, whose explicit form depends on the choice of gauge. As a first application, we show that the asymptotic behaviour near the cosmological initial singularity can be given a simple geometrical description in terms of the local past attractor on the boundary of the scale-invariant dynamical state space. The analysis suggests the name ``asymptotic silence'' to describe the evolution of the gravitational field near the cosmological initial singularity.

gr-qc↗

arXiv2001

General relativistic analysis of peculiar velocities

We give a careful general relativistic and (1+3)-covariant analysis of cosmological peculiar velocities induced by matter density perturbations in the presence of a cosmological constant. In our quasi-Newtonian approach, constraint equations arise to maintain zero shear of the non-comoving fundamental worldlines which define a Newtonian-like frame, and these lead to the (1+3)-covariant dynamical equations, including a generalized Poisson-type equation. We investigate the relation between peculiar velocity and peculiar acceleration, finding the conditions under which they are aligned. In this case we find (1+3)-covariant relativistic generalizations of well-known Newtonian results.

gr-qc↗

arXiv2000

On the propagation of jump discontinuities in relativistic cosmology

A recent dynamical formulation at derivative level $\ptl^{3}g$ for fluid spacetime geometries $({\cal M}, {\bf g}, {\bf u})$, that employs the concept of evolution systems in first-order symmetric hyperbolic format, implies the existence in the Weyl curvature branch of a set of timelike characteristic 3-surfaces associated with propagation speed $|v| = \sfrac{1}{2}$ relative to fluid-comoving observers. We show it is the physical role of the constraint equations to prevent realisation of jump discontinuities in the derivatives of the related initial data so that Weyl curvature modes propagating along these 3-surfaces cannot be activated. In addition we introduce a new, illustrative first-order symmetric hyperbolic evolution system at derivative level $\ptl^{2}g$ for baryotropic perfect fluid cosmological models that are invariant under the transformations of an Abelian $G_{2}$ isometry group.

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