arXiv subjects

Artur Alho

Publications and source records attributed to Artur Alho.

At least 19 recordsLinked to original sources

arXiv2026

Dynamics of quadratic f(R) cosmology with a perfect fluid: Jordan and Einstein frames

We investigate the global dynamics of the field equations of (pure) quadratic theories of gravity which generalise Einstein's theory in spatially flat homogeneous and isotropic cosmological models with a perfect fluid. We introduce global and regular 3-dimensional dynamical systems' formulations, on both the Jordan frame and the conformally related Einstein frame. The analysis in the Jordan frame explores the monotonicity properties of the interior flow which, together with the characterisation of the orbit structure on the 2-dimensional invariant boundaries and the desingularisation of non-hyperbolic fixed points, provides a global description of the flow and its limit sets. In the Einstein frame, the analysis uses the skew-product structure of the Einstein state space and the characterisation of the orbit structure on the 2-dimensional invariant boundaries. Furthermore, by obtaining asymptotic expansions we identify the solutions that are global conformally mapped from the Jordan frame to the Einstein frame and those that are not.

gr-qc↗

arXiv2025

Quintessence: Quadratic potentials

Arguably one can use a canonical scalar field $φ$, minimally coupled to gravity, with quadratic potentials $V = Λ\pm \frac12 m^2φ^2$ to explore some general features of slow-roll and hilltop thawing quintessence, respectively. For each of these two potentials, and pressure-free matter, we introduce a regular unconstrained dynamical system on a compact state space, where the formulation for the hilltop case is new. Together with a derivation of monotonic functions in the two global state space settings, this enables us to obtain global results and to introduce figures that illustrate the global solution spaces of these models, in which we situate the observationally viable quintessence solutions.

gr-qc↗

arXiv2025

Quintessence

Recent observations suggest that the accelerated expansion of the Universe at late times is caused by a temporally changing dark energy component, rather than the constant one in the standard $Λ$CDM scenario. In this context quintessence, i.e. a canonical scalar field minimally coupled to gravity, plays a prominent role. There are, however, three main types of quintessence models: thawing quintessence, scaling freezing quintessence, and tracking quintessence. Dynamical systems reformulations of the field equations for a broad set of scalar field potentials, including some new ones, allow us to use dynamical systems methods to derive global and asymptotic features, visualised in bounded state space pictures clearly illustrating the relationships and properties of the different types of quintessence, clarifying initial data issues, and yielding simple and accurate approximations.

gr-qc↗

arXiv2025

New simple and accurate quintessence approximations

We derive new approximations for quintessence solutions that are simpler and an order of magnitude more accurate than anything available in the literature, which from an observational perspective \emph{makes numerical calculations superfluous}. For example, our tracking quintessence approximation yields $\sim 0.1\%$ maximum relative errors of $H(z)/H_0$ and $Ω_\mathrm{m}(z)$ for the observationally viable inverse power law scalar field potentials, and similarly for viable thawing quintessence models using two slow-roll parameters. The approximations are trivially computed from the scalar field potential and as an application we give \emph{analytic} expressions for the CPL parameters calculated from an arbitrary scalar field potential for thawing and tracking quintessence models.

gr-qc↗

arXiv2024

Spherically symmetric elastic bodies in general relativity

The purpose of this review it to present a renewed perspective of the problem of self-gravitating elastic bodies under spherical symmetry. It is also a companion to the papers [Phys. Rev. D105, 044025 (2022)], [Phys. Rev. D106, L041502 (2022)], and [arXiv:2306.16584 [gr-qc]], where we introduced a new definition of spherically symmetric elastic bodies in general relativity, and applied it to investigate the existence and physical viability, including radial stability, of static self-gravitating elastic balls. We focus on elastic materials that generalize fluids with polytropic, linear, and affine equations of state, and discuss the symmetries of the energy density function, including homogeneity and the resulting scale invariance of the TOV equations. By introducing invariant characterizations of physical admissible initial data, we numerically construct mass-radius-compactness diagrams, and conjecture about the maximum compactness of stable physically admissible elastic balls.

gr-qc↗

arXiv2024

Scale invariant elastic stars in General Relativity

We present a model of relativistic elastic stars featuring scale invariance. This implies a linear mass-radius relation and the absence of a maximum mass. The most compact spherically symmetric configuration that is radially stable and satisfies all energy and causality conditions has a slightly smaller radius than the Schwarzschild light ring radius. To the best of our knowledge, this is the first material compact object with such remarkable properties in General Relativity, which makes it a unique candidate for a black-hole mimicker.

gr-qc↗

arXiv2024

Quintessential $α$-attractor inflation: A dynamical systems analysis

The equations for quintessential $α$-attractor inflation with a single scalar field, radiation and matter in a spatially flat FLRW spacetime are recast into a regular dynamical system on a compact state space. This enables a complete description of the solution space of these models. The inflationary attractor solution is shown to correspond to the unstable center manifold of a de Sitter fixed point, and we describe connections between slow-roll and dynamical systems approximations for this solution, including Padé approximants. We also introduce a new method for systematically obtaining initial data for quintessence evolution by using dynamical systems properties; in particular, this method exploits that there exists a radiation dominated line of fixed points with an unstable quintessence attractor submanifold, which plays a role that is reminiscent of that of the inflationary attractor solution for inflation.

gr-qc↗

arXiv2024

Tracking Quintessence

Tracking quintessence, in a spatially flat and isotropic space-time with a minimally coupled canonical scalar field and an asymptotically inverse power-law potential $V(φ)\proptoφ^{-p}$, $p>0$, as $φ\rightarrow0$, is investigated. This is done by introducing a new three-dimensional \emph{regular} dynamical system, which enables a rigorous explanation of the tracking feature: 1) The dynamical system has a tracker fixed point $\mathrm{T}$ with a two-dimensional stable manifold that pushes an open set of nearby solutions toward a single tracker solution originating from $\mathrm{T}$. 2) All solutions, including the tracker solution and the solutions that track/shadow it, end at a common future attractor fixed point that depends on the potential. Thus, the open set of solutions that shadow the tracker solution share its properties during the tracking quintessence epoch. We also discuss similarities and differences of underlying mechanisms for tracking, thawing and scaling freezing quintessence, and, moreover, we illustrate with state space pictures that all of these types of quintessence exist simultaneously for certain potentials.

gr-qc↗

arXiv2022

Quintessence from a state space perspective

We use dynamical systems methods to study quintessence models in a spatially flat and isotropic spacetime with matter and a scalar field with potentials for which $λ(φ)=-V_{,φ}/V$ is bounded, thereby going beyond the exponential potential for which $λ(φ)$ is constant. The scalar field equation of state parameter $w_φ$ plays a central role when comparing quintessence models with observations, but with the dynamical systems used to date $w_φ$ is an indeterminate, discontinuous, function on the state space in the asymptotically matter dominated regime. Our first main result is the introduction of new variables that lead to a \emph{regular} dynamical system on a \emph{bounded} three-dimensional state space on which $w_φ$ is a \emph{regular} function. The solution trajectories in the state space then provide a visualization of different types of quintessence evolution, and how initial conditions affect the transition between the matter and scalar field dominated epochs; this is complemented by graphs $w_φ(N)$, where $N$ is the $e$-fold time, which enables characterizing different types of quintessence evolution.

gr-qc↗

arXiv2022

Dynamics of interacting monomial scalar field potentials and perfect fluids

Motivated by cosmological models of the early universe we analyse the dynamics of the Einstein equations with a minimally coupled scalar field with monomial potentials $V(ϕ)=\frac{(λϕ)^{2n}}{2n}$, $λ>0$, $n\in\mathbb{N}$, interacting with a perfect fluid with linear equation of state $p_\mathrm{pf}=(γ_\mathrm{pf}-1)ρ_\mathrm{pf}$, $γ_\mathrm{pf}\in(0,2)$, in flat Robertson-Walker spacetimes. The interaction is a friction-like term of the form $Γ(ϕ)=μϕ^{2p}$, $μ>0$, $p\in\mathbb{N}\cup\{0\}$. The analysis relies on the introduction of a new regular 3-dimensional dynamical systems' formulation of the Einstein equations on a compact state space, and the use of dynamical systems' tools such as quasi-homogeneous blow-ups and averaging methods involving a time-dependent perturbation parameter. We find a bifurcation at $p=n/2$ due to the influence of the interaction term. In general, this term has more impact on the future (past) asymptotics for $p n/2$). For $p n/2$ the future asymptotics is similar to the case with no interaction. Finally, we show that irrespective of the parameters, an inflationary quasi-de-Sitter solution always exists towards the past, and therefore the cases with $p\leq(n-2)/2$ may provide new cosmological models of quintessential inflation.

gr-qc↗

arXiv2022

Compactness bounds in General Relativity

A foundational theorem due to Buchdahl states that, within General Relativity (GR), the maximum compactness $\mathcal{C}\equiv GM/(Rc^2)$ of a static, spherically symmetric, perfect fluid object of mass $M$ and radius $R$ is $\mathcal{C}=4/9$. As a corollary, there exists a compactness gap between perfect fluid stars and black holes (where $\mathcal{C}=1/2$). Here we generalize Buchdahl's result by introducing the most general equation of state for elastic matter with constant longitudinal wave speeds and apply it to compute the maximum compactness of regular, self-gravitating objects in GR. We show that: (i) the maximum compactness grows monotonically with the longitudinal wave speed; (ii) elastic matter can exceed Buchdahl's bound and reach the black hole compactness $\mathcal{C}=1/2$ continuously; (iii) however, imposing subluminal wave propagation lowers the maximum compactness bound to $\mathcal{C}\approx0.462$, which we conjecture to be the maximum compactness of \emph{any} static elastic object satisfying causality; (iv) imposing also radial stability further decreases the maximum compactness to $\mathcal{C}\approx 0.389$. Therefore, although anisotropies are often invoked as a mechanism for supporting horizonless ultracompact objects, we argue that the black hole compactness cannot be reached with physically reasonable matter within GR and that true black hole mimickers require either exotic matter or beyond-GR effects.

gr-qc↗

arXiv2022

Compact elastic objects in general relativity

We introduce a rigorous and general framework to study systematically self-gravitating elastic materials within general relativity, and apply it to investigate the existence and viability, including radial stability, of spherically symmetric elastic stars. We present the mass-radius ($M-R$) diagram for various families of models, showing that elasticity contributes to increase the maximum mass and the compactness up to $\approx 22\%$, thus supporting compact stars with mass well above two solar masses. Some of these elastic stars can reach compactness as high as $GM/(c^2R)\approx 0.35$ while remaining stable under radial perturbations and satisfying all energy conditions and subluminal wave propagation, thus being physically realizable models of stars with a light ring. We provide numerical evidence that radial instability occurs for central densities larger than that corresponding to the maximum mass, as in the perfect-fluid case. Elasticity may be a key ingredient to build consistent models of exotic ultracompact objects and black-hole mimickers, and can also be relevant for a more accurate modelling of the interior of neutron stars.

gr-qc↗

arXiv2022

Cosmological global dynamical systems analysis

We consider a dynamical systems formulation for models with an exponential scalar field and matter with a linear equation of state in a spatially flat and isotropic spacetime. In contrast to earlier work, which only considered linear hyperbolic fixed point analysis, we do a center manifold analysis of the non-hyperbolic fixed points associated with bifurcations. More importantly though, we construct monotonic functions and a Dulac function. Together with the complete local fixed point analysis this leads to proofs that describe the entire global dynamics of these models, thereby complementing previous local results in the literature.

gr-qc↗

arXiv2021

Self-gravitating static balls of power-law elastic matter

We study a class of power-law stored energy functions for spherically symmetric elastic bodies that includes well-known material models, such as the Saint Venant-Kirchhoff, Hadamard, Signorini and John models. We identify a finite subclass of these stored energy functions, which we call Lamé type, that depend on no more material parameters than the bulk modulus $κ>0$ and the Poisson ratio $-1<ν\leq1/2$. A general theorem proving the existence of static self-gravitating elastic balls for some power-law materials has been given elsewhere. In this paper numerical evidence is provided that some hypotheses in this theorem are necessary, while others are not.

math-ph↗

arXiv2020

Dynamical systems in perturbative scalar field cosmology

We derive a new \emph{regular} dynamical system on a 3-dimensional \emph{compact} state space describing linear scalar perturbations of spatially flat Robertson-Walker geometries for relativistic models with a minimally coupled scalar field with an exponential potential. This enables us to construct the global solution space, illustrated with figures, where known solutions are shown to reside on special invariant sets. We also use our dynamical systems approach to obtain new results about the comoving and uniform density curvature perturbations. Finally we show how to extend our approach to more general scalar field potentials. This leads to state spaces where the state space of the models with an exponential potential appears as invariant boundary sets, thereby illustrating their role as building blocks in a hierarchy of increasingly complex cosmological models.

gr-qc↗

arXiv2020

Static self-gravitating Newtonian elastic balls

The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials.

math-ph↗

arXiv2020

Global dynamics of Yang-Mills field and perfect-fluid Robertson-Walker cosmologies

We apply a new global dynamical systems formulation to flat Robertson-Walker cosmologies with a massless and massive Yang-Mills field and a perfect-fluid with linear equation of state as the matter sources. This allows us to give proofs concerning the global dynamics of the models including asymptotic source-dominance towards the past and future time directions. For the pure massless Yang-Mills field, we also contextualize well-known explicit solutions in a global (compact) state space picture.

gr-qc↗

arXiv2019

Multi-body spherically symmetric steady states of Newtonian self-gravitating elastic matter

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

math-ph↗