Artur Carlos Ferreira Alho
@camgsd.tecnico.ulisboa.pt
University of Lisbon
CAMGSD
RESEARCH, TEACHING, or OTHER INTERESTS
Mathematical Physics, Physics and Astronomy, Applied Mathematics, Analysis
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Scopus Publications
Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo
IOP Publishing
Abstract 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 (2022 Phys. Rev. D 105 044025, 2022 Phys. Rev. D 106 L041502) 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 physically admissible initial data, we numerically construct mass-radius-compactness diagrams, and conjecture about the maximum compactness of stable physically admissible elastic balls.
Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo
American Physical Society (APS)
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.
Artur Alho and Claes Uggla
IOP Publishing
Abstract 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.
Artur Alho, Claes Uggla, and John Wainwright
Elsevier BV
Artur Alho, Vitor Bessa, and Filipe C. Mena
Springer Science and Business Media LLC
AbstractMotivated by cosmological models of the early universe we analyse the dynamics of the Einstein equations with a minimally coupled scalar field with monomial potentials $$V(\\phi )=\\frac{(\\lambda \\phi )^{2n}}{2n}$$ V ( ϕ ) = ( λ ϕ ) 2 n 2 n , $$\\lambda >0$$ λ > 0 , $$n\\in {\\mathbb {N}}$$ n ∈ N , interacting with a perfect fluid with linear equation of state $$p_{\\textrm{pf}}=(\\gamma _{\\textrm{pf}}-1)\\rho _{\\textrm{pf}}$$ p pf = ( γ pf - 1 ) ρ pf , $$\\gamma _{\\textrm{pf}}\\in (0,2)$$ γ pf ∈ ( 0 , 2 ) , in flat Robertson–Walker spacetimes. The interaction is a friction-like term of the form $$\\Gamma (\\phi )=\\mu \\phi ^{2p}$$ Γ ( ϕ ) = μ ϕ 2 p , $$\\mu >0$$ μ > 0 , $$p\\in {\\mathbb {N}}\\cup \\{0\\}$$ p ∈ N ∪ { 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.
Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo
American Physical Society (APS)
A foundational theorem due to Buchdahl states that, within General Relativity (GR), the maximum compactness C ≡ GM/ ( Rc 2 ) of a static, spherically symmetric, perfect fluid object of mass M and radius R is C = 4 / 9. As a corollary, there exists a compactness gap between perfect fluid stars and black holes (where 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 C = 1 / 2 continuously; (iii) however, imposing subluminal wave propagation lowers the maximum compactness bound to C ≈ 0 . 462, which we conjecture to be the maximum compactness of any static elastic object satisfying causality; (iv) imposing also radial stability further decreases the maximum compactness to C ≈ 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.
Artur Alho, Woei Chet Lim, and Claes Uggla
IOP Publishing
Abstract 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.
Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo
American Physical Society (APS)
Artur Alho, José Natário, Paolo Pani, and Guilherme Raposo
American Physical Society (APS)
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 ≈ 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 2 R ) ≈ 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.
Artur Alho, Simone Calogero, and Astrid Liljenberg
International Press of Boston
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 Lam\\'e type, that depend on no more material parameters than the bulk modulus $\\kappa>0$ and the Poisson ratio $-1<\\nu\\leq1/2$. 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.
Artur Alho, Claes Uggla, and John Wainwright
IOP Publishing
Abstract We derive a new regular dynamical system on a three-dimensional 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.
A. Alho and S. Calogero
Springer Science and Business Media LLC
Artur Alho, Vitor Bessa, and Filipe C. Mena
AIP Publishing
We apply a new global dynamical system 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 toward 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.
A. Alho and S. Calogero
Springer Science and Business Media LLC
Artur Alho, Claes Uggla, and John Wainwright
IOP Publishing
The observational success and simplicity of the ΛCDM model, and the explicit analytic perturbations thereof, set the standard for any alternative cosmology. It therefore serves as a comparison ground and as a test case for methods which can be extended and applied to other cosmological models. In this paper we introduce dynamical systems and methods to describe linear scalar and tensor perturbations of the ΛCDM model, which serve as pedagogical examples that show the global illustrative powers of dynamical systems in the context of cosmological perturbations. We also study the asymptotic properties of the shear and Weyl tensors and discuss the validity of the perturbations as approximations to the Einstein field equations. Furthermore, we give a new approximation for the linear growth rate, f(z) = d ln δ/d ln a = Ω6/11m − 1/70(1−Ωm)5/2, where z is the cosmological redshift, Ωm = Ωm(z), while a is the background scale factor, and show that it is much more accurate than the previous ones in the literature.
Artur Alho, Grigorios Fournodavlos, and Anne T. Franzen
World Scientific Pub Co Pte Lt
We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.
A. Alho and S. Calogero
Elsevier BV
Artur Alho and Claes Uggla
American Physical Society (APS)
We study flat Friedmann-Lemaitre-Robertson-Walker alpha-attractor E- and T-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimension ...
Artur Alho, Filipe C. Mena, and Juan A. Valiente Kroon
International Press of Boston
A frame representation is used to derive a first order quasi-linear symmetric hyperbolic system for a scalar field minimally coupled to gravity. This procedure is inspired by similar evolution equations introduced by Friedrich to study the Einstein-Euler system. The resulting evolution system is used to show that small nonlinear perturbations of expanding Friedman-Lemaitre-Robertson-Walker backgrounds, with scalar field potentials satisfying certain future asymptotic conditions, decay exponentially to zero, in synchronous time.
Artur Alho, Sante Carloni, and Claes Uggla
IOP Publishing
We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in f(R)-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, f(R) = R + α R2, α > 0, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to f(R) cosmology and discuss their advantages and disadvantages.
Artur Alho and Claes Uggla
American Physical Society (APS)
This paper treats nonrelativistic matter and a scalar field phi with a monotonically decreasing potential minimally coupled to gravity in flat Friedmann-Lemaitre-Robertson-Walker cosmology. The fie ...
Artur Alho, Juliette Hell, and Claes Uggla
IOP Publishing
We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat Friedmann–Lemaître–Robertson–Walker cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ‘attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Padé approximants and compare with the slow-roll approximation.
A. Alho, S. Calogero, M.P. Machado Ramos, and A.J. Soares
Elsevier BV
Artur Alho and Claes Uggla
AIP Publishing
We consider the familiar problem of a minimally coupled scalar field with quadratic potential in flat Friedmann-Lemaitre-Robertson-Walker cosmology to illustrate a number of techniques and tools, which can be applied to a wide range of scalar field potentials and problems in, e.g., modified gravity. We present a global and regular dynamical systems description that yields a global understanding of the solution space, including asymptotic features. We introduce dynamical systems techniques such as center manifold expansions and use Pade approximants to obtain improved approximations for the “attractor solution” at early times. We also show that future asymptotic behavior is associated with a limit cycle, which shows that manifest self-similarity is asymptotically broken toward the future and gives approximate expressions for this behavior. We then combine these results to obtain global approximations for the attractor solution, which, e.g., might be used in the context of global measures. In addition, we elucidate the connection between slow-roll based approximations and the attractor solution, and compare these approximations with the center manifold based approximants.