Relativistic structure formation models and gravitoelectromagnetism

In the framework of Lagrangian perturbation theory in general relativity we discuss the possibility to split the Einstein equations, written in terms of spatial Cartan coframes within a 3+1 foliation of spacetime, into gravitoelectric and gravitomagnetic parts. While the former reproduces the full hierarchy of the Newtonian perturbation solutions, the latter contains non-Newtonian aspects like gravitational waves. This split can be understood and made unique through the Hodge decomposition of Cartan coframe fields.Comment: 6 pages; contribution to the proceedings of MG14, Parallel Session DE

Théorie Lagrangienne Relativiste de la Formation des Grandes Structures : description Intrinsèque des Perturbations et Gravitoélectromagnétisme

The dynamics of structure formation in the Universe is usually described by Newtonian numerical simulations and analytical models in the frame of the Standard Model of Cosmology. The structures are then defined on a homogeneous and isotropic background. Such a description has major drawbacks since, to be self-consistent, it entails a large amount of dark components in the content of the Universe. To address the problem of dark matter and dark energy, we will neither suppose that exotic sources contribute to the content of the Universe, nor that General Relativity is obsolete. We will develop a more realistic description of structure formation in the frame of General Relativity and thus no longer assume that the average model is a homogeneous-isotropic solution of the Einstein equations, as claimed by the Standard Model of Cosmology. During my work under the supervision of Thomas Buchert, I contributed to the development of the perturbative formalism that enables a more realistic description of spacetime dynamics. In the framework of the intrinsic Lagrangian approach, which avoids defining physical quantities on a flat background, I contributed to the building of relativistic solutions to the gravitoelectric part of the Einstein equations from the generalization of the Newtonian perturbative solutions. Moreover, the gravitoelectromagnetic approach I worked with has provided a new understanding of the dynamics of the analytical solutions to the field equations. Finally, treating globally the spatial manifold, I used powerful mathematical tools and theorems to describe the impact of topology on the dynamics of gravitational wavesLa dynamique de formation des structures de l'Univers est habituellement décrite dans le cadre du modèle standard de Cosmologie. Cependant, pour que les observations cosmologiques soient cohérentes avec le modèle standard, il est nécessaire de supposer l'existence d'une grande proportion d'éléments de nature inconnue dans le contenu de l'Univers. Pour tenter de résoudre cette énigme, nous ne considèrerons pas d'autres sources dans le contenu de l'Univers que celles ordinaires et resterons dans le cadre de la Relativité Générale. Nous développerons néanmoins une description plus réaliste de la formation de structures dans le cadre de la théorie d'Einstein. Ainsi, contrairement au modèle standard de Cosmologie, nous ne supposerons pas que l'Univers moyenné est une solution homogène et isotrope des équations d'Einstein. Lors de mon travail sous la direction de Thomas Buchert, j'ai participé au développement d'un formalisme perturbatif permettant une description plus réaliste de la dynamique de l'espace-temps. J'ai également contribué à l'obtention de solutions relativistes à la partie gravitoélectrique des équations d'Einstein en généralisant les solutions perturbatives newtoniennes. Ces travaux ont été réalisés dans le cadre d'une approche lagrangienne intrinsèque, évitant ainsi de définir les grandeurs physiques sur un fond plat. L'approche gravitoélectromagnétique que j'ai adoptée m'a permis une interprétation nouvelle et performante des solutions des équations d'Einstein. Enfin, j'ai étudié l'impact de la topologie sur la dynamique des ondes gravitationelles à l'aide d'une description globale de l'hypersurface spatiale, permise par des théorèmes mathématiques puissant

Lagrangian theory of structure formation in relativistic cosmology III: gravitoelectric perturbation and solution schemes at any order

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the first-order trace solutions that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given together with extensions and applications for accessing the nonperturbative regime. We furnish here construction rules to obtain from Newtonian solutions the gravitoelectric class of relativistic solutions, for which we give the complete perturbation and solution schemes at any order of the perturbations. By construction, these schemes generalize the complete hierarchy of solutions of the Newtonian Lagrangian perturbation theory.Comment: 17 pages, a few minor extensions to match the published version in PR

Nonperturbative collapse models for collisionless self-gravitating flows

Structure formation in the Universe has been well-studied within the Eulerian and Lagrangian perturbation theories, where the latter performs substantially better in comparison with N-body simulations. Standing out is the celebrated Zel'dovich approximation for dust matter. In this work, we recall the description of gravitational noncollisional systems and extend both the Eulerian and Lagrangian approaches by including, possibly anisotropic, velocity dispersion. A simple case with plane symmetry is then studied with an exact, nonperturbative approach, and various approximations of the derived model are then compared numerically. A striking result is that linearized Lagrangian solutions outperform models based on Burgers' equation in the multi-stream regime in comparison with the exact solution. These results are finally extended to a 3D case without symmetries, and master equations are derived for the evolution of all parts of the perturbations.Comment: 20 pages, 5 figures, submitted to PR

On the Lagrangian Theory of Structure Formation in Relativistic Cosmology : intrinsic Perturbation Approach and Gravitoelectromagnetism

La dynamique de formation des structures de l'Univers est habituellement décrite dans le cadre du modèle standard de Cosmologie. Cependant, pour que les observations cosmologiques soient cohérentes avec le modèle standard, il est nécessaire de supposer l'existence d'une grande proportion d'éléments de nature inconnue dans le contenu de l'Univers. Pour tenter de résoudre cette énigme, nous ne considèrerons pas d'autres sources dans le contenu de l'Univers que celles ordinaires et resterons dans le cadre de la Relativité Générale. Nous développerons néanmoins une description plus réaliste de la formation de structures dans le cadre de la théorie d'Einstein. Ainsi, contrairement au modèle standard de Cosmologie, nous ne supposerons pas que l'Univers moyenné est une solution homogène et isotrope des équations d'Einstein. Lors de mon travail sous la direction de Thomas Buchert, j'ai participé au développement d'un formalisme perturbatif permettant une description plus réaliste de la dynamique de l'espace-temps. J'ai également contribué à l'obtention de solutions relativistes à la partie gravitoélectrique des équations d'Einstein en généralisant les solutions perturbatives newtoniennes. Ces travaux ont été réalisés dans le cadre d'une approche lagrangienne intrinsèque, évitant ainsi de définir les grandeurs physiques sur un fond plat. L'approche gravitoélectromagnétique que j'ai adoptée m'a permis une interprétation nouvelle et performante des solutions des équations d'Einstein. Enfin, j'ai étudié l'impact de la topologie sur la dynamique des ondes gravitationelles à l'aide d'une description globale de l'hypersurface spatiale, permise par des théorèmes mathématiques puissantsThe dynamics of structure formation in the Universe is usually described by Newtonian numerical simulations and analytical models in the frame of the Standard Model of Cosmology. The structures are then defined on a homogeneous and isotropic background. Such a description has major drawbacks since, to be self-consistent, it entails a large amount of dark components in the content of the Universe. To address the problem of dark matter and dark energy, we will neither suppose that exotic sources contribute to the content of the Universe, nor that General Relativity is obsolete. We will develop a more realistic description of structure formation in the frame of General Relativity and thus no longer assume that the average model is a homogeneous-isotropic solution of the Einstein equations, as claimed by the Standard Model of Cosmology. During my work under the supervision of Thomas Buchert, I contributed to the development of the perturbative formalism that enables a more realistic description of spacetime dynamics. In the framework of the intrinsic Lagrangian approach, which avoids defining physical quantities on a flat background, I contributed to the building of relativistic solutions to the gravitoelectric part of the Einstein equations from the generalization of the Newtonian perturbative solutions. Moreover, the gravitoelectromagnetic approach I worked with has provided a new understanding of the dynamics of the analytical solutions to the field equations. Finally, treating globally the spatial manifold, I used powerful mathematical tools and theorems to describe the impact of topology on the dynamics of gravitational wave

Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves

International audienceThe relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the perturbation and solution schemes that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given to any order of the perturbations, together with extensions and applications for accessing the nonperturbative regime. We here discuss more in detail the general first-order scheme within the Cartan formalism including and concentrating on the gravitational wave propagation in matter. We provide master equations for all parts of Lagrangian-linearized perturbations propagating in the perturbed spacetime, and we outline the solution procedure that allows one to find general solutions. Particular emphasis is given to global properties of the Lagrangian perturbation fields by employing results of Hodge–de Rham theory. We here discuss how the Hodge decomposition relates to the standard scalar-vector-tensor decomposition. Finally, we demonstrate that we obtain the known linear perturbation solutions of the standard relativistic perturbation scheme by performing two steps: first, by restricting our solutions to perturbations that propagate on a flat unperturbed background spacetime and, second, by transforming to Eulerian background coordinates with truncation of nonlinear terms

Brain-imaging evidence for compression of binary sound sequences in human memory

According to the language of thought hypothesis, regular sequences are compressed in human memory using recursive loops akin to a mental program that predicts future items. We tested this theory by probing memory for 16-item sequences made of two sounds. We recorded brain activity with functional MRI and magneto-encephalography (MEG) while participants listened to a hierarchy of sequences of variable complexity, whose minimal description required transition probabilities, chunking, or nested structures. Occasional deviant sounds probed the participants’ knowledge of the sequence. We predicted that task difficulty and brain activity would be proportional to the complexity derived from the minimal description length in our formal language. Furthermore, activity should increase with complexity for learned sequences, and decrease with complexity for deviants. These predictions were upheld in both fMRI and MEG, indicating that sequence predictions are highly dependent on sequence structure and become weaker and delayed as complexity increases. The proposed language recruited bilateral superior temporal, precentral, anterior intraparietal and cerebellar cortices. These regions overlapped extensively with a localizer for mathematical calculation, and much less with spoken or written language processing. We propose that these areas collectively encode regular sequences as repetitions with variations and their recursive composition into nested structures

Nonperturbative collapse models for collisionless self-gravitating flows

International audienceStructure formation in the Universe has been well-studied within the Eulerian and Lagrangian perturbation theories, where the latter performs substantially better in comparison with N-body simulations. Standing out is the celebrated Zel'dovich approximation for dust matter. In this work, we recall the description of gravitational noncollisional systems and extend both the Eulerian and Lagrangian approaches by including, possibly anisotropic, velocity dispersion. A simple case with plane symmetry is then studied with an exact, nonperturbative approach, and various approximations of the derived model are then compared numerically. A striking result is that linearized Lagrangian solutions outperform models based on Burgers' equation in the multi-stream regime in comparison with the exact solution. These results are finally extended to a 3D case without symmetries, and master equations are derived for the evolution of all parts of the perturbations

Brain-imaging evidence for compression of binary sound sequences in human memory

According to the language-of-thought hypothesis, regular sequences are compressed in human memory using recursive loops akin to a mental program that predicts future items. We tested this theory by probing memory for 16-item sequences made of two sounds. We recorded brain activity with functional MRI and magneto-encephalography (MEG) while participants listened to a hierarchy of sequences of variable complexity, whose minimal description required transition probabilities, chunking, or nested structures. Occasional deviant sounds probed the participants’ knowledge of the sequence. We predicted that task difficulty and brain activity would be proportional to the complexity derived from the minimal description length in our formal language. Furthermore, activity should increase with complexity for learned sequences, and decrease with complexity for deviants. These predictions were upheld in both fMRI and MEG, indicating that sequence predictions are highly dependent on sequence structure and become weaker and delayed as complexity increases. The proposed language recruited bilateral superior temporal, precentral, anterior intraparietal, and cerebellar cortices. These regions overlapped extensively with a localizer for mathematical calculation, and much less with spoken or written language processing. We propose that these areas collectively encode regular sequences as repetitions with variations and their recursive composition into nested structures

Humans parsimoniously represent auditory sequences by pruning and completing the underlying network structure

International audienceSuccessive auditory inputs are rarely independent, their relationships ranging from local transitions between elements to hierarchical and nested representations. In many situations, humans retrieve these dependencies even from limited datasets. However, this learning at multiple scale levels is poorly understood. Here, we used the formalism proposed by network science to study the representation of local and higher-order structures and their interaction in auditory sequences. We show that human adults exhibited biases in their perception of local transitions between elements, which made them sensitive to high-order network structures such as communities. This behavior is consistent with the creation of a parsimonious simplified model from the evidence they receive, achieved by pruning and completing relationships between network elements. This observation suggests that the brain does not rely on exact memories but on a parsimonious representation of the world. Moreover, this bias can be analytically modeled by a memory/efficiency trade-off. This model correctly accounts for previous findings, including local transition probabilities as well as high-order network structures, unifying sequence learning across scales. We finally propose putative brain implementations of such bias