Inhomogeneous cosmology : an answer to the Dark Matter and Dark Energy problems? Alexandre Alles To cite this version: Alexandre Alles. Inhomogeneous cosmology : an answer to the Dark Matter and Dark Energy problems?. Astrophysics [astro-ph]. Université Claude Bernard - Lyon I, 2014. English. NNT : 2014LYO10165. tel-01266465 HAL Id: tel-01266465 https://tel.archives-ouvertes.fr/tel-01266465 Submitted on 2 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Num´ero d’ordre : 165 - 2014 Ann´ee 2014 THESE` DE L’UNIVERSITEDELYON´ D´elivr´ee par L’UNIVERSITE´ CLAUDE BERNARD LYON 1 ECOLE´ DOCTORALE DE PHYSIQUE ET ASTROPHYSIQUE DE LYON (PHAST, ED52) pour l’obtention du Diplomeˆ de Doctorat (arrˆet´edu7aoˆut 2006) soutenue publiquement le 22 Septembre 2014 par M. Alexandre Alles Titre : Inhomogeneous cosmology, an answer to the Dark Matter and Dark Energy problems? Jury : Alexandre Arbey, MCF. Examinateur Thomas Buchert,Pr. Directeur Aldo Deandrea,Pr. Examinateur Boudewijn F. Roukema,Pr. Rapporteur et Examinateur Pierre Salati,Pr. Pr´esident David L. Wiltshire,Pr. Rapporteur – Centre de Recherche Astrophysique de Lyon (CRAL - UMR 5574) – – Universite´ Claude Bernard Lyon 1 (UCBL) – – Ecole Doctorale de Physique et Astrophysique de Lyon (ED PHAST 52) – Inhomogeneous cosmology: an answer to the Dark Matter and Dark Energy problems? Author: Alexandre Alles September 21, 2014 Supervisor: Pr. Thomas Buchert Centre de Recherche Astrophysique de Lyon 9 avenue Charles Andre´ 69561 Saint Genis Laval cedex http://www-obs.univ-lyon1.fr/ Acknowledgements I wish to thank Thomas Buchert for his supervision and presence. He was very available during these three years of research. It was very interesting to work with him on such a technical subject and I hope I could continue to work on such subjects. I address special thanks to the two conscientious proofreaders of this thesis, Xavier Roy and Aurore Hutzler, who spent a lot of time improving the quality of this work. I thank all my interlocutors during this thesis, for their help, their visions of science and the technical details about their specific fields they gave to me: Nathaniel Obadia, Fosca Al Roumi, Martin France, Boud Roukema, Maxime Trebitsch, L´eo Brunswic and every person I met during this scientific adventure. In a more general way I thank all the CRAL staff at the Observatory and ENS. Naturally, I cordially thanks the two reviewers of this thesis, Boud Roukema and David Wiltshire for the time they spent to improve this piece of work. I also thank Alexandre Arbey, Aldo Deandrea, Boud Roukema and Pierre Salati to devote time to be part of the jury of this thesis. I address additional thanks to Pierre Salati who will preside this jury, for the discussions we had during my thesis and his permanent good mood. I thank the Lyon University and the doctoral school of physics and astrophysics (ED52, Phast) for their financial support and the chance to be part of the life of a doctoral school. I want to thank every person I socialised with during my Ph.D. thesis. I will not try the dangerous game to name everyone, each concerned people will recognise himself: I think about the people I met at the ENS Lyon before and during my thesis (friends and teachers), a special thought to the “R´esidents Foyer” and the “Foyer” of the ENS Lyon, the team and customers of Ukronium 1828 (it’s a great shop of recreational games in Lyon, you have to go!), the LARP and steampunk community, people I worked with at the University during my teachings and also from the Observatory. I also thank the Highschool and “Pr´epa” teachers who left their mark on my academic path: Christian Alix, Guillaume Haberer, Gilbert Rosset and others. The last but not the least, I warmly thank my family and my friends from Lyon and elsewhere in the world who supported me during these three years. 5 Additional thanks to the composers, musicians and singers who offered to the world their art. As said Nietzsche: “Without music, life would be a mistake” 6 Abstract & R´esum´e Abstract The standard model of cosmology describes the formation of large scale struc- tures in the late Universe within a quasi–Newtonian theory. This model requires the presence of unknown compounds of the Universe, Dark Matter and Dark En- ergy, to properly fit the observations. These two quantities, according to the Standard Model, represent almost 95% of the content of the Universe. Although the dark components are searched for by the scientific community, there exist several alternatives which try to deal with the problem of the large scale structures. Inhomogeneous theories describe the impact of the kinematical fluctuations on the global behaviour of the Universe. Or some theories proposed to go beyond general relativity. During my Ph.D. thesis, I developed key–elements of a fully relativistic La- grangian theory of structure formation. Assuming a specific space–time slicing, I solved the first order system of equations to obtain solutions which describe the matter evolution within the perturbed geometry, and I developed higher order schemes and their correspondences with the Lagrangian perturbation solutions in the Newtonian approach. I also worked on some applications of these results like the description of a silent Universe or the Weyl curvature hypothesis and the prob- lem of gravitational entropy. Further objectives are the description of physical observables and the development of direct applications. Next step of the devel- opment is an interaction between theoretical and numerical approaches, a study which would require strong cooperation with observers. 7 . R´esum´e Le Mod`ele Standard de la cosmologie d´ecrit la formation des structuresa ` grande ´echelle dans l’Univers r´ecent dans un cadre quasi–newtonien. Ce mod`ele requiert la pr´esence de composantes inconnues, la Mati`ere Noire et l’Energie´ Noire, afin de v´erifier correctement les observations. Ces deux quantit´es repr´esentent `a elles seules pr`es de 95% du contenu de l’Univers. Bien que ces composantes sombres soient activement recherch´ees par la commu- naut´e scientifique, il existe plusieurs alternatives qui tentent de traiter le probl`eme des structuresagrande´ ` echelle. Les th´eories inhomog`enes d´ecrivent l’impact des fluctuations cin´ematiques sur le comportement global de l’Univers. D’autres th´eories proposent ´egalement d’aller au-del`adelarelativit´eg´en´erale. Durant cette th`ese, j’ai mis au point des ´el´ements cl´es d’une th´eorie lagrangien- ne totalement relativiste de la formation des structures. Supposant un feuilletage particulier de l’espace–temps j’ai r´esolu le syst`eme d’´equations du premier ordre afin d’obtenir des solutions d´ecrivant l’´evolution de la mati`ere dans un espace `a la g´eom´etrie perturb´ee. J’ai ´egalement d´evelopp´eunsch´ema de r´esolution pour les ordres sup´erieurs de perturbation ainsi que leurs ´equivalent newtoniens. Une autre partie de ce travail de th`ese consiste en le d´eveloppement de quelques appli- cations directes : la description d’un Univers silencieux ou l’hypoth`ese de courbure de Weyl et le probl`eme de l’entropie gravitationnelle. Les objectifs `a plus ou moins court terme seraient d’obtenir la description d’observables physiques and le d´eveloppement d’autres applications. Cette ´etapeded´eveloppement sera une interaction entre approches th´eorique et num´erique et requerra de se rapprocher fortement des observateurs. 8 Contents Abstract & R´esum´e 7 I Introduction 13 1 A historical overview ......................... 13 2 The standard model of cosmology .................. 17 3 Homogeneous cosmology: Friedmann equations ........... 24 4 Large–scale structure formation ................... 27 5 Observations and initial data ..................... 28 II Newtonian perturbation theory 33 1 Introduction .............................. 33 2 Newtonian model expressions .................... 35 2.1 Euler–Newton system ..................... 37 2.2 Lagrange–Newton system ................... 38 3 Perturbative development of the Lagrange–Newton system .... 41 3.1 Perturbation scheme ...................... 42 3.2 Solution scheme ........................ 47 3.3 Explicit solutions scheme ................... 49 3.4 Einstein–de Sitter explicit solutions ............. 51 4 Concluding remarks .......................... 58 5 The motivation of a relativistic perturbation theory ........ 59 III Lagrangian relativistic perturbation theory 61 1 Introduction .............................. 62 2 Equations of motion and constraints ................. 64 2.1 Notations and technicalities .................. 64 2.2 Newtonian theory ....................... 65 2.3 Einstein theory in the Lagrangian frame ........... 67 2.4 Gravitoelectric set of equations in the Minkowski Restriction 69 3 General first order perturbation and solution schemes ....... 71 3.1 First order perturbation scheme ............... 71 9 CONTENTS 3.2 First order equations ...................... 76 3.3 Solution building ........................ 81 4 Comparison with other works .................... 85 4.1 Solving