Springer Theses Recognizing Outstanding Ph.D. Research Jérôme Gleyzes Dark Energy and the Formation of the Large Scale Structure of the Universe Springer Theses Recognizing Outstanding Ph.D. Research Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected foritsscientificexcellenceandthehighimpactofitscontentsforthepertinentfield of research. For greater accessibility to non-specialists, the published versions includeanextendedintroduction,aswellasaforewordbythestudent’ssupervisor explainingthespecialrelevanceoftheworkforthefield.Asawhole,theserieswill provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. 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More information about this series at http://www.springer.com/series/8790 é ô J r me Gleyzes Dark Energy and the Formation of the Large Scale Structure of the Universe Doctoral Thesis accepted by the Institute for Theoretical Physics-CEA Saclay, France 123 Author Supervisor Dr. Jérôme Gleyzes Dr. Filippo Vernizzi Jet Propulsion Laboratory CEA, IPhT Pasadena,CA Gif-sur-Yvette USA France ISSN 2190-5053 ISSN 2190-5061 (electronic) SpringerTheses ISBN978-3-319-41209-2 ISBN978-3-319-41210-8 (eBook) DOI 10.1007/978-3-319-41210-8 LibraryofCongressControlNumber:2016943810 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland ’ Supervisor s Foreword In the last 30 years, cosmology has witnessed much progress mostly spurred by a positive interplay between theoretical developments and observational discoveries. Thanks to high-precision data such as the cosmic microwave background aniso- tropies, we have been able to reconstruct the evolution of the universe and of the structuresthatweobservetodaywithanexquisiteaccuracy.Forinstance,weknow that large-scale structures formed by the gravitational collapse of dark matter aroundsomesmallinitialinhomogeneities.Accordingtoourcurrentunderstanding, the initial seeds of these inhomogeneities were generated during inflation, an early universephasecharacterisedbyanacceleratedexpansion,triggeredbythevacuum energy of a scalar field. The quantum fluctuations of this field, converted into density perturbations, are imprinted in the observable sky. By studying their sta- tistical distribution we have been able to confirm inflationary predictions. Despite these confirmations, inflation remains a paradigm awaiting for a convincing proof and lacking robust connections with know high energy physics. Even more mysterious, the current accelerated expansion of the universe, dis- covered in 1998, has triggered enormous interest in both the communities of observational cosmologists and theoretical physicists. The simplest explanation, a cosmological constant is consistent with observations. However, given the theo- retical difficulties traditionally associated to this possibility, most of the efforts of the scientific community are now devoted to rule it out. Alternatively, the acceleration may be due to some dynamical component called dark energy or to some modification in the laws of gravity on very large scales. Many models have been proposed, each of them leading to specific effects on the evolution of struc- tures formation. As for inflation, the knowledge of the statistical properties of the large-scalestructuresandtheirevolutionmaybecriticaltohelptobetterunderstand theoriginofthecurrentacceleration.Forthisreason,themajorscienceagenciesare currently planning large field cosmic surveys. The main scientific driver of these ground based and space telescopes is a precise determination of the statistical properties of the cosmic fields and their evolution, with the primary goal of con- straining dark energy and the origin of cosmological perturbations. v vi Supervisor’sForeword JérômeGleyzesstartedtoworkonhisPh.D.thesisinthisscientificcontext.He first focused his research on bridging theoretical models of dark energy with observations. It should be emphasised that the number and variety of proposed modelsofdarkenergyandmodifiedgravityisratherimpressive,whichrepresentsa challenge for future observations. Jérôme developed an approach to characterise most models in a unified way, in terms of a few number of parameters corre- spondingtoparticularobservationaleffectsoncosmologicalscales.Thisisbasedon the construction of a general theory of cosmological perturbations around a cos- mological background in terms of all possible Lagrangian operators satisfying certain symmetries, dictated by the class of models under consideration. This approachhasrapidlybecomepopularinthescientificcommunityunderthenameof effective fieldtheory ofdark energy andwill likelyplay animportant role inyears tocomeinattemptstoconstraindeviationsfromgeneralrelativityoncosmological scales. Indevelopingthisapproach,toavoidmodelswithinstabilitiesJérômerestricted to the quadratic operators leading at most to two derivatives in the equations of motion for the propagating degrees of freedom. However, he discovered that the quadraticoperatorssatisfyingthese properties were onemore than those needed to describe the so-called Horndeski theories. This came as a surprise, because such theories, developed by Horndeski in the seventies and recently rediscovered, were long believed to be the most general ones being free from dangerous instabilities. BycompletingtheLagrangianofthequadraticoperatorsatthefullnon-linearlevel, he was able to construct theories beyond Horndeski. These theories allow higher derivativesintheequationsofmotions.However,duetotheirdegeneracytheyonly contain propagating degrees offreedom whose order of derivatives is never higher thantwo,asrequiredforahealthytheorywithoutinstabilities.Thisdiscoveryledto a very rich activity in the literature, with many studies of the phenomenological consequences of theories beyond Horndeski and several theoretical developments on their extensions. In 2014, triggered by the exciting—but unfortunately incorrect—conclusions that the BICEP2 telescope had observed primordial gravitational waves from inflation, Jérôme turned to the study of inflationary predictions. As I mentioned above, current observations confirm inflation. However, incontestable evidence could only come from observing primordial tensor modes. In this context, Jérôme demonstratedthatthepredictionsforthegravitationalwavespectrumfrominflation arecompletelyrobust.Inparticular,despitesomeclaimsintheliterature,heshowed thatitisnotpossible toalterthestandard predictionsofinflationbymodifyingthe speedofpropagationoftensors.Incontrasttowhathappensforscalarfluctuations, a scale-invariant spectrum of tensor fluctuations can only arise in inflation. Thus, the measurement of the gravitational wave amplitude would unambiguously determine the energy scale of inflation. Finally, the last part of the thesis is devoted to the so-called consistency rela- tions, relations between correlation functions of the cosmic fields (e.g. the dark matter density contrast or the galaxy number density), valid in the limit in which one of the wavelength modes is much longer than the others. These relations are Supervisor’sForeword vii non-perturbative for the short-scale modes; in other words they automatically incorporate short-scale baryonic effects and the bias between galaxies and dark matter. For this reason, they can be employed as a test of standard cosmology. Moreover, Jérôme showed that since they are based on the equivalence principle, theirbreakingcanbeusedtotestthepresenceofafifthforce,whicharisesinsome modified gravity models. In conclusion, Jérôme’s thesis covers many aspects of modern cosmology. It provides an innovative and pedagogical introduction to each of them and contains cutting-edge results, rare tobe found in aunique researchmanuscript. It should be clearthatJérôme’sresearchhasalreadymadealargeimpactinthecommunityand will surely continue to do so in the future. Gif-sur-Yvette, France Dr. Filippo Vernizzi April 2016 Acknowledgements This Ph.D. has been a great adventure, where I interacted with a number of excellent people, for which I am extremely grateful. A special mention to the students at ICTP, like Marko and Gabriele, whom I visited numerous times. To Michele, I say thank you for sharing the work, writing notes, and being there to compare our codes. Ihavelearntalotfrommanygreatscientists.Inparticular,Iwouldliketothank ClaudiadeRham,AndrewTolley,Justin KhouryandMarkTroddenfor thetime I had with them where our interactions were very fruitful. I would also like to give many thanks to Pedro Ferreira and Tessa Baker. My visits to Oxford were always enriching. Finally, I would like to give special thanks to the people that played a crucial roleduringmyPh.D.andhaveshapedtheresearcherIamtoday:PaoloCreminelli, David Langlois, Federico Piazza. The most important of them is Filippo Vernizzi, myadvisor.Thanksforalwaysbeingtheretoanswermyquestions,fortreatingme like an equal, for giving me so many opportunities to travel and present our work, and for guiding me through many aspects of the world of physicists. ix Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Homogeneous Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Friedmann-Lemaître-Robertson-Walker Metric . . . . . . 2 1.1.2 Comoving Distance and Redshift . . . . . . . . . . . . . . . . . . 3 1.1.3 The Friedmann Equations. . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.4 Observations and the Discovery of Dark Energy. . . . . . . . 5 1.2 The Large Scale Structure of the Universe . . . . . . . . . . . . . . . . . 8 1.2.1 Growth of Perturbation in ΛCDM. . . . . . . . . . . . . . . . . . 10 1.2.2 Galaxy Surveys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 This Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 The Effective Field Theory of Dark Energy. . . . . . . . . . . . . . . . . . . 21 2.1 The Unitary Gauge Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 ADM Formalism and the Effective Field Theory of Dark Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Background Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 The Quadratic Action . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Going from Models to the EFT of DE. . . . . . . . . . . . . . . . . . . . 32 2.4 Stability and Theoretical Consistency. . . . . . . . . . . . . . . . . . . . . 35 2.5 Evolution of Cosmological Perturbations . . . . . . . . . . . . . . . . . . 37 2.5.1 Vector Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5.2 Tensor Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.3 Scalar Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Beyond Horndeski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1 Horndeski Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 General Considerations on Higher Order Derivatives . . . . . . . . . . 55 xi