Doctoral Thesis in Theoretical Physics Classically Consistent Theories of Interacting Spin-2 Fields Angnis Margreta Schmidt-May Oskar Klein Centre for Cosmoparticle Physics and Cosmology, Particle Astrophysics and String Theory Department of Physics Stockholm University SE-106 91 Stockholm Stockholm, Sweden 2013 Cover image: Still life of partially massless Bimetric Theory. ISBN 978-91-7447-791-7 (pp. i{xx, 1{134) pp. i{xx, 1{134 c Angnis Schmidt-May, 2013 Printed by Universitetsservice US-AB, Stockholm, Sweden, 2013. Typeset in pdfLATEX Daß ich erkenne, was die Welt im Innersten zusammenh¨alt. J. W. von Goethe in Faust To my parents. iii iv Abstract This doctoral thesis comprises a study of recently developed nonlinear theories for massive spin-2 fields that go by the names of Massive Gravity and Bimetric Theory. As we demonstrate in a Hamiltonian analysis, the construction of these theories avoids the notorious Boulware-Deser ghost instability that generically plagues nonlinear spin-2 interactions and renders their descrip- tion inconsistent. We explore the mass spectrum of ghost-free Bimetric Theory and reformulate its action in terms of nonlinear mass eigenstates. A generalized version of the theory with multiple vielbein interactions is reviewed and then reexpressed in terms of metrics. Homogeneous and isotropic solutions to the equations of motion in Bimetric Theory are de- rived and discussed in the context of cosmology. We furthermore single out particular values for the parameters in the bimetric interaction po- tential for which the theory could potentially contain an additional gauge redundancy. This establishes a unique candidate for a nonlinear theory of a partially massless spin-2 field. Finally, we reveal a close connection of Bimetric Theory to a certain class of higher-curvature theories for gravity and offer a solution to their ghost problems. In particular we find that, in four space-time dimensions, the bimetric candidate for nonlinear partial masslessness is closely related to Conformal Gravity. v vi Svensk sammanfattning Denna doktors-avhandling utg¨orsav studier inom de nyligen konstruer- ade icke-linj¨arateorierna f¨ormassiva spin-2 f¨alt,Massiv gravitation samt Bimetrisk teori. Vi visar genom Hamiltonsk analys att dessa teorier ¨arfria fr˚andet s˚akallade Boulware-Deser sp¨oket, som annars generiskt ˚aterfinns i teorier f¨oricke-linj¨araspin-2 v¨axelverkningar och d¨arf¨org¨ors˚adana teorier inkonsistenta. Vi utforskar den bimetriska teorins masspektrum och omformulerar sedan dess verkansfunktional i termer av icke-linj¨ara massegentillst˚and.En ¨overblick ges ¨over konstruktionen av en generalis- ering av den bimetriska teorin med multipla vielbeins och visar n¨aroch hur denna teori ist¨allet kan formuleras med metriker. Vi finner ¨aven ho- mogena och isotropiska l¨osningar i den bimetriska teorin och diskuterar dessa ur ett kosmologiskt perspektiv. F¨orett speciellt val av parame- trarna i den bimetriska v¨axelverkanspotentialen visar vi att en ny gauge- symmetry kunde vara realiserad och den resulterande modellen skulle s˚aledesutg¨oraen icke-linj¨arteori f¨orett partiellt massl¨ostspin-2 f¨alt. Slutligen p˚avisarvi en stark koppling mellan den bimetriska teorin och en klass av h¨ogre-derivata teorier och kan d¨arigenomerbjuda en l¨osning p˚ade senares sp¨ok-problem. Speciellt finner vi att den bimetriska kan- didaten f¨oren partiellt massl¨osteori har en stark koppling till Konform gravitation. vii viii List of Accompanying Papers Paper I S.F. Hassan, R. Rosen & A. Schmidt{May. Ghost-free Mas- sive Gravity with a General Reference Metric, JHEP 02, 026 (2012) arXiv:1109.3230. Paper II M. von Strauss, A. Schmidt{May, J. Enander, E. M¨ortsell& S.F. Hassan. Cosmological Solutions in Bimetric Gravity and their Observational Tests, JCAP 03, 042 (2012) arXiv:1111.1655. Paper III S.F. Hassan, A. Schmidt{May & M. von Strauss. Proof of Consistency of Nonlinear Massive Gravity in the St¨uckelberg For- mulation, Phys. Lett. B 715, 335-339 (2012) arXiv:1203.5283. Paper IV S.F. Hassan, A. Schmidt{May & M. von Strauss. Metric Formulation of Ghost-Free Multivielbein Theory, To be submitted, arXiv:1204.5202. Paper V S.F. Hassan, A. Schmidt{May & M. von Strauss. On Consis- tent Theories of Massive Spin-2 Fields Coupled to Gravity, JHEP 05, 086 (2013) arXiv:1208.1515. Paper VI S.F. Hassan, A. Schmidt{May & M. von Strauss. On Par- tially Massless Bimetric Gravity, Phys. Lett. B 726, 834-838 (2013) arXiv:1208.1797. Paper VII S.F. Hassan, A. Schmidt{May & M. von Strauss. Bimet- ric Theory and Partial Masslessness with Lanczos-Lovelock Terms in Arbitrary Dimensions, Class. Quant. Grav. 30, 184010 (2013) arXiv:1212.4525. Paper VIII S.F. Hassan, A. Schmidt{May & M. von Strauss. Higher Derivative Gravity and Conformal Gravity From Bimetric and Par- tially Massless Bimetric Theory, To be submitted, arXiv:1303.6940. ix x Acknowledgments I am grateful to my advisor, Sayed Fawad Hassan, for his support, his patience and his encouraging guidance. Thank you for giving me the opportunity to work in an extremely interesting and active field of research and for always taking the time for explanations and discussions. This work would not have been possible without my other collabora- tors, Mikael von Strauss as well as Jonas Enander, Edvard M¨ortselland Rachel Rosen, whom I thank for sharing their knowledge and enthusiasm with me. Furthermore, I would like to express my gratitude to Marcus Berg and Igor Buchberger at Karlstad University for their hospitality and for giving me the opportunity to work on a subject unrelated to my PhD research. My thanks also go to my former office mates, Irina Galstyan and Stefan Sj¨ors,for numerous interesting conversations and for being supportive friends. Moreover, I owe gratitude to all my colleagues in the CoPS group and the Oskar-Klein-Center for providing a constantly inspiring and friendly environment. Finally, I thank Anja Fath, Dominik Werder as well as my family, Hilke, Jenss and Alice Schmidt-May for their love and support and for being there when I need them to take my mind away from physics. xi xii Preface Thesis plan This doctoral thesis consists of two major parts: a detailed presentation of the scientific results gathered during my PhD education and the cor- responding eight articles published or submitted for publication. Instead of discussing the papers in strict chronological order of their appearance, I try to present their results in the logically most coherent way. The first part of the thesis is further divided into eight chapters whose contents are briefly summarized in the following. Chapter1 motivates this work both from a cosmological and a field ∗ theoretical point of view and furthermore contains a short introduc- tion into the subject of massless and massive spin-2 fields. Chapter2 introduces the recently developed ghost-free nonlinear ∗ theory for massive spin-2 fields and outlines its consistency proofs contained in Paper I and Paper III. Chapter3 discusses the more general setup of ghost-free Bimetric ∗ Theory. In particular, the focus lies on its mass spectrum and its formulation in terms of nonlinear mass eigenstates as presented in Paper V and Paper VII. Chapter4 reviews the consistent theory for multiple interacting ∗ spin-2 fields in terms of vielbeins and develops an equivalent de- scription in terms of metrics, following Paper IV. Chapter5 derives the homogeneous and isotropic solutions in Bi- ∗ metric Theory found in Paper II and summarizes their implications for cosmology. xiii Chapter6 singles out a special parameter choice for which the bi- ∗ metric equations of motion can exhibit an extra gauge symmetry. This analysis, which was first performed in Paper V, identifies a candidate for the nonlinear theory of a partially massless spin-2 field in four space-time dimensions. Furthermore, the chapter re- views Paper VII which generalizes the results to higher dimensions by including Lanczos-Lovelock terms into the bimetric action. Chapter7 summarizes the results of Paper VIII which demonstrates ∗ the close relation between ghost-free Bimetric Theory and a cer- tain class of higher-curvature theories for gravity. In particular, a connection between the nonlinear partially massless theory and Conformal Gravity is drawn. Chapter8 contains a summary of the main results and gives an ∗ outlook on possible future developments in the field. AppendixA states the geometric conventions employed in this work ∗ along with some identies used frequently in the computations. xiv Contribution to papers Paper I I was involved in the discussions that lead to the development of the con- sistency proof and checked all calculations independently myself. Paper II The majority of the work was done by the article's first author. The computations were checked analytically by me and I contributed actively to the discussions of the results. I also wrote an appendix. Paper III The proof was developed in discussions to which all authors contributed equally. I wrote most of the text. Paper IV Most of the calculations were performed in collaboration by all authors at the board; a large part of the article was written by me. Paper V The computations were done by Mikael von Strauss and me. The writing of the text was split equally among the authors. Paper VI I played a major role in developing the main idea in this article and also wrote most of the text. Paper VII Together with Mikael von Strauss, I performed the analytical calculations for this paper. I also wrote a large part of the paper's second half. Paper VIII After developing the idea for this work together, all three authors carried out the computations independently. I wrote up most of the technical parts. Angnis Schmidt-May Stockholm, September 2013 xv xvi Contents Abstractv Svensk sammanfattning vii List of Accompanying Papers ix Acknowledgments xi Preface xiii Contents xvii I Detailed Presentation of Scientific Results1 1 Introduction3 1.1 Historical development....................5 1.2 The cosmological constant problem.............6 1.3 Massless and massive field theories.............8 1.3.1 Spin-1........................