MODIFIED ACTIONS FOR GRAVITY: THEORY AND PHENOMENOLOGY Thesis submitted for the degree of “Doctor Philosophiæ” October 2007 CANDIDATE SUPERVISORS Thomas P. Sotiriou Stefano Liberati and John Miller International School for Advanced Studies Via Beirut 2-4, 34014 Trieste, Italy. E-mail: [email protected] ii iii The important thing is not to stop questioning. Albert Einstein iv Abstract This thesis is devoted to the study of gravitational theories which can be seen as modifications or generalisations of General Relativity. The motivation for con- sidering such theories, stemming from Cosmology, High Energy Physics and As- trophysics is thoroughly discussed (cosmological problems, dark energy and dark matter problems, the lack of success so far in obtaining a successful formulation for Quantum Gravity). The basic principles which a gravitational theory should follow, and their geometrical interpretation, are analysed in a broad perspective which highlights the basic assumptions of General Relativity and suggests possi- ble modifications which might be made. A number of such possible modifications are presented, focusing on certain specific classes of theories: scalar-tensor theo- ¢¡¤£¦¥ ¢¡¤£¦¥ ries, metric ¢¡¤£¦¥ theories, Palatini theories, metric-affine theories and Gauss–Bonnet theories. The characteristics of these theories are fully explored and attention is payed to issues of dynamical equivalence between them. Also, cosmo- logical phenomenology within the realm of each of the theories is discussed and it is shown that they can potentially address the well-known cosmological problems. A number of viability criteria are presented: cosmological observations, Solar Sys- tem tests, stability criteria, existence of exact solutions for common vacuum or matter configurations etc. Finally, future perspectives in the field of modified grav- ity are discussed and the possibility for going beyond a trial-and-error approach to modified gravity is explored. v vi Collaborations The research presented in this thesis was mainly conducted in SISSA-International School for Advanced Studies between November 2004 and October 2007. This thesis is the result of the authors own work, as well as the outcome of scientific collaborations stated below, except where explicit reference is made to the results of others. The content of this thesis is based on the following research papers published in refereed Journals or refereed conference proceedings: 1. “The nearly Newtonian regime in Non-Linear Theories of Gravity” T. P. Sotiriou Gen. Rel. Grav. 38 1407 (2006) [arXiv:gr-qc/0507027] 2. “Unification of inflation and cosmic acceleration in the Palatini formal- ism” T. P. Sotiriou Phys. Rev. D 73, 063515 (2006) [arXiv:gr-qc/0509029] 3. “Constraining f(R) gravity in the Palatini formalism” T. P. Sotiriou Class. Quant. Grav. 23, 1253 (2006) [arXiv:gr-qc/0512017] 4. “Metric-affine f(R) theories of gravity” T. P. Sotiriou and S. Liberati Ann. Phys. 322, 935 (2007) [arXiv:gr-qc/0604006] 5. “f(R) gravity and scalar-tensor theory” T. P. Sotiriou Class. Quant. Grav. 23, 5117 (2006) [arXiv:gr-qc/0604028] 6. “The metric-affine formalism of ¢¡¤£¦¥ gravity” T. P. Sotiriou and S. Liberati J. Phys. Conf. Ser. 68, 012022 (2007) [arXiv:gr-qc/0611040] Talk given (by T. P. S.) at the 12th Conference on Recent Developments in Gravity (NEB XII), Nafplio, Greece, 29 Jun-2 Jul 2006 vii viii 7. “Curvature scalar instability in f(R) gravity.” T. P. Sotiriou, Phys. Lett. B 645, 389 (2007) [arXiv:gr-qc/0611107] 8. “The significance of matter coupling in ¢¡¤£¦¥ gravity” T. P. Sotiriou, Proceedings of the 11th Marcel Grossman Meeting in press Talk given at the 11th Marcel Grossman Meeting, Berlin, Germany, 23-29 Jul 2006 9. “Post-Newtonian expansion for Gauss-Bonnet gravity.” T. P. Sotiriou and E. Barausse, Phys. Rev. D 75, 084007 (2007) [arxiv:gr-qc/0612065] 10. “A no-go theorem for polytropic spheres in Palatini f(R) gravity.” E. Barausse, T. P. Sotiriou and J. C. Miller, Submitted to Phys. Rev. Lett. [arXiv:gr-qc/0703132] 11. “Theory of gravitation theories: a no-progress report.” T. P. Sotiriou, V. Faraoni and S. Liberati, Submitted to Int. J. Mod. Phys. D [arXiv:0707.2748 [gr-qc] ] Invited paper in the Special Issue: Invited Papers and Selected Essays from the Annual Essay Competition of the Gravity Research Foundation for the Year 2007 Notation An attempt has been made to keep the basic notation as standard as possible. How- ever, the use of non-metric connections did require the use of some non-standard notation. The following list will hopefully be a useful tool for clarifying these non- standard notation. In general, the notation, standard or not, is always defined at its first occurence in the text and in all places that ambiguities may arise, irrespec- tivelly of whether it has been included in this guide. The signature of the metric ¡¡ ¥ £ £ £ ¢ ¢ ¢ ¥ is assumed to be and the speed of light ¤ is taken to be equal to throughout this thesis. In order to lighten the notation, in some cases a coordinate ¦ system is used in which ¦¨§©¤ § ¥ , where is Newton’s gravitational constant. ¥ However, for clarity ¦ is not set to be equal to throught the text. : Lorentzian metric : Determinant of : General Affine Connection : Levi–Civita Connection : Covariant derivative with respect to : Covariant derivative with respect to ¥ ¡ : Symmetrization over the indices and : Anti-symmetrization over the indices and ! " : Non-metricity ( ) # %$ " : Cartan torsion tensor ( ) '& £ )( ( : Riemann tensor of £ £ ( " : Ricci tensor of ( ) £ £ " : Ricci scalar of ( ) ( *+ ( : Riemann tensor constructed with * ( * " : * * " : #-, : Matter action 0 . 398': ; 1 2435¡67 /" : Stress-energy tensor 5 < 0 ; 5¡67 1 243 /" : Hypermomentum 8': 5>=? @ : Scalar field (generic) A : Matter fields (collectively) ix x Preface The terms “modified gravity” and “alternative theory of gravity” have become stan- dard terminology for theories proposed for describing the gravitational interaction which differ from the most conventional one, General Relativity. Modified or al- ternative theories of gravity have a long history. The first attempts date back to the 1920s, soon after the introduction of Einstein’s theory. Interest in this research field, which was initially driven by curiosity or a desire to challenge the then newly introduced General Theory of Relativity, has subsequently varied depending on circumstances, responding to the appearance of new motivations. However, there has been more or less continuous activity in this subject over the last 85 years. When the research presented in this thesis began, interest in modified gravity was already at a high point and it has continued increasing further until the present day. This recent stimulus has mostly been due to combined motivation coming from the well-known cosmological problems related to the accelerated expansion of the universe and the feedback from High Energy Physics. Due to the above, and even though the main scope of this thesis is to present the research conducted by the author during the period November 2004 - Octo- ber 2007, a significant effort has been made so that this thesis can also serve as a guide for readers who have recently developed an interest in this field. To this end, special attention has been paid to giving a coherent presentation of the mo- tivation for considering alternative theories of gravity as well as to giving a very general analysis of the foundations of gravitation theory. Also, an effort has been made to present the theories discussed thoroughly, so that readers less familiar with this subject can be introduced to them before gradually moving on to their more complicated characteristics and applications. The outline of this thesis is as follows: In the Introduction, several open is- sues related to gravity are discussed, including the cosmological problems related to dark matter and dark energy, and the search for a theory of Quantum Gravity. Through the presentation of a historical timeline of the passage from Newtonian gravity to General Relativity, and a comparison with the current status of the latter in the light of the problems just mentioned, the motivations for considering alter- native theories of gravity are introduced. Chapter 2 is devoted to the basic princi- ples which gravitation theories should follow. The Dicke framework, the various forms of the Equivalence Principle and the so-called metric postulates are criti- cally reviewed and the assumptions that lead to General Relativity are examined. xi xii PREFACE Additionally, the ways of relaxing these assumptions are explored together with the resulting theories. In Chapter 3, we focus on specific theories: scalar-tensor theory, metric, Palatini and metric-affine ¢¡¤£¦¥ gravity and Gauss–Bonnet gravity, and their theoretical characteristics are thoroughly presented. Chapter 4 contains a discussion about the possible dynamical equivalence between these theories, while in Chapter 5 their cosmological phenomenology is presented. Attention is paid to their ability to address the well-known cosmological problems and to their cosmo- logical viability. Chapter 6 is devoted to the study of the weak and strong gravity regimes in these modified theories of gravity. The Newtonian and post-Newtonian limits, stability issues, non-vacuum solutions, etc. are discussed as criteria for the viability of these theories. Finally, Chapter 7 contains the conclusions of this work, as well as suggestions and remarks about future work in the field of modified grav- ity. A number of people have contributed in this thesis in various ways. First and foremost, I would like to thank my PhD advisors, Stefano Liberati and John Miller, for their constant support during the course of this work. It is difficult for me to imagine having better advisors than Stefano and John, to whom I am truly grateful, not only for their guidance but also for standing by me in all my choices and for the impressive amount of patience they have exhibited during the course of these three years. Special thanks to John for his untiring correction of my spelling, grammar and (ab)use of the English language.