Modified Gravity with Torsion Carina Suzana Negreanu Queens’ College 2018 September A dissertation submitted for the degree of Doctor of Philosophy Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. Throughout this dissertation the plural pronoun ‘we’ is used for stylistic reasons and should be taken to refer to either the singular author, or to the author and her thesis advisors. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. It does not exceed the prescribed word limit (60,000 words) for the relevant Degree Committee. iii Summary Title: Modified Gravity with Torsion Author: Carina Suzana Negreanu We are at a point in time when alternative gravitational theories are beginning to be constrained by high precision cosmological and astrophysical data. The work in this thesis focuses on applications of a particular modified gravity theory, the extended Weyl Gauge Theory (eWGT), that was recently developed by Prof. Lasenby and Prof. Hobson. The applications corresponds to subsets of the full theory that are applicable to different cosmological and astrophysical sectors. We start by investigating a simplified scenario that simulates a famous alternative theory of gravity, Weyl2 Gravity. Recently a couple of issues have been raised regarding the validity of the theory. Starting from a gauge theory perspective we bring a fresh contribution to the debate. We argue against the classical formulation by showing that the theory cannot support astrophysical matter (introduced by a perfect fluid). Furthermore we extend the theory and show that even if we allow torsion to be present we cannot reach a physical setup. In this process we have discovered interesting properties of the torsion field that could play an important role in generalised cosmological setups. In the next application we consider a cosmology dictated by a Riemann2 Lagrangian that can accommodate only radiation. We find new physical behaviour in the perturbed regime that discretises the power spectrum. We prove that the setup admits gravitational waves. Finally, we construct a new Lagrangian theory for spinning fluids. We show that it is compatible with current literature for a flat space time. Considering its extensibility we believe that it can be widely used in future research. v Acknowledgements This work would not have been possible without the help of a lot of people. I would like to start by thanking my supervisors, Prof. Anthony Lasenby and Prof. Mike Hobson, for the continuous support and encouragement. When I started my PhD I thought that the measure of success was given by the number of citations. My supervisors taught me that there are significantly more dimensions and helped me become an independent researcher that values scientific integrity. I am proud to say that I have been part of the Cavendish Astrophysics Group. I got the chance to meet great scientists who take pride in their work and are diligently trying to improve the world. I would like to thank everyone for the great conversations that have undeniably changed my worldview. My PhD experience would not be complete without the great times I spent with my of- ficemates Do Young Kim, Yun-Cherng Lin, Sonke Hee, Zoe Ye and ‘honorary’ officemates Lukas Hergt, Ed Higson and Will Handley. I will always cherish our endless conversations on politics, chinese calligraphy, ethics, gastronomy and math. Another thesis could have been written from all the encouraging messages my friends have sent me throughout the years. Do Young Kim and Jordan Boddy have helped me through the hardest times and I hope that one day I can return the favour. Last but by no means least, thanks go to my mom, my dad and my husband for almost unbelievable support. This thesis is dedicated to them. vii Contents Declaration iii Summary v Acknowledgements vii Contents ix 1 Introduction1 1.1 A brief encounter with Gauge Theories of Gravity ............... 4 1.2 Torsion Review.................................. 10 2 Weyl-squared Gravity 17 2.1 Introduction.................................... 17 2.2 A different perspective .............................. 23 2.3 Our framework .................................. 24 2.4 Looking at vacuum................................ 28 2.5 Adding Matter................................... 32 2.6 Introducing Torsion................................ 42 2.7 Cosmological Torsion............................... 44 2.8 Real Torsion ................................... 51 2.9 Conclusion .................................... 65 3 Riemann-squared Gravity 67 3.1 Introduction.................................... 67 3.2 Perturbation theory in Flat-Λ cosmology..................... 74 3.3 Perturbation Theory in Riemann-squared Gravity ................ 80 3.4 Gravitational Waves ............................... 97 3.5 Conclusion ....................................102 4 A new Lagrangian formalism for a perfect fluid with spin 105 4.1 Introduction....................................105 4.2 Non-spinning relativistic fluids..........................108 4.3 Spinning relativistic fluids ............................113 4.4 Applying the Theory ...............................125 4.5 Conclusions....................................136 5 Conclusions and Future Work 137 ix A Derivations for the Spinning Fluid Formalism 141 A.1 Looking at S˙(D˙ ) .................................141 A.2 Equations for the Spin bivector..........................142 A.3 Identity for Riemann construction ........................144 A.4 Identity for the Ω equation ............................145 Bibliography 147 x Chapter 1 Introduction The General Theory of Relativity (GR) has undoubtedly revolutionised our understanding of the Universe and from 1915 until this day Einstein’s field equations are still the norm when it comes to describing the behaviour of space-time on macroscopic scales. The considerable success of the theory has not stopped alternatives being proposed in order to overcome the worrying shortcomings (e.g. the need for Dark Matter and Dark Energy in our cosmological model) and in time a class of theories labelled “modified gravity theories” emerged. A more formal definition can be found in [1]: “ the effect of gravity on matter is tightly constrained to be mediated by interactions of the matter fields with a single rank-2 tensor. The term “gravitational theory” can then by functionally defined by the set of field equations obeyed by the rank-2 tensor, and any other non-matter fields it interacts with. If these equations are anything other than Einstein’s equations, then it is considered to be a modified theory of gravity”. There is a vast number of theories that satisfy this definition and each have merits and shortcomings, without an obvious candidate to replace (or generalize) Einstein’s theory. In our work we actually employ a theory that lies outside this sphere, where we describe gravity as a gauge field and our interactions are not defined by a single rank-2 tensor. To ensure some uniformity we start this thesis by making a general introduction to the main considerations behind theories of gravity. This description will be heavily based on a recent review of theories of gravitation, [1] and the staple book in Gauge Theories of Gravity [2]. 1 2 Chapter 1. Introduction The premise behind theories of gravity Any viable relativistic theory of gravity has to satisfy the foundational requirements (e.g. universality of free fall and isotropy of space) as well as be compatible with observational results. In this section we will briefly outline the current results and their importance in formulating a theory of gravity. We refer the reader to [1] for a more detailed description. When looking at the foundational requirements the equivalence principle plays an important role. In the following, we will employ the definitions of Weak Equivalence Principle (WEP) - once an initial position and velocity have been prescribed all uncharged, freely falling test particles follow the same trajectories; Einstein Equivalence Principle (EEP) - the WEP is valid and “the outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.” (Einstein); Strong Equivalence Principle (SEP) - the EEP is valid and “the gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitu- tion.”(Einstein) From Eo¨tvo¨s-type experiments we have a tight constraint on the relative difference in accelerations of two bodies to be of order 10−13 [3], which validates the Weak Equivalence Principle. Moreover, it is worth noting that by having a theory that satisfies the WEP the test proposed by Einstein in 1916 involving the gravitational redshifting of light is automatically passed: since energy-momentum is conserved in a closed system then it is only a test of WEP, and the Eo¨tvo¨s-type experiments