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The Pennsylvania State University the Graduate School Eberly College of Science The Pennsylvania State University The Graduate School Eberly College of Science GRAVITATIONAL RADIATION WITH A POSITIVE COSMOLOGICAL CONSTANT A Dissertation in Physics by Béatrice Bonga © 2017 Béatrice Bonga Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 The dissertation of Béatrice Bonga was reviewed and approved∗ by the following: Abhay Ashtekar Eberly Professor of Physics Dissertation Advisor, Chair of Committee Eugenio Bianchi Assistant Professor of Physics Sarah Shandera Assistant Professor of Physics Ping Xu Professor of Mathematics Richard Robinett Professor of Physics Director of Graduate Studies ∗Signatures are on file in the Graduate School. ii Abstract Gravitational radiation is well-understood in spacetimes that are asymptotically flat. However, our Universe is currently expanding at an accelerated rate, which is best described by including a positive cosmological constant, Λ, in Einstein’s equations. Consequently, no matter how far one recedes from sources generating gravitational waves, spacetime curvature never dies and is not asymptotically flat. This dissertation provides first steps to incorporate Λ in the study of gravitational radiation by analyzing linearized gravitational waves on a de Sitter background. Since the asymptotic structure of de Sitter is very different from that of Minkowski spacetime, many conceptual and technical difficulties arise. The limit Λ → 0 can be discontinuous: Although energy carried by gravitational waves is always positive in Minkowski spacetime, it can be arbitrarily negative in de Sitter spacetime. Additionally, many of the standard techniques, including 1/r expansions, are no longer applicable. We generalize Einstein’s celebrated quadrupole formula describing the power radiated on a flat background to de Sitter spacetime. Even a tiny Λ brings in qualitatively new features such as contributions√ from pressure quadrupole moments. Nonetheless, corrections induced by Λ are O( Λtc) with tc the characteristic time scale of the source and are negligible for current gravitational wave observatories. We demonstrate this explicitly for a binary system in a circular orbit. Radiative modes are encoded in the transverse-traceless part of the spatial components of a gravitational perturbation. When Λ = 0, one typically extracts these modes in the wave zone by projecting the gravitational perturbation onto the two-sphere orthogonal to the radial direction. We show that this method for waves emitted by spatially compact sources on Minkowski spacetime generically does not yield the transverse-traceless modes; not even infinitely far away. However, the difference between the transverse-traceless and projected modes is non-dynamical and disappears from all physical observables. When one is interested in ‘Coulombic’ information not captured by the radiative modes, the projection method does not suffice. This is, for example, important for angular momentum carried by gravitational waves. This result relies on Bondi-type expansions for asymptotically flat spacetimes. Therefore, the projection method is not applicable to de Sitter spacetimes. iii Table of Contents List of Figures vi Preamble vii Acknowledgments viii Chapter 1 Introduction 1 1.1 Gravitational waves in our expanding Universe . 1 1.2 Even a tiny Λ can cast a long shadow . 6 1.2.1 Linearized gravity off de Sitter spacetime . 10 Chapter 2 General formula for power emitted in de Sitter spacetimes 17 2.1 Introduction . 17 2.2 Preliminaries . 20 2.3 The retarded solution and quadrupole moments . 26 2.3.1 The late time and post-Newtonian approximations . 27 2.3.2 Expressing the approximate solutions in terms of quadrupole moments . 32 2.3.3 The tail term and its properties . 35 2.4 Time-varying quadrupole moment and energy flux . 40 + 2.4.1 I and the perturbed electric part Eab of Weyl curvature . 40 2.4.2 Fluxes across I+ ........................ 43 2.4.3 Properties of fluxes across I+ . 47 2.5 Discussion . 54 Chapter 3 Power radiated by a binary system in a de Sitter Universe 60 3.1 Introduction . 60 3.2 Preliminaries: Physical set-up . 62 iv 3.3 Power radiated . 65 3.4 Discussion . 70 Chapter 4 On the conceptual confusion in the notion of transversality 72 4.1 Introduction . 72 4.2 Null infinity and the Peeling behavior . 76 4.2.1 Future null infinity I+ ..................... 77 4.2.2 Peeling . 78 4.3 Asymptotic conditions on potentials . 80 T t 4.4 Aa versus Aa .............................. 86 4.4.1 The two notions . 86 4.4.2 Comparison . 88 4.4.3 Fluxes of energy-momentum and angular momentum . 89 4.4.4 Soft charges and electromagnetic memory . 91 4.4.5 Summary . 92 4.5 Discussion . 94 Chapter 5 Conclusion and outlook 97 5.1 Summary . 98 5.2 Looking forward . 101 Appendix A Transverse-traceless decomposition of the quadrupole moment 105 A.1 General properties TT decomposition . 105 A.2 Transverse-traceless decomposition of the quadrupole moment . 108 Appendix B Point charges and dipoles 112 Appendix C Various investigations in early Universe cosmology 116 C.1 Spatially closed Universe . 118 C.2 Robustness of predictions of a quantum gravity extension to inflation 121 C.3 Cosmic variance in quasi-single field inflation . 125 Bibliography 130 v List of Figures 1.1 Contrasting the conformal diagram of an asymptotically flat space- time with an asymptotically de Sitter spacetime . 11 1.2 Illustration of the time-like and space-like character of the de Sitter time translation . 14 2.1 A time changing quadrupole emitting gravitational waves in de Sitter spacetime . 21 3.1 Physical set-up of a binary system in a circular orbit with fixed physical distance . 66 3.2 Embedding of a binary system living on a de Sitter background in Minkowski spacetime . 67 3.3 Relation between power on I and the cosmological horizon in a de Sitter spacetime . 71 C.1 Ratio of the power spectrum of primordial fluctuations in the closed FLRW model with Ωk = −0.005 to that in the flat model . 120 C.2 Deviation from the consistency relation (r + 8nt = 0) for a spatially closed Universe with Ωk = −0.005 . 121 C.3 Comparison of LQC corrections to the tensor-to-scalar ratio between the Starobinsky and quadratic potential . 124 vi Preamble “At this point we must say a few words about the famous lambda. The field-equations, in their most general form, contain a term multiplied by a constant, which is denoted by the Greek letter λ (lambda), and which is sometimes called the “cosmological constant”. This is a name without any meaning, which was only conferred upon because it was thought appropriate that it should have a name, and because it appeared to have something to do with the constitution of the universe; but it must not be inferred that, since we have given it a name, we know what it means. We have, in fact, not the slightest inkling of what its real significance is. It is put in the equations in order to give the greatest possible degree of mathematical generality, but, so far as its mathematical function is concerned, it is entirely undetermined: it may be positive or negative, it might also be zero. Purely mathematical symbols have no meaning by themselves; its is the privilege of pure mathematician, to quote Bertrand Russell, not to know what they are talking about. They — the symbols — only get a meaning by the interpretation that is put on the equations when they are applied to the solution of physical problems. It is the physicist, and not the mathematician, who must know what he is talking about. ” Willem de Sitter, in Kosmos, a course of six lectures on the development of our insight into the structure of the Universe delivered for the Lowell Institute in Boston in November 1931 [1] vii Acknowledgments My dissertation would not be complete without thanking all the wonderful people who made this venture possible. They did not only contribute to the completion of this dissertation, but they made the process and my years at Penn State more instructive, inspiring and enjoyable than I could have ever imagined. First and foremost, I would like to thank my advisor Abhay Ashtekar. As a researcher, his broad knowledge of physics is admirable while his enthusiasm and passion for the field have been nothing short of infectious. As a supervisor, he has guided me through this journey with incredibly valuable feedback combined with exceptional kindness and patience. His academic professionalism has provided me with a great example: for I cannot help but aspire to continuously push myself to become a better researcher. I would also like to thank the other members of my committee, consisting of Eugenio Bianchi, Ping Xu and Sarah Shandera. Eugenio often informed me about the latest arXiv papers or interesting facts through regular chats during lunch and social events, while Ping served as a valuable outside member. Sarah broadened my horizon through our collaboration on cosmic variance in quasi-single field inflation. Her cosmological point of view added to my overall understanding of physics. Another collaborator whom I ought to thank is Aruna Kesavan. I have had the good fortune of working with her extensively and I fondly look back at the great team we formed. Aruna was dearly missed during the final year of my PhD. In the first year of my PhD, I connected with Brajesh Gupt during a conference in Warsaw. Initially we were merely acquaintances, but due to a twist in fate he became my next-door-office member and as result of many enticing discussions one of my collaborators. I am not only thankful for the many physics discussions we had, but I am also grateful for the times we spend talking about travel, food, culture and brewing beer. In a later stage, we were joined by Nelson Yokomizo whose broad knowledge helped to a better understanding of many physics concepts.
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