This Thesis Has Been Submitted in Fulfilment of the Requirements for a Postgraduate Degree (E.G
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This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. Testing Gravity In The Local Universe Ryan McManus I V N E R U S E I T H Y T O H F G E R D I N B U Doctor of Philosophy The University of Edinburgh May 2018 2 Abstract General relativity (GR) has stood as the most accurate description of gravity for the last 100 years, weathering a barrage of rigorous tests. However, attempts to derive GR from a more fundamental theory or to capture further physical principles at high energies has led to a vast number of alternative gravity theories. The individual examination of each gravity theory is infeasible and as such a systematic method of examining modified gravity theories is a necessity. Studying generic classes of gravity theories allows for general statements about observables to be made independent of explicit models. Take, for example, those models described by the Horndeski action, the most general class of scalar- tensor theory with at most second-order derivatives in the equations of motion, satisfying theoretical constraints. But these constraints alone are not enough for a given modified gravity model to be physically viable and hence worth studying. In particular, observations place incredibly tight constraints on the size of any deviation in the solar system. Hence, any modified gravity would have to mimic GR in such a situation. To accommodate this requirement, many models invoke screening mechanisms which suppress deviations from GR in regions of high density. But these mechanisms really upon non-linear effects and so studying them in complex models is mathematically complex. To constrain the space of actions of Horndeski type to those which pass solar- system tests, a set of conditions on the four free functions of the Horndeski action are derived which indicate whether a specific model embedded in the action possesses a GR limit. For this purpose, a new and surprisingly simple scaling method is developed, identifying dominant terms in the equations of motion by considering formal limits of the couplings that enter through the new terms in the modified gravity action. Solutions to the dominant terms identify regimes where nonlinear terms dominate and Einstein’s field equations are recovered to leading order. Together with an efficient approximation of the scalar field profile, one can i determine whether the recovery of Einstein’s field equations can be attributed to a genuine screening effect. The parameterised post-Newtonian (PPN) formalism has enabled stringent tests of static weak-field gravity in a theory-independent manner. This is through parameterising common perturbations of the metric found when performing a post-Newtonian expansion. The framework is adapted by introducing an effective gravitational coupling and defining the PPN parameters as functions of position. Screening mechanisms of modified gravity theories can then be incorporated into the PPN framework through further developing the scaling method into a perturbative series. The PPN functions are found through a combination of the scaling method with a post-Newtonian expansion within a screened region. For illustration, we show that both a chameleon and cubic galileon model have a limit where they recover GR. Moreover, we find the effective gravitational constant and all PPN functions for these two theories in the screened limit. To examine how the adapted formalism compares to solar-system tests, we also analyse the Shapiro time delay effect for these two models and find no deviations from GR insofar as the signal path and the perturbing mass reside in a screened region of space. As such, tests based upon the path light rays such as those done by the Cassini mission do not constrain these theories. Finally, gravitational waves have opened up a new regime where gravity can be tested. To this end, we examine how the generation of gravitational waves are affected by theories of gravity with screening to second post-Newtonian (PN) order beyond the quadrupole. This is done for a model of gravity where the black hole binary lies in a screened region, while the space between the binary’s neighbourhood and the detector is described by Brans-Dicke theory. We find deviations at both 1.5 and 2 PN order. Deviations of this size can be measured by the Advanced LIGO gravitational wave detector highlighting that our calculation may allow for constraints to be placed on these theories. We model idealised data from the black hole merger signal GW150914 and perform a best fit analysis. The most likely value for the un-screened Brans-Dicke parameter is found to be ! = −1:42, implying on large scales gravity is very modified, incompatible with cosmological results. ii Lay Summary The scientific study of gravity spans centuries, with work stretching from Galileo to Newton resulting in the development of the classical theory of gravity. The classical theory stood strong until the discovery of electrodynamics, highlighting problems with Newtonian gravity, namely the absolute nature of space and time. These problems were overcome by Einstein. First, the discovery of special relativity describing how both time and space where intertwined. This is through different observers, who when traveling at different speeds would measure the length of a given rod and the time taken for a given clock to tick differently from each other. However, they would both agree on a function of both length and time, called the proper time. The proper time is the time taken for the clock to tick if one was not moving relative to it, hence the name. This conception of space-time proved fundamental to formulating a theory of gravity which melded with special relativity, called general relativity (GR). The intrinsic idea of GR is that space is curved and observers travel along the straight lines of curved space. But an observer at cannot tell if they are in free fall or accelerating due to gravity without comparison to other regions of space. Gravity, too, is relative. However, the ideas that form general relativity are not unique to general relativity. One can find other models of gravity which satisfy the many of the assumptions of GR but introduce new gravitational forces, exotic types of matter or extra dimensions. This thesis concerns itself with the study models which contain new gravitational forces. The motivation for this choice is that it has recently been found that the rate that the universe is expanding is accelerating. This would imply that there exists a "dark energy" in the universe causing this acceleration. In GR, this is explained through the use of a cosmological constant; a ubiquitous energy which is constant iii across all of space-time. However, this energy has to be much smaller than predicted from any fundamental theory of physics to match observations. A common aim of modern modified gravity physics is to explain this late time acceleration dynamically without a cosmological constant. However, gravitational effects span from laboratory scales up to the size of the universe, and so any theory would need to satisfy all existing gravity measurements. This thesis examines which modified gravity theories can be made to pass solar system tests but allow for deviations on cosmological scales, and effects such theories are expected to have in the solar system not found in GR. Finally, extreme gravitational events such as the mergers of black holes emit gravitational radiation similar to how light is emitted by the motion of charged particles. With the recent detection of gravitational waves, new tests of gravity are available. We look at a particular class of gravity theories and examine how the gravitational radiation is changed relative to GR, and if such a change is potentially measurable. iv Declaration I declare that this thesis was composed by myself, that the work contained herein is my own except where explicitly stated otherwise in the text, and that this work has not been submitted for any other degree or professional qualification except as specified. Parts of this work have been published in [127]. Parts of this work have been published in [128]. (Ryan McManus, May 2018 ) v vi Acknowledgements This thesis is the culmination of many years of work which has seen me drastically evolve as both as a student, researcher, and person. There is no doubt that I could not have achieved this without the support of too many people to mention, but I will try my best. I primarily dedicate this work to my Mum, who has supported me throughout my life. I am forever grateful for all you have done for me. To my Nan and Granddad for always taking care of me and helping to ensure that I could walk this path. To my cousin Kyle, for being a friend I can always rely upon and pushing me to be better at everything. I have spent the last four years being masterfully guided by my supervisors who I would like to thank deeply. To Dr.