¡ ¢ £ ¤ ¥ ¦ § ¨ © Faculté des Sciences Département de Physique Unité de Physique Théorique et de Physique Mathématique Light deflection experiments as a test of relativistic theories of gravitation Dissertation présentée en vue de l’obtention du grade de Docteur en Sciences (Groupe des Sciences Physiques) par Sophie PIREAUX Supervisor: Professor Jean-Marc GERARD September 2002 Composition of the jury: Professor Jean-Pierre ANTOINE, Unité de Physique Théorique et Mathématique (FYMA), Université catholique de Louvain (UCL), BELGIUM, Professor Jean-Marc GERARD, Unité de Physique Théorique et Mathématique (FYMA), Université catholique de Louvain (UCL), BELGIUM, Professor Jacques WEYERS, Unité de Physique Théorique et Mathématique (FYMA), Université catholique de Louvain (UCL), BELGIUM, Professor Jan GOVAERTS, Unité de Physique Nucléaire (FYNU), Université catholique de Louvain (UCL), BELGIUM, Professor Jean SURDEJ, Institut d’Astrophysique et de Géophysique, Université de Liège (Ulg), BELGIUM, Professor François MIGNARD, Centre d’Etudes et de Recherches en Géodynamique et Astrométrie (CERGA), Observatoire de la Côte d’Azur (OCA), FRANCE. Acknowledgements We wish to thank our nearest and dearest for their support and their encouragements ; Professor Jean Govaerts, Professor Jean Surdej and the late Professor Jacques Demaret, for their readiness to spare some of their time answering our questions. Also the help from those persons who kindly read this work and helped improve the English is highly appreciated. Finally , we are in debt to the members of our jury (and particularly to our supervisor , Professor Jean-Marc Gérard) for their judicious comments; those initiated further reflec- tions and teased our curiosity. Contents Contents 1 Introduction 5 Refractive versus gravitational light deflection 5 Why alternative theories to General Relativity? 7 Importance of light deflection tests 9 Light bending as a test for alternative theories of gravitation 9 The structure and main results of this thesis 11 Chapter 1: Selected alternative theories to General Relativity (GR) 13 1.1 Scale Invariant Tensor Scalar theory (SITS) 13 1.2 Tensor Scalar theories (TS/BD/STRINGS) 14 1.2.1 Jordan frame 14 1.2.2 Einstein frame 16 1.3 Minimal Tensor Scalar theory (MTS) 18 1.4 Large Extra-Dimension theories (LED) 18 1.4.1 Metric reduction and Brans-Dicke effective action 20 1.4.2 Effective field content of the gravitational sector in 4-dimensions 22 1.4.3 Weak field limit 23 1.5 The Weyl theory (W) 24 1.6 Eddington-Robertson-Schiff parameters 27 1.6.1 General 27 1.6.2 MTS isotropic solution 28 1.6.3 SITS isotropic solution 28 1.6.4 TS ( or BD, STRINGS) isotropic solutions 28 1.6.5 Eddington-Robertson-Schiff parameters for selected theories 30 Chapter 2: TS cosmological estimation of the PN parameter γ 33 2_____________________________________________________________________________________________ 2.1 Cosmological equations 33 2.1.1 Formulation in terms of the Hubble parameter 33 2.1.2 A notion of quintessence 34 2.2 The case of a null cosmological constant 35 2.2.1 Conditions for an attractor mechanism towards GR 36 2.2.2 Solutions that admit GR as an attractor 38 2.2.3 A confrontation with SNIa results 40 2.2.4 Solutions in a flat radiation universe 42 2.2.5 Solutions in a flat dust universe 44 2.2.6 Values for the integration constants 49 2.2.7 Our model and the attractor mechanism 50 2.2.8 Evaluation of the observables 51 2.3 The case of a non null cosmological constant 55 2.3.1 Solutions that admit GR as an attractor 55 2.3.2 A confrontation with SNIa results 56 2.3.3 Impact of the cosmological constant on γ-predictions 56 2.4 Impact of the scalar potential and initial conditions on γ-predictions 59 2.5 Summary of the main results 63 2.5.1 Main properties of our model 63 2.5.2 Predicted value of the PN parameter γ 63 Chapter 3: Light deflection angle in the solar system: general considerations 65 3.1 First order measurement 65 3.1.1 Why the solar system, and especially the Sun? 65 3.1.2 Present and future constraints on γ 66 3.2 Second order measurement 67 3.2.1 Relevance of the impact parameter b 67 3.2.2 Measurability of the impact parameter b and asymptotic deflection angle α 72 3.2.3 Other effects: non sphericity of the gravitational source, angular momentum 74 3.2.4 Orders of magnitude of the different contributions 78 3.2.5 Further remarks on ‘‘corrective’’ effects 83 3.2.6 Current state of the art 84 Chapter 4: Testing alternative theories in the solar system using the light deflection angle? 85 4.1 Relevant (P)PN parameters at first and second order in light deflection 85 _____________________________________________________________________________________________3 4.2 Theories leading to first order corrections 86 4.2.1 TS theories 86 4.2.2 LED theories 87 4.3 Theories leading to second order corrections 93 4.3.1 Effective geodesic potential for light 93 4.3.2 Asymptotic weak field second order and finite distance deflection angle for MTS/SITS theories 95 4.4 Theories leading to a modification of the Newtonian potential 96 4.4.1 Effective geodesic potential for the Weyl theory 96 4.4.2 Asymptotic deflection angle for the Weyl theory 99 4.5 Summary of the main results 109 4.5.1 TS theories 109 4.5.2 LED theories 109 4.5.3 MTS/SITS theories 109 4.5.4 The Weyl theory 109 Chapter 5: Gravitational lensing/microlensing 111 5.1 Present and future of microlensing/gravitational lensing 111 5.2 Brief summary of the phenomenon 112 5.2.1 Asymptotic first and second order light deflection angle 112 5.2.2 Lens equation 112 5.2.3 Amplifications 115 5.2.4 Microlensing 116 5.3 Testing alternative theories of gravitation with microlensing/gravitational lensing? 117 5.3.1 Parameters of the simulations 117 5.3.2 TS theories 118 5.3.3 MTS/SITS theories 123 5.3.4 The Weyl theory 133 5.4 Summary of the main results of this chapter 143 5.4.1 TS theories 143 5.4.2 MTS/SITS theories 143 5.4.3 The Weyl theory 143 Conclusions and perspectives 145 TS theories 145 MTS/SITS theories 147 4_____________________________________________________________________________________________ LED theories 148 The Weyl theory 149 General 151 Perspectives 151 Bibliography 153 APPENDIX A: Conventions I APPENDIX B: Fact sheets of γ-tests through light deflection and other experiments III In the optical waveband IV In the radio waveband XVIII Other experiments measuring γ XXIV APPENDIX C: Light deflection angle in the framework of field theory XXV Feynman rules from the linearized LED gravitational action XXV Light deflection process in LED theories XXVIII Cross-section of LED light - massive scalar-field scattering XXXII Light deflection angle in LED theories XXXIII Introduction 1 Refractive versus gravitational light deflection Everyone is familiar with the fact that light is refracted as it travels through different media (glass, water, air...). This well-known phenomenon accounts for many different observations: a stick halfway in water appears to be bent, the fish is not exactly where you see it, the mirages in the desert form far from the real objects, etc. Mirages are explained by light rays propagating through layers of warm air of different refractive index, and bending towards regions of colder (i.e. denser) air. This refractive light deflection may significantly affect our view of distant earth objects. For example, we could see two (or more) images (one direct and one inverted lower image) of the lights of a distant car travelling on a sun heated road (Figure 0.1). Furthermore, the images are often deformed, or the car may seem to be closer as its lights appear brighter. Indeed, the so-called atmospheric lensing causes the multiplication of images, image deformation, or light amplification of distant unresolved objects. Any of these ‘‘optical illusions’’ are simply due to the fact that light is deflected if the medim is inhomogeneous. As we are aware that the stick in the water is not truly bent, and as, when we aim at a fish, we slightly compensate for the offset, we all know that the image of the sky on the desert sand must not be mistaken for a pound of water. Mankind even took advantage of refractive light bending, and among others, built optical lens telescopes to observe the sky. In 1915, Albert Einstein linked the fate of space-time with the presence of matter. Indeed, in General Relativity (GR), space-time can be compared with a mattress which is flat when no object lies on it. The presence of massive bodies on that mattress deforms (or we should say forms) it; the heavier the body, the more pronounced the dip. Consequently, a small marble originally travelling freely on a straight line on its surface, will now be following a curved trajectory. The closer the marble rolls from the massive body, the more deflected it will be from its originally straight path. So do the photons (particles of light) as they follow curves, called geodesics, of our space-time shaped by the presence of gravitational masses (Figure 0.2). The effect of space-time on light trajectories in General Relativity is exactly the same as for an inhomogeneous medium in classical optics. The only difference is that General Relativity is simpler in a sense because there is no dispersion since the effective deflection index does not depend on the light frequency. In analogy with refractive light deflection, what is called gravitational light deflection leads us to see a light source, like a star, slightly offset from its true position (just like the fish in water).