Living Rev. Relativity, 15, (2012), 10 LIVINGREVIEWS http://www.livingreviews.org/lrr-2012-10 in relativity Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions Beno^ıtFamaey Observatoire Astronomique de Strasbourg CNRS, UMR 7550, France and AIfA, University of Bonn, Germany email: [email protected] http://astro.u-strasbg.fr/~famaey/ Stacy S. McGaugh Department of Astronomy University of Maryland, USA and Case Western Reserve University, USA email: [email protected] http://astroweb.case.edu/ssm/ Accepted on 30 April 2012 Published on 7 September 2012 Abstract A wealth of astronomical data indicate the presence of mass discrepancies in the Universe. The motions observed in a variety of classes of extragalactic systems exceed what can be explained by the mass visible in stars and gas. Either (i) there is a vast amount of unseen mass in some novel form { dark matter { or (ii) the data indicate a breakdown of our understanding of dynamics on the relevant scales, or (iii) both. Here, we first review a few outstanding challenges for the dark matter interpretation of mass discrepancies in galaxies, purely based on observations and independently of any alternative theoretical framework. We then show that many of these puzzling observations are predicted by one single relation { Milgrom's law { involving an acceleration constant a0 (or a characteristic surface density Σy = a0=G) on the order of the square-root of the cosmological constant in natural units. This relation can at present most easily be interpreted as the effect of a single universal force law resulting froma modification of Newtonian dynamics (MOND) on galactic scales. We exhaustively reviewthe current observational successes and problems of this alternative paradigm at all astrophysical scales, and summarize the various theoretical attempts (TeVeS, GEA, BIMOND, and others) made to effectively embed this modification of Newtonian dynamics within a relativistic theory of gravity. This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 Germany License. http://creativecommons.org/licenses/by-nc-nd/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck Institute for Gravitational Physics, Am M¨uhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351. This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 Germany License: http://creativecommons.org/licenses/by-nc-nd/3.0/de/. Figures that have been previously published elsewhere may not be reproduced without consent of the original copyright holders. 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Contents 1 Introduction 5 2 The Missing Mass Problem in a Nutshell7 3 A Brief Overview of the ΛCDM Cosmological Model 11 3.1 Dark Energy (Λ)..................................... 11 3.2 Cold Dark Matter (CDM)................................ 12 4 Some Challenges for the ΛCDM Model 13 4.1 Coincidences....................................... 13 4.2 Unobserved predictions................................. 14 4.3 Unpredicted observations................................ 18 4.3.1 Baryonic Tully{Fisher relation......................... 18 4.3.2 The role of surface density............................ 24 4.3.3 Mass discrepancy-acceleration relation..................... 27 4.3.4 Renzo's rule.................................... 31 5 Milgrom's Empirical Law and \Kepler Laws" of Galactic Dynamics 35 5.1 Milgrom's law and the dielectric analogy........................ 35 5.2 Galactic Kepler-like laws of motion........................... 37 6 Milgrom's Law as a Modification of Classical Dynamics: MOND 42 6.1 Modified inertia or modified gravity: Non-relativistic actions............ 42 6.1.1 Modified inertia................................. 43 6.1.2 Bekenstein{Milgrom MOND.......................... 44 6.1.3 QUMOND.................................... 46 6.2 The interpolating function................................ 50 6.3 The external field effect................................. 54 6.4 MOND in the solar system............................... 56 6.5 MOND in rotationally-supported stellar systems................... 57 6.5.1 Rotation curves of disk galaxies......................... 57 6.5.2 The Milky Way.................................. 67 6.5.3 Disk stability and interacting galaxies..................... 69 6.5.4 Tidal dwarf galaxies............................... 73 6.6 MOND in pressure-supported stellar systems..................... 75 6.6.1 Elliptical galaxies................................. 75 6.6.2 Dwarf spheroidal galaxies............................ 78 6.6.3 Star clusters................................... 80 6.6.4 Galaxy groups and clusters........................... 81 7 Relativistic MOND Theories 87 7.1 Scalar-tensor k-essence.................................. 88 7.2 Stratified theory..................................... 90 7.3 Original Tensor-Vector-Scalar theory.......................... 90 7.4 Generalized Tensor-Vector-Scalar theory........................ 91 7.5 Bi-Scalar-Tensor-Vector theory............................. 92 7.6 Non-minimal scalar-tensor formalism.......................... 92 7.7 Generalized Einstein-Aether theories.......................... 93 7.8 Bimetric theories..................................... 95 7.9 Dipolar dark matter................................... 96 7.10 Non-local theories and other ideas........................... 97 8 Gravitational Lensing in Relativistic MOND 100 8.1 Strong lensing by galaxies................................ 100 8.2 Weak lensing by galaxies................................. 103 8.3 Strong and weak lensing by galaxy clusters...................... 103 8.4 Weak lensing by large-scale structure.......................... 107 9 MOND and Cosmology 108 9.1 Expansion history.................................... 108 9.2 Large-scale structure and Cosmic Microwave Background.............. 109 10 Summary and Discussion 117 11 Acknowledgements 126 References 127 List of Tables 1 Values predicted from the Besan¸conmodel of the Milky Way in MOND as seen by a Newtonist........................................ 69 2 Observational tests of MOND.............................. 125 Modified Newtonian Dynamics (MOND) 5 1 Introduction Two of the most tantalizing mysteries of modern astrophysics are known as the dark matter and dark energy problems. These problems come from the discrepancies between, on one side, the observations of galactic and extragalactic systems (as well as the observable Universe itself in the case of dark energy) by astronomical means, and on the other side, the predictions of general relativity from the observed amount of matter-energy in these systems. In short, what astronomical observations are telling us is that the dynamics of galactic and extragalactic systems, as well as the expansion of the Universe itself, do not correspond to the observed mass-energy as they should if our understanding of gravity is complete. Thus, this indicates either (i) the presence of unseen (and yet unknown) mass-energy, or (ii) a failure of our theory of gravity, or (iii) both. The third case is a priori the most plausible, as there are good reasons for there being more particles than those of the standard model of particle physics [257] (actually, even in the case of baryons, we suspect that a lot of them have not yet been seen and, thus, literally make up unseen mass, in the form of \missing baryons"), and as there is a priori no reason that general relativity should be valid over a wide range of scales, where it has never been tested [45], and where the need for a dark sector actually prevents the theory from being tested until this sector has been detected by other means than gravity itself1. However, either of the first two cases could be the dominant explanation of the discrepancies in a given class of astronomical systems (or even in all astronomical systems), and this is actually testable. For instance, as far as (ii) is concerned, if the mass discrepancies in a class of systems are mostly caused by some subtle change in gravitational physics, then there should be a clear signature of a single, universal force law at work in this whole class of systems. If instead there is a distinct dark matter component in these, the kinematics of any given system should