Varying constants, Gravitation and Cosmology Jean-Philippe Uzan Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Universit´ePierre et Marie Curie, 98 bis bd Arago, 75014 Paris (France) and Department of Mathematics and Applied Mathematics, Cape Town University, Rondebosch 7701 (South Africa) and National Institute for Theoretical Physics (NITheP), Stellenbosch 7600 (South Africa). email: [email protected] http//www2.iap.fr/users/uzan/ September 29, 2010 Abstract Fundamental constants are a cornerstone of our physical laws. Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to mat- ter. This will induce a violation of the universality of free fall. It is thus of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy. We thus detail the relations between the constants, the tests of the local posi- tion invariance and of the universality of free fall. We then review the main experimental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, Solar system observations, meteorites dating, quasar absorption spectra, stellar physics, pul- sar timing, the cosmic microwave background and big bang nucleosynthesis. At each step we arXiv:1009.5514v1 [astro-ph.CO] 28 Sep 2010 describe the basics of each system, its dependence with respect to the constants, the known systematic effects and the most recent constraints that have been obtained. We then describe the main theoretical frameworks in which the low-energy constants may actually be varying and we focus on the unification mechanisms and the relations between the variation of differ- ent constants. To finish, we discuss the more speculative possibility of understanding their numerical values and the apparent fine-tuning that they confront us with. 1 Contents 1 Introduction 4 2 Constants and fundamental physics 5 2.1 Aboutconstants ................................... .. 5 2.1.1 Characterizingthe fundamental constants . ...... 6 2.1.2 Constantsandmetrology . .. .. .. .. .. .. .. .. .. .. .. 10 2.2 The constancy of constants as a test of general relativity . ............ 14 2.2.1 General relativity . 14 2.2.2 Varying constants and the universality of free fall . ...... 17 2.2.3 Relations with cosmology . 19 3 Experimental and observational constraints 21 3.1 Atomicclocks....................................... 23 3.1.1 Atomicspectraandconstants. .. 23 3.1.2 Experimentalconstraints . 24 3.1.3 Physical interpretation . 28 3.1.4 Futureevolutions................................. 29 3.2 TheOklophenomenom ................................. 30 3.2.1 Anaturalnuclearreactor . .. .. .. .. .. .. .. .. .. .. .. 30 3.2.2 Constraining the shift of the resonance energy . ...... 31 3.2.3 From the resonance energy to fundamental constants . ......... 34 3.3 Meteoritedating .................................... 36 3.3.1 Long lived α-decays ............................... 36 3.3.2 Long lived β-decays ............................... 38 3.3.3 Conclusions.................................... 39 3.4 Quasarabsorbtionspectra. ..... 40 3.4.1 Generalities.................................... 40 3.4.2 Alkali doublet method (AD) . 42 3.4.3 Manymultipletmethod(MM) ......................... 43 3.4.4 Single ion differential measurement (SIDAM) . .. 47 i 2 3.4.5 H -21 cm vs UV: x = αEM gp/µ ......................... 47 i 2 3.4.6 H vs molecular transitions: y gpαEM .................... 48 2 1.57≡ 3.4.7 OH - 18 cm: F = gp(αEM µ) ......................... 49 ′ 2 3.4.8 Far infrared fine-structure lines: F = αEM µ .................. 49 1.85 3.4.9 “Conjugate” satellite OH lines: G = gp(αEM µ) .............. 50 3.4.10 Molecular spectra and the electron-to-proton mass ratio . ......... 50 3.4.11 Emissionspectra ................................. 53 3.4.12 Conclusionandprospects . 53 3.5 Stellarconstraints .................................. .. 56 3.6 CosmicMicrowaveBackground . ... 59 3.7 21cm ........................................... 62 3.8 Bigbangnucleosynthesis.............................. ... 63 3.8.1 Overview ..................................... 63 3.8.2 Constantseverywhere... 65 3.8.3 From BBN parameters to fundamental constants . ...... 68 3.8.4 Conclusion .................................... 70 2 4 The gravitational constant 70 4.1 Solarsystemsconstraints . .. .. .. .. .. .. .. .. .. .. .. .. .... 71 4.2 Pulsartiming....................................... 72 4.3 Stellarconstraints .................................. .. 74 4.3.1 Ages of globular clusters . 74 4.3.2 Solar and stellar sysmology . 75 4.3.3 Late stages of stellar evolution and supernovae . ..... 76 4.3.4 Newdevelopments ................................ 77 4.4 Cosmologicalconstraints. .... 77 4.4.1 Cosmicmicrowavebackground . 77 4.4.2 BBN........................................ 78 5 Theories with varying constants 79 5.1 Introducing new fields: generalities . ..... 80 5.1.1 The example of scalar-tensor theories . ... 80 5.1.2 Makingotherconstantsdynamical . .. 82 5.2 High-energytheoriesandvaryingconstants . ........ 83 5.2.1 Kaluza-Klein ................................... 83 5.2.2 Stringtheory ................................... 84 5.3 Relationsbetweenconstants. ..... 86 5.3.1 Implication of gauge coupling unification . 86 5.3.2 Masses and binding energies . 89 5.3.3 Gyromagneticfactors .............................. 91 5.4 Modelswithvaryingconstants . ... 91 5.4.1 String dilaton and Runaway dilaton models . 91 5.4.2 TheChameleonmechanism ........................... 94 5.4.3 Bekenstein and related models . 94 5.4.4 Otherideas .................................... 96 6 Spatial variations 97 6.1 Localscales....................................... 97 6.1.1 Generalities.................................... 97 6.1.2 Solarsystemscales................................ 98 6.1.3 MilkyWay .................................... 99 6.2 Cosmologicalscales.................................. .. 99 6.3 Implication for the universality of free fall . ...... 101 7 Why are the constants just so? 103 7.1 Universeandmultiverseapproaches . ..... 103 7.2 Fine-tunings and determination of the anthropic range . ......... 104 7.3 Anthropicpredictions ................................ 105 8 Conclusions 107 A Notations 108 A.1 Constants ........................................ 108 A.2 Sensibility coefficients . 108 A.3 Backgroundcosmologicalspacetime. ...... 109 3 1 Introduction Fundamental constants appear everywhere in the mathematical laws we use to describe the phe- nomena of Nature. They seem to contain some truth about the properties of the physical world while their real nature seem to evade us. The question of the constancy of the constants of physics was probably first addressed by Dirac [154, 155] who expressed, in his “Large Numbers hypothesis”, the opinion that very large (or small) dimensionless universal constants cannot be pure mathematical numbers and must not occur in the basic laws of physics. He suggested, on the basis of this numerological principle, that these large numbers should rather be considered as variable parameters characterizing the state of 2 −42 the universe. Dirac formed five dimensionless ratios among which δ H0~/mpc 2h 10 2 −2 −4 ≡ ∼ × and ǫ Gρ0/H0 5h 10 and asked the question of which of these ratio is constant as the universe≡ evolves.∼ Usually,×δ and ǫ vary as the inverse of the cosmic time. Dirac then noticed that αG /µαEM , representing the relative magnitude of electrostatic and gravitational forces between a 2 2 proton and an electron, was of the same order as H0e /mec = δαEM µ representing the age of the universe in atomic units so that his five numbers can be “harmonized” if one assumes that αG and δ vary with time and scale as the inverse of the cosmic time. This argument by Dirac is indeed not a physical theory but it opened many doors in the investigation on physical constants, both on questioning whether they are actually constant and on trying to understand the numerical values we measure. First, the implementation of Dirac’s phenomenological idea into a field-theory framework was proposed by Jordan [267] who realized that the constants have to become dynamical fields and proposed a theory where both the gravitational and fine-structure constants can vary (Ref. [492] provides some summary of some earlier attempts to quantify the cosmological implications of Dirac argument). Fierz [196] then realized that in such a case, atomic spectra will be spacetime- dependent so that these theories can be observationally tested. Restricting to the sub-case in which only G can vary led to definition of the class of scalar-tensor theories which were further explored by Brans and Dicke [66]. This kind of theory was further generalized to obtain various functional dependencies for G in the formalization of scalar-tensor theories of gravitation (see e.g. Ref. [123]). Second, Dicke [149] pointed out that in fact the density of the universe is determined by its age, this age being related to the time needed to form galaxies, stars, heavy nuclei... This led him to formulate that the presence of an observer in the universe places constraints on the physical laws that can be observed. In fact, what is meant by observer is the existence of (highly?) organized systems and this principle can be seen as a rephrasing of the question “why is the universe the way it is?” (see Ref. [251]). Carter [81, 82], who actually coined the term “anthropic principle” for it, showed that the numerological coincidences found by Dirac