arXiv:hep-ph/0205340v1 30 May 2002 CONTENTS emphasized. are relativity general of discussed. theo tests are alternative the results K with other these including of and implications reviewed, theories cosmological also string are theories, variation Klein leadi such mass models of theoretical proton The to prediction electron spectra. the absorption of quasar var and in a constant of structure detections fine the of the particularl claims recent results, the different understanding investig the constan of an compare given Such to a hypotheses. importance constrains underlying (2 of the it measurement, out why each point to possible of (3) as basics clearly the reviewed as provide are show to ratio (1) mass proton aims to review int electron strong the and and and weak of couplings definition variation gravitational, the the their constant, on of constraints structure various constants, the fundamental metrology, of with di a role After the nature. on of constants fundamental the of variation observatio variation: motivations their theoretical and constants fundamental The I.Fn tutr constant structure Fine III. V rvttoa constant Gravitational IV. I Generalities II. .Introduction I. hsatcedsrbstevroseprmna onso t on bounds experimental various the describes article This .Fo ia ueooia rnil oanthropic to principle numerological Dirac From A. .Oeve ftemtos7 methods the of Overview C. Metrology B. .Paeayadselrobt 26 25 19 orbits stellar and Planetary B. arguments geophysical and Paleontological A. constraints Cosmological C. .Gooia osrit 8 constraints Geological A. .EuvlnePicpe22 constraints Stellar C. Principle Equivalence D. .Aoi spectra Atomic B. arguments .Sotnosfiso 11 4. fission Spontaneous 3. .EpnigErh25 25 Pulsars 3. system Solar 2. works Early 1. 19 Earth Expanding 14 16 temperature 2. surface Earth 1. Conclusion 3. Nucleosynthesis 2. background microwave Cosmic 1. observations Astrophysical 3. experiments Laboratory 2. 1. Conclusion 5. .TeOl hnmnn8 phenomenon 2. Oklo The 1. aoaor ePyiu herqe NSUR82,Univer 8627, Th´eorique, CNRS-UMR Physique 9 de 2435, Laboratoire CNRS-FRE GReCO, Paris, de d’Astrophysique Institut Uzan Jean-Philippe β α α -decay EM -decay dpnec faoi pcr 13 spectra atomic of -dependence ∗ h links The nview in y eractions gt the to ng isand ries ainof iation scussion h fine the to is ation This . aluza- and t ratio link to ) he 25 10 29 28 27 26 22 20 12 12 11 1 8 4 4 5 1 nSlrsse clsadgooia iesae n one and scales time geological and variations scales huge system undergone Solar not on Obvi- have constants precise. thus more the estab- is ously, and as more It is especially become variation possibility measurements applied. this the a investigate be to such to importance if have of would and corrections universe distorted lished a our the to of lead Ignoring could view constants time. varying look-back of the possibility measures where cosmology redshift and astronomy the impor- This in an particular space. plays in constants and role tant the time the of on constancy if depend of only not hypothesis sense does nature makes of which laws approach scientific the of as conceive. to and in easier change Earth much smooth is on a constants distance physical but the the else) somewhere of power square the physical another as of the behaving form force of the gravitation inverse of Newtonian change a a (e.g. Indeed, imagine laws to assumption. It difficult this is differ. question Heavens it to in and natural Earth however of on is point laws the Aristotelian spacetime which the in in with point view contrasts one This the from that differ another. stating not to and do universe physics the of in laws place living particular not are a we in that states which principle, Copernician INTRODUCTION I. References Acknowledgments Conclusions VII. I hoeia oiain 38 motivations Theoretical VI. .Ohrconstants Other V. oprn n erdcn xeiet sas root a also is experiments reproducing and Comparing the on considerably relied physics of development The i´ ai u,baiet20 -10 ra ee,France cedex, Orsay F-91405 bˆatiment 210, Sud, sit´e Paris i,B rg,704Prs France. Paris, 75014 Arago, Bd bis, 8 .Ohrivsiain 42 40 38 45 36 problem? 35 constant cosmological new A D. investigations Other C. theories Superstring B. theories Kaluza-Klein 37 A. of variation the constant constrain cosmological to the Attempts of case F. particular The factor E. gyromagnetic ratio Proton mass proton D. to Electron C. interaction Strong B. interaction Weak A. .Csooia osrit 30 constraints Cosmological D. iesoflcntns38 constants dimensionful .Nucleosynthesis 2. CMB 1. a ttsand status nal 48 46 45 34 32 31 30 30 . is looking for tiny effects. Besides, the question of the tion value, three angles and a phase of the Cabibbo- values of the constants is central to physics and one can Kobayashi-Maskawa matrix, a phase for the QCD vac- hope to explain them dynamically as predicted by some uum and three coupling constants gS ,gW ,g1 for the gauge high-energy theories. Testing for the constancy of the group SU(3) SU(2) U(1) respectively. Below the Z × × constants is thus part of the tests of general relativity. mass, g1 and gW combine to form the electro-magnetic Let us emphasize that this latter step is analogous to the coupling constant transition from the Newtonian description of mechanics − 5 −2 − in which space and time were just a static background in g 2 = g + g 2. (1) EM 3 1 W which matter was evolving to the relativistic description where spacetime becomes a dynamical quantity deter- The number of free parameters indeed depends on the mined by the Einstein equations (Damour, 2001). physical model at hand (see Weinberg, 1983). This is- Before discussing the properties of the constants of na- sue has to be disconnected from the number of required ture, we must have an idea of which constants to consider. fundamental dimensionful constants. Duff, Okun and First, all constants of physics do not play the same role, Veneziano (2002) recently debated this question, respec- and some have a much deeper one than others. Follow- tively arguing for none, three and two (see also Wignall, ing Levy-Leblond (1979), we can define three classes of 2000). Arguing for no fundamental constant leads to fundamental constants, class A being the class of the con- consider them simply as conversion parameters. Some stants characteristic of particular objects, class B being of them are, like the Boltzmann constant, but some oth- the class of constants characteristic of a class of physi- ers play a deeper role in the sense that when a physi- cal phenomena, and class C being the class of universal cal quantity becomes of the same order of this constant constants. Indeed, the status of a constant can change new phenomena appear, this is the case e.g. of ¯h and c with time. For instance, the velocity of light was a ini- which are associated respectively to quantum and rela- tially a type A constant (describing a property of light) tivistic effects. Okun (1991) considered that only three which then became a type B constant when it was real- fundamental constants are necessary, the underlying rea- ized that it was related to electro-magnetic phenomena son being that in the international system of units which and, to finish, it ended as a type C constant (it enters has 7 base units an 17 derived units, four of the seven many laws of physics from electromagnetism to relativity base units are in fact derived (Ampere, Kelvin, mole and including the notion of causality...). It has even become candela). The three remaining base units (meter, second a much more fundamental constant since it has been cho- and kilogram) are then associated to three fundamental sen as the definition of the meter (Petley, 1983). A more constants (c, ¯h and G). They can be seen as limiting conservative definition of a fundamental constant would quantities: c is associated to the maximum velocity and thus be to state that it is any parameter that can not be ¯h to the unit quantum of angular momentum and sets calculated with our present knowledge of physics, i.e. a a minimum of uncertainty whereas G is not directly as- free parameter of our theory at hand. Each free param- sociated to any physical quantity (see Martins 2002 who eter of a theory is in fact a challenge for future theories argues that G is the limiting potential for a mass that to explain its value. does not form a black hole). In the framework of quan- How many fundamental constants should we consider? tum field theory + general relativity, it seems that this The set of constants which are conventionally consid- set of three constants has to be considered and it al- ered as fundamental (Flowers and Petley, 2001) con- lows to classify the physical theories (see figure 1). How- ever, Veneziano (1986) argued that in the framework of sists of the electron charge e, the electron mass me, the string theory one requires only two dimensionful funda- proton mass mp, the reduced Planck constant ¯h, the velocity of light in vacuum c, the Avogadro constant mental constants, c and the string length λs. The use of ¯h seems unnecessary since it combines with the string NA , the Boltzmann constant kB , the Newton constant tension to give λs. In the case of the Goto-Nambu ac- G, the permeability and permittivity of space, ε0 and −2 tion S/¯h = (T/¯h) d(Area) λs d(Area) and the µ0. The latter has a fixed value in the SI system of ≡ −2 −7 −1 Planck constant is just given by λ . In this view, ¯h has unit (µ0 = 4π 10 H m ) which is implicit in the s × not disappeared butR has been promotedR to the role of a definition of the Ampere; ε0 is then fixed by the rela- −2 UV cut-off that removes both the infinities of quantum tion ε0µ0 = c . The inclusion of NA in the former list has been debated a lot (see e.g. Birge, 1929). To field theory and singularities of general relativity. This compare with, the minimal standard model of particle situation is analogous to pure quantum gravity (Novikov physics plus gravitation that describes the four known and Zel’dovich, 1982) where ¯h and G never appear sepa- rately but only in the combination ℓ = G¯h/c3 so that interactions depends on 20 free parameters (Cahn, 1996; Pl only c and ℓ are needed. Volovik (2002) considered the Hogan, 2000): the Yukawa coefficients determining the Pl p masses of the six quark (u,d,c,s,t,b) and three lepton analogy with quantum liquids. There, an observer knows (e,µ,τ) flavors, the Higgs mass and vacuum expecta- both the effective and microscopic physics so that he can judge whether the fundamental constants of the effective

2 theory remain fundamental constants of the microscopic for the velocity of light, the reduced Planck constant, the theory. The status of a constant depends on the consid- Newton constant, the masses of the electron, proton and ered theory (effective or microscopic) and, more interest- neutron, and the charge of the electron. We also define ingly, on the observer measuring them, i.e. on whether e2 this observer belongs to the world of low-energy quasi- q2 (9) particles or to the microscopic world. ≡ 4πε0 Resolving this issue is indeed far beyond the scope and the following dimensionless ratios of this article and can probably be considered more as q2 an epistemological question than a physical one. But, αEM 1/137.03599976(50) (10) as the discussion above tends to show, the answer de- ≡ ¯hc ∼ G m2 c pends on the theoretical framework considered [see also α F p 1.03 10−5 (11) Cohen-Tannoudji (1985) for arguments to consider the W ≡ ¯h3 ∼ × Boltzmann constant as a fundamental constant]. A more g2(E) α (E) s (12) pragmatic approach is then to choose a theoretical frame- S ≡ ¯hc work, so that the set of undetermined fixed parameters Gm2 α p 5 10−39 (13) is fully known and then to wonder why they have the G ≡ ¯hc ∼ × values they have and if they are constant. m µ e 5.44617 10−4 (14) We review in this article both the status of the ex- ≡ mp ∼ × perimental constraints on the variation of fundamental x g α2 µ 1.62 10−7 (15) constants and the theoretical motivations for considering ≡ p EM ∼ × 2 −4 such variations. In section II, we recall Dirac’s argument y gpαEM 2.977 10 (16) ≡ ∼ × that initiated the consideration of time varying constants which characterize respectively the strength of the and we briefly discuss how it is linked to anthropic ar- electro-magnetic, weak, strong and gravitational forces guments. Then, since the fundamental constants are en- and the electron-proton mass ratio, gp 5.585 is the tangled with the theory of measurement, we make some proton gyro-magnetic factor. Note that the≃ relation (12) very general comments on the consequences of metrology. between two quantities that depend strongly on energy; In Sections III and IV, we review the observational con- this will be discussed in more details in Section V. We straints respectively on the variation of the fine structure introduce the notations and of gravitational constants. Indeed, we have to keep ¯h in mind that the obtained constraints depend on under- a0 = =0.5291771 A˚ (17) lying assumptions on a certain set of other constants. We mecαEM summarize more briefly in Section V, the constraints on 1 E = m c2α2 = 13.60580eV (18) other constants and we give, in Section VI, some hints − I 2 e EM of the theoretical motivations arising mainly from grand EI 7 −1 R∞ = =1.0973731568549(83) 10 m (19) unified theories, Kaluza-Klein and string theories. We − hc × also discuss a number of cosmological models taking these respectively for the Bohr radius, the hydrogen ionization variations into account. For recent shorter reviews, see energy and the Rydberg constant. Varshalovich et al. (2000a), Chiba (2001), Uzan (2002) While working in cosmology, we assume that the uni- and Martins (2002). verse is described by a Friedmann-Lemaˆıtre spacetime Notations: In this work, we use SI units and the follow- 2 2 2 i j ing values of the fundamental constants today1 ds = dt + a (t)γij dx dx , (20) − c = 299, 792, 458m s−1 (2) where t is the cosmic time, a the scale factor and γ the · ij ¯h =1.054571596(82) 10−34 J s (3) metric of the spatial sections. We define the redshift as × · −11 3 −1 −2 a ν G =6.673(10) 10 m kg s (4) 1+ z 0 = e (21) × −· · a ν m =9.10938188(72) 10 31 kg (5) ≡ 0 e × −27 where a0 is the value of the scale factor today while νe mp =1.67262158(13) 10 kg (6) × and ν are respectively the frequencies at emission and m =1.67492716(13) 10−27 kg (7) 0 n × today. We decompose the Hubble constant today as e =1.602176462(63) 10−29 C (8) × H−1 =9.7776 109 h−1 yr (22) 0 × where h =0.68 0.15 is a dimensionless number, and the density of the universe± today is given by 1see http://physics.nist.gov/cuu/Constants/ for an up −26 2 −3 to date list of the recommended values of the constants of ρ0 =1.879 10 Ωh kg m . (23) nature. × ·

3 II. GENERALITIES and Rees (1979) then showed how one can scale up from atomic to cosmological scales only by using combinations A. From Dirac numerological principle to anthropic of αEM , αG and me/mp. arguments Dicke (1961, 1962b) brought Mach’s principle into the discussion and proposed (Brans and Dicke, 1961) a the- The question of the constancy of the constants of ory of gravitation based on this principle. In this the- physics was probably first addressed by Dirac (1937, ory the gravitational constant is replaced by a scalar 1938, 1979) who expressed, in his “Large Numbers hy- field which can vary both in space and time. It follows pothesis”, the opinion that very large (or small) dimen- that, for cosmological solutions, G t−n, H t−1 and sionless universal constants cannot be pure mathemat- ρ tn−2 where n is expressible in terms∝ of an∝ arbitrary ical numbers and must not occur in the basic laws of ∝ −1 parameter ωBD as n =2+3ωBD /2. Einstein gravity is physics. He suggested, on the basis of this numerological −n recovered when ωBD . This predicts that αG t principle, that these large numbers should rather be con- −1 → ∞ ∝ and δ t whereas αEM , αW and ǫ are kept constant. sidered as variable parameters characterizing the state This kind∝ of theory was further generalized to obtain var- of the universe. Dirac formed the five dimensionless ra- ious functional dependences for G in the formalization of 2 −42 tios αEM , αW , αG , δ H0¯h/mpc 2h 10 and scalar-tensor theories of gravitation (see e.g. Damour and ǫ Gρ /H2 5h−2 ≡10−4 and asked∼ the× question of ≡ 0 0 ∼ × Esposito-Far`ese, 1992). which of these ratio is constant as the universe evolves. The first extension of Dirac’s idea to non-gravitational Usually, only δ and ǫ vary as the inverse of the cosmic forces was proposed by Jordan (1937, 1939) who still time (note that with the value of the density chosen by considered that the weak interaction and the proton to Dirac, the universe is not flat so that a t and ρ t−3). ∝ ∝ electron mass ratio were constant. He realized that the Dirac then noticed that αG µ/αEM , representing the rel- constants has to become dynamical fields and used the ative magnitude of electrostatic and gravitational forces action between a proton and an electron, was of the same or- 2 der as H e2/m c2 = δα /µ representing the age of the φ φ 0 e EM S = √ gd4xφη R ξ ∇ F 2 , (24) universe in atomic time so that the five previous num- − − φ − 2 Z "   # bers can be “harmonized” if one assumes that αG and δ vary with time and scale as the inverse of the cosmic η and ξ being two parameters. Fierz (1956) realized that time2. This implies that the intensity of all gravitational with such a Lagrangian, atomic spectra will be space- effects decrease with a rate of about 10−10 yr−1 and that dependent. But, Dirac’s idea was revived after Teller ρ t−2 (since ǫ is constant) which corresponds to a (1948) argued that the decrease of G contradicts paleon- flat∝ universe. Kothari (1938) and Chandrasekhar (1939) tological evidences [see also Pochoda and Schwarzschild were the first to point out that some astronomical con- (1964) and Gamow (1967c) for evidences based on the sequences of this statement may be detectable. Similar nuclear resources of the Sun]. Gamow (1967a, 1967b) ideas were expressed by Milne (1937). proposed that αEM might vary as t in order to save Dicke (1961) pointed out that in fact the density of the the, according to him, “elegant” idea of Dirac (see also universe is determined by its age, this age being related Stanyukovich, 1962). In both Gamow (1967a, 1967b) to the time needed to form galaxies, stars, heavy nuclei... and Dirac (1937) theories the ratio αG /αEM decreases as −1 −1 This led him to formulate that the presence of an ob- t . Teller (1948) remarked that αEM ln H0tPl so −1 ∼ − server in the universe places constraints on the physical that αEM would become the logarithm of a large num- laws that can be observed. In fact, what is meant by ber. Landau (1955), de Witt (1964) and Isham et al. observer is the existence of (highly?) organized systems (1971) advocated that such a dependence may arise if and the anthropic principle can be seen as a rephrasing of the Planck length provides a cut-off to the logarithmic the question “why is the universe the way it is?” (Hogan, divergences of quantum electrodynamics. In this latter −1 −1 2000). Carter (1974, 1976, 1983), who actually coined class of models αEM 1/ ln t, αG t , δ t and αW ∝ ∝ ∝ the term “anthropic principle”, showed that the numero- and ǫ remain constant. Dyson (1967), Peres (1967) and logical coincidence found by Dirac can be derived from then Davies (1972) showed, using respectively geological physical models of stars and the competition between the data of the abundance of rhenium and osmium and the weakness of gravity with respect to nuclear fusion. Carr stability of heavy nuclei, that these two hypothesis were ruled out observationally (see Section III for details on the experimental results). Modern theories of high en- ergy physics offer new arguments to reconsider the varia- 2 The ratio δαEM /µ represents roughly the inverse of the tion of the fundamental constants (see Section VI). The number of times an electron orbits around a proton during most important outcome of Dirac’s proposal and of the the age of the universe. Already, this suggested a link be- following assimilated theories [among which a later ver- tween micro-physics and cosmological scales. sion of Dirac (1974) theory in which there is matter cre-

4 ation either where old matter was present or uniformly ters without giving any explanation for their values. throughout the universe] is that the hypothesis of the Rozental (1988) also argued that the existence of hy- constancy of the fundamental constants can and must be drogen and the formation of complex elements in stars checked experimentally. (mainly the possibility of the reaction 3α 12C) set A way to reconcile some of the large numbers is to constraints on the values of the strong coupling→ constant. consider the energy dependence of the couplings as de- The production of 12C in stars requires a triple tuning: termined by the renormalization group (see e.g. Itzkyson (i) the decay lifetime of 8Be, of order 10−6 s, is four orders and Zuber, 1980). For instance, concerning the fine struc- of magnitude longer than the time for two α particles to ture constant, the energy-dependence arises from vacuum scatter, (ii) an excited state of the carbon lies just above polarization that tends to screen the charge. This screen- the energy of 8Be+ α and finally (iii) the energy level of ing is less important at small distance and the charge ap- 16O at 7.1197 MeV is non resonant and below the energy pears bigger so that the effective coupling constant grows of 12C+α, of order 7.1616 MeV, which ensures that most with energy. It follows from this approach that the three of the carbon synthetized is not destroyed by the capture gauge groups get unified into a larger grand unification of an α-particle (see Livio et al., 2000). Oberhummer et group so that the three couplings αEM , αW and αS stem al. (2000) showed that outside a window of respectively from the same dimensionless number αGUT . This might 0.5% and 4% of the values of the strong and electromag- explain some large numbers and answer some of Dirac netic forces, the stellar production of carbon or oxygen concerns (Hogan, 2000) but indeed, it does not explain will be reduced by a factor 30 to 1000 (see also Pochet et the weakness of gravity which has become known as the al, 1991). Concerning the gravitational constant, galaxy 4 hierarchy problem. formation require αG < 10 . Other such constraints on Let us come back briefly to the anthropic considera- the other parameters listed in the previous section can tions and show that they allow to set an interval of ad- be obtained. missible values for some constants. Indeed, the anthropic principle does not tell whether the constants are varying or not but it gives an insight on how special our uni- B. Metrology verse is. In such an approach, one studies the effect of small variations of a constant around its observed value The introduction of constants in physical law is closely and tries to find a phenomenon highly dependent on this related to the existence of systems of units. For instance, constant. This does not ensure that there is no other set Newton’s law states that the gravitational force between of constants (very different of the one observed today) two masses is proportional to each mass and inversely for which an organized universe may exist. It just tells proportional to their separation. To transform the pro- about the stability in a neighborhood of the location of portionality to an equality one requires the use of a quan- tity with dimension of m3 kg−1 s−2 independent of the our universe in the parameter space of physical constants. · · Rozental (1988) argued that requiring that the lifetime separation between the two bodies, of their mass, of their −2 2 32 composition (equivalence principle) and on the position of the proton τp αEM (¯h/mpc ) exp(1/αEM ) 10 yr ∼ ∼ 17 (local position invariance). With an other system of units is larger than the age of the universe tu c/H0 10 s ∼ ∼ this constant could have simply been anything. implies that αEM < 1/80. On the other side, if we be- lieve in a grand unified theory, this unification has to take The determination of the laboratory value of constants relies mainly on the measurements of lengths, frequen- place below the Planck scale implying that αEM > 1/170, this bound depending on assumptions on the particle con- cies, times,... (see Petley, 1985 for a treatise on the mea- tent. Similarly requiring that the electromagnetic repul- surement of constants and Flowers and Petley, 2001, for sion is much smaller than the attraction by strong in- a recent review). Hence, any question on the variation of teraction in nuclei (which is necessary to have nuclei) constants is linked to the definition of the system of units and to the theory of measurement. The choice of a base leads to αEM < 1/20. The thermonuclear reactions in 2 units affects the possible time variation of constants. stars are efficient if kB T αEM mpc and the temper- ∼ The behavior of atomic matter is mainly determined ature of a star of radius RS and mass MS can roughly by the value of the electron mass and of the fine structure be estimated as kB T GMsmp/RS , which leads to the −3∼ constant. The Rydberg energy sets the (non-relativistic) estimate αEM 10 . One can indeed think of many other examples∼ to put such bounds. From the previous atomic levels, the hyperfine structure involves higher considerations, we retain that the most stringent is powers of the fine structure constant, and molecular modes (including vibrational, rotational...modes) depend

1/170 < αEM < 1/80. (25) on the ratio me/mp. As a consequence, if the fine struc- ture constant is spacetime dependent, the comparison be- It is difficult to believe that these arguments can lead to tween several devices such as clocks and rulers will also much sharper constraints. They are illustrative and give be spacetime dependent. This dependence will also differ a hint that the constants may not be “random” parame-

5 from one clock to another so that metrology becomes both where k2 is another dimensionless constant and F2 a func- device and spacetime dependent. tion of a sufficient number of fundamental constants to be Besides this first metrologic problem, the choice of consistent with the initial base units. The time variation units has implications on the permissible variations of of X is given by certain dimensionful constant. As an illustration, we fol- d ln X d ln k1 d ln F1 d ln k2 d ln F2 low Petley (1983) who discusses the implication of the = + = + . dt dt dt dt dt definition of the meter. The definition of the meter via a prototype platinum-iridium bar depends on the inter- Since only dk1/dt or dk2/dt can be measured, it is nec- atomic spacing in the material used in the construction essary to have chosen a system of units, the constancy of the bar. Atkinson (1968) argued that, at first order, it of which is assumed (i.e. that either dF1/dt = 0 or mainly depends on the Bohr radius of the atom so that dF2/dt = 0) to draw any conclusion concerning the time this definition of the meter fixes the combination (17) variation of X, in the same way as the description of a as constant. Another definition was based on the wave- motion needs to specify a reference frame. length of the orange radiation from krypton-86 atoms. To illustrate the importance of the choice of units and It is likely that this wavelength depends on the Rydberg the entanglement between experiment and theory while constant and on the reduced mass of the atom so that it measuring a fundamental constant, let us sketch how one 2 2 determines me in the SI system (following Mohr and Tay- ensures that mec αEM /2¯h is constant. The more recent definition of the meter as the length of the path traveled lor, 2001), i.e. in kilogram (see figure 2). The kilo- by light in vacuum during a time of 1/299792458 of a gram is defined from a platinum-iridium bar to which second imposes the constancy of the speed of light3 c. we have to compare the mass of the electron. The key to this measurement is to express the electron mass as Identically, the definitions of the second as the duration 2 m = 2hR∞/α c. From the definition of the second, of 9,192,631,770 periods of the transition between two e EM hyperfine levels of the ground state of cesium-133 or of R∞ is determined by precision laser-spectroscopy on hy- the kilogram via an international prototype respectively drogen and deuterium and the theoretical expression for 2 2 4 the 1s-2s hydrogen transition as ν = (3/4)R∞c[1 µ + impose that m c α /¯h and mp are fixed. e EM 11α2 /48+(56α3 )/(9π) ln α +...] arising from− QED. Since the definition of a system of units and the value EM EM EM of the fundamental constants (and thus the status of their The fine structure constant is determined by compar- constancy) are entangled, and since the measurement of ing theory and experiment for the anomalous magnetic any dimensionful quantity is in fact the measurements of moment of the electron (involving again QED). Finally, a ratio to standards chosen as units, it only makes sense the Planck constant is determined by a Watt balance to consider the variation of dimensionless ratios. comparing a Watt electrical power to a Watt mechanical In theoretical physics, we often use the fundamental power (involving classical mechanics and classical elec- constants as units (see McWeeny, 1973 for the relation tromagnetism only: h enters through the current and between natural units and SI units). The international voltage calibration based on two condensed-matter phe- system of units (SI) is more appropriate to human size nomena: Josephson and quantum Hall effects so that it measurements whereas natural systems of units are more involves the theories of these two effects). appropriate to the physical systems they refer to. For As a conclusion, let us recall that (i) in general, the val- instance ¯h, c and G allows to construct the Planck mass, ues of the constants are not determined by a direct mea- time and length which are of great use as units while surement but by a chain involving both theoretical and studying high-energy physics and the same can be done experimental steps, (ii) they depend on our theoretical understanding, (iii) the determination of a self-consistent from ¯h, e, me and ε0 to construct a unit mass (me), 2 2 3 4 set of values of the fundamental constants results from an length (4πε0h /mee ) and time (2ε0h /πmee ). A phys- ical quantity can always be decomposed as the product of adjustment to achieve the best match between theory and a label representing a standard quantity of reference and a defined set of experiments (see e.g., Birge, 1929) (iv) a numerical value representing the number of times the that the system of units plays a crucial role in the mea- standard has to be taken to build the required quantity. surement chain, since for instance in atomic units, the It follows that a given quantity X that can be expressed mass of the electron could have been obtained directly from a mass ratio measurement (even more precise!) and as X = k1F1(m, kg, s,...) with k1 a dimensionless quan- (v) fortunately the test of the variability of the constants tity and F1 a function of the base units (here SI) to some does not require a priori to have a high-precision value power. Let us decompose X as X = k2F2(¯h, e, c, . . .) of the considered constant. In the following, we will thus focus on the variation of dimensionless ratios which, for instance, characterize the 3Note that the velocity of light is not assigned a fixed value relative magnitude of two forces, and are independent of directly, but rather the value is fixed as a consequence of the the choice of the system of units and of the choice of stan- definition of the meter. dard rulers or clocks. Let us note that some (hopeless)

6 attempts to constraint the time variation of dimensionful mine the law of evolution and the links between the con- constants have been tried and will be briefly discussed in stants. Long time scale experiments allow to test a slow Section V.F. This does not however mean that a physical drift evolution while short time scale experiments enable theory cannot have dimensionful varying constants. For to test the possibility of a rapidly varying constant. instance, a theory of varying fine structure constant can The next step is to convert the bound on the varia- be implemented either as a theory with varying electric tion of some measured physical quantities (decay rate, charge or varying speed of light. cross section,...) into a bound on some constants. It is clear that in general (for atomic and nuclear methods at least) it is impossible to consider the electromagnetic, C. Overview of the methods weak and strong effects independently so that this latter step involves some assumptions. Before going into the details of the constraints, it is Atomic methods are mainly based on the comparison worth taking some time to discuss the kind of experi- of the wavelengths of different transitions. The non rel- ments or observations that we need to consider and what ativistic spectrum depends mainly on R∞ and µ, the we can hope to infer from them. 2 fine structure on R∞αEM and the hyperfine structure on As emphasized in the previous section, we can only 2 gpR∞αEM . Extending to molecular spectra to include measure the variation of dimensionless quantities (such rotational and vibrational transitions allows to have ac- as the ratio of two wavelengths, two decay rates, two cess to µ. It follows that we can hope to disentangle the cross sections ...) and the idea is to pick up a physical observations of the comparisons of different transitions system which depends strongly on the value of a set of to constrain on the variation of (αEM ,µ,gp). The excep- constants so that a small variation will have dramatic tion is CMB which involves a dependence on αEM and me effects. The general strategy is thus to constrain the mainly due to the Thomson scattering cross section and spacetime variation of an observable quantity as precisely the ionization fraction. Unfortunately the effect of these as possible and then to relate it to a set of fundamental parameters have to be disentangled from the dependence constants. on the usual cosmological parameters which render the Basically, we can split all the methods into three interpretation more difficult. classes: (i) atomic methods including atomic clocks, The internal structure and mass of the proton and neu- quasar absorption spectra and observation of the cos- tron are completely determined by strong gauge fields mic microwave background radiation (CMBR) where one and quarks interacting together. Provided we can ignore compares ratios of atomic transition frequencies. The the quark masses and electromagnetic effects, the whole CMB observation depends on the dependence of the re- structure is only dependent on an energy scale ΛQCD . It combination process on αEM ; (ii) nuclear methods in- follows that the stability of the proton greatly depends cluding nucleosynthesis, α- and β-decay, Oklo reactor for on the electromagnetic effects and the masses mu and md which the observables are respectively abundances, life- of the up and down quarks. In nuclei, the interaction of times and cross sections; and (iii) gravitational methods hadrons can be thought to be mediated by pions of mass including the test of the violation of the universality of 2 mπ mp(mu + md). Since the stability of the nucleus free fall where one constrains the relative acceleration of mainly∼ results from the balance between this attractive two bodies, stellar evolution. . . nuclear force, the nucleon degeneracy pressure and the These methods are either experimental (e.g. atomic Coulomb repulsion, it will mainly involve mu, md, αEM . clocks) for which one can have a better control of the sys- Big bang nucleosynthesis depends on G (expansion tematics, observational (e.g. geochemical, astrophysical rate), GF (weak interaction rates), αS (binding of light and cosmological observations) or mixed (α- and β-decay, elements), αEM (via the electromagnetic contribution to universality of free fall). This sets the time scales on mn mp but one will also have to take into account the which a possible variation can be measured. For instance, contribution− of a possible variation of the mass of the in the case of the fine structure constant (see Section III), quarks, mu and md). Besides, if mn mp falls below one expects to be able to constrain a relative variation − me the β-decay of the neutron is no longer energetically −8 −5 of αEM of order 10 [geochemical (Oklo)], 10 [astro- possible. The abundance of helium is mainly sensitive to −3 −2 physical (quasars)], 10 10 [cosmological methods], the freeze-out temperature and the neutron lifetime and −13 −14 − 10 10 [laboratory methods] respectively on time heavier element abundances to the nuclear rates. scales− of order 109 yr, 109 1010 yr, 1010 yr and 1 12 − − All nuclear methods involve a dependence on the mass months. This brings up the question of the comparison of the nuclei of charge Z and atomic number A and of the compatibility of the different measurements since one will have to take into account e.g. the rate of m(A, Z)= Zm + (A Z)m + E + E , p − n S EM change of αEM which is often assumed to be constant. In general, this requires to specify a model both to deter- where ES and EEM are respectively the strong and elec- tromagnetic contributions to the binding energy. The

7 α δmc2 Bethe-Weiz¨acker formula gives that δ~a = ln α. (31) − m δα ∇ Z(Z 1)   E = 98.25 − α MeV. (26) EM A1/3 EM In the most general case, for non-relativistically moving body, If we decompose mp and mn as (see Gasser and 2 Leutwyler, 1982) m(p,n) = u3 + b(u,d)mu + b(d,u)md + α δmc α α˙ ~v δ~a = ∇ + . (32) B(p,n)αEM where u3 is the pure QCD approximation of − m δα α α c2 the nucleon mass (bu, bd and B(n,p)/u3 being pure num-     bers), it reduces to It reduces to Eq. (30) in the appropriate limit and the additional gradient term will be produced by local matter m(A, Z) = (Au3 + ES ) (27) sources. This anomalous acceleration is generated by the + (Zbu + Nbd)mu + (Zbd + Nbu)md change in the (electromagnetic, gravitational,...) binding Z(Z 1) energy (Dicke, 1964; Dicke, 1969; Haugan, 1979; Eardley, + ZB + NB + 98.25 − MeV α , p n A1/3 EM 1979; Nordtvedt, 1990). Besides, the α-dependence is a   priori composition-dependent (see e.g. Eq. 27). As a con- with N = A Z, the neutron number. This depends sequence, any variation of the fundamental constants will − on our understanding of the description of the nucleus entail a violation of the universality of free fall: the total and can be more sophisticated. For an atom, one would mass of the body being space dependent, an anomalous have to add the contribution of the electrons, Zme. The force appears if energy is to be conserved. The variation form (27) depends on strong, weak and electromagnetic of the constants, deviation from general relativity and quantities. The numerical coefficients B(n,p) are given violation of the weak equivalence principle are in general explicitly by (Gasser and Leutwiller, 1982) expected together, e.g. if there exists a new interaction mediated by a massless scalar field. B α =0.63 MeV B α = 0.13 MeV. (28) p EM n EM − Gravitational methods include the constraints that can It follows that it is in general difficult to disentangle the be derived from the test of the theory of gravity such as effect of each parameter and compare the different meth- the test of the universality of free fall, the motion of the ods. For instance comparing the constraint on µ obtained planets in the Solar system, stellar and galactic evolu- from electromagnetic methods to the constraints on α tions. They are based on the comparison of two time S scales, the first (gravitational time) dictated by grav- and GF from nuclear methods requires to have some the- oretical input such as a theory to explain the fermion ity (ephemeris, stellar ages,. . . ) and the second (atomic masses. Moreover, most of the theoretical models pre- time) is determined by any system not determined by dict a variation of the coupling constants from which one gravity (e.g. atomic clocks,. . . ) (Canuto and Goldman, has to infer the variation of µ etc... 1982). For instance planet ranging, neutron star bina- For macroscopic bodies, the mass has also a negative ries observations, primordial nucleosynthesis and paleon- contribution tological data allow to constraint the relative variation of G respectively to a level of 10−12 10−11, 10−13 10−12, G ρ(~r)ρ(~r′) −12 −10 − − ∆m(G)= d3~rd3~r′ (29) 10 , 10 per year. −2c2 ~r ~r′ Attacking the full general problem is a hazardous and Z | − | dangerous task so that we will first describe the con- from the gravitational binding energy. As a conclusion, straints obtained in the literature by focusing on the fine from (27) and (29), we expect the mass to depend on all structure constant and the gravitational constant and the coupling constant, m(α , α , α , α , ...). EM W S G we will then extend to some other (less studied) com- This has a profound consequence concerning the mo- binations of the constants. Another and complementary tion of any body. Let α be any fundamental constant, approach is to predict the mutual variations of different assumed to be a scalar function and having a time vari- constants in a given theoretical model (see Section VI). ation of cosmological origin so that in the privileged cos- mological rest-frame it is given by α(t). A body of mass m moving at velocity ~v will experience an anomalous ac- III. FINE STRUCTURE CONSTANT celeration 1 dm~v d~v ∂ ln m A. Geological constraints δ~a = α~v.˙ (30) ≡ m dt − dt ∂α 1. The Oklo phenomenon Now, in the rest-frame the body, α has a spatial depen- dence α[(t′ + ~v.~r′/c2)/ 1 v2/c2] so that, as long as Oklo is a prehistoric natural fission reactor that oper- v c, α = (α/c ˙ 2)~v. The− anomalous acceleration can ated about 2 109 yr ago during (2.3 0.7) 105 yr p thus≪ be∇ rewritten as in the Oklo uranium× mine in Gabon. This± phenomenon×

8 −E/k T was discovered by the French Commissariat `al’Energie´ 1 2 2E e B σˆ(E ,T )= σ (E) √EdE Atomique in 1972 (see Naudet, 1974, Maurette, 1976 and r v √π (n,γ) m (k T )3/2 0 r n B Petrov, 1977 for early studies and Naudet, 2000 for a Z (35) general review) while monitoring for uranium ores. Two billion years ago, uranium was naturally enriched (due where the velocity v = 2200m s−1 corresponds to an 235 238 0 to the difference of decay rate between U and U) energy E = 25.3 meV and the effective· neutron flux is 235 0 and U represented about 3.68% of the total uranium similarly given by (compared with 0.72% today). Besides, in Oklo the con- − centration of neutron absorbers which prevent the neu- E/kB T ˆ 2 2E e √ trons from being available for the chain fission was low; φ = v0 3/2 EdE. (36) √π mn (k T ) water played the role of moderator and slowed down fast Z r B neutrons so that they can interact with other 235U and The samples of the Oklo reactors were exposed the reactor was large enough so that the neutrons did not (Naudet, 1974) to an integrated effective fluence φˆdt escape faster than they were produced. of about 1021 neutron cm−2 = 1 kb−1. It implies that R From isotopic abundances of the yields, one can ex- any process with a cross· section smaller than 1 kb can tract informations about the nuclear reactions at the time be neglected in the computation of the abundances; this 144 148 the reactor was operational and reconstruct the reaction includes neutron capture by 62 Sm and 62 Sm. On the 235 rates at that time. One of the key quantity measured is other hand, the fission of 92 U, the capture of neutron 149 147 143 149 the ratio 62 Sm/62 Sm of two light isotopes of samarium by 60 Nd and by 62 Sm with respective cross sections which are not fission products. This ratio of order of 0.9 σ5 0.6 kb, σ143 0.3 kb and σ149 70 kb are the domi- in normal samarium, is about 0.02 in Oklo ores. This low nant≃ processes. It∼ follows that the equations≥ of evolution 149 value is interpreted by the depletion of 62 Sm by thermal for the number densities N147, N148, N149 and N235 of 147 148 149 235 neutrons to which it was exposed while the reactor was 62 Sm, 62 Sm, 62 Sm and 92 U are (Damour and Dyson, active. 1996; Fujii et al., 2000) Shlyakhter (1976) pointed out that the capture cross 149 dN section of thermal neutron by 62 Sm 147 = σˆ φNˆ +ˆσ φNˆ (37) dt − 147 147 f235 235 n + 149Sm 150Sm + γ (33) dN 62 −→ 62 148 =σ ˆ φNˆ (38) dt 147 147 is dominated by a capture resonance of a neutron of en- dN 149 = σˆ φNˆ +ˆσ φNˆ (39) ergy of about 0.1 eV. The existence of this resonance is a dt − 149 149 f235 235 consequence of an almost cancellation between the elec- dN235 ∗ tromagnetic repulsive force and the strong interaction. = σ N235 (40) dt − 5 To obtain a constraint, one first needs to measure the 149 neutron capture cross section of 62 Sm at the time of the where the system has to be closed by using a modified ∗ reaction and to relate it to the energy of the resonance. absorption cross section σ = σ5(1 C) (see references 5 − One has finally to translate the constraint on the varia- in Damour and Dyson, 1996). This system can be inte- tion of this energy on a constraint on the time variation grated under the assumption that the cross sections are of the considered constant. constant and the result compared with the natural abun- The cross section of the neutron capture (33) is dances of the samarium to extract the value ofσ ˆ149 at strongly dependent on the energy of a resonance at the time of the reaction. Shlyakhter (1976) first claimed E = 97.3 meV and is well described by the Breit-Wigner thatσ ˆ149 = 55 8 kb (at cited by Dyson, 1978). Damour r ± formula and Dyson (1996) re-analized this result and found that 57 kb σˆ149 93 kb. Fujii et al. (2000) found that g π ¯h2 Γ Γ ≤ ≤ 0 n γ σˆ149 = 91 6 kb. σ(n,γ)(E)= 2 2 (34) ± 2 mnE (E Er) +Γ /4 By comparing this measurements to the current value − of the cross section and using (35) one can transform it where g (2J + 1)(2s + 1)−1(2I + 1)−1 is a statistical 0 ≡ into a constraint on the variation of the resonance energy. factor which depends on the spin of the incident neutron This step requires to estimate the neutron temperature. s = 1/2, of the target nucleus I and of the compound It can be obtained by using informations from the abun- nucleus J; for the reaction (33), we have g0 =9/16. The dances of other isotopes such as lutetium and gadolinium. total width Γ Γn +Γγ is the sum of the neutron par- ≡ Shlyakhter (1976) deduced that ∆Er < 20meV but as- tial width Γn = 0.533 meV (at Er) and of the radiative sumed the much too low temperature| | of T = 20o C. partial width Γγ = 60.5 meV. Dyson and Damour (1996) allowed the temperature to The effective absorption cross section is defined by vary between 180o C and 700o C and deduced the con- servative bound 120 meV < ∆E < 90 meV and Fujii − r

9 et al. (2000) obtained two branches, the first compatible he concluded that ∆αEM /αEM 20∆gS /gS , leading to −17 −1 ∼ with a null variation ∆Er =9 11 meV and the second α˙ EM /αEM < 10 yr . Irvine (1983a,b) quoted the ± | | −17 −1 indicating a non-zero effect ∆Er = 97 8 meV both bound α˙ EM /αEM < 5 10 yr . The analysis of Sis- for T = 200 400o C and argued that− the± first branch terna and| Vucetich| (1990)× used, according to Damour was favored. − and Dyson (1996) an ill-motivated finite-temperature de- Damour and Dyson (1996) related the variation of Er scription of the excited state of the compound nucleus. to the fine structure constant by taking into account Most of the studies focus on the effect of the fine structure 149 that the radiative capture of the neutron by 62 Sm cor- constant mainly because the effects of its variation can responds to the existence of an excited quantum state be well controlled but, one would also have to take the ef- 150Sm (so that E = E∗ E m ) and by assuming fect of the variation of the Fermi constant, or identically 62 r 150 − 149 − n that the nuclear energy is independent of αEM . It follows αW , (see Section V.A). Horv´ath and Vucetich (1988) in- that the variation of αEM can be related to the difference terpreted the results from Oklo in terms of null-redshift of the Coulomb energy of these two states. The com- experiments. putation of this latter quantity is difficult and requires to be related to the mean-square radii of the protons in the isotopes of samarium and Damour and Dyson (1996) 2. α-decay showed that the Bethe-Weiz¨acker formula (26) overesti- The fact that α-decay can be used to put constraints mates by about a factor the 2 the αEM -sensitivity to the resonance energy. It follows from this analysis that on the time variation of the fine structure constant was pointed out by Wilkinson (1958) and then revived by ∆Er Dyson (1972, 1973). The main idea is to extract the αEM 1.1 MeV, (41) ∆αEM ≃− αEM -dependence of the decay rate and to use geological samples to bound its time variation. which, once combined with the constraint on ∆Er, im- A The decay rate, λ, of the α-decay of a nucleus Z X of plies charge Z and atomic number A − − 7 7 A+4 A 4 0.9 10 < ∆αEM /αEM < 1.2 10 (42) X X+ He (45) − × × Z+2 −→ Z 2 − − corresponding to the range 6.7 10 17 yr 1 < is governed by the penetration of the Coulomb barrier −17 −1 − × α˙ EM /αEM < 5.0 10 yr ifα ˙ EM is assumed constant. described by the Gamow theory and well approximated × This tight constraint arises from the large amplification by between the resonance energy ( 0.1 eV) and the sensi- −4πZα c/v ∼ λ Λ(α , v)e EM (46) tivity ( 1 MeV). Fujii et al. (2000) re-analyzed the data ≃ EM and included∼ data concerning gadolinium and found the −17 −1 where v is the escape velocity of the α particle and where favored resultα ˙ EM /αEM = ( 0.2 0.8) 10 yr which − ± × Λ is a function that depends slowly on αEM and v. It corresponds to follows that the variation of the decay rate with respect ∆α /α = ( 0.36 1.44) 10−8 (43) to the fine structure constant is well approximated by EM EM − ± × d ln λ c 1 d ln ∆E and another branchα ˙ /α = (4.9 0.4) 10−17 yr−1. 4πZ 1 (47) EM EM ± × dα ≃− v − 2 d ln α The first bound is favored given the constraint on the EM  EM  temperature of the reactor. Nevertheless, the non-zero where ∆E 2mv2 is the decay energy. Consider- result cannot be eliminated, even using results from ing that the≡ total energy is the sum of the nuclear en- gadolinium abundances (Fujii, 2002). Note however that ergy Enuc and of the Coulomb energy EEM /80 MeV −1/3 ≃ spliting the analysis in two branches seems to be at odd Z(Z 1)A αEM and that the former does not depend et al − with the aim of obtaining a constraint. Olive . (2002) on αEM , one deduces that refined the analysis and confirmed the previous results. − d ln ∆E ∆E 1 Earlier studies include the original work by Shlyakhter f(A, Z) (48) (1976) who found that α˙ /α < 10−17 yr−1 corre- d ln α ≃ 0.6 MeV EM EM EM   sponding to | | with f(A, Z) (Z + 2)(Z + 1)(A + 4)−1/3 ≡ ∆α /α < 1.8 10−8. (44) Z(Z 1)A−1/3 . It follows that the sensitivity of the EM EM  | | × decay− rate− on the fine structure constant is given by In fact he stated that the variation of the strong interac-  tion coupling constant was given by ∆g /g ∆E /V d ln λ S S ∼ r 0 s where V0 50 MeV is the depth of a square potential ≡ d ln αEM well. Arguing≃ that the Coulomb force increases the av- c 0.3 MeV 4πZ α f(A, Z) 1 . (49) erage inter-nuclear distance by about 2.5% for A 150, ≃ v EM ∆E − ∼   

10 238 This result can be qualitatively understood since an in- Gold (1968) studied the fission of 92 U with a decay −17 −1 crease of αEM induces an increase in the height of the time of 7 10 yr . He obtained a sensitivity (49) Coulomb barrier at the nuclear surface while the depth of s = 120.× Ancient rock samples allow to conclude, of the nuclear potential below the top remains the same. after comparison of rock samples dated by potassium- It follows that the α particle escapes with greater en- argon and rubidium-strontium, that the decay time of 238 ergy but at the same energy below the top of the barrier. 92 U has not varied by more than 10% in the last 2 Since the barrier becomes thinner at a given energy be- 109 yr. Indeed, the main uncertainty comes from the× low its top, the penetrability increases. This computation dating of the rock. Gold (1968) concluded on that basis indeed neglects the effect of a variation of αEM on the nu- that cleus that can be estimated to be dilated by about 1% if −4 α increases by 1%. ∆αEM /αEM < 4.66 10 (54) EM | | × Wilkinson (1958) considered the most favorable α- −13 −1 238 which corresponds to α˙ EM /αEM < 2.3 10 yr if decay reaction which is the decay of 92 U | | × one assumes thatα ˙ EM is constant. This bound is indeed 238 235 4 92 U 90 Th + 2He (50) comparable, in order of magnitude, to the one obtained → by α-decay data. for which ∆E 4.27 MeV (s 540). By comparing the ≃ ≃ Chitre and Pal (1968) compared the uranium-lead and geological dating of the Earth by different methods, he potassium-argon dating methods respectively governed 238 235 concluded that the decay constant λ of U, U and by α- and β- decay to date stony meteoric samples. Both 232 Th have not changed by more than a factor 3 or 4 methods have different α -dependence (see below) and 9 EM during the last 3 4 10 years from which it follows they concluded that − × −12 −1 that α˙ EM /αEM < 2 10 yr and thus | | × ∆α /α < (1 5) 10−4. (55) ∆α /α < 8 10−3. (51) | EM EM | − × | EM EM | × Dyson (1972) argued on similar basis that the decay rate This bound is very rough but it agrees with Oklo on of 238U has not varied by more than 10% in the past comparable time scale. This constraint was revised by 92 2 109 yr so that Dyson (1972) who claimed that the decay rate has not × changed by more than 20%, during the past 2 109 years, −3 ∆αEM /αEM < 10 . (56) which implies × | | ∆α /α < 4 10−4. (52) | EM EM | × 4. β-decay These data were recently revisited by Olive et al. (2002). Using laboratory and meteoric data for 147Sm (∆E Dicke (1959) stressed that the comparison of the 2.31 MeV, s 770) for which ∆λ/λ was estimated to be≃ rubidium-strontium and potassium-argon dating meth- of order 7.5 ≃10−3 they concluded that × ods to uranium and thorium rates constrains the varia- −5 tion of α . He concluded that there was no evidence to ∆αEM /αEM < 10 . (53) EM | | rule out a time variation of the β-decay rate. Peres (1968) discussed qualitatively the effect of a fine structure constant increasing with time arguing that the 3. Spontaneous fission nuclei chart would have then been very different in the past since the stable heavy element would have had N/Z α-emitting nuclei are classified into four generically in- ratios much closer to unity (because the deviation from dependent decay series (the thorium, neptunium, ura- unity is mainly due to the electrostatic repulsion between nium and actinium series). The uranium series is the protons). For instance 238U would be unstable against longest known series. It begins with 238U, passes a sec- 92 double β-decay to 238Pu. One of its arguments to claim ond time through Z =92 (234U) as a consequence of an 92 that α has almost not varied lies in the fact that 208Pb α-β-decay and then passes by five α-decays and finishes EM existed in the past as 208Rn, which is a gas, so that the by an α-β-β-decay to end with 206Pb. The longest lived 82 lead ores on Earth would be uniformly distributed. member is 238U with a half-life of 4.47 109 yr, which 92 As long as long-lived isotopes are concerned for which four orders of magnitude larger than the× second longest the decay energy ∆E is small, we can use a non- lived elements. 238U thus determines the time scale of 92 relativistic approximation for the decay rate the whole series. p± The expression of the lifetime in the case of sponta- λ =Λ± (∆E) (57) neous fission can be obtained from Gamow theory of α- − decay by replacing the charge Z by the product of the respectively for β -decay and electron capture. Λ± are charges of the two fission products. functions that depend smoothly on αEM and which can

11 −4 thus be considered constant, p+ = ℓ + 3 and p− =2ℓ +2 ∆αEM /αEM < 9 10 (63) are the degrees of forbiddenness of the transition. For | | × during the past 3 109 years. In a re-analysis (Dyson, high-Z nuclei with small decay energy ∆E, the exponent × p becomes p =2+ 1 α2 Z2 and is independent of ℓ. 1972) he concluded that the rhenium decay-rate did not EM 9 It follows that the sensitivity− (49) becomes change by more than 10% in the past 10 years so that p −6 d ln ∆E ∆αEM /αEM < 5 10 . (64) s = p . (58) | | × d ln αEM 187 Using a better determination of the decay rate of 75 Re 187 The second factor can be estimated exactly as in based on the growth of Os over a 4-year period into Eq. (48) for α-decay but with f(A, Z) = (2Z + a large source of osmium free rhenium, Lindner et al. 1)A−1/3[0.6 MeV/∆E], the , + signs corresponding± re- (1986) deduced that spectively to β-decay and electron− capture. ∆α /α = ( 4.5 9) 10−4 (65) The laboratory determined decay rates of rubidium to EM EM − ± × strontium by β-decay over a 4.5 109 yr period. This was recenlty updated (Olive et al.×, 2002) to take into account the improvements 87Rb 87Sr+ν ¯ + e− (59) 37 −→ 38 e in the analysis of the meteorite data which now show that and to potassium to argon by electron capture the half-life has not varied by more than 0.5% in the past 4.6 Gyr (i.e. a redshift of about 0.45). This implies that 40 − 40 K+ e Ar + νe (60) 19 −→ 18 ∆α /α < 3 10−7. (66) | EM EM | × are respectively 1.41 10−11 yr−1 and 4.72 10−10 yr−1. The decay energies× are respectively ∆E =× 0.275 MeV We just reported the values of the decay rates as used and ∆E = 1.31 MeV so that s 180 and s 30. at the time of the studies. One could want to update Peebles and Dicke (1962) compared≃ these − laboratories≃ − de- these constraints by using new results on the measure- termined values with their abundances in rock samples ments on the decay rate,. . . . Even though, they will not, after dating by uranium-lead method and with meteorite in general, be competitive with the bounds obtained by data (dated by uranium-lead and lead-lead). They con- other methods. These results can also be altered if the neutrinos are massive. cluded that the variation of αEM with αG cannot be ruled out by comparison to meteorite data. Later, Yahil (1975) used the concordance of the K-Ar and Rb-Sr geochemical 5. Conclusion ages to put the limit

∆α /α < 1.2 (61) All the geological studies are on time scales of order of | EM EM | the age of the Earth (typically z 0.1 0.15 depending ∼ − over the past 1010 yr. on the values of the cosmological parameters). The case of the decay of osmium to rhenium by electron The Oklo data are probably the most powerful geo- emission chemical data to study the variation of the fine structure constant but one has to understand and to model care- 187Re 187Os+¯ν + e− (62) fully the correlations of the variation of α and g as 75 −→ 76 e W S well as the effect of µ (see the recent study by Olive et was first considered by Peebles and Dicke (1962). They al., 2002). This difficult but necessary task remains to noted that the very small value of its decay energy be done. ∆E 2.5 keV makes it a very sensitive indicator of ≃ The β-decay results depend on the combination s 2 the variation of αEM . In that case p 2.8 so that α α and have the advantage not to depend on G. s 18000. It follows that a change of≃ about 10−2% EM W ≃ − They may be considered more as historical investigations of αEM will induce a change in the decay energy of order than as competitive methods to constraint the variation of the keV, that is of the order of the decay energy itself. of the fine structure constant, especially in view of the With a time decreasing αEM , the decay rate of rhenium Oklo results. The dependence and use of this method on will have slowed down and then osmium will have become αS was studied by Broulik and Trefil (1971) and Davies unstable. Peebles and Dicke (1962) did not have reliable (1972) (see section V.B). laboratory determination of the decay rate to put any constraint. Dyson (1967) compared the isotopic analysis of molybdenite ores, the isotopic analysis of 14 iron me- B. Atomic spectra teorites and laboratory measurements of the decay rate. Assuming that the variation of the decay energy comes The previous bounds on the fine structure constant as- entirely from the variation of αEM , he concluded that sume that other constants like the Fermi constant do not

12 vary. The use of atomic spectra may offer cleaner tests and 2p1/2, where we have added in indices the total elec- since we expect them to depend mainly on combinations tron angular moment quantum number J. The second 2 of αEM , µ and gp. correction arises from the (v/c) -relativistic terms and is We start by recalling some basics concerning atomic of the form spectra in order to desribe the modelling of the spec- P4 tra of many-electron systems which is of great use while Wrel = 3 2 (71) studying quasar absorption spectra. We then focus on −8mec laboratory experiments and the results from quasar ab- and it is easy to see that its amplitude is also of order sorption spectra. 2 Wrel αEM H0. The third and last correction, known as the Darwin∼ term, arises from the fact that in the Dirac equation the interaction between the electron and the 1. α -dependence of atomic spectra EM Coulomb field is local. But, the non-relativist approx- As an example, let us briefly recall the spectrum of imation leads to a non-local equation for the electron the hydrogen atom (see e.g. Cohen-Tannoudji et al., spinor that is sensitive to the field on a zone of order of r 1977). As long as we neglect the effect of the spins and we the Compton wavelength centered in . It follows that work in the non-relativistic approximation, the spectrum 2 π¯h q2 is simply obtained by solving the Schr¨odinger equation W = δ(r). (72) D m2c2 with Hamiltonian e

2 2 P e The average in an atomic state is of order WD = 2 2 2 2 2 2 4 2 h i H0 = (67) 0 π¯h q /(2mec ) ψ( ) mec αEM αEM H0. In con- 2me − 4πε0r | | ∼ ∼ 2 clusion all the relativistic corrections are of order αEM 2 ∼ the eigenfunctions of which is of the form ψnlm = (v/c) . The energy of a fine structure level is Rn(r)Ylm(θ, φ) where n is the principal quantum num- E m c2 n 3 ber. This solution has an energy E = m c2 I e α4 + ... nlJ e − n2 − 2n4 J +1/2 − 4 EM   EI me En = 2 1 (68) 4 − n − mp and is independent of the quantum number l.   A much finer effect, referred to as hyperfine structure, independently of the quantum numbers l and m satisfy- arises from the interaction between the spins of the elec- ing 0 l

Since r is of order of the Bohr radius, it follows that 4This is valid to all 2 WS.O. α H0. The splitting is indeed small: for in- EM order in αEM and the Dirac equation directly gives EnlJ = ∼ −5 − stance, it is of order 4 10 eV between the levels 2p −2 1/2 3/2 2 2 − − 2 − 2 × mec 1+ αEM n J 1/2+ (J + 1/2) αEM . h p
i 13 level in a series of hyperfine levels labelled by F [ J ω = ω + q x + q y (76) ∈ | − 0 1 2 I , I + J]. For instance for the level 2s1/2 and 2p1/2, we | (0) 2 (0) 4 have J =1/2 and F can take the two values 0 and 1, for with x [αEM /αEM ] 1 and y [αEM /αEM ] 1. As ≡ − ≡ et al. − the level 2p3/2, J = 3/2 and F =1 or F = 2 etc...(see an example, let us cite the result of Dzuba (1999b) figure 4 for an example). This description neglects the for Fe II quantum aspect of the electromagnetic field; one effects of the coupling of the atom to this field is to lift the 6d J =9/2 ω = 38458.9871+ 1394x + 38y degeneracy between the levels 2s1/2 and 2p1/2. This is J =7/2 ω = 38660.0494+ 1632x +0y called the Lamb effect. 6f J = 11/2 ω = 41968.0642+ 1622x +3y In more complex situations, the computation of the J =9/2 ω = 42114.8329+ 1772x +0y spectrum of a given atom has to take all these effects into account but the solution of the Schr¨odinger equa- J =7/2 ω = 42237.0500+ 1894x +0y tion depends on the charge distribution and has to be 6p J =7/2 ω = 42658.2404+ 1398x 13y (77) − performed numerically. −1 The easiest generalization concerns hydrogen-like with the frequency in cm for transitions from the atoms of charge Z for which the spectrum can be ob- ground-state. An interesting case is Ni II (Dzuba et al., 2 2 2001) which has large relativistic effects of opposite signs tained by replacing e by (Ze) and mp by Amp. For an external electron in a many-electron atoms, the elec- 2f J =7/2 ω = 57080.373 300x tron density near the nucleus is given (see e.g. Dzuba − 2 3 et al., 1999a) by Z Z/(n∗a ) where Z is the effective 6d J =5/2 ω = 57420.013 700x a 0 a − charge felt by the external electron outside the atom, 6f J =5/2 ω = 58493.071+ 800x. (78) n∗ an effective principal quantum number defined by 2 2 En∗ = EI Za/n∗. It follows that the relativistic cor- Such results are particularly useful to compare with spec- rections− to the energy level are given by tra obtained from quasar absorption systems as e.g. in the analysis by Murphy et al. (2001c). EI 2 2 2 n∗ Za Za In conclusion, the key point is that the spectra of atoms ∆En∗,l,J = Z Z α 1 . n4 a EM J +1/2 − Z − 4Z depend mainly on µ, α and g and contain terms both ∗    EM p in α2 and α4 and that typically Such a formula does not take into account many-body EM EM effects and one expects in general a formula of the form H = α2 H + α4 W + g µ2α4 W , (79) 2 2 EM 0 EM fine p EM hyperfine ∆En∗,l,J = En∗Z αEM [1/J +1/2 C(Z,J,l)]/n∗. Dzuba et al. (1999b) developed a method− to compute the atomic so that by comparinge differentf kind of transitionsf in dif- spectra of many-electrons atoms including relativistic ef- ferent atoms there is hope to measure these constants fects. It is based on many-body perturbation theory despite the fact that αS plays a role via the nuclear mag- (Dzuba et al., 1996) including electron-electron correla- netic moment. We describe in the next section the lab- tions and use a correlation-potential method for the atom oratory experiments and then turn to the measurement (Dzuba et al., 1983). of quasar absorption spectra. Laboratory measurements can provide these spectra (0) but only for αEM = αEM . In order to detect a variation of αEM , one needs to compute them for different values 2. Laboratory experiments of αEM . Dzuba et al. (1999a) describe the energy levels within one fine-structure multiplet as Laboratory experiments are based on the comparison either of different atomic clocks or of atomic clock with 2 4 ultra-stable oscillators. They are thus based only on the αEM αEM E = E0 + Q1 1 + Q2 1 quantum mechanical theory of the atomic spectra. They  α(0) −   α(0) −  EM ! EM ! also have the advantage to be more reliable and repro-  2   4  ducible, thus allowing a better control of the systematics α α L S EM L S 2 EM and a better statistics. Their evident drawback is their + K1 . (0) + K2( . ) (0) (75) αEM ! αEM ! short time scales, fixed by the fractional stability of the least precise standards. This time scale is of order of a where E0, Q1 and Q2 describe the configuration center. L S month to a year so that the obtained constraints are re- The terms in . induce the spin-orbit coupling, sec- stricted to the instantaneous variation today, but it can ond order spin-orbit interaction and the first order of the be compensated by the extreme sensibility. They involve Breit interaction. Experimental data can be fitted to get the comparison of either ultra-stable oscillators to differ- K1 and K2 and then numerical simulations determine Q1 ent composition or of atomic clocks with different species. and Q2. The result is conveniently written as Solid resonators, electronic, fine structure and hyperfine

14 2 structure transitions respectively give access to R∞/α , 8 1 2 z d∆n EM ν = I + α g Z 1 F (α Z) 2 2 alkali EM I 3 rel EM R∞, R∞α and g µR∞α . 3 2 n − dn EM p EM   ∗   Turneaure and Stein (1974) compared cesium atomic (1 δ)(1 ǫ)µR∞c, (86) clocks with superconducting microwave cavities oscilla- − − tor. The frequency of the cavity-controlled oscillators where z is the charge of the remaining ion once the va- was compared during 10 days that one of a cesium beam. lence electron has been removed and ∆n = n n∗. The − −1 − The relative drift rate was ( 0.4 3.4) 10 14 day . term (1 δ) is the correction to the potential with respect The dimensions of the cavity− depends± on× the Bohr ra- to the Coulomb− potential and (1 ǫ) a correction for the dius of the atom while the cesium clock frequency de- finite size of the nuclear magnetic− dipole moment. It is 2 pends on gpµαEM (hyperfine transition). It follows that estimated that δ 4% 12% and ǫ 0.5%. Frel(αEM Z) 3 ≃ − ≃ νCe /νcavity gpµαEM so that is the Casimir relativistic contribution to the hyperfine ∝ structure and one takes advantage of the increasing im- d 3 −12 −1 portance of Frel as the atomic number increases (see fig- ln gpµα < 4.1 10 yr . (80) dt EM × ure 5). It follows that
Godone et al. (1993) compared the frequencies of ce- d ν α˙ EM d ln Frel(αEM Z) sium and magnesium atomic beams. The cesium clock, ln alkali = , (87) dt ν α d ln α used to define the second in the SI system of units, is H EM EM hyperfine transition based on the F =3, mF =0 F = where νH is the frequency of a H maser and when com- 2 133 → 4, mF = 0 in the ground-state 6 s1/2 of Ce with fre- paring two alkali atoms quency given, at lowest order and neglecting relativistic d ν α˙ d ln F d ln F and quantum electrodynamic corrections, by ln alkali1 = EM rel rel . (88) dt ν α d ln α − d ln α 3 2 alkali2 EM  EM 1 EM 2 32cR∞Zs αEM ν = gI µ 9.2 GHz, (81) Ce 3n3 ∼ The comparison of different alkali clocks was performed and the comparison of Hg+ ions with a cavity tuned H where Zs the effective nuclear charge and gI the cesium maser over a period of 140 days led to the conclusion that nucleus gyromagnetic ratio. The magnesium clock is based on the frequency of the fine structure transition −14 −1 α˙ EM /αEM < 3.7 10 yr . (89) 24 | | × 3p1 3p0, ∆mj = 0 in the meta-stable triplet of Mg → This method constrains in fact the variation of the quan- 4 2 cR∞Z α tity α g /g . One delicate point is the evaluation of ν = s EM 601 GHz. (82) EM p I Hg 6n3 ∼ the correction function and the form used by Prestage et 2 al. (1995) [Frel 1 + 11(ZαEM ) /6+ ...] differs with It follows that ∼ 2 the 1s [Frel 1 + 3(ZαEM ) /2 + ...] and 2s [Frel ∼2 ∼ d ν d 1 + 17(ZαEM ) /8+ ...] results for hydrogen like atoms ln Ce = ln (g µ) 1 10−2 . (83) dt ν dt I × ± (Breit, 1930). Hg  
Sortais et al. (2001) compared a rubidium to a ce- The experiment led to the bound sium clock over a period of 24 months and deduced that −15 −1 d ln(νRb /νCs )/dt = (1.9 3.1) 10 yr , hence improv- d ± × ln(g µ) < 5.4 10−13 yr−1 (84) ing the uncertainty by a factor 20 relatively to Prestage dt p × et al. (1995). Assuming gp constant, they deduced

− − after using the constraint d ln(gp/gI )/dt < 5.5 15 1 α˙ EM /αEM = (4.2 6.9) 10 yr (90) 10−14 yr−1 (Demidov et al., 1992). When combined with× ± × the astrophysical result by Wolfe et al. (1976) on the con- if all the drift can be attributed to the Casimir relativistic 2 straint of gpµαEM (see Section V.D) it is deduced that correction Frel. All the results and characteristics of these experiments −13 −1 α˙ EM /αEM < 2.7 10 yr . (85) are summed up in table III. Recently, Braxmaier et al. | | × (2001) proposed a new method to test the variability of We note that relativistic corrections were neglected. αEM and µ using electromagnetic resonators filled with Prestage et al. (1995) compared the rates of differ- a dielectric. The index of the dielectric depending on ent atomic clocks based on hyperfine transitions in alkali both αEM and µ, the comparison of two oscillators could atoms with different atomic numbers. The frequency of lead to an accuracy of 4 10−15 yr−1. Torgerson (2000) the hyperfine transition between I 1/2 states is given × ± proposed to compare atom-stabilized optical frequency by (see e.g. Vanier and Audoin, 1989) using an optical resonator. On an explicit example us- ing indium and thalium, it is argued that a precision of

15 −18 α˙ EM /αEM 10 /t, t being the time of the experiment, 7. Atmospheric dispersion across the spectral direc- can be reached.∼ tion of the spectrograph slit can stretch the spec- Finally, let us note that similar techniques were used trum. It was shown that this can only mimic a to test local Lorentz invariance (Lamoreaux et al., 1986, negative ∆αEM /αEM (Murphy et al., 2001b). Chupp et al., 1989) and CPT symmetry (Bluhm et al., 2002). In the former case, the breakdown of local Lorentz 8. The presence of a magnetic field will shift the en- invariance would cause shifts in the energy levels of atoms ergy levels by Zeeman effect. and nuclei that depend on the orientation of the quantiza- 9. Temperature variations during the observation will tion axis of the state with respect to a universal velocity change the air refractive index in the spectrograph. vector, and thus on the quantum numbers of the state. 10. Instrumental effects such as variations of the intrin- sic instrument profile have to be controlled. 3. Astrophysical observations The effect of these possible systematic errors are dis- The observation of spectra of distant astrophysical ob- cussed by Murphy et al. (2001b). In the particular case jects encodes information about the atomic energy levels of the comparison of hydrogen and molecular lines, Wik- at the position and time of emission. As long as one lind and Combes (1997) argued that the detection of the sticks to the non-relativistic approximation, the atomic variation of µ was limited to ∆µ/µ 10−5. A possibility ≃ transition energies are proportional to the Rydberg en- to reduce the systematics is to look at atoms having rel- ergy and all transitions have the same αEM -dependence, ativistic corrections of different signs (see Section III.B) so that the variation will affect all the wavelengths by the since the systematics are not expected, a priori, to simu- same factor. Such a uniform shift of the spectra can not late the correlation of the shift of different lines of a mul- be distinguished from a Doppler effect due to the motion tiplet (see e.g. the example of Ni II Dzuba et al., 2001). of the source or to the gravitational field where it sits. Besides the systematics, statistical errors were important The idea is to compare different absorption lines from in early studies but have now enormously decreased. different species or equivalently the redshift associated An efficient method is to observe fine-structure dou- with them. According to the lines compared one can blets for which extract information about different combinations of the 2 4 α Z R∞ constants at the time of emission (see table I). EM −1 ∆ν = 3 cm , (91) While performing this kind of observations a number 2n of problems and systematic effects have to be taken into ∆ν being the frequency splitting between the two lines account and controlled: of the doublet andν ¯ the mean frequency (Bethe and 2 Salpeter, 1977). It follows that ∆ν/ν¯ αEM and thus 1. Errors in the determination of laboratory wave- 2 ∝ ∆ ln λ z/∆ ln λ 0 = [1 + ∆αEM /αEM ] . It can be inverted lengths to which the observations are compared, | | ¯ to give ∆αEM /αEM as a function of ∆λ and λ as 2. while comparing wavelengths from different atoms ∆α 1 ∆λ ∆λ one has to take into account that they may be lo- EM (z)= / 1 . (92) α 2 λ¯ λ¯ − cated in different regions of the cloud with different  EM   z  0  velocities and hence with different Doppler redshift. As an example, it takes the following form for Si IV (Var- 3. One has to ensure that there is no light blending. shalovich et al., 1996a) 4. The differential isotopic saturation has to be con- ∆α ∆λ EM (z)=77.55 0.5. (93) trolled. Usually quasars absorption systems are ¯ αEM λ z − expected to have lower heavy element abundances     (Prochoska and Wolfe, 1996, 1997, 2000). The spa- Since the observed wavelengths are redshifted as λobs = tial inhomogeneity of these abundances may also λem(1 + z) it reduces to play a role. ∆α ∆z EM (z)=77.55 . (94) 5. Hyperfine splitting can induce a saturation similar αEM 1+¯z to isotopic abundances.   As a conclusion, by measuring the two wavelengths of 6. The variation of the velocity of the Earth during the doublet and comparing to laboratory values, one can the integration of a quasar spectrum can induce measure the time variation of the fine structure constant. differential Doppler shift, This method has been applied to different systems and

is the only one that gives a direct measurement of αEM .

16 Savedoff (1956) was the first to realize that the fine and it is based on the measurement of the difference of wave- hyperfine structures can help to disentangle the redshift lengths which can be measured much more accurately effect from a possible variation of αEM and Wilkinson than (broader) emission lines and (2) these transitions (1958) pointed out that “the interpretation of redshift correspond to transitions from a single level and are thus of spectral lines probably implies that atomic constants not affected by differences in the radial velocity distribu- have not changed by less than 10−9 parts per year”. tions of different ions. They used data on 1414 absorption Savedoff (1956) used the data by Minkowski and Wol- doublets of C IV, N V, O VI, Mg II, Al III and Si IV and son (1956) of the spectral lines of H, N II, O I, O II, obtained Ne III and N V for the radio source Cygnus A of redshift ∆α /α = (2.1 2.3) 10−3 (99) z 0.057. Using the data for the fine-structure doublet EM EM ± × of∼ N II and Ne III and assuming that the splitting was at z 3.2 and d ln α /dz < 5.6 10−4 between z =0.2 proportional to α2 (1 + z) led to ∼ | EM | × EM and z = 3.7 at 2σ level. In these measurements Si IV, ∆α /α = (1.8 1.6) 10−3. (95) the most widely spaced doublet, is the most sensitive to EM EM ± × a change in αEM . The use of a large number of systems Bahcall and Salpeter (1965) used the fine structure split- allows to reduce the statistical errors and to obtain a red- ting of the O III and Ne III emission lines in the spectra shift dependence after averaging over the celestial sphere. of the quasi-stellar radio sources 3C 47 and 3C 147. Bah- Note however that averaging on shells of constant red- call et al. (1967) used the observed fine structure of Si II shift implies that we average over a priori non-causally and Si IV in the quasi-stellar radio sources 3C 191 to connected regions in which the value of the fine structure deduce that constant may a priori be different. This result was fur- ther constrained by Varshalovitch and Potekhin (1994) ∆α /α = ( 2 5) 10−2 (96) EM EM − ± × who extended the catalog to 1487 pairs of lines and got at a redshift z =1.95. Gamow (1967) criticized this mea- ∆α /α < 1.5 10−3 (100) surements and suggested that the observed absorption | EM EM | × lines were not associated with the quasi-stellar source but at z 3.2. It was also shown that the fine structure split- were instead produced in the intervening galaxies. But ting∼ was the same in eight causally disconnected regions Bahcall et al. (1967) showed on the particular example at z =2.2ata3σ level. of 3C 191 that the excited fine structure states of Si II Cowie and Songaila (1995) improved the previous anal- were seen to be populated in the spectrum of this object ysis to get and that the photon fluxes required to populate these −4 states were orders of magnitude too high to be obtained ∆αEM /αEM = ( 0.3 1.9) 10 (101) in intervening galaxies. − ± × Bahcall and Schmidt (1967) then used the absorption for quasars between z = 2.785 and z = 3.191. Var- lines of the O III multiplet of the spectra of five radio shalovich et al. (1996a) used the fine-structure doublet galaxies with redshift of order z 0.2 to improve the of Si IV to get former bound to ∼ ∆α /α = (2 7) 10−5 (102) EM EM ± × ∆α /α = (1 2) 10−3, (97) EM EM ± × at 2σ for quasars between z = 2.8 and z = 3.1 (see also considering only statistical errors. Varshalovich et al., 1996b). Wolfe et al. (1976) studied the spectrum of Varshalovich et al. (2000a) studied the doublet lines of AO 0235+164, a BL Lac object with redshift z 0.5. Si IV, C IV and Ng II and focused on the fine-structure From the comparison of the hydrogen hyperfine∼ fre- doublet of Si IV to get + quency with the resonance line for Mg , they obtained −5 2 ∆α /α = ( 4.5 4.3[stat] 1.4[syst]) 10 a constraint on g µα (see Section V.D). From the EM EM p EM − ± ± × comparison with the Mg+ fine structure separations they (103) + constrained gpµαEM , and the Mg fine structure doublet splitting gave for z = 2 4. An update of this analysis (Ivanchik et al., 1999) with− 20 absorption systems between z = 2 and −2 ∆αEM /αEM < 3 10 . (98) z =3.2 gave | | × Potekhin and Varshalovich (1994) extended this method −5 ∆αEM /αEM = ( 3.3 6.5[stat] 8[syst]) 10 . (104) based on the absorption lines of alkali-like atoms and − ± ± × compared the wavelengths of a catalog of transitions Murphy et al. (2001d) used the same method with 21 2s1/2 2p3/2 and 2s1/2 2p1/2 for a set of five ele- Si IV absorption system toward 8 quasars with redshift ments.− The advantages of− such a method are that (1) z 2 3 to get ∼ −

17 −5 ∆αEM /αEM = ( 0.5 1.3) 10 (105) al. (1999). Over the whole sample (z =0.5 1.8) it gives − ± × the constraint − hence improving the previous constraint by a factor 3. −5 ∆αEM /αEM = ( 0.7 0.23) 10 . (110) Recently Dzuba et al. (1999a,b) and Webb et al. − ± × (1999) introduced a new method referred to as the many Webb et al. (2001) re-analyzed their initial sample multiplet method in which one correlates the shift of the and included new optical QSO data to have 28 absorp- tion systems with redshift z =0.5 1.8 plus 18 damped absorption lines of a set of multiplets of different ions. It − is based on the parametrization (76) of the computation Lyman-α absorption systems towards 13 QSO plus 21 of atomic spectra. One advantage is that the correlation Si IV absorption systems toward 13 QSO . The analysis between different lines allows to reduce the systematics. used mainly the multiplets of Ni II, Cr II and Zn II and An improvement is that one can compare the transitions Mg I, Mg II, Al II, Al III and Fe II were also included. from different ground-states and using ions with very dif- One improvement compared with the analysis by Webb ferent atomic mass also increases the sensitivity because et al. (1999) is that the “q” coefficient of Ni II, Cr II and the difference between ground-states relativistic correc- Zn II in Eq. (76) vary both in magnitude and sign so that tions can be very large and even of opposite sign (see the lines shift in opposite directions. The data were reduced example of Ni II by Dzuba et al., 2001). to get 72 individual estimates of ∆αEM /αEM spanning a Webb et al. (1999) analyzed one transition of the Mg II large range of redshift. From the Fe II and Mg II sample doublet and five Fe II transitions from three multiplets. they obtained The limit of accuracy of the method is set by the fre- − ∆α /α = ( 0.7 0.23) 10 5 (111) quency interval between Mg II 2796 and Fe II 2383 which EM EM − ± × induces a fractional change of ∆α /α 10−5. Using EM EM ∼ for z =0.5 1.8 and from the Ni II, Cr II and Zn II they the simulations by Dzuba et al. (1999a,b) it can be de- got − duced that a change in αEM induces a large change in the −5 spectrum of Fe II and a small one for Mg II (the mag- ∆αEM /αEM = ( 0.76 0.28) 10 (112) − ± × nitude of the effect being mainly related to the atomic for z =1.8 3.5ata4σ level. The fine-structure of Si IV charge). The method is then to measure the shift of the − Fe II spectrum with respect to the one of Mg II. This com- gave parison increases the sensitivity compared with methods ∆α /α = ( 0.5 1.3) 10−5 (113) using only alkali doublets. Using 30 absorption systems EM EM − ± × toward 17 quasars they obtained for z =2 3. This series− of results is of great importance since all ∆α /α = ( 0.17 0.39) 10−5 (106) EM EM other constraints are just upper bounds. Note that they − ± × −5 ∆αEM /αEM = ( 1.88 0.53) 10 (107) are incompatible with both Oklo (z 0.1) and mete- − ± × orites data (z 0.45) if the variation∼ is linear with respectively for 0.6 redshifts larger than 1 and of physics. Among the first questions that arise, it is particularly in the range z 0.9 1.2. The summary interesting to test whether this variation is compatible of these measurements are depicted∼ − on figure 7. A pos- with other bounds (e.g. test of the universality of free sible explanation is a variation of the isotopic ratio but fall), to study the level of detection needed by the other the change of 26Mg/24Mg would need to be substantial experiments knowing the level of variation by Webb et to explain the result (Murphy et al., 2001b). Calibra- al. (2001), to sort out the amplitude of the variation of tion effects can also be important since Fe II and Mg II the other constants and to be ensure that no systematic lines are situated in different order of magnitude of the effects has been forgotten. For instance, the fact that spectra. Mg II and Fe II are a priori not in the same region of Murphy et al. (2001a) extended this technique of fit- the cloud was not modelled; this could increase the er- ting of the absorption lines to the species Mg I, Mg II, rors even if it is difficult to think that it can mimic the Al II, Al III, Si II, Cr II, Fe II, Ni II and Zn II for observed variation of α . If one forgets the two points 49 absorption systems towards 28 quasars with redhsift EM arising from HI 21 cm and molecular absorption systems z 0.5 3.5 and got ∼ − (hollow squares in figure 7), the best fit of the data of −5 figure 7 does not seem to favor today’s value of the fine ∆αEM /αEM = ( 0.2 0.3) 10 (108) − ± × −5 structure constant. This could indicate an unknown sys- ∆αEM /αEM = ( 1.2 0.3) 10 (109) − ± × tematic effect. Besides, if the variation of αEM is mono- respectively for 0.5

18 C. Cosmological constraints and Peebles (1968) and taking into account only the re- combination of hydrogen, the equation of evolution of the 1. Cosmic microwave background ionization fraction takes the form dx B B The Cosmic Microwave Background Radi- e = β (1 x )exp 1 − 2 n x2 . dt C − e − k T − R p e ation (CMBR) is composed of the photons emitted at   B   2 the time of the recombination of hydrogen and helium Bn = EI /n is the energy of the nth hydrogen atomic when the universe was about 300,000 years old [see e.g. level, β−is the ionization coefficient, the recombination Hu and Dodelson (2002) or Durrer (2002) for recent re- coefficient, the correction constantR due to the redshift C views on CMBR physics]. This radiation is observed to of Ly-α photons and to 2-photon decay and np = ne is be a black body with a temperature T = 2.723 K with the number of proton. β is related to by the principle small anisotropies of order of the µK. The temperature of detailed balance so that R fluctuation in a direction (ϑ, ϕ) is usually decomposed on 2πm k T B a basis of spherical harmonics as β = e B exp 2 . (117) R h2 −k T    B  m=+ℓ δT The recombination rate to all other excited levels is (ϑ, ϕ)= aℓmYℓm(ϑ, ϕ). (114) T 3/2 ∗ ∞ ℓ m=−ℓ 2 8π kB T n y dy X X B /kB T = 2 (2l+1)e σnl y 2 R c 2πme Bn/k T e 1 The angular power spectrum miltipole Cℓ = alm is   Xn,l Z B − the coefficient of the decomposition of the angularh| corre-| i where σnl is the ionization cross section for the (n,l) ex- lation function on Legendre polynomials. Given a model cited level of hydrogen. The star indicates that the sum of structure formation and a set of cosmological parame- needs to be regularized and the αEM -, me-dependence ters, this angular power spectrum can be computed and of the ionization cross section is complicated to ex- compared to observational data in order to constraint tract. It can however be shown to behave as σnl this set of parameters. −1 −2 ∝ αEM me f(hν/B1). Prior to recombination, the photons are tightly coupled Finally, the factor is given by to the electrons, after recombination they can be consid- C ered mainly as free particles. Changing the fine structure 1+ KΛ2s(1 xe) = − (118) constant modifies the strength of the electromagnetic in- C 1+ K(β +Λ2s)(1 xe) teraction and thus the only effect on CMB anisotropies − where Λ is the rate of decay of the 2s excited level arises from the change in the differential optical depth of 2s to the ground state via 2 photons; it scales as m α8 . photons due to the Thomson scattering e EM The constant K is given in terms of the Ly-α photon 2 3 λα = 16π¯h/(3meαEM c) by K = npλα/(8πH) and scales τ˙ = xenecσT (115) −3 −6 as me αEM . which enters in the collision term of the Boltzmann equa- Changing αEM will thus have two effects: first it tion describing the evolution of the photon distribution changes the temperature at which the last scattering hap- function and where xe is the ionization fraction (i.e. the pens and secondly it changes the residual ionization after number density of free electrons with respect to their recombination. Both effects influence the CMB temper- et al. total number density ne). The first dependence of the ature anisotropies [see Kaplinghat (1999) and Bat- optical depth on the fine structure constant arises from tye et al. (2001) for discussions]. The last scattering can the Thomson scattering cross-section given by roughly be determined by the maximum of the visibility function g =τ ˙ exp( τ) which measures the differential − 8π ¯h2 probability for a photon to be scattered at a given red- σ = α2 (116) T 2 2 EM shift. Increasing α shifts g to higher redshift at which 3 mec EM the expansion rate is faster so that the temperature and and the scattering by free protons can be neglected since xe decrease more rapidly, resulting in a narrower g. This −4 me/mp 5 10 . The second, and more subtle depen- induces a shift of the C spectrum to higher multipoles ∼ × ℓ dence, comes from the ionization fraction. Recombina- and an increase of the values of the Cℓ. The first ef- tion proceeds via 2-photon emission from the 2s level or fect can be understood by the fact that pushing the last via the Ly-α photons which are redshifted out of the res- scattering surface to a higher redshift leads to a smaller onance line (Peebles, 1968) because recombination to the sound horizon at decoupling. The second effect results ground state can be neglected since it leads to immediate from a smaller Silk damping. reionization of another hydrogen atom by the emission of Hannestad (1999) and then Kaplinghat et al. (1999) a Ly-α photon . Following Ma and Bertschinger (1995) implemented these equations in a Boltzmann code, tak- ing into account only the recombination of hydrogen and

19 neglecting the one of helium, and showed that com- Battye et al. (2001) checked that the helium recombina- 5 6 ing satellite experiments such as MAP and Planck tion was negligible in the range of ∆αEM considered. should provide a constraint on αEM at recombination In conclusion, strong constraints on the variation of −13 −1 with a precision α˙ EM /αEM 7 10 yr , which cor- αEM can be obtained from the CMB only if the cosmolog- | |≤ × −2 −3 responds to a sensitivity ∆αEM /αEM 10 10 ical parameters are independently known. This method at a redshift of about z| 1, 000.| ∼ Avelino−et al. is thus non competitive unless one has strong bounds on ∼ (2000) studied the dependence of the position of the Ωb and h (and the result will always be conditional to the

first acoustic peak on αEM . Hannestad (1999) chose the model of structure formation) and assumptions about the underlying ΛCDM model (Ω, Ωb, Λ,h,n,Nν,τ,αEM ) = variation of other constants such as the electron mass, (0) (1, 0.08, 0, 0.5, 1, 3, 0, αEM) and performed a 8 parameters gravitational constant are made. fit to determine to which precision the parameters can be extracted. Kaplinghat et al. (1999) worked with the pa- 2. Nucleosynthesis rameters (h, Ωb, Λ,Nν , Yp, αEM ). They showed that the precision on ∆α /α varies from 10−2 if the maximum EM EM 4 observed CMB multipole is of order 500-1000 to 10−3 if The amount of He produced during the big bang nu- one observes multipoles higher than 1500. cleosynthesis is mainly determined by the neutron to pro- Avelino et al. (2000) claim that BOOMERanG and ton ratio at the freeze-out of the weak interactions that interconvert neutrons and protons. The result of Big MAXIMA data favor a value of αEM smaller by a few percents in the past (see also Martins et al., 2002) and Bang nucleosynthesis (BBN) thus depends on G, αW , Battye et al. (2001) showed that the fit to current CMB αEM and αS respectively through the expansion rate, the neutron to proton ratio, the neutron-proton mass differ- data are improved by allowing ∆αEM = 0 and pointed out that the evidence of a variation of the6 fine structure ence and the nuclear reaction rates, besides the standard constant can be thought of as favoring a delayed recom- parameters such as e.g. the number of neutrino fami- bination model (assuming Ω = 1 and n = 1). Avelino et lies. The standard BBN scenario (see e.g. Malaney, 1993, al. (2001) then performed a joint analysis of nucleosyn- Reeves, 1994) proceeds in three main steps: thesis and CMB data and did not find any evidence for a 1. for T > 1 MeV, (t < 1 s) a first stage during variation of α at one-sigma level at either epoch. They EM which the neutrons, protons, electrons, positrons consider Ω and ∆α as independent and the marginal- b EM an neutrinos are kept in statistical equilibrium by ization over one of the two parameters lead to the (rapid) weak interaction

0.09 < ∆αEM < 0.02 (119) − − − n p + e +¯νe, n + νe p + e , ←→+ ←→ at 68% confidence level. Martins et al. (2002) concluded n + e p +¯νe. (121) that MAP and Planck will allow to set respectively a ←→ 2.2% and 0.4% constraint at 1σ if all other parameters As long as statistical equilibrium holds, the neutron are marginalized. Landau et al. (2001) concluded from to proton ratio is the study of BOOMERanG, MAXIMA and COBE data −Q/k T in spatially flat models with adiabatic primordial fluctu- (n/p)=e B (122) ations that, at 2σ level, 2 where Q (mn mp)c = 1.29 MeV. The abun- 0.14 < ∆α < 0.03. (120) ≡ − − EM dance of the other light elements is given by (Kolb and Turner, 1993) All these works assume that only αEM is varying but, as can been seen from Eqs. (114-118), one has to assume − ζ(3) A 1 the constancy of the electron mass. Battye et al. (2001) Y = g 2(3A−5)/2A5/2 A A √π show that the change in the fine structure constant and   in the mass of the electron are degenerate according to 3(A−1)/2 kB T − − A A 1 Z A Z B /kB T ∆α 0.39∆m but that this degeneracy was broken 2 η Yp Yn e , (123) EM e mNc for multipoles≈ higher than 1500. The variation of the   gravitational constant can also have similar effects on where gA is the number of degrees of freedom of the the CMB (Riazuelo and Uzan, 2002). All the works also A nucleus Z X, mN is the nucleon mass, η the baryon- assume the α -dependence of to be negligible and 2 EM R photon ratio and BA (Zmp +(A Z)mn mA)c the binding energy. ≡ − − 2. Around T 0.8 MeV (t 2 s), the weak inter- 5 ∼ ∼ http://map.gsfc.nasa.gov/ actions freeze out at a temperature Tf determined 6http://astro.estec.esa.nl/SA-general/Projects/Planck/ by the competition between the weak interaction

20 rates and the expansion rate of the universe and that ∆G/G < 0.25 (see Section IV) and ∆GF /GF < | −2 | | | thus determined by Γw (Tf ) H(Tf ) that is 6 10 (see Section V.A). ∼ ×First, the radiative and Coulomb corrections for the 2 5 2 G (k T ) GN∗(k T ) (124) weak reactions (121) have been computed by Dicus et al. F B f ∼ B f (1982) and shown to have a very small influence on the p abundances. where GF is the Fermi constant and N∗ the num- ber of relativistic degrees of freedom at Tf . Be- The constraints on the variation of these quantities low Tf , the number of neutrons and protons change were first studied by Kolb et al. (1986) who calculated 4 only from the neutron β-decay between Tf to TN the dependence of primordial He on G, GF and Q. They 0.1 MeV when p + n reactions proceed faster than∼ studied the influence of independent changes of the for- their inverse dissociation. TN is determined by de- mer parameters and showed that the helium abundance manding that the relative number of photons with was mostly sensitive in the change in Q. Other abun- energy larger that the deuteron binding energy, dances are less sensitive to the value of Q, mainly be- E , is smaller than one, i.e. so that n /n cause 4He has a larger binding energy; its abundances D γ p ∼ exp(ED/TN) 1. is less sensitive to the weak reaction rate and more to ∼ the parameters fixing the value of (n/p). To extract the 3. For 0.05 MeV

21 ∆Yp ∆Tf ∆Q . (130) captures matrix elements are proportional to αEM . The Yp ≃ Tf − Q most secure constraint arising from D/H measurements and combining with CMB data to determine ΩB gives They used this to study the constraints on GF (see Sec- −2 tion V.A). ∆αEM /αEM = (3 7) 10 (135) Bergstr¨om et al. (1999) extended the original work by ± × Kolb et al. (1986) by considering other nuclei. They at 1σ level. assumed the dependence of Q on α Ichikawa and Kawasaki (2002) included the effect of EM the quark mass and by considering a joint variation of

Q (1.29 0.76∆αEM /αEM ) MeV (131) the different couplings as it appears from a dilaton. Q ≃ − then takes the form that relies on a change of quark masses due to strong and Q = aα Λ + b(y y )v (136) electromagnetic energy binding. Since the abundances of EM QCD d − u other nuclei depend mostly on the weak interaction rates, where a and b are two parameters and y , y the Yukawa they studied the dependence of the thermonuclear rates d u couplings. The neutron lifetime then behaves as on αEM . In the non-relativistic limit, it is obtained as 5 −1 the thermal average of the cross section times the relative τn = (1/vye )f (Q/me), (137) velocity times the number densities. The key point is that for charged particles the cross section takes the form with f is a known function. Assuming that all the cou- plings vary due to the effect of a dilaton, such that the S(E) Higgs vacuum expectation value v remains fixed, they σ(E)= e−2πη(E) (132) E constrained the variation of this dilaton and deduced

−4 where η(E) arises from the Coulomb barrier and is given ∆αEM /αEM = ( 2.24 3.75) 10 . (138) in terms of the charges and the reduced mass µ of the − ± × two particles as In all the studies, one either assumes all other con- stants fixed or a functional dependence between them, µc2 as inspired from string theory. The bounds are of the η(E)= α Z Z . (133) EM 1 2 2E same order of magnitude that the ones obtained from the r CMB; they have the advantage to be at higher redshift The factor S(E) has to be extrapolated from experimen- but suffer from the drawback to be model-dependent. tal nuclear data which allows Bergstr¨om et al. (1999) to determine the αEM -dependence of all the relevant reac- tion rates. Let us note that the αEM -dependence of the 3. Conclusion reduced mass µ and of S(E) were neglected; the latter Even if cosmological observations allow to test larger one is polynomial in αEM (Fowler et al., 1975). Keep- ing all other constants fixed, assuming no exotic effects time scales, it is difficult to extract tight constraints on and taking a lifetime of 886.7 s for the neutron, it was the variation of the fine structure constant from them. deduced that The CMB seems clean at first glance since the effect of the fine structure constant is well decoupled from the −2 ∆αEM /αEM < 2 10 . (134) effect of the weak and strong coupling constants. Still, | | × it is entangled with assumption on G. Besides, it was −10 7 In the low range of η 1.8 10 the Li abundance shown that degeneracy between some parameters exists ∼ × 4 does not depend strongly on αEM and the one of He has and mainly between the fine structure constant, the elec- to be used to constrain αEM . But it has to be noted that tron to proton mass ratio, the baryonic density and the 4 the observational status of the abundance of He is still dark energy equation of state (Huey et al., 2001). a matter of debate and that the theoretical prediction of Nucleosynthesis is degenerate in the four fundamental its variation with αEM depends on the model-dependent coupling constants. In some specific models where the −10 ansatz (131). For the high range of η 5 10 , the variation of these constants are linked it allows to con- 7 ∼ × variation of Li with αEM is rapid, due to the exponential straint them and definitively the helium abundance alone Coulomb barrier and limits the variation of αEM . cannot constraint the fine structure constant. Nollet and Lopez (2002) pointed out that Eq. (132) does not contain all the αEM -dependence. They argue that (i) the factor S depends linearly on αEM , (ii) when D. Equivalence Principle a reaction produces two charged particles there should be an extra αEM contribution arising from the fact that The equivalence principle is closely related to the de- the particles need to escape the Coulomb potential, (iii) velopment of the theory of gravity from Newton’s the- the reaction energies depend on αEM and (iv) radiative ory to general relativity (see Will, 1993 and Will, 2001

22 for reviews). Its first aspect is the weak equivalence fraction of gravitational self-energy and (ii) a difference principle stating that the weight of a body is propor- of composition (the core of the Earth having a larger tional to its mass or equivalently that the trajectory of Fe/Ni ratio than the Moon). This makes this test sensi- any freely falling body does not depend on its internal titive both to self-gravity and to non-gravitational forms structure, mass and composition. Einstein formulated of energy. The experiment by Baessler et al., (1999) lifts a stronger equivalence principle usually referred to as the degeneracy by considering miniature “Earth” and Einstein equivalence principle stating that (1) the weak “Moon”. equivalence principle holds, (2) any non-gravitational ex- As explained in Section II.C, if the self-energy depends periment is independent of the velocity of the laboratory on position, the conservation of energy implies the exis- rest-frame (local Lorentz invariance) and (3) of when an tence of an anomalous acceleration. In the more general where it is performed (local position invariance). case where the long range force is mediated by a scalar If the Einstein equivalence principle is valid then grav- field φ, one has to determine the dependence mi(φ) of ity can be described as the consequence of a curved space- the different particles. If it is different for neutron and time and is a metric theory of gravity, an example of proton, then the force will be composition dependent. which are general relativity and the Brans-Dicke (1961) At the Newtonian approximation, the interaction poten- theory. This statement is not a “theorem” but there are tial between two particles is of the form (Damour and a lot of indications to back it up (see Will, 1993, 2001). Esposito-Far`ese, 1992) Note that superstring theory violates the Einstein equiv- −r/λ m1m2 alence principle since it introduces additional fields (e.g. V (r)= G 1+ α12e (140) − r dilaton, moduli...) that have gravitational-strength cou-   plings which violates of the weak equivalence principle. A with α12 f1f2 and fi defined as ≡ time variation of a fundamental constant is in contradic- ∂ ln m (φ) tion with Einstein equivalence principle since it violates f M i (141) i ≡ 4 ∂φ the local position invariance. Dicke (1957, 1964) was probably the first to try to use the result of E¨otv¨os et al. where M −2 8πG/¯hc is the four dimensional Planck 4 ≡ (1922) experiment to argue that the strong interaction mass. The coefficient α12 is thus not a fundamental con- constant was approximatively position independent. All stant and depends a priori on the chemical composition new interactions that appear in the extension of standard of the two test masses. It follows that physics implies extra scalar or vector fields and thus an f f f m expected violation of the weak equivalence principle, the η = ext| 1 − 2| M f ∂ ln 1 . (142) 12 1+ f (f + f )/2 ≃ 4 ext φ m only exception being metric theories such as the class of ext 1 2 2 tensor-scalar theories of gravitation in which the dilaton To set any constraint, one has to determine the couples universally to all fields and in which one can have functions fi(φ), which can only be made in a model- a time variation of gravitational constant without a vio- dependent approach [see e.g. Damour (1996) for a dis- lation of the weak equivalence principle (see e.g. Damour cussion of the information that can be extracted in a and Esposito-Far`ese, 1992). model-independent way]. For instance, if φ couples to a The difference in acceleration between two bodies of charge Q the additional potential is expected to be of the different composition can be measured in E¨otv¨os-type ex- form periments (E¨otv¨os et al., 1922) in which the acceleration of various pairs of material in the Earth gravitational Q1Q2 −r/λ V (r)= fQ e (143) field are compared. The results of this kind of laboratory − r experiments are presented as bounds on the parameter η with fQ being a fundamental constant (fQ > 0 for scalar exchange and fQ < 0 for vector exchange). It follows ~a1 ~a2 η 2| − |. (139) that α12 depends explicitly of the composition of the two ≡ ~a + ~a | 1 2| bodies as

The most accurate constraints on η are η = ( 1.9 Q1 Q2 −12 − ± α12 = ξQ (144) 2.5) 10 between beryllium and copper (Su et al., µ1 µ2 1994)× and η < 5.5 10−13 between Earth-core-like | | × where µ m /m and ξ = f /Gm2 . Their relative and Moon-mantle-like materials (Baessler et al., 1999). i ≡ i H Q Q H The Lunar Laser Ranging (experiment) gives the bound acceleration in an external field ~gext is η = (3.2 4.6) 10−13 (Williams et al., 1996) and ± × − Q Q1 Q2 η = (3.6 4) 10 13 (M¨uller and Nordtvedt, 1998; ∆~a12 = ξQ ~gext. (145) µ µ − µ M¨uller et al.± , 1999).× Note however that, as pointed by  ext  1 2  Nordtvedt (1988, 2001a), the LLR measurement are am- For instance, in the case of a fifth force induced by a dila- biguous since the Earth and the Moon have (i) a different ton or string moduli, Damour and Polyakov (1994a,b)

23 showed that there are three charges B = N + Z, D = Dvali and Zaldarriaga (2002) [followed by a re-analysis N Z and E = Z(Z 1)B1/3 representing respectively by Chiba and Khori (2001), Wetterich (2002)] expanded − − the baryon number, the neutron excess and a term pro- αEM around it value today as portional to the nuclear Coulomb energy. The test of the 2 equivalence principle results in an exclusion plot in the φ φ αEM = αEM (0) + λ + 2 (150) plane (ξ , λ) (see Figure 6). M4 O M Q  4  To illustrate the link between the variation of the constants and the tests of relativity, let us considered from which it follows, from Webb et al. (2001) measure, that λ∆φ/M 10−7 during the last Hubble time. The the string-inspired model developed by Damour and 4 ∼ Polyakov (1994a, 1994b), in which the fine structure change of the mass of the proton and of the neutron due constant is given in terms of a function of the four di- to electromagnetic effects was obtained from Eqs. (27-28) −1 but with neglecting the last term. The extra-Lagrangian mensional dilaton as αEM = BF (φ). The QCD mass scale can be expressed in terms of the string mass scale, for the field φ is thus M 3 1017 GeV [see Section VI.B for details and s ∼ × φ Eq. (276)]. In the chiral limit, the (Einstein-frame) δL = λ (Bppp¯ + Bnnn¯) . (151) M hadron mass is proportional to the QCD mass scale so 4 that, A test body composed of nn neutrons and np protons will be characterized by a sensitivity Ms 1 ∂ ln αEM fhadron ln + . (146) ≃− mhadron 2 ∂φ λ   fi = (νpBp + νnBn) (152) mN With the expected form ln B (φ)= κ(φ φ )2/2 (see F − − m Section VI.B), the factor of the r.h.s. of the previous where νn (resp. νp) is the ratio of neutrons (resp. equation is of order 40κ(φ φm). The exchange of the protons) and where it has been assumed that mn − 8 Earth ∼ scalar field excitation induces a deviation from general mp mN. Assuming that ν 1/2 and us- ∼ n,p ∼ relativity characterized, at post-Newtonian level, by the ing that the compactness of the Moon-Earth system −3 Eddington parameters ∂ ln(mEarth/mMoon)/∂ ln αEM 10 , one gets η12 10−3λ2. Dvali and Zaldarriaga∼ (2002) obtained the same∼ 2 2 −2 −1 1 γEdd 2(40κ) (φ0 φm) , (147) result by considering that ∆νn,p 6 10 10 . This − ≃ 2 − 2 −5 ∼ × − β 1 (40κ) (φ0 φm) /2. (148) implies that λ< 10 which is compatible with the vari- Edd − ≃ − −2 ation of αEM if ∆φ/M4 > 10 during the last Hubble Besides, the violation of the universality of free fall is period. given by η12 = δˆ1 δˆ2 with From cosmological investigations one can show that − 2 (∆φ/M4) (ρφ + Pφ)/ρtotal. If φ dominates the matter B D ∼ δˆ = (1 γ ) c + c content of the universe, ρtotal, then ∆φ M4 so that 1 Edd 2 D −7 ∼ − µ 1 µ 1 λ 10 whereas if it is sub-dominant ∆φ M4 and      ∼ − ≪ E λ 10 7. In conclusion +0.943 10−5 (149) ≫ × µ  1 10−7 <λ< 10−5. (153) obtained from the expression (27) for the mass. In this expression, the third term is expected to dominate. This explicits the tuning on the parameter λ. We see on this example that the variation of the con- An underlying approximation is that the φ-dependence stants, the violation of the equivalence principle and post- arises only from the electromagnetic self-energy. But, in Newtonian deviation from general relativity have to be general, one would expect that the dominant contribu- considered together. tion to the hadron mass, the QCD contributions, also in- Similarly, in an effective 4-dimensional theory, the only duces a φ-dependence (as in the Damour and Polyakov, consistent approach to make a Lagrangian parameter 1994a,b approach). time dependent is to consider it as a field. The Klein- In conclusion, the test of the equivalence principle Gordon equation for this field (φ¨+3Hφ˙ + m2φ + ... = 0) offers a very precise test of the variation of constants < −13 implies that φ is damped as φ˙ a−3 if its mass is much (Damour, 2001). The LLR constrain η 10 , i.e < −14 −2 ∼ smaller than the Hubble scale.∝ Thus, in order to be vary- ~aEarth ~aMoon 10 cm.s , implies that on the size | − | ∼ ing during the last Hubble time, φ has to be very light with typical mass m H 10−33 eV. This is analo- ∼ 0 ∼ gous to the case of quintessence models (see Section V.E 8 for details). As a consequence, φ has to be very weakly For copper νp = 0.456, for uranium νp = 0.385 and for lead νp = 0.397. coupled to the standard model fields. To illustrate this,

24 −33 −32 −1 of the Earth orbit ln αEM < 10 10 cm . Ex- 1. Earth surface temperature tending this measurement|∇ to| the∼ Hubble− size leads to the −4 −5 estimate ∆αEM /αEM < 10 10 . This indicates that Teller (1948) first emphasized that Dirac hypothesis if the claim by Webb et∼ al. (2001)− is correct then it should may be in conflict with paleontological evidence. His ar- induce a detectable violation of the equivalence princi- gument is based on the estimation of the temperature at ple by coming experiments such as MICROSCOPE9 and the center of the Sun T⊙ GM⊙/R⊙ using the virial ∝ STEP10 will test it respectively at the level η 10−15 theorem. The luminosity of the Sun is then propor- and η 10−18. Indeed, this is a rough estimate in∼ which tional to the radiation energy gradient times the mean ∼ α˙ EM is assumed to be constant, but this is also the con- free path of a photon times the surface of the Sun, that 7 7 −2 7 5 clusion indicated by the result by Dvali and Zaldarriaga is L⊙ T⊙R⊙M , hence concluding that L⊙ T⊙M⊙. ∝ ⊙ ∝ (2002) and Bekenstein (1982). Computing the radius of the Earth orbit in Newtonian Let us also note that this constraint has been discarded mechanics, assuming the conservation of angular momen- by a some models (see Section VI.C) and particularly tum (so that GM⊙REarth is constant) and stating that −5 while claiming that a variation of αEM of 10 was re- the Earth mean temperature is proportional to the fourth alistic (Sandvik et al., 2002; Barrow et al., 2001) [see root of the energy received, he concluded that however the recent discussion by Magueijo et al. (2002)]. T G2.25M 1.75. (154) Earth ∝ ⊙

IV. GRAVITATIONAL CONSTANT If M⊙ is constant and G was 10% larger 300 million years ago, the Earth surface temperature should have been 20% As pointed by Dicke and Peebles (1965), the impor- higher, that is close to the boiling temperature. This tance of gravitation on large scales is due to the short was in contradiction with the existence of trilobites in range of the strong and weak forces and to the fact that the Cambrian. the electromagnetic force becomes weak because of the Teller (1948) used a too low value for the age of the global neutrality of the matter. As they provide tests of universe. Gamow (1967a) actualized the numbers and the law of gravitation (planetary motions, light deflec- showed that even if it was safe at the Cambrian era, there tion,. . . ), space science and cosmology also offer tests of was still a contradiction with bacteria and alga estimated to have lived 4 109 years ago. It follows that the constancy of the gravitational constant. × Contrary to most of the other fundamental constants, ∆G/G < 0.1. (155) as the precision of the measurements increased, the dis- | | parity between the measured values of G also increased. Eichendorf and Reinhardt (1977) re-actualized Teller’s This led the CODATA11 in 1998 to raise the relative un- argument in light of a new estimate of the age of the uni- certainty for G from 0.013% to 0.15% (Gundlach and verse and new paleontological discoveries to get G/G˙ < Merkowitz, 2000). 2.0 10−11 yr−1 (cited by Petley, 1985). | | When× using such an argument, the heat balance of the atmosphere is affected by many factors (water vapor A. Paleontological and geophysical arguments content, carbon dioxide content, circulatory patterns,. . . ) is completely neglected. This renders the extrapolation Dicke (1964) stressed that the Earth is such a complex during several billion years very unreliable. For instance, system that it would be difficult to use it as a source the rise of the temperature implies that the atmosphere of evidence for or against the existence of a time varia- is at some stage mostly composed of water vapor so that tion of the gravitational constant. He noted that among its convective mechanism is expected to change in such the direct effects, a weakening of the gravitational con- a way to increase the Earth albedo and thus to decrease stant induces a variation of the Earth surface tempera- the temperature! ture, an expansion of the Earth radius and a variation of the length of the day (Jordan, 1955, and then Murphy and Dicke, 1964; Hoyle, 1972). 2. Expanding Earth

Egeyed (1961) first remarked that paleomagnetic data could be used to calculate the Earth paleoradius for dif- 9http://sci2.esa.int/Microscope/ ferent geological epochs. Under the hypothesis that the 10http://einstein.stanford.edu/STEP/ area of continental material has remained constant while 11 The CODATA is the COmmittee on Data for Science and the bulk of the Earth has expanded, the determination Technology, see http://www.codata.org/. of the difference in paleolatitudes between two sites of known separation give a measurement of the paleoradius. Creer (1965) showed that data older than 3 108 years ×

25 form a coherent group inr ˙Earth and Wesson (1973) con- where k is a constant and G0 the gravitational constant cluded from a compilation of data that the expansion was today, can be integrated. For bounded orbits, the so- most probably of 0.66 mm per year during the last 3 109 lution describes a growing ellipse with constant eccen- years. × tricity, e, pericenter argument, ω and a linearly growing 2 Dicke (1962c, 1964) related the variation of the Earth semi-latus rectum p(t) = (l /G0m)(k +t)/(k +t0), where radius to a variation of the gravitational constant by l is the constant angular momentum, of equation

∆ ln r = 0.1∆ ln G. (156) p(t) Earth − r = . (161) 1+ e cos(θ ω) McElhinny et al. (1978) re-estimated the paleoradius of − the Earth and extended the analysis to the Moon, Mars Similarly, Lynden-Bell (1982) showed that the equations and Mercury. Starting from the hydrostatic equilibrium of motion of the N-body problem can be transformed to equation the standard equation if G varies as t−1. It follows that in the Newtonian limit, the orbital pe- dP ρ(r)M(r) = G , (157) riod of a two-body system is dr − r2 2πl 1 G2m2 where M(r) is the mass within radius r, they generalized P = 1+ (162) Dicke’s result to get (Gm)2 (1 e2)3/2 O c2l2 −    in which the correction terms represent the post- ∆ ln rEarth = α∆ ln G (158) − Newtonian corrections to the Keplerian relationship. It where α depends on the equation of state P (ρ), e.g. is typically of order 10−7 and 10−6 respectively for So- α = 1/(3n 4) for a polytropic gas, P = Cρn. In the lar system planetary orbits and for a binary pulsar. It − case of small planets, one can work in a small gravita- follows that tional self-compression limit and set P = K0(ρ/ρ0 1). − P˙ l˙ G˙ m˙ Eq. (157)then gives α = (2/15)(∆ρ/ρ0), ∆ρ being the =3 2 2 . (163) density difference between the center and surface. This P l − G − m approximation is poor for the Earth and more sophis- Only for the orbits of bodies for which the gravitational ticated model exist. They give α = 0.085 0.02, Earth self-energy can be neglected does the previous equation α = 0.032, α = 0.02 0.05 and α± = Mars Mercury Moon reduce to 0.004 0.001. Using the observational± fact that the Earth ± has not expanded by more than 0.8% over the past 4 108 P˙ G˙ 9 × = 2 . (164) years, the Moon of 0.06% over the past 4 10 years and P − G Mars of 0.6%, they concluded that × This leads to two observable effects in the Solar system G/G˙ < 8 10−12 yr−1. (159) (Shapiro, 1964; Counselman and Shapiro, 1968). First, − × ∼ the scale of the Solar system changes and second, if G Despite any real evidence in favor of an expanding Earth, evolves adiabatically as G = G0 +G˙ 0(t t0), there will be the rate of expansion is also limited by another geophys- a quadratically growing increment in the− mean longitude ical aspect, i.e. the deceleration of the Earth rotation. of each body. Dicke (1957) listed out some other possible conse- For a compact body, the mass depends on G as well quences on the scenario of the formation of the Moon as other post-Newtonian parameters. At first order in and on the geomagnetic field but none of them enable the post-Newtonian expansion, there is a negative con- to put serious constraints. The paleontological data give tribution (29) to the mass arising from the gravitational only poor limits on the variation of the gravitational con- binding energy and one cannot neglectm ˙ in Eq. (163). stant and even though the Earth kept a memory of the This is also the case if other constants are varying. early gravitational conditions, this memory is crude and geological data are not easy to interpret. 1. Early works

B. Planetary and stellar orbits Early works mainly focus on the Earth-Moon system and try to relate a time variation of G to a variation of Vinti (1974) studied the dynamics of two-body system the frequency or mean motion (n = 2π/P ) of the Moon in Dirac cosmology. He showed that the equation of mo- around the Earth. Arguments on an expanding Earth tion also raised interests in the determination of the Earth d2~r k + t ~r rotation rate. One of the greatest problem is to evaluate = G 0 m (160) dt2 − 0 k + t r3 and subtract the contribution of the spin-down of the

26 G G Earth arising from the friction in the seas due to tides tidal effect to a variation of G,n ˙ Moon/2nMoon = ( 8 raised by the Moon [Van Flandern (1981) estimated that 5) 10−9 century−2 to claim that − ± ′′ −2 × n˙ tidal = ( 28.8 1.5) century ] and a contribution from − ± G/G˙ = ( 8 5) 10−11 yr−1. (169) the Moon recession. − ± × The determination of ancient rotation rates can rely In a new analysis, Van Flandern (1981) concluded that on paleontological data, ancient eclipse observations as n˙ G /nG = (3.2 1.1) 10−11 yr−1 hence that G was well as measurements of star declinations (Newton, 1970, Moon Moon increasing as ± × 1974). It can be concluded from these studies that there were about 400 days in a year during the Devonian. In- G/G˙ = (3.2 1.1) 10−11 yr−1 (170) deed, this studies are entailed by a lot of uncertainties, for ± × instance, Runcorn (1964) compared telescope observa- which has the opposite sign. In this comparison the time tion from the 17th century to the ancient eclipse records scale of the atomic time is 20 years and the one of the and found a discrepancy of a factor 2. As an example, ephemeris 200 years but is less precise. It follows that Muller (1978) studied eclipses from 1374BC to 1715AD the comparison is not obvious and that these results are to conclude that far from being convincing. In this occultation method, one has to be sure that the proper motions of the stars G/G˙ = (2.6 15) 10−11 yr−1 (165) ± × are taken into account. One also has to assume that (1) the mass of the planets are not varying (see Eq. 163), and Morrison (1973) used ephemeris from 1663 to 1972 which can happen if e.g. the strong and fine structure including 40,000 Lunar occultations from 1943 to 1972 constants are varying (2) the fine structure constant is to deduce that not varying while comparing with atomic time and (3) G/G˙ < 2 10−11 yr−1. (166) the effect of the changing radius of the Earth was not | | × taken into account. Paleontological data such as the growth rhythm found in fossil bivalves and corals also enable to set constraint on the Earth rotational history and the Moon orbit (Van 2. Solar system Diggelen, 1976) [for instance, in the study by Scrutton Monitoring the separation of orbiting bodies offers a (1965) the fossils showed marking so fine that the phases possibility to constrain the time variation of G. This of the Moon were mirrored in the coral growth]. Blake accounts for comparing a gravitational time scale (set by (1977b) related the variation of the number of sidereal the orbit) and an atomic time scale and it is thus assumed days in a sidereal year, Y = n /n , and in a sidereal E S that the variation of atomic constants is negligible on the month, M = n /n , (n , n and n being respectively E M E S M time of the experiment. the orbital frequencies of the motion of the Earth, of the Shapiro et al. (1971) compared radar-echo time de- Moon around the Earth and of the Earth around the Sun) lays between Earth, Venus and Mercury with a cesium to the variation of the Newton constant and the Earth atomic clock between 1964 and 1969. The data were fit- momentum of inertia I as ted to the theoretical equation of motion for the bodies ∆Y ∆M ∆I ∆G in a Schwarzschild spacetime, taking into account the (γ 1) γ = +2 (167) − Y − M I G perturbations from the Moon and other planets. They concluded that with γ = 1.9856 being a calculated constant. The fossil data represent the number of Solar days in a tropical year G/G˙ < 4 10−10 yr−1. (171) and in a synodic month which can be related to Y and | | × M so that one obtains a constraint on ∆I/I + 2∆G/G. The data concerning Venus cannot be used due to impre- Attributing the variation of I to the expansion of the cision in the determination of the portion of the planet Earth (Wesson, 1973), one can argue that ∆I/I repre- reflecting the radar. This was improved to sents only 10-20% of the r.h.s of (167). Blake (1977b) G/G˙ < 1.5 10−10 yr−1 (172) concluded that | | × G/G˙ = ( 0.5 2) 10−11 yr−1. (168) by including Mariner 9 and Mars orbiter data (Reasen- − ± × berg and Shapiro, 1976, 1978). The analysis was further Van Flandern (1971, 1975) studied the motion of the extended (Shapiro, 1990) to give Moon from Lunar occultation observations from 1955 to G/G˙ = ( 2 10) 10−12 yr−1. (173) 1974 using atomic time which differs from the ephemeris − ± × time relying on the motion of the Earth around the Sun. The combination of Mariner 10 an Mercury and Venus He attributed the residual acceleration after correction of ranging data gives (Anderson et al., 1991)

27 G/G˙ = (0.0 2.0) 10−12 yr−1. (174) 3. Pulsars ± × The Lunar laser ranging (LLR) experiment has mea- Contrary to the Solar system case, the dependence sured the position of the Moon with an accuracy of about of the gravitational binding energy cannot be neglected 1 cm for thirty years. This was made possible by the while computing the time variation of the period (Dicke, American Appolo 11, 14 and 15 missions and Soviet- 1969; Eardley, 1975; Haugan, 1979). Here two ap- French Lunakhod 1 and 4 which landed retro-reflectors proaches can be followed; either one sticks to a model on the Moon that reflect laser pulse from the Earth (e.g. scalar-tensor gravity) and compute all the effects in (see Dickey et al. (1994) for a complete description). this model or one has a more phenomenological approach Williams et al. (1976) deduced from the six first years of and tries to put some model-independent bounds.

LLR data that ωBD > 29 so that Eardley (1975) followed the first route and discussed the effects of a time variation of the gravitational con- ˙ < −11 −1 G/G 3 10 yr . (175) stant on binary pulsar in the framework of the Brans- | | ∼ × Dicke theory. In that case, both a dipole gravitational M¨uller et al. (1991) used 20 years of data to improve this radiation and the variation of G induce a periodic varia- result to tion in the pulse period. Nordtvedt (1990) showed that G/G˙ < 1.04 10−11 yr−1, (176) the orbital period changes as | | × P˙ 2(m c + m c )+3(m c + m c ) G˙ the main error arising from the Lunar tidal acceleration. = 2+ 1 1 2 2 1 2 2 1 P − m + m G Dickey et al. (1994) improved this constraint to  1 2  (183) G/G˙ < 6 10−12 yr−1 (177) | | × where c δ ln m /δ ln G. He concluded that for the et al. i i and Williams (1996) with 24 years of data con- pulsar PSR≡ 1913+16 (m m and c c ) one gets cluded that 1 ≃ 2 1 ≃ 2 P˙ G˙ G/G˙ < 8 10−12 yr−1 (178) = [2+5c] , (184) | | × P − G Reasenberg et al. (1979) considered the 14 months the coefficient c being model dependent. As another data obtained from the ranging of the Viking spacecraft −10 application, he estimated that cEarth 5 10 , and deduced that ω > 500 which implies −8 −6 ∼ − × BD cMoon 10 and cSun 4 10 justifying the ap- proximation∼ − (164) for the∼ Solar − × system. G/G˙ < 10−12 yr−1. (179) − Damour et al. (1988) used the timing data of the bi- nary pulsar PSR 1913+16. They implemented the effect Hellings et al. (1983) using all available astrometric data of the time variation of G by considering the effect on and in particular the ranging data from Viking landers P˙ /P and making use of the transformation suggested by on Mars deduced that Lynden-Bell (1982) to integrate the orbit. They showed, − − ˙ ˙ G/G˙ = (2 4) 10 12 yr 1. (180) in a theory-independent way, that G/G = 0.5δP/P , | | ± × where δP˙ is the part of the orbital period derivative− that The major contribution to the uncertainty is due to the is not explained otherwise (by gravitational waves radi- modeling of the dynamics of the asteroids on the Earth- ation damping). It has to be contrasted with the result Mars range. Hellings et al. (1983) also tried to attribute (184) by Nordtvedt (1990). They got their result to a time variation of the atomic constants. G/G˙ = (1.0 2.3) 10−11 yr−1. (185) Using the same data but a different modeling of the as- ± × teroids, Reasenberg (1983) got Damour and Taylor (1991) reexamined the data of G/G˙ < 3 10−11 yr−1 (181) PSR 1913+16 and the upper bound | | × G/G˙ < (1.10 1.07) 10−11 yr−1. (186) which was then improved by Chandler et al. (1993) to ± × − − Kaspi et al. (1994) used data from PSR B1913+16 and G/G˙ < 10 11 yr 1. (182) | | PSR B1855+09 respectively to get All these measurements allow to test more than just G/G˙ = (4 5) 10−12 yr−1 (187) the time variation of the gravitational constant and offer ± × a series of tests on the theory of gravitation and constrain and PPN parameters, geodetic precession etc... (see Will, 1993). G/G˙ = ( 9 18) 10−12 yr−1, (188) − ± ×

28 the latter case being more “secure” since the orbiting A side effect of the change of luminosity is a change in companion is not a neutron star. the depth of the convection zone. This induces a mod- All the previous results concern binary pulsar but iso- ification of the vibration modes of the star and partic- lated can also be used. Heintzmann and Hillebrandt ularly to the acoustic waves, i.e p-modes, (Demarque et (1975) related the spin-down of the pulsar JP1953 to a al., 1994). Demarque et al. (1994) considered an ansatz time variation of G. The spin-down is a combined ef- in which G t−β and showed that β < 0.1 over the last fect of electromagnetic losses, emission of gravitational 4.5 109 years,∝ which corresponds| to| waves, possible spin-up due to matter accretion. As- × G/G˙ < 2 10−11 yr−1. (194) suming that the angular momentum is conserved so that × I/P =constant, one deduces that Guenther et al. (1995) also showed that g-modes could

P˙ d ln I G˙ provide even much tighter constraints but these modes = . (189) are up to now very difficult to observe. Nevertheless, P G d ln G G   they concluded, using the claim of detection by Hill and The observational spin-down can be decomposed as Gu (1990), that ˙ −12 −1 P˙ P˙ P˙ P˙ G/G < 4.5 10 yr . (195) = + + . (190) ×

P obs P mag P GW P G Guenther et al. (1998) compared the p-mode spectra

predicted by different theories with varying gravitational Since P˙ /P and P˙ /P are positive definite, it follows mag GW constant to the observed spectrum obtained by a network that P˙ /P P˙ /P so that a bound on G˙ can be in- obs G of six telescopes and deduced that ferred if the main≥ pulse period is the period of rotation. Heintzmann and Hillebrandt (1975) modeled the pul- G/G˙ < 1.6 10−12 yr−1. (196) n × sar by a polytropic (P ρ ) white dwarf and deduced ∝ The standard Solar model depends on few parameters that d ln I/d ln G =2 3n/2 so that − and G plays a important role since stellar evolution is G/G˙ < 10−10 yr−1. (191) dictated by the balance between gravitation and other | | interactions. Astronomical observations determines very Mansfield (1976) assumed a relativistic degenerate, zero accurately GM⊙ and a variation of G with GM⊙ fixed 2 2 temperature polytropic star and got induces a change of the pressue (P = GM⊙/R⊙) and 3 density (ρ = M⊙/R⊙). Ricci and Villante (2002) studied G/G˙ < 5.8+1 10−11 yr−1 (192) − −1 × the effect of a variation of G on the density and pressure profile of the Sun and concluded that present data cannot ˙ at a 2σ level. He also noted that a positive G induces constrain G better than 10 2%. a spin-up counteracting the electromagnetic spin-down The late stages of stellar− evolution are governed by the which can provide another bound if an independent esti- 3/2 −2 Chandrasekhar mass (¯hc/G) mn mainly determined mate of the pulsar magnetic field can be obtained. Gold- by the balance between the Fermi pressure of a degen- man (1990), following Eardley (1975), used the scaling erate electron gas and gravity. Assuming that the mean relations N G−3/2 and M G−5/2 to deduce that ∝ ∝ neutron star mass is given by the Chandrasekhar mass, 2d ln I/d ln G = 5 + 3d ln I/d ln N. He used the data ˙ ˙ − one expects that G/G = 2MNS /3MNS . Thorsett (1996) from the pulsar PSR 0655+64 to deduce used the observations of− five neutron star binaries for − − which five Keplerian parameters can be determined (the G/G˙ < (2.2 5.5) 10 11 yr 1. (193) − − × binary period Pb, the projection of the orbital semi-major axis a1 sin i, the eccentricity e, the time and longitude of the periastron T0 and ω) as well as the relativistic ad- C. Stellar constraints vance of the angle of the periastronω ˙ . Assuming that the neutron star masses vary slowly as M = M (0) M˙ t , NS NS − NS NS In early works, Pochoda and Schwarzschild (1964), that their age was determined by the rate at which Pb Ezer and Cameron (1966) and then Gamow (1967c) stud- is increasing (so that tNS 2Pb/P˙b) and that the mass ≃ ied the Solar evolution in presence of a time varying grav- follows a normal distribution, Thorsett (1996) deduced itational constant. They came to the conclusion that un- that, at 2σ, der Dirac hypothesis, the original nuclear resources of the − − G/G˙ = ( 0.6 4.2) 10 12 yr 1. (197) Sun would have been burned by now. This results from − ± × the fact that an increase of the gravitational constant is Analogously, the Chandrasekhar mass sets the charac- equivalent to an increase of the star density (because of teristic of the light curves of supernovae (Riazuelo and the Poisson equation). Uzan, 2002).

29 Garcia-Berro et al. (1995) considered the effect of a (1) The variation of G modifies the Friedmann equa- variation of the gravitational constant on the cooling of tion and therefore the age of the Universe (and, hence, white dwarfs and on their luminosity function. As first the sound horizon). For instance, if G is larger at earlier pointed out by Vila (1976), the energy of white dwarfs time, the age of the Universe is smaller at recombina- is entirely of gravitational and thermal origin so that a tion, so that the peak structure is shifted towards higher variation of G will induce a modification of their energy angular scales. balance. Restricting to cold white dwarfs with luminosity (2) The amplitude of the Silk damping is modified. At smaller than ten Solar luminosity, the luminosity can be small scales, viscosity and heat conduction in the photon- related to the star binding energy B and gravitational baryon fluid produce a damping of the photon perturba- energy, Egrav , as tions (Silk, 1968). The damping scale is determined by the photon diffusion length at recombination, and there- dB G˙ fore depends on the size of the horizon at this epoch, and L = + E (198) − dt G grav hence, depends on any variation of the Newton constant throughout the history of the Universe. which simply results from the hydrostatic equilibrium. (3) The thickness of the last scattering surface is mod- Again, the variation of the gravitational constant inter- ified. In the same vein, the duration of recombination is venes via the Poisson equation and the gravitational po- modified by a variation of the Newton constant as the tential. The cooling process is accelerated if G/G˙ < 0 expansion rate is different. It is well known that CMB which then induces a shift in the position of the cut-off anisotropies are affected on small scales because the last in the luminosity function. Garcia-Berro et al. (1995) scattering “surface” has a finite thickness. The net ef- concluded that fect is to introduce an extra, roughly exponential, damp- ˙ +1 −11 −1 ing term, with the cutoff length being determined by the G/G < 3−3 10 yr . (199) − × thickness of the last scattering surface. When translating The result depends on the details of the cooling theory redshift into time (or length), one has to use the Fried- and on whether the C/O white dwarf is stratified or not. mann equations, which are affected by a variation of the A time variation of G also modifies the main sequence Newton constant. The relevant quantity to consider is time of globular clusters (Dicke 1962a; Roeder, 1967). the visibility function g. In the limit of an infinitely thin Del’Innocenti et al. (1996) calculated the evolution of last scattering surface, τ goes from to 0 at recombi- low mass stars and deduced the age of the isochrones. nation epoch. For standard cosmology,∞ it drops from a The principal effect is a modification of the main se- large value to a much smaller one, and hence, the visibil- quence evolutionary time scale while the appearance ity function still exhibits a peak, but is much broader. of the color-magnitude diagram remained undistorted Chen and Kamionkowski (1999) studied the CMB within the observational resolution and theoretical un- spectrum in Brans-Dicke theory and showed that CMB certainties. Since the globular clusters must be younger experiments such as MAP will be able to constrain these than the universe, and assuming that their age was be- theories for ωBD < 100 if all parameters are to be de- tween 8 and 20 Gyr, they concluded termined by the same CMB experiment, ωBD < 500 if all parameters are fixed but the CMB normalization and ˙ −11 −1 G/G = ( 1.4 2.1) 10 yr . (200) ω < 800 if one uses the polarization. For the Planck − ± × BD mission these numbers are respectively, 800, 2500 and This analysis was also applied to clusters of galaxies by 3200. Dearborn and Schramm (1974). In that case a lower As far as we are aware, no complete study of the impact gravitational constant allows the particle to escape from of the variation of the gravitational constant (e.g. in the cluster since the gravitational binding energy also scalar-tensor theory) on the CMB has been performed decreases. They deduced that the decrease of G that yet. Note that, to compute the CMB anisotropies, one allows the existence of clusters at the present epoch is needs not only the value of G at the time of decoupling G/G˙ < 4 10−11 yr−1. (201) but also its complete time evolution up to now, since it − × will affect the integrated Sachs-Wolfe effect.

D. Cosmological constraints 2. Nucleosynthesis

1. CMB As explained in details in section III.C.2, changing the value of the gravitational constant affects the freeze-out A time-dependent gravitational constant will have temperature Tf. A larger value of G corresponds to a mainly three effects on the CMB angular power spectrum higher expansion rate. This rate is determined by the (Riazuelo and Uzan, 2002):

30 combination Gρ and in the standard case the Friedmann density but also influences the weak interaction rates so equations imply that Gρt2 is constant. The density ρ that it cannot be absorbed in a variation of G. It was is determined by the number N∗ of relativistic particles shown that a higher gravitational constant can be bal- at the time of nucleosynthesis so that nucleosynthesis al- anced by a higher electron-neutrino degeneracy so that lows to put a bound on the number of neutrinos Nν . the range of (electron chemical potential, G) was wider. Equivalently, assuming the number of neutrinos to be Damour and Pichon (1999) extended these investiga- three, leads to the conclusion that G has not varied from tions by considering a two-parameter family of scalar- more than 20% since nucleosynthesis. But, allowing for tensor theories of gravitation involving a non-linear scalar a change both in G and Nν allows for a wider range of field-matter coupling function. They concluded that even variation. Contrary to the fine structure constant the in the cases where before BBN the scalar-tensor theory role of G is less involved. was far from general relativity, BBN enables to set quite Steigmann (1976) used nucleosynthesis to put con- small constraints on the observable deviations from gen- straints on the Dirac theory. Barrow (1978) assumed eral relativity today. that G t−n and obtained from the helium abundances Let us also note the work by Carroll and Kaplinghat that 5∝.9 10−3 25, which is a much osynthesis is approximated as H(T ) = (T/1 MeV) H1 in smaller bounds that the ones obtained today. Yang et order to infer the constraints on (α, H1). This a simple al. (1979) included the computation of the deuterium way to compare an alternative to cosmology with data. and lithium. They improved the result by Barrow (1978) −3 to n < 5 10 which corresponds to ωBD > 50 and also pointed× out that the constraint is tighter if there are V. OTHER CONSTANTS extra-neutrinos. It was further improved by Rothman and Matzner (1982) to n < 3 10−3 implying Up to now, we have detailed the results concerning the | | × two most studied constants, αG and αEM . But, as we em- −13 −1 G/G˙ < 1.7 10 yr . (203) phasized, if αEM is varying one also expects a variation of × other constants such as α and α . There are many the- S W oretical reasons for that. First, in Kaluza-Klein or string Accetta et al. (1990) studied the dependence of the abun- dances of D, 3He, 4He and 7Li upon the variation of G inspired models, all constants are varying due either to and concluded that the dilaton or the extra-dimensions (see Section VI for details). 0.3 < ∆G/G < 0.4 (204) Another argument lies in the fact that if we believe in − grand unified theories, there exists an energy scale Λ − − GUT which roughly corresponds to 9 10 3 380 when Nν =3 (see also Damour and Gundlach, 1991, and Serna et al., the QCD scale ΛQCD by 1992) and ω > 50 when N = 2. BD ν 2π Kim and Lee (1995) calculated the allowed value for α (E)= (206) S −β ln(E/Λ ) the gravitational constant, electron chemical potential 0 QCD and entropy consistent with observations up to lithium- with β0 = 11+2nf /3, nf being the number of quark 7 and argued that beryllium-9 and bore-11 abundances flavors. It follows− that are very sensitive to a change in G. Kim et al. (1998) further included neutrino degeneracy. The degeneracy ∆ΛQCD E ∆αS = ln . (207) of the electron-neutrino not only increases the radiation Λ Λ α QCD  QCD  S

31 The time variation of αS is thus not the same at all en- Section V.D). Calmet and Fritzsch (2002) considered ergy scales. In the chiral limit, in which the quarks are different scenarios: (i) ΛGUT is constant and αGUT time- massless, the proton mass is proportional to the QCD dependent, (ii) only ΛGUT is time-dependent and (iii) energy scale, m Λ , so that a change in α (or in both are varying. They concluded that the most “in- p ∝ QCD S αGUT ) induces a change in µ and we have teresting” situation, in view of the variation of αEM and µ, is the second case. Langacker et al. (2002) pointed ∆mp/mp = ∆ΛQCD /ΛQCD . (208) out that changes in the quark masses and in the Higgs vacuum expectation value were also expected and they The energy-scale evolution of the three coupling con- parameterized the effects of the variation of αGUT on stants in a 1-loop approximation takes the form the electroweak and Yukawa sector. They assumed that

b E αGUT was the vacuum expectation value of a slowly vary- α−1(E)= α−1 i ln (209) i GUT ing scalar field. They concluded that ∆ΛQCD /ΛQCD − 2π ΛGUT ∼   34∆αEM /αEM (with a precision of about 20% on the nu- where the numerical coefficients depend on the choice merical factor) and that a variation of the fine structure of the considered gauge group. For instance bi = of the magnitude of the one observed by Webb et al. −4 (41/10, 19/16, 7) in the standard model (SM) and (2001) would imply ∆mp/mp 2.5 10 . They also − − ≃− × −5 b = (33/5, 1, 3) in its minimal supersymmetric exten- argued that ∆x/x 32∆αEM /αEM 8 10 , which is i ∼− ∼ × sion. In the− case of supersymmetric models (SUSY), consistent with current bounds, if one assumes the vari- Eq. (209) has to be replaced by ation of the proton gyromagnetic factor to be negligible. Earlier, Sisterna and Vucetich (1990) tried to determine SUSY −1 −1 bi E the compatibility of all the bounds by restricting their αi (E)= αGUT ln Θ(E ΛSUSY ) study to (ΛQCD , αEM , GF , G) and then included the u, d " − 2π ΛGUT # −   and s quark masses (Sisterna and Vucetich, 1991). SM − b E Since we do not have the theories of electroweak and + α 1(Λ ) i ln Θ(Λ E). i SUSY − 2π Λ SUSY − supersymmetry breaking as well as the ones for the gen- "  SUSY # eration of fermion masses, the correlations between dif- Using (209), one can work out the variation of all cou- ferent low-energy observables remain model-dependent. plings once the grand unified group is chosen, assuming But, in this unification picture, one is abale to derive or not supersymmetry. stronger constraints. For instance Olive et al. (2002) In the string-inspired model by Damour and Polyakov expressed the constraints from α-, β-decays and Oklo in

(1994a,b), Eq. (146) [obtained from Eq. (276)] implies fonction of ∆ΛQCD /ΛQCD ∆v/v 50∆αEM /αEM to | − | ∼ − − that give respectively the tighter constraints < 10 7, < 10 9 and < 10−10. The goal of this section is to discuss the ∆m /m 40∆α /α , (210) hadron hadron ≃ EM EM constraints on some of these constantss which are of im- portance while checking for consistency. as was first pointed out by Taylor and Veneziano (1988). Recently Calmet and Fritzsch (2001,2002), Dent and Fairbairn (2001) and Langacker et al. (2002) tried to A. Weak interaction work out these relationships in different models and confirmed the order of magnitude (210). Calmet and Most of the studies on the variation either of GF or αW Fritzsch (2001) computed low energy effects of a time concern BBN, Oklo, CMB and geochemichal dating. varying fine structure constant within a GUT-like theory The Fermi constant can be expressed in terms of gW with a constraint of the form (205) and focus their anal- and of the mass of the boson W, MW , as ysis on the nucleon mass. Nevertheless, they assumed g2 that the mechanisms of electroweak and supersymmetry W GF = 2 . (211) breaking as well as fermion mass generation were left un- 8MW changed and thus that quarks, leptons, W and Z masses In the standard model, M 2 is simply the product of g2 /4 do not vary. But, as seen from Eqs (211-212) below, W W by the Higgs vacuum expectation value v2 φ 2, so that one cannot vary gW with MW being fixed without vary- ≡ h i ing the Higgs vacuum expectation value which induces G =1/2v2. (212) a variation of the mass of the fermions. On this basis F they concluded that the result by Webb et al. (2001) Thus, at tree level, GF is actually independent of the on the fine structure constant implies that ∆mp/mp SU(2) coupling and is a direct measurement of the mag- −4 ≃ 38∆αEM /αEM 4 10 (keeping the Planck mass nitude of the electroweak symmetry breaking. Note that ≃ − × −4 constant) and that ∆y/y 121∆αEM /αEM 3 10 , a change in v is related to a change in the Yukawa cou- which is above the current∼− observational constraints∼ × (see plings.

32 Kolb et al. (1985) considered the effect of the vari- Planck mass constant) for a maximum multipole of ℓ ations of the fundamental constants on nucleosynthesis. 500 2500 if α is assumed constant. The degeneracy∼ − EM As detailed in Section III.C.2, they found the dependence with αEM is broken at high multipole so that one can hope of the helium abundance on G, GF and Q, the variation to detect a 1% variation with a maximum multipole of of which were related to the variation of αEM (then re- ℓ> 1500. lated to the size of extra-dimensions). Kolb et al. (1985) From Oklo data, Shlyakhter (1976) argued that the did not considered changes in GF due to the variation in weak interaction contribution to the total energy of the −5 2 MW and assumed that δGF δgW . Since G, αEM and nucleus is of order 10 (mπ/mp) so that ∆gW /gW ∝ 6 ∼ GF where related to the volume of the extra-space, this 5 10 ∆gS /gS to conclude that study gives no bound on the variation of G . × F −3 Dixit and Sher (1988) pointed out that the relation be- ∆αW /αW < 4 10 . (217) | | × tween G and g in the work by Kolb et al. (1985) was F W But in fact, the change in α is much more difficult ill-motivated and that the only way to vary G was to W F to model than the change in α . Damour and Dyson vary v. Changing v has four effects on BBN: it changes EM (1996) used the estimate by Haugan and Will (1976) for (1) all weak interaction rates, (2) me, (3) the quark masses and hence Q and (4) the pion mass which affects the weak interaction contribution to the nuclear ground 150 149 the strong nuclear force and the binding of the deuteron. state energy of samarium E( Sm) E( Sm) 5.6 eV to conclude that, if no subtle cancellation− appears,≃ Using results on the dependence of me and mp on αEM and v (Gasser and Leutwyler, 1982) they got ∆α /α < 0.02. (218) | W W | Q ∆α ∆v =1.293 0.9 EM +2.193 . (213) Concerning geolochemical data (see Section V.A), 1 MeV − α v EM Dyson (1972) pointed out that all β-decay rates are pro- 2 Besides, a change of 1% of the quark masses changes the portional to αW so that all constraints are in fact de- s 2 pion mass by 0.5% which implies that the deuteron bind- pendent on the combination αEM αW , s being defined ing energy changes also by 0.5% (Davies, 1972). They in Eq. (49). The degeneracy can be lifted by compar- 187 concluded that the helium abundance was given by ing different nuclei, e.g. 75 Re (sRe = 18000) and 40K (s = 30). The constancy of the decay− rates of 19 K − Yp =0.240 0.31∆v/v +0.38∆αEM /αEM (214) these two nuclei have approximatively the same accuracy. − From the constancy of the ratio and deduced that ∆v/v < 0.032 if αEM is fixed and λRe λRe ∆αEM ∆αEM /αEM < 0.026 if v is fixed. They also noted that ∆ = (s s ) λ Re K λ α the changes in Q, me and GF induced by v tend to cancel K − K EM making the change in α , appearing only in Q, larger. EM 10 Scherrer and Spergel (1993) followed the same way and within a few parts in 10 per year, one can deduce that, focused on two cases: (1) that the Yukawa couplings are independently of any assumption on αW , fixed so that both G and the fermion masses vary in F ∆α /α < 2 10−5 (219) parallel and (2) that the Yukawa couplings vary so that | EM EM | × G changes while the fermion masses are kept constant. F and thus that Considering the abundances of helium they deduced, as- −1 suming αEM fixed, that ∆α /α < 10 (220) | W W | 0.22 < ∆G /G < 0.01 (215) 9 − F F during the last 10 years. Wilkinson (1958) studied the variation of αW by using in the first case and pleochroic haloes, that is spheres formed by α-ray tracks around specks of uranium-bearing mineral in mica. The 0.01 < ∆G /G < 0.09 (216) − F F intensities of the haloes of different radii give a picture of in the second case. the natural radioactive series integrated over geological To finish with cosmological constraints, a change in G time from which one can deduce the proportion of differ- F ent daughter-activities in the decay chain from uranium induces a change in me which can be constrained by the CMB. The electron mass appears in the expression of the to lead. This series contain elements undergoing both α- Thomson cross section (116) and on the binding energy of and β-decay. For instance Ac branches 1.2% by α-decay 9 hydrogen (117) which induces a change in the ionization and the rest by β-decay. From 10 years old samples, fraction. Kujat and Scherrer (2000) implemented these Wilkinson (1958) deduced that changes as in Section III.C and showed that the upper −2 −3 ∆αW /αW < 10. (221) limit on ∆me/me is of order 10 10 (keeping the | | −

33 Let us also point out some works (Agrawal et al., model, the stability of a nucleus can be discussed by 1998a,b) in which the mass scale of the standard model comparing the Coulomb repulsion between protons to and the scale of electroweak symmetry breaking are con- the strong interaction attraction modeled by a surface strained by mean of the anthropic principle. Passarino tension T proportional to αS . With increasing atomic (2001) investigated the effects of the time variation of weight, the individual nucleons become progressively the Higgs vacuum expectation value and showed that more weakly bound as the Coulomb force dominates. A the classical equation of motion for the Higgs field in nucleus is stable against spontaneous fission if the standard model accepts time dependent solution. Z2 40π r2 < 0 T. (222) A 3 e2 B. Strong interaction If αS /αEM was larger in the past some unstable nuclei There is a very small number of works addressing this would have been stable. The idea is thus to find unsta- ble nuclei with long half-life which do not occur naturally. issue. Due to the strong energy dependence of αS , it The variation of αS /αEM would make them stable in the makes more sense to constraint the variation of ΛQCD . It has a lot of implications on the stability properties of past but this must have occured roughly more than about nuclei and it follows that most of the constraints arise ten times their lifetime since otherwise they will be in de- from nuclear considerations. Let us remind that in the tectable abundances. Assuming that αEM is fixed, they 244 −3 chiral limit, all dimensional parameters are proportional concluded from data on 94 Pu that ∆αS /αS < 1.7 10 8 | | × to Λ so that all dimensionless ratios will be, in this on a time scale of about 7.6 10 yr. Unfortunately, four QCD × 244 limit, pure numbers and thus insensitive to a change of month later it was reported that 94 Pu occurs naturally the strong interaction. But, quark masses will play an on Earth (Hoffmann et al., 1971) hence making the ar- important role in the variation of dimensionless ratios gument invalid. Davies (1972) argued that the binding 2 and have to be taken into account. energy is expected to vary as αS (contrary to the ansatz A change in the strong interaction affects the light el- by Broulik and Trefil, 1971) so that the previous bound becomes ∆α /α < 8.5 10−4. ements and (1) the most weakly bound nucleus, namely | S S | × the deuteron, can be unbind if it is weaker(2) there may Barrow (1987) studied the effect of the change of αS exist stable dineutron and diproton if it is stronger [and on the BBN predictions in Kaluza-Klein and superstring hydrogen would have been burned catastrophically at the theories in which all the couplings depends on the com- beginning of the universe, (Dyson, 1971)], (3) the rate of pactification radius. Assuming that the deuteron binding the proton capture (p+n D +γ) is altered and (4) the energy, probably the most sensitive parameter of BBN, → neutron lifetime changes. All these effects influence the scales as αS , he concluded that nucleosynthesis (Barrow, 1987). It will also be a catas- t /τ G−1/2α2G2 (223) trophe if the deuteron was not stable (by affecting the N n ∝ S F the hydrogen burning properties in stars). which affects the helium abundances from Eqs. (126-127). Most of the early studies considered these stability Flambaum and Shuryak (2001) discussed the effects of properties by modelling the nuclear force by a Yukawa the variation of α and took the quarks masses into ac- approximation of the form V (r) g2 exp( m r). In the S S π count. They expressed their results in terms of the pa- following of this section, the cited∼ bounds− refer to suh rameter δπ δ ln(mπ/Λ )= δ ln( (mu + md)/Λ ) a definition of α . Davies (1972) studied the stability ≡ QCD QCD S where the pion mass mπ determines the range of the nu- of two-nucleon systems in terms of α and α assum- p EM S clear force. They concluded that δπ < 0.005 between ing that α remains fixed and concluded that the dipro- | | W BBN and today. ton is not bounded if ∆α /α ∆α /α < 0.034. S S − EM EM As detailed in Section III.A.1, Shlyakhter (1976) ar- Rozental (1980) assumed that the depth of the poten- gued that the change in the energy of the resonance is tial well in the deuteron is proportional to αS , to state related to a change in gS by that a decrease of αS of 10-15% would make it unsta- ble. An increase of αS would render the diproton stable ∆g /g ∆Er/V0 (224) −1 S S ∼ so that ∆αS /αS < 10 at nucleosynthesis. A previ- | | −9 ous and more detailed analysis by Davies (1972) yield and deduced that ∆gS/gS < 1.9 10 . Clearly, this −2 | | × ∆αS /αS < 4 10 and Pochet et al. (1991) concluded analysis is not very reliable. Flambaum and Shuryak | | × −2 that ∆αS /αS < 4 10 for the deuteron to be stable (2001) estimated the variation of the resonance energy | | × −1 and ∆αS /αS < 6 10 for the diproton to be unstable. due to a variation of the pion mass and concluded that | | × 8 −10 Concerning high-Z nuclei, Broulik an Trefil (1971) ∆Er/Er 3 10 δπ so that δπ < 7 10 . used the liquid drop model of the nucleus and the ob- Flambaum∼ × and Shuryak| | (2001)| | also argued× that in the served half lives and abundances of transuramium ele- worst case all strong interaction phenomena depend on ments to constraint the variation of αS /αEM . In this ΛQCD +KmS where K is some universal constant and mS

34 the strange quark mass but a real study of the effect of ∆µ/µ < 2 10−4 (228) | | × mS on all hadronic masses remains to be done. It also follows that the proton gyromagnetic factor can be time at z = 2.811. Cowie and Songaila (1995) observed the dependent and constrained by observations such as those same quasar and deduced that presented in Section V.D. ∆µ/µ = (0.75 6.25) 10−4 (229) ± ×

C. Electron to proton mass ratio at 95% C.L. from the data on 19 absorption lines. Var- shalovich and Potekhin (1995) calculated the coefficient An early limit on the variation of 12 µ was derived by Kij with a better precision and deduced that Yahil (1975) who compared the concordance of K-Ar and ∆µ/µ < 2 10−4 (230) Rb-Sr geochemical ages and deduced that ∆µ/µ < 1.2 | | × 10 | | over the past 10 yr. at 95% C.L.. Lanzetta et al. (1995) and Varshalovich et As first pointed out by Thomson (1975) molecular ab- al. (1996b) used 59 transitions for H2 rotational levels in sorption lines can provide a test of the variation of µ. The PKS 0528-250 and got energy difference between two adjacent rotational levels in a diatomic molecule is proportional to Mr−2, r being +5.5 −5 ∆µ/µ = ( 8.3−6.6) 10 (231) the bond length and M the reduced mass, and that the − × vibrational transition of the same molecule has, in first at 1.6σ level and approximation, a √M dependence. For molecular hydro- −4 gen M = m /2 so that comparison of an observed vibro- ∆µ/µ = ( 1 1.2) 10 (232) p − ± × rotational spectrum with its present analog will thus give at 2σ level. These results were confirmed by Potekhin et information on the variation of mp and mn. Compar- ing pure rotational transitions with electronic transitions al. (1998) using 83 absorption lines to get gives a measurement of µ. − ∆µ/µ = ( 7.5 9.5) 10 5 (233) Following Thompson (1975), the frequency of vibra- − ± × tion-rotation transitions is, in the Born-Oppenheimer ap- at a 2σ level. proximation, of the form More recently, Ivanchik et al. (2001) measured, with the VLT, the vibro-rotational lines of molecular hydrogen ν EI (c + c /√µ + c /µ) (225) ≃ elec vib rot for two quasars with damped Lyman-α systems respec- where celec , cvib and crot are some numerical coefficients. tively at z = 2.3377 and 3.0249 and also argued for the Comparing the ratio of wavelengths of various electronic- detection of a time variation of µ. Their most conserva- vibration-rotational lines in quasar spectrum and in the tive result is (the observational data were compared to laboratory allow to trace the variation of µ since, at low- two experimental data sets) est order, Eq. (225) implies ∆µ/µ = ( 5.7 3.8) 10−5 (234) 2 − ± × ∆Eij (z) ∆µ ∆µ =1+ Kij + 2 . (226) ∆Eij (0) µ O µ at 1.5σ and the authors cautiously point out that addi-   tional measurements are necessary to ascertain this con- Varshalovich and Levshakov (1993) used the observations clusion. 1.5σ is not really significant and this may not of a damped Lyman-α system associated with the quasar survive further extended analysis. The result is also de- PKS 0528-250 (which is believed to have molecular hy- pendent on the laboratory dataset used for the compar- drogen in its spectrum) of redshift z =2.811 and deduced ison since it gave ∆µ/µ = ( 12.2 7.3) 10−5 with that another dataset. − ± × As in the case of Webb et al. (1999, 2001), this mea- ∆µ/µ < 4 10−3. (227) | | × surement is very important in the sense that it is a A similar analysis was first tried by Foltz et al. (1988) but non-zero detection that will have to be compared with their analysis did not take into account the wavelength- other bounds. The measurements by Ivanchik et al. to-mass sensitivity and their result hence seems not very (2001) is indeed much larger than one would expect reliable. Nevertheless, they concluded that from the electromagnetic contributions. As seen above, the change in any unified theory, the changes in the

masses are expected to be larger than the change in αEM . Typically, we expect ∆µ/µ ∆Λ /Λ ∆v/v ∼ QCD QCD − ∼ 12 (30 40)∆α /α , so that it seems that the detection In the literature µ refers either to me/mp or to its inverse. − EM EM In the present work we choose the first definition and we har- by Webb et al. (2001) is too large by a factor of order 10 monize the results of the different articles. to be compatible with it.

35 Wiklind and Combes (1997) observed the quasar they concluded that PKS 1413+135 with redshift z =0.247 and used different ∆x/x = (5 10) 10−5 (241) transitions from the same molecule to constrain the varia- ± × tion of µ. They compared different lines of HCO+, HCN, CO and showed that the redshift difference are likely to at redshift of z =0.5. They also got a constraint on the be dominated by the velocity difference between the two variation of gpµ by comparing the separation of Mg II doublet to hydrogen to get ∆g µ/g µ < 6 10−2. Wolfe species which limits the precision of the measurements to | p p | × ∆µ/µ 10−5 at 3σ level. In one source (B3 1504+377) and Davis (1979) used the 21 cm absorption lines of neu- they observed∼ a discrepancy of ∆µ/µ 10−4 tral hydrogen in front of the quasar QSO 1331+170 at Pagel (1977, 1983) used another method∼ to constrain µ a redshift z 2.081. They determined that the cloud was at redshift∼ z 1.755. The agreement between the based on the measurement of the mass shift in the spec- ∼ tral lines of heavy elements. In that case the mass of 21 cm and optical redshifts is limited by the error in the the nucleus can be considered as infinite contrary to the determination of the optical redshift. They concluded case of hydrogen. A variation of µ will thus influence that the redshift determined from hydrogen [see Eq. (68)]. He − ∆x/x 2 10 4 (242) compared the redshifts obtained from spectrum of hydro- | |≤ × gen atom and metal lines for quasars of redshift ranging at a redshift z 1.755 and from another absorber at from 2.1 to 2.7. Since redshift z 0.524∼ around the quasar AO 0235+164 gives ∼ ∆µ −4 ∆z zH zmetal = (1+ z) , (235) ∆x/x 2.8 10 . (243) ≡ − 1 µ0 | |≤ × − he obtained that Tubbs and Wolfe (1980) used a set of four quasars among which MC3 1331+17 for which z21 = 1.77642 −5 ± ∆µ/µ < 4 10−1 (236) 2 10 is known with very high precision and deduced | | × that× at 3σ level. This result is unfortunately not conclusive ∆x/x < 2 10−4. (244) because usually heavy elements and hydrogen belong to | | × different interstellar clouds with different radial velocity. Cowie and Songaila (1995) used the observations of C0 absorption and fine structure to get the better optical −5 D. Proton gyromagnetic factor redshift zopt =1.77644 2 10 which enables them to improve the constraint± to ×

As seen in Section III.B, the hyperfine structure in- −6 2 ∆x/x = (7 11) 10 . (245) duces a splitting dependent on gpµαEM . The ratio be- ± × tween the frequency ν of the hyperfine 21 cm absorption 21 Besides the uncertainty in the determination of the opti- transition an optical resonance transition of frequency cal redshift, since the 21 cm optical depth depends sen- ν mainly depends on opt sitively on spin temperature while resonance-line opti- cal depths do not, the two regions of absorption need ν /ν α2 g µ x. (237) 21 opt ∝ EM p ≡ not coincide. This induces an uncertainty ∆z = (1 + ± By comparing the redshift of the same object determined z)(∆vopt /c) into Eq. (237) [see e.g. Wolfe and Davis from optical data and the 21 cm transition, one deduces (1979) for a discussion]. that Drinkwater et al. (1998) compared the hydrogen hy- perfine structure to molecular absorption for three sys- tems at redshift z = 0.24, 0.67 and 0.68 and used CO ∆z = zopt z21 = (1+ z)∆x/x (238) − 2 absorption lines. This allows to constrain y gpαEM Savedoff (1956) used the spectrum of Cygnus A and and they got ≡ deduced that ∆y/y < 5 10−6. (246) ∆x/x = (3 7) 10−4 (239) | | × ± × Assuming that the change in gp and αEM are not cor- −6 at z 0.057. Wolfe et al. (1976) discovered a BL Lac ob- related they deduced that ∆gp/gp < 5 10 and ∼ −6 | | × ject (AO 0235+164) having the same redshift determined ∆αEM /αEM < 2.5 10 . Varshalovich and Potekhin either by the 21 cm absorption line or by the ultraviolet |(1996) used| the CO× and hyperfine hydrogen redshift to- doublet of Mg+. Using that ward PKS 1413+135 (z =0.247) to get

ν /ν g µα2 (1 3µ + ...) (240) ∆y/y = ( 4 6) 10−5 (247) H Mg ∝ p EM − − ± ×

36 and PKS 1157+0.14 (z =1.944) (referred to as “dark energy”) able to explain the cosmo- logical observations. −5 ∆y/y = (7 10) 10 (248) Among all the proposals (see e.g. Bin´etruy 2000 and ± × Carroll, 2000 for a review) quintessence seems to be a Murphy et al. (2001c) improved the precision of this promising mechanism. In these models, a scalar field is measurement by fitting Voigt profiles to the H 21 cm rolling down a runaway potential decreasing to zero at profile instead of using published redshifts and got infinity hence acting as a fluid with an effective equation − of state in the range 1 w 1 if the field is mini- ∆y/y = ( 0.2 0.44) 10 5 (249) − ≤ ≤ − ± × mally coupled. Runaway potentials such as exponential at z =0.25 and potential and inverse power law potentials 4+α α ∆y/y = ( 0.16 0.54) 10−5 (250) V (φ)= M /φ , (252) − ± × with α > 0 and M a mass scale, arise in models where at z =0.68. With the same systems Carrilli et al. (2001) supersymmetry is dynamically broken (Bin´etruy, 1999) found and in which flat directions are lifted by non-perturbative ∆y/y < 1.7 10−5 (251) effects. | | × One of the underlying motivation to replace the cosmo- both at z =0.25 and z =0.68. Murphy et al. (2001c) ar- logical constant by a scalar field comes from superstring gued that one can estimate the velocity to 1.2 km s−1 in- models in which any dimensionful parameter is expressed stead of the 10km s−1 assumed by Carrilli et al.· (2001) in terms of the string mass scale and the vacuum expec- so that their results· in fact lead to ∆y/y = (1 0.03) tation value of a scalar field. However, the requirement 10−5 at z = 0.25 and ∆y/y = (1.29 0.08) ±10−5 at× of slow roll (mandatory to have a negative pressure) and z =0.68. ± × the fact that the quintessence field dominates today im- ply, if the minimum of the potential is zero, that (i) it is very light, roughly of order 10−33 eV (Carroll, E. The particular case of the cosmological constant 1998) and that (ii) the vacuum expectation∼ value of the quintessence field today is of order of the Planck mass. The cosmological constant has also been loosing its It follows that coupling of this quintessence field leads to status of constant. In this section, we briefly review the observable long-range forces and time dependence of the observations backing up this fact and then describe the constant of nature. theoretical models in favor of a time dependent cosmolog- Carroll (1998) considered the effect of the coupling of ical constant and some links with the variation of other this very light quintessence field to ordinary matter via fundamental constants. an interaction of the form βi(φ/M) i and to the elec- µν L The combination of recent astrophysical and cosmolog- tromagnetic field as φF Fµν . Chiba and Kohri (2001) ical observations [among which the luminosity distance- also argued that an ultra-light quintessence field induces redshift relation up to z 1 from type Ia supernovae ∼ a time variation of the couplinge constant if it is coupled (Riess et al., 1998; Perlmutter et al., 1998), the cos- to ordinary matter and studied a coupling of the form µν mic microwave background temperature anisotropies (de φF Fµν . Dvali and Zaldarriaga (2002) showed that it Bernardis et al., 2000) and gravitational lensing (Mellier, will be either detectable as a quintessence field or by tests 1999)] seems to indicate that the universe is accelerating of the equivalence principle, as also concluded by Wet- and that about 70% of the energy density of the universe terich (2002). is made of a matter with a negative pressure (i.e. having It was proposed that the quintessence field is also an equation of state w P/ρ < 0). ≡ the dilaton (Uzan, 1999; Banerjee and Pavon, 2001; There are many different candidates to account for this Esposito-Far`ese and Polarski, 2001; Riazuelo and Uzan, exotic type of matter. The most simple solution would 2000; Gasperini et al., 2002). The same scalar field drives be a cosmological constant (for which w = 1) but one − the time variation of the cosmological constant and of will then have to face the well known cosmological con- the gravitational constant and it has the property to also stant problem (Weinberg, 1989), i.e. the fact that the have tracking solutions (Uzan, 1999). value of this cosmological constant inferred from the cos- Another motivation for considering the link between mological observations is extremely small — about 120 a dynamical cosmological constant and the time varia- order of magnitude — compared with the energy scales of tion of fundamental constants comes from the origin of high energy physics (Planck, GUT and even electroweak the inverse power law potential. As shown by Bin´etruy scales). Another way is to argue that there exists a (yet (1999), it can arise from supersymmetry breaking by non unknown) mechanism which makes the cosmological con- perturbative effects such as gaugino condensation. The stant strictly vanish and to find another matter candidate same kind of potential was also considered by Vayonakis

37 (1988) while discussing the variation of the fundamental emitted by two quasars. Their idea is based on the re- couplings in the framework of 10-dimensional supergrav- mark that a prism is sensitive to the energy E of the pho- ity. ton and a diffraction grating to its wavelength λ so that The variation of fundamental constants has also other any difference in the comparison of the wavelengths of a implications on the measurement of the cosmological con- particular spectral line could be attributed to a change stant. Riazuelo and Uzan (2002) considered the effect of in ¯h. Their study led to a null result in terms of experi- the variation of the gravitational constant on supernovae mental errors. data. Besides changing the luminosity distance-redshift Noerdlinger (1973) [and later Blake (1977a)] tried to relation, the variation of G changes the standard picture, measure Eλ. His argument was that the intensity of the according to which type Ia supernovae are standard can- Rayleigh-Jeans tail of the Planck spectrum of the CMB dles, in two ways. First the thermonuclear energy release photon determines kB T whereas the turnover point of the proportional to the synthetized nickel mass is changing spectrum determines hν/kB T . It follows that one can (and hence the maximum of the light curve); second the determine the value of hc at the time of recombination, time scale of the supernovae explosion and thus the width leading to the constraint ∆ ln hc < 0.3. of the light curve is also changed. Riazuelo and Uzan Further works were performed| by| Solheim et al. (1973) (2002) derived the modified magnitude-redshift relation and Baum and Florentin-Nielsen (1976) who compared to include the effect of the variation of G, using a one- the light of nearby and distant galaxies in order to test zone analytical model for the supernovae and was con- the constancy of Eλ. Bekenstein (1979) demonstrated firmed by numerical simulations (Gazta˜naga et al., 2002). that these experiments were meaningless since the con- Barrow and Magueijo (2001) considered the effect of a stancy of Eλ was interpreted as the constancy of ¯hc but time dependent fine structure constant on the interpreta- that this latter fact was implicitely assumed in the two tion of the supernovae data. Their study was restricted experiments since the wave vector and momentum of the to a class of varying speed of light theories (see Sec- photon were both parallely propagated. This is only pos- tion VI.C) which have cosmological solutions very similar sible if their proportionality factor ¯hc is constant, hence to quintessence. But, only the effect on the Hubble di- ensuring the null result of the two experiments. agram was studied and the influence of the change of the fine structure constant on the thermonuclear burst of the supernovae, and hence on its light curve, was not VI. THEORETICAL MOTIVATIONS considered at all. Up to now there is no observational evidence of a time One general feature of extra-dimensional theories, such variation of the cosmological constant. The measurement as Kaluza-Klein and string theories, is that the “true” of the equation of state of the dark energy can be hoped constants of nature are defined in the full higher di- to be possible very soon, the best candidate method being mensional theory so that the effective 4-dimensional con- the use of large-scale structure growth and weak gravi- stants depends, among other things, on the structure and tational lensing (Benabed and Bernardeau, 2001). But, sizes of the extra-dimensions. Any evolution of these sizes it seems that the variation of constants and the dark en- either in time or space, would lead to a spacetime depen- ergy are somehow related (Dvali and Zaldarriaga, 2002; dence of the effective 4-dimensional constants. Chiba and Khori, 2001; Banks et al., 2002; Wetterich, We present in Sections VI.A and VI.B some results 2002; Fujii, 2002), at least they share the properties to concerning Kaluza-Klein theories and string theories. We be very light and to appear in many models with a run- end in Section VI.C by describing some phenomenological away potential. approaches initiated by Bekenstein (1982).

F. Attempts to constrain the variation of dimensionful A. Kaluza-Klein theories constants The aim of the early model by Kaluza (1921) and Klein As emphasized in Section I, considering the variation (1926) to consider a 5-dimensional spacetime with one 1 of dimensionful constants is doubtful and seems meaning- spatial extra-dimension S (assumed to be of radius RKK ) less but such attempts have nevertheless been performed. was to unify electromagnetism and gravity (for a review We briefly review and comment them. These investiga- see e.g. Overduin and Wesson, 1997). Starting from the tions were mainly motivated by the construction of cos- 5-dimensional Einstein-Hilbert Lagrangian mological models alternative to the big bang scenario and 1 in which the redshift needs to have another interpreta- S = d5x√ g M 3R , (253) 5 2 − 5 5 5 tion. Z Bahcall and Salpeter (1965) proposed to look for a time we decompose the 5-dimensional metric as variation of the Planck constant by comparing the light

38 2 µ ν 2σ µ 2 ds5 = gµν dx dx +e (Aµdx + dy) . (254) SU(3) SU(2) U(1)]. Indeed the two main problems of these theories× is× that one cannot construct chiral fermions This form still allows 4-dimensional reparametrizations in four dimensions by compactification on a smooth man- of the form y′ = y + λ(xµ) provided that A′ = A µ µ − ifold with such a procedure and that gauge theories in five ∂µλ so that gauge transformations arise from the higher dimensions or more are not renormalisable. dimensional coordinate transformations group. Any field The expression for the structure constants at lower en- φ can be decomposed as ergy are obtained by the renormalisation group (Mar-

µ (n) µ iny/R ciano, 1987; Wu and Wang, 1986) φ(x ,y)= φ (x )e KK . (255) n∈Z 1 m X α−1(mc2)= α−1(m c2) C ln KK i i KK − π ij m The 5-dimensional Klein-Gordon equation for a massless j j X  field becomes mj θ(m m ) ln (260) − − j m µ (n) 2 (n) µ φ = (n/R ) φ (256) i ∇ ∇ KK where the sum is over all leptons, quarks, gluons... and (n) the C are constants that depend on the spin and group so that φ has a mass mn = n/RKK . At energies small ij −1 representation (Georgi et al., 1974). Note however that with respect to mKK = RKK , only y-independent fields remain and the physics is 4-dimensional. The effective this relation is obtained by considering the renormaliza- action for the massless fields is obtained from the relation tion group in four dimensions and does not take into −σ σ 2σ 2 R5 = R4 2e ∆e e F /4 with Fµν = ∂µAν ∂ν Aµ account the contribution of the Kaluza-Klein modes in so that − − − loops. Chodos and Detweiler (1980) illustrated the effect of S = π d4x√ ge2σR M 3 [R ∂ σ∂µσ the fifth dimension by considering a 5-dimensional vac- 4 − KK 5 4 − µ Z uum solution of the Kasner form 1 2σ µν 2pi e Fµν F . (257) t 2 −4 ds2 = dt2 + dxi (261)  − t i=1..4  0  The field equations do not determine the compactifica- X
2 tion radius and only the invariant radius ρ = RKK exp(σ) with pi = pi = 1 and assuming compact spatial distinguishes non-equivalent solutions (one can set RKK sections 0 xi < L. In order to ensure local isotropy P ≤ P to unity without loss of generality). and homogeneity, they choose the solution p1 = p2 = Setting A = R A , the covariant derivative is p3 = 1/2 and p4 = 1/2 so that the universe has four µ KK µ − ∂µ + ipyAµ = ∂µ + inAµ so that the charges are inte- macroscopic spatial dimensions at the time t0 and looks gers. The 4-dimensionale Yang-Mills coupling, identified spatially 3-dimensional at a time t t0 with a small 1/2≫ 2 2 compact dimension of radius (T0/t) L. Considering Aµ as the coefficient 1/4geYM of F , and gravitational con- stant are given by− as a small metric perturbation, they deduced that e 2 3 −2 2 2 αEM /αG = t/t0 (262) M4 =2πρM5 , 4gYM = M4 ρ /2, (258) 2πρ being the volume of the extra-space. Note that as hence offering a realization of Dirac large number hy- long as one considers vacuum as in Eq. (254), there is pothesis. Freund (1982) studied (4 + D) Kaluza-Klein a conformal undeterminacy that has to be lifted when cosmologies starting both in a (4 + D)-dimensional Ein- adding matter fields. This generalizes to the case of D stein gravity or a (4 + D)-dimensional Brans-Dicke grav- extra-dimensions (see e.g. Cremmer and Scherk, 1977 ity. and Forg´acs and Horv´ath, 1979 for the case of two extra- Using the expressions (259-260), Marciano (1984) re- dimensions) to lated the time dependence of the different couplings and restricted his discussion to the cases where K˙ i andm ˙ j −D 2 G ρ , αi(mKK )= Ki(D)Gρ (259) vanish. In the case whereα ˙ i(mKK ) = 0 (as studied in ∝ Chodos and Detweiler, 1980) one can relate the time vari- where the constants Ki depends only on the dimension ation of the gravitational and fine structure constant as and topology of the compact space (Weinberg, 1983b) ˙ so that the only fundamental constant of the theory is α˙ EM αEM 5 G = C1j + C2j . (263) M . A theory on where is a D- α − 2π 3 G 4+D 4 D D EM j   dimensional compact spaceM generates× M a low-energyM quan- X tum field theory of the Yang-Mills type related to the In the case whereα ˙ (m ) = 0 (as studied in Freund, i KK 6 isometries of D [for instance Witten (1981) showed 1982), it was shown that the time variation of αS is en- that for D = 7,M it can accommodate the Yang-Mills group hanced at low energy so that constraints on the time

39 variation of me/mp provide a sensitive test. It is also N = 1 and the quantization of the left-moving modes im- claimed that in the case of an oscillating mKK the am- poses that the gauge group is either SO(32) or E8 E8 plitude of the oscillations will be damped by radiation depending on the fermionic boundary conditions.× The in our 3-dimensional spacetime due to oscillating charges effective tree-level action is and that experimental bounds can be circumvented. Kolb et al. (1985) used the variation (259) to constrain S = d10x√ g e−2Φ M 8 R +4✷Φ 4( Φ)2 H − 10 H 10 − ∇ the time variation of the radius of the extra-dimensions Z 6   during primordial nucleosynthesis (see section III.C.2) MH AB FABF + ... . (264) but their ansatz concerning the variation of G was ill- − 4 F  motivated. They deduced ∆RKK /RKK < 1%. Barrow | | −2 When compactified on a 6-dimensional Calabi-Yau space, (1987) took the effects of the variation of αS RKK (see Section V.B) and deduced from the helium abundances∝ the effective 4-dimensional action takes the form that for ∆RKK /RKK < 0.7% and ∆RKK /RKK < 1.1% 2 2 | | | | 4x 8 φ 1 V6 respectively for D = 2 and D = 7 Kaluza-Klein theory SH = d √ g4φ MH R4 + ∇ ∇ −10 − " ( φ − 6 V6 ) and that ∆RKK /RKK < 3.4 10 from the Oklo data.     | | × Z It follows that the radius of the extra-dimensions has M 6 H F 2 + ... (265) to be stabilized but no satisfactory and complete mech- − 4 anism has yet been found. Li and Gott (1998) consid-  −2Φ ered a 5-dimensional Kaluza-Klein inflationary scenario where φ V6e couples identically to the Einstein and which is static in the internal dimension and expand- Yang-Mills≡ terms. It follows that ing in the other dimensions and solve the 5-dimensional 2 8 −2 6 semi-classical Einstein equations including the Casimir M4 = MH φ, gYM = MH φ (266) effect. In particular, it was deduced that the effective 4- dimensional cosmological constant is related to the fine at tree-level. Note that to reach this conclusion, one has 7 2 to assume that the matter fields (in the ‘dots’ of Eq. (265) structure constant by GΛeff = (15g∗/2048π )αEM . are minimally coupled to g4, see e.g. the discussion by Maeda, 1988). B. Superstring theories The strongly coupled SO(32) heterotic string theory is equivalent to the weakly coupled type I string the- There exist five anomaly free, supersymmetric pertur- ory. Type I superstring admits open strings, the bound- bative string theories respectively known as type I, type ary conditions of which divide the number of supersym- IIA, type IIB, SO(32) heterotic and E E heterotic the- metries by two. It follows that the tree-level effective 8× 8 ories (see e.g. in Polchinski, 1997). One of the definitive bosonic action is N = 1, D = 10 supergravity which predictions of these theories is the existence of a scalar takes the form, in the string frame, field, the dilaton, that couples directly to matter (Tay- lor and Veneziano, 1988) and whose vacuum expectation S = d10x√ g M 6e−Φ e−ΦM 2R I − 10 I I 10 value determines the string coupling constant (Witten, Z 1984). There are two other excitations that are common F 2  + ... (267) to all perturbative string theories, a rank two symmetric − 4  tensor (the graviton) gµν and a rank two antisymmetric tensor Bµν . The field content then differs from one theory where the dots contains terms describing the dynamics of to another. It follows that the 4-dimensional couplings the dilaton, fermions and other form fields. At variance are determined in terms of a string scale and various dy- with (264), the field Φ couples differently to the gravi- namical fields (dilaton, volume of compact space, . . . ). tational and Yang-Mills terms because the graviton and When the dilaton is massless, we expect three effects: (i) Yang-Mills fields are respectively excitation of close and a scalar admixture of a scalar component inducing devia- open strings. It follows that MI can be lowered even tions from general relativity in gravitaional effects, (ii) a to the weak scale by simply having exp Φ small enough. variation of the couplings and (iii) a violation of the eak Type I theories require D9-branes for consistancy. When equivalence principle. Our purpose is to show how the 4- V6 is small, one can use T-duality (to render V6 large, dimensional couplings are related to the string mass scale, which allows to use a quantum field theory approach) to the dilaton and the structure of the extra-dimensions and turn the D9-brane into a D3-brane so that mainly on the example of heterotic theories. 10x −2Φ 8 The two heterotic theories originate from the fact that SI = d √ g10e MI R10 − left- and right-moving modes of a closed string are inde- Z 1 pendent. This reduces the number of supersymmetry to d4x√ g e−Φ F 2 + ... (268) − − 4 4 Z

40 where the second term describes the Yang-Mills fields Vayonakis (1988) considered the same model but localized on the D3-brane. It follows that assumed that supersymmetry is broken by non- perturbative effects such as gaugino condensation. In 2 −2Φ 8 −2 −Φ M4 =e V6MI , gYM =e (269) this model, and contrary to the work by Maeda (1988), φ is stabilized and the variation of the constants arises at tree-level. If one compactifies the D9-brane on a 6- mainly from the variation of R in a runaway potential. dimensional orbifold instead of a 6-torus, and if the brane Damour and Polyakov (1994a, 1994b) argued that the is localized at an orbifold fixed point, then gauge fields effective action for the massless modes taking into ac- couple to fields Mi living only at these orbifold fixed count the full string loop expansion is of the form points with a (calculable) tree-level coupling ci so that

2 −2Φ 8 −2 −Φ 4x 2 ˆ ✷ ˆ 2 M =e V M , g =e + c M . (270) S = d gˆ Ms Bg(Φ)R +4BΦ(Φ) ˆΦ ( Φ) 4 6 I YM i i − − ∇ Z h n h io p k ¯ The coupling to the field ci is a priori non universal. At B (Φ) Fˆ2 B (Φ)ψˆD/ˆψˆ + ... (273) − F 4 − ψ strong coupling, the 10-dimensional E8 E8 heterotic  theory becomes M-theory on R10 S1/Z× (Hoˇrava and 2 in the string frame, M being the string mass scale. The Witten, 1996). The gravitational field× propagates in the s functions B are not known but can be expanded as 11-dimensional space while the gauge fields are localized i on two 10-dimensional branes. B (Φ) = e−2Φ + c(i) + c(i)e2Φ + c(i)e4Φ + ... (274) At one-loop, one can derive the couplings by including i 0 1 2 Kaluza-Klein excitations to get (see e.g. Dudas, 2000) in the limit Φ , so that these functions can ex- → −∞ b hibit a local maximum. After a conformal transforma- −2 6 a 2 −3/4 1/2 g = MH φ (RMH ) + ... (271) tion (g = CB gˆ , ψ = (CB ) B ψˆ), the action YM − 2 µν g µν g ψ in Einstein frame takes the form when the volume is large compared to the mass scale and d4x k in that case the coupling is no more universal. Otherwise, S = √ g R 2( φ)2 B (φ)F 2 16πG − − ∇ − 4 F one would get a more complicated function. Obviously, Z  the 4-dimensional effective gravitational and Yang-Mills ψD/ψ¯ + ... (275) couplings depend on the considered superstring theory, − on the compactification scheme but in any case they de- from which it follows that the Yang-Mills coupling be- −2 pend on the dilaton. haves as gYM = kBF (φ). This also implies that the QCD Wu and Wang (1986) studied the cosmological behav- mass scale is given by ior of the theory (264) assuming a 10-dimensional metric 2 2 2 −1/2 −8π kBF /b of the form diag( 1, R (t) g˜ (x), R (t) g˜ (y)) where ΛQCD Ms(CBg) e (276) − 3 ij 6 mn ∼ R3 and R6 are the scale factors of the external and in- ternal spaces. The rate of evolution of the size of the where b depends on the matter content. It follows that internal space was related to the time variation of the the mass of any hadron, proportional to ΛQCD in first ap- gravitational constant. The effect of a potential for the proximation, depends on the dilaton, mA(Bg,BF ,...). size of the internal space was also studied. With the anstaz (274), mA(φ) can exhibit a minimum Maeda (1988) considered the (N = 1,D = 10)- φm that is an attractor of the cosmological evolution that supergravity model derived from the heterotic super- drives the dilaton towards a regime where it decouples string theory in the low energy limit and assumed that from matter. But, one needs to assume for this mecha- the 10-dimensional spacetime is compactified on a 6- nism to apply, and particularly to avoid violation of the torus of radius R(xµ) so that the effective 4-dimensional equivalence principle at an unacceptable level, that all theory described by (265) is of the Brans-Dicke type the minima are the same, which can be implemented by with ω = 1. Assuming that φ has a mass µ, setting Bi = B. Expanding ln B around its maximum φm − as ln B κ(φ φ )2/2, Damour and Polyakov (1994a, and couples to the matter fluid in the universe as ∝− − m 10x 1994b) constrained the set of parameters (κ, φ0 φm) us- Smatter = d √ g10 exp( 2Φ) matter (g10), the re- − duced 4-dimensional− matter action− L is ing the different observational bounds. This toy model R allows to address the unsolved problem of the dilaton S = d4x√ gφ (g). (272) stabilization and to study all the experimental bounds matter − Lmatter together. Z Damour, Piazza and Veneziano (2002a,b) extended The cosmological evolution of φ and R can then be this model to a case where the coupling functions have computed and Maeda (1988) deduced thatα ˙ /α EM EM a smooth finite limit for infinite value of the bare string 1010(µ/1eV)−2 yr−1. In this approach, there is an ambi-≃ coupling, so that B = C + (e−φ). The dilaton runs guity in the way to introduce the matter fluid. i i O

41 −1 away toward its attractor at infinity during a stage of in- Fµν = ǫ [µ ǫAν] (277) flation. The amplitude of residual dilaton interaction is ∇ related to the amplitude of the primordial density fluctu- and the electromagnetic action is given
by ations and it can induce a variation of the fundamental 1 S = − F F µν √ gd4x. (278) constants, provided it couples to dark matter or dark en- EM 16π µν − ergy. It is concluded that, in this framework, the largest Z allowed variation of α is of order 2 10−6, which is The dynamics of ǫ can be shown to derive from the action EM × reached for a violation of the universality of free fall of 1 ¯hc ∂ ǫ∂µǫ −12 µ 4x of order 10 . Sǫ = − √ gd (279) 2 ℓ2 ǫ−2 − Kolb et al. (1985) argued that in 10-dimensional su- Z perstring models, G R−6 and α R−2 to deduce where ℓ is length scale which needs to be small enough to ∝ EM ∝ that ∆R/R < 0.5%. This was revised by Barrow (1987) be compatible with the observed scale invariance of elec- | | −15 −16 who included the effect of αS to deduce that helium tromagnetism (ℓPl <ℓ< 10 10 cm around which abundances impose ∆R/R < 0.2%. Recently Ichikawa electromagnetism merges with the− weak interaction). Fi- | | and Kawasaki (2002) considered a model in which all nally, the matter action for point particles of mass m 2 µ −1 3 i the couplings vary due to the dilaton dynamics and con- takes the form Sm = [ mc + (e/c)u Aµ]γ δ (x strain the variation of the dilaton field from nucleosyn- xi(τ))d4x where γ is the Lorentz− factor and τ the proper− thesis as 1.5 10−4 < √16πG∆φ< 6.0 10−4. From time. P R − × × the Oklo data, Barrow (1987) concluded that ∆R/R < Varying the total action gives the electromagnetic − | | 1.5 10 10. equation To× conclude, superstring theories offer a theoretical ǫ−1F µν =4πjν (280) framework to discuss the value of the fundamental con- ∇µ stants since they become expectation values of some and the equation for the dynamics
of ǫ fields. This is a first step towards their understanding but yet, no complete and satisfactory mechanism for the ℓ2 ∂σ 1 ✷ǫ = ǫ F F µν (281) stabilization of the extra-dimension and dilaton is known. ¯hc ∂ǫ − 8π µν   with σ = mc2γ−1δ3(xi xi(τ))/√ g. The Maxwell C. Other investigations equation (280) is the same− as electromagnetism− in a material mediumP with dielectric constant ǫ−2 and per- Independently of string theory, Bekenstein (1982) for- meability ǫ2 [this was the original description proposed mulated a framework to incorporate a varying fine struc- by Fierz (1955) and Lichn´erowicz (1955); see also Dicke ture constant. Working in units in which ¯h and c are (1964)]. constant, he adopted a classical description of the elec- On cosmological scales, it can be shown that the dy- tromagnetic field and made a set of assumptions to obtain namical equation for ǫ can be cast under the form a reasonable modification of Maxwell equations to take 2 into account the effect of the variation of the elementary 3 . 3 ℓ 2 a ǫ/ǫ˙ = a ζ ρmc (282) charge [for instance to take into account the problem of − ¯hc
charge conservation which usually derived from Maxwell where ζ = (10−2) is a dimensionless (and approxi- equations]. His eight postulates are that (1) for a con- matively constant)O measuring the fraction of mass in stant α electromagnetism is described by Maxwell the- EM Coulomb energy for an average nucleon compared with ory and the coupling of the potential vector Aν to mat- the free proton mass and ρm is the matter density. Since ter is minimal, (2) the variation of α results from dy- −3 EM ρm a , Eq. (282) can be integrated to relate (˙ǫ/ǫ)0 namics, (3) the dynamics of electromagnetism and αEM ∝ to ℓ/ℓPl and the cosmological parameters. In order to can be obtained from an invariant action that is (4) lo- integrate this equation, Bekenstein assumed that ζ was cally gauge invariant, (5) electromagnetism is causal and constant, which was a reasonable assumption at low red- (6) its action is time reversal invariant, (7) the short- shift. Livio and Stiavelli (1998) extended this analysis est length scale is the Planck length and (8) gravitation and got ζ = 1.2 10−2(X +4/3Y ) where X and Y are is described by a metric theory which satisfies Einstein the mass fraction× of hydrogen and helium. equations. Replacing the quantity in the brackets of the r.h.s. of Assuming that the charges of all particles vary in the Eq. (281) by ζρ c2 with ζ = (10−2), the static form µ µ m same way, one can set e = e0ǫ(x ) where ǫ(x ) is a di- Eq. (281) is analogous to the standardO Poisson equation mensionless universal field (it should be invariant under so that ln ǫ is proportional to the gravitational potential ǫ constant ǫ through a redefinition of e0). The elec- tromagnetic→ tensor× generalizes to ζ ℓ ln ǫ = 2 Φ (283) 4πc ℓPl

42 from which it follows that a test body of mass m and of where φ(0) is the value of the field today. It then follows electromagnetic energy EEM experiences an acceleration that the cosmological evolution will drive the system to- −1 of ~a = Φ M (∂EEM /∂ǫ) ǫ. ward a state in which φ is almost stabilized today but From−∇ the confrontation− of the∇ results of the spatial and allowing for cosmological variation of the constants of cosmological variation of ǫ Bekenstein (1982) concluced, nature. In their two-parameter extension, Livio and Sti- given his assumptions on the couplings, that αEM “is a avelli (1998) found that only variations of ∆αEM /αEM of parameter, not a dynamical variable”. This problem was 8 10−6 and 9 10−7 respectively for z < 5 and z < 1.6 recently by passed by Olive and Pospelov (2001) who were× compatible× with Solar system experiments. generalized the model to allow additional coupling of a The formalism developed by Bekenstein (1982) was 2 scalar field ǫ = BF (φ) to non-baryonic dark matter (as also applied to the the strong interaction (Chamoun et b c first proposed by Damour et al., 1990) and cosmological al., 2000, 2001) by simply adding a term fabcAµAν to a constant, arguing that in certain classes of dark matter describe the gluon tensor field Gµν , fabc being the struc- models, and particularly in supersymmetric ones, it is ture constants of the non-Abelian group. It was also im- natural to expect that φ would couple more strongly to plemented in the braneworld context (e.g. Youm, 2001) dark matter than to baryon. For instance, supersym- and Magueijo et al. (2001) studied the effect of a varying metrizing Bekenstein model, φ will get a coupling to the fine structure constant on a complex scalar field undergo- −1 kinetic term of the gaugino of the form M∗ φχ∂χ¯ so that, ing an electromagnetic U(1) symmetry breaking in this assuming that the gaugino is a large fraction of the sta- framework. Armend´ariz-Pic´on (2002) derived the most ble lightest supersymmetric particle, then the coupling general low energy action including a real scalar field that to dark matter would be of order 103 104 times larger. is local, invariant under space inversion and time rever- Such a factor could almost reconcile the− constraint aris- sal, diffeomormism invariant and with a U(1) gauge in- ing from the test of the universality of free fall with the variance. This form includes the previous form (284) of order of magnitude of the cosmological variation. This Bekenstein’s theory as well as scalar-tensor theories and generalization of Bekenstein model relies on an action of long wavelength limit of bimetric theories. the form Recently Sandvik et al. (2001) claimed to have gener- alized Bekenstein model by simply redefining aµ ǫAµ, 1 2 4 ≡ S = M R√ gd x fµν ∂[µaν] and ψ ln ǫ so that the covariant derivative −2 4 − ≡ ≡ Z becomes Dµ ∂µ +ie0aµ. It follows that the total action ≡ 1 2 µ 1 µν 4 including the Einstein-Hilbert action for gravity the ac- + M∗ ∂µφ∂ φ BF (φ)Fµν F √ gd x 2 − 4 − tions (278) and (279) for the modified electromagnetism Z   and normal matter takes the form ¯ 1 4 + N [iD/ m B i (φ)]N + χ∂χ¯ √ gd x i − i N i 2 − Z   4x −2ψ X S = √ gd grav + mat + ψ + EM e (285) 2 1 T 4x − L L L L M4 BΛ(φ)Λ + MχBχ(φ)χ χ √ gd (284) Z − 2 −
  µ Z with ψ = (ω/2)∂µψ∂ ψ so that the Einstein equa- µ L − where the sum is over proton [D/ = γ (∂µ ie0Aµ)] and tion are the “standard” Einstein equations with an ad- µ − neutron [D/ = γ ∂µ]. The functions B can be expanded ditive stress-energy tensor for the scalar field ψ. Indeed, (since one focuses on small variations of the fine structure Bekenstein (1982) did not take into account the effect of 2 constant and thus of φ) as BX =1+ ζX φ + ξX φ /2. It ǫ (or ψ) in the Friedmann equation and studied only the 2 follows that αEM (φ)= e0/4πBF (φ) so that ∆αEM /αEM = time variation of ǫ in a matter dominated universe. In 2 2 ζF φ + (ξF 2ζF )φ /2. This framework extends the anal- that sense Sandvik et al. (2002) extended the analysis ysis by Bekenstein− (1982) to a 4-dimensional parameter by Bekenstein (1982) by solving the coupled system of space (M∗, ζF , ζm, ζΛ). It contains the Bekenstein model Friedmann and Klein-Gordon equations. They studied −4 2 (ζF = 2, ζΛ = 0, ζm 10 ξF ), a Jordan-Brans-Dicke numerically in function of ζ/ωSBM (with ωSBM = ℓPl /ℓ) model− (ζ = 0, ζ = ∼2 2/2ω + 3, ξ = 1/√4ω + 6), and showed that cosmological and astrophysical data can F Λ − m − a string-like model (ζ = √2, ζ = √2, ζ = √2/2) be explained with ωSBM = 1 if ζ ranges between 0.02% F p − Λ m so that ∆/αEM /αEM = 3) and supersymmetrized Beken- and 0.1% (that is about one order of magnitude smaller stein model (ζF = 2, ζχ = 2, ζm = ζχ so that than Bekenstein’s value based on the argument that dark ∆α /α 5/ω).− In all the− models, the universal- matter has to be taken into account). An extension of the EM EM ∼ ity of free fall sets a strong constraint on ζF /√ω (with discussion of the cosmological scenarios was performed 2 ω M∗/2M4 ) and the authors showed that a small set in Barrow et al. (2002c) and it was shown that αEM of≡ models was compatible with the cosmological variation is constant during the radiation era, then evolves loga- and the equivalence principle tests. rithmically with the cosmic time during the matter era The constraint arising from the universality of free fall and then tends toward a constant during a curvature or can be fulfilled if one sets by hand B 1 [φ φ(0)]2 cosmological constant era. The scalar-tensor case with F − ∝ −

43 both varying G and αEM was considered by Barrow et al. which the speed of light is varying. Earlier related at- (2002a,b). tempts were investigated by Moffat (1993a,1993b). Al- Sandvik et al. (2002), following Barrow and O’Toole brecht and Magueijo (1998) postulated that the Fried- (2001), estimated the spatial variations to be of order mann equations are kept unchanged from which it fol- −4 2 ∆ ln αEM 4.8 10 GM/c r (Magueijo, 2001) to con- lows that the matter conservation has to be changed and ∼ × −8 clude that on cosmological scale ∆ ln αEM 10 if get a term proportional toc/c ˙ . The flatness and hori- GM/c2r 10−4, as expected on cosmological∼ scales zon problems are not solved by a period of accelerated ∼ 2 if (δT/T )CMB GM/c r. On the Earth orbit scale, expansion so that, contrary to inflation, it does not offer ∼ −23 this leads to the rough estimate ln αEM 10 any explanation for the initial perturbations (see however 10−22 cm−1 which is about ten|∇ orders of| ∼ magnitude− Harko and Mak, 1999). Albrecht and Magueijo (1998) higher than the constraint arising from the test of the considered an abrupt change in the velocity of light as universality of free fall. Nevertheless, Magueijo et al. may happen during a phase transition. It was extended (2002) re-analyzed the violation of the universality of free to scenarios in which both c and G were proportional to fall and claimed that the theory is still compatible with some power of the scale factor by Barrow (1999) (see also equivalence principle tests provided that ζm < 1 for dark Barrow and Magueigo, 1999a,b). The link between this ∼ matter. This arises probably from the fact that only αEM theory and Bekenstein theory was investigated by Bar- is varying while other constants are fixed so that the dom- row and Magueijo (1998). Magueijo et al., (2002) inves- inant factor in Eq. (146) is absent. tigated the test of universality of free fall. A Lagrangian formulation would probably requires the introduction of Wetterich (2002) considered the effect of the scalar an “ether” vector field to break local Lorentz invariance field responsible for the acceleration of the universe (the as was used in e.g. Lubo et al. (2002). “cosmon”) on the couplings arising from the coupling Clayton and Moffat (1999) implemented a varying of the cosmon to the kinetic term of the gauge field as speed of light model by considering a bimetric the- 2 ZF (φ)F /4. Focusing on grand unified theory, so that ory of gravitation in which one metric gµν describe the 2 one gets a coupling of the form = ZF (φ)Tr(F )/4+ standard gravitational vacuum whereas a second metric L µ iZψ(φ)ψD/ψ¯ and assuming a runaway expotenial poten- gµν + βψµψν , β being a dimensionless constant and ψ a tial, he related the variation of αEM , αS , the nucleon dynamical vector field, describes the geometry in which masses to the arbitrary function ZF and to the φ- matter is propagating (see also Bekenstein, 1993). When dependent electroweak scale so that the different bounds choosing ψµ = ∂µφ this reduces to the models developed can be discussed in the same framework. by Clayton and Moffat (2000, 2001). Some cosmological Chacko et al. (2002) proposed that the variation implications were discussed by Moffat (2001, 2002) but of the fine structure constant could be explained by a no study of the constraints arising from Solar system ex- late second order phase transition at z 1 3 (that periments have been taken into account. Note that Dirac is around T 10−3 eV) inducing a change∼ − in the (1979) also proposed that a varying G can be reconciled vacuum expectation∼ value of a scalar field. This can with Einstein theory of gravity if the space metric was be implemented for instance in supersymmetric theo- different from the “atomic” metric. Landau and Vucetich ries with low energy symmetry breaking scale. This will (2000) investigated the constraints arising from the viola- induce a variation of the masses of electrically charge tion of the charge conservation. Other realizations arise particle and. From the renormalization group equa- from the brane world picture in which our universe is −1 −1 tion αEM = αEM (Λ) + i(bi+1/2π) ln(mi+1/mi) and as- a 3-dimensional brane embedded in a higher dimensional −1 suming that αEM (Λ) was fixed, one would require that spacetime. Kiritsis (1999) showed that when a test brane −2P δmi/mi = (10 ) to explain the observations by is moving in a black hole bulk spacetime (Kehagias and Webb et al. (2001),O so that the masses have to increase. Kiritsis, 1999) the velocity of light is varying as the dis- NoteP that it will induce a time variation of the Fermi con- tance between the brane and the black hole. Alexander stant. Such models can occur in a large class of super- (2000) generalizes this model (see also Steer and Parry, symmetric theories. Unfortunately, it is yet incomplete 2002) by including rotation and expansion of the bulk so and its viability depends on the exitence of an adjustment that the speed of light gets stabilized at late time. Carter mechanism for the cosmological constant. But, it offers et al. (2001) nevertheless showed that even if a Newton- new way of thinking the variation of the constants at odd like force is recovered on small scales such models are very with the previous analysis involving a rolling scalar field. constrained at the post-Newtonian level. Brane models allowing for the scalar field in the bulk naturally predicts Motivated by resolving the standard cosmological puz- a time variable gravitational constant (see e.g. Brax and zles (horizon, flatness, cosmological constant, entropy, Davis, 2001). homogeneity problems) without inflation, Albrecht and Magueijo (1998) introduced a cosmological model in

44 D. A new cosmological constant problem? the dilaton and/or the dynamics of the internal space. But indeed, independently of these motivations, the un- The question of the compatibility between an ob- derstanding of the value of the fundamental constants of served variation of the fine structure constant and par- nature and the discussion of their status of constant re- ticle physics models was put forward by Banks et al. mains a central question of physics in general: question- (2002). As seen above, in the low energy limit, the change ing the free parameters of a theory accounts to question of the fine structure constant can be implemented by cou- the theory itself. It is a basic and direct test of the law µν pling a scalar field to the photon kinetic term F Fµν , of gravity. but this implies that the vacuum energy computed in this As we have shown, proving that a fundamental con- low energy limit must depend on αEM . Estimating that stant has changed is not an obvious task mainly because observations usually entangle a set of constants and be- ∆Λ Λ4∆α , (286) vac ∼ EM cause the bounds presented in the literature often as- 28 4 sume the constancy of a set of parameters. But, in GUT, leads to a variation of order ∆Λvac 10 (eV) for a vari- −4 ∼ Kaluza-Klein and string inspired models, one expects all ation ∆αEM (10 ) and for Λ = ΛQCD 100 MeV. Indeed, this∼ contrasts O with the average energy∼ density the couplings to vary simultaneously. Better analysis of the degeneracies as started by Sisterna and Vucetich of the universe of about 104 (eV)4 during the matter era so that the universe was dominated by the cosmological (1990, 1991) (see also Landau and Vucetich, 2002) are constant at z 3, which is at odds with observations. It really needed before drawing definitive conclusions but was thus concluded∼ that this imposes that such analysis are also dependent in the progresses in our understanding of the fundamental interactions and par- −28 ticularly of the QCD theory and on the generation of the ∆αEM /αEM < 10 . (287) | | fermion masses. Contrary to the standard cosmological constant problem, Other progresses require (model-dependent) investiga- the vacuum zero-point energy to be removed is time- tions of the compatibility of the different bounds. It dependent and one can only remove it for a fixed value has also to be remembered that arguing about the non- of αEM . Whereas the cosmological constant problem in- existence of something to set constraints (e.g. Broulik volves the fine tuning of a parameter, this now implies and Trefil, 1971) is very dangerous. On an observational the fine tuning of a function! point of view, one needs to further study the systemat-

It follows that a varying αEM cannot be naturally ex- ics (and remember some erroneous claims such as those plained in a field theory approach. A possible way out by Van Flandern, 1975) and to propose new experiments would be to consider that the field is in fact an axion (see e.g. Karshenboim, 2000, 2001 who proposed experi- (see Carroll, 1998; Choi, 2000; Banks and Dine, 2001). ments based on the hyperfine structure of deuterium an Some possible links with Heisenberg relations and quan- ytterbium-171 as well as atoms with magnetic moment; tum mechanics were also investigated by Ra˜nada (2002). Braxmaier et al., 2001; Torgerson, 2000 who proposed to Besides, the resolution of the cosmological constant prob- compare optical frequency references; Sortais et al., 2001 lem may also provide the missing elements to understand who improved the sensitivity of frequency standards, the the variation of the constants. Both preoblems can be coming satellite experiments ACES, MICROSCOPE and hoped to be solved by string theory. STEP...). On local scales, the test of the universal- ity of free fall sets drastic constraints and one can hope to use similar methods on cosmological scales from the VII. CONCLUSIONS measurements of weak gravitational lensing (Uzan and Bernardeau, 2001) or from structure formation (Mar- The experimental and observational constraints on the tins et al., 2002). The complementarity between local variation of the fine structure, gravitational constants, of experiments and geo-astrophysical observations is neces- the electron to proton mass ration and different combi- sary since these methods test different time-scales and nations of the proton gyromagnetic factor and the two are mainly sensitive either to rapid oscillations or a slow previous constants, as well as bounds on α are summa- W drift of the constants. rized in tables II, IV and V. The recent astrophysical observations of quasars tend The developments of high energy physics theories such to show that both the fine structure constant and the as multi-dimensional and string theories provide new mo- electron to proton mass ratio have evolved. These two tivations to consider the time variation of the fundamen- measurements are non-zero detections and thus very tal constants. The observation of the variability of these different in consequences compared with other bounds. constants constitutes one of the very few hope to test di- They draw the questions of their compatibility with the rectly the existence of extra-dimensions and to test these bounds obtained from other physical systems such as e.g. high energy-physics models. In the long run, it may help the test of the universality of free fall and Oklo but also to discriminate between different effective potentials for

45 on a more theoretical aspect of the understanding of such G a late time variation which does not seem to be natural from a field theory point of view. Theoretically, one ex- pects all constants to vary and the level of their variation NG NQG is also worth investigating. One would need to study the implication of these measurements for the other experi- GR ments and try to determine their expected level of detec- TOE tion. Both results arise from the observation of quasar absorption spectra; it is of importance to ensure that all systematics are taken into account and are confirmed by independent teams, using e.g. the VLT which offers a NM h better signal to noise and spectral resolution. QM The step from the standard model+general relativity SR to string theory allows for dynamical constants and thus QFT starts to address the question of why the constants have 1/c the value they have. Unfortunately, no complete and satisfactory stabilization mechanism is known yet and FIG. 1. The cube of physical theories as presented by Okun we have to understand why, if confirmed, the constants (1991). At the origin stands the part of Newtonian mechan- are still varying and whether such a variation induces a ics (NM) that does not take gravity into account. NG, QM and SR then stand for Newtonian gravity, quantum mechanics new cosmological constant problem. and special relativity which respectively introduce the effect of The study of the variation of the constants offers a new one of the constants. Special relativity ‘merges’ respectively link between astrophysics, cosmology and high-energy with quantum mechanics and Newtonian gravity to give quan- physics complementary to primordial cosmology. It is tum field theory (QFT) and general relativity (GR). Bringing deeply related to the test of the law of gravitation, both quantum mechanics and Newtonian gravity together leads to of the deviations from general relativity and the violation non-relativistic quantum gravity and all theories together give of the weak equivalence principle. But yet much work is the theory of everything (TOE). [From Okun (1991)]. needed both to disentangle the observations and to relate them to theoretical models.

ACKNOWLEDGMENTS Precision laser spectroscopy + second

R 8 Atomic and QED theory It is a pleasure to thank Robert Brandenberger, Michel Cass´e, Thibault Damour, Emilian Dudas, Nathalie Deru- Watt balance 2h R elle, Gilles Esposito-Far`ese, Patrick Peter and Patrick 8 h kilogram m e + Petitjean for their numerous comments and sugges- α2c condensed matter theory tions to improve this text. I want also to thank Pierre Bin´etruy, Francis Bernardeau, Philippe Brax, α Electron in Penning trap + meter Brandon Carter, Christos Charmousis, C´edric Deffayet, QED Ruth Durrer, Gia Dvali, Bernard Fort, Eric Gourgoul- FIG. 2. Sketch of the experimental and theoretical chain lead- hon, Christophe Grojean, Joseph Katz, David Langlois, ing to the determination of the electron mass. Note that, as

Roland Lehoucq, J´erˆome Martin, Yannick Mellier, Ji- expected, the determination of αEM requires no dimensional had Mourad, Kenneth Nordtvedt, Keith Olive, Simon input. [From Mohr and Taylor (2001)]. Prunet, Alain Riazuelo, Christophe Ringeval, Christophe Salomon, Aur´elien Thion, Gabriele Veneziano, Filippo Vernizzi for discussions on the subject. This work was initially motivated by the questions of Ren´eCuillierier M and the monday morning discussions of the Orsay cos- r mology group. I want to thank Patricia Flad for her help p in gathering the literature. θ A b O c=ae F P a

46 FIG. 3. The standard orbital parameters. a and b is the semi- +ξ [2σ Constraints] 10-1 major and semi-minor axis, c = ae the focal distance, p the semi-latus rectum, θ the true anomaly. F is the focus, A 10-2 (a) Laboratory -3 Free Fall and B the periastron and apoastron (see e.g. Murray and 10 Eötvös

2 2 2 12345678 − 12345678 12345678

Dermott, 2000). It is easy to check that b = a (1 e ) and -4 12345678 10 12345678

12345678

2 12345678

− 12345678 that p = a(1 e ) and one defines the frequency or mean 12345678

12345678 -5 12345678 10 12345678

12345678

12345678

motion as n = 2π/P where P is the period. 12345678

+ξ +ξ 12345678 1986 B 12345678 -6 12345678 10 12345678

12345678

12345678

12345678

Terrestrial 12345678 1996

12345678

12345678 10-7 Eötvös 12345678 12345678

12345678

(Local Source) 12345678

12345678

12345678 10-8 12345678 1s Terrestrial 10-9 Eötvös (Entire Earth) Solar Eötvös 10-10 10-3 10-1 101 103 105 107 109 1011 1013 1015 λ ( m )

2 4 -m c α / 10 1 e 8 Laboratory Inverse-Square (b) 10 -1 Laboratory Free Fall Eötvös 10 -3

12345

12345 +ξ 10 -5 12345 I 12345

12345

12345 F=0 -7 Terrestrial 10 12345 Eötvös 12345 Solar Eötvös 1s 1/2 (Local Source)

12345

-9 12345 10 Terrestrial 2 Eötvös 12345 A h (Entire Earth) 10 -11 -3 -1 1 3 5 7 9 11 13 F=1 10 10 10 10 10 10 10 10 10 FIG. 4. Hyperfine structure of the n = 1 level of the hydro- λ ( m ) gen atom. The fine structure Hamiltonian induces a shift of FIG. 6. Constraints on the coupling ξB (upper panel) and ξI − 2 4 mec αEM /8 of the level 1s. J can only take the value +1/2. (lower panel) respectively to N + Z and N − Z as a function The hyperfine Hamiltonian (74) induces a splitting of the level of the length scale λ. The shaded regions are excluded at 2σ. 1s1/2 into the two hyperfine levels F = 0 and F = +1. The [From Fischbach and Talmadge (1997)]. transition between these two levels corresponds to the 21 cm ray with Ah2 = 1, 420, 405, 751.768 ± 0.001 Hz and is of first importance in astronomy.

FIG. 5. The correction function Frel . [From Prestage et al.

(1995)]. FIG. 7. ∆αEM /αEM as a function of the look-back time com- puted with the cosmological parameters (Ωm, ΩΛ)=(0.3, 0.7) and h = 0.68. Squares refer to the data by (Murphy et al. (2001c) assuming gp constant; triangles refer to Si IV systems by Murphy et al. (2001d); dots correspond to Mg II and Fe II systems for redshidts smaller than 1.6 (Webb et al., 2001) and higher redshifts come from Murphy et al. (2001a). [From Webb et al. (2001)].

47 REFERENCES Bekenstein, J.D., 1979, Comments Astrophys. 8, 89. Bekenstein, J.D., 1982, Phys. Rev. D 25, 1527. Bekenstein, J.D., 1993, Phys. Rev. D 48, 3641. Accetta, F.S., L.M. Krauss, and P. Romanelli, 1990, Phys. Benabed, K., and F. Bernardeau, 2001, Phys. Rev. D 64, 248 Lett. B , 146. 083501. Agrawal, V., S.M. Barr, J. F. Donoghue, and D. Seckel, 1998a, Bergstr¨om, L., S. Iguri, and H. Rubinstein, 1999, Phys. Rev. 57 Phys. Rev. D , 5480. D 60, 045005. Agrawal, V., S.M. Barr, J. F. Donoghue, and D. Seckel, 1998b, Bethe, H.A., and E.E. Salpeter, 1977, Quantum mechanics of 80 Phys. Rev. Lett. , 1822. one- and two-electron atoms, Plenum, New York. 59 Albrecht, A., and J. Magueijo, 1999, Phys. Rev. D , 043516. Bin´etruy, P., 1999, Phys. Rev. D 60, 063502. 0011 Alexander, S.H.S., 2000, JHEP , 017. Bin´etruy, P., 2000, Int. J. Theor. Phys. 39, 1859. Anderson, J.D., J.K. Campbell, R.F. Jurgens, E.L. Lau, X.X. Birge, R.T., 1929, Rev. Mod. Phys. 1, 1. Newhall, M.A. Slade III, and E.M. Standish Jr., 1991, in 181 th Blake, G.M., 1977a, Month. Not. R. Astron. Soc. , 47p. Proceedings of the 6 Marcel Grossmann meeting on gen- Blake, G.M., 1977b, Month. Not. R. Astron. Soc. 181, 41p. eral relativity, Kyoto, june 1991, edited by H. Sato and T. Bluhm, R., V.A. Kosteleck´y, C.D. Lane, and N. Russel, 2002, Nakamura (World Scientific, Singapore 1992), 353. Phys. Rev. Lett. 88, 090801. Armend´ariz-Pic´on, C., 2002, preprint astro-ph/0205187. Brans, C., and R.H. Dicke, 1961, Phys. Rev. 124, 925. 170 Atkinson, A., 1968, Phys. Rev. , 1193. Brax, P., and Davis, A.C., 2001, JHEP 0105, 007 Avelino, P.P., S. Esposito, G. Mangano, C.J. Martins, A. Mel- Braxmaier, C., O. Pradl, H. M¨uller, A. Peters, J. Mlynek, V. chiorri, G. Miele, O. Pisanti, G. Rocha, and T.P. Viana, Loriette, and S. Schiller, 2001, Phys. Rev. D 64, 042001. 64 2001, Phys. Rev. D , 103505. Breakiron, L., 1993, in Proc. of the 25th Annual Precise Time Avelino, P.P., C.J.A. Martins, and G. Rocha, 2000, Phys. Interval Applications and Planning Meeting (NASA Conf. 483 Lett. B , 210. Publ. No. 3267), 401. Avelino, P.P., C.J. Martins, G. Rocha, and P. Viana , 2000, Breit, G., 1930, Phys. Rev. 35, 1477. 62 Phys. Rev. D , 123508. Broulik, B., and J.S. Trefil, 1971, Nature (London) 232, 246. Baessler, S., B.R. Heckel, E.G. Adelberger, J.H. Gundlach, Cahn, R.N., 1996, Rev. Mod. Phys. 68, 951. 83 U. Schmidt, and H.E. Swanson, 1999, Phys. Rev. Lett. , Calmet, X., and H. Fritzsch, 2001, preprint hep-ph/0112110. 3585. Calmet, X., and H. Fritzsch, 2001, preprint hep-ph/0204258. 142 Bahcall, J.N., and E.E. Salpeter, 1965, Astrophys. J. , Campbell, B.A., and K.A. Olive, 1995, Phys. Lett. B 345, 1677. 429-434. 19 Bahcall, J.N., and M. Schmidt, 1967, Phys. Rev. Lett. , Canuto, V.M., and I. Goldman, 1982, Nature (London) 296, 1294. 709. Bahcall, J.N., W.L. Sargent, and M. Schmidt, 1967, Astro- Carilli, C.L., K.M. Menten, J.T.Stocke, E. Perlman, R. Ver- 149 phys. J. , L11. meulen, F. Briggs, A.G. de Bruyn, J. Conway, and C.P. 18 Banerjee, N., and D. Pavon, Class. Quant. , 593. Moore, 2000, Phys. Rev. Lett. 85, 5511. 0110 Banks, T., and M. Dine, 2001, JHEP , 012. Carr, B.J., and M.J. Rees, 1979, Nature (London) 278, 605. Banks, T., M. Dine, and M.R. Douglas, 2002, Phys. Rev. Lett. Carroll, S.M., 1998, Phys. Rev. Lett. 81, 3067. 88 , 131301. Carroll, S.M., 2001, Living Rev. Rel. 4, 1. 184 Barrow, J.D., 1978, Month. Not. R. Astron. Soc. , 677. Carroll, S.M., and M. Kaplinghat, 2001, preprint astro- 35 Barrow, J.D., 1987, Phys. Rev. D , 1805. ph/0108002. 59 Barrow, J.D., 1999, Phys. Rev. D , 043515. Carter, B., 1974, in Confrontation of cosmological theories Barrow, J.D., and C. O’Toole, 2001, Month. Not. R. Astron. with observational data, IAU Symposia 63, edited by M. 322 Soc. , 585. Longair (Reidel, Dordrecht) 291. 443 Barrow, J.D., and J. Magueijo, 1998, Phys. Lett. B , 104. Carter, B., 1976, in Atomic physics and fundamental con- Barrow, J.D., and J. Magueijo, 1999a, Class. Quant. Grav. stants 5, edited by J.H. Sanders and A.H. Wapstra, (Plenum 16 , 1435. press, New York) 650. 447 Barrow, J.D., and J. Magueijo, 1999b, Phys. Lett. B , Carter, B., 1983 Phil. Trans. R. Soc. London A 310, 347. 246. Carter, B., J.-P. Uzan, R. Battye, and A. Mennim, 2001, Barrow, J.D., and J. Magueijo, 2001, Astrophys. J. Lett. (in Class. Quant. Grav 18, 4871. press). Casas, J.A., J. Garcia-Bellido, and N. Quiros, 1992a, Mod. Barrow, J.D., J. Magueijo, and H.B. Sandvik, 2002a, preprint Phys. Lett. A 7, 447. astro-ph/0202129. Casas, J.A., J. Garcia-Bellido, and N. Quiros, 1992b, Phys. Barrow, J.D., J. Magueijo, and H.B. Sandvik, 2002b, preprint Lett. B 278, 94. astro-ph/0204357.. Chacko, Z., C. Grojean, and M. Perelstein, 2002, preprint Barrow, J.D., H.B. Sandvik, and J. Magueijo, 2002c, Phys. hep-ph/0204142. 65 Rev. D , 06350. Chamoun, N., S.J. Landau, and H. Vucetich, 2000, preprint Battye, R.A., R. Crittenden, and J. Weller, 2001, Phys. Rev. astro-ph/0009204. 63 D , 0453505. Chamoun, N., S.J. Landau, and H. Vucetich, 2001, Phys. Lett. Baum, W.A., and R. Florentin-Nielsen, 1976, Astrophys. J. B 504, 1. 209 , 319. Chandler, J.F., R.D. Reasenberg, I.I. Shapiro, 1993, Bull.

48 Am. Astron. Soc. 25, 1233. Dent, T., and M. Fairbairn, 2001, preprint hep-ph/0112279. Chandrasekhar, S., 1937, Nature (London) 139, 757. de Witt, B.S., 1964, Phys. Rev. Lett. 13, 114. Chen, X., and M. Kamionkowski, 1999, Phys. Rev. D 60, Dicke, R.H., 1957, Rev. Mod. Phys. 29, 355. 104036. Dicke, R.H., 1959, Nature (London) 183, 170. Chiba, T., 2001, preprint gr-qc/0110118. Dicke, R.H., 1961, Nature (London) 192, 440. Chiba, T., and Khori, 2001, preprint hep-ph/0111086. Dicke, R.H., 1962a, Rev. Mod. Phys. 34 110. Chodos, A., and S. Detweiler, 1980, Phys. Rev. D 21, 2167. Dicke, R.H., 1962b, Phys. Rev. 125, 2163. Choi, K., 2000, phys. Rev. Lett. 85, 4434. Dicke, R.H., 1962c, Science 138, 653. Chupp, T.E., R.J. Hoare, R.A. Loveman, E.R. Oteiza, J.M. Dicke, R.H., 1964, in Relativity, Groups and Topology, Lec- Richardson, M.E. Wagshul, and A.K. Thompson, 1989, tures delivered at Les Houches 1963, edited by C. DeWitt Phys. Rev. Lett. 63, 1541. and B. DeWitt, (Gordon and Breach, New York). Clayton, M.A., and J.W. Moffat, 1998, Phys. Lett. B 460, Dicke, R.H., 1969, Gravitation and the universe, American 263. Philosophical Society (Philadelphia). Clayton, M.A., and J.W. Moffat, 2000, Phys. Lett. B 477, Dicke, R.H., and P.J.E. Peebles, 1965, Space Sci. Rev. 4, 419. 275. Dickey, J.O., et al., 1994, Science 265, 482. Clayton, M.A., and J.W. Moffat, 2001, Phys. Lett. B 506, Dicus, D.A., E.W. Kolb, A.M. Gleeson, E.C.G. Sudarshan, 177. V.L. Teplitz, and M.S. Turner, 1982, Phys. Rev. D 26, 2694. Cohen-Tannoudji, C., B. Diu, and F. Lalo¨e, 1986, M´ecanique Dirac, P.A.M., 1937, Nature (London) 139, 323. Quantique, Ed. Hermann. Dirac, P.A.M., 1938, Proc. Roy. Soc. London A 165, 198. Cohen-Tannoudji, G., 1995, Les constantes universelles, Coll. Dirac, P.A.M., 1974, Proc. Roy. Soc. London A 338, 439. Questions de sciences (Hachette, Paris) Dirac, P.A.M., 1979, Proc. Roy. Soc. London A 365, 19. Coley, A.A., 1990, Astron. Astrophys. 233, 305. Dixit, V.V., and M. Sher, 1988, Phys. Rev. D 37, 1097. Counselman, C.C., and I.I. Shapiro, 1968, Science 162, 352. Drinkwater, M.J., J.K. Webb, J.D. Barrow, and V.V. Flam- Cowie, L.L., and A. Songaila, 1995, Astrophys. J. 453, 596. baum, 1998, Month. Not. R. Astron. Soc. 295, 457. Creer, K.M., 1965, Nature (London) 206, 539. Dudas, E., 2000, Class. Quant. Grav. 17, R41. Cremmer, E., and J. Scherk, 1977, Nuc. Phys. B 118, 61. Duff, M.J., L.B. Okun, and G. Veneziano, 2002, JHEP 0203, Damour, T., 1996, Class. Quant. Grav. 13, A33. 37. Damour, T., 2001, in the Proceedings of the workshop Mis- Durrer, R, 2002, J. Phys. Stud., in press. sions spatiales en physique fondamentale (Chatillon, 18-19 Dvali, G., and M. Zaldarriaga, 2002, Phys. Rev. Lett. 88, Jan. 2001), Comptes Rendus de l’Academie des Sciences 091303. (Paris), edited by C. Borde and P. Touboul. Dyson, F.J., 1967, Phys. Rev. Lett. 19, 1291. Damour, T., and F.J. Dyson, 1996, Nuc. Phys. B 480, 37. Dyson, F.J., 1971, Sc. Am. 225, 51. Damour, T., and G. Esposito-Far`ese, 1992, Class. Quant. Dyson, F.J., 1972, in Aspects of Quantum Theory, edited by Grav. 9, 2093. A. Salam and E.P. Wigner (Cambridge University Press, Damour, T., and C. Gundlach, 1991, Phys. Rev. D 43, 3873. Cambridge) 213-236. Damour, T., and B. Pichon, 1999, Phys. Rev. D 59, 123502. Dyson, F.J., 1978, in Current Trends in the Theory of Fields, Damour, T., and A.M. Polyakov, 1994a, Nuc. Phys. B 423, Proceedings of the Symposium in honor of P.A.M. Dirac, 532. Tallahassee, Florida, 1978, AIP Conf. Proc. No. 48, edited Damour, T., and A.M. Polyakov, 1994b, Gen. Rel. Grav. 26, by J.E. Lannutti and P.K. Williams (American Institute of 1171. Physics, New York) 163. Damour, T., and J.H. Taylor, 1991, Astrophys. J. 366, 501. Dzuba, V.A., V.V. Flambaum, and O.D. Sushkov, 1983, J. Damour, T., G.W. Gibbons, and C. Gundlach, 1990, Phys. Phys. B 16, 715. Rev. Lett. 64, 123. Dzuba, V.A., V.V. Flambaum, and M.G. Kozlov, 1996, Phys. Damour, T., G.W. Gibbons, and J.H. Taylor, 1988, Phys. Rev. A 54, 3948. Rev. Lett. 61, 1151. Dzuba, V.A., V.V. Flambaum, and J.K. Webb, 1999a, Phys. Damour, T., F. Piazza, and G. Veneziano, 2002a, preprint Rev. A 59 230. gr-qc/0204094. Dzuba, V.A., V.V. Flambaum, and J.K. Webb, 1999b, Phys. Damour, T., F. Piazza, and G. Veneziano, 2002b, preprint Rev. Lett.82, 888.. hep-th/0205111. Dzuba, V.A., V. V. Flambaum, M. T. Murphy, and J. K. Davies, P.C.W., 1972, J. Phys. A 5, 1296. Webb, 2001, Phys. Rev. A 63, 042509. Dearborn, D.S., and D.N. Schramm, 1974, Nature (London) Eardley, D.M., 1975, Astrophys. J. 196, L59. 247, 441. Egyed, L., 1961, Nature (London) 190, 1097. Del’Innocenti, S., G. Fiorentini, G.G. Raffelt, B. Ricci, and E¨otv¨os, R.V., D. Pek´ar, and E. Fekete, 1922, Ann. Phys. A. Weiss, 1996, Astron. Astrophys. 312, 345. (Leipzig) 68, 11. de Bernardis, P., 2000, Nature (London) 404, 955. Esposito-Far`ese, G., and D. Polarski, 2001, Phys. Rev. D 63, Demarque, P., L.M. Krauss, D.B. Guenther, and D. Nydam, 063504. 1994, Astrophys. J. 437, 870. Ezer, D., and A.G.W. Cameron, 1966, Can. J. Phys. 4, 593. Demidov N.A., et al., 1992, in Proceedings of the 6th European Fierz, M., 1956, Helv. Phys. Acta 29, 128. frequency and time forum, Noordwijk, NL, 1992, (European Fischbach, E., and C. Talmadge, 1996, Proc. of the XXXIth Space Agency, Noordwjik) 409. Rencontres de Moriond, Les Arcs (France) 20-27 january

49 1996. Isham, C.J., A. Salam, and J. Strathdee, 1971, Phys. Rev. D Flambaum, V.V., and E.V. Shuryak, 2001, preprint hep- 3, 1805. ph/0201303. Itzykson, C., and J-B. Zuber, 1980, Quantum field theory, Flowers, J.L., and B.W. Petley, 2001, Rept. Prog. Phys. 64, MacGraw-Hill, New York. 1191. Ivanchik, A.V., A.Y. Potekhin, and D.A. Varshalovich, 1999, Foltz, C.B., F.H. Chaffee, and J.H. Black, 1988, Astrophys. Astron. Astrophys. 343, 439. J. 324, 1988. Ivanchik, A.V., E. Rodriguez, P. Petitjean, and D. Var- Forg´acs, P., and Z. Horv´ath, 1979, Gen. Rel. Grav. 10, 931. shalovich, 2001, preprint astro-ph/0112323. Fowler, W.A., G.R. Caughlan, and B.A. Zimmerman, 1975, Jordan, P., 1937, Naturwiss. 25, 513. Annu. Rev. Astron. Astrophys. 13, 69. Jordan, P., 1939, Z. Physik 113, 660. Freund, P.G.O., 1982, Nuc. Phys. B 209, 146. Jordan, P., 1955, Schwerkraft und Weltall, (Vieweg, Braun- Fujii, Y., A. Iwamoto, T. Fukahori, T. Ohnuki, M. Nakagawa, schweig). H. Hidaka, Y. Oura, and P. M¨oller, 2000, Nuc. Phys. B 573 Kaluza, T., 1921, Sitzungsber. Preuss. Akad. Wiss. Phys. 377. Math. Kl. LIV, 966. Fujii, Y., 2002, preprint astro-ph/0204069. Kaplinghat, M., R.J. Scherrer, and M.S. Turner, 1999, Phys. Gamow, G., 1967a, Phys. Rev. Lett. 19, 759. Rev. D 60, 023516. Gamow, G., 1967b, Phys. Rev. Lett. 19, 913. Kaspi, V.M., J.H. Taylor, and M.F. Riba, 1994, Astrophys. Gamow, G., 1967c, Proc. Natl. Acad. Sci. U.S. 57, 187. J. 428, 713. Gasperini, M., F. Piazza, and G. Veneziano, 2002, Phys. Rev. Kehagias, A., and E. Kiritsis, 1999, JHEP 9911, 022. D 65, 023508. Khare, P., 1986, Phys. Rev. D 34, 1936. Gasser, J., and H. Leutwyler, 1982, Phys. Rep. 87, 77 Kim. J.B., and H.K. Lee, 1995, Astrophys. J. 448, 510. Gazta˜naga, E., E. Garcia-Berro, J. Isern, E. Bravo, and I. Kim. J.B., J.H. Kim, and H.K. Lee, 1998, Phys. Rev. D 58, Dominguez, 2002, Phys. Rev. D 65, 023506. 027301. Georgi, H. Quinn H., and S. Weinberg, 1974, Phys. Rev. Lett. Kiritsis, E., 1999, JHEP 9910, 010. 33, 451. Karshenboim, S.G., 2000, Can. J. Phys. 78, 639. Gold, R., 1968, Phys. Rev. Lett. 20, 219. Karshenboim, S.G., 2001, in Laser Physics at the Limits, Godone, A., C. Novero, P. Tavella, and K. Rahimullah, 1993, edited by H. Figger, D. Meschede and C. Zimmermann, Phys. Rev. Lett. 15, 2364. (Springer-Verlag, Berlin, Heidelberg), 165. Goldman, I., 1990, Month. Not. R. astr. Soc. 244, 184. Klein, O., 1926, Z. Phys. 37, 875. Guenther, L.M. Krauss, and P. Demarque, 1998, Astrophys. Kolb, E.W., M.J. Perry, and T.P. Walker, 1986, Phys. Rev. J. 498, 871. D 33, 869-871. Guenther, D.B., K. Sills, P. Demarque, and L.M. Krauss, Kothari, D.S., 1938, Nature (London) 142, 354. 1995, Astrophys. J. 445, 148. Kujat, J., and R.J. Scherrer, 2000, Phys. Rev. D 62, 023510. Gundlach, J.H., and S.M. Merkowitz, 2000, Phys. Rev. Lett. Lamoreaux, S.K., J.P. Jacobs, B.R. Heckel, F.J. Raab, and 85, 2869. E.N. Fortson, 1986, Phys. Rev. Lett. 57, 3125. Hannestad, S., 1999, Phys. Rev. D 60, 023515. Landau, L.D., 1955, in Niels Bohr and the development of Harko, T., and M.K. Mak, 1999, Class. Quant. Grav. 16, physics, edited by W. Pauli (Pergamon press, London). 2741. Landau, S., and H. Vucetich, 2000, preprint astro- Haugan, M.P., and C.M. Will, 1976, Phys. Rev. Lett. 37, 1. ph/0007108. Haugan, M.P.,1979, Ann. Phys. (NY) 118, 156. Landau, S., and H. Vucetich, 2002, Astrophys. J. 570, 463. Heintzmann, H., and H. Hillebrandt, 1975, Phys. Lett. A 54, Landau, S., D.D. Harai, and M. Zaldarriaga, 2001, Phys. Rev. 349. D 63, 083505. Hellings, R.W., P.J. Adams, J.D. Anderson, M.S. Keesey, E.L. Langacker, P., G. Segr`e, and M.J. Strassler, 2002, Phys. Lett. Lau, E.M. Standish, V.M. Canuto, and I. Goldman, 1983, B 528, 121 Phys. Rev. Lett. 51, 1609. Lanzetta, K.M., et al., 1995, in proc. of the XVIIyh Texas Hill, H.A., and Y.-M. Gu, 1990, Sci. China A 37, 854. symposiumon relativistic astrophysics, edited by W. Voges. Hoffmann, D.C., F.O. Lawrence, J.L. Mewherter, and F.M. Levshakov, S.A., 1992, ESO conf. Proc. 40, 139. Rourke, 1971, Nature (London) 234, 132. Levy-Leblond, J.M., 1979, in Problems in the foundations of Hogan, C.J., 2000, Rev. Mod. Phys. 72, 1149. physics, LXXII corso, (societa Italiana di Fisica Bologna), Hoˇrava, P., and E. Witten, 1996, Nuc. Phys. B 460, 506. 237-263. Horv´ath, J.E., and H. Vucetich, 1988, Phys. Rev. D 37, 931. Li, L.-X., and R. Gott III, 1998, Phys. Rev. D 58, 103513. Hoyle, F., 1972, Q. Jl. R. astr. Soc. 13, 328. Lichn´erowicz, A., 1955, Th´eories relativistes de la gravita- Hu, W., and S. Dodelson, 2002, Ann. Rev. Astron. Astrophys. tion et de l’´electromagn´etisme, Ed. Masson and Cie, Paris, (2002) in press. France. Huey, G., S. Alexander, and L. Pogosian, 2001, preprint astro- Lindner, M., D.A. Leich, R.J. Borg, G.P. Russ, J.M. Bazan, ph/0110562. D.S. Simons, and A.R. Date, 1986, Nature (London) 320, Ichikawa, K., and M. Kawasaki, 2002, preprint hep- 246. ph/0203006. Livio, M., and M. Stiavelli, 1998, Astrophys. J. 507, L13. Irvine, J.M., 1983a, Phil. Trans. R. Soc. London a 310, 239. Livio, M., D. Hollowell, A. Weiss, Nature (London) 340, 281. Irvine, J.M., 1983b, Contemp. Phys. 24, 427. Lubo, M., M. Lemoine, J. Martin, and J.-P. Uzan, 2002, Phys.

50 Rev. D 65, 023510. Newton, R.R., 1970, Ancient Astronomical observations and Lynden-Bell, D., 1982, Observatories 102, 86. Acceleration of the Earth and Moom, John Hopkins Press Ma, C.-P., and E. Bertschinger, 1995, Astrophys. J. 455, 7. (Baltimore). Magueijo, J., 2000, Phys. Rev. D 62, 103521. Newton, R.R., 1974, Month. Not. R. astr. Soc. 169, 331. Magueijo, J., 2001, Phys. Rev. D 63, 043502. Noerdlinger, P.D., 1973, Phys. Rev. Lett. 30, 761. Magueijo, J., H.B. Sandvik, and T.W.B. Kibble, 2001, Phys. Nollet, K.M., and R.E. Lopez, 2002, preprint astro- Rev. D 64, 023521. ph/0204325. Magueijo, J., J.D. Barrow, and H.B. Sandvik, 2002, preprint Nordtvedt, K., 1988, Phys. Rev. D 37, 1070. astro-ph/0202374. Nordtvedt, K., 1990, Phys. Rev. Lett. 65, 953. Malaney, R.A., and G.J. Mathews, 1993, Phys. Rep. 229, Nordtvedt, K., 2001, Class. Quant. Grav. 18, L133. 145. Novikov, I.D., and Ya. B. Zel’dovich, 1982, The structure Mansfield, 1976, Nature (London) 261, 560. and evolution of the universe, part V, University of Chicago Marciano, W.J., 1984, Phys. Rev. Lett. 52, 489. Press. Martins, C.J.A.P., 2002, preprint atro-ph/0205504. Oberhummer, H., A. Cs´ot´o, and H. Schlattl, 2000, Science Martins, C.J.A.P., A. Melchiorri, R. Trotta, R. Bean, G. 289, 88. Rocha, P.P. Avelino, and P.T.P. Viana, 2002, preprint atro- Okun, L.B., 1991, Usp. Fiz. Nauk. 161, 177 [Sov. Phys. Usp. ph/0203149. 34, 818]. Maurette, M. 1976, Ann. Rev. Nuc. Sci. 26, 319. Olive, K., and M. Pospelov, 2001, preprint hep-ph/0110377. Mellier, Y., 1999, Ann. Rev. Astron. Astrophys. 37, 127. Olive, K., M. Pospelov, Y.-Z. Qian, A. Coc, M. Cass´e, and E. McElhinny, M.W., S.R. Taylor, and D.J. Stevenson, 1978, Vangioni-Flam, 2002, preprint hep-ph/0205269. Nature (London) 271, 316. Overduin, J.M., and P.S. Wesson, 1997, Phys. Rep. 283, 303. McWeeny, R., 1973, Nature (London) 243, 196. Pagel, B.E.J., 1977, Month. Not. R. Astron. Soc. 179, 81. Milne, E.A., 1935, Proc. Roc. Soc. London A 158, 324. Pagel, B.E.J., 1983, Phil. Trans. R. Soc. Lond. A 310, 245. Minkowski, R., and O.C. Wilson, 1956, Astrophys. J. 123, Passarino, G., 2001, preprint hep-ph/0108524. 373. Peebles, P.J.E., 1968, Astrophys. J. 153, 1. Moffat, J., 1993a, Int. J. Phys. D 2, 351. Peebles, P.J., and R.H. Dicke, 1962, Phys. Rev. 128 2006. Moffat, J., 1993b, Found. Phys. 23, 411. Peres, A., 1967, Phys. Rev. Lett. 19, 1293. Moffat, J., 2001, preprint astro-ph/0109350. Perlmutter, S., et al., 1998, Nature (London) 391, 51. Moffat, J., 2002, preprint gr-qc/0202012. Petley, B.W., 1983, Nature (London) 303, 373. Mohr, J.-P., and B.N. Taylor, 2001, Phys. Today 54(3), 29. Petley, B.W., 1985, The fundamental physical constants and Morrison, L.V., 1973, Nature (London) 241, 519. the frontiere of measurement, (Adam Hilger, Bristol and Muller, P.M., 1978, in On the measurement of Cosmolog- Philadelphia). ical Variations of the gravitaional constant, edited by L. Petrov, Y.V., 1977, Sov. Phys. Usp. 20, 937. Halphern, University of Florida, p. 93. Pochet, T., J.M. Pearson, G. Beaudet, and H. Reeves, 1991, M¨uller, J., and K. Nordtvedt, 1998, Phys. Rev. D 58, 062001. Astron. Astrophys, 243, 1. M¨uller, J., M. Schneider, K. Nordtvedt, and D. Vokrouhlicky, Pochoda, P., and M. Schwarzschild, 1963, Astrophys. J. 139, 1999, in Proceedings of the 8th Marcel Grossman Meeting 587. on General Relativity, Jerusalem, 1997 (World Scientific, Polchinsky, J., 1997, Superstring Theory, Cambridge Univer- Singapore). sity Press, UK. M¨uller, J., M. Schneider, M. Soffel, and H. Ruder, 1991, As- Potekhin, A.Y., and D.A. Varshalovich, 1994, Astron. Astro- trophys. J. 382, L101. phys. Suppl. Ser. 104, 89. Murphy, C.T., and R.H. Dicke, 1964, Proc. Am. Phil. Soc. Potekhin, A.Y., A.V. Ivanchik, D.A. Varshalovich, K.M. 108, 224. Lanzetta, J.A. Baldwin, G.M. Williger, and R.F. Carswell, Murphy, M.T., J.K. Webb, V.V. Flambaum, C.W. Churchill, 1998, Astrophys. J. 505, 523. J.X. Prochaska, 2001b, Month. Not. R. Astron. Soc. 327, Prestage, J.D., R.L. Tjoelker, and L. Maleki, Phys. Rev. Lett. 1223. 74, 3511. Murphy, M.T., J.K. Webb, V.V. Flambaum, M.J. Drinkwa- Prochoska, J.X., and A.M. Wolfe, 1996, Astrophys. J. 470, ter, F. Combes, and T. Wiklind, 2001, Month. Not. R. As- 403. tron. Soc. 327, 1244. Prochoska, J.X., and A.M. Wolfe, 1997, Astrophys. J. 474, Murphy, M.T., J.K. Webb, V.V. Flambaum, V.A. Dzuba, 140. C.W. Churchill, J.X. Prochaska, J.D. Barrow, and A.M. Prochoska, J.X., and A.M. Wolfe, 2000, Astrophys. J. 533, Wolfe, 2001, Month. Not. R. Astron. Soc. 327, 1208. L5. Murphy, M.T., J.K. Webb, V.V. Flambaum, J.X. Prochaska, Ra˜nada, A.F., 2002, preprint astro-ph/0202224. and A.M. Wolfe, 2001, Month. Not. R. Astron. Soc. 327, Reasenberg, R.D., and I.I. Shapiro, 1976, in Atomic masses 1237. and fundamental constants, Vol. 5, edited by J.H. Sanders Murray, C., and S. Dermott, 2000, Solar System Dynamics, and A.H. Wapstra (Plenum, New York), 643. Cambridge University Press. Reasenberg, R.D., and I.I. Shapiro, 1978, in On the mea- Naudet, R., 1974, Bull. Inf. Sci. Tech. (Paris) 193, 1. surements of cosmological variations of the gravitational Naudet, R., 2000, Oklo, des r´eacteurs nucl´eaires fossiles: constant, edited by L. Halperin (University of Florida, ´etude physique, Editions du CEA (France). Gainesville), 71.

51 Reasenberg, R.D., 1983, Phil. Trans. R. Soc. Lond. A 310, Van Flandern, T.C., 1975, Month. Not. R. Astron. Soc. 170, 227. 333. Reeves, H., 1994, Rev. Mod. Phys. 66, 193. Van Flandern, T.C., 1981, in Precision measurements and Riazuelo, A., and J.-P. Uzan, 2000, Phys. Rev. D 62, 083506. fundamental constants II, NBS circular No. 617 (U.S. GPO., Riazuelo, A., and J.-P. Uzan, 2002, Phys. Rev. D 65 (to ap- Whashington DC) 625. pear); preprint astro-ph/0107386. Vanier, J., and C. Audouin, 1989, The quantum physics Ricci, B., and F.L. Villante, 2002, preprint astro-ph/0204482. of atomic frequency standards, Adam Hilger (Bristol and Riess, A.G., et al., 1998, Astron. J. 116, 1009. Philadelphia). Roeder, R.C., 1967, Astrophys. J. 149, 131. Varshalovich, D.A., and S.A. Levshakov, 1993, J. Exp. Theor. Rothman, T., and R. Matzner, 1982, Astrophys. J. 257, 450. Phys. Lett, 58, 231. Rozental, I.L., 1980, Usp. Fiz. Nauk. 131, 239; [Sov. Phys. Varshalovich, D.A., A.Y. Potekhin, 1994, Astron. Lett. 20, Usp. 23 (1980) 296]. 771. Rozental, I.L., 1988, Big Bang, Big Bounce (Springer-Verlag, Varshalovich, D.A., A.Y. Potekhin, 1995, Space Sci. Rev. 74, Berlin). 259. Runcorn, S.K., 1964, Nature (London) 204, 823. Varshalovich, D.A. and A.Y. Potekhin, 1996, Astron. Lett. Sandvik, H.B., J.D. Barrow, and J. Magueijo, 2002, Phys. 22, 1. Rev. Lett. 88, 031302. Varshalovich, D.A., V.E. Panchuk, A.V. Ivanchik, 1996b, As- Savedoff, M.P., 1956, Nature (London) 176, 688. tron. Lett. 22, 6. Scherrer, R.J., and D.N. Spergel, 1993, Phys. Rev. D 47, Varshalovich, D.A., A.Y. Potekhin, A.V. Ivanchik, V.E. 4774. Panchuk, and K.M. Lanzetta, 1996a, Proceedings of the Sec- Scrutton, C.T., 1965, Paleontology 7, 552. ond International Sakharov Memorial Conference (Moscow, Serna, A., and R. Dom´ınguez-Tenreiro, 1992, Astophys. J. 19-26 May, 1996). 389, 1. Varshalovich, D.A., A.Y. Potekhin, and A.V. Ivanchik, Serna, A., R. Dom´ınguez-Tenreiro, and G. Yepes, 1992, Asto- 2000b, Comments At. Mol. Phys. (to appear), preprint phys. J. 391, 433. physics/0004068. Shapiro, I.I., 1964, Phys. Rev. Lett. 13, 789. Varshalovich, D.A., A.Y. Potekhin, and A.V. Ivanchik, 2000a, Shapiro, I.I., 1990, in General Relativity and Gravitation, in X-ray and Inner-Shell Processes, AIP Conf. Proc. (AIP, edited by N. Ashby, D.F. Bartlett, and W. Wyss, Cambridge Melville, 2000), edited by R.W. Dunford, D.S. Gemmel, E.P. University Press. Kanter, B. Kraessig, S.H. Southworth, and L. Young, 506, Shapiro, I.I., W.B. Smith, and M.B. Ash, 1971, Phys. Rev. 503. Lett. 26, 27. Vayanokis, C.E., 1988, Phys. Lett. B 213, 419. Shlyakhter, A.I., 1976, Nature (London) 264, 340. Veneziano, G., 1986, Europhysics Lett. 2, 199. Silk, J., 1968, Astrophys. J. 151, 459. Vila, S.C., 1976, Astrophys. J. 206, 213. Sisterna, P., and H. Vucetich, 1990, Phys. Rev. D 41 1034. Vinti, J.P., 1974, Month. Not. R. Soc. 169, 417. Sisterna, P., and H. Vucetich, 1991, Phys. Rev. D 44 3096. Volovik, G.E., 2002, preprint physics/0203075. Solheim, J.-E., T.G. Barnes III, and H.J. Smith, 1976, Astro- Webb, J.K., V.V. Flambaum, C.W. Churchill, M.J. Drinkwa- phys. J. 209, 330. ter, J.D. Barrow, 1999, Phys. Rev. Lett. 82, 884. Sortais, Y., et al., 2001, Physica Scripta T95, 50. Webb, J.K., M.T. Murphy, V.V. Flambaum, V.A. Dzuba, Stanyukovich, K.P., 1962, Dokl. Akad. Nauk. 147, 1348 [Sov. J.D. Barrow, C.W. Churchill, J.X. Pochaska, and A.M. Phys. Dokl. 7, 1150 (1962)]. Wolfe, 2001, Phys. Rev. Lett. 87, 091301. Steer, D., and M.P. Parry, 2002, preprint hep-th/0201121. Weinberg, S., 1983a, Phil. Trans. R. Soc. London A 310, 249. Steigmann, G., 1976, Nature (London) 261, 479. Weinberg, S., 1983b, Phys. Lett. B 125, 265. Su, Y., B.R. Heckel, E.G. Adelberger, J.H. Gundlach, M. Har- Weinberg, S., 1989, Rev. Mod. Phys. 61, 1. ris, G.L. Smith, and H.E. Swanson, 1994, Phys. Rev. D 50, Wesson, P.S., 1973, Q. Jl. astr. Soc. 14, 9. 3614. Wetterich, C., 2002, preprint hep-ph/0203266. Taylor, T.R., and G. Veneziano, 1988, Phys. Lett. B 213, 450. Wignall, J.W.G., 2000, Int. J. Mod. Phys. A 15, 875-892. Teller, E., 1948, Phys. Rev. 73, 801. Wiklind, T., and F. Combes, 1997, Astron. Astrophys. 328, Thompson, R., 1975, Astron. Lett. 16, 3. 48. Thorsett, S.E., 1996, Phys. Rev. Lett. 77, 1432. Wilkinson, D.H., 1958, Phil. Mag. 3, 582. Torgerson, J.R., 2000, preprint physics/0012054. Will, C.M., 1993, Theory and experiment in gravitaional Tubbs, A.D., and A.M. Wolfe, 1980, Astrophys. J. 236, L105. physics, (Cambridge University Press). Turneaure, J.P., and S.R. Stein, 1974, in Atomic masses and Will, C.M., 2001, Living Rev. Relativity 4, 4. fundamental constants, Vol. 5, edited by J.H. Sanders and Williams, P.J., et al., 1976, Phys. Rev. Lett. 36, 551. A.H. Wapstra (Plenum, New York) 636. Williams, P.J., X.X. Newhall, and J.O. Dickey, 1996, Phys. Uzan, J.-P., 1999, Phys. Rev. D 59, 123510. Rev. D 53, 6730-6739. Uzan, J.-P., 2002, Pour la Science (to appear, july). Witten, E., 1981, Nuc. Phys. B 186, 412. Uzan, J.-P., and F. Bernardeau, 2001, Phys. Rev. D 64, Witten, E., 1984, Phys. Lett. B 149, 351. 083004. Wolfe, A.M., R.L. Brown and M.S. Roberts, 1976, Phys. Rev. Van Diggelen, J., 1976, Nature (London) 262, 275. Lett. 37, 179. Van Flandern, T.C., 1971, Astron. J. 76, 81. Wolfe, A.M., and M.M. Davis, 1979, Astron. J. 84, 699.

52 Wu, Y.S., and Z. Wang, 1986, Phys. Rev. Lett. 57, 1978. Yahil, 1975, in The interaction between science and philoso- phy, edited by Y. Elkana, Humanities Press, 27. Yang, J., D.N. Schramm, G. Steigmann, and R.T. Rood, 1979, Astrophys. J. 227, 697. Youm, D., 2001, preprint hep-th/0108237.

53 TABLE I. Comparison of absorption lines and the combinations of the fundamental constants that can be constrained.

Comparison constant

fine structure doublet αEM 2 hyperfine H vs optical gpµαEM

hyperfine H vs fine-structure gpµαEM rotational vs vibrational modes of molecules µ 2 rotational modes vs hyperfine H gpαEM fine structure doublet vs hyperfine H gp

TABLE II. Summary of the constraints on the variation of the fine structure constant ∆αEM /αEM . Reference Constraint Redshift Time (109 yr) Method − (Savedoff, 1956) (1.8 ± 1.6) × 10 3 0.057 CygnusA(NII,NeIII) − (Wilkinson, 1958) (0 ± 8) × 10 3 3-4 α-decay − (Bahcall et al., 1967) (−2 ± 5) × 10 2 1.95 QSO(SiII,SiIV) (Bahcall and Schmidt, 1967) (1 ± 2) × 10−3 0.2 radio galaxies (O III) − (Dyson, 1967) (0 ± 9) × 10 4 3 Re/Os − (Gold, 1968) (0 ± 4.66) × 10 4 2 Fission ± +2 × −4 (Chitre and Pal, 1968) (0 3−2) 10 1 Fission − (Dyson, 1972) (0 ± 4) × 10 4 2 α-decay − (Dyson, 1972) (0 ± 1) × 10 3 2 Fission (Dyson, 1972) (0 ± 5) × 10−6 1 Re/Os − (Shlyakhter, 1976) (0 ± 1.8) × 10 8 1.8 Oklo − (Wolfe et al., 1976) (0 ± 3) × 10 2 0.524 QSO (Mg I) − (Irvine, 1983a) (0 ± 9) × 10 8 1.8 Oklo − (Lindner et al., 1986) (−4.5 ± 9) × 10 6 4.5 Re/Os − (Kolb et al., 1986) (0 ± 1) × 10 4 108 BBN − (Potekhin and Varshalovich, 1994) (2.1 ± 2.3) × 10 3 3.2 QSO(CIV,SiIV,...) − (Varshalovich and Potekhin, 1994) (0 ± 1.5) × 10 3 3.2 QSO(CIV,SiIV,...) − (Cowie and Songaila, 1995) (−0.3 ± 1.9) × 10 4 2.785 − 3.191 QSO − (Prestage et al., 1995) (0 ± 1.42) × 10 14 0 140 days Atomic cloks (Damour and Dyson, 1996) (0.15 ± 1.05) × 10−7 1.8 Oklo − (Varshalovich et al., 1996a) (2 ± 7) × 10 5 2.8 − 3.1 QSO(SiIV) − (Bergstr¨om et al., 1999) (0 ± 2) × 10 2 108 BBN (Webb et al., 1999) (−0.17 ± 0.39) × 10−5 0.6 − 1 QSO(MgII,FeII) − (Webb et al., 1999) (−1.88 ± 0.53) × 10 5 1 − 1.6 QSO(MgII,FeII) − (Ivanchik et al., 1999) (−3.3 ± 6.5 ± 8) × 10 5 2 − 3.5 QSO(SiIV) (Fujii et al., 2000) (−0.36 ± 1.44) × 10−7 1.8 Oklo − (Varshalovich et al., 2000a) (−4.5 ± 4.3 ± 1.4) × 10 5 2 − 4 QSO(SiIV) − (Avelino et al., 2001) (−3.5 ± 5.5) × 10 2 103 CMB (Landau et al., 2001) (−5.5 ± 8.5) × 10−2 103 CMB − (Webb et al., 2001) (−0.7 ± 0.23) × 10 5 0.5 − 1.8 QSO(FeII,MgII) − (Webb et al., 2001) (−0.76 ± 0.28) × 10 5 1.8 − 3.5 QSO(NiII,CrII,ZnII) (Webb et al., 2001) (−0.5 ± 1.3) × 10−5 2 − 3 QSO(SiIV) − (Murphy et al., 2001a) (−0.2 ± 0.3) × 10 5 0.5 − 1 QSO(MgI,MgII,...) − (Murphy et al., 2001a) (−1.2 ± 0.3) × 10 5 1 − 1.8 QSO(MgI,MgII,...) (Murphy et al., 2001a) (−0.7 ± 0.23) × 10−5 0.5 − 1.8 QSO(MgI,MgII,...) − (Murphy et al., 2001d) (−0.5 ± 1.3) × 10 5 2 − 3 QSO(SiIV) − (Sortais et al., 2001) (8.4 ± 13.8) × 10 15 24 months Atomic clock (Nollet and Lopez, 2002) (3 ± 7) × 10−2 108 BBN − (Ichikawa and Kawasaki, 2002) (−2.24 ± 3.75) × 10 4 108 BBN − (Olive et al., 2002) (0 ± 1) × 10 7 1.8 Oklo (Olive et al., 2002) (0 ± 3) × 10−7 ∼ 0.45 4.6 Re/Os − (Olive et al., 2002) (0 ± 1) × 10 5 α-decay

54 TABLE III. The different atomic clock experiments. We recall the transitions which are compared and the constraint on the time variation obtained. SCO refers to superconductor cavity oscillator and the reference to (Breakiron, 1993) is cited in Prestage et al. (1995). fs and hfs refer respectively to fine structure and hyperfine structure.

− Reference Experiment Constant Duration Limit (yr 1) 3 × −12 (Turneature and Stein, 1974) hfs of Cs vs SCO gpµαEM 12 days < 1.5 10 −13 (Godone et al., 1983) hfs of Cs vs fs of Mg gpµ 1 year < 2.5 × 10 × −14 (Demidov, 1992) hfs of Cs vs hfs of H αEM gp/gI 1 year < 5.5 10 × −14 (Breakiron, 1993) hfs of Cs vs hfs of H αEM gp/gI < 5 10 + × −14 (Prestage et al., 1995) hfs of HG vs hfs of H αEM gp/gI 140 days < 2.7 10 (Sortais et al., 2001) hfs of Cs vs hfs of Rb 24 months (4.2 ± 6.9) × 10−15

∗ TABLE IV. Summary of the constraints on the time variation of the Newton constant G. The constraints labelled by refer to bounds on the rate of decrease of G (that is −G/G˙ < . . .).

− Reference Constraint (yr 1) Method − (Teller, 1948) (0 ± 2.5) × 10 11 Earth temperature − (Shapiro at al., 1971) (0 ± 4) × 10 10 Planetary ranging (Morison, 1973) (0 ± 2) × 10−11 Lunar occultations ∗ − (Dearborn and Schramm, 1974) < 4 × 10 11 Clusters of galaxies − (van Flandern, 1975) (−8 ± 5) × 10 11 Lunar occultations (Heintzmann and Hillebrandt, 1975) (0 ± 1) × 10−10 Pulsar spin-down − (Reasenberg and Shapiro, 1976) (0 ± 1.5) × 10 10 Planetary ranging ∗ − (Mansfield, 1976) < (−5.8 ± 1) × 10 11 Pulsar spin-down (Williams et al., 1996) (0 ± 3) × 10−11 Planetary ranging − (Blake, 1977b) (−0.5 ± 2) × 10 11 Earth radius − (Muller, 1978) (2.6 ± 1.5) × 10 11 Solar eclipses (McElhinny et al., 1978) ∗ < 8 × 10−12 Planetary radii − (Barrow, 1978) (2 ± 9.3)h × 10 12 BBN ∗ − (Reasenberg et al., 1979) < 10 12 Viking ranging − (van Flandern, 1981) (3.2 ± 1.1) × 10 11 Lunar occultation − (Rothman and Matzner, 1981) (0 ± 1.7) × 10 13 BBN − (Hellings et al., 1983) (2 ± 4) × 10 12 Viking ranging − (Reasenberg, 1983) (0 ± 3) × 10 11 Viking ranging − (Damour et al., 1988) (1.0 ± 2.3) × 10 11 PSR 1913+16 − (Shapiro, 1990) (−2 ± 10) × 10 12 Planetary ranging ∗ − (Goldman, 1990) < (3.85 ± 1.65) × 10 11 PSR 0655+64 (Accetta et al., 1990) (0 ± 9) × 10−13 BBN − (M¨uller et al., 1991) (0 ± 1.04) × 10 11 Lunar laser ranging − (Anderson et al., 1991) (0.0 ± 2.0) × 10 12 Planetary ranging (Damour and Taylor, 1991) (1.10 ± 1.07) × 10−11 PSR 1913+16 − (Chandler, 1993) (0 ± 1) × 10 11 Viking ranging − (Dickey et al., 1994) (0 ± 6) × 10 12 Lunar laser ranging (Kaspi et al., 1994) (4 ± 5) × 10−12 PSR B1913+16 − (Kaspi et al., 1994) (−9 ± 18) × 10 12 PSR B1855+09 − (Demarque et al., 1994) (0 ± 2) × 10 11 Heliosismology (Guenther et al., 1995) (0 ± 4.5) × 10−12 Heliosismology ∗ +3 × −11 (Garcia-Berro et al., 1995) < (3−1) 10 White dwarf − (Williams et al., 1996) (0 ± 8) × 10 12 Lunar laser ranging − (Thorsett, 1996) (−0.6 ± 4.2) × 10 12 Pulsar statistics − (Del’Innocenti et al., 1996) (−1.4 ± 2.1) × 10 11 Globular clusters − (Guenther et al., 1998) (0 ± 1.6) × 10 12 Heliosismology

55 ≡ ≡ 2 TABLE V. Summary of the constraints on the variation of the constant k. We use the notation µ me/mp, x αEM gpµ and ≡ 2 y αEM gp. Reference Constant Constraint redshift Time (109 yr) Method (Yahil, 1975) µ (0 ± 1.2) 10 Rb-Sr,K-Ar (Pagel, 1977) µ (0 ± 4) × 10−1 2.1-2.7 QSO − (Foltz et al., 1988) µ (0 ± 2) × 10 4 2.811 QSO − (Varshalovich and Levshakov 1993) µ (0 ± 4) × 10 3 2.811 QSO (Cowie and Songaila, 1995) µ (0.75 ± 6.25) × 10−4 2.811 QSO − (Varshalovich and Potekhin, 1995) µ (0 ± 2) × 10 4 2.811 QSO − (Varshalovich et al., 1996a) µ (0 ± 2) × 10 4 2.811 QSO − (Varshalovich et al., 1996b) µ (−1 ± 1.2) × 10 4 2.811 QSO − (Potekhin et al., 1988) µ (−7.5 ± 9.5) × 10 5 2.811 QSO − (Ivanchik et al., 2001) µ (−5.7 ± 3.8) × 10 5 2.3 − 3 QSO − (Savedoff, 1956) x (3 ± 7) × 10 4 0.057 Cygnus A − (Wolfe et al., 1976) x (5 ± 10) × 10 5 ∼ 0.5 QSO(MgI) − (Wolfe and Davis, 1979) x (0 ± 2) × 10 4 1.755 QSO (Wolfe and Davis, 1979) x (0 ± 2.8) × 10−4 0.524 QSO − (Tubbs and Wolfe, 1980) x (0 ± 1) × 10 4 1.776 QSO − (Cowie and Songaila, 1995) x (7 ± 11) × 10 6 1.776 QSO − (Varshalovich and Potekhin, 1996) y (−4 ± 6) × 10 5 0.247 QSO − (Varshalovich and Potekhin, 1996) y (−7 ± 10) × 10 5 1.94 QSO (Drinkwater et al., 1998) y (0 ± 5) × 10−6 0.25, 0.68 QSO − (Carrilli et al., 2001) y (0 ± 3.4) × 10 5 0.25, 0.68 QSO − (Murphy et al., 2001c) y (−0.2 ± 0.44) × 10 5 0.25 QSO (Murphy et al., 2001c) y (−0.16 ± 0.54) × 10−5 0.68 QSO −2 (Wolfe et al., 1976) gpµ (0 ± 0.68) × 10 0.524 QSO 3 ± × −16 (Turneature and Stein, 1994) gpµαEM (0 9.3) 10 12 days Atomic clocks −13 (Godone et al., 1993) gpµ (0 ± 5.4) × 10 1 year Atomic clocks ± × 1 (Wilkinson, 1958) αW (0 1) 10 1 Fission ± × −1 (Dyson, 1972) αW (0 1) 10 1 β-decay ± × −3 (Shlyakhter, 1976) αW (0 4) 10 1.8 Oklo ± × −2 (Damour and Dyson, 1996) αW (0 2) 10 1.8 Oklo

56