Precision Gravity Tests and the Einstein Equivalence Principle
G. M. Tino, L. Cacciapuoti, S. Capozziello, G. Lambiase, F. Sorrentino
aa r X i v : . [ g r- q c ] A p r Precision Gravity Tests and the Einstein Equivalence Principle
G. M. Tino, L. Cacciapuoti, S. Capozziello, , a G. Lambiase, , a F. Sorrentino Dipartimento di Fisica e Astronomia and LENS LaboratoryUniversit`a di Firenze and INFN-Sezione di Firenzevia Sansone 1, Sesto Fiorentino, Italy European Space Agency, Keplerlaan 1 - P.O. Box 299, 2200 AG Noordwijk ZH, TheNetherlands Dipartimento di Fisica ”E. Pancini” , Universit`a di Napoli Federico II, via Cinthia9, I-80126, Napoli, Italy. a INFN, Sezione di Napoli, via Cinthia 9, I-80126 Napoli, Italy Dipartimento di Fisica E.R: Caianiello, Universit`a di Salerno, Via Giovanni PaoloII 132, I-84084, Fisciano (SA), Italy. a INFN, Sezione di Napoli, Gruppo collegato di Salerno, Via Giovanni Paolo II 132,I-84084 Fisciano (SA), Italy INFN Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, ItalyJune 16, 2020
Abstract
General Relativity is today the best theory of gravity addressing a wide range of phenomena.Our understanding of physical laws, from cosmology to local scales, cannot be properly formulatedwithout taking into account its concepts, procedures and formalim. It is based on one of themost fundamental principles of Nature, the Equivalence Principle, which represents the core of theEinstein theory of gravity describing, under the same standard, the metric and geodesic structure ofthe spacetime. The confirmation of its validity at different scales and in different contexts representsone of the main challenges of modern physics both from the theoretical and the experimental pointsof view.A major issue related to this principle is the fact that we actually do not know if it is valid ornot at quantum level. We are simply assuming its validity at fundamental scales. This question iscrucial in any self-consistent theory of gravity.Furthermore, recent progress on relativistic theories of gravity, including deviations from Gen-eral Relativity at various scales, such as extensions and alternatives to the Einstein theory, haveto take into account, besides the Equivalence Principle, new issues like Dark Matter and DarkEnergy, as well as the validity of fundamental principles like local Lorentz and position invari-ance. The general trend is that high precision experiments are conceived and realized to test bothEinstein’s theory and its alternatives at fundamental level using established and novel methods.For example, experiments based on quantum sensors (atomic clocks, accelerometers, gyroscopes, ravimeters, etc.) allow to set stringent constraints on well established symmetry laws (e.g. CPTand Lorentz invariance), on the physics beyond the Standard Model of particles and interactions,and on General Relativity and its possible extensions.In this review, we discuss precision tests of gravity in General Relativity and alternative theo-ries and their relation with the Equivalence Principle. In the first part, we discuss the EinsteinEquivalence Principle according to its weak and strong formulation. We recall some basic topicsof General Relativity and the necessity of its extension. Some models of modified gravity are pre-sented in some details. This provides us the ground for discussing the Equivalence Principle alsoin the framework of extended and alternative theories of gravity. In particular, we focus on thepossibility to violate the Equivalence Principle at finite temperature, both in the frameworks ofGeneral Relativity and of modified gravity. Equivalence Principle violations in the Standard ModelExtension are also discussed. The second part of the paper is devoted to the experimental testsof the Equivalence Principle in its weak formulation. We present the results and methods used inhigh-precision experiments, and discuss the potential and prospects for future experimental tests. ontents G . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.11 Violation of the weak equivalence principle and quintessence . . . . . . . . . . . . . . . 292.12 Equivalence principle in screening mechanisms . . . . . . . . . . . . . . . . . . . . . . . 312.13 Long-range forces and spin-gravity coupling terms . . . . . . . . . . . . . . . . . . . . . 322.14 The equivalence principle in Poincare Gauge Theory and Torsion . . . . . . . . . . . . . 332.15 The violation of the equivalence principle for charged particles . . . . . . . . . . . . . . 332.16 Equivalence principle violation via quantum field theory . . . . . . . . . . . . . . . . . 342.17 Equivalence principle violation via modified geodesic equation . . . . . . . . . . . . . . 362.18 Application to the Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.19 Application to the Brans-Dicke metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.20 Standard Model Extensions and the Weak Equivalence Principle . . . . . . . . . . . . . 392.21 Strong Equivalence principle in modified theories of gravity . . . . . . . . . . . . . . . . 41 Introduction
General Relativity (GR) relies on the assumption that space and time are entangled into a uniquestructure, i.e. the spacetime. It is assigned on a pseudo-Riemannian manifold endowed with a Lorenziansignature. Dynamics has to reproduce, in the absence of a gravitational field, the Minkowski spacetime.GR, as an extension of classical mechanics, has to match some minimal requirements to be considereda self-consistent theory of gravitation: it has to reproduce results of Newton’s physics in the weak-energylimit, hence it must be able to explain dynamics related to the planetary motion and the gravitatingstructures such as stars, galaxies, clusters of galaxies. Moreover, it has to pass observational tests inthe Solar System. These facts constitute the experimental foundation on which GR is based. They areusually called the ”classical tests of GR” [1, 2].Beside the above ”mechanical issues”, GR has to explain Galactic dynamics, considering baryonicconstituents, like stars, planets, dust and gas, radiation. These components are tied together by theNewtonian potential, which is supposed to work at any Galactic scales. Also, GR has to address the largescale structure formation and dynamics. At cosmological scales, GR is required to address dynamicsof the whole universe and correctly reproduce cosmological parameters like the Hubble expansion rate,the density parameters and the accelerated (decelerated) behavior of cosmic fluid. Cosmological andastrophysical observations actually probe only the standard baryonic matter, the radiation and theattractive overall interaction of gravity acting at all scales.Furthermore, starting from Galileo, the free-fall of different bodies is assumed to be independent ofthe nature of massive bodies on the Earth. The free-fall acceleration is unique and implies that gravi-tational and inertial mass ratio is identical for different bodies. This experimental result is one of thefoundations of Einstein’s GR as well as of any metric theory of gravity. After Galileo’s experiment fromthe leaning tower of Pisa, the free-fall acceleration uniqueness has been verified in many experiments,as widely discussed in the second part of this review. Summarizing, we can say that GR is based onfour main assumptions:The ”
Relativity Principle ” - there is no preferred inertial frames, i.e. all frames (acceleratedor not) are good frames for Physics.The ”
Equivalence Principle ” (EP) - inertial effects are locally indistinguishable from grav-itational effects (which means the equivalence between the inertial and the gravitationalmasses). In other words, any gravitational field can be locally cancelled.The ”
General Covariance Principle ” - field equations must be ”covariant” in form, i.e. theymust be invariant under the action of any spacetime diffeomorphisms.The ”
Causality Principle ” - each point of space-time admits a universal notion of past,present and future.On these bases, Einstein postulated that, in a four-dimensional spacetime manifold, the gravitationalfield is described by the metric tensor ds = g µν dx µ dx ν , with the same signature of Minkowski metric.The metric coefficients are the physical gravitational potentials. Moreover, spacetime is curved by thedistribution of energy-matter sources, e.g., the distribution of celestial bodies.An important historical remark is necessary at this point. E. Kretschmann, in 1917 [3], criticizedthe General Covariance Principle. In demanding General Covariance, he asserted that Einstein placedno constraint on the physical content of his theory. Kretschmann stressed that any spacetime theorycould be formulated in a generally covariant way without any physical principle. In formulating GR,Einstein used tensor calculus. Kretschmann pointed out that this calculus allowed for general covariantformulations of theories while Einstein discussed general covariance as the form invariance of theory’s4quations as soon as the spacetime coordinates are transformed. This can be considered as a ”passive”point of view of General Covariance: if we have some system of fields, we can change our spacetimecoordinate system as we like and the new descriptions of the fields in the new coordinate systemswill still solve the theory’s equations. The answer by Einstein was that the form invariance of thetheory’s equations also allows a second version, the so-called ”active” General Covariance. It involvesno transformation of the spacetime coordinate system. In fact, active General Covariance gives riseto the generation of new solutions of the equations of the theory in the same coordinate system onceone solution is given. According to this approach, General Covariance Principle can be considered aphysical principle.The above principles require that the spacetime structure has to be determined by either one orboth of the two following fields: a Lorentzian metric g and a linear connection Γ, assumed by Einsteinto be torsionless because, at that time, the spin of particles was not considered a possible source forthe gravitational field. The physical meaning of these two fields is the following: The metric g fixes thespacetime causal structure, that is the light cones. According to this statement, metric relations, i.e.clocks and rods, are possible. On the other hand, the connection Γ fixes the free-fall of objects, thatis the local inertial observers according to the Equivalence Principle. Both fields, of course, have tosatisfy some compatibility relations like the requirement that photons follow null geodesics of Γ. Thismeans that Γ and g can be independent, a priori , but they are constrained, a posteriori , according tosome physical restrictions which impose that Γ has to be the Levi-Civita connection of g . However, inmore general approaches, Γ and g can be independent [4].Despite the self-consistency and the solid foundation, there are several issues for GR, both fromthe theoretical and the experimental (observational) points of view. The latter clearly points out thatGR is not capable of addressing Galactic, extra-galactic and cosmic dynamics unless a huge quantityof some exotic forms of matter-energy is considered. These ingredients are usually called dark matter and dark energy and constitute up to 95% of the total amount of cosmic matter-energy [5, 6].On the other hand, instead of changing the source side of the Einstein field equations, a ”geometricalview” can be taken into account to fit the missing matter-energy of the observed Universe. In sucha case, the dark side could be addressed by extending GR including further geometric invariants intothe Hilbert - Einstein Action besides the Ricci curvature scalar R . These effective Lagrangians canbe justified at the fundamental level considering the quantization of fields on curved spacetimes [6].However, at the present stage of the research, there is no final probe discriminating between darkmatter-energy picture and extended (alternative) gravity . Furthermore, the bulk of observations to beconsidered is very large and then an effective Lagrangian or a single new particle, addressing the wholephenomenology at all astrophysical and cosmic scales, would be very difficult to find.An important discussion is related to the choice of the dynamical variables. In formulating GR,Einstein assumed that the metric g is the fundamental object to describe gravity. The connectionΓ αµν = n αµν o g is assumed, by construction, with no dynamics. Only g has dynamics. This meansthat the metric g determines, at the same time, the causal structure (light cones), the measurements(rods and clocks) and the free fall of test particles (geodesic structure). Spacetime is given by a coupleof mathematical objects {M , g } constituted by a Riemann manifold and a metric. Einstein realizedthat gravity induces free fall and that the EP selects an object that cannot be a tensor because theconnection Γ can be switched off and set to zero at least in a point. According to this consideration,Einstein was obliged to choose the Levi - Civita connection determined by the metric structure.Alternatively, in the Palatini formalism a (symmetric) connection Γ and a metric g are assumed and An important remark is useful at this point. With the term
Extended Gravity , we mean any class of theories by whichit is possible to recover Einstein GR as a particular case or in some post-Einstenian limit as in the case of f ( R ) gravity.With Alternative Gravity , we mean a class of theories which considers different approach with respect to GR, for examplethe Teleparallel Equivalent Gravity considering the torsion scalar instead of curvature scalar to describe dynamics. {M , g, Γ } where the metric determines causal structure while Γ determines the free fall. This means that, in thePalatini formalism, connections are differential equations determining dynamics. From this point ofview, Γ is the Levi-Civita connection of g as an outcome of the field equations.The connection is the fundamental field in the Lagrangian while the metric g enters the Lagrangianas the need to define lengths and distances to make experiments. It defines the causal structure buthas no dynamical role. As a consequence, there is no reason to assume g as the potential of Γ.With this consideration in mind, we discuss here the role of the EP in the debate of theories ofgravity both from a theoretical and experimental point of view.This review is organized as follows. In Section 2 we discuss the different formulations of the EP.After summarizing the main topics of GR and Quantum Field Theory (QFT) in curved spacetimes, wediscuss metric theories of gravity considering possible extensions and modifications of GR. Motivations,both theoretical and experimental, suggesting generalizations of GR, are considered. Specifically, thesetheories have been introduced to account for shortcomings of GR, both at early and late phases of theUniverse evolution: Cosmological Inflation, Dark Matter and Dark Energy represent the main issues ofthis debate. From the other side, GR is not a fundamental theory of physics because it should requirethe inclusion of quantum effects. It is then natural to ask whether the Equivalence Principle still holds inthe framework of any modified gravity approach aimed to enclose quantum physics under the standardof gravitational interaction. According to this requirement, we discuss the possibility to violate the EPby considering QFT at finite temperature. Besides, violations of the EP also occur in the framework ofthe extensions of Standard Model of particles. Section 3 is essentially devoted to experimental tests. Wepresent a wide class of experiments aiming to test the EP, in particular its weak formulation, with a highaccuracy. These include free falling tests, measurements based on Earth-to-Moon and Earth-to-satellitedistances, cold atoms and particles interferometry tests, spin-gravity coupling tests, matter-antimattertests. Conclusions are drawn in Section 4. The EP is related to the above considerations and plays a relevant role to discriminate among concurringtheories of gravity. In particular, the role of g and Γ are related to the validity of EP. Specifically, precisemeasurements of EP could say if Γ is only Levi - Civita or if a more general connection, disentangledfrom g , is necessary to describe gravitational dynamics. Furthermore, possible violation of EP canput in evidence other dynamical fields like torsion discriminating among the fundamental structure ofspacetime that can be Riemannian or not.Summarizing, the relevance of EP comes from the following points: • Competing theories of gravity can be discriminated according to the validity of EP; • EP holds at classical level but it could be violated at quantum level; • EP allows to investigate independently geodesic and causal structure of spacetime.If it is violatedat fundamental level, such structures could be independent.From a theoretical point of view, EP constitutes the foundation of metric theories. The first formulationof EP comes out from the formulation of gravity by Galileo and Newton, i.e. the Weak EquivalencePrinciple (WEP) which states that the inertial mass m i and the gravitational mass m g of physical objectsare equivalent. The WEP implies that it is impossible to distinguish, locally, between the effects ofa gravitational field from those experienced in uniformly accelerated frames using the straightforwardobservation of the free fall of physical objects. 6he first generalization of WEP states that Special Relativity is locally valid. Einstein obtained,in the framework of Special Relativity, that mass can be reduced to a manifestation of energy andmomentum. As a consequence, it is impossible to distinguish between an uniform acceleration and anexternal gravitational field, not only for free-falling objects, but whatever is the experiment. Accordingto this observation, Einstein EP states: • The WEP is valid. • The outcome of any local non-gravitational test experiment is independent of the velocity offree-falling apparatus. • The outcome of any local non-gravitational test experiment is independent of where and when itis performed in the universe.One can define a ”local non-gravitational experiment” as that performed in a small-size of a free-fallinglaboratory. Immediately, it is possible to realize that gravitational interaction depends on the curvatureof spacetime. It means that the postulates of metric gravity theories have to be satisfied. Hence thefollowing statements hold: • Spacetime is endowed with a metric g µν that constitutes the dynamic variables. • The world lines of test bodies are geodesics of the metric. • In local freely falling frames, i.e. the local Lorentz frames, the non-gravitational laws of physicsare those of Special Relativity.One of the predictions of this principle is the gravitational red-shift, experimentally probed by Poundand Rebka [1]. It is worth noticing that gravitational interactions are excluded from WEP and EinsteinEP.To classify extended and alternative theories of gravity, the gravitational WEP and the StrongEquivalence Principle (SEP) is introduced. The SEP extends the Einstein EP by including all the lawsof physics. It states: • WEP is valid for self-gravitating bodies as well as for test bodies (gravitational WEP). • The outcome of any local test is independent of the velocity of the free-falling apparatus. • The outcome of any local test is independent of where and when it is performed in the universe.The Einstein EP is recovered from SEP as soon as the gravitational forces are neglected. Several authorsclaim that the only theory coherent with SEP is GR and then WEP has to be deeply investigated.A very important issue is the consistency of EP with respect to Quantum Mechanics. GR is not theonly gravity theory and several alternatives have been investigated starting from the 60’s [6]. Some ofthem are effective descriptions coming from quantum field theories on curved spacetime. Considering thespacetime as special relativistic at a background level, gravitation can be treated as a Lorentz-invariantperturbation field on the background. Assuming the possibility of GR extensions and alternatives, twodifferent classes of experiments can be conceived: • Tests for the foundations of gravity according to the various formulations of EP. • Tests of metric theories where spacetime is endowed with a metric tensor and where the EinsteinEP is assumed valid. 7he difference between the two classes of experiments consists in the fact that EP can be postulated ”apriori” or ”recovered” from the self-consistency of the theory. What is clear is that, for several funda-mental reasons, extra fields are necessary to describe gravity with respect to other interactions. Suchfields can be scalar fields or higher-order corrections of curvature and torsion invariants [6].Accordingto these reasons, two sets of field equations can be considered: The first couples the gravitational fieldto non-gravitational fields, i.e. the matter distribution, the electromagnetic fields, etc. The second setof equations considers dynamics of non-gravitational fields. In the framework of metric theories, theselaws depend only on the metric and this is a consequence of the Einstein EP. In the case where gravi-tational field equations are modified with respect to the Einstein ones, and matter field are minimallycoupled with gravity, we are dealing with the
Jordan frame . In the case where Einstein field equationsare preserved and matter field are non-minimally coupled, we are dealing with the
Einstein frame . Bothframes are conformally related but the very final issue is to understand if passing from one frame to theother (and vice versa) is physically significant. Clearly, EP plays a fundamental role in this discussion.In particular, the main question is if EP is valid in any case or it is violated at quantum level.
As discussed before, GR provides a comprehensive description of space, time, gravity, and matter underthe same standard at macroscopic level. Einstein formulated it in such a way that space and time aredynamical and entangled quantities determined by the distribution and motion of matter and energy.As a consequence, GR is related to a new conception of the universe which can be considered as adynamical system where precise physical measurements are possible.In this perspective, cosmology is not only a philosophical branch of knowledge but can be legitimatelyincorporated into science. Investigating scientifically the universe has led, along the last century, tothe formulation of the Standard Big Bang Model [7]which, in principle, matched most of the availablecosmological observations until more or less twenty years ago.Deapite these successes, several shortcomings of Einstein’s theory emerged recently at ultravioletand infrared scales and scientists considered the hypothesis whether GR is the only fundamental the-ory of gravitational interaction. This new point of view comes from cosmology (infrared scales) andquantum field theory (ultraviolet scales). In the first case, the Big Bang singularity, the flatness, hori-zon, and monopole problems [8] led to the conclusion that a cosmological model based on GR andthe Standard Model of particles is inadequate to describe the universe in extreme energy-curvatureregimes. Furthermore, GR cannot work as a fundamental theory of gravity if a quantum description ofspacetime is required. The Einstein theory is essentially a classical description. Due to these reasons,and, in particular due to the lack of a self-consistent quantum theory of gravity, various alternative andextensions of GR were proposed. The general approach is to formulate, at least, a semiclassical effectivetheory where GR and its positive results can be recovered in some limit (e.g. the weak field limit or theSolar System scales). A fruitful approach is the so-called
Extended Theories of Gravity (ETGs) whichhave recently become a paradigm to study the gravitational interaction. Essentially they are based oncorrections and extensions of Einstein’s GR. The paradigm consistsin adding higher order curvatureinvariants and minimally or non-minimally coupled scalar fields into the dynamics. In this sense, wecan deal with effective gravity actions emerging from quantum field theory [6, 9].Other reasons to modify GR are related to the issue of incorporating Mach’s principle into thetheory. GR is only partially Machian and allows solutions that are explicitly anti-Machian, e.g. theG¨odel solution [10] or exact pp -waves [11].Mach’s principle states that local inertial frames are determined by the average motions of distantastronomical objects [12]. This implies that the gravitational coupling can be determined by the sur-rounding matter distribution and, therefore, becomes a spacetime function which can assume the formof a scalar field. As a consequence, inertia and Equivalence Principle are concepts that have to be8evised. Brans-Dicke theory [13] is the first alternative to GR and the first attempt to fully incorporatethe Mach principle. It is considered the prototype of alternative theories of gravity and a straightfor-ward GR extension. The gravitational “constant” is assumed ”variable” and corresponds to a scalarfield non-minimally coupled to geometry. This approach constitute a more satisfactory implementationof Mach’s principle than GR [13–15].Furthermore, any scheme unifying fundamental interactions with gravity, such as superstrings, su-pergravity, or Grand-Unified Theories (GUTs) produces effective actions where non-minimal couplingsto the geometry are present. Also higher order curvature invariants are present in general. They emergeas loop corrections in high-curvature regimes. This scheme has been adopted in quantizing matter fieldson curved spacetimes and the result is that interactions between quantum fields and background geom-etry, or gravitational self-interactions give rise to corrections in the Hilbert-Einstein Lagrangian [16].Furthermore, these corrections are unavoidable in the effective quantum gravity actions [17] and thenGR extensions are necessary. All these models do not constitute a self-consistent quantum gravitytheory, but are useful working schemes towards it.To summarize, higher order curvature invariants like R , R µν R µν , R µναβ R µναβ , R (cid:3) R , R (cid:3) k R , ornon-minimally couplings between matter fields and geometry such as φ R have to be added to thegravitational Lagrangian as soon as quantum corrections are introduced. For example, these termsoccur in the low-energy limit of string Lagrangian or in Kaluza-Klein theories where extra spatialdimensions are taken into account [18].Moreover, from a conceptual viewpoint, there is no a priori reason to restrict the gravitationalLagrangian to a function, linear in the Ricci scalar R , minimally coupled to matter [19]. This conceptis in agreement with the idea that there are no “exact” laws of physics. It this case, the effectiveLagrangians of physical interactions would be given by the average quantities arising from the stochasticbehaviour of fields at a microscopic level. This approach means that the local gauge invariances andthe conservation laws are approximated and emerge only in the low-energy limit. In this perspective,fundamental constants of physics can be assumed variable.Furthermore, besides fundamental physics motivations, ETGs are interesting in cosmology becausethey exhibit inflationary behaviours able to overcome shortcomings of Standard Big Bang model. Therelated inflationary scenarios are realistic and match current observations coming from the cosmicmicrowave background (CMB) [20, 21]. It can be been shown that, by conformal transformations, thehigher order and non-minimally coupled terms correspond to Einstein gravity plus one or more than onescalar field(s) minimally coupled to the curvature [22–24]. Specifically,after conformal transformations,higher order and non-minimally coupled terms appear as equivalent scalar fields in the Einstein fieldequations. For example, in the Jordan frame, a term like R in the Lagrangian gives fourth orderfield equations, R (cid:3) R gives sixth order equations [25], R (cid:3) R yields eighth order equations [26], andso on. After a conformal transformation, second order derivative terms corresponds to a scalar field: specifically, fourth order gravity is conformally equivalent to Einstein gravity plus a scalar field; sixthorder gravity is conformally equivalent to GR plus two scalar fields; and so on [27].Furthermore, it is also possible to show that f ( R ) gravity to the Einstein theory minimally coupledto an ideal fluid [28]. This feature is useful if multiple inflationary events are necessary for structureformation. In fact, an early stage could produce large-scale structure with very long wavelengths whichafter give rise to the observed clusters of galaxies. A later stage could select smaller scales observed asgalaxies today [25]. The underlying philosophy is that any inflationary era is related to the dynamicsof a related scalar field. Finally, these extended schemes could solve the graceful exit problem, avoidingthe shortcomings of other inflationary models [29].The revision of early cosmological scenarios, leading to inflation, can imply that a new approach isnecessary also at late epochs: ETGs play a fundamental role also in today observed universe. In fact, Dynamics of any of these scalars fields is determined by a second order Klein-Gordon equation.
Concordance Model or Λ-Cold Dark Matter (ΛCDM) model.The Hubble diagram of type Ia Supernovae (hereafter SNeIa) was the first evidence that the universeis today undergoing an accelerated expansion phase. Furthermore, balloon-born experiments [30] deter-mined the location of the first two Doppler peaks in the spectrum of CMB anisotropies. These featuresstrongly suggest a spatially flat universe also if some recent data could question this result. If combinedwith constraints on matter density parameter Ω M , these data point out that the universe is dominatedby an un-clustered fluid, with negative pressure, usually referred to as dark energy . Such a fluid drivesthe accelerated expansion. This picture has been strengthened by other precise measurements on CMBspectrum and by the extension of the SNeIa Hubble diagram to redshifts higher than one.A huge amount of papers appeared following these observational results. They present several modelsattempting to explain the cosmic acceleration. The simplest explanation is the well-known cosmologicalconstant Λ. Although this ingredient provides the best-fit to most of the available astrophysical data [31],the ΛCDM model fails in explaining why the value of Λ is so tiny (120 orders of magnitude lower) ifcompared with the typical vacuum energy density predicted by particle physics, and why its presentvalue is comparable to the matter density. This constitutes the so-called coincidence problem .A possible solution is replacing the cosmological constant with a scalar field ϕ rolling slowly down aflat section of a potential V ( ϕ ) and giving rise to the models known as quintessence [32, 33]. Also if itis successfully fitting data with many models, the quintessence approach to dark energy is still plaguedby the coincidence problem since the dark energy and dark matter densities evolve in a different wayand reach comparable values only during a very short time of the history of the universe. In particular,they coincide right at present era. In other words, the quintessence is tracking matter and evolves in thesame way for a long time; at late times, it changes its behaviour and no longer tracks the dark matterbut dominates dynamics as a cosmological constant. This is, specifically, the quintessence coincidenceproblem.The origin of this quintessence scalar field is one of the big mystery of modern cosmology. Althoughseveral models have been proposed, a great deal of uncertainty is related to the choice of the scalar fieldpotential V ( ϕ ) necessary to achieve the late-time acceleration of the universe. The elusive nature ofdark energy has led many authors to look for a different explanation of the cosmic acceleration withoutintroducing exotic components. It is worth stressing that the present-day cosmic acceleration requiresa negative pressure that has to dominate dynamics after the matter era. However, we do not anythingabout the nature and the number of the cosmic fluids filling the universe. This consideration suggests usthat the accelerated expansion could be explained with a single cosmic fluid characterized by an equationof state acting like dark matter at high densities, and like dark energy at low densities. The relevantfeature of these models, referred as Unified Dark Energy (UDE) or
Unified Dark Matter (UDM) models,is that the coincidence problem is naturally solved. Examples are the Chaplying gas [34], tachyonfields [35], and condensate cosmology [36]. These models are extremely interesting because they can beinterpreted both in the framework of UDE models or as two-fluid models representing the dark matterand the dark energy regime. A main feature of this approach is that a generalized equation of state canbe easily obtained and the fit of observational data can be achieved.There is another approach to face the problem of the cosmic acceleration. As reported in [37], itis possible that the observed acceleration is not related to another cosmic ingredient, but rather thesignal of a breakdown, at infra-red scales, of the law of gravitation as a scale invariant interaction.From this view point, modifying the Einstein-Friedmann equations, fitting the astrophysical data withmodels containing only standard matter and without exotic fluids is another approach. Examples inthis direction are the Cardassian model [38] and Dvali-Gabadadze-Porrati (DGP) cosmology [39]. Inthe same conceptual framework, it is possible to find alternative approaches where a quintessentialbehavior is obtained by incorporating effective models coming from fundamental physics and giving riseto extended gravity actions. For example, a cosmological constant can be recovered as a consequence10f non-vanishing torsion fields. Also in this case, it is possible to build up models consistent with theSNeIa Hubble diagram and the Sunyaev - Zel’dovich effect in galaxy clusters [40]. SNeIa data can also befitted by including higher-order curvature invariants. These models provide a cosmological componentwith negative pressure which is originated by the geometry of the universe. According to this picture,we do not need new particle counterparts to address the phenomenology.The amount of cosmological models which are viable candidates to explain the observed acceleratedexpansion is too wide to be reported here. This overabundance points out that only a few numberof cosmological tests is available to discriminate between competing approaches, so it is clear thatthere is a high degeneracy of models. It s important to stress that both SNeIa Hubble diagram andangular size-redshift relation of compact radio sources are distance-based probes of the cosmologicalmodel and, therefore, systematic errors and biases could be iterated. According to this consideration,it is interesting to search for tests based on time-dependent observables. For example, we can take intoaccount the lookback time to distant objects. This quantity discriminates among different cosmologicalmodels. The lookback time is estimated as the difference between the age of the universe and the ageof a given object at redshift z . This estimate becomes realistic when the object is a galaxy observed inmore than one photometric band because its color is determined by the age as a consequence of stellarevolution. Hence, it is possible to obtain the galaxy age by measuring its magnitude in different bandsand then using stellar evolutionary codes to best reproduce the observed colors.In general, in the case of weak-field limit, which essentially coincides with Solar System scales, ETGsare expected to reproduce GR which is precisely tested at these scales [1]. Even this limit is a matterof debate because several theories do not reproduce exactly the Einstein theory in its Newtonian limitbut, in some sense, generalize it giving rise to Yukawa-like corrections to the Newtonian potential whichcould be physically relevant already at Galactic scales [41–46].As a general remark, relativistic gravity theories give rise to corrections to the weak-field gravita-tional potentials which, at the post-Newtonian level and in the Parametrized Post-Newtonian (PPN)formalism, constitute a test bed for these theories [1]. Furthermore, the gravitational lensing astron-omy [47] provide additional tests over small, large, and very large scales which can provide measurementson the variation of the Newton constant [48], the potential of galaxies, clusters of galaxies, and otherfeatures of gravitating systems. In principle, such data can be capable of confirming or ruling out anyalternative to GR.In ETGs, the Einstein field equations can be modified in two ways: i) the geometry part can benon-minimally coupled to some scalar field, and/or ii) higher than second order derivatives of the metriccan appear. In the former case, we deal with scalar-tensor theories of gravity; in the latter, with higherorder theories of gravity. Combinations of non-minimally coupled and higher order components canalso emerge.From the mathematical viewpoint, the problem of reducing more general theories to the Einsteintheory has been widely discussed. Through a Legendre transformation on the metric, higher ordertheories with Lagrangians satisfying some regularity conditions assume the form of GR with (possiblymultiple) scalar field(s) as sources the gravitational field ( e.g. , [19, 49, 50]). The formal equivalencebetween models with variable gravitational coupling and Einstein gravity via conformal transformationsis also well known [51]. This gave rise to the debate on whether the mathematical equivalence betweendifferent conformal representations is also a physical equivalence [52, 53].Another important issue is the Palatini approach: this problem was first proposed by Einsteinhimself, but it was called the Palatini approach because the Italian mathematician Attilio Palatiniformalized it [4]. The main idea of this formalism is considering the connection Γ µαβ as independentof the metric g µν . It is well known that the Palatini formulation of GR is equivalent to the metricformulation [7]. This result follows from the fact that the field equations for connection Γ µαβ , also ifassumed independent of the metric, yield the Levi-Civita connection of g µν in GR. Therefore, the Palatinivariational principle in the Einstein theory gives exactly the same field equations of the metric variational11rinciple. However, the situation changes if we consider ETGs formulated as functions of curvatureinvariants, such as f ( R ), or as scalar-tensor theories. There, the Palatini and the metric variationalprinciples give rise to different field equations that could describe different physics. The relevance ofthe Palatini formulation has been recently highlighted according to cosmological applications [54].Another crucial problem is related to the Newtonian potential in alternative gravity and its relationswith the conformal factor [55]. From a physical point of view, considering the metric and the connectionas independent fields corresponds to decoupling the metric structure of spacetime from the geodesicstructure (with the connection being, in general, different from the Levi-Civita connection of the metric.The causal structure of spacetime is governed by g µν , while the spacetime trajectories of particles aregoverned by Γ µαβ .The decoupling of causal and geodesic structures enlarges the spacetime geometry and generalizesthe metric formalism. This metric-affine structure can be immediately translated, by means of thePalatini field equations, into a bi-metric structure. In addition to the physical metric g µν , a secondmetric h µν is present which is related, in the case of f ( R ) gravity, to the connection. As a matterof fact, the connection Γ µαβ turns out to be the Levi-Civita connection of this second metric h µν andprovides the geodesic structure of spacetime.If we consider non-minimal couplings in gravitational Lagrangian, the further metric h µν is relatedto the coupling. According to the Palatini formalism, non-minimal couplings and scalar fields enteringthe evolution of the gravitational field are related by the metric structure of spacetime The Newton theory of gravity was the issue that Einstein needed to recover in the weak field limit andslow motion. In Newton formulation, space and time are absolute entities and require particles to move,in a preferred inertial frame, along curved trajectories, the curvature of which ( i.e. , the acceleration) isa function of the intensity of the sources through the “forces”. According to this requirements, Einsteinpostulated that the gravitational forces have to be described by the curvature of the metric tensor g µν related to the line element ds = g µν dx µ dx ν of a four-dimensional spacetime manifold. This metric hasthe same signature of the Minkowski metric (the Lorentzian signature here assumed to be ( − , + , + , +)).Einstein postulated also that spacetime curves onto itself and that curvature is locally determined bythe distribution of the sources, that is by the four-dimensional generalization of the matter stress-energytensor (another rank-two symmetric tensor) T ( m ) µν of continuum mechanics.Once a metric g µν is assigned, curvature is given by the Riemann (or curvature) tensor R αβµν = Γ ναβ,µ − Γ νβµ,α + Γ σαµ Γ νσβ − Γ σβµ Γ νσα (1)where the commas denote partial derivatives. Its contraction R αµ ≡ R αβµβ , (2)is the Ricci tensor , while the contraction R ≡ R µµ = g µν R µν (3)is the Ricci curvature scalar of g µν . Einstein initially derived the field equations R µν = κ T ( m ) µν , where κ = 8 πG (in units in which c = 1) is the gravitational coupling constant. These equations turned outto be inconsistent as pointed out by Levi-Civita. Furthermore Hilbert stressed that they do not derivefrom an action principle [57]. In fact, there is no action reproducing them exactly through a variation. Due to these features, the Palatini approach could play a main role in clarifying the physical aspects of conformaltransformations [56]. T ( m ) µν = ( P + ρ ) u µ u ν + P g µν , (4)where u µ is the four-velocity of the particles, P and ρ the pressure and energy density of the fluid, respec-tively, the continuity equation requires T ( m ) µν to be covariantly constant, i.e. , to satisfy the conservationlaw ∇ µ T ( m ) µν = 0 , (5)where ∇ α denotes the covariant derivative operator of the metric g µν . In fact, ∇ µ R µν does not vanish,except in the trivial case R ≡
0. Einstein concluded that the field equations are G µν = κ T ( m ) µν (6)where G µν ≡ R µν − g µν R (7)is the Einstein tensor of g µν . These equations can be derived also by minimizing an action containing R and satisfy the conservation law (5) since ∇ µ G µν = 0 , (8)holds as a contraction of the Bianchi identities that the curvature tensor of g µν satisfies [7].Specifically, the Lagrangian that, if varied, produces the field equations (6) is the sum of a “matter”Lagrangian density L ( m ) , whose variational derivative is T ( m ) µν = δ L ( m ) δg µν , (9)and of the gravitational ( Hilbert-Einstein ) Lagrangian density L HE = √− g g µν R µν = √− g R , (10)where g is the determinant of the metric g µν .Einstein’s choice was arbitrary but it was certainly the simplest. As clarified by Levi-Civita in 1919,curvature is not a purely metric notion but it is also related to the linear connection of parallel transportand covariant derivative. In some sense, this idea is the precursor of “gauge-theoretical framework” [58],following the pioneering work by Cartan of 1925.After, it was clarified that the principles of relativity, equivalence, and covariance, together withcausality, require only that the spacetime structure can be determined by a Lorentzian metric g µν anda linear connection Γ αµν , assumed to be torsionless for the sake of simplicity. The metric fixes thecausal structure of spacetime (the light cones) as well as its metric relations measured by clocks androds and the lenghts of four-vectors. The connection determines the laws of free fall, that is the four-dimensional spacetime trajectories followed by locally inertial observers. These observers must satisfysome compatibility relations including the requirement that photons follow null geodesics, so that Γ αµν and g µν can a priori be independent, but constrained a posteriori by the physics. These physicalconstraints, however, do not necessarily impose that Γ αµν is the Levi-Civita connection of g µν [6].13 .3 Quantum Gravity motivations A challenge of modern physics is constructing a theory capable of describing the fundamental interac-tions of nature under the same standard. This goal has led to formulate several unification schemeswhich attempt to describe gravity together with the other interactions. All these schemes describe thefields under the conceptual apparatus of Quantum Mechanics. This is based on the assumption thatthe states of physical systems are described by vectors in a Hilbert space H and the physical fieldsare linear operators defined on domains of H . Till now, any attempt to incorporate gravity into thisscheme is failed or revealed unsatisfactory. The main problem is that gravitational field describes, atthe same time, the gravitational degrees of freedom and the spacetime background where these degreesof freedom are defined.Owing to the difficulties of building up a self-consistent theory unifying interactions and particles,the two fundamental theories of modern physics, GR and Quantum Mechanics, have been criticallyre-analyzed. On the one hand, we assume that matter fields (bosons and fermions) come out fromsuperstructures ( e.g. , Higgs bosons or superstrings) that, undergoing certain phase transitions, generatethe known particles. On the other hand, it is assumed that the geometry interacts directly with quantummatter fields which back-react on it. This interaction necessarily modifies the standard gravitationaltheory, that is the Hilbert-Einstein Lagrangian. This fact leads to the ETGs.From the point of view of cosmology, the modifications of GR provide inflationary scenarios ofremarkable interest. In any case, a condition that such theories have to respect in order to be physicallyacceptable is that GR is recovered in the low-energy limit.Although conceptual progresses have been made assuming generalized gravitational theories, atthe same time mathematical difficulties have increased. The corrections into the Lagrangian enlargethe (intrinsic) non-linearity of the Einstein equations, making them more difficult to study becausedifferential equations of higher order than second are often obtained and because it is extremely difficultto separate geometry from matter degrees of freedom. To overcome these difficulties and try to simplifythe field equations, one often looks for symmetries of dynamics and identifies conserved quantities which,often, allow to find out exact solutions.The necessity of quantum gravity was recognized at the end of 1950s, when physicists tried todeal with all interactions at a fundamental level and describe them under the standard of quantumfield theory. The first attempts to quantize gravity adopted the canonical approach and the covariantapproache, which had been already applied with success to electromagnetism. In the first approach ap-plied to electromagnetism, one takes into account electric and magnetic fields satisfying the Heisenberguncertainty principle and the quantum states are gauge-invariant functionals, generated by the vectorpotential, defined on 3-surfaces labeled with constant time. In the second approach, one quantizesthe two degrees of freedom of the Maxwell field without (3+1) decomposition of the metric, while thequantum states are elements of a Fock space [59]. These procedures are fully equivalent. The formerallows a well-defined measure, whereas the latter is more convenient for perturbative calculations suchas the computation of the S -matrix in Quanrum Electrodynamics (QED).These methods have been adopted also in GR, but several difficulties arise in this case due to thefact that GR cannot be formulated as a quantum field theory on a fixed Minkowski background. Tobe specific, in GR the geometry of background spacetime cannot be given a priori: spacetime is itselfthe dynamical variable. To introduce the notions of causality, time, and evolution, one has to solveequations of motion and build up the related spacetime. For example, to know if a particular initialcondition will give rise to a black hole, it is necessary to evolve it by solving the Einstein equations.Then, taking into account the causal structure of the solution, one has to study the asymptotic metric atfuture null infinity in order to assess whether it is related, in the causal past, with that initial condition.This problem become intractable at quantum level. Due to the uncertainty principle, in non-relativisticquantum mechanics particles do not move along well-defined trajectories and one can only calculate14he probability amplitude ψ ( t, x ) that a measurement at a given time t detects a particle at the spatialpoint x . In the same way, in quantum gravity, the evolution of an initial state does not provide a givenspacetime (that is a metric). In absence of a spacetime, how is it possible to introduce basic conceptsas causality, time, scattering matrix, or black holes?Canonical and covariant approaches provide different answers to these issues. The first is based onthe Hamiltonian formulation of GR and is adopting a canonical quantization procedure. The canonicalcommutation relations are those that lead to the Heisenberg uncertainty principle; in fact, the commu-tation of operators on a spatial 3-manifold at constant time is assumed, and this 3-manifold is fixed inorder to preserve the notion of causality. In the limit of asymptotically flat spacetime, motions relatedto the Hamiltonian have to be interpreted as time evolution (in other words, as soon as the backgroundbecomes the Minkowski spacetime, the Hamiltonian operator assumes again its role as the generator oftime translations). The canonical approach preserves the geometric structure of GR without introducingperturbative methods.On the other hand, the covariant approach adopts quantum field theory concepts. The basic idea isthat the shortcomings mentioned above can be circumvented by splitting the metric g µν into a kinematicpart η µν (usually flat) and a dynamical part h µν . That is g µν = η µν + h µν . (11)The background geometry is given by the flat metric tensor and it is the same of Special Relativityand standard quantum field theory. It allows to define concepts of causality, time, and scattering.The procedure of quantization is applied to the dynamical field, considered as a little deviation of themetric from the flat background. Quanta are particles with spin two, i.e. gravitons , which propagatein MInkowski spacetime and are defined by h µν . Substituting g µν into the GR action, it follows thatthe gravitational Lagrangian contains a sum whose terms contains a different orders of approximation,the interaction of gravitons and, eventually, terms describing matter-graviton interaction (if matter ispresent). These terms are analyzed by the standard perturbation approach of quantum field theory.These quantization approaches were both developed during the 1960s and 1970s. In the canonicalapproach, Arnowitt, Deser, and Misner developed the Hamiltonian formulation of GR using methodsproposed by Dirac and Bergmann. In this scheme, the canonical variables are the 3-metric on the spatial3-manifolds obtained by foliating the 4-dimensional manifold. It is worth noticing that this foliation isarbitrary. Einstein’s field equations give constraints between the 3-metrics and their conjugate momentaand the evolution equation for these fields is the so-called Wheeler-DeWitt (WDW) equation . In thispicture, GR is the dynamical theory of the 3-geometries, that is the geometrodynamics . The mainproblems arising from this formulation are that the quantum equations involve products of operatorsdefined at the same spacetime point and, furthermore, they give rise to distributions whose physicalmeaning is unclear. In any case, the main question is the absence of the Hilbert space of states and, asconsequence, the probabilistic interpretation of the quantities is not exactly defined.The covariant quantization is much similar to the physics of particles and fields, because, in somesense, it has been possible to extend QED perturbation methods to gravitation. This allowed theanalysis of mutual interaction between gravitons and of the matter-graviton interactions. The Feynmanrules for gravitons and the demonstration that the theory might be, in principle, unitary at any orderof expansion was achieved by DeWitt.Further progress was reached by the Yang-Mills theories, describing the strong, weak, and electro-magnetic interactions of particles by means of symmetries. These theories are renormalizable becauseit is possible to give the particle masses through the mechanism of spontaneous symmetry breaking.According to this principle, it is natural to try to consider gravitation as a Yang-Mills theory in thecovariant perturbation approach and search for its renormalization. However, gravity does not fit intothis scheme; it turns out to be non-renormalizable if we consider the graviton-graviton interactions15two-loops diagrams) and graviton-matter interactions (one-loop diagrams). In any case, the covariantmethod allows to construct a gravity theory which is renormalizable at one-loop level in the perturbationseries [16].Due to the non-renormalizability of gravity at higher orders, the validity of the approach is restrictedto the low-energy domain, that is, to infrared scales, while it fails at ultraviolet scales. This impliesthat the theory of gravity is unknown near or at Planck scales. On the other hand, sufficiently farfrom the Planck epoch, GR and its first loop approximation describe quite well gravitation. In thiscontext, it makes sense to add higher order and non-minimally coupled terms to the Hilbert-Einsteinaction. Furthermore, if the free parameters are chosen appropriately, the theory has a better ultravioletbehaviour and it is asymptotically free. Nevertheless, the Hamiltonian of these theories is not boundedfrom below and they are unstable. Specifically, unitarity is violated and probability is not conserved.Another approach to the search for quantum gravity comes from the study of the electroweakinteraction. Here, gravity is treated neglecting the other fundamental interactions. The unification ofelectromagnetism and weak interaction suggests that it could be possible to obtain a consistent theory ifgravity is coupled to some kind of matter. This is the basic idea of Supergravity. In this kind of theories,divergences due to the bosons (in this case the 2-spin gravitons) are cancelled exactly by those due tothe fermions. In this picture, it is possible to achieve a renormalized theory of gravity. Unfortunately,the approach works only up to two-loop level and for matter-gravity couplings. The correspondingHamiltonian is positive-definite and the theory is unitary. However, including higher order loops, theinfinities appear and renormalizabilty is lost.Perturbation methods are also adopted in string theories. In this case, the approach is different fromthe previous one since particles are replaced by extended objects which are the fundamental strings. Thephysical particles, including the spin two gravitons, correspond to excitations of the strings. The onlyfree parameter of the theory is the string tension and the interaction couplings are uniquely determined.As a consequence, string theory contains all fundamental physics and it is considered a possible
Theoryof Everything . String theory is unitary and the perturbation series converges implying finite terms. Thisfeature follows from the characteristic that strings are intrinsically extended objects, so that ultravioletdivergencies, appearing at small scales or at large transferred impulses, are naturally cured. This meansthat the natural cutoff is given by the string length, which is of Planck size l P . At larger scales than l P ,the effective string action can be written as non-massive vibrational modes, that is, in terms of scalarand tensor fields. This constitutes the tree-level effective action . This approach leads to an effectivetheory of gravity non-minimally coupled with scalar fields, which are the so-called dilaton fields .In conclusion, we can summarize the previous considerations: 1) a unitary and renormalizabletheory of gravity does not yet exists . 2) In the quantization program of gravity, two approaches areused: the covariant approach and the perturbation approach . They do not lead to a self-consistentquantum gravity. 3) In the low-energy regime, with respect to the Planck energy, GR can be improvedby introducing, into the Hilbert-Einstein action, higher order terms of curvature invariants and non-minimal couplings between matter and gravity. The approach leads, at least at one-loop level, to aconsistent and renormalizable theory. Higher order terms in the perturbative series imply an infinite number of free parameters. At the one-loop level, it issufficient to renormalize only the effective constants G eff and Λ eff which, at low energy, reduce to the Newton constant G N and the cosmological constant Λ. It is worth to mention that recently it has been shown that an infinite derivative theory of covariant gravity, whichis motivated from string theory, see [60, 61], can be made ghost free and also singularity free [62, 63] (see Refs. [64–68] forsome applications). .4 Emergent gravity and thermodynamics of spacetime Recently, several theoretical approaches towards the so-called emergent gravity theories have been pro-posed. The main idea is that, given the lack of experimental data for quantum gravity at high-energies,it is worth approaching gravity from low-energies considering some effective theories. Gravity emergesfrom fundamental constituents, as a sort of “atoms of spacetime”, with metric and affine-connectionbeing collective variables similar to hydrodynamics, where a fluid description emerges from an aggregateof microscopic particles. Emergent gravity attempts the reconstruction of microscopic system underly-ing classical gravity. It is possible to constrain the microscopic features of the fundamental constituentsby requiring that the emergent gravity is similar to GR in weak field limit. This picture, however,questions the principles constituting the foundations of gravitational theories.A related research line is that of analogue models : if gravity emerges as a collective system made ofmicroscopic quantum constitutents, it could be possible to model it with the help of physical systemswhere an effective metric and a connection govern the dynamics. For example one can study the Hawkingradiation coming from black holes adopting acoustic analogues (the so called “dumb holes”) [69–71], orBose-Einstein condensates (see [72] for a review of analogue models). If an effective metric is generated,it is a kinematic, in the sense that field equations are not generated by it. However, some results areable to generate a theory of scalar gravity [73] and progresses are possible in this direction. A standardfeature for emergent spacetimes is that they exhibit Lorentz invariance at low-energies. The Lorentzsymmetry is broken in the ultraviolet limit where the fundamental quantum constituents of gravitycannot be avoided.We mention these approaches here because they question the foundations of gravitational theoryand do not state that GR is the only theory to be reproduced at large scale in coarse-graining: themessage is that different theories with similar features are possible as well.Another approach is based on the idea that gravity could be reproduced through a sort of spacetimethermodynamics . This means that the Einstein field equations should be derived through local thermo-dynamics at equilibrium. Using thermodynamics on the Rindler horizons associated to the worldlinesof physical observers and assuming the relation S = A/ f ( R ) gravity, it is then necessary to consider near-equilibrium thermodynamicsin order to derive the field equations. This demonstrates that GR is just a state of gravity correspond-ing to a given thermodynamic equilibrium and, when this equilibrium is perturbed, near-equilibriumconfigurations correspond to alternative theories of gravity. According to this approach, this justify thestudy of ETGs.A result with a conceptual similar meaning is found in scalar-tensor cosmologies: they should relaxto GR during the evolution of the universe at recent epochs. This is another hint that GR could be onlya particular state of equilibrium, while an entire spectrum of theories should be considered at higherenergy excitations.These results are very speculative and require further studies; however, they stress the necessity tothink about gravity outside of the strict GR scheme and hint to the fact that much more work needsto be done before claiming for a self-consistent theory of gravity also at lower energies.17 .5 Kaluza-Klein theories The attempt to construct a unified theory of GR and electromagnetism was first proposed by Kaluza [74](for a review, see [75–78]). He showed that the electromagnetism and the gravitation interactions can bedescribed by making use of a single metric tensor if an additional spatial dimension is introduced. In aUniverse with 5-dimensions, the element line reads ds = G AB ( x, y ) dx A dx B , where x A = ( x µ , y ), being y the additional dimension (here A, B = 0 , , , ,
4, and x µ , with µ = 0 , , ,
3, the usual four dimensionalcoordinates). In matrix form, the 5-dimensional metric tensor assumes the form G AB = (cid:18) g µν g µ g ν g (cid:19) .From the metric tensor, one construct all geometric quantities such as the Riemann tensor, the Riccitensor and scalar curvature and then the field equations. The components of the metric tensor aretypically written in the following form: G = φ , G µ = κφA µ , G µν = g µν + κ φA µ A ν . The fields g µν ( x, y ), A µ ( x, y ), and φ ( x, y ) transform as a tensor, a vector, and a scalar under diffeomorphisms(four-dimensional general coordinate transformations), respectively. The field φ is the dilaton field. Isit then natural to write down the The Einstein-Hilbert action in Kaluza-Klein five-dimensional gravity S HE = R d XR , where κ represents the five-dimensional coupling constant while R the five-dimensional scalar curvature. The field equations of gravity and electromagnetism can be derived fromthe usual variational principles.The extra dimension y is imposed to be become compact [79]. Hence y must satisfy the boundarycondition y = y + 2 πR . This implies that the fields F A ( x, y ) = { g µν ( x, y ) , A µ ( x, y ) , φ ( x, y ) } are periodicin y and may be expanded in a Fourier series as follows F A = + ∞ X n = −∞ F An e iny/R , where R is the radius of the compactified dimension. The equations of motion are (cid:3) F A = (cid:18) (cid:3) + n R (cid:19) F A = 0 , where (cid:3) = (cid:3) − ∂ y ∂ y and (cid:3) = ∂ µ ∂ µ is the usual 4-dimensional D’Alembert operator. Comparingwith the Klein-Gordon equation, one infers that only the massless zero modes n = 0 is observable atour present energy, while all the excited states (Kaluza-Klein states) have a mass and charge given by m ∼ | n | /R and q ∼ κn/R [17], with n the mode of excitation. In 4-dimensions, all these excited stateswould appear with mass or momentum ∼ O ( n/R ). The natural radius of compactification is the Plancklength R = l P l = 1 /M P l .Concerning the number of degree of freedom present in the Kaluza-Klein theory, owing to the factthat the metric is a 5 × .6 Quantum field theory in curved spacetime In this Section we point out that any attempt to formulate quantum field theory on curved spacetimenecessarily leads to modifying the Hilbert-Einstein action. This means adding terms containing non-linear invariants of the curvature tensor or non-minimal couplings between matter and the curvatureoriginating in the perturbative expansion.At high energies, a desription of matter as a hydrodynamical perfect fluid is inadequate: an accuratedescription asks for quantum field theory formulated on a curved spacetime. Since, at scales comparableto the Compton wavelength of particles, matter has to be quantized, one can adopt a semiclassicaldescription of gravitaty where the Einstein field equations assume the form G µν ≡ R µν − g µν R = < T µν > , (12)where the Einstein tensor G µν is on the left hand side while the right hand contains the expectationvalue of quantum stress-energy tensor which is the source of the gravitational field. Specifically, if | ψ > is a quantum state, then < T µν > ≡ < ψ | ˆ T µν | ψ > , where ˆ T µν is the quantum operator associated withthe classical energy-momentum tensor of the matter field with a regularized expectation value.If the background is curved, , quantum fluctuations of matter fields give, even in absence of classicalmatter and radiation, non-vanishing contributions to < T µν > like it happens in QED [16]. If matterfields are free, massless and conformally invariant, these corrections are < T µν > = k H µν + k H µν . (13)Here k , are numerical coefficients and (1) H µν = 2 R ; µν − g µν (cid:3) R + 2 R στ R στ g µν − g µν R , (14) (3) H µν = R σµ R νσ − RR µν − g µν R στ R στ + 14 g µν R . (15) (1) H µν is a tensor derived by varying the local action, (1) H µν = 2 √− g δδg µν (cid:0) √− g R (cid:1) . (16)It is divergence free, that is (1) H νµ ; ν = 0.Infinities coming from < T µν > are removed by introducing an infinite number of counterterms inthe Lagrangian density of gravitation. The procedure yields a renormalizable theory. For example, oneof these terms is CR √− g , where with C indicates a parameter that diverges logarithmically. Eq. (12)cannot be generated by a finite action because the gravitational field would be completely renormal-izable, that is, it would eliminate a finite number of divergences to make gravitation similar to QED.On the contrary, one can only construct a truncated quantum theory of gravity. The parameter usedfor the expansion in loop is the Planck constant ~ . It follows that the truncated theory at the one-looplevel contains all terms of order ~ , that is the first quantum correction. Some points have to be stressednow: 1) Matter fields are free and, if the Equivalence Principle is valid at quantum level, all forms ofmatter couple in the same way to gravity. 2) The intrinsic non-linearity of gravity naturally arises,and then a number of loops are nrcessary to take into account self-interactions interactions betweenmatter and gravitation. In view ofremoving divergences at one-loop order, one has to renormalize thegravitational coupling G eff and the cosmological constant Λ eff . One-loop contributions of < T µν > arethe quantities introduced above, that is (1) H µν and (3) H µν . Furthermore, one has to consider (2) H µν = 2 R σµ ; νσ − (cid:3) R µν − g µν (cid:3) R + R σµ R σν − R στ R στ g µν . (17)19n a conformally flat spacetime, one has (2) H µν =
13 (1) H µν [16], so that only the first and third termsof H µν are present in (13). The tensor (3) H µν is conserved only in conformally flat spacetimes and itcannot be obtained by varying a local action. The trace of the energy-momentum tensor is null forconformally invariant classical fields while, one finds that the expectation value of the tensor (13) hasnon-vanishing trace. This result gives rise to the so-called trace anomaly [16].By summing all the geometric terms in < T ρρ > ren , deduced by the Riemann tensor and of the sameorder, one derives the right hand side of (13). In the case in which the background metric is conformallyflat, it can be expressed in terms of Eqs. (14) and (15). We conclude that the trace anomaly, relatedto the geometric terms emerges because the one-loop approach formulates quantum field theories oncurved spacetime. Masses of the matter fields and their mutual interactions can be neglected in the high curvature limitbecause R ≫ m . On the other hand, matter-graviton interactions generate non-minimal couplings inthe effective Lagrangian. The one-loop contributions of such terms are comparable to those given bythe trace anomaly and generate, by conformal transformations, the same effects on gravity.The simplest effective Lagrangian taking into account these corrections is L NMC = − ∇ α ϕ ∇ α ϕ − V ( ϕ ) − ξ Rφ , (18)where ξ is a dimensionless constant. The stress-energy tensor of the scalar field results modified ac-cordingly but a conformal transformation can be found such that the modifications related to curvatureterms can be cast in the form of a matter-curvature interaction. The same argument holds for the traceanomaly. Some Grand Unification Theories lead to polynomial couplings of the form 1 + ξφ + ζ φ thatgeneralize the one in (18). An exponential coupling e − αϕ R between a scalar field (dilaton) ϕ and theRicci scalar appears in the effective Lagrangian of strings.Field equations derived by varying the action L NMC are (cid:0) − κξφ (cid:1) G µν = κ (cid:26) ∇ µ φ ∇ ν φ − g µν ∇ α φ ∇ α φ − V g µν + ξ (cid:2) g µν (cid:3) (cid:0) φ (cid:1) − ∇ µ ∇ ν (cid:0) φ (cid:1)(cid:3)(cid:9) , (19) (cid:3) φ − dVdφ − ξRφ = 0 . (20)The non-minimal coupling of the scalar field is similar to that derived for the 4-vector potential ofcurved space in Maxwell theory. See below Eq. (36).Several motivations can be provided for the non-minimal coupling in the Lagrangian L NMC . Anonzero ξ is generated by first loop corrections even if it does not appear in the classical action [16,83–86].Renormalization of a classical theory with ξ = 0 shifts this coupling constant to a value which is small[87,88]. It can, however, affect drastically an inflationary cosmological scenario and determine its successor failure [89–92]. A non-minimal coupling is expected at high curvatures [85, 86]. Furthermore, non-minimal coupling solves potential problems of primordial nucleosynthesis [93] and, besides, the absenceof pathologies in the propagation of ϕ -waves requires conformal coupling for all non-gravitational fields[94–98] . Eqs. (14) and (15) can include terms containing derivatives of the metric of order higher than fourth (fourth orderbeing the R term) if all possible Feynman diagrams are included. For example, corrections such as R (cid:3) R or R (cid:3) R canbe present in (3) H µν implying equations of motion that contain sixth order derivatives of the metric. Also these termscan be treated by making use of conformal transformations [25]. Note, however, that the distinction between gravitational and non-gravitational fields becomes representation-dependent in ETGs, together with the various formulations of the EP [99]. ξ = 1 / ξ as a free parameter to fix problems of specific inflationary scenarios [92, 116]. Cos-mological reheating with strong coupling ξ >> ξ is not, in general, a free parameter but depends on the particular scalar field ϕ consid-ered [85, 86, 91, 92, 116, 127–129]. Let us take into account higher order theories and their relations to scalar-tensor gravity [130]. Thefirst straightforward generalizaion of GR is L = √− g f ( R ) , (21)The variation with respect to g µν yields the field equations f ′ ( R ) R µν − f ( R ) g µν − ∇ µ ∇ ν f ′ ( R ) + g µν (cid:3) f ′ ( R ) = 0 , (22)with f ′ ≡ df ( R ) /dR . Equation (22) is a fourth-order field equations (in metric formalism). It isconvenient to introduce the new set of variables p = f ′ ( R ) = f ′ ( g µν , ∂ σ g µν , ∂ σ ∂ ρ g µν ) , (23)˜ g µν = p g µν . (24)This choice links the Jordan frame variable g µν to the Einstein frame variables ( p, ˜ g µν ), where p is someauxiliary scalar field. The term “Einstein frame” comes from the fact that the transformation g → ( p, ˜ g )allows to recast Eqs. (22) in a form similar to the Einstein field equations of GR. In absence of matter,hence T ( m ) µν = 0, the Einstein equations in are˜ G µν = 1 p (cid:20) p ,µ p ,ν −
34 ˜ g µν ˜ g αβ p ,α p ,β + 12 ˜ g µν ( f ( R ) − Rp ) (cid:21) . (25)These equation can be rewritten in a more attractive way by defining ϕ = q ln p , which implies˜ G µν = (cid:20) ϕ ,µ ϕ ,ν −
12 ˜ g µν ϕ ,σ ϕ ,σ − ˜ g µν V ( ϕ ) (cid:21) , (26)where V ( ϕ ) = Rf ′ ( R ) − f ( R )2 f ′ ( R ) | R = R ( p ( ϕ )) . (27)The curvature R = R ( p ( ϕ )) is inferred by inverting the relation p = f ′ ( R ) (provided f ′′ ( R ) = 0). Thefield equation (26) can be obtained from the Lagrangian (21) rewritten in terms of ϕ and the tildedquantities L = p − ˜ g (cid:18)
12 ˜ R −
12 ˜ g µν ϕ µ ϕ ν − V ( ϕ ) (cid:19) . (28)which has the same form of Einstein gravity minimally coupled to a scalar field in presence of a self-interaction potential. Equation Eq. (28) clearly suggests that why the set of variables (˜ g µν , p ) is calledEinstein frame [52, 53, 131]. 21 comment is in order. As we have seen, in the vacuumm, one can pass from the Einstein frame tothe Jordan frame. However, in the presence of matter fields, a caution is required since particles andphotons have to be dealt in different ways. In the case of photons, their worldlines are geodesics bothin the Jordan frame and in the Einstein frame. This is not the case for massive particles since theirgeodesic in the Jordan frame are no longer transformed into geodesic in the Einstein frame, and vice-versa, and therefore, the two frames are not equivalent. The consequence is that the physical meaningof conformal transformations is not straightforward, although the mathematical transformations are, inprinciple, always possible. These considerations extend to any higher-order or non-minimally coupledtheory. As we have mentioned, in the previous Sections, the formulation of the EP is the equivalence betweeninertial and gravitational mass m I = m G (Galieleo’s experiment), which implies that all bodies fall withthe same acceleration, independently of their mass and internal structure, in a given gravitational field(universality of free fall or WEP). A more precise statement of WEP is [1]“ If an uncharged body is placed at an initial event in spacetime and given an initial velocity there,then its subsequent trajectory will be independent of its internal structure and composition ”This formulation of WEP was enlarged by Einstein adding a new fundamental part: according towhich in a local inertial frame (the freely-falling elevator) not only the laws of mechanics behave in itas if gravity were absent, but all physical laws (except those of gravitational physics) have the samebehaviour. The current terminology refers to this principle as the
Einstein Equivalence Principle (EEP).A more precise statement is [1]“
The outcome of any local non-gravitational test experiment is independent of the velocity of the(free falling) apparatus and the outcome of any local non-gravitational test experiment is independentof where and when in the universe it is performed ”.From the EEP it follows that the gravitational interaction must be described in terms of a curvedspacetime, that is the postulates of the so-called metric theories of gravity have to be satisfied [1]:1. spacetime is endowed with a metric g µν ;2. the world lines of test bodies are geodesics of that metric;3. in local freely falling frames (called local Lorentz frames ), the non-gravitational laws of physicsare those of Special Relativity.These definitions characterize the most fundamental feature of GR, hence the Equivalence Principle,as well as the physical properties that allow to discriminate between GR and other metric theories ofgravity, In the ETGs some additional features arise because these defintions depend on the conformalrepresentation of the theory adopted. More precisely, in scalar-tensor gravity, massive test particles inthe Jordan frame follow geodesics, satisfying the WEP, but the same particles deviate from geodesicmotion in the Einstein frame (a property referred to as non-metricity of the theory). This differenceshows that the EP is formulated in a representation-dependent way [99]. This serious shortcoming hasnot yet been addressed properly; for the moment we proceed ignoring this problem.In what follows we shall discuss some specific features related to the Equivalence Principle: A “local non-gravitational experiment” is defined as an experiment performed in a small size freely falling laboratory,in order to avoid the inhomogeneities of the external gravitational field, and in which any gravitational self-interactioncan be ignored. For example, the measurement of the fine structure constant is a local non-gravitational experiment,while the Cavendish experiment is not. Let us assume that WEP is violated. Let us assume, for example, that the inertial masses ( m Ii )in a system differ from the passive ones, m P i = m Ii (cid:18) A η A E A m Ii c (cid:19) , (29)where E A is the internal energy of the body connected to the A-interaction and η A is a dimen-sionless parameter quantifying the violation of the WEP. It is then convenient to introduce anew dimensionless parameter (the E¨otv¨os ratio ) considering, for example, two bodies moving withaccelerations a i = (cid:18) A η A E A m Ii c (cid:19) g ( i = 1 ,
2) ; (30)where g is now the acceleration of gravity. Then we define the E¨otvos ratio as η = 2 | a − a || a + a | = Σ A η A (cid:18) E A m I c − E A m I c (cid:19) . (31)The measured value of η provides information on the WEP-violation parameters η A . Experimen-tally, the equivalence between inertial and gravitational masses is strongly confirmed [1]. • The minimal coupling prescriptions. In electrodynamics the interaction is introduced replacingthe partial derivative with the covariant derivative ∂ µ → D µ ≡ ∂ µ + ieA µ [132] (see also [59]). Asimilar scheme is used to introduce the gravitational interaction η µν → g µν , ∂ µ → ∇ µ , √− η d x → √− g d x , (32)Here η µν is the flat Minkowski metric and g µν is the Riemannian one, while η and g are theirdeterminants [133–135].Consider the Maxwell equations in a curved spacetime F αβ ; β = 4 πJ α , F αβ ; γ + F βγ ; α + F γα ; β = 0 , (33)and the four-vector potential A µ related to the Maxwell field by F αβ = ∇ α A β − ∇ β A α . In thisframework, however, a problem arises. Using the above-mentioned rule one obtains two possibleequations from the first of eqs. (33): A β ; α ; β − A α ; β ; β = 4 πJ α , (34)or A β ; α ; β − A α ; β ; β + R αβ A β = 4 πJ α ; (35)while the second of eqs. (33) yields, using the Lorentz gauge ∇ µ A µ = 0,( △ dR A ) α = 4 πJ α , (36)where ( △ dR A ) α = − (cid:3) A α + R αβ A β (37)and △ dR is the de Rham vector wave operator. Now the question is: both Maxwell equations forthe four-potential A µ are obtained using the “comma goes to semicolon” rule, but which is thecorrect one? The answer is: the one obtained using the de Rham operator. As consequence, wesee that “correspondence rules” are not sufficient to write down equations in curved space from23nown physics in flat space when second derivatives are involved (that is, in most situations ofphysical interest). In such cases, extra caution is needed The minimal coupling prescription here discussed is connected with the mathematical formulationof the EEP (actually, to implement the EEP one needs to put in special-relativistic form the lawsunder consideration and then proceed to find the general-relativistic formulation, switching ongravity. In other words, we have to apply minimal coupling prescriptions with the caveat alreadydiscussed). • The last point is strictly related with the scalar-tensor theories of gravity, do these theories satisfythe EEP?To address this question one has to generalize the above two principles and introduce new concepts.Following Will [1], one introduces the notion of “purely dynamical metric theory”, i.e. a theoryin which the behaviour of each field is influenced to some extent by a coupling to at least one ofthe other fields in the theory [1]. In this respect, GR is a purely dynamical theory, as well as theBrans-Dicke theory since the equations for the metric involve the scalar field, and vice-versa.In these theories, the calculations of the metric is done in two stages: 1) the assignment ofboundary conditions “far” from the local system; 2) infer the solutions of equations for the fieldsgenerated by the local system. Owing to the coupling of the metric with fields (for given boundaryconditions), the latter will influence the metric. This implies that local gravitational experimentscan depend on where the lab is located in the universe, as well as on its velocity relative to theexternal world. One of the consequence of such a new physical scenario is that in a Brans-Dicketheory, and more generally in Scalar Tensor Theories, the gravitational coupling “constant” turnsout to depend on the asymptotic value of the scalar field.All these considerations are strictly related to the
Strong Equivalence Principle (SEP) [1]: (i) “WEP is valid for self-gravitating bodies as well for test bodies;(ii) the outcome of any local test experiment is independent of the velocity of the (freely falling)apparatus;(iii) the outcome of any local test experiment is independent on where and when in the universeit is performed” [1].The SEP differs from the EEP because it includes the self-gravitating interactions of bodies (suchas planets or stars), and because of experiments involving gravitational forces ( e.g. , the Cavendishexperiment). SEP reduces to the EEP when gravitational forces are ignored. In connectionwith the SEP, many authors have conjectured that the only theory compatible with the StrongEquivalence Principle is GR (that is
SEP −→ GR − only). The Schiff conjecture represents one of the most important topic related to the foundations of thegravitational physics. Its original formulation asserts that every theory of gravity that satisfies theWEP and is relativistic necessarily satisfies the EEP, and is consequently a metric theory of gravity .Hence
W EP ⇒ EEP . Later, Will proposed a slight modification of Schiff conjecture: every theory of As stressed, for example, in [15], such a prescription does not work for interactions which do not have a “Minkowskian”counterpart. These interactions are expressed in terms of the Riemann tensor or some function of it and occur, for example,in the study of the free fall of a particle with spin: the corresponding equations of motion (Papapetrou equations) involvea contribution in which the spin tensor couples to the Riemann tensor [15]. Such a contribution can not be obtained fromthe prescriptions given above. This motion is described by the corrected geodesic equation [7]. ravity that satisfies WEP and the principle of universality of gravitational red shift (UGH) necessarilysatisfies EEP . Hence in such a case W EP + U GR → EEP .Let us discuss in some details these topics. Notice that the correctness Schiff’s conjecture implies thatthe E¨otv¨os and the gravitational red-shift experiments would provide a direct empirical confirmationof the EEP, with the consequence that gravity can be interpreted as a geometrical (curved space-time)phenomenon. The relevance of such a fundamental aspect of the gravitational physics led todifferent mathematical approaches to prove the Schiff conjecture. These frameworks encompass allmetric theories, as well as non metric theories of gravity. Lightman and Lee [136, 137] proved Schiff’sconjecture in the framework of the so called
T Hǫµ formalism. They consider the motion of a chargedparticles (electromagnetic coupling) in a static spherically symmetric gravitational field U = GM/rS
T Hǫµ = − X a m a Z dt p T − Hv a + X a e a Z dtv µa A µ ( x µa ) + 12 Z d x (cid:18) ǫ E + B µ (cid:19) , where m a , e a , v µa ≡ dx µa dt represent the mass, the charge and the velocity of the particle a . The parameters T Hǫµ do depend on the gravitational field U , that is they essentially account for the response of theelectromagnetic fields to the external potential, and may vary from theory to theory. A metric theorymust satisfy the relation ǫ = µ = r HT for all U . In the case of non-metric theories, the parameters T Hǫµ may depend on the species of particles or on the field coupling to gravity. The metric is givenby ds = T ( r ) dt − H ( r )( dr + r d Ω). Lightmann and Lee showed in [136] that the rate of fall of a testbody made up of interacting charged particles does not depend on the structure of the body (WEP) if and only if ǫ = µ = r HT . This implies W EP ⇒ EEP , satisfying hence the Schiff conjecture. Willgeneralized the Dirac equation in
T Hǫµ formalism, and computed the gravitational red-shift experiencedby different atomic clocks showing that the red-shift is independent on the nature of clacks (Universalityof Gravitational Red-shift (
U GR )) if and only if ǫ = µ = r HT [138]. Therefore U GR ⇒ EEP , verifyingin such a way another aspect of the Schiff conjecture (see also [139]).W.-T. Ni was able to provide a counterexample to Schiff’s conjecture by considering the couplingbetween a pseudoscalar field φ with the electromagnetism field L φF ∼ φε αβγδ F αβ F γδ , where ε αβγδ is thecompletely anti-symmetric Levi-Civita symbol [140]. In [141–143] the Schiff conjecture is analyzed inthe framework of gravitational non-minimally coupled theories. More specifically, the total Lagrangiandensity considered is given by L NMC = R πG + L M + L I ( ψ A , g µν ), where L I ( ψ A , g µν ) is the Lagrangiandensity of some field ψ A non-minimally coupled to gravity [142, 143], while L I = χ αβγδ R αβγδ in [141],where χ αβγδ depends on matter, for example χ αβγδ = ¯ ψσ αβ ψ ¯ ψσ γδ ψ, ψ αµ ψ βν − ψ βµ ψ αν , where ψ is aspin-half field and ψ αβ is a (nongravitational) spin-2 field. Both results show that these gravitationaltheories are in general, incompatible with Schiff’s conjecture.These counterexample indicate that a rigorous proof of such a conjecture is impossible. However,some powerful arguments of plausibility can be formulated. One of them is based upon the assumptionof energy conservation [144]. Following [145], consider a system in a quantum state | A i that decays ina state | B i , with the emission of a photon with frequency ν . The quantum system falls a height H inan external gravitational field gH = ∆ U , so that the system in state B falls with acceleration g B andthe photon frequency is shifted to ν ′ . Assuming a violation of the WEP, the acceleration g A and g B ofthe system A and B are g A = g (cid:18) αE A m A (cid:19) , g A = g (cid:18) αE A m A (cid:19) , E B − E A = hν ν is frequency of thequantum emitted by the system | A i . The conservation of energy implies that there must be a cor-responding violation of local position invariance in the frequency shift given by ν ′ − νν = (1 + α )∆ U ,where ν ′ is the frequency of the quantum at the bottom of the trajectory. The E¨otv¨os parameter is (for m A ∼ m B ∼ m ) η = | g B − g A || g B + g A | ≃ α ( E A − E B ) m . The Schiff conjecture is still nowadays an argument of a strong scientific debate and deep scrutiny. G Following Bondi [12] there are, at least in principle, two entirely different ways of measuring the rota-tional velocity of Earth. The first is a purely terrestrial experiment ( e.g. , a Foucault pendulum), whilethe second is an astronomical observation consisting of measuring the terrestrial rotation with respect tothe fixed stars. In the first type of experiment the motion of the Earth is referred to an idealized inertialframe in which Newton’s laws are verified. However, a unique general relativistic approach to definerotations has been introduced by Pirani considering the boucing photons [146, 147] (see also [148]). Inthe second kind of experiment the frame of reference is connected to a matter distribution surroundingthe Earth and the motion of the latter is referred to this matter distribution. In this way we facethe problem of Mach’s principle, which essentially states that the local inertial frame is determined bysome average motion of distant astronomical objects [12, 15]. Trying to incorporate Mach’s principleinto metric gravity, Brans and Dicke constructed a theory alternative to GR [13]. Taking into accountthe influence that the total matter has at each point (constructing the “inertia”), these two authorsintroduced, together with the standard metric tensor, a new scalar field of gravitational origin as theeffective gravitational coupling. This is why the theory is referred to as a “scalar-tensor” theory; actu-ally, theories in this spirit had already been proposed years earlier by Jordan, Fierz, and Thiery (see thebook [150]). An important ingredient of this approach is that the gravitational “constant” is actuallya function of the total mass distribution, that is of the scalar field, and is actually variable. In thispicture, gravity is described by the Lagrangian density L BD = √− g (cid:20) ϕR − ωϕ ∇ µ ϕ ∇ µ ϕ + L ( m ) (cid:21) , (38)where ω is the dimensionless Brans-Dicke parameter and L ( m ) is the matter Lagrangian including allthe non-gravitational fields. As stressed by Dicke [51], the Lagrangian (38) has a property similarto one already discussed in the context of higher order gravity. Under the conformal transformation g µν → ˜ g µν = Ω g µν with Ω = √ G ϕ , the Lagrangian (38) is mapped into L = p − ˜ g (cid:16) ˜ R + G ˜ L ( m ) + G ˜ L (Ω) (cid:17) , (39)where ˜ L (Ω) = − (2 ω + 3)4 πG Ω ( ∇ α √ Ω)( ∇ α √ Ω) , (40)and ˜ L ( m ) is the conformally transformed Lagrangian density of matter. In this way the total matterLagrangian ˜ L tot = ˜ L ( m ) + ˜ L (Ω) has been introduced. The field equations are now written in te form of An interesting discussion on this topic, also connected with different theories of space, both in philosophy and inphysics, is found in Dicke’s contribution “The Many Faces of Mach” in
Gravitation and Relativity [149]. This discussionpresents also the problematic position that Einstein had on Mach’s principle. R µν −
12 ˜ g µν ˜ R = G ˜ τ µν , (41)where the stress-energy tensor is now the sum of two contributions,˜ τ µν = T ( m ) µν + Λ µν (Ω) . (42)Dicke noted that this new (tilded, or Einstein frame) form of the scalar-tensor theory has certainadvantages over the theory expressed in the previous (non-tilded, or Jordan frame) form; the Einsteinframe representation, being similar to the Einstein standard description is familiar and easier to handlein some respects. But, in this new form, Brans-Dicke theory also exhibits unpleasant features. If weconsider the motion of a spinless, electrically neutral, massive particle, we find that in the conformallyrescaled world its trajectory is no longer a geodesic. Only null rays are left unchanged by the conformalrescaling. This is a manifestation of the fact that the rest mass is not constant in the conformallytransformed world and the equation of motion of massive particles is modified by the addition of anextra force proportional to ∇ µ Ω [51]. Photon trajectories, on the other hand, are not modified becausethe vanishing of the photon mass implies the absence of a preferred physical scale and photons staymassless under the conformal rescaling, therefore their trajectories are unaffected.This new approach to gravitation has increased the relevance of theories with varying gravitationalcoupling. They are of particular interest in cosmology since, as we discuss in detail in the followingchapters, they have the potential to circumvent many shortcomings of the standard cosmological model.We list here the Lagrangians of this type which are most relevant for this review. • The low-energy limit of the bosonic string theory [151–153] produces the Lagrangian L = √− g e − φ ( R + 4 g µν φ µ φ ν − Λ) . (43) • The general scalar-tensor Lagrangian is L ST = √− g (cid:20) f ( ϕ ) R − ω ( ϕ )2 g αβ ∇ α φ ∇ β φ − V ( ϕ ) (cid:21) , (44)where f ( ϕ ) and ω ( ϕ ) are arbitrary coupling functions and V ( ϕ ) is a scalar field potential. Theoriginal Brans-Dicke Lagrangian is contained as the special case f ( ϕ ) = ϕ, ω ( ϕ ) = ω /ϕ (with ω a constant), and V ( ϕ ) ≡ • A special case of the previous general theory is that of a scalar field non-minimally coupled tothe Ricci curvature, which has received so much attention in the literature to deserve a separatemention, L NMC = √− g (cid:20)(cid:18) πG − ξ (cid:19) R − g µν ∇ µ φ ∇ ν φ − V ( ϕ ) (cid:21) , (45)where ξ is a dimensionless non-minimal coupling constant. This explicit non-minimal couplingwas originally introduced in the context of classical radiation problems [154] and, later, confor-mal coupling with ξ = 1 / λϕ theory on a curved spacetime [16,155]. The corresponding stress-energy tensor (“improved energy-momentum tensor”) and the relevant equations will be discussed later. In particular, the theoryis conformally invariant when ξ = 1 / V ≡ V = λϕ [16, 134, 155, 156].All these theories exhibit a non-constant gravitational coupling. The Newton constant G N is replacedby the effective gravitational coupling G eff = 1 f ( ϕ ) , (46)27n eq. (44) which, in general, is different from G N (we use φ as the generic function describing theeffective gravitational coupling). In string theory or with non-minimally coupled scalars, such functionsare specified in (43) and (45). In particular, in spatially homogeneous and isotropic cosmology, thecoupling G eff can only be a function of the epoch, i.e. , of the cosmological time.We stress that all these scalar-tensor theories of gravity do not satisfy the SEP because of theabove mentioned feature: the variation of G eff implies that local gravitational physics depends on thescalar field via φ . We have then motivated the introduction of a stronger version of the EquivalencePrinciple, the SEP. General theories with such a peculiar aspect are called non-minimally coupledtheories . This generalizes older terminology in which the expression “non-minimally coupled scalar”referred specifically to the field described by the Lagrangian L NMC of (45), which is a special case of(44).Let us consider, as in (44), a general scalar-tensor theory in presence of “standard” matter with totalLagrangian density φR + L ( φ ) + L ( m ) , where L ( m ) describes ordinary matter. The dynamical equationsfor this matter are contained in the covariant conservation equation ∇ ν T ( m ) µν = 0 for the matter stress-energy tensor T ( m ) µν , which is derived from the variation of the total Lagrangian with respect to g µν . Inother words: concerning standard matter, everything goes as in GR ( i.e. , η µν → g µν , ∂ µ → ∇ µ ) followingthe minimal coupling prescription. What is new in these theories is the way in which the scalar and themetric degrees of freedom appear: now there is a direct coupling between the scalar degree of freedomand a function of the tensor degree of freedom (the metric) and its derivatives (specifically, with theRicci scalar of the metric R ( g, ∂g, ∂ g )). Then, confining our analysis to the cosmological arena, weface two alternatives. The first is lim t →∞ G eff ( φ ( t )) = G N ; (47)this is the case in which standard GR cosmology is recovered at the present time in the history of theuniverse. The second possibility occurs if the gravitational coupling is not constant today, i.e. , G eff isstill varying with the epoch and ˙ G eff /G eff | now (in brief ˙ G/G ) is non-vanishing.In many theories of gravity, then, it is perfectly conceivable that G eff varies with time: in somesolutions G eff does not even converge to the value observed today. What do we know, from the obser-vational point of view, about this variability? There are three main avenues to analyze the variabilityof G eff : the first is lunar laser ranging (LLR) monitoring the Earth-Moon distance; the second isinformation from solar astronomy; the third consists of data from binary pulsars. The LLR consistsof measuring the round trip travel time and thus the distances between transmitter and reflector, andmonitoring them over an extended period of time. The change of round trip time contains informationabout the Earth-Moon system. This round trip travel time has been measured for more than twenty-five years in connection with the Apollo 11, 14, 15, and the Lunakhod 2 lunar missions. Combiningthese data with those coming from the evolution of the Sun (the luminosity of main sequence stars isquite sensitive to the value of G ) and the Earth-Mars radar ranging, the current bounds on ˙ G/G allowat most 0 . × − to 1 . × − per year [157]. The third source of information on G -variability isgiven by binary pulsars systems. In order to extract data from this type of system (the prototype isthe famous binary pulsar PSR 1913+16 of Hulse and Taylor [158]), it has been necessary to extend thepost-Newtonian approximation, which can be applied only to a weakly (gravitationally) interacting n -body system, to strongly (gravitationally) interacting systems. The order of magnitude of ˙ G/G allowedby these strongly interacting systems is 2 × − yr − [157].A general remark is necessary at this point. According to the Mach Principle, gravity can beconsidered as an average interaction given by the distribution of celestial bodies. This means that thesame gravitational coupling can be related to the spacetime scale, then supposing a variation of G N isan issue to make more Machian the theory. From an experimental point of view, this fact reflects onthe uncertainties of the measurements of G N and it could constitute a test for any alternative theoryof gravity with respect to GR. 28inally it is worth noticing that there exist also Higgs-scalar-tensor theories (see for example, [159–161]) where inertia and gravity are strongly related. Such theories have been introduced to solvethe issues raised in the Brans-Dicke theory where the observational results, coming from the Mercuryperihelion shift, are not matched. In view of this shortcoming, Dicke postulated the existence of amass-quadrupole momentum giving rise to an oblateness correction of the Sun shape. Since this featurewas not detected, Higgs-scalar-tensor theories were deemed necessary. In the previous Sections, we have pointed out that over the last years several, observations led tothe conclusions that the observed Universe is dominated by some form of (homogeneously distributed)DE. In modified gravity the DE can be described by introducing one or more than one scalar fieldscoupled (minimally or non-minimally) to gravity. A candidate for DE is quintessence (the energy densityassociated a scalar field that evolves slowly in time) [162–166]. In this scenario, fundamental couplingconstants do depend on time even in late cosmology [162, 167–170]. This because, as we have seen,it is usual that in modified models of gravity the fundamental coupling constants may depend on thescalar field, that vary during the Universe evolution. Clearly, an observation of a possible time-variationof fundamental constants could be a signal in favour of quintessence, and more generally, of modifiedtheories of gravity, since no such time dependence would be connected to DE in the case in which thelatter is described by a cosmological constant. It is expected that, in a quintessence scenario, the gauge couplings may vary owing to the couplingbetween the field φ ( x ) and the kinetic term for the gauge fields in a GUT [20]. For example, for theelectromagnetic field one has L F = Z F ( φ ( x )) F , where F = F µν F µν . Such a coupling preserves allsymmetries and makes the renormalized gauge coupling g ∼ Z − / F dependent on time through theevolution of the field φ ( x ) [177]. As argued in [177], the coupling of the field φ ( x ) with matter induces anew gravity-like force that does depend on the composition of the test bodies. In this respect a violationof the equivalence principle arises [178].Along these lines, very recently it have been proposed new and general models in which a lightscalar field (playing the role of scalar Dark Matter) is introduced in the gravity action (similar to Eq.(72)). In the most and simplest general case, in fact, the light scalar field couples non-universally tothe standard matter fields, leading as a consequence to a violation of the Einstein equivalence principle(EEP). As discussed in the previous Sections, the scalar fields are predicted in high dimensional theories,in particular in string theory with the dilaton and the moduli fields [151,179,180]. It is worth to mentionthat these models based on light scalar field provide galactic and cosmological predictions for low masses,ranging from 10 − eV to 10 − eV (see for example, Refs. [181–184]. Here we recall the total action inwhich a microscopic modeling for the coupling between the scalar field and standard matter has beenconveniently introduced [185, 186] S = Z d x [ L NMC + L SM + L int ] , (48)where L NMC is the Lagrangian density (72), L SM the Lagrangian density of the Standard Model, andfinally L int is the Lagrangian density of the interaction, which can be of two form [185–187] L int = φ a " d ( a ) e µ F − d ( a ) g β g ( F A ) − X i = e,u,d (cid:0) d ( a ) m i + γ m i d ( a ) g (cid:1) m i ¯ ψ i ψ i (49) A low value of the electromagnetic fine structure constant α em was reported [171] for absorption lines in the light fromdistant quasars. The data are consistent with a variation ∆ α em /α em ≃ − . × − for a cosmological red-shift z ≈ a = 1 and a = 2 correspond to the linear and quadratic [187–189] coupling between scalar andmatter field, respectively, while F µν and F Aµν are the electromagnetic and the gluon strength tensors, µ the magnetic permeability, g the QCD gauge coupling, β the β function for the running of g , m i themass of the fermions (electron and light quarks u, d ), γ m i the anomalous dimension giving the energyrunning of the masses of the QCD coupled fermions, and finally d a are the constants characterizing theinteraction between the light scalar field and the different matter sectors. The main consequence of themodel based on (48) is that the constants of nature turn out to be linearly or quadratically dependingon the scalar field [185, 186]. For the electromagnetic fine structure constant α EM , the masses m i of thefermions, and the QCD energy scale Λ , one obtains α EM ( φ ) = α EM " d ( a ) e φ a a m i ( φ ) = m i " d ( a ) m i φ a a i = e, u, d Λ ( φ ) = Λ " d ( a ) g φ a a with a = 1 , A and B located at the same position in a gravitationalfield generated by a body C , is∆ a ≡ a A − a B = − [ α A ( φ ) − α B ( φ )][ ∇ φ + v ˙ φ ] , (50)where α A,B = ∂ ln m A,B ( φ ) ∂φ and v the particle velocity. Using the expressions for the scalar field derivedin the case of a spherically symmetric extended body with radius R and constant matter density withmass M , one infers the explicit expression for the E¨otv¨os parameter η (Eq. (64)) [190] η = 2 | a A − a B || a A + a B | = ∆˜ a (1) s (1) C e − r/λ φ (cid:16) rλ φ (cid:17) (linear coupling)∆ ˜ α (2) s (2) C φ (cid:16) − s (2) C GM C r (cid:17) (quadratic coupling) (51)Here ∆ ˜ α ( a ) = ˜ α ( a ) A − ˜ α ( a ) A , with a = 1 , α ( a ) is a combination of the coefficients d ( a ) e,m,g and the dilatoniccharges associated to the bodies A and B , s (1) C = 3 ˜ α (1) C x cosh x − sinh xx , with x = Rλ φ ( λ φ = m − φ isthe Compton wavelength of the scalar field), and s (2) C = ˜ α (2) C J ± ( y ), with y = q | ˜ α (2) C | GM A /R A and J ± = ± y − tanh yy , and finally φ is the amplitude of the scalar field.An interesting aspect of these results is that in the neighborhood region of a central body and in thelimit of strong coupling, for the quadratic coupling Eq. (51) assumes the form η ≃ ∆ ˜ α (2) s (2) C φ hR C + h ,where h is the altitude with respect to the radius R C . On the other hand, for small coupling and farfrom the gravitational source, one gets η ≃ s (2) C ∆ ˜ α (2) φ , that is the E¨otv¨os parameter is independenton the location of the two masses. As argued in [190], this particular forms of the E¨otv¨os parametercould be potentially tested in dedicated experiments.30inally we comment the possibility to violate the Einstein equivalence principle by measuring thefrequency ratio between two clocks located at the same position and working on different atomic tran-sition. Defining Y = X A /X B , where X A,B are the specific transitions for each clocks, one finds [190] Y ( t, x ) Y = K + ∆ κ (1) h φ cos( ωt − k · x + δ ) − s (1) A GM A r e − r/λ φ i (linear coupling) K + ∆ κ (2) φ (cid:20)(cid:16) − s (2) A GM A r (cid:17) + cos(2 ωt + 2 δ ) (cid:16) − s (2) A GM C r (cid:17) (cid:21) (quadratic coupling)(52)where K is an unobservable constant and k ( a ) , a = 1 , d ( a ) e,m,g . As extensively discussed in the previous Sections, the introduction of the extended theories of gravityhave been motivated by the necessity to explain the observed cosmic acceleration, hence to provide a”geometric” interpretation of the DE. In these models, gravity is modified on large distances. However,although modifications to GR must be relevant on large scales, they are strongly constrained in SolarSystem (in what follows we shall refer to [191]). In fact, any deviation is subdominant in Solar Systemtests by a factor . − , and the latter is further reduced in some specific theories (in [192, 193] isdiscussed the case of theory that predicts strong violations of the weak equivalence principle for whichdeviations are constrained by a factor . − ). As an example of extended theories of gravity, consideronce again the Brans-Dicke gravity (the scalar field φ couples to gravity and is parameterized by theparameter ω BD ). In the non-relativistic limit, one finds the equation of motion for φ ∇ φ = − πGρ ω BD (53)from which one derives the PPN parameter | γ − | = (2 + ω BG ) − . The Cassini bound | γ − | < . × − [194] implies ω BD > × . From (53) it follows that the effective coupling to matteris α eff ∼ /ω BD . − . As a consequence, any Brans-Dicke like modifications of GR must besubdominant on all scales by a factor ∼ , hence such theories are cosmologically irrelevant. Asimilar conclusions follows if one assumes that the scalar field is massive, so that the field equation(53) gets modified a ( ∇ + m ) φ = − πGαρ , yielding, for a a static, spherically symmetric body, aYukawa-like potential V ( r ) = GMr (cid:0) α e − mr (cid:1) (experiments constrained Yukawa-like potentials ondistances ranging from the Earth-Moon scale [194, 195] to micron scales [196, 197], so that m > ( µ m) − is required to evade Solar system tests).These two examples show that solar system tests constraint these models with the consequencethat they do not have any cosmological relevance because the force must either be too weak, or tooshort ranged. Such difficulties are avoided by screening mechanisms by nonlinear modifications ofthe Poisson equation. The modifications are such that deviations from GR in the Solar system aredynamically suppressed, without requiring a fine-tuning of the mass or the coupling to matter. Screeningmechanisms studied in literature are: • Chameleon screening [198, 199] (the mass of the field changes dynamically mediating short rangedforces in the Solar System but may have effects on cosmological scales). • Symmetron screening [198, 199] (the coupling to matter varies dynamically so that it is uncoupledin the Solar System and may induces deviations from GR on cosmological scales).31
Vainshtein’s mechanism [200] (nonlinear kinetic terms alter the field profile sourced by massivebodies. In such a case fifth forces are highly suppressed in the Solar System, while on cosmologicalscales, theories that exhibit this mechanism can self-accelerate without a cosmological constant,which makes them interesting alternatives to ΛCDM cosmologies).An interesting aspect of screening mechanisms, is that they may violate the equivalence principle[201] (see also [202]). For example, in chameleon theories one can define a scalar charge for an object [201] Q i = M i (cid:18) − M i ( r s ) M i (cid:19) so that the force on an object due to an externally applied chameleon field is F Ch = αQ i ∇ φ ext (this isanalogous to the gravitational charge M so that F grav = M ∇ φ extN where φ extN is an external Newtonianpotential). Two objects of different masses and internal compositions will have different scalar chargesand will therefore fall at different rates in an externally applied chameleon field, signifying a breakdownof the weak equivalence principle (WEP). The chameleon force between two bodies, A and B, is [203] F AB = GM A M B r (cid:0) αQ A Q B e − m eff r (cid:1) and as a result of this the PPN parameter γ is γ = 21 + 2 αQ A Q B e − m eff r − In this Section we discuss the possibility that the spin of particles can be present in gravitationalpotentials. There are essentially some reasons for searching long-range forces that are depending onspin of particles: 1) The role of spin in gravitation (see for example [207–209]). 2) The interactionassociated with the exchange of a light or massless pseudoscalar boson or similar interactions [210–215].In fact, new particles predicted in theories that extend the standard model may induce modificationsto spin-spin interaction between fermions [216]. As an example, we recall the pseudoscalar fields, suchas the axion [215], and the axial-vector fields, such as paraphotons [217] and extra Z bosons [216, 218],the first associated with theories with spontaneously broken symmetries [210–212], the latter in newgauge theories (these new particles, predicted also in string theories [219], are typically introduced toexplain the DE [220, 221] and the DM [222]). 3) A number of Kaluza-Klein theories [223, 224] andsupersymmetric theories [225], in the low-energy limit, predict couplings in which the spins of particlesare involved.As an example we report the Yukawa-like potential between fermions in the case in which theyexchange a (new) vector or axial vector A [215, 216] V A ( r ) = ξ A s · s e − r/λ r , (54) It is worth to recall that gravitational interactions between two objects that do not conserve the discrete symmetrieswere proposed in [208] U ( r ) = GMr (cid:20) α s (1) · ˆr r + α s (1) · v r + α µ ˆr · v (cid:21) , where α , , are generic coefficients, M is the total mass, µ the reduced mass, r the relative displacement, v the relativevelocity, and s (1) is the intrinsic spin of one of the objects (see also [226, 227]). ξ A = g ( e ) A g ( e ) A π is the dimensionless axial-vector coupling constant between the electrons, s and s represent the spins of the electrons, and r is the inter-particle separation, while the dipole-dipolepotential between electrons corresponding to the exchange of a new axion-like pseudoscalar particle P is [215, 216] V P ( r ) = ξ P e − r/λ m e (cid:20) s · s (cid:18) π δ ( r ) + 1 λr + 1 r (cid:19) − ( s · ˆr )( s · ˆr ) (cid:18) λ r + 3 λr + 3 r (cid:19)(cid:21) , (55)where ξ P = g ( e ) P g ( e ) P π is the dimensionless pseudoscalar coupling constant between the electrons, m e themass of the electron mass, and ˆr = r /r .The general analysis of long-range forces between macroscopic objects (polarized spin medium)mediated by light particles that include spin and velocity terms have been performed in [215, 216].Experiments aimed to tests such new terms will be discussed in next Sections. As we have seen the equivalence principle sates that the effect of gravity on matter is locally equivalent tothe effect of a non-inertial reference frame in special relativity. The dynamical content of the equivalenceprinciple can be understood by considering an inertial frame in M , in which matter field φ is describedby the Lagrangian L M ( φ ; ∂ i φ ). Passing to a noninertial frame, L M transforms into √− gL M ( φ ; ∇ i φ ),with ∇ i = e µi ( ∂ µ + ω µ ) the covariant derivative (this is the minimal substitution discussed in theprevious Section). The gravitational field (equivalent to the non-inertial reference frame) appears inthe quantities √− g and ∇ i , and can be eliminated on the whole spacetime by reducing to the globalinertial frame, while for real gravitational fields one has that they can be eliminated only locally. Forintroducing a real gravitational field, hence, Einstein replaced M with a Riemann space V . However,also a Riemann-Cartan space U could have been chosen [228].Another formulation of the Equivalence Principle asserts that the effect of gravity on matter canbe locally eliminated by a suitable choice of reference frame, and matter behaves following the laws ofSpecial Relativity [228], i.e. at any point P in spacetime an orthonormal reference frame e i can be chosensuch that ω ij µ = 0 and e µi = δ µi at P . The important consequence of this statement is that it holds notonly in GR (i.e. V ), but also in Poincare Gauge Theory (i.e. U ) [229, 230]. The Equivalence Principleis not violated in manifolds with torsion, fitting in natural way into a U geometry of spacetime. It holdsin V , as well as in T . Notes however that in more general geometries, characterized by a symmetry ofthe tangent space higher than the Poincare group, the usual form of the Equivalence Principle can beviolated, and local physics differs from Special Relativity [228, 231]. Let us discuss now the tests of Universality of Free Fall (UFF) for charged particles. The interest forthese studies follows from the fact that, in some frameworks, a violation of the UFF is related withcharge non-conservation [232]. Considering a connection of UFF and Universality of the GravitationalRed-shift (UGR) [169], the most favourable model for a violation of the UGR is a time dependent fine-structure constant caused by a time-varying electron charge. Therefore, tests of the UFF for chargedmatter can be interpreted as UGR tests, too. In this Section, we follow the notation in [228]. A space ( L ; g ) with the most general metric compatible linearconnection Γ is called Riemann-Cartan space U . If the torsion vanishes, a U becomes a Riemannian space V of GR;if, alternatively, the curvature vanishes, a U becomes Weitzenbock’s teleparallel space T . The condition R αβγχ = 0transforms a V into a Minkowski space M , and T αβγ = 0 transforms a T into an M .
33o test the validity of the EP is analogue to test the minimal coupling procedure, hence to searchfor an anomalous coupling of the gravitational field (as an extension of the standard minimal couplingprocedure, previously discussed). In the non-relativistic regime, the Hamiltonian of a charged particlein a gravitational field is given by H = − ~ m (cid:18) ∇ + iq ~ c A (cid:19) + mU + κqU + λq U , (56)where κ and λ are free parameters with dimensions [ κ ] = [ mass/charge ] and [ λ ] = [ mass/charge ]respectively. In the Hamiltonian (56), one can define an effective mass m eff = ( m + κq ) U that canbe interpreted as the charge dependence of the gravitational mass. Since the charge of a particle isrelated to spacetime symmetries through the CPT theorem, the problem of violation of EP for chargedparticles assumes a particular interest for anomalous charge couplings.The stability of an non-pointlike electron requires an effective dependence on the square of theelectron charge [233]. Furthermore, the generalized Maxwell equations, in general, violate the UFF ina way in which appears once more the square of the charge [140]. According to these considerations,it makes sense to take into account a general model having chargedependent inertial and gravitationalmasses [234]. One can choose the parameters in such a way that neutral systems, made up of boundcharged particles, exactly fulfill the UFF while isolated charged particles may violate it. Thus, one canintroduce an E¨otv¨os coefficient that depends on the charge of particles, i.e. η = η + κ q m − κ q m .Here only the linear charge dependence is considered and η indicates the ordinary E¨otv¨os parameterfor the masses. These considerations then suggest a comparison between the free fall of a chargedand a neutral particle described by η = η + κ qm . By shielding all electromagnetic fields, neutral andcharged particles, without internal structure, must fall following the same path. Let us note that, inthis framework, experiments in space seem to be favoured in order to reduce the disturbances inducedby the stray fields [234]. In this Section we discuss the EP violation in a QFT and GR framework [235,236] (for modified gravity,see for example Refs. [237, 238] and Ref. [239] for the generalized uncertainty principle). The systemconsists of an electron with mass m (the renormalized mass of the particle when the temperatureis zero) in thermal equilibrium with a photon heat bath. The aim of the analysis is the evaluation ofelectron’s gravitational and inertial mass in the low-temperature limit (namely, T ≪ m ). The presenceof a non-zero temperature is crucial since m g = m i for T = 0.The gravitational and inertial masses are derived by adopting a Foldy–Wouthuysen transforma-tion [240] on the Dirac equation which allows to derive a Schr¨odinger equation (non-relativistic limit ofparticles with spin half) in which the expression for the mass is easily recognizable.In order to operationally define the inertial mass, one applies an electric field to charged particleand study the consequent acceleration [235, 236]. One has therefore to evaluate the finite temperature(radiative) corrections to the electromagnetic vertex. After the renormalization procedure and takinginto account the finite temperature contributions, one obtains [235, 236] (cid:16) /p − m − α π /I (cid:17) ψ = e Γ µ A µ ψ. (57)Here we ahve used the notation /a ≡ γ µ a µ , α is the fine-structure constant, γ µ are the Dirac matrices, A µ is the electromagnetic four-potential, and the quantity I µ is defined as I µ = 2 Z d k n B ( k ) k k µ ω p k − p · k , (58)34ith k µ = ( k , k ) and where ω p and p are connected by ω p = p m + | p | . In Eq. (58) n B ( k ) representsthe Bose-Einstein distribution: n B ( k ) = 1 e βk − , (59)where β = 1 /k B T , with k B being the Boltzmann constant. Finally, Γ µ accounts for the finite tempera-ture corrections to the electromagnetic vertexΓ µ = γ µ (cid:18) − α π I E (cid:19) + α π I µ . (60)Applying the Foldy–Wouthuysen transformation, Eq. (57) reduces to a Schr¨odinger-like equation i ∂ψ s ∂t = m + απT m + | p | (cid:16) m + απT m (cid:17) + eφ + p · A + A · p (cid:16) m + απT m (cid:17) + . . . ψ s (61)= Hψ s To identify the inertial mass one calculates the acceleration a = − [ H, [ H, r ]] = e E m + απT m from which one identifies the inertial mass m i = m + απT m . (62)This relation shows that the difference between the inertial mass of an electron at finite temperatureand m is due exclusively to the thermal radiative correction of Eq. (62). The fact that the inertialmass m i increases with T is expected since it represents the increased inertia needed to travel throughthe background heat bath.An analogous procedure can be also performed for the gravitational mass m g . Calculations ofRefs. [235, 236] rely on the weak field approximation, i.e., to first order in the gravitational field (seeEq. (11)), and consider the radiative corrections calculated in flat space. The Dirac equation that takesinto account the gravitational interaction reads [235, 236] (cid:16) /p − m − α π /I (cid:17) ψ = 12 h µν τ µν ψ, (63)where τ µν is the renormalized stress-energy tensor while h µν = 2 φ g diag (1 , , , φ g the gravita-tional potential. Once again, a Foldy–Wouthuysen transformation yields the Schr¨odinger-like equation i ∂ψ s ∂t = m + απT m + | p | (cid:16) m + απT m (cid:17) + (cid:18) m − απT m (cid:19) φ g ψ s , (64)= H g ψ s The calculation of the acceleration gives a = − [ H g , [ H g , r ]] = m − απT / m m + απT / m m g = (cid:18) m − απT m (cid:19) . (65)Clearly, there is no difference between m g and m i at zero temperature, so that only radiative correctionsrender the violation of the equivalence principle feasible. In principle, this result would yield a violationof the equivalence principle in an E¨otv¨os-type experiment, although at accessible temperatures the effectis small. In fact, from Eqs. (62) and (65) one gets m g m i = 1 − απT m , (66)in the first-order approximation in T . At temperature of the order T ∼ ∼ − .We point once more that these results hold in the approximation T ≪ m e . Equation (66) is a directconsequence of the fact that Lorentz invariance of the finite temperature vacuum is broken, which meansthat it is possible to define an absolute motion through the vacuum (i.e. the one at rest with the heatbath). The case of gravitational coupling of leptons in a medium has been studies in [241, 242]. Let us now discuss a different method, proposed in [243], that reproduces the previous results, inparticular Eq. (66).The starting point is the analysis of a charged test particle of renormalized mass at zero temperature m in thermal equilibrium with a photon heat bath in the low-temperature limit T ≪ m . The dispersionrelation reads [235] E = r m + | p | + 23 απT , (67)which can be easily identified with the first-order correction in T that descends from the finite tem-perature analysis. The stress-energy tensor T µν related to the test particle, whose world line can becontained in a narrow “world tube” in which T µν is non-vanishing. The conservation equation for thestress-energy tensor can be integrated over a three-dimensional hyper-surface Σ and defined as: Z Σ d x ′ √− gT µν ( x ′ ) = p µ p ν E , (68)where p µ is the four-momentum and E = p the energy, given by E = R Σ d x ′ √− gT ( x ′ ) . Theseequations hold in the limit where the world tube radius goes to zero [244].As shown in [235], the source of gravity, at finite temperature and in weak-field approximation,turns out to be (in the rest frame of the heat bath)Ξ µν = T µν − απ T E δ µ δ ν T , (69)where Ξ µν contains not only the information on the Einstein tensor G µν , but also thermal correctionsto it Eq. (69) is explicitly derived after the choice of the privileged reference frame at rest with the heat bath. Thelatter give rises to a Lorentz invariance violation of the finite temperature vacuum. In fact, in the tangent space (flatspace), one cannot consider a Minkowski vacuum anymore owing to the fact that it is replaced by a thermal bath. Asa consequence, Lorentz group is no longer the symmetry group of the local tangent space to the Riemannian manifold,even though general covariance still holds there. According to this, one can proceed keeping in mind that the situationunder investigation is slightly different from the usual GR scheme [243]. µν = T µν − απ T E e µ ˆ0 e ν ˆ0 T ˆ0ˆ0 , (70)where e µ ˆ0 denotes the vierbein field and the hatted indexes are the ones related to the flat tangentspace. The Einstein field equations are hence given by G µν = Ξ µν . The Bianchi identity ∇ ν G µν = 0implies ∇ ν T µν = ∇ ν (cid:18) απ T E e µ ˆ0 e ν ˆ0 T ˆ0ˆ0 (cid:19) , (71)so that, using . x µ ≡ dx µ /ds and E = m . x ˆ0 = m . x ρ e ˆ0 ρ one gets [243] .. x µ + Γ µαν . x α . x ν = 23 απT . x ν ∂ ν e µ ˆ0 mE − e µ ˆ0 (cid:16) .. x ν e ˆ0 ν + . x ν . x β ∂ β e ˆ0 ν (cid:17) E + Γ µαν e α ˆ0 e ν ˆ0 m . (72)Eq. (72) represents a generalization of the geodesic equation to the case in which the temperature isnon-vanishing. We now analyze Eq. (72) for the Schwarzschild geometry. The metric tensor is given by g µν = diag (cid:0) e ν , − e λ , − r , − r sin θ (cid:1) , (73)where e ν = e − λ = 1 − φ = 1 − Mr .
For our purpose, we shall consider only radial motion ( . ϑ = . ϕ = 0). The non-vanishing vierbeins forthe Schwarzschild metric are e = e − ν , e = e − λ .The geodesic equation for µ = 0 is (here ′ ≡ ∂/∂r ) .. t + ν ′ . r . t = − απT " . rν ′ mE + .. t + . r . tν ′ E e ν e − ν , (74)and since Em = . x ˆ0 = . x α e ˆ0 α = . t e ν/ , Eq. (74) can be cast in the form (cid:18) απT E (cid:19) (cid:16) .. t + . ν . t (cid:17) = 0 . (75)The radial contribution can be computed involving Eq. (72) for µ = 1 .. r + ν ′ (cid:18) . t e ν − λ − . r − απT m e − λ (cid:19) = 0 . (76)An integration of Eq. (76) gives [243] e λ . r − e ν . t − απT m ν = const . (77)37he constant is determined from the condition of normalization of the 4-velocity . x µ . x µ = − e λ . r − e ν . t = − . (78)In the limit of vanishing gravitational field (namely, ν, λ → r → ∞ ), Eq. (78) reduces to . r ∞ − . t ∞ = −
1, which compared with Eq. (77), implies e λ . r − e ν . t − απT m ν = − . (79)In the weak-field approximation and owing to Eq. (79), it is immediate to find that Eq. (76) resultsmodified as: .. r = − Mr (cid:18) − απT m (cid:19) . (80)To first-order approximation in T , as in the previous QFT treatment, one obtains m g m i = 1 − απT m , which is exactly Eq. (66). In the case of the Brans-Dicke action, the action reads S BD = Z d x √− g (cid:18) ϕR − ω ϕ g µν ∂ µ ϕ∂ ν ϕ + L matter ( ψ ) (cid:19) . (81)where ϕ = 116 πG eff , (82)and such a result is traduced in the introduction of a new “effective” gravitational constant that hasto be identified with the scalar field. Here one assumes that ϕ is spatially uniform, and it must varyslowly with cosmic time (this is consistent with experimental data)Field equations derived from Eq. (81) are2 ϕG µν = T µν + T ϕµν − g µν ∇ µ ∇ ν ) ϕ, (83)and (cid:3) ϕ = ζ T, (84)where ζ − = 6+4 ω and T = g µν T µν . The symbol (cid:3) denotes the usual D’Alembert operator. In Eq. (83), T µν and T ϕµν are extracted by varying L matter and the kinetic term of S BD , respectively. As expected,field equations for the metric tensor becomes the ones derived by GR in the limit ϕ = const = 1 / πG .The field equations admit a static and isotropic solution so that the line element is: ds = e v dt − e u (cid:2) dr + r (cid:0) dϑ + sin ϑd Φ (cid:1)(cid:3) , (85)with e v = e α − Br Br ! λ , e u = e β (cid:18) Br (cid:19) − Br Br ! λ − C − λ , (86)38ith α , β , B , C and λ being constants that can be connected to the free parameter of the theory ω . Since it is a scalar-tensor theory, a solution for ϕ must also be found; in the considered case, theoutcome turns out to be ϕ = ϕ − Br Br ! − Cλ , (87)where ϕ is another constant.Repeating the previous analysis leading to (80), for BD theory one gets .. r = − v ′ (cid:26) (cid:0) e − v − (cid:1) (cid:18) λBr − C (cid:19) − απT m (cid:20) v − (cid:18) λBr − C (cid:19) v (cid:21)(cid:27) e − u . (88)From Eq. (88), one observes that there is not only the radiative correction to the ratio m g /m i ,but also another contribution which exclusively depends on ω and that correctly vanishes in the limit ω → ∞ , that is when GR is recovered. The evaluation of the second quantity of Eq. (88) allows to puta lower bound to the parameter of the Brans-Dicke theory. In fact, in the weak field regime, imposing | ( m g − m i ) /m i | < − [245] and using [246] α = β = 0; C = −
12 + ω ; B = GM λ λ = r ω + 32 ω + 4 . (89)one infers ω > GMr · , (90)which is the final expression for the lower bound of the Brans-Dicke parameter in the weak-field ap-proximation. For the Earth M ⊕ = 5 . · Kg ; R ⊕ = 6 . · m. so that [247] ω > . · , (91)that is similar to a bound recently obtained [145], which gives ω > · . For the sake of completeness,it is useful to look at a table that contains a prediction of the most reliable bounds for ω [248].Table 1: This table includes expected bounds on the parameter ω from different experiments (see [248]and references therein). Detector System Expected bound on ω aLIGO (1 . M ⊙ ∼ . M ⊙ ∼ Einstein Telescope (1 . M ⊙ ∼ eLISA (1 . M ⊙ ∼ LISA (1 . M ⊙ ∼ DECIGO (1 . M ⊙ ∼ Cassini Solar System ∼ As stated before, the EEP asserts that in any local Lorentz frame about any point in spacetime, the lawsof physics are described by the special relativity (including the standard model of particle physics) [133].39s widely believed, general relativity and the standard model can be considered as the low energylimit of some fundamental theory of physics at high energy scales, that, in turn, might give rise toviolations of EEP at some scale [249–251], although its exact form is not well defined. In this framework,the standard model extension (SME) [251] represent a flexible and widely applied [252] context fordescribing violations of EEP. The SME is an effective field theory that extend the standard modelaction by adding new terms that break local Lorentz invariance and other tenets of EEP [253]. In thismodel, the energy conservation, gauge invariance, and general covariance are preserved. As in othermodels [249], EEP violation in the SME can manifest in different ways (for example, it may be stronglysuppressed in normal matter relative to antimatter [253]).In the framework of SME, in Ref. [254] the authors show that EEP violation in antimatter can beconstrained by means of tests in which bound systems of normal matter are used. More specifically,an anomaly that violates the WEP for free particles generates anomalous gravitational redshifts in theenergy of systems in which they are bound. For a nuclear shell model one can estimate the sensitivityof a variety of atomic nuclei to EEP violation for matter and antimatter.Focusing on conventional matter (made up of protons, neutrons, and electrons), the spin-independentviolations of EEP in the SME acting on a test particle of mass m w are described by the action [253](see also [252, 255]) S = − Z " m w p [ g µν − c w ) µν ] dx µ dx ν (˜ c w ) + (˜ a w ) µ dx µ , (92)where the superscript w = p, n, e (for proton, neutron, or electron) indicates the type of particle in ques-tion, g µν is the metric tensor, dx µ is the interval between two points in spacetime. The (˜ c w ) µν tensor de-scribes a fixed background field that modifies the effective metric that the particle experiences, and thus,its inertial mass relative to its gravitational mass. The four vector (˜ a weff ) µ = ((1 − αU )(˜ a weff ) , (˜ a weff ) j ),where U is the Newtonian potential, represents the particles coupling to a field with a nonmetric inter-action α with gravity. As (˜ a weff ) µ is CPT odd [251], this term enters with opposite sign in the action foran antiparticle ˜ w . Both (˜ c w ) µν and (˜ a w ) µ vanish if general relativity is valid. For convenience, Eq. (92)includes an unobservable scaling of the particle mass by 1 + (˜ c w ) . Consider the isotropic subset ofthe model [253], i.e. (˜ c w ) µν is diagonal and traceless, and the spatial terms in the vector (˜ a weff ) µ vanish.In the nonrelativistic, Newtonian limit, the single particle Hamiltonian produced by the action (92) isgiven by H = 12 m w v − m wg U (93)where the effective gravitational mass m wg is given by m wg = m w (cid:20) −
23 (˜ c w ) + 2 αm w (˜ a weff ) (cid:21) . (94)Experimentally observable EEP violations are proportional to the particles gravitational to inertial massratio m wg m w = 1 −
23 (˜ c w ) + 2 αm w (˜ a weff ) ≡ β w (95)and are described by the parameter β w [255]. From Eq. (95), it follows that (˜ c w ) and (˜ a weff ) areresponsible for violations of the WEP, an aspect of EEP [145], since they produce particle-dependentrescalings of the effective gravitational potential. In addition, EEP violation is not apparent in thenonrelativistic motion of a free particle if α (˜ a weff ) = m w (˜ c w ) , although it remains manifest in themotion of the antiparticle ˜ w , for which β ˜ w = − αm w (˜ a weff ) − (˜ c w ) , a limit discussed in [253]. Theantimatter anomaly β ˜ w does contribute to tests involving nongravitationally bound systems of matter,owing to the anomalous gravitational red-shift produced by (˜ c w ) in the energies of bound systems (fordetails, see [255]). 40 .21 Strong Equivalence principle in modified theories of gravity As we have discussed in the previous Sections, in modified or alternative theories of gravity, GeneralRelativity is generalized including extra degrees of freedoms, such as scalar, vector or tensor fields,higher orders terms in the scalar invariants, and so on [1]. Typically, in these models the new degreeof freedoms couple non-minimally with, referring to Section 2, scalar curvature. More explicitly, this isthe case of the Brans-Dicke theory, the prototype of scalar tensor-theories, in which the scalar field φ couples minimally to scalar curvature R , so that the action reads (81). The effects of the non-minimalcoupling is, in some regime, to generate new (gravitational) interactions among masses, modifying in adifferent way the values of the perturbations of the metric (weak-field approximation) h = − GM G /r ,related to the gravitational mass M G , and h ij , related to the inertial mass M I [256, 257], i.e. M I = πG R d x √− g ( h ii,j − h ij,i ) dS j . In General Relativity, since h ij = − GM G /r ( h = h ij , one gets M I = M G , while in Brans-Dicke theory (and hence in more general theories of gravity), since h = h ij (weak-field limit of (86)), one gets M G = M I + f ( ω, E φ ), where f ( ω, E φ ) depends on the parameter ω and the self-energy of the scalar field E φ [256, 257]. The WEP has been experimentally verified to remarkable accuracy. This is made possible by the factthat the universality of free fall (UFF) can be tested in null experiments, as the physical quantity ofinterest is the relative acceleration between two freely falling proof masses. If the gravitational mass m g of a body differs from its inertial mass m I , the acceleration a of the body in a gravitational field g is given by a = ( m g /m I ) g . Experiments determine upper limits to the differential acceleration | a − a | between two freely falling test masses of different composition. Possible violations of WEP are thenquantified by the E¨otv¨os parameter defined in Eq. (31) η = 2 (cid:12)(cid:12)(cid:12)(cid:12) a − a a + a (cid:12)(cid:12)(cid:12)(cid:12) . (96)Tests with increasing accuracy correspond to decreasing upper limits on η . As long as UFF is valid,the differential acceleration and thus η must be null within experimental uncertainties. As for any nullexperiment, no specific model is required to obtain the physical quantity of interest to be comparedwith the measured signal.Various kinds of null experiments are possible to test WEP, differing in the magnitude of the potentialsignal and in the impact of noise sources and systematic effects. In the following of this paper we describepast, ongoing and future WEP test experiments by grouping them into three main classes. Section 3.1describes experiments in which the test masses are macroscopic bodies. In section 3.2 we present UFFtests by the observation of celestial bodies and their movement with respect to each other. In section 3.3we discuss experiments with microscopic test masses, i.e. atoms, molecules, and elementary particles.WEP tests can be also classified according to different criteria. Sections 3.1 and 3.3 include groundlaboratory tests as well as experiments in space. Macroscopic proof masses in ground experiments canbe either suspended or left in free fall. The differences between experimental classes are discussed inthe following sections.It is worth mentioning that other experiments, that strictly speaking cannot be considered as testsof WEP, deeply rely on it for their validity. Relevant examples are the measurement of the Newtoniangravitational constant G performed with freely falling samples [258, 259] or the comparison of differentgravimeters for metrological purposes [260]. 41 .1 Lab experiments with macroscopic masses Laboratory WEP tests based on macroscopic masses are either performed with freely falling massesor with suspended masses. The latter class of experiments compares the acceleration experienced bytwo masses of different composition as they fall in the gravity field of the Earth. In this case, thesignal to be detected, namely a non-zero differential acceleration resulting from a WEP violation, ismaximum as it is proportional, via the E¨otv¨os parameter, to the full gravitational acceleration of theEarth. Unfortunately, the typical free fall time on Earth cannot be longer than a few seconds tokeep the height of the instrument within a reasonable size. This imposes a major limitation to themeasurement sensitivity. In addition, free-fall experiments are very much dependent from the initialconditions (position and velocity) of the test masses as they are released and therefore to externalperturbations acting on the instrument.Experiments with suspended masses are done, with a few exceptions, using a torsion balance, withtest masses of different composition suspended at the opposite ends of the beam. When the beam isoriented along the East-West direction, the differential acceleration responsible for a WEP violation isproportional to the centrifugal acceleration, which provides a driving signal for a WEP violation aboutthree orders of magnitude smaller than in a free-fall experiments. Despite the lower signal, torsionbalances are today providing the best laboratory tests of the Weak Equivalence Principle due to thelong measurement time at equilibrium and to the excellent control of systematic effects that they canoffer by spinning the instrument around its axis.Experiments with suspended and freely falling macroscopic masses are described in the next sections.
As a cornerstone of mechanical theories, WEP has been experimentally investigated since the dawn ofmodern age. First experimental tests of the UFF date back to the early 1600’s, when Galileo Galileicompared the oscillation periods of two simple pendulums with different composition [261]. Consideringthat the two masses are in free fall along the tangent to the trajectory of their respective oscillation,Galilei managed to test the UFF with an accuracy at the 10 − level [262]. Newton repeated theexperiment to test the equivalence of inertial and gravitational mass with similar precision [263], andtwo centuries later Bessel improved it to an accuracy of 2 × − [264] with a more precise determinationof the pendulum length, and by comparing many different materials including gold, silver, lead, quartz,marble, clay, loadstone, water. Pendulum WEP tests were also performed on radioactive materials inthe early XX century: Thomson [265] reached a 5 × − precision for radon, and Southerns [266],achieved a 5 × − precision for uranium oxide. Further evolution of this method led to the remarkableprecision of 3 × − in the experiment of H. H. Potter in 1923 [267]. Simple pendulum experimentsare intrinsecally limited by the large impact of dissipative damping forces from suspension and fromair, as well as by geometrical asymmetries between the two pendulums to be compared. Moreover, withincreasing precision the anharmonic terms of the pendulum dynamics become relevant, and the perioddepends on the amplitude which then has to be controlled with high precision.A breakthrough occurred in the late 19th century, due to the intuition of E¨otv¨os to employ aCavendish torsion balance (more precisely, Boy’s modification) to compare inertial and gravitationalmass in a null experiment. E¨otv¨os’ first series [268], published in 1890, reached a precision of 5 × − .Two decades later, with D. Pekar and E. Fekete, E¨otv¨os improved it to 3 × − [269]. In E¨otv¨os’experiments, the inertial acceleration is given by the centrifugal force due to the Earth’s rotation,while the gravitational acceleration is the component of g necessary to compensate it. Two massesof different composition are suspended at opposite ends of the torsion balance beam; the centrifugalforces on the two weights due to the Earth’s rotation are balanced against a component of the Earth’sgravitational field. A WEP violation would produce a rotation of the torsion balance: if the ratio ofpassive gravitational mass to inertial mass should differ from one test mass to the other, there would42e a torque tending to twist the torsion balance. When the beam is oriented along the East-Westdirection, the differential acceleration responsible for a WEP violation is proportional to the centrifugalacceleration a c = Ω ⊕ R ⊕ cos θ sin θ , where R ⊕ and = Ω ⊕ are the radius of our planet and the angularvelocity of its rotation motion, and θ is the latitude at the instrument location. For θ = π/
4, thecentrifugal acceleration amounts to 16 . providing a driving signal for a WEP violation about600 times smaller than in a free-fall experiments. One major limitation is given by the fact that apotential violation would produce a static (DC) signal. The effect of a non-zero signal can be detectedby exchanging the position of the two masses, so that the sense of the twist from WEP violation wouldbe reversed.Improved versions of E¨otv¨os’ experiment were designed to produce an AC signal from potentialWEP violations, by spinning the instrument around its axis. The spin motion introduces a modulationof the WEP violating signal without intervening on the balance configuration [270,271]. This is achievedmainly in two different ways: by locking the torsion balance on the gravitational field of Sun at equilib-rium with the inertia of the Earth that rotates around it, so that the violation signal is modulated bythe Earth’s spin with a 24 h period; or by actively rotating the torsion balance around the suspensionwire to up-convert possible violation signals from DC to the rotation frequency. Main advantage ofthe former method is the natural modulation of the potential signal without potential systematics andtechnical noises from active mechanical rotation of the apparatus. The main advantage of the lattermethod is the higher modulation frequency of the signal, allowing to remove many low-frequency noisesources and systematics. In particular, mechanical losses due to internal damping are lower at higherfrequencies, and up-conversion brings the signal to a region of reduced thermal noise. Combinations ofdifferent up-conversions have also been designed, e.g. with rotating torsion balances in the field of theSun.Earth rotation offers a natural platform for spinning a torsion balance with daily period againstthe Sun. If the beam is aligned with the north-south direction, a WEP-violating differential accelera-tion would produce a maximum torque when the Sun is at the astronomical horizon. The horizontalcomponent of the gravitational acceleration toward the Sun is at most 6 mm/s . Thus the signal forUFF tests in the gravitational field of the Sun is smaller than for tests in the Earth’s field by about afactor 3 /
8. Dicke’s torsion balance experiment provided the first UFF test in the field of the Sun [270],reaching 10 − , followed by Braginsky and Panov down to 10 − . The latter experiment provided thebest estimate of the E¨otv¨os parameter for nearly 30 years. A variant of the torsion balance is obtainedby replacing the suspension wire with a so-called fluid fiber, introduced by Keiser and Faller at theend of the 1970s [272]. In this kind of setup, test masses made of hollow metal bodies float on fluids,and their position is controlled by an electrostatic system. A potential WEP violation, inducing adifferential acceleration between the solid and the liquid, is measured on the control signal needed tokeep the test masses in constant position. WEP tests with fluid fibers were performed in 1979 with anaccuracy of 10 − [272] and in 1982 with an accuracy of 4 × − [273]. The potential accuracy of suchmethod was estimated at levels between 10 − and 10 − [274]. A similar experiment was performedby Thieberger in 1986 by observing the horizontal drift of a hollow copper sphere floating freely inwater [275]. The driving horizontal gravitational force was generated by placing the setup near a steepcliff; a differential acceleration between copper and water would result in a drift velocity of the sphere.Indeed Thieberger measured a net differential acceleration, indicating a potential WEP violation with η < . × − arising from a fifth force. A comparative experiment of the same kind was performedwith a more symmetric setup by Bizzeti et al. in 1989 [276]; no WEP violation was found, up to anaccuracy of 2 . × − .A disadvantage with the Sun as source is a weaker driving signal as compared to that in the field of theEarth. Spinning a torsion balance by means of a uniformly rotating turntable allows the Earth to be usedas the attractor [277]. Moreover, driving force modulation can be kept at higher frequencies, reducingthe thermal noise [278] and disentangling the WEP violating signal from other effects, e.g. temperature43ariations, that naturally occur at the diurnal frequency, including thermal effects, microseismicity, localmass motions. Such effects originate from the Sun through radiative heating of the Earth’s surface andatmosphere, with a typical thermal time delay, rather than by gravitational interaction. Though Earthtidal forces have no daily periodicity and can be neglected, gravity gradients originating from solar tidesoccur mostly at twice the diurnal frequency, resulting in a spurious WEP violation of the order 10 − for a balance arm of 15 cm.Torsion balance experiments provided the best limits on potential WEP violations for ground testsso far (see [277], Table 3). Such experiments have confirmed UFF both in the field of the Sun, up toabout 10 − , and in the field of the Earth, up to about 10 − , as well as in the field of local sourcemasses, up to about 5 × − .The most precise torsion balance to date was realised by the so called E¨ot-Wash research group.A first experiment in 1989 provided a WEP test in the gravitational field of the Earth with 1 × − accuracy [279]. In the same year, a test with 5 . × − accuracy was done in the field of a local massdistribution by placing the torsion balance near a river lock, resulting in a 12 min periodic modulationof ∼ × kg of water with known distribution as an attractor for copper and lead test masses [280].The precision was improved to 1 . × − in 1994 with test masses made of Beryllium and of anAluminum/Copper alloy in the field of the Earth, and later to 1 . × − with Si and Al+Cu testmasses in the field of the Sun. In 1999 the E¨ot-Wash group measured the differential acceleration of aCu test mass toward a Pb attractor to be a Cu − a Pb < (1 . ± . × − m/s . Comparing to thecorresponding gravitational acceleration of 9 . × − m/s this leads to η < − [281]. An experimentperformed in 2001 in the field of the Sun [282] improved the result of Braginsky and Panov to η < − .In 2008 the same group obtained η < (0 . ± . × − with Be and Ti test masses in the field of theEarth [283]. The latter result represents the most accurate WEP test on ground.Another recent torsion balance experiment provided a WEP test at the 10 − level on chiral masses[284], using a pair of lef-handed and right-handed quartz crystals.Current experiments with torsion balances are mainly limited by systematic effects arising fromgravity gradients coupling to geometrical asymmetries in the the torsion pendulum, and thus producingdifferential directions for the forces on the test bodies. Several environmental parameters can produceeffects that mimic a WEP-violating signal. Tilts of the rotation axis with respect to local vertical, couplethe pendulum to gravity gradients; the same applies to temperature fluctuations, thermal gradients,and magnetic fields. The main bias terms can be subtracted to some extent using the method describedin [285, 286]. For each driving term, the corresponding parameter is modulated with large amplitudeto calibrate its effect on the WEP-violating signal; calibration factors and measured parameters arecombined in post-processing of the actual WEP data to correct for the contribution of bias driving terms.Gravity gradients can be measured with a gradiometer and compensated with a suitable configurationof local source masses. Rotating the compensation system by 180 ◦ about its vertical principal axisdoubles the effect of ambient gradient. This allows to determine the corresponding systematic error onthe torsion balance WEP test, which is measured from the ratio of the torsion balance and gradiometersignals in the two compensator positions. Additional sources of systematic errors originate from fibretwisting due to residual tilts of the setup or wobbles of the rotary axis in combination with asymmetriesin the upper suspension point. Such effects can be corrected by carefully measuring the residual tilt, e.gwith a dual-axis tilt sensor placed above the upper attachment of the fibre and beneath the pendulum,and controlling the rotation axis to be along the vertical direction. Temperature gradients and magneticeffects are usually mitigated by multi-stage passive shielding. Changes in the balance spinning frequency ω s are another source of systematic errors. A spurious signal would be proportional to the component ofthe rate variation at the Fourier frequency ω s . As the corresponding torque scales as ω s , the effect canbe measured by operating the torsion pendulum at different spinning frequencies, and can be mitigatedby choosing the lowest spinning rate compatible with the technical noise floor.A modern variant of Galileo’s simple pendulum WEP test in the field of the Sun has been proposed44ecently, [287]. The experiment is based on a differential accelerometer with zero baseline, measuringthe relative acceleration of two test masses of different materials suspended on a pendulum. Ensuringa precise centering of the test masses the system should provide a high degree of attenuation of thelocal seismic noise. With a cryogenic differential accelerometer under vacuum, the experiment shouldprovide a WEP test with 10 − precision. Unlike in torsion balances, mass drop WEP tests are done by leaving two test masses in free fall at thesame time, and measuring their relative displacement as a signature of differential acceleration. Thismethod was never used in high-precision experiments until the late 1980s. The legendary UFF testby Galileo was indeed never done by dropping masses from the Pisa leaning tower, but rather usingpendulumsFree fall experiments in evacuated tubes, which are nowadays popular in science teaching and out-reach, date back to the 17th century, when Boyle performed free fall tests with feathers or pieces ofpaper [288]. Similar experiments with coins and feathers are reported in 1717, when Desaguliers demon-strated the UFF to King George I and to the Royal Society led by Newton [289]. In 1971 AstronautScott performed a UFF test on the Moon during the Apollo 15 mission, by dropping an Aluminumhammer and a falcon feather from a height of about 1.6 m and observing them hit the ground simulta-neously [290].A revival of UFF tests with freely falling test masses occurred during the 1980’s, in the attemptto improve E¨otv¨os’ WEP tests [291]. After Dicke’s and Braginsky’s experiments it was clear thatsubstantial progress required a rotation of the torsion balance; however the control of systematic effectsin rotating the setup in the laboratory, that was necessary for tests in the field of the Earth, wasconsidered extremely challenging. On the contrary, the recent progress in laser interferometry rangingmade mass drop tests more attractive. Moreover, free fall tests have the advantage of a much higherdriving acceleration ( g ≃ . in mass drop tests versus g < . · − m/s on the torsion balance).A first precision mass drop WEP test was performed by Worden in 1982 at the 10 − uncertainty level,with a test mass constrained to 1D motion by means of a magnetic bearing [292]. This result wasimproved in 1984 by Sakuma with an accuracy of 1 × − [293]. In 1986, by observing the rotation ofa freely falling disc made of two halves of different materials, Cavasinni et al. confirmed the validityof WEP with 1 × − accuracy [294]. A high-precision mass drop experiment with two separate testmasses was performed for the first time in 1987 by Niebauer et al. [295]. Measuring the position of freelyfalling Uranium and Copper test masses with an interferometer, they proved the WEP up to 5 × − .The same experiment with different materials was repeated by Kuroda and Mio in 1989, reaching anaccuracy of 1 × − [296, 297]. The disc experiment of [294] was repeated in 1992 by Carusotto et al.with Aluminum and Copper, reaching an accuracy of 7 . × − [298, 299].So far, the precision of drop tests was limited to a few parts in 10 , in spite of the 600-fold largerdriving signal strength compared to torsion balances. The improvements over E¨otv¨os’ result was limitedto one order of magnitude, in contrast to the much more sophisticated technologies employed. Morerecently this has been overwhelmed by the E¨ot-Wash rotating balance. As discussed in [295, 298, 299]the main limitations of mass drop experiments are from errors in initial conditions at release couplingto the gravity gradient of the Earth to provide a differential acceleration error that mimics a violationsignal. In experiments with separate test masses, a laser interferometer tacks the differential trajectory δx ( t ) = δx + δv t (1 + γ v t /
6) + γ h t / δX , δv , γ v , γ h represent the initial differential displacement, initial differential velocity, verticalgravity gradient, and horizontal gravity gradient. The effect of vertical gradient is partly removed byfitting the measured trajectory with Eq. 97. Further mitigation of the systematic error from vertical45elocity differences is obtained by alternating the order in which the objects are dropped. Errors arisingfrom the horizontal gravity gradient are mitigated by alternating the position of the two masses. Forexperiments with disk test masses, a major systematic error arises from the disk precession aroundits angular momentum. The effect can be partly corrected by measuring the two components of theangular momentum in the disk plane just after the release.A way to improve drop tests is by increasing the free fall time, since the effect of a violation increasesquadratically with time. Experiments on balloons [300] and sounding rockets [301] have been proposed,which would allow a free fall time of several tens of seconds. The effect of gravity gradients, whichwould be a major source of systematic errors, can be separated from potential WEP violation signalsby spinning the system around a horizontal axis: in such way the signal from gravity gradient appearsat twice the rotation frequency while a violation signal would be at the rotation frequency.In principle there is still more potential in mass drop experiments, especially with a longer free falltime and sensors with a higher resolution. Similar tests are possible on ground in facilities such asthe drop tower of the ZARM center at the University of Bremen, where a free fall time of 4.74 s isachievable (9.3 s using the catapult). A mass drop WEP test with 10 − accuracy was performed in 2001with highly sensitive SQUID sensors [302]. In free fall experiments the control of starting conditions forpositions and velocities of the test masses is crucial. An Electrostatic Positioning System (EPS) wasdeveloped to this purpose [303]. For optimal conditions an accuracy of 10 is expected with this setup.Another free fall experiment is the project Principle Of the Equivalence Measurement (POEM) atthe Harvard- Smithsonian Center for Astrophysics [304, 305]. Two test masses in 0.5 m distance in aco-moving vacuum chamber were bounced on a kind of trampoline 0.9 m up and down several times.To test the WEP, the shifting between the test masses is measured. In principle, with a time averageover several bounces a sensitivity of 5 × − can be reached and an improvement on ground up to1 × − is possible.Free fall experiments with microscopic test masses enable in principle a much better control oversystematic effects. They are discussed in section 3.3. Testing the Weak Equivalence Principle in a ground based laboratory has some obvious limitations thatcan be overcome by going to space.As discussed in the previous sections, experiments with freely falling masses exploit the full grav-itational acceleration of the Earth to maximize the strength of a WEP violation; unfortunately, theirsensitivity is hampered by the short measurement time achievable in a ground-based laboratory andtheir accuracy is limited by the poor control of systematic effects depending on the initial conditions(position and velocity) of the test masses. On the contrary, experiments based on masses suspendedon a torsion balance allow for a long integration time and provide a much better control of systematiceffects, but their sensitivity is limited by the driving gravitational signal being three orders of magnitudelower than the Earth gravity acceleration.The next evolutionary step of these instruments is clearly space. The laboratory inside a freelyfalling spacecraft is indeed the ideal environment to push WEP tests to their ultimate limits.Under weightlessness conditions, the classical free fall experiment can still benefit from a WEPviolating signal proportional to the local acceleration of gravity (at the spacecraft height). More impor-tantly, the experiment can now be executed in a compact apparatus, with the test masses still in thespacecraft reference frame, where their relative motion can be observed over a long and unperturbedmeasurement time. The reduced volume of the instrument, compared to the much larger drop towers orfree-fall capsules used on Earth, allows to better control the experiment against external perturbations.A torsion balance in space would as well benefit from the full gravitational acceleration as a drivingsignal, gaining almost three orders of magnitude on the effect to be measured with respect to an Earth-46ased experiment having the same sensitivity to differential accelerations.Concentric test masses are the common denominator of all the instruments proposed for a space testof the Weak Equivalence Principle. This is the case for the space missions that will be described in thissection, MICROSCOPE, STEP and GG, which is the natural evolution of a torsion balance for space.Such a design is possible in space because of the extremely small coupling forces needed to control themasses position under weightlessness conditions. Small coupling constants directly translate into higherinstrument sensitivity to differential accelerations and therefore to WEP violations.More importantly, a spinning spacecraft can be used to introduce a modulation of the differentialacceleration resulting from a WEP violation, both for a free-fall and a torsion balance experiment, todistinguish the WEP violating signal from other effects appearing at different frequencies. In this case,as the platform is rotating with the instrument itself, the mass distribution in the immediate vicinityof the test bodies does not introduce any signal modulation as soon as there are no moving parts orchanges in the mass distribution of the spacecraft.Still, gravity gradients remain one of the predominant sources of systematic error imposing an ad-hocdesign of the experimental setup and the test masses. Test masses of different shape couple differently togravity gradients due their different multipole moments. This effect produces a differential accelerationcompeting with a violation of the Weak Equivalence Principle. Test masses shall therefore be designedto approach the shape of a gravitational monopole or to have matching gravitational multipole moments[306]. As a consequence, manufacturing processes shall ensure precise control on the shape of the massesand the material itself shall be selected to be highly homogeneous and easily machinable [307]. On theother hand, the gravity environment generated by the spacecraft surrounding the test masses, andthus primarily interacting with them, can also be controlled. As already demonstrated by the LISAPathfinder mission, a protocol-based measurement of the mass and the distance of all satellite partscan ensure a balance of the gravitational accelerations at the sub nms − level [308]. Such techniqueshave proven to be very effective in reducing the systematic errors introduced by gravity gradients.Differential accelerometers based on a nested test mass design are also affected by the radiometereffect. The infrared radiation of the Earth is absorbed by the satellite and consequently by the instru-ment housing, thus producing a temperature gradient that depends on the satellite’s orientation withrespect to the Earth-to-satellite direction. Due to the residual gas around the test masses, this temper-ature gradient is responsible for a differential acceleration that is directly proportional to the pressureof the residual gas and to the temperature gradient and that cannot be distinguished from a WEPviolating signal. The thermal design of the spacecraft and the instrument head as well as the design ofthe vacuum system enclosing the test masses is therefore important to minimize this effect. In STEP,where cryogenic temperatures are reached in a He dewar, the residual gas pressure and the temperaturegradients can be better controlled. In [309] the radiometer effect is calculated for MICROSCOPE,STEP and GG and discussed with respect to the specific design of the three instruments.In space, the test masses of the differential acceleration sensor can accumulate charges due to theinteraction with high energy charged particles travelling through the solar system. In the presence ofstray charges, the source mass interacts with the caging mechanism and readout system via Coulombforces introducing noise and bias on its position and on the measurement signal readout. Differentmethods can be used to discharge the masses and counteract this effect. MICROSCOPE stray chargesare managed via a thin (0.7 µ m) gold wire connected to a sole plate and driven by a control voltage [307].A different discharging system has been demonstrated in space by the LISA Pathfinder mission. In thiscase, an ultra-violet lamp illuminating both the test masses and the surrounding environment is usedto generate a current of photoelectrons [310] that can be tuned to null the charge of the test mass itself.In this way, the corresponding noise and bias can be reduced to negligible levels.Finally, external perturbations can be accurately controlled in space. Stabilization loops can beimplemented to reduce temperature fluctuation at the instrument head below 100 µ K [311]. TheNewtonian noise, generated by fluctuations of terrestrial gravity and representing one of the most47mportant limitations of ground-based tests of the Weak Equivalence Principle, is totally absent on aspacecraft. In space, other perturbations, such as air drag or solar radiation pressure, can introducenoise and bias affecting the WEP test. However, several drag free systems have already demonstratedtheir ability to reduce residual accelerations below 3 × − ms − Hz − / , as in the MICROSCOPEmission [311], or even better, down in the 10 − ms − Hz − / regime, as in LISA Pathfinder [312, 313].The MICROSCOPE (MICROSatellite `a train´ee Compens´ee pour l’Observation du Principe d’Equiv-alence) mission provides the most accurate test of the Weak Equivalence Principle [307, 311]. Thesatellite was launched from Kourou on 25 April 2016 on a Soyuz rocket and injected into a dawn-duskSun-synchronous orbit with an altitude of 710 km. The spacecraft embarks two differential accelerom-eters, each of them based on two hollow cylinders. They are aligned along the symmetry axis, preciselycentered, and kept in their equilibrium position by capacitive electrodes. The two differential accelerom-eters only vary for the test masses composition: Pt:Rh alloy for both cylinders of the reference sensorunit (SUREF); Pt:Rd and Ti:Al:V for the inner and outer test mass of the unit sensitive to WEPviolations (SUEP). The test masses have been precisely machined to a relative difference between themomenta of inertia smaller than 10 − and a density homogeneity better than 0.1%, thus reducing differ-ential accelerations due to gravity gradients to negligible levels [307]. The same set of electrodes providesmeasurement and control of both the position and the attitude of the test masses. They are machinedon a silica substrate to ensure high position stability. The voltage applied at the electrodes, which isproportional to the force exerted on the test masses to keep them centered, represents the main dataoutput of the instrument from which the differential acceleration between the test masses is extracted.Once the test masses are correctly aligned, if General Relativity holds, zero differential accelerationat both the SUREF and SUEP sensor heads shall be read. The magnetic environment is controlledby a magnetic shield surrounding the complete payload and modelled by a finite element calculation.The tight housing allows the sensors to operate in the 10 − Pa regime, where the radiometer effect isstrongly reduced. Radiometer effect and radiation pressure disturbances are kept below the dampingintroduced by the 7 µ m wire connecting the test masses and the cage to control electrical chargingeffects [314]. Cold gas thrusters actuated by the accelerometers’ measurements reduce the effect of airdrag and, more generally, of non-gravitational forces acting on the spacecraft. The drag-free controlsystem relies on the linear and angular accelerations measured at one of the test masses. Residualaccelerations below 3 × − ms − Hz − / could be measured in closed-loop configuration. This result isabout a factor 10 better than originally specified. Star tracker measurements are also used to determinethe spacecraft attitude. To increase the modulation frequency of the gravitational signal provided bythe Earth, the satellite is rotated ( ∼ . × − ms − Hz − / and 1 . × − ms − Hz − / , respectively, at the modulation frequency of a few mHz expected for theWEP violation. This floor level is limited by the damping noise of the thin gold wire connected to thetest masses to control charging effects. Systematic errors are currently dominated by thermal effects,which could be evaluated to < × − ms − . The instrument sensitivity was determined by applyingtemperature variations both at the electronics and at its baseplate. Measurements revealed that thesensor unit temperature coefficient is 2 orders of magnitude higher than expected. This issue is stillunder investigation. On the positive side, the temperature stability of the instrument baseplate and theelectronics was measured to be better than 20 µ K over 120 orbits, about 2 orders of magnitude smallerthan initial estimates, thus mitigating temperature-related effects. The contribution of self-gravity andstray magnetic fields to the measurement error was estimated from finite element models and foundnegligible. After analyzing the data corresponding to 120 satellite orbits, an E¨otv¨os parameter of[ − ± ± × − could be estimated for titanium and platinum [311], improving by oneorder of magnitude previous results obtained from torsion balance [277] and lunar laser ranging [315]experiments. This measurement also establishes new constraints to modifications of the Newton’s law48f gravity by a Yukawa-like coupling and improves existing constraints on WEP violations by a lightscalar field [316]. The MICROSCOPE mission has been decommissioned on 18 October 2018, afteraccumulating about 1900 orbits of science data on the SUEP sensor, 900 on the SUREF sensor, and 300orbits for calibration. This also includes 750 orbits of measurements for characterizing the on-boardtemperature sensors and further reduce the systematic effects due to temperature variations. After thefirst results reported in 2017, the complete data set delivered by the MICROSCOPE mission is stillunder scrutiny to improve both the statistical and systematic error on the WEP test, hopefully goingbelow the 1 × − accuracy level.The STEP (Satellite Test of the Equivalence Principle) mission concept is similar to MICROSCOPE,but it relies on a completely different technology, which is expected to push the accuracy of WEP testsdown to 1 part in 10 [317, 318]. The STEP mission was selected for a phase A study in 1990. Anengineering model of the accelerometer to test the technology was built in 2004. The payload is com-posed of 4 differential accelerometers (DAs) operating simultaneously with the following combinationof test masses: Be and Pt:Ir for DA1; Be and Nb for DA2; Nb and Pt:Ir for DA3; Be and Pt:Ir forDA4. DA1 to DA3 measurements allow to check that the sum of the differential acceleration measuredat the 3 sensor heads between the three materials - Be, Pt:Ir, and Nb - is zero thus providing control onmeasurements systematics. DA1 and DA4 differ for the shape of the test masses and for their couplingto Earth and spacecraft gravity. As for MICROSCOPE, the differential accelerometers are composed oftwo hollow masses with cylindrical symmetry, precisely centred and aligned along their axis. STEP DAsare arranged in a helium dewar and operated at 2 K. The cryogenic environment is providing very goodthermal and mechanical stability for the DAs operation, ultra-high vacuum and reduced thermal noisefrom gas damping, excellent shielding from external magnetic fields, reduced radiation pressure effectsdue to temperature gradients. More importantly, it allows to use SQUID technology for high-sensitivityposition readout and for generating the weak reaction force that centers the test masses along the axialdirection. The gas generated by boiling helium is used by thrusters to stabilize the spacecraft againstnon gravitational accelerations. In addition, when in drag-free, common mode accelerations can bemeasured with respect to the spacecraft reference frame to 10 − − − ms − , well below the MOND(MOdified Newtonian Dynamics) acceleration scale a . Based on this performance, STEP has recentlybeen proposed for a test of MOND theories and of the Strong Equivalence Principle [319]. As discussedin section 3.1.2, alternative WEP tests in microgravity are also possible on ground laboratories such asthe Bremen drop tower [302].Galileo Galilei (GG) is an alternative proposal designed to test the Einstein WEP to better than1 part in 10 [320, 321]. Differently from MICROSCOPE and STEP, GG can be considered as thespace version of a beam balance. The test masses are two hollow cylinders of different composition thatare weakly coupled by means of mechanical suspensions. Once properly set into equilibrium by piezoactuators, the beam of the balance is aligned along the symmetry axis of the cylinders, thus defining theplane orthogonal to this direction as the sensitive plane of the instrument. This configuration providesa rejection of common mode acceleration noise as high as 10 , thus drastically relaxing the level of dragcontrol required at the spacecraft. Rapid rotation of the instrument is important to reduce the thermalnoise in the detection of WEP violations and to efficiently decouple it from systematic effects appearingat different frequencies. In GG, the spin axis of the spacecraft coincides with the symmetry axis of theinstrument. Therefore, after initial spin up, the spacecraft co-rotates with the cylindrical test massesaround the principal axis of the system and it is passively stabilized to very fast rotation rates ( ∼ ∼ × − ms − (at1 . × − Hz upconverted by rotation to 0.2 Hz), currently limited by Newtonian noise, mainly tilt,acting on the ball bearings [322]. An optimized design based on low noise air bearings, low couplingjoints, and a laser interferometer readout system is under study to push the instrument performancedown to the 10 − − − regime. Lunar Laser Ranging (LLR) experiments are performed since 1969, when the first array of cornercube reflectors was positioned on the Moon by Apollo 11. A review of LLR tests of gravity con befound in [323]. To date, 5 arrays of retro-reflectors are operational on the Moon surface and routinelyused for ranging experiments: Apollo 11, 14 and 15, Lunokhod 1 and 2. Among them, Apollo 15is the one with the largest lidar cross section and therefore the most widely used for LLR (about75% of normal point data). In a LLR measurement, a short laser pulse is fired by a ground-basedSatellite Laser Ranging (SLR) station towards one of the Moon corner cube reflector arrays. The backreflection is collected by the SLR station and the interval between the fire time and the reception time isrecorded. Round-trip travel time measurements are then fitted to a model of the solar system ephemerisincluding tidal effects, relativistic effects, propagation in the atmosphere, plate motion, etc. The SLRstations mostly contributing to LLR data are the Observatoire de la Cˆote dAzur (OCA) in France,the McDonald Laser Ranging System (MLRS) and the Apache Point Observatory Lunar Laser-rangingOperation (APOLLO), both in the US. Thanks to the 3.5 m diameter telescope and to the array ofhigh-efficiency avalanche detectors, the APOLLO station has today reached millimeter ranging precisionand accuracy to the Moon [324–326]. To this level, effects like regolith motion, thermal expansion ofthe retro-reflectors array, oceans and atmosphere loading effects start to become relevant.Earth and Moon are two celestial bodies freely falling in the gravitational field of the Sun (primarybody). If the Universality of Free Fall principle is violated, their accelerations towards the Sun aredifferent, thus introducing a polarization of the lunar orbit [327]. This effect manifests itself with theappearance of a modulation of the Earth-Moon distance (LLR measurements) along the Earth-Sundirection at the synodic period (29.53 day). For a relative differential acceleration between the Earthand the Moon of ∆ a/a , the perturbation δr to the Earth-Moon distance expressed in meters is givenby [328] δr = − . × ∆ aa cos D [m] , (98)where D is the synodic angle. After the first LLR test of the Equivalence Principle in 1976 [329, 330],the accuracy of relative differential acceleration measurements ∆ a/a between Earth and Moon hasprogressively improved to 1 . × − [315,331,332], to recently reach 7 × − [326] and 5 × − [333].Such results could be obtained after modelling the effects of the gravitational interaction of the Sunand the planets on the Moon, now treated as an extended body. High order terms of the Earth-Moongravitational interaction and the effect of solid Earth tides on the Moon orbit were also improved.Celestial bodies have non negligible gravitational self-energy. This was already clear in 1968 [327],when Nordtvedt proposed to use the Earth-Moon system to test the Strong Equivalence Principle. Inthis case, the relative differential acceleration responsible for a violation of the Universality of Free Fallprinciple can be expressed as ∆ aa = η CD + η SEP (cid:18) U E M E c − U M M M c (cid:19) , (99)where η CD is the composition-dependent violation parameter, η SEP is the Nordtvedt parameter mea-suring SEP violations, U and M represent the gravitational self-energy of the test body and its mass.50herefore, to exclude any cancellation effect between a composition-dependent WEP violation and anequal and opposite SEP violation, an independent test of the Weak Equivalence Principle based on testmasses having similar composition to the Earth and Moon interior, but with negligible gravitationalself-energy is required. The experiment, performed in 1999 with the torsion balance apparatus of theWashington group, confirmed the validity of WEP for two test bodies reproducing the Earth and theMoon composition to 1 . × − [245, 282]. Combined with with LLR measurements, this test can beused to constrain SEP violations. The best estimate of the Nordtvedt parameter based on laser rangingmeasurements of the Earth-Moon system has reached an uncertainty of 1 . × − [333].The Universality of Free Fall can also be tested by tracking satellites orbiting around the Earth,e.g. LAGEOS, LAGEOS II and LARES [334]. As also discussed in [335], the sensitivity of a test basedon the Earth-LAGEOS system in the gravitational field of the Sun is a factor 300 worse than for theEarth-Moon system. Indeed, LAGEOS and LARES satellites are much closer to the Earth comparedto the Moon. As a consequence, the effects of the Sun gravitational potential on the Moon orbit aresignificantly stronger than for a satellite orbiting the Earth at low altitude. Even if not competitivewith LLR experiments, SLR tests still remain of interest to evaluate the impact of different systematicerrors in the final result. As an example, non-gravitational perturbations, which play a major role inthe determination of the LAGEOS and LARES orbits, are completely negligible for the Moon. On thecontrary, some gravitational perturbations (tidal effects) are important for the Moon and negligible foran Earth-orbiting satellite. Finally, SLR measurements could also be combined with LLR measurementsin a grand-fit procedure to better estimate common parameters thus improving LLR and interplanetaryranging [336].Similar tests can be performed by ranging other gravitating bodies in the solar system. The MES-SENGER mission with its Doppler tracking measurements collected over 7 years allows the precisedetermination of Mercury’s ephemeris. This wealth of data has recently been used to test the StrongEquivalence Principle with reduced uncertainty [337]. Spacecraft and planet orbits are numericallyintegrated to provide a global solution from which parameters relevant for General Relativity, planetaryphysics, and heliophysics can be extracted. This analysis is today constraining the Nordtvedt parameterto 7 × − . This section is devoted to a review of the tests of the WEP with microscopic particles, mainly atoms.Based on recent advances in cold atom optics, atomic sensors, namely atom interferometers [338,339]and atomic clocks [340, 341], established themselves as new powerful tools for precision measurementsand fundamental tests in physics [342].Atom interferometry enabled the realization of precision tests of the WEP that were previouslyperformed only with macroscopic classical masses. As will be clear from the data reported in thisreview, the sensitivity of atomic experiments did not reach yet the one of the classical experiments butpredictions are that similar or even higher levels of sensitivity will be obtained both in Earth laboratoriesand in experiments in space.An important advantage of using atoms is, in a properly designed apparatus, the control of possiblesystematics thanks to the well known and reproducible properties of the atoms, the possibility ofrealizing an atomic probe of extremely small size and precisely controlling its position, the potentialimmunity from stray field effects, and the availability of different states and different isotopes that insome cases allows the rejection of common-mode spurious effects and/or a cross-check of the results.Perhaps still more important is that new kinds of tests are possible that exploit the specific quantumfeatures of atomic probes: qualitatively new experiments can be performed with test masses havingwell-defined properties in terms of, e.g., proton and neutron number, spin, internal quantum state,51osonic or fermionic nature.In the final part of this section, tests with neutrons, with charged particles, and with anti-matterare also described because of their fundamental interest but the precision achieved so far in these casesis still much lower compared to the other tests.It can be expected that in the future the development of matter-wave interferometry with moleculeswill enable also the comparison of the free fall for such systems with different conformations, differentinternal states, different chiralities; this will not be discussed in the present review because sensitivitiesare still too low to be significant in this context but preliminary results and a discussion of futureprospects can be found in [343].It should be noticed that all experiments so far were performed with systems consisting of particlesof the first elementary particle family and that direct tests for particles of the second and the thirdfamilies are missing until now (see, e.g., [344] and references therein).
The idea of an atom interferometer can be easily understood from the analogy with an optical interfer-ometer: using suitable atom optics made of material structures or, more often, laser light, an atomicwave packet is split, reflected and recombined: at the output, interference can be observed. In a moregeneral view, it can be considered as a quantum interference effect arising from the different pathsconnecting the initial and final states of a system. Any effect affecting in a different way the differentpaths will produce a change in the interference pattern at the output; by detecting this change, theeffect can be measured.In most experiments, the best performances have been achieved using atom interferometry schemesin which the wavepackets of freely falling samples of cold atoms are split and recombined using laserpulses in a Raman [345, 346] or Bragg [347, 348] configuration. The effect of gravity leads to a phasechange ∆ φ = kgT where k is the effective wave-vector of the light used to split and recombine thewave packet, g is gravity acceleration, and T is the time of free-fall of the atom between the laser pulses.This corresponds indeed to the free-fall distance measured in terms of the laser wavelength.Other schemes were developed to measure g based on Bloch oscillations ( [349] and referencestherein). In this case, the atoms are not falling freely under the effect of gravity but the combinedeffect of gravity and the periodical potential produced by the laser standing wave leads to oscillationsin momentum space with a frequency ν BO = mgλ/ h , where m is the atomic mass, λ is the wavelengthof the laser producing the lattice and h is Planck’s constant. By measuring the frequency of the Blochoscillation ν BO , g is determined. This method can also be interpreted as the measurement of thegravitational potential difference between adjacent lattice wells which are separated by λ/
2. A fewwells are filled with ultracold atoms so that the gravimeter has a sub-millimiter size down to a fewmicrometers. For this reason, it was also proposed as a method to test the 1 /r Newtonian law forgravity at micrometric distances [349, 350].Atom interferometers enabled precise measurements of several physical effects; in particular, ingravitational physics, the measurement of gravity acceleration [346, 351–358], gravity gradient [355,359–364] and curvature [365, 366], determination of the gravitational constant G [259, 360, 367–372],investigation of gravity at microscopic distances [349, 350, 373], search for dark energy, chameleon andtheories of modified gravity [374–376], applications to geodesy, geophysics, engineering prospecting,inertial navigation [377–379].In [380], experiments measuring g with different atom interferometry methods [349, 351, 381] werereinterpreted as measurements of the Einstein’s gravitational redshift, thus claiming an improvementin precision by 4 orders of magnitude with respect to the Gravity Probe A test reported in [382]. Thispaper started a controversy on such an interpretation and on the nature of the phase shift measurementin an atom interferometer [255, 380, 383–390]. 52tom interferometers and optical atomic clocks were proposed for the detection of gravitationalwaves on ground and in space [391–400] and the first prototypes are presently under construction[401–403].Experiments in space based on cold atom sensors were proposed since long [404–407], the requiredtechnology development is in progress [408], and proof-of-principle experiments were recently performed[409].After the early observation of free fall of atoms using long beams of potassium and cesium atoms[410], several experiments have compared the free fall of different atoms to test the WEP, as describedin detail in the following: Rb vs Rb [411–413], K vs Rb [414], the bosonic Sr vs the fermionic Sr [415], atoms in different spin orientations [415, 416]. The relative accuracy of these measure-ments, reaching so far 10 − - 10 − , is expected to improve by several orders of magnitude in the nearfuture thanks to the rapid progress of atom-optical elements based on multi-photon momentum trans-fer [417, 418] and of large-scale facilities providing a few seconds of free fall during the interferometersequence [419–421]. Experiments testing the free fall of anti-hydrogen are in progress [422–424]. WEPtests with a precision ∼ − using atom interferometers in space were proposed [404–406].Concerns have been raised on the potential of atom interferometry for high precision tests of theWEP [425]. A scheme to overcome these possible limitations was proposed in [426] and experimentallydemonstrated in [427, 428]. In different experiments, gravity acceleration for atoms has been compared with the one for macroscopicmasses.Already in the first demonstration of a high-precision atom interferometry gravimeter with Cs atoms,a classical gravimeter based on a Michelson optical interferometer with a vertical arm containing afreely falling corner-cube was used for comparison. The atom gravimeter was realized with a Ramaninterferometry scheme. An uncertainty of ∆ g/g = 3 × − was achieved with a free-fall time 2 T =320 ms. The comparison between the two gravimeters was interpreted as the demonstration that themacroscopic glass mirror falls with the same acceleration, to within 7 parts in 10 , as the quantum-mechanical Cs atom [351].In [260,429], comparisons between a mobile Raman atom gravimeter with Rb and classical absolutegravimeters were performed with comparable uncertainties.A conceptually different scheme was used in [430]. The experiment was based on Bloch oscillationsof Sr atoms in a vertical optical lattice. In order to increase the sensitivity, in this work a methodto measure the frequency of higher harmonics of the Bloch frequency was adopted. The value of theacceleration measured with this atomic sensor was compared with the one obtained in the same labwith a classical FG5 gravimeter. The two values agreed within 140 ppb.
Experiments were performed testing WEP for different isotopes of an atomic species. Compared tothe experiments discussed in the following in which different atomic species are compared, these aresomehow simpler; the similar masses and nearby transition frequencies make the apparatus and thecontrol of systematics less complex.In [411], an atom interferometer based on the diffraction of atoms from standing optical wavesacting as effective absorption gratings was used to compare the two stable isotopes of rubidium, Rband Rb, with a relative accuracy of 1 . × − . In this work, a test for a possible difference of the freefall acceleration as a function of relative orientation of nuclear and electron spin was also performedwith a differential accuracy of 1 . × − by comparing interference patterns for Rb atoms in two53ifferent hyperfine ground states (see sect. 3.3.5). A comparable precision for the differential free fallmeasurement of Rb and Rb was later obtained in [412] using Raman atom interferometry.About one order of magnitude improvement in precision was obtained in [413]. A four-wave double-diffraction Raman transition scheme was used for the simultaneous dual-species atom interferometer tocompare Rb and Rb. The value obtained for the E¨otv¨os parameter is η = (2 . ± . × − ).Ongoing experiments in large baseline interferometers aim to a final precision in the 10 − rangeand beyond [419–421]. Limiting factors due to the gravity gradients were discussed [425] and possiblesolutions were proposed in [426] and demonstrated in [427, 428]. In [428] the gravity gradient compen-sation in a long duration and large momentum transfer dual-species interferometer with Rb and Rballowed to reach a relative precision of ∆ g/g ≈ × − / shot or 3 × − / √ Hz that makes such aWEP test realistically feasible at the 10 − level.In [415], the WEP was tested for the bosonic and fermionic isotopes of strontium atoms, namely, Sr and Sr. As in [430], gravity acceleration for the two isotopes was determined by measuringthe frequency of the Bloch oscillations for the atoms in a vertical optical lattice. By detecting thecoherent delocalization of matter waves induced by an amplitude modulation of the lattice potential ata frequency corresponding to a multiple of the Bloch frequency, the limit obtained in this work for theE¨otv¨os parameter is η = (0 . ± . × − ). As discussed in the following, the results of this experimentare also relevant as a WEP test for bosons vs fermions and for the search of spin-gravity coupling. Recently, experiments were performed to test WEP with different atomic species. This requires thedevelopment of more complex experimental setups and a more difficult control of systematics. Thetheoretical framework to interpret the experimental results can be found in Refs. [431–433].The possibility of a test using rubidium and potassium atoms was discussed in [434]. The first resultswere reported in [414]; in this work, Rb and K were compared using two Raman interferometers.The result was an E¨otv¨os ratio η = (0 . ± . × − ) mainly limited by the quadratic Zeeman effect andthe wave front curvature of the Raman beams. The choice of atomic species in this paper was comparedwith others in terms of sensitivity to possible violations of the EP predicted by a dilaton model [431]and by standard-model extensions [432].The ongoing activity for a test with rubidium and ytterbium atoms in a 10-m baseline atom inter-ferometer was discussed in [433] with the goal to reach an E¨otv¨os ratio in the 10 − − − range. While it can be argued that some of the experiments with atoms described above are not qualitativelydifferent from the ones performed with macroscopic classical systems as far as the physics which is testedis concerned, the experiments described in this section take full advantage of the quantum nature ofthe atoms as probes of the gravitational interaction. • Atoms in different energy eigenstates and in superposition statesThe mass-energy relation E = mc in special relativity implies that the internal energy of asystem affects its mass. It is then of interest to verify the validity of the equivalence of the inertialand gravitational mass for systems in different internal quantum states. This was theoreticallydiscussed in [435, 436] and possible experimental tests with atoms in different internal states wereproposed. In particular, the importance of tests involving atoms in superpositions of the internalenergy eigenstates was highlighted because this corresponds to a genuine quantum test. Anotherpossible experimental test of the quantum formulation of the equivalence principle was proposedin [437]. 54 first experimental test of the equivalence principle in this quantum formulation was reportedin [438]. A Bragg atom interferometer was used to compare the free fall of Rb atoms preparedin two hyperfine states | i = | F = 1 , m F = 0 i and | i = | F = 2 , m F = 0 i , and in their coherentsuperposition | s i = ( | i + e iγ | i ) / √
2. In order to increase the measurement sensitivity, theatom interferometer was operated at the 3rd Bragg diffraction order, corresponding to 6 ~ k totalmomentum transfer between the atoms and the radiation field.The comparison of the free-fall acceleration for atoms in the | i and | s i states led to the firstexperimental upper bound of 5 × − for the parameter corresponding to a violation of the WEPfor a quantum superposition state.Based on models in which WEP violations increase with the energy difference between the internallevels [431], in this paper the prospect to use states with an energy separation larger than thehyperfine splitting was also proposed considering optically separated levels in strontium for whichthe relevant atom interferometry schemes were already demonstrated [439–441].The comparison of gravity acceleration for atoms in the | i and | i hyperfine states led to anE¨otv¨os ratio η − = (1 . ± . × − ) that corresponds to an improvement by about two ordersof magnitude with respect to the previous limit set in [411]. A further improvement by a factorof 5 in the precision of this test was recently reported in [442], approaching the 10 − level. • Atoms in entangled statesIn [443] a quantum test of the WEP with entangled atoms was proposed. In the proposed ex-periment, a measurement of the differential gravity acceleration between the two atomic specieswould be performed by entangling two atom interferometers operating on the two species. Theexample of Rb and Rb was analyzed in detail showing that an accuracy better than 10 − onthe E¨otv¨os parameter can be achieved.Although no theoretical model is available predicting a WEP violation in the presence of entan-glement, this is clearly a case of a purely quantum system to be further investigated.The free fall of particles in quantum states without a classical analogue and in particular forSchr¨odinger cat states in the configuration space was studied theoretically in [444]. • Atoms in different spin statesAs discussed above (see in particular Sect. 2.13), spin-gravity coupling and torsion of space-timewere extensively investigated theoretically [6, 445, 446].Different experiments were performed using macroscopic test masses [284, 286, 446], atomic mag-netometers [447, 448], and by measuring hyperfine resonances in trapped ions [449]. The dif-ferential free-fall experiments with atoms in different hyperfine states are also relevant in thisframe [411, 438, 442].Recently, experiments were performed using atom interferometry to search for the coupling of theatomic spin with gravity.In [415], the experimental comparison of the gravitational interaction for the bosonic isotope ofstrontium Sr, which has zero total spin in its ground state, with that of the fermionic isotope Sr, which has a half-integer nuclear spin I = 9 /
2, was performed based on the measurement ofthe frequency of Bloch oscillations for the atoms in a vertical optical lattice under the effect ofEarth’s gravity.A modified gravitational potential including a possible violation of WEP and the presence of aspin-dependent gravitational mass was considered in the form V g,A ( z ) = (1 + β A + kS z ) m A gz ,where m A is the rest mass of the atom, β A is the anomalous acceleration generated by a nonzerodifference between gravitational and inertial mass due to a coupling with a field with nonmetricinteraction with gravity, k is a model-dependent spin-gravity coupling strength, and S z is theprojection of the atomic spin along gravity direction.55s already described above, the Bloch frequency corresponds to the site-to-site energy differenceinduced by the gravitational interaction; by measuring the frequency of Bloch oscillations for Srand Sr an E¨otv¨os parameter (0 . ± . × − was obtained. Since the frequency of Blochoscillations depends on the mass of the particle, in the analysis of the data the m /m massratio was taken into account which is known with a relative precision ∼ − . The analysisof the Bloch resonance spectrum for Sr provided an upper limit for the spin-gravity couplingstrength k = (0 . ± . × − . This result also sets a bound for an anomalous acceleration anda spin-gravity coupling for the neutron either as a difference in the gravitational mass dependingon the spin direction or as a coupling to a finite-range interaction [446, 447].In [416], a Mach-Zehnder-type Raman atom interferometer was used to compare the gravityacceleration of freely-falling Rb atoms in different Zeeman sublevels m F = +1 and m F = − . ± . × − . The data were also interpreted as providing an upperlimit of 5 . × − m − for a possible gradient field of the spacetime torsion.In [343], based on recent advances of matter-wave interferometry with large molecules, the prospectof a test of WEP for molecules with opposite chiralities was proposed.It should be noted that a complete analysis connecting theoretically the models tested in thedifferent experiments performed so far in this frame is still missing. • Atoms in a Bose-Einstein condensatePossible differences in the gravitational interaction for bosons and fermions were investigatedtheoretically [450] and tested experimentally [415].Violations of the WEP for atoms in a quantum state such as a Bose-Einstein condensate werediscussed in ( [451, 452] and references therein). Since in quantum physics particles are describedby an extended wave packet, the validity of the WEP which refers to point-like particles can bequestioned. A model based on spacetime fluctuations allows to predict a possible difference in theobserved free fall for different particles because the different spatial extensions of the wavefunctionof particles of different masses would lead to an averaging of the metric fluctuations over differentspatial volumes. Also, the metric fluctuations would produce decoherence.Such elusive effects, if ever observable, would require atom interferometers with extremely highsensitivity, that is, a very long evolution time. For this and other scientific goals, the technol-ogy needed to perform experiments in microgravity is being developed [303, 406, 409, 453, 454] asdescribed in detail in the following.
The ultimate performance of atomic sensors for WEP tests can be reached in a space-based laboratory.In space atoms can rely on a very quiet environment where Newtonian noise is absent and microvibra-tions and non-gravitational accelerations can be reduced to very low levels. Very long and unperturbedfree fall conditions can be obtained, allowing atomic wavepackets to evolve, sense the space-time metric,and record its signature in their phase. At the same time, very long and unperturbed interaction timesbetween the atomic ensemble and the interrogation fields can be achieved. This is translating into asignificant increase of the instrument sensitivity and a better control of the systematic errors.As an example, the phase accumulated in a Mach-Zehnder interferometer, ∆ φ = kgT , is directlyproportional to the square of the free evolution time T between the three laser pulses of the interfer-ometry sequence. The typical duration of an atom interferometry sequence on the ground is 2 T ≈ T = 10 s or longercan be achieved, improving the instrument sensitivity by a factor 100 or more with respect to a similarinstrument operated on the ground.Achieving a free evolution time of 2 T ≈
10 s on the ground would require an atomic fountainapparatus with several hundred meters of free-fall length, showing another important aspect of atom-based sensors designed for space compared to their laboratory counterpart, i.e. the compactness. Inspace, atoms interrogation can take place in a small vacuum chamber with a typical size of a few liters.This volume can be better controlled against external perturbations, such as temperature, magneticfields, etc. As an example, the development of large size mu-metal shields to accurately control theexternal magnetic field along the free evolution trajectories of a long atomic fountain (10 m or longer)remains a non negligible technology challenge.Finally, a technique to counteract the effect of gravity gradients has recently been developed [426]and experimentally demonstrated [427], reducing to a negligible level one important source of instabilityand systematic error in precision measurements by atom interferometry.STE-QUEST (Space-Time Explorer and QUantum Equivalence Space Test) is a mission designedto test different aspects of the Einstein Equivalence principle in space [406]. The STE-QUEST sci-entific objectives include an atom interferometry test of the Weak Equivalence Principle, an absolutemeasurement of the Einstein’s gravitational redshift, and tests of Standard Model Extension (SME).Here, we will only focus on the WEP test. The on-board instrument dedicated to this measurementis a differential atom interferometer. Originally designed to compare the free fall of the 85 and 87rubidium isotopes, the instrument has recently been re-adapted to operate on potassium and rubidiumthat, due to the larger difference in neutrons and protons, are expected to provide higher sensitivity inthe detection of a WEP violation [455]. The two atomic ensembles would be cooled down to very coldtemperatures (100 pK regime) and simultaneously interrogated in the atom interferometry sequence byusing the double-diffraction technique [456]. The simultaneous interrogation provides rejection ratio ofcommon mode acceleration noise (e.g. air drag and mechanical vibrations), which can vary from 10 − for Rb- Rb simultaneous interferometers [408] to < − for the Rb- K couple. The requirementson the control of non-gravitational accelerations acting on the spacecraft are therefore very modest fora Rb- Rb differential interferometer and significantly more stringent for the Rb- K one, but stillwell within the available technology as demonstrated in the MICROSCOPE [311] and LISA Pathfindermissions [313]. A design description of the STE-QUEST differential atom interferometer can be foundin [457]. The expected error budget is presented in [408]. The instrument will be able to measuredifferential accelerations down to 8 × − ms − corresponding to a WEP test at the 1 × − level.A similar instrument has also been proposed for a WEP test on the International Space Station(ISS) [458]. The ISS is a harsh environment for what concerns non gravitational accelerations, rotations,tilt noise, and mechanical vibrations. The instrument is therefore designed to ensure optimal control onsystematic errors and high rejection of common mode effects. The differential accelerometer comparesthe free fall acceleration of Rb and Rb atomic samples in a symmetric configuration with twoseparate source regions. Bragg lasers tuned to the wavelength for which the two rubidium isotopes havethe same polarizability are used to simultaneously interrogate the atomic samples in the interferometricsequence. This approach ensures a very high suppression of laser noise and common mode accelerationnoise. The instrument will be accommodated on a rotating platform to control gravity gradient effects.The SAGE (Space Atomic Gravity Explorer) mission proposal [407] has the scientific objective toinvestigate gravitational waves, dark matter, and other fundamental aspects of gravity such as the WEPas well as the connection between gravitational physics and quantum physics using optical atomic clocksand atom interferometers based on ultracold strontium atoms.Several experiments and test activities are currently in progress to demonstrate the maturity ofatom-based sensors for space operation and to evaluate the ultimate stability and accuracy that can bereached in differential acceleration measurements for WEP tests.57n 23 January 2017, the MAIUS-1 experiment was launched in a sounding rocket to a height of243 km. During the lift-off phase and the 360 min of free-fall conditions, 110 experiments involvingatoms cooling and manipulation were performed. They include laser cooling and trapping of atoms,observation of the BEC phase transition, BEC transport on the atom chip, and study of BEC collectiveoscillations under weightlessness conditions [409]. This experiment demonstrates the building blocks offuture atom interferometry experiments in space.The Cold Atom Lab (CAL) is a multiuser facility launched to the ISS in 21 May 2018. The CALinstrument is designed to produce ultracold atomic samples of Rb, and K [459] down to quantumdegeneracy. In the microgravity environment of the ISS, it is possible to decompress the atomic trapsto very low levels thus achieving ultra-low densities and picokelvin temperatures. The experiment willtest different atomic sources for atom interferometry in weightlessness conditions.Significant progress has also been achieved by making use of microgravity facilities available on theground, in particular the Bremen drop tower and the zero-gravity parabolic airplane.Mach-Zehnder interferometry experiments on a Bose-Einstein condensate have been performed inthe Bremen drop tower [460]. The drop tower capsule was operated both in drop and catapult mode,providing a free fall duration of 4.7 s and 9.4 s, respectively. The atom interferometer could thendemonstrate a shot-noise limited resolution of 6 . × − ms − in the drop mode and 5 . × − ms − in the catapult mode. With this performance, a sensitivity of a few parts in 10 for a WEP test shouldbe possible in less than 100 drops (also see [452]). Unfortunately, the study of systematic effects of aWEP test would result very unpractical in the drop tower facility.A WEP test on Rb and K has been performed in the microgravity conditions of an airplanein parabolic flight. The E¨otv¨os ratio was measured to 3 . × − , limited by the noisy accelerationenvironment (10 − g Hz − / ). This result, certainly not competitive with respect to other WEP tests,remains important as it demonstrates the possibility of using correlated interference fringes to performa WEP test with an accuracy two orders of magnitude below the level of ambient vibration noise. Theexperiment could therefore confirm the expected rejection to common mode vibration for a Rb- Kdifferential interferometer [434, 461].
As for the atoms, the first low-precision measurements of gravity acceleration for neutrons were per-formed by measuring the drop of collimated beams of thermal neutrons [462, 463]. They were alsointerpreted as tests of the universality of free fall. In [463], a test of a possible dependence of neutronacceleration on the two vertical neutron-spin projections ± was performed finding no difference withinthe experimental sensitivity.After the first observation of gravitationally induced interference in a neutron interferometer [464],tests of WEP were performed using neutron interferometers reaching a precision of 10 − [465, 466], anaccelerated interferometer [467], and by a slow neutron gravity refractometer with a quoted relativeuncertainty of 3 × − [468, 469].More recently, gravity acceleration for neutrons was measured with a cold neutron interferometer[470] and with a spin-echo spectrometer [471] with a relative precision of 10 − .In [472] prospects to achieve a relative precision ∆ g/g ∼ − using a three phase-grating moir´elarge area neutron interferometer were discussed proposing also a measurement of the value of the grav-itational constant G with neutrons. 58 .3.8 Tests with antimatter and with charged particles In principle, gravitation for antimatter may obey different laws than for ordinary matter. Measuringand comparing the gravitational properties of matter and antimatter may probe different aspects ofSME [253] and quantum vacuum [473]. Theoretical considerations based on energy conservation in thegravitational field and on arguments from QFT constrain the validity of the WEP for antimatter up toan accuracy of 10 − [474, 475]. Nevertheless, these arguments are indirect and need some experimentalvalidation by comparing the effect of gravity on antiparticles on and their corresponding ordinary matterparticles.Experiments to test gravity on elementary particles and antiparticles were proposed and developedsince the 1960’s. Electrically charged antiparticles (e.g. positrons and antiprotons) can be eitherobserved in beams under free fall conditions, or trapped within a combination of magnetic and electricfields, as in Penning traps [476]. In 1967 Witteborn and Fairbank observed the time-of-flight distributionof electrons and positrons in free fall inside a drift tube. Two decades later, an experiment was performedat the CERN Low Energy Antiproton Ring to measure the gravitational acceleration of antiprotons [477].¯ p were collected and cooled in a Penning trap, then released in a vertical drift tube. However bothfreely falling and trap-based systems of charged antiparticles are affected by errors from residual strayelectric and magnetic fields [478] which make any gravitational measurement extremely difficult. Thestatic electric field mg/q required to compensate gravity acceleration is only 56 pV/m for positrons, andabout 100 nV/m for antiprotons. Even in case of perfect shielding from stray fields, the gravitationalsag of electrons in a drift tube produces charge density anisotropies resulting in a electric forces of thesame order of gravity [479]. WEP tests with electrons under weightless conditions have been proposedto get rid of gravity-induced electric fields [234]. On the other hand, current experiments to testWEP on anti-matter are focusing on neutral systems such as positronium [480], muonium [344, 481],antihydrogen [422, 482–485], and on particle-antiparticle pairs [486], for which the effect of stray fieldsis strongly suppressed.The first evidence of antihydrogen was achieved at the European Organization for Nuclear Research(CERN) in 1995 [487] and confirmed two years later at Fermilab [488]. The recombination of electron-positron pairs produced from the collision of relativistic antiprotons with Xenon targets formed ¯ H atomsat relativistic energies, unsuitable for precision measurements. A breakthrough occurred when ¯ H atthermal energies (few hundreds K) was first produced in 2002 by the ATHENA experiment [489], viathree-body reaction by mixing trapped antiprotons (¯ p ) with positrons ( e + ) at low energies. This result,shortly followed by a similar achievement from the ATRAP experiment [490] opened the possibility totest WEP on neutral antimatter.However the gravitational force is so weak that a WEP experiment requires further cooling of anti-matter down to cryogenic temperatures. Several second-generation experiments with antimatter havebeen then developed at CERN, where the only intense source of low energy antiprotons is availableworldwide, i.e. the Antiproton Decelerator (AD) currently under upgrade to Extra Low ENergy An-tiproton ring (ELENA). Such experiments must face several challenges. Neutral antimatter particlesare produced from their charged constituents. This requires complex experiments combining advancedmethods from high-energy physics for particle beams optics and detectors, with advanced methods forion trapping and atom optics. Moreover neutral antiparticles are produced at much lower rates and atmuch higher temperatures than in the typical quantum sensors described in section 3.3. The variousexperiments at AD developed different methods to produce ¯ H at rates and temperatures suited forprecision measurements.The ALPHA experiment, designed to perform high resolution spectroscopy on ¯ H , generates anti-hydrogen by three-body recombination between trapped, evaporatively cooled antiprotons and trappedpositrons. The low-energy tail of the ¯ H distribution is captured in a magnetic trap at a rate of about10 atoms on cycles of 4 minutes [491]. ALPHA performed a preliminary measurement of the Earths59ravitational effect on magnetically trapped ¯ H . The resulting gravitational acceleration of ¯ H wasconstrained to within 100 times the g value for matter [492].The GBAR experiment aims to generate ultracold antihydrogen through the anti-ion ¯ H + [482, 483].The ¯ H + ion is produced via two cascaded charge exchange processes from the interaction of ¯ p witha positronium target, then of the generated ¯ H with the same target. ¯ H + ion can be sympatheticallycooled with laser cooled Be + ions down to µ K temperatures. The excess positron can then be laserdetached in order to recover the neutral ¯ H with very low temperature. A high-intensity positron sourcehas been developed for ¯ H + production.The projects Antimatter Experiment: Gravity, Interferometry, Spectroscopy (AEgIS) [422], whichis operating since 2012, has been designed to measure the gravitational acceleration with matter-waveinterferometry on a pulsed ¯ H beam at sub-kelvin temperatures. In AEgIS, antihydrogen atoms are pro-duced via a charge exchange reaction between Rydberg-excited positronium atoms and cold antiprotonswithin an electromagnetic trap. The resulting Rydberg antihydrogen atoms will be horizontally accel-erated by an electric field gradient (Stark effect), then they will pass through a moir´e deflectometer.The vertical deflection caused by the Earth’s gravitational field will provide a Weak Equivalence Prin-ciple test for antimatter. Detection will be undertaken via a position sensitive detector. Around 10 antihydrogen atoms are needed for the gravitational measurement to be completed. The generation ofantihydrogen via charge-exchange process was already demonstrated in the ATRAP experiment [493],where Ps were excited toward a Rydberg state by collisions with laser-excited cesium atoms. AEgISrather plans to directly laser excite positronium.An alternative to testing UFF on bound antimatter systems is the search for mass differences onparticle-antiparticle pairs. In particular, neutral kaon is the only system where particle-antiparticledifferences are detected; this is explained as arising from CP-violating terms in the K − ¯ K massmatrix. In [486] upper limits on possible K − ¯ K mass difference were determined from the analysisof data on tagged K and ¯ K decays into π + π − from the CPLEAR experiment over three years. Theresults are in agreement with the Equivalence Principle to a level of 6.5, 4.3 and 1 . × − respectively,for scalar, vector and tensor potentials originating from the Sun. Such determination of mass differencefor kaon is ten orders of magnitude more precise than for p − ¯ p [494].High precision gravity measurements will require the application of interferometric methods. Whilethe most precise quantum sensors are based on light pulse matter-wave interferometry, see section 3.3,such method is not readily applied on antimatter systems. This is mostly due to the comparablyhigh temperatures currently achievable and to the extremely short wavelength of resonance opticaltransitions in ¯ H and in ps . Inertial sensing with Talbot-Lau interferometry [495] allows to work withlow-intensity, weakly coherent beams. This method has been recently demonstrated on a beam of lowenergy positrons [496]. General Relativity and metric theories of gravity are based on the validity of the Equivalence Principle,according to which the gravitational acceleration is (locally) indistinguishable from acceleration causedby mechanical (apparent) forces. The consequence of the Equivalence Principle is that gravitationalmass is equal to inertial mass, m g = m I . This identity was already pointed out by Galileo and Newton,but Einstein recognizes it as a fundamental aspect involving also accelerations and forces and thenelevating it to a principle. Equivalence Principle allowed Einstein to construct a theory capable ofexplaining gravity and acceleration under the same physical standard. Based on this assumption,he stated the following fundamental postulated: in a free falling frame, all non-gravitational lawsof physics (hence not only the mechanical ones) behave as if gravity was absent. More generally,Equivalence Principle asserts that objects with different (internal) composition are subject to the same60cceleration when moving in a gravitational field. This new principle of nature led Einstein to therevolutionary interpretation of gravitation: gravity can be described as a curvature effect of space-time.as a consequence, the Einstein Equivalence Principle plays a crucial role in all metric theories of gravity,as well as in the Standard Model of particle physics which is not in conflict with GR.As we have seen in this review, the Equivalence Principle essentially encodes the local Lorentz invari-ance (clock rates are independent of the clocks velocities), the local position invariance (the universalityof red-shift) and the universality of free fall (all free falling point particles follow the same trajectoriesindependently of their internal structure and composition). The first to two principle, i.e. the Lorentzand position invariance, are hence related to the local properties of physics, so that they can be testedby using atomic clocks and measurements of spectroscopy, while the third one, the universality of freefalling point-like particles, can be tested by tracking trajectories, hence freely falling test masses asdiscussed in the experimental part of this Review paper.The Equivalence Principle can be formulated in two different forms: the weak and the strong form.The weak form of the Equivalence Principle states that the gravitational properties of the interactionof particle physics of the Standard Model, hence the strong and electro-weak interactions, obey the EP.As we have pointed out, the equality m g = m i implies that, in an external gravitational field, different(and neutral) test particles undergo to the same free fall acceleration, and in a free falling inertial frameonly tidal forces may appear (apart the latter, free falling bodies behave as if the gravity is absent).However, it is worth noticing that in many extensions of the Standard Model, new interactions (quantumexchange forces) are introduced, and, in general, they may violate the weak equivalence principle owingto the coupling with generalized charges, rather than mass/energy as happens in gravity.The second form, the strong Equivalence Principle, is such that it extends the weak one includingthe gravitational energy. In GR, the strong Equivalence Principle is fulfilled thanks to the gravitationalstress-energy tensor, while it can be violated in some extension of GR (as, for example, scalar tensortheories discussed above where a scalar field is present in the gravitational interaction and can benon-minimally coupled with geometry).As a final remark, an important issue has to be discussed. It is related to the parameterized post-Newtonian formalism [1, 497] and the Equivalence Principle. In a nutshell, the PPN approximation isa method for obtaining the motion of the system in terms of higher powers of the small parameters(gravitational potential and velocity square) with respect the ones given by Newtonian mechanics. Therelevant aspect is that this formalism allows to describe the motion of M q celestial bodies that is commonto many theories of gravity. The acceleration of a body can be written in the form ¨ r p = ¨ r GRp + δ ¨ r P P Np ,where ¨ r GRp is the usual acceleration derived in GR, while ¨ r P P Np is the PPN corrections [1, 497]¨ r P P Np = X q = p GM q ( r q − r p ) | r q − r p | ( (cid:20) M g M I (cid:21) q − ! − β + γ − c X r = p GM r | r r − r p | − β − c X s = q GM s | r s − r q | + γ − c | ˙ r q − ˙ r p | + ˙ GG ( t − t ) ) + 2( γ − c X q = p GM q | r p − r q | (cid:20) ¨ r q + [( r p − r q ) · ( ˙ r p − ˙ r q )]( ˙ r p − ˙ r q ) | r p − r q | (cid:21) Some comments are in order: ) The correction ¨ r P P Np vanishes in the case of GR, that is when thePPN parameter γ and β assume the values: γ = 1 = β . ) The expression for ¨ r P P Np does contain thevariation of the gravitational constant G , the ˙ G/G -term, typical of scalar tensor theories of gravity; )In ¨ r P P Np a term related to the strong equivalence principle appears, i.e. h M g M I i −
1, which is typicallyexpressed in the form h M g M I i = 1 + η (cid:0) Emc (cid:1) , where the parameter η , depending on PPN parameters as η = 4 β − γ −
3, encodes deviations from GR, and it is therefore related to the violation of the strongEquivalence Principle ( η = 0 in GR), while m and E are the mass and gravitational self-energy of the61ody, respectively. Taking, for example, a uniform sphere of radius R one gets (cid:20) Emc (cid:21) = − G mc Z d x d x ′ ρ ( x ) ρ ( x ′ ) | x − x ′ | = − GMc R . (100)This relation shows that, for Solar System it is (cid:2)
Emc (cid:3) ⊙ ∼ − , while for bodies of lab test one gets (cid:2) Emc (cid:3) lab ∼ − . This suggests that planet-size bodies are required for testing the strong EquivalencePrinciple with a certain confidence level.Finally, we discussed the possibility to violate the Equivalence Principle considering systems at fi-nite temperature. Although the Equivalence Principle asserts (in the weak form) that the gravitationalacceleration is identical for all bodies, i.e. m I = m g , the latter equality can be violated in quantum fieldtheory considering a finite temperature framework. In fact, as shown in a series of seminal papers [235],a fraction of the mass of a particle arises through the finite-temperature component of the radiativecorrections . This result is a consequence of the Lorentz non-invariance of the finite temperature vac-uum. According to this result, theories at finite temperature could be the straightforward way to testEquivalence Principle at fundamental quantum level.In conclusion, various issues in modern physics, both from gravitational and particle physics sectors,predict violations of the Equivalence Principle. Given the importance of this question, the experimentalchallenge is to look for frameworks where possible violations could manifest. Besides, one has to improvethe limits of experimental tests. As we have shown in this review, matching experiments from laboratoryand satellites is an important complement to probing fundamental physics at very high precision level,and, in turn, possible results could open new and unexpected scenarios. G.M.T. acknowledges useful discussions with Enrico Iacopini and Marco Tarallo. S.C. and G.L. thankMaurizio Gasperini for useful discussions on the presented topics and
Istituto Nazionale di Fisica Nu-cleare (INFN)- iniziativa specifica
MOONLIGHT2. This work was supported by the Italian Ministry ofEducation, University and Research (MIUR) under the PRIN 2015 project “Interferometro AtomicoAvanzato per Esperimenti su Gravit`a e Fisica Quantistica e Applicazioni alla Geofisica”.
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