Tests of Lorentz Symmetry in the Gravitational Sector - CERN

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Tests of Lorentz Symmetry in the Gravitational Sector - CERN

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universe
Review
Tests of Lorentz Symmetry in the Gravitational Sector
Aurélien Hees 1,*, Quentin G. Bailey 2, Adrien Bourgoin 3, Hélène Pihan-Le Bars 3, Christine Guerlin 3,4 and Christophe Le Poncin-Lafitte 3
1 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA 2 Physics Department, Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA; [email protected] 3 SYRTE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06,
LNE, 61 avenue de l’Observatoire, 75014 Paris, France; [email protected] (A.B.); [email protected] (H.P.-L.B.); [email protected] (C.G.); [email protected] (C.L.P.-L.) 4 Laboratoire Kastler Brossel, ENS-PSL Research University, CNRS, UPMC-Sorbonne Universités, Collège de France, 75005 Paris, France * Correspondence: [email protected]; Tel.: +1-310-825-8345
Academic Editors: Lorenzo Iorio and Elias C. Vagenas Received: 15 October 2016; Accepted: 22 November 2016; Published: 1 December 2016
Abstract: Lorentz symmetry is one of the pillars of both General Relativity and the Standard Model of particle physics. Motivated by ideas about quantum gravity, unification theories and violations of CPT symmetry, a significant effort has been put the last decades into testing Lorentz symmetry. This review focuses on Lorentz symmetry tests performed in the gravitational sector. We briefly review the basics of the pure gravitational sector of the Standard-Model Extension (SME) framework, a formalism developed in order to systematically parametrize hypothetical violations of the Lorentz invariance. Furthermore, we discuss the latest constraints obtained within this formalism including analyses of the following measurements: atomic gravimetry, Lunar Laser Ranging, Very Long Baseline Interferometry, planetary ephemerides, Gravity Probe B, binary pulsars, high energy cosmic rays, . . . In addition, we propose a combined analysis of all these results. We also discuss possible improvements on current analyses and present some sensitivity analyses for future observations.
Keywords: experimental tests of gravitational theories; Lorentz and Poincaré invariance; modified theories of gravity; celestial mechanics; atom interferometry; binary pulsars

1. Introduction
The year 2015 was the centenary of the theory of General Relativity (GR), the current paradigm for describing the gravitational interaction (see e.g., the Editorial of this special issue [1]). Since its creation, this theory has passed all experimental tests with flying colors [2,3] ; the last recent success was the discovery of gravitational waves [4], summarized in [5]. On the other hand, the three other fundamental interactions of Nature are described within the Standard Model of particle physics, a framework based on relativistic quantum field theory. Although very successful so far, it is commonly admitted that these two theories are not the ultimate description of Nature but rather some effective theories. This assumption is motivated by the construction of a quantum theory of gravitation that has not been successfully developed so far and by the development of a theory that would unify all the fundamental interactions. Moreover, observations requiring the introduction of Dark Matter and Dark Energy also challenge GR and the Standard Model of particle physics since they cannot be explained by these two paradigms altogether [6]. It is therefore extremely important to test our current description of the four fundamental interactions [7].
Lorentz invariance is one of the fundamental symmetry of relativity, one of the corner stones of both GR and the Standard Model of particle physics. It states that the outcome of any local

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experiment is independent of the velocity and of the orientation of the laboratory in which the experiment is performed. If one considers non-gravitational experiments, Lorentz symmetry is part of the Einstein Equivalence Principle (EEP). A breaking of Lorentz symmetry implies that the equations of motion, the particle thresholds, etc. . . may be different when the experiment is boosted or rotated with respect to a background field [8]. More precisely, it is related to a violation of the invariance under “particle Lorentz transformations” [8] which are the boosts and rotations that relate the properties of two systems within a specific oriented inertial frame (or in other words they are boosts and rotations on localized fields but not on background fields). On the other hand, the invariance under coordinates transformations known as “observer Lorentz transformations” [8] which relate observations made in two inertial frames with different orientations and velocities is always preserved. Considering the broad field of applicability of this symmetry, searches for Lorentz symmetry breaking provide a powerful test of fundamental physics. Moreover, it has been suggested that Lorentz symmetry may not be a fundamental symmetry of Nature and may be broken at some level. While some early motivations came from string theories [9–11], breaking of Lorentz symmetry also appears in loop quantum gravity [12–15], non commutative geometry [16,17], multiverses [18], brane-world scenarios [19–21] and others (see for example [22,23]).
Tests of Lorentz symmetry have been performed since the time of Einstein but the last decades have seen the number of tests increased significantly [24] in all fields of physics. In particular, a dedicated effective field theory has been developed in order to systematically consider all hypothetical violations of the Lorentz invariance. This framework is known as the Standard-Model Extension (SME) [8,25] and covers all fields of physics. It contains the Standard Model of particle physics, GR and all possible Lorentz-violating terms that can be constructed at the level of the Lagrangian, introducing a large numbers of new coefficients that can be constrained experimentally.
In this review, we focus on the gravitational sector of the SME which parametrizes deviations from GR. GR is built upon two principles [2,26,27]: (i) the EEP; and (ii) the Einstein field equations that derive from the Einstein-Hilbert action. The EEP gives a geometric nature to gravitation allowing this interaction to be described by spacetime curvature. From a theoretical point of view, the EEP implies the existence of a spacetime metric to which all matter minimally couples [28]. A modification of the matter part of the action will lead to a breaking of the EEP. In SME, such a breaking of the EEP is parametrized (amongst others) by the matter-gravity coupling coefficients a¯µ and c¯µν [29,30]. From a phenomenological point of view, the EEP states that [2,27]: (i) the universality of free fall (also known as the weak equivalence principle) is valid; (ii) the outcome of any local non-gravitational experiment is independent of the velocity of the free-falling reference frame in which it is performed; and (iii) the outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed. The second part of Einstein theory concerns the purely gravitational part of the action (the Einstein-Hilbert action) which is modified in SME to introduce hypothetical Lorentz violations in the gravitational sector. This review focuses exclusively on this kind of Lorentz violations and not on breaking of the EEP.
A lot of tests of GR have been performed in the last decades (see [2] for a review). These tests rely mainly on two formalisms: the parametrized post-Newtonian (PPN) framework and the fifth force formalism. In the former one, the weak gravitational field spacetime metric is parametrized by 10 dimensionless coefficients [27] that encode deviations from GR. This formalism therefore provides a nice interface between theory and experiments. The PPN parameters have been constrained by a lot of different observations [2] confirming the validity of GR. In particular, three PPN parameters encode violations of the Lorentz symmetry: the α1,2,3 PPN coefficients. In the fifth force formalism, one is looking for a deviation from Newtonian gravity where the gravitational potential takes the form of a Yukawa potential characterized by a length λ and a strength α of interaction [31–34]. These two parameters are very well constrained as well except at very small and large distances (see [35]).
The gravitational sector of SME offers a new framework to test GR by parametrizing deviations from GR at the level of the action, introducing new terms that are breaking Lorentz symmetry. The idea

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is to extend the standard Einstein-Hilbert action by including Lorentz-violating terms constructed by contracting new fields with some operators built from curvature tensors and covariant derivatives with increasing mass dimension [36]. The lower mass dimension (dimension 4) term is known as the minimal SME and its related new fields can be split into a scalar part u, a symmetric trace free part sµν and a traceless piece tκλµν. In order to avoid conflicts with the underlying Riemann geometry, the Lorentz violating coefficients can be assumed to be dynamical fields and the Lorentz violation to arise from a spontaneous symmetry breaking [37–42]. The Lorentz violating fields therefore acquire a non-vanishing vacuum expectation value (denoted by a bar). It has been shown that in the linearized gravity limit the fluctuations around the vacuum values can be integrated out so that only the vacuum expectation values of the SME coefficients influence observations [39]. In the minimal SME, the coefficient u¯ corresponds to a rescaling of the gravitational constant and is therefore unobservable and the coefficients t¯κλµν do not play any role at the post-Newtonian level, a surprising phenomenon known as the t-puzzle [43,44]. The s¯µν coefficients lead to modifications from GR that have thoroughly been investigated in [39]. In particular, the SME framework extends standard frameworks such as the PPN or fifth force formalisms meaning that “standard” tests of GR cannot directly be translated into this formalism.
In the last decade, several measurements have been analyzed within the gravitational sector of the minimal SME framework: Lunar Laser Ranging (LLR) analysis [45,46], atom interferometry [47,48], planetary ephemerides analysis [49,50], short-range gravity [51], Gravity Probe B (GPB) analysis [52], binary pulsars timing [53,54], Very Long Baseline Interferometry (VLBI) analysis [55] and Cˇ erenkov radiation [56]. In addition to the minimal SME, there exist some higher order Lorentz-violating curvature couplings in the gravity sector [43] that are constrained by short-range experiments [57–59], Cˇ erenkov radiation [30,56] and gravitational waves analysis [60,61]. Finally, some SME experiments have been used to derive bounds on spacetime torsion [62,63]. A review for these measurements can be found in [30]. The classic idea to search for or to constrain Lorentz violations in the gravitational sector is to search for orientation or boost dependence of an observation. Typically, one will take advantage of modulations that will occur through an orientation dependence of the observations due to the Earth’s rotation, the motion of satellites around Earth (the Moon or artificial satellites), the motion of the Earth (or other planets) around the Sun, the motion of binary pulsars, . . . The main goal of this communication is to review all the current analyses performed in order to constrain Lorentz violation in the pure gravitational sector.
Two distinct procedures have been used to analyze data within the SME framework. The first procedure consists in deriving analytically the signatures produced by the SME coefficients on some observations. Then, the idea is to fit these signatures within residuals obtained by a data analysis performed in pure GR. This approach has the advantage to be relatively easy and fast to perform. Nevertheless, when using this postfit approach, correlations with other parameters fitted in the data reduction are completely neglected and may lead to overoptimistic results. A second way to analyze data consists in introducing the Lorentz violating terms directly in the modeling of observables and in the global data reduction. In this review, we highlight the differences between the two approaches.
In this communication, a brief theoretical review of the SME framework in the gravitational sector is presented in Section 2. The two different approaches to analyze data within the SME framework (postfit analysis versus full modeling of observables within the SME framework) are discussed and compared in Section 3. Section 4 is devoted to a discussion of the current measurements analyzed within the SME framework. This discussion includes a general presentation of the measurements, a brief review of the effects of Lorentz violation on each of them, the current analyses performed with real data and a critical discussion. A “grand fit” combining all existing analyses is also presented. In Section 5, some future measurements that are expected to improve the current analyses are developed. Finally, our conclusion is presented in Section 6.

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2. The Standard-Model Extension in the Gravitational Sector
Many of the tests of Lorentz and CPT symmetry have been analyzed within an effective field theory framework which generically describes possible deviations from exact Lorentz and CPT invariance [8,25] and contains some traditional test frameworks as limiting cases [64,65]. This framework is called, for historical reasons, the Standard-Model Extension (SME). One part of the activity has been a resurgence of interest in tests of relativity in the Minkowski spacetime context, where global Lorentz symmetry is the key ingredient. Numerous experimental and observational constraints have been obtained on many different types of hypothetical Lorentz and CPT symmetry violations involving matter [24]. Another part, which has been developed more recently, has seen the SME framework extended to include the curved spacetime regime [37]. Recent work shows that there are many ways in which the spacetime symmetry foundations of GR can be tested [29,39].
In the context of effective field theory in curved spacetime, violations of these types can be described by an action that contains the usual Einstein-Hilbert term of GR, a matter action, plus a series of terms describing Lorentz violation for gravity and matter in a generic way. While the fully general coordinate invariant version of this action has been studied in the literature, we focus on a limiting case that is valid for weak-field gravity and can be compactly displayed. Using an expansion of the spacetime metric around flat spacetime, gµν = ηµν + hµν, the effective Lagrange density to quadratic order in hµν can be written in a compact form as

c3

L = LEH + 32πG hµνs¯αβGαµνβ + ...,

(1)

where LEH is the standard Einstein-Hilbert term, Gαµνβ is the double dual of the Einstein

tensor linearized in hµν, G the bare Newton constant and c the speed of light in a vacuum.

The Lorentz-violating effects in this expression are controlled by the 9 independent coefficients in

the traceless and dimensionless s¯µν [39]. These coefficients are treated as constants in asymptotically

flat cartesian coordinates. The ellipses represent additional terms in a series including terms that

break CPT symmetry for gravity; such terms are detailed elsewhere [43,56,60] and are part of the

so-called nonminimal SME expansion. Note that the process by which one arrives at the effective

quadratic Lagrangian (1) is consistent with the assumption of the spontaneous breaking of local Lorentz

symmetry, which is discussed below.

Also of interest are the matter-gravity couplings. This form of Lorentz violation can be realized in

the classical point-mass limit of the matter sector. In the minimal SME the point-particle action can be

written as

SMatter = dλ c −m −(gµν + 2cµν)uµuν − aµuµ ,

(2)

where the particle’s worldline tangent is uµ = dxµ/dλ [29]. The coefficients controlling local Lorentz violation for matter are cµν and aµ. In contrast to s¯µν, these coefficients depend on the type of point mass (particle species) and so they can also violate the EEP. When the coefficients s¯µν, cµν, and aµ vanish perfect local Lorentz symmetry for gravity and matter is restored. It is also interesting to mention that this action with fixed (but not necessarily constant) aµ and cµν represents motion in a Finsler geometry [66,67].
It has been shown that explicit local Lorentz violation is generically incompatible with Riemann geometry [37]. One natural way around this is assumption of spontaneous Lorentz-symmetry breaking. In this scenario, the tensor fields in the underlying theory acquire vacuum expectation values through a dynamical process. Much of the literature has been devoted to studying this possibility in the last decades [9,38,68–78], including some original work on spontaneous Lorentz-symmetry breaking in string field theory [10,11]. For the matter-gravity couplings in Equation (2), the coefficient fields cµν, and aµ are then expanded around their background (or vacuum) values c¯µν, and a¯µ. Both a modified spacetime metric gµν and modified point-particle equations of motion result from the spontaneous breaking of Lorentz symmetry. In the linearized gravity limit these results rely only on the vacuum

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values c¯µν, and a¯µ. The dominant signals for Lorentz violation controlled by these coefficients are revealed in the calculation of observables in the post-Newtonian limit.
Several novel features of the post-Newtonian limit arise in the SME framework. It was shown in Ref. [39] that a subset of the s¯µν coefficients can be matched to the PPN formalism [2,27], but others lie outside it. For example, a dynamical model of spontaneous Lorentz symmetry breaking can be constructed from an antisymmetric tensor field Bµν that produces s¯µν coefficients that cannot be reduced to an isotropic diagonal form in any coordinate system, thus lying outside the PPN assumptions [78]. We can therefore see that the SME framework has a partial overlap with the PPN framework, revealing new directions to explore in analysis via the s¯µν, c¯µν, and a¯µ coefficients. The equations of motion for matter are modified by the matter-gravity coefficients for Lorentz violation c¯µν and a¯µ, which can depend on particle species, thus implying that these coefficients also control EEP violations. One potentially important class of experiments from the action (2) concerns the Universality of Free Fall of antimatter whose predictions are discussed in [29,79]. In addition, the post-Newtonian metric itself receives contributions from the matter coefficients c¯µν and a¯µ. So for example, two (chargeless) sources with the same total mass but differing composition will yield gravitational fields of different strength.
For solar-system gravity tests, the primary effects due to the nine coefficients s¯µν can be obtained from the post-Newtonian metric and the geodesic equation for test bodies. A variety of ground-based and space-based tests can measure these coefficients [80–82]. Such tests include Earth-laboratory tests with gravimeters, lunar and satellite laser ranging, studies of the secular precession of orbital elements in the solar system, and orbiting gyroscope experiments, and also classic effects such as the time delay and bending of light around the Sun and Jupiter. Furthermore, some effects described by the Lagrangian (1) can be probed by analyzing data from binary pulsars and measurements of cosmic rays [56].
For the matter-gravity coefficients c¯µν and a¯µ, which break Lorentz symmetry and EEP, several experiments can be used for analysis in addition to the ones already mentioned above including ground-based gravimeter and WEP experiments. Dedicated satellite EEP tests are among the most sensitive where the relative acceleration of two test bodies of different composition is the observable of interest. Upon relating the satellite frame coefficients to the standard Sun-centered frame used for the SME, oscillations in the acceleration of the two masses occur at a number of different harmonics of the satellite orbital and rotational frequencies, as well as the Earth’s orbital frequency. Future tests of particular interest include the currently flying MicroSCOPE experiment [83,84].
While the focus of the discussion to follow are the results for the minimal SME coefficients s¯µν, recent work has also involved the nonminimal SME coefficients in the pure-gravity sector associated with mass dimension 5 and 6 operators. One promising testing ground for these coefficients is sensitive short-range gravity experiments. The Newtonian force between two test masses becomes modified in the presence of local Lorentz violation by an anisotropic quartic force that is controlled by a subset of coefficients from the Lagrangian organized as the totally symmetric (k¯eff)jklm, which has dimensions of length squared [43]. This contains 14 measurable quantities and any one short-range experiment is sensitive to 8 of them. Two key experiments, from Indiana University and Huazhong University of Science and Technology, have both reported analysis in the literature [57,58] . A recent work combines the two analyses to place new limits on all 14, a priori independent, (k¯eff)jklm coefficients [59]. Other higher mass dimension coefficients play a role in gravitational wave propagation [60] and gravitational Cˇ erenkov radiation [56].
To conclude this section, we ask: what can be said about the possible sizes of the coefficients for Lorentz violation? A broad class of hypothetical effects is described by the SME effective field theory framework, but it is a test framework and as such does not make specific predictions concerning the sizes of these coefficients. One intriguing suggestion is that there is room in nature for violations of spacetime symmetry that are large compared to other sectors due to the intrinsic weakness of gravity. Considering the current status of the coefficients s¯µν, the best laboratory limits are at the 10−10–10−11 level, with improvements of four orders of magnitude in astrophysical tests on these coefficients [56]. However, the limits are at the 10−8 m2 level for the mass dimension 6 coefficients

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(k¯eff)jklm mentioned above. Comparing this to the Planck length 10−35 m, we see that symmetry breaking effects could still have escaped detection that are not Planck suppressed. This kind of “countershading” was first pointed out for the a¯µ coefficients [85], which, having dimensions of mass, can still be as large as a fraction of the electron mass and still lie within current limits.
In addition, any action-based model that breaks local Lorentz symmetry either explicitly or spontaneously can be matched to a subset of the SME coefficients. Therefore, constraints on SME coefficients can directly constrain these models. Matches between various toy models and coefficients in the SME have been achieved for models that produce effective s¯µν, c¯µν, a¯µ, and other coefficients. This includes vector and tensor field models of spontaneous Lorentz-symmetry breaking [29,39,75–78], models of quantum gravity [12,65] and noncommutative quantum field theory [17]. Furthermore, Lorentz violations may also arise in the context of string field theory models [86].

3. Postfit Analysis Versus Full Modeling
Since the last decade, several studies aimed to find upper limits on SME coefficients in the gravitational sector. A lot of these studies are based on the search of possible signals in post-fit residuals of experiments. This was done with LLR [45], GPB [52], binary pulsars [53,54] or Solar System planetary motions [49,50]. However, two new works focused on a direct fit to data with LLR [46] and VLBI [55], which are more satisfactory.
Indeed, in the case of a post-fit analysis, a simple modeling of extra terms containing SME coefficients are least square fitted in the residuals, attempting to constrain the SME coefficients of a testing function in residual noise obtained from a pure GR analysis, where of course Lorentz symmetry is assumed. It comes out correlations between SME coefficients and other global parameters previously fitted (masses, position and velocity. . . ) cannot be assessed in a proper way. In others words, searching hypothetical SME signals in residuals, i.e., in noise, can lead to an overestimated formal error on SME coefficients, as illustrated in the case of VLBI [55], and without any chance to learn something about correlations with other parameters, as for example demonstrated in the case of LLR [46]. Let us consider the VLBI example to illustrate this fact. The VLBI analysis is described in Section 4.2. Including the SME contribution within the full VLBI modeling and estimating the SME coefficient s¯TT altogether with the other parameters fitted in standard VLBI data reduction leads to the estimate s¯TT = (−5 ± 8) × 10−5. A postfit analysis performed by fitting the SME contribution within the VLBI residuals obtained after a pure GR analysis leads to s¯TT = (−0.6 ± 2.1) × 10−8 [55]. This example shows that a postfit analysis can lead to results with overoptimistic uncertainties and one needs to be extremely careful when using such results.

4. Data Analysis
In this section, we will review the different measurements that have already been used in order to constrain the SME coefficients. The different analyses are based on quite different types of observations. In order to compare all the corresponding results, we need to report them in a canonical inertial frame. The standard canonical frame used in the SME framework is a Sun-centered celestial equatorial frame [64], which is approximately inertial over the time scales of most observations. This frame is asymptotically flat and comoving with the rest frame of the Solar System. The cartesian coordinates related to this frame are denoted by capital letters

XΞ = (cT, XJ) = (cT, X, Y, Z) .

(3)

The Z axis is aligned with the rotation axis of the Earth, while the X axis points along the direction from the Earth to the Sun at vernal equinox. The origin of the coordinate time T is given by the time when the Earth crosses the Sun-centered X axis at the vernal equinox. These conventions are depicted in Figure 2 from [39].

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In the following subsections, we will present the different measurements used to constrain the SME coefficients. Each subsection contains a brief description of the principle of the experiment, how it can be used to search for Lorentz symmetry violations, what are the current best constraints obtained with such measurements and eventually how it can be improved in the future.
4.1. Atomic Gravimetry
The most sensitive experiments on Earth searching for Lorentz Invariance Violation (LIV) in the minimal SME gravity sector are gravimeter tests. As Earth rotates, the signal recorded in a gravimeter, i.e., the apparent local gravitational acceleration g of a laboratory test body, would be modulated in the presence of LIV in gravity. This was first noted by Nordtvedt and Will in 1972 [87] and used soon after with gravimeter data to constrain preferred-frame effects in the PPN formalism [88,89] at the level of 10−3.
This test used a superconducting gravimeter, based on a force comparison (the gravitational force is counter-balanced by an electromagnetic force maintaining the test mass at rest). While superconducting gravimeters nowadays reach the best sensitivity on Earth, force comparison gravimeters intrinsically suffer from drifts of their calibration factor (with e.g., aging of the system). Development of other types of gravimeters has evaded this drawback: free fall gravimeters. Monitoring the motion of a freely falling test mass, they provide an absolute measurement of g. State-of-the art free fall gravimeters use light to monitor the mass free fall. Beyond classical gravimeters that drop a corner cube, the development of atom cooling and trapping techniques and atom interferometry has led to a new generation of free fall gravimeters, based on a quantum measurement: atomic gravimeters.
Atomic gravimeters use atoms in gaseous phase as a test mass. The atoms are initially trapped with magneto-optical fields in vacuum, and laser cooled (down to 100 nK) in order to control their initial velocity (down to a few mm/s). The resulting cold atom gas, containing typically a million atoms, is then launched or dropped for a free fall measurement. Manipulating the electronic and motional state of the atoms with two counterpropagating lasers, it is possible to measure, using atom interferometry, their free fall acceleration with respect to the local frame defined by the two lasers [90]. This sensitive direction is aligned to be along the local gravitational acceleration noted zˆ; the atom interferometer then measures the phase ϕ = kazˆ T2, where T is half the interrogation time, k 2(2π/λ) with λ the laser wavelength, and azˆ is the free fall acceleration along the laser direction. The free fall time is typically on the order of 500 ms, corresponding to a free fall distance of about a meter. A new “atom preparation—free fall—detection” cycle is repeated every few seconds. Each measurement is affected by white noise, but averaging leads to a typical sensitivity on the order of or below 10−9 g [91–93].
Such an interferometer has been used by H. Müller et al. in [47] and K. Y. Chung et al. in [48] for testing Lorentz invariance in the gravitational sector with Caesium atoms, leading to the best terrestrial constraints on the s¯µν coefficients. The analysis uses three data sets of respectively 2.5 days for the first two and 10 days for the third, stretched over 4 years, which allows one to observe sidereal and annual LIV signatures. The gravitational SME model used for this analysis can be found in [39,47,48]; its derivation will be summarized hereunder. Since the atoms in free fall are sensitive to the local phase of the lasers, LIV in the interferometer observable could also come from the pure electromagnetic sector. This contribution has been included in the experimental analysis in [48]. Focusing here on the gravitational part of SME, we ignore it in the following.
The gravitational LIV model adjusted in this test restricts to modifications of the Earth-atom two-body gravitational interaction. The Lagrangian describing the dynamics of a test particle at a point on the Earth’s surface can be approximated by a post-Newtonian series as developed in [39]. At the Newtonian approximation, the two bodies Lagrangian is given by
L = 21 mV2 + GN MRm 1 + 12 s¯tJK Rˆ J Rˆ K − 32 s¯TJ VcJ − s¯TJ Rˆ J VcK Rˆ k , (4)

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where R and V are the position and velocity expressed in the standard SME Sun-centered frame and Rˆ = R/R with R = |R|. In addition, we have introduced GN the observed Newton constant measured by considering the orbital motion of bodies and defined by (see also [39,50] or Section IV of [52])

GN = G 1 + 5 s¯TT ,

(5)

3

and the 3-dimensional traceless tensor
s¯tJK = s¯JK − 13 s¯TTδJK . (6)
From this Lagrangian one can derive the equations of motion of the free fall mass in a laboratory frame (see the procedure in Section V.C.1. from [39]). It leads to the modified local acceleration in the presence of LIV [39] given by

azˆ = g

1 − 1 i4s¯TT + 1 i4s¯zˆzˆ



ω⊕2

R⊕

sin2

χ



gi4s¯Tzˆ

βz⊕ˆ



3gi1 s¯ T J

J
β⊕

,

(7)

6

2

where

g

=

GN M⊕/R2⊕, ω⊕

is the Earth’s angular velocity,

β⊕

=

V⊕ c



10−4

is the Earth’s boost,

R⊕

is

the Earth radius, M⊕ is the Earth mass and χ the colatitude of the lab whose reference frame’s zˆ

direction is the sensitive axis of the instrument as previously defined here. This model includes the

shape of the Earth through its spherical moment of inertia

I⊕

which appears in i⊕

=

I⊕ M R2

, i1

=

1+

13 i⊕

⊕⊕

and i4 = 1 − 3i⊕. In [48], Earth has been approximated as spherical and homogeneous leading to

i⊕

=

12 ,

i1

=

7 6

and

i4

=

− 12 .

The sensing direction of the experiment precesses around the Earth rotation axis with sidereal

period, and the lab velocity varies with sidereal period and annual period. At first order in V⊕ and

ω⊕ and as a function of the SME coefficients, the LIV signal takes the form of a harmonic series with

sidereal and annual base frequencies (denoted resp. ω⊕ and Ω) together with first harmonics. The time dependence of the measured acceleration azˆ from Equation (7) arises from the terms involving the zˆ

indices. It can be decomposed in frequency according to [39]

δazˆ = ∑ Cl cos (ωlt + φl) + Dl sin (ωlt + φl) .

(8)

azˆ

l

The model contains seven frequencies l ∈ {Ω, ω⊕, 2ω⊕, ω⊕ ± Ω, 2ω⊕ ± Ω}. The 14 amplitudes Cl and Dl are linear combinations of 7 s¯µν components: s¯JK, s¯TJ and s¯XX − s¯YY which can be found in Table 1 of [48] or Table IV from [39].
In order to look for tiny departures from the constant Earth-atom gravitational interaction, a tidal model for azˆ variations due to celestial bodies is removed from the data before fitting to Equation (8). This tidal model consists of two parts. One part is based on a numerical calculation of the Newtonian tide-generating potential from the Moon and the Sun at Earth’s surface based on ephemerides. It uses here the Tamura tidal catalog [94] which gives the frequency, amplitude and phase of 1200 harmonics of the tidal potential. These arguments are used by a software (ETGTAB) that calculates the time variation of the local acceleration in the lab and includes the elastic response of Earth’s shape to the tides, called “solid Earth tides”, also described analytically e.g., by the DDW model [95]. A previous SME analysis of the atom gravimeter data using only this analytical tidal correction had been done, but it led to a degraded sensitivity of the SME test [47]. Indeed, a non-negligible contribution to azˆ is not covered by this non-empirical tidal model: oceanic tide effects such as ocean loading, for which good global analytical models do not exist. They consequently need to be adjusted from measurements. For the second analysis, reported here, additional local tidal corrections fitted on altimetric data have been removed [96] allowing to improve the statistical uncertainty of the SME test by one order of magnitude.

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After tidal subtraction, signal components are extracted from the data using a numerical Fourier transform (NFT). Due to the finite data length, Fourier components overlap, but the linear combinations of spectral lines that the NFT estimates can be expressed analytically. Since the annual component ωl = Ω has not been included in this analysis, the fit provides 12 measurements. From there, individual constraints on the 7 SME coefficients and their associated correlation coefficients can be estimated by a least square adjustment. The results obtained are presented in Table 1.

Table 1. Atom-interferometry limits on Lorentz violation in gravity from [48]. The correlation coefficients can be derived from Table III of [48].

s¯TX s¯TY s¯TZ s¯XX − s¯YY s¯XY s¯XZ s¯YZ

Coefficient
(−3.1 ± 5.1) × 10−5 (0.1 ± 5.4) × 10−5 (1.4 ± 6.6) × 10−5 (4.4 ± 11) × 10−9 (0.2 ± 3.9) × 10−9 (−2.6 ± 4.4) × 10−9 (−0.3 ± 4.5) × 10−9

Correlation Coefficients

1

0.05

1

0.11 −0.16 1

−0.82 0.34 −0.16 1

−0.38 −0.86 0.10 −0.01 1

−0.41 0.13 −0.89 0.38 0.02 1

−0.12 −0.19 −0.89 0.04 0.20 0.80 1

All results obtained are compatible with null Lorentz violation. As expected from boost suppressions in Equation (7) and from the measurement uncertainty, on the order of a few 10−9 g [97], typical limits obtained are in the 10−9 range for purely spatial s¯µν components and 4 orders of magnitude weaker for the spatio-temporal components s¯TJ. It can be seen e.g., with the purely spatial components that these constraints do not reach the intrinsic√limit of acceleration resolution of the instrument (which has a short term stability of 11 × 10−9 g/ Hz) because the coefficients are still correlated. Their marginalized uncertainty is broadened by their correlation.
Consequently, improving the uncertainty could be reached through a better decorrelation, by analyzing longer data series. In parallel, the resolution of these instruments keeps increasing and has nowadays improved by about a factor 10 since this experiment. However, increasing the instrument’s resolution brings back to the question of possible accidental cancelling in treating “postfit” data. Indeed, it should be recalled here that local tidal corrections subtracted prior to analysis are based on adjusting a model of ocean surface from altimetry data. In principle, this observable would as well be affected by gravity LIV; fitting to these observations thus might remove part of SME signatures from the atom gravimeter data. This was mentioned in the first atom gravimeter SME analysis [47]. The adjustment process used to assess local corrections in gravimeters is not made directly on the instrument itself, but it always involves a form of tidal measurement (here altimetry data, or gravimetry data from another instrument in [98]). All LIV frequencies match to the main tidal frequencies. Further progress on SME analysis with atom gravimeters would thus benefit from addressing in more details the question of possible signal cancelling.
4.2. Very Long Baseline Interferometry
VLBI is a geometric technique measuring the time difference in the arrival of a radio wavefront, emitted by a distant quasar, between at least two Earth-based radio-telescopes. VLBI observations are done daily since 1979 and the database contains nowadays almost 6000 24 h sessions, corresponding

Universe 2016, 2, 30

10 of 40

to 10 millions group-delay observations, with a present precision of a few picoseconds. One of the principal goals of VLBI observations is the kinematical monitoring of Earth rotation with respect to a global inertial frame realized by a set of defining quasars, the International Celestial Reference Frame [99], as defined by the International Astronomical Union [100]. The International VLBI Service for Geodesy and Astrometry (IVS) organizes sessions of observation, storage of data and distribution of products, in particular the Earth Orientation parameters. Because of this precision, VLBI is also a very interesting tool to test gravitation in the Solar System. Indeed, the gravitational fields of the Sun and the planets are responsible of relativistic effects on the quasar light beam through the propagation of the signal to the observing station and VLBI is able to detect these effects very accurately. By using the complete VLBI observations database, it was possible to obtain a constraint on the γ PPN parameter at the level of 1.2 × 10−4 [101,102]. In its minimal gravitational sector, SME can also be investigated with VLBI and obtaining a constrain on the s¯TT coefficient is possible.
Indeed, the propagation time of a photon emitted at the event (cTe, Xe) and received at the position Xr can be computed in the SME formalism using the time transfer function formalism [103–107] and is given by [39,80]

T

(Xe, Te, Xr)

=

Tr

− Te

=

Rer c

+

2

GN M c3

1 − 23 s¯TT − s¯TJ NeJr

ln

Re − Ner .Xe Rr −Ner.Xr

(9)

+ GN M
c3

s¯TJ PeJr − s¯JK NeJr PeKr

Re−Rr + GN M

Re Rr

c3

s¯TJ NeJr + s¯JK PˆeJr PˆeKr − s¯TT

(Nr.Ner − Ne.Ner)

where the terms a1 and a2 from [80] are taken as unity (which corresponds to using the harmonic gauge, which is the one used for VLBI data reduction), Re = |Xe|, Rr = |Xr|, Rer = |Xr − Xe| with the central body located at the origin and where we introduce the following vectors

K = Xe , Re

Xij Xj − Xi Nij ≡ Rij = |Xij| ,

Ni = Xi , |Xi|

Per = Ner × (Xr × Ner),

and Pˆer = Per , (10) |Per |

and where GN is the observed Newton constant measured by considering the orbital motion of bodies and is defined in Equation (5). This equation is the generalization of the well-known Shapiro time delay including Lorentz violation. The VLBI is actually measuring the difference of the time of arrival of a signal received by two different stations. This observable is therefore sensitive to a differential time delay (see [108] for a calculation in GR). Assuming a radio-signal emitted by a quasar at event (Te, Xe) and received by two different VLBI stations at events (T1, X1) and (T2, X2) (all quantities being expressed in a barycentric reference frame), respectively, the VLBI group-delay ∆τ(SME) in SME formalism can be written [55]
∆τ(SME) = 2 GNc3M (1 − 23 s¯TT) ln RR12 ++ KK..XX12 + 23 GNc3M s¯TT (N2.K − N1.K) , (11)
where we only kept the s¯TT contribution (see Equation (7) from [55] for the full expression) and we use the same notations as in [108] by introducing three unit vectors

K = Xe , N1 = X1 , and N2 = X2 .

(12)

|Xe|

|X1|

|X2|

Ten million VLBI delay observations between August 1979 and mid-2015 have been used to estimate the s¯TT coefficient. First, VLBI observations are corrected from delay due to the radio wave crossing of dispersive media by using 2 GHz and 8 GHz recordings. Then, we used only the 8 GHz delays and the Calc/Solve geodetic VLBI analysis software, developed at NASA Goddard Space Flight Center and coherent with the latest standards of the International Earth Rotation and Reference Systems Service [109]. We added the partial derivative of the VLBI delay with respect to s¯TT from Equation (11) to the software package using the USERPART module of Calc/Solve. We turned to a
Lorentz SymmetrySmeSectorLorentz ViolationModel