Tests of Gravity with Future Space-Based Experiments
TTests of Gravity with Future Space-Based Experiments
Jeremy Sakstein Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania 209 S. 33rd St., Philadelphia, PA 19104, USA ∗ Future space-based tests of relativistic gravitation—laser ranging to Phobos, accelerometers inorbit, and optical networks surrounding Earth—will constrain the theory of gravity with unprece-dented precision by testing the inverse-square law, the strong and weak equivalence principles, andthe deflection and time delay of light by massive bodies. In this paper, we estimate the boundsthat could be obtained on alternative gravity theories that use screening mechanisms to suppressdeviations from general relativity in the solar system: chameleon, symmetron, and galileon models.We find that space-based tests of the parameterized post-Newtonian parameter γ will constrainchameleon and symmetron theories to new levels, and that tests of the inverse-square law usinglaser ranging to Phobos will provide the most stringent constraints on galileon theories to date. Weend by discussing the potential for constraining these theories using upcoming tests of the weakequivalence principle, and conclude that further theoretical modeling is required in order to fullyutilize the data. I. Introduction
We are in a golden age for testing relativistic theories ofgravitation. The recent discovery of gravitational wavesfrom merging binary black holes [1] has tested generalrelativity (GR) in the strong field regime for the firsttime [9], and, on cosmological scales, ongoing surveyssuch as DES, as well as future surveys such as Euclid,LSST, SKA, and WFIRST will test the theory of grav-ity on cosmological distance scales [10]. On Earth, ad-vances in table-top experiments such as torsion-balanceexperiments [11], optically levitated microspheres [12],and atom interferometry [13] have probed new potentialgravitational interactions at micron distances, and forcesas weak as 10 − N .From a theoretical viewpoint, there has been a resur-gence in the study of modified gravity models drivenby the mysterious acceleration of the cosmic expansion:dark energy [14–17]. Typically, cosmologically relevantmodifications of gravity are difficult to reconcile with so-lar system tests of GR, either because they require strongcouplings to matter or because they have force ranges oforder the size of the universe. This has led the com-munity to focus on a narrow class of models that in-clude screening mechanisms [16, 18–20]. Screening mech-anisms use non-linear effects to suppress deviations fromGR in the solar system while allowing them to be rele-vant on larger, cosmological scales. For this reason, thefree parameters (masses and couplings) do not need tobe tuned to evade solar system tests.Complementary to the tests mentioned above, the nextgeneration of space-based tests —accelerometers in or- ∗ [email protected] See [2–8] for tests of cosmological infra-red modifications ofgravity using the recent simultaneous observation of both grav-itational waves (GW170817) and a gamma ray burst (GRB170817A) from merging neutron stars by the LIGO/Virgo andFermi collaborations. See [21–23] for recent reviews. bit, laser networks surrounding the Sun and Earth, andlaser ranging to Mars—will constrain relativistic gravityto unprecedented levels in the solar system by measuringthe parameters γ , β , and δ appearing in the parame-terized post-Newtonian (PPN) metric, testing the strongand weak equivalence principles, constraining the time-variation of Newton’s constant, and by looking for de-viations in the inverse-square law. The purpose of thispaper is to explore the implications of these future mis-sions for three theories of gravity that exhibit differentscreening mechanisms: chameleon [24, 25], symmetron[26], and galileon [27] theories.We will proceed as follows: In the next section, we willmotivate screening mechanisms and introduce the threementioned above. Next, in section III we will brieflyreview the current missions that have tested gravity inspace, and the proposed missions that we will use in thiswork to forecast the projected bounds on the model’sparameter space. In section IV we will present the cur-rent and projected bounds, and discuss their implicationsfor the models, and for other tests of screening mecha-nisms. We will also discuss other future tests that maybe useful for testing screening mechanisms but that wewill not forecast for here due to uncertainties in the the-oretical modeling; we discuss these in order to highlighthow a dedicated effort towards a better modeling of thesesystems could improve the current bounds on screenedmodified gravity models. We conclude in section V. InAppendix A we provide a brief derivation of the PPNparameter for chameleon and symmetron theories. II. Screening MechanismsA. Why Screening?
The study of scalar-tensor theories has been motivatedby the cosmological observation of dark energy, the mys-terious driving mechanism for the acceleration of the cos-mic expansion. Indeed, one proposed explanation is that a r X i v : . [ a s t r o - ph . C O ] M a r gravity is modified on large distances. In order to be rel-evant today, any modification must necessarily be as im-portant as general relativity but this cannot be the casein the solar system because deviations are constrained tobe subdominant by a factor of 10 − or more depend-ing on the specific theory . As an example, considerBrans-Dicke gravity, which describes a new scalar field φ coupled to gravity and is parameterized by a single pa-rameter ω BD . In the non-relativistic limit, one finds aPoisson-like equation for φ : ∇ φ = − πG ω BD ρ, (1)which gives a contribution to the PPN parameter γ | γ − | = 12 + ω BD . (2)In order to satisfy the Cassini bound | γ − | < . × − [30] one needs to take ω BD > α eff ∼ /ω BD < ∼ − . This implies that any Brans-Dicke-like modifications of GR must be subdominant tothe Einstein-Hilbert term on all scales by at least a factorof 10 . Such a requirement means that any such theoriesare cosmologically irrelevant.One reason that solar system tests are so constrainingfor Brans-Dicke-like theories is that they contain mass-less scalars, and hence fit into the PPN form due to theresultant 1 /r potentials. One can try to circumvent thisissue but introducing a mass for the scalar so that itsequation of motion is( ∇ + m ) φ = 8 παGρ, (3)in which case the total potential sourced by a static,spherically symmetric body is of the Yukawa form V ( r ) = GMr (1 + 2 α e − mr ) . (4)Yuakawa forces have been searched for extensively at dis-tances ranging from the Earth-Moon distance [31, 32] tomicron-scales [11, 33, 34], and so the mass m > ( µ m) − in order to evade these tests. Again, such a scalar canhave nothing to say about cosmological-scale physics.One common issue with the previous two models isthat solar system tests automatically preclude any rel-evance for cosmology because the force must either betoo weak, or too short-ranged. Screening mechanismscircumvent this problem by introducing non-linear mod-ifications of the Poisson equation that dynamically sup-press deviations from GR in the solar system without theneed to fine-tune the mass or the coupling to matter. Inthis paper we will consider three well-studied screeningmechanisms: For example, a theory that predicts strong violations of the weakequivalence principle will be constrained to levels of O (10 − )[28, 29]. • Chameleon Screening:
This dynamicallychanges the mass of the field so that it mediatesa short ranged force in the solar system but mayinfluence cosmology on Mpc scales. • Symmetron Screening:
This dynamically variesthe coupling to matter so that it is essentially un-coupled in the solar system but can source devia-tions from GR on linear cosmological scales. • The Vainshtein Mechanism:
This uses non-linear kinetic terms to alter the field profile sourcedby massive bodies so that fifth-forces are highly-suppressed in the solar system. On cosmologi-cal scales, theories that exhibit this mechanismcan self-accelerate without a cosmological con-stant, which makes them interesting alternativesto ΛCDM. The fifth-forces can also modify the dy-namics of linear and non-linear perturbations [35–37].We now proceed to discuss each of these briefly in turn.Our discussion will be far from comprehensive and theinterested reader is directed to references [20, 38–41] formore details . B. Screening Mechanisms
1. Chameleon Screening
Chameleon screening [24, 25] uses a non-linear poten-tial to make the field’s mass a function of the environ-mental density. It’s equation of motion is ∇ φ = − n Λ n φ n +1 + αρM pl , (5)the right hand side of which can be derived from an ef-fective potential V eff = Λ n φ n + αφρM pl . (6)The mass-scale Λ can vary over many orders of mag-nitude, but it is often compared to the dark energyscale Λ DE = 2 . . The location of the Unpublished lecture notes can be found at the following url. Note that we have switched to a dimensionful scalar in keepingwith the conventions in the literature. This is why there is afactor of αρ/M pl rather than 8 παGρ as in equation (3), whichused a dimensionless scalar to ensure that the equation had asimilar form to the Poisson equation in GR. As an example, many authors consider a generalized potentialof the form V ( φ ) = Λ exp(Λ n /φ n ) = Λ + Λ n /φ n + · · · ,which would give a common origin for the cosmological constantand the chameleon. Note that the chameleon cannot acceleratecosmologically without a cosmological constant [42]. minimum of the effective potential is density-dependent φ min ( ρ ) = (cid:18) nM pl Λ n αρ (cid:19) n +1 , (7)and hence so is the effective mass of the field about saidminimum m = V (cid:48)(cid:48) eff ( φ ) = n ( n + 1)Λ n +4 (cid:18) αρnM pl Λ n +4 (cid:19) n +2 n +1 . (8)Since the cosmological and terrestrial density vary by 29orders of magnitude, the parameters can be chosen suchthat the chameleon force in laboratory experiments issub-micron. Current experimental searches [20, 40] im-ply that the chameleon cannot drive the cosmic accelera-tion [42] but the chameleon force can still be relevant forcosmology on smaller (Mpc) scales.Astrophysically, the chameleon profile of a spherically-symmetric object of mass M and radius R is not sourcedby the object’s mass but rather by the mass inside a shellnear the surface, a phenomenon that has been dubbed thethin-shell effect . This is depicted in figure 1. The rea-son for this is the following: deep inside the object, thefield minimizes its effective potential corresponding to theambient density but, as one moves away from the center,the field must eventually roll in order to begin to asymp-tote towards the minimum at the density of the mediumin which the object is immersed (galactic densities orcosmological densities depending on the situation). Thefield can only roll once the density is low enough so thatits effective mass is light enough. The radius at whichthis happens is typically called the screening radius r s ,and only the mass inside the screening radius sources amodification of the Newtonian potential, which is givenby V ( r ) = GMr (cid:20) α (cid:18) − M ( r s ) M (cid:19) e − m eff r (cid:21) . (9)Objects for which r s ≈ R have drastically suppressedYuakawa forces since M ( r s ) ≈ M whereas those where r s (cid:28) R have strong enhancements. These two situationsare referred to as screened and unscreened respectively.The screening radius of an object can be determined fromthe relation [20, 39, 43, 44] φ BGmin αM pl = 4 πG (cid:90) Rr s r (cid:48) ρ ( r (cid:48) ) d r (cid:48) , (10)where φ BGmin is the asymptotic (background) value of thefield far from the object. If equation (10) has no solutionsthen r s = 0 and the object is fully unscreened.Chameleon theories violate the weak equivalence prin-ciple [38]. Indeed, one can define a scalar charge for anobject Q i = M i (cid:18) − M i ( r i s ) M i (cid:19) (11) FIG. 1. Chameleon screening. Only the mass inside thescreening radius r s contributes to the fifth-force. so that the force on an object due to an externally appliedchameleon field is F cham = αQ i ∇ φ ext (this is analogousto the ‘gravitational charge’ M so that F grav = M ∇ Φ extN where Φ extN is an externally applied Newtonian potential).Two objects of different masses and internal compositionswill have different scalar charges and will therefore fall atdifferent rates in an externally applied chameleon field,signifying a breakdown of the weak equivalence principle(WEP). The chameleon force between two bodies, A and B , is [45, 46] F AB = − GM A M B r (cid:0) α Q A Q B e − m eff r (cid:1) (12)and as a result of this the PPN parameter γ is (see Ap-pendix A for the derivation of this result and [20, 44, 47,48] for other approaches) γ = 2 (cid:2) α Q A Q B e − m eff r (cid:3) − − . (13)In this formula, body A is the body responsible for thedeflection/time delay of light and body B is a separatebody used to measure the mass of body A . For example,for light bending by the Sun one would take A as the Sunand B as the Earth. See Appendix A for more details.
2. Symmetron Screening
The symmetron model [26] screens in a similar fashionto the chameleon—in the sense that it utilizes a mecha-nism similar to the thin-shell effect—but differs on howthis is achieved. Instead of having a large mass insidethe screening radius, the symmetron has a light mass(in all environments) but an environmentally-dependentcoupling to matter that becomes zero. It’s equation ofmotion is ∇ φ = d V eff d φ (14)where the effective potential is V eff ( φ ) = − µ (cid:18) − ρµ M (cid:19) φ + λ φ , (15)which is sketched in figure 2. This represents a field witha tachyonic mass µ and a field-dependent coupling tomatter α ( φ ) = M pl φM . (16)The effective potential can have two shapes dependingon the magnitude of the ambient density: either there isa single minimum at φ = 0 when ρ > ρ (cid:63) ≡ µ M (17)or there are two degenerate minima at φ = φ ± ≈ ± µ √ λ (18)when ρ < ρ (cid:63) . In both cases, the mass about this mini-mum is m = V (cid:48)(cid:48) eff ( φ ) ∼ µ so that the mass does notvary significantly with density. When ρ > ρ (cid:63) the couplingvanishes identically since φ = 0 whereas when ρ < ρ (cid:63) thecoupling is α ≡ | α ( φ ± ) | = µM pl √ λM . (19)The screening then works as follows: Given a sphericalobject embedded in a larger background of lower density( ρ < ρ (cid:63) ), the field will lie at φ = 0 at the center (providedthe density ρ > ρ (cid:63) at some radius) and will remain hereuntil the screening radius, at which point it begins toasymptote to φ ± . When r < r s the coupling is zero andthere is no fifth-force but when r > r s the coupling is α (given in (19)) and one finds, outside the object, V ( r ) = GMr (cid:20) α (cid:18) − M ( r s ) M (cid:19) e − µr (cid:21) . (20)Like Chameleons, symmetrons also violate the WEPand one has precisely the same scalar charge as definedin equation (11) so that the force between two bodies isgiven by equation (12) with m eff = µ and α → α . ThePPN parameter γ is, similarly, γ = 2 (cid:2) α Q A Q B e − µr (cid:3) − − . (21)where one again takes r to be the typical length-scale ofthe experiment. The screening radius can be found byevaluating [20] M = (cid:90) Rr s r (cid:48) ρ ( r (cid:48) ) d r (cid:48) . (22)If there is no solution then r s = 0 and the object isfully unscreened. For the Sun, this is the case when M s > ∼ . × GeV and for the Earth one finds this isthe case when M s > ∼ . × GeV. ●● FIG. 2. The symmetron effective potential.
3. Vainshtein Screening: Galileons
The Vainshtein mechanism [49] is very general and isfound ubiquitously in theories of massive gravity [50],braneworld models [51], and general scalar-tensor theo-ries [52–59]. In this paper, we will illustrate it by consid-ering two simple and well-studied models that have be-come quintessential paradigms for the Vainshtein mech-anism, the cubic galileon [27] ∇ φ + r c (cid:2) ( ∇ φ ) − ∇ i ∇ j φ ∇ i ∇ j φ (cid:3) = 8 παGρ, (23)and the quartic galileon ∇ φ + r c (cid:2) ( ∇ φ ) − ∇ φ ∇ i ∇ j φ ∇ i ∇ j φ +2 ∇ i ∇ j φ ∇ j ∇ k φ ∇ k ∇ i φ (cid:3) = 8 παGρ. (24)The new parameter r c is referred to as the crossoverscale . In each case, one can see that the left hand sideof the equation of motion contains a Poisson term and anon-linear term. The relative importance of each term isdetermined by the Vainshtein radius r = (cid:40) αGM r c , cubic galileon √ αGM r c , quartic galileon . (25)When r (cid:29) r V the Poisson term dominates so that thefield is Brans-Dicke-like and one has O ( α ) fifth-forces.On the other hand, when r (cid:28) r V the non-linear termsare dominant and one finds a total force F = GMr (cid:20) α (cid:18) rr V (cid:19) p (cid:21) , (26) Note that we have taken the scalar to be dimensionless in con-trast to the chameleon and symmetron scalars in order to matchthe conventions of reference [60]. The cubic galileon is a certain limit of five-dimensional braneworld models and this scale determines when the extra dimensionis important. In the case of the quartic galileon there is no analogbut we use the same symbol for the new parameter for the sakeof consistency. where p = 3 / p = 2 for thequartic. One can see that deviations from the inverse-square law are highly suppressed by powers of r/r V fordistances inside the Vainshtein radius. For the solarsystem, the relevant Vainshtein radius is that of theSun, which is of order 100 pc, and so deviations fromGR are highly-screened in the solar system. Cosmolog-ically, galileons can self-accelerate without the need fora cosmological constant provided that r c ∼ r c are not dom-inant cosmologically but represent new and interestingpotential modifications of gravity that have yet to bewell-constrained. Unlike chameleons and symmetrons,galileons obey the weak equivalence principle i.e. Q = M [38], although non-linear effects mean that this may notbe the case for two or more extended bodies in close prox-imity [62]. III. Present and Future Space-Based Experiments
In this section we briefly describe the experiments wewill use to constrain the screened modified gravity mod-els presented above. In particular, we will indicate therelevant tests for these theories and state the predictedprecision with which the appropriate parameters can bemeasured.
A. Cassini
Primarily a mission to study the physics of Saturn, theCassini satellite and Earth were in conjunction in 2002,allowing for a test of GR using the Shapiro time delayeffect. Signals sent from Earth to the satellite (en routeto it’s future host planet at the time) that passed the Sunwith different impact parameters were able to overcomethe noise due to the solar corona and allow an accuratemeasurement of the time delay. The resulting constrainton the PPN parameter γ , | γ − | < . × − , is currentlythe strongest bound on this parameter to date. B. Lunar Laser Ranging
Laser ranging to five reflectors placed on the moon dur-ing the Apollo and Lunokhod missions can measure therelative distance between the Earth and the Moon withmm precision. This is achieved by measuring the round-trip time for short laser pulses aimed at these reflectors,a technique known as lunar laser ranging (LLR). The in-credibly high precision has allowed for tests of generalrelativity at the Earth-Moon distance (10 cm). In par-ticular, the time-variation of Newton’s constant has beenmeasured to ˙ G/G < × − yr − , the inverse-squarelaw has been verified with a precision δV /V < . × − ,and the equivalence principle has been tested to 10 − [63–65]. In the latter case, the bound refers to the differ- ential acceleration between the Earth and Moon towardsthe Sun ( a ⊕ − a (cid:36) ) /a N , where a N is the Newtonian ac-celeration. C. Phobos Laser Ranging
Building on the success of LLR, it has been proposedto land a pulsed laser transponder on the surface of Pho-bos [66]. The resulting Phobos laser ranging (PLR) pro-gram would be able to achieve mm-level accuracies at theEarth-Mars distance (1.5 AU). Earth and Mars would bein conjunction after 1.5 years, with a second and thirdconjunction in three and six years respectively. At con-junction, the laser pulses would pass close to the Sunand experience a strong Shapiro time delay effect dueto the warping of space-time. This would allow for con-straints on the PPN parameter γ to 10 − –10 − levels(the latter could be achieved after three conjunctions).Additionally, the time-variation of G could be measuredto ˙ G/G < ∼ − yr − and the inverse-square law couldbe tested to 10 − at the Earth-Mars distance. Notonly does this allow for tests of general relativity onsmall scales but theories such as galileons that predictforces that increase with distance could be constrainedto new levels. The equivalence principle could be testedto the 10 − level using the Earth-Mars-Sun-Jupiter sys-tem [67]. D. LATOR
The Laser Astrometric Test of Relativity (LATOR)[68–74] aims to place two microsatellites in heliocentricorbits on the far side of the Sun with orbital radius 1AU. Lasers placed on the satellites will send light signalsto an optical interferometer placed on the internationalspace station (ISS). The line of sight of each microsatel-lite passes at close but different distances to the Sun sothat the entire configuration forms a triangle. If therewere no warping of space-time by the Sun then the ge-ometry of this triangle would be exactly Euclidean butthe warping leads to departures from this, which man-ifests as a deflection of the laser signal. Measuring theamount by which the properties of the triangle deviatefrom their Euclidean values therefore probes the PPNparameter γ , which will be measured to an accuracy | γ − | ∼ . × − . E. GTDM
The gravitational time delay measurement (GTDM)experiment [75] proposes to measure the Shapiro timedelay effect using a similar configuration to LATOR, thedifference being that laser ranging between two drag-freesatellites, one at the L1 Lagrange point of the Earth-Sunsystem and one in a LATOR-type orbit, is to be used.
Cassini
PLRLATOR
BEACON - - - E o .. t - Wash
AstrophysicsInterferometry P r e c i s i on A t o m i c M ea s u r e m en t s Len s i ng ●● Casimir ●● Microspheres ●● E o .. t - Wash
AstrophysicsInterferometry P r e c i s i on A t o m i c M ea s u r e m en t s Len s i ng ●● Casimir ●● Microspheres ●● BEACON - - - - - FIG. 3. The potential exclusion range for n = 1 chameleon models. Left Panel : The region excluded in the Λ– α plane byfuture space based tests of the PPN parameter γ ; the exclusion range due to each experiment is indicated in the figure. Inthe case of PLR, the green region indicates the constraints that could be obtained if three conjunctions are achieved; thecorresponding region for one and two conjunctions are shown using the dotted and dashed green lines respectively. Note thatwe have normalized the chameleon mass-scale Λ to the dark energy scale so that a value of zero indicates that the chameleon anddark energy may have a common origin. Right Panel : Comparison with other chameleon bounds coming from the experimentslabelled in the figure. In this case we have normalized Λ in units of eV and have translated the bounds into M c = M pl /α inorder to conform with conventions in the experimental literature. The black dotted line shows the dark energy scale and thecolored arrows indicate lower bounds on M c coming from neutron bouncing and interferometry experiments. See [20] for adescription of each of these experiments, and the resulting bounds. This will measure the PPN parameter γ to | γ − | < × − . We will not consider GTDM here since thiswill not be as strong as the LATOR constraint, whichoperates at the same distances. F. BEACON
Rather than measuring the space-time curvaturesourced by the Sun, the Beyond Einstein Advanced Co-herent Optical Network (BEACON) [76] will attempt tomeasure the space-time warping by the Earth. Foursmall satellites will be placed in circular orbits aroundthe Earth (at distances of order 8 × km) in a trape-zoidal configuration. Laser transceivers on each satellitewill allow the distances between any two satellites to bemeasured with high precision. The laser beams of op-posite satellites passes close to the Earth and thereforethe signal picks up a time delay due to the warping ofspace-time. Modulating the position of one spacecraftrelative to the others changes the impact parameter andwill therefore allow a measurement of the PPN parameter γ , which will be measured to an accuracy | γ − | = 10 − . IV. Potential ConstraintsA. Chameleons and Symmetrons
Since Chameleons and Symmetrons screen in a quali-tatively similar manner, we will consider potential testsof both simultaneously. Whereas chameleons and sym-metrons do give rise to deviations in the inverse-squarelaw, this is only the case for a narrow range of parame-ters (where the effective mass is of order the Earth-Moonor Earth-Mars distance for LLR and PLR respectively).Furthermore, it is likely that the field sourced by Marswill environmentally-screen Phobos (see [38] for a discus-sion on this) and a modeling of this effect is very com-plicated due to the high degree of non-linearity in thesystem [45, 46]. For this reason, tests using the PPN pa-rameter γ are cleaner and cover a larger range of param-eter space; we will therefore focus on the bounds thatfuture tests of this parameter place on chameleon andsymmetron theories.The tests described above will constrain γ using thespace-time warping due to either the Sun (CASSINI,PLR, LATOR) or the Earth (BEACON) using either theShapiro time delay effect or by measuring the deflectionof light. The PPN parameter γ is given in equations (13)and (21) for chameleon and symmetron models respec-tively. In the case of Cassini, PLR, and LATOR, body A is the Sun and body B is the Earth since its orbitaldynamics are used to measure the Sun’s mass. For BEA-CON, body A is the Earth and body B is the LAGEOSsatellite, which has been used to make a measurementof the geocentric gravitational constant [77]. In practice,the LAGEOS satellite is fully unscreened (its Newtonianpotential GM/Rc is of order 10 − ) and hence acts likea point particle. For this reason Q ≈ γ will exceed the projected bound coming fromeach experiment. In the case of the Sun, we use the solardensity profile of [78]. We take the dark matter densityin the solar neighborhood to be ρ DM = 0 .
324 GeV/cm (6 × − g/cm ) [79], which sets the background valueof the field φ BGmin for chameleons given a set of parameters.For the Earth, we assume a mean density of 5 .
51 g/cm and for the LAGEOS satellite we calculate the mean den-sity ρ = 3 M/ (4 πR ) ≈ .
45 g/cm (assumed constant)using the mass (407 kg) and radius (60 cm).In the left panel of figure 3 we show the potential re-gions of parameter space that could be excluded for n = 1chameleon models and compare these with current exper-imental constraints taken from [20]. Note that it is com-mon in the experimental literature to write α = M pl /M c and so we do the same here for comparative purposes.One can see that only BEACON will be competitive withcurrent experimental searches. It is interesting that theregion constrained by BEACON is the region where Λ isof order the dark energy scale. BEACON therefore hasthe possibility to rule out n = 1 chameleon models thatmay have a common origin with dark energy.In figure 4 we show the region of symmetron param-eter space that could potentially be excluded. We fo-cus on models with µ = 8 × − eV so that the forcerange is larger than one AU. Note that the parameterrange in is very different those considered in laboratorytests [13, 80–83]. Symmetrons are far less constrainedthan chameleons and there are typically large gaps inthe parameter space separating constraints from labora-tory and astrophysical tests [20]. In order to be able toprobe screened fifth-forces in a laboratory setting, theCompton wavelength must be of order (or smaller than)the width of the walls of the vacuum chamber in whichthe experiment is performed. Chameleons can vary theirmass over many orders of magnitude and therefore differ-ent laboratory tests can probe a complementary range ofparameter space whereas symmetrons have a fixed massof O ( µ ) and hence the range of parameters that can beprobed is limited. The parameters we consider in figure3 are adapted to the solar system rather than laboratory tests.One can see from figure 4 that the same region thatis constrained by solar system tests is also probed by as-trophysical tests using distance indicators and rotationcurves [39, 43, 84–87]. In this case, solar system testsconstrain a complementary region of parameter space toastrophysical tests, and therefore future space-based testswill cover a currently unconstrained region. Note that itis not possible to extend the astrophysical bounds to ar-bitrarily small values of M because these tests requiredwarf galaxies in cosmic voids to be unscreened. Accord-ing to equation 22, these galaxies would become screenedat small M and, in fact, the minimum value of M con-strained in the figure is close to the threshold for theonset of screening in dwarf galaxies. B. Cubic and Quartic Galileons
Galileon theories produce deviations in the inverse-square law (see equation (26)) which can be tested withlaser ranging. Since the galileon force generated by theEarth scales with distance to some positive power, itis stronger at the Earth-Mars distance than the Earth-Moon distance. PLR will therefore improve the boundsover the current LLR bounds [88, 89]. According to equa-tion (26), one has δVV = (cid:18) rr V (cid:19) p , (27)with r V given in equation (25) and where p = 3 / , r should betaken to be the Earth-Mars distance in the case of PLR.Demanding that δV /V is less than the bound reportedfrom LLR and the predicted sensitivity of PLR (both10 − at the Earth-Moon and Earth-Mars distance re-spectively) we obtain the bounds (LLR) and predictedimprovements (PLR) shown in figure 5.There have been few tests of galileon theories on smallscales to date, due mainly to the efficiency of the Vain-shtein mechanism. LLR yielded the strongest constraintsuntil they were overtaken recently by tests using super-massive black holes (SMBHs) [60]. Galileons predictviolations of the strong equivalence principle (SEP) sothat black holes do not couple to external galileon fieldswhereas non-relativistic matter does [90, 91]. The accel-eration of a galaxy infalling into a massive cluster receivesa large but subdominant contribution from the galileonfield of the cluster that the SMBH at its center doesnot feel. For this reason, as the galaxy falls towards thecenter of the cluster the black hole begins to lag behindand is eventually stabilized by the restoring force of the This is because extended distributions do not screen as efficientlyas point sources in galileon theories [60].
CassiniPLR
LATORBEACONCassiniPLR
LATORBEACON A s t r oph ys i cs - - - - - - FIG. 4. Current and potential constraints on the symmetronmodel with µ = 8 × − eV from space based tests of thePPN parameter γ . The shaded regions correspond to thecurrent or future experiment indicated in the figure. Thebounds from astrophysical searches are also shown. baryons left at the galactic center. This results in an ob-servable offset that can be as large as O (kpc). Reference[60] used the lack of an offset in the central SMBH of M87(located in the Virgo cluster) to place the constraintsthat we also show in figure 5. One can see that theseare stronger than the current LLR bounds but that thebounds from PLR would supersede these. The boundsfrom PLR would therefore be the strongest bounds ongalileons gravity models on small scales. C. Other Tests of Gravity in Space
We end by discussing other possibilities for testingscreened modified gravity in space that we will not fore-cast here due to technical difficulties with the theoreticalmodeling. Our goal is to point out that future effort inthis direction could yield fruitful results.
1. Tests of the Time-Variation of Newton’s Constant
All three of the screening mechanisms mentioned herepredict that Newton’s constant is time-dependent. Thisis primarily because the asymptotic value of the field isgiven by the cosmological or galactic field value (depend-ing on the parameters one chooses). For chameleon andsymmetron models, the galaxy (and therefore the Sun)must be unscreened for the former case to be relevantand, in the latter case, the time-dependence of the field is very model-dependent and requires detailed N-bodysimulations or excursion set methods to predict [92]. Forthis reason, it is difficult to use the time-variation of G to constrain symmetron and chameleon models.Galileon models predict a strong variation in the time-dependence of G [93, 94]. Again, this time-dependencecomes from the cosmological boundary conditions thatmake the coupling to matter α time-dependent. Thistime-dependence is fixed by the cosmology of the galileonand therefore constrains a combination of the fundamen-tal model parameters, as well as ˙ φ , H , and Ω m . Sincewe are not interested in these parameters here we willnot attempt to forecast how the improved bounds on˙ G/G will impact galileon cosmology, but, for complete-ness, we will make a few pertinent observations. First,galileon models with a direct coupling to matter (thatwe study here) have not been well-studied in the con-text of cosmology; some references [95–97] have studiedthe theoretical cosmology but no fiducial model has beenproposed. Second, those that have been studied differfrom the models studied in this work in that they pro-duce self-acceleration by having a phantom quadratic ki-netic term (i.e the term ∇ φ → −∇ φ in the equationsof motion (23) and (24)). These models are in tensionwith (but not excluded by) current cosmological data [98]and are strongly ruled out (except for very fine-tunedcases) by the recent LIGO/Virgo-Fermi observation ofgravitational waves and a gamma ray burst from merg-ing neutron stars, which constrains the difference in thespeed of gravitons and photons to the 10 − level [2–8]. Finally, many different theories of gravity, includ-ing massive gravity [50, 99, 100] and Horndeski theories(and their generalizations) [101–104] reduce to the samegalileon theories considered in this work on solar systemscales but give very different cosmologies. For this rea-son, constraining the time-variation of G does not con-strain the fundamental parameters α and r c consideredhere. In many of these theories, the matching of smallscales to a cosmological backgrounds presents a separatetechnical challenge because one needs to make sure touse a metric for the solar system that is consistent withcosmological asymptotics [105–108].
2. Tests of Other PPN Parameters
In addition to γ , some of the tests mentioned abovewill also probe the PPN parameters β and δ (for the firsttime in this case). The resulting bounds on these param-eters will be weaker than for γ , and, since the relevantcombination of parameters is the same for these addi-tional parameters, no new information will be gained byconstraining them . This conclusion may be different This is only true for screened modified gravity theories and isa result of the Newtonian scalar field ( O ( v /c )) being highly- LLR PLR SMBH
200 500 1000 2000 50000.0010.0100.100110
Cubic Galileon
LLR PLR SMBH
200 500 1000 2000 50000.1110100100010 Quartic GalileonFIG. 5. Current and potential constraints on the cubic (left panel) and quartic (right panel) galileon models. The blue regionsare the current bounds from LLR and the region above the black dashed line is excluded by the SMBH tests of reference [60].The red region shows the parameter space that could be potentially ruled out with PLR. for theories that include a disformal coupling to matter[110].
3. Tests of the Equivalence Principle
The Strong Equivalence Principle:
The Earth-Sun-Mars-Jupiter system allows for a novel test of theSEP [67] that could be performed using PLR [66]. Sincechameleon and symmetron models violate the WEP, thissystem could be used to test these theories. In partic-ular, all of these bodies will have different thin-shells(and hence scalar charges Q ) and will therefore fall to-wards any given body at a different rate. The analysisby [67] involved integrating the equations of motion forthe four-body system including higher-order effects suchas tidal forces. Such an analysis would be more diffi-cult for chameleon and symmetron models due to theirnon-linear nature. Any analytic solution would requiredifferent approximations because the regime of validitydepends on the model parameters [45, 46]—for example,if the mass of the field becomes of order the separationbetween two bodies then superposition no longer holds—and deviations from spherical symmetry (including tidaleffects) are harder (but not impossible [111]) to model.In practice, it is likely that a numerical integration ofthe field equations will be necessary to find the resultingconstraints.Similarly, whereas the WEP is satisfied for a single ex-tended object in galileon theories, two or more extended suppressed. More general scalar-tensor theories will source post-Newtonian fields that are sensitive to higher-order corrections to α , which will be additionally constrained by measuring β and δ .Additionally, Vainshtein screened theories have additional termsin the metric that are not captured by the PPN expansion alone[109] and it may be possible to constrain these. objects may violate the WEP due to the failure of super-position that results from the high degree of non-linearityin the equations of motion. It is likely that the theoreticalmodeling of this four-body system would be even harderfor galileon theories. Their equations are harder to solveand may have multiple branches of solutions, deviationsfrom spherical symmetry are poorly understood (exceptin other symmetric situations [112]), and it is not clearthat perturbation theory works for these models [113]. The Weak Equivalence Principle:
There are sev-eral proposed experiments that will measure the WEP us-ing accelerometers (of various design) orbiting the Earth[114–116]. The perpetual free-fall of these accelerometerswill allow for longer experiments. The satellite test of theequivalence principle (STEP) [116] will reach a precisionof 10 − , five orders of magnitude stronger than the cur-rent bounds from LLR (10 − ). In all cases, these exper-iments consist of a capsule in orbit around Earth withtwo test-bodies (typically cylinders that are designed toresemble spheres to high multipole moments) that free-fall towards the Earth, and accelerometers designed tomeasure any difference between the free-fall rates. It isunlikely that these experiments will constrain galileontheories because the capsule is a point mass to a goodapproximation but chameleons and symmetrons are verysensitive to the precise geometry of experimental cham-bers (see [20] for a review of experimental tests). Thefact that these accelerometers operate with a highly non-symmetric geometry makes the theoretical modeling ofthe field profile very difficult and a numerical treatmentwould be necessary in order to make predictions. The pa-per that originally introduced chameleons estimated therange of parameters for which a STEP-like experimentwould be unscreened [25] by demanding that the capsulehas no thin-shell, but going beyond this would requireconsiderably more effort and so we do not attempt thishere. It may be that a dedicated vacuum chamber in0space with a geometry specifically tailored to optimizethe chameleon and symmetron WEP violations wouldprovide more stringent results than a detailed reanaly-sis of the current generation of planned and proposedaccelerometers. V. Conclusions
In this work we have explored the implications thatcurrent, planned and, proposed space-based tests of rel-ativistic gravitation have for theories of gravity that in-clude screening mechanisms . Screening mechanisms usenon-linear equations of motion to dynamically suppressdeviations from GR in the solar system without the needto tune the theory parameters to negligible values. Theytherefore allow the theories to be relevant on cosmologi-cal scales, potentially allowing them to address the darkenergy mystery. We have examined three well-studiedand common paradigms for screening: chameleon, sym-metron, and galileon models. (The latter models areparagons for Vainshtein screening, which occurs in a verybroad class of scalar-tensor theories.)In the case of chameleon and symmetron models, whichscreen in a qualitatively similar manner using the thin-shell effect (see section II), we have argued that space-based tests of the PPN parameter γ using either laserranging to Phobos (PLR) or optical networks (LATORand BEACON) will provide the best constraints. Onlythe potential BEACON bounds on chameleon models willprobe into the region of parameter space not yet cov-ered by current tests. BEACON has the ability to fill inthe remaining region around the dark energy scale wherechameleons and dark energy may have a common origin.The bounds on symmetrons will be complementary tocurrent bounds from astrophysical probes. For galileonmodels, the strongest constraints would come from test-ing the inverse-square law at interplanetary distances us-ing PLR. In particular, tests of the inverse-square lawwould provide the strongest constraints on galileon mod-els to date.Finally, we have discussed whether or not the next gen-eration of experiments aimed at testing the strong andweak equivalence principles in space could provide newand improved constraints. Ascertaining how strong thesewould be (if at all) is difficult due to uncertainties in thetheoretical modeling of the both the four-body field pro-file and dynamics of the Earth-Sun-Jupiter-Mars system,and the proposed accelerometers that will be placed inorbit around Earth. In the former case, one simply needsto numerically solve the non-linear equations. In the lat-ter, it is likely that only a small range of parameters canbe probed, and it may be more fruitful to have a specifi-cally designed vacuum chamber in space.To paraphrase the paper that first introducedchameleons [24]: 14 years later, screened modified gravityis still awaiting surprise tests for gravity in space. Acknowledgments
The author is indebted to the anonymous refereefor their detailed reviewing of this manuscript, and toJustin Khoury for numerous useful discussions. I wouldlike to thank Bhuvnesh Jain, Mark Trodden, AlexanderVikman, and especially Slava Turyshev for comments onthe manuscript. JS is supported by funds provided tothe Center for Particle Cosmology by the University ofPennsylvania.
A. Light Bending and Time Delay in Chameleonand Symmetron Theories
The purpose of this Appendix is to calculate the PPNparameter γ that is relevant for chameleon and sym-metron theories. We briefly review the pertinent the-oretical aspects of these theories before deriving a valuefor γ . The reader is referred to [20] and references thereinfor further details.Chameleon and symmetron models are both scalar-tensor theories defined in the Einstein frame by S = (cid:90) d x √− g (cid:34) M R ( g ) − ∇ µ φ ∇ µ φ − V ( φ ) (cid:35) + S m [˜ g ] , (A1)where the Jordan frame metric ˜ g µν is a Weyl rescalingof the Einstein frame metric g µν by a conformal factor A ( φ ) ˜ g µν = A ( φ ) g µν . (A2)The coupling α ( φ ) = M pl d ln A ( φ )d φ (A3)and the specific model is set by the choice of A ( φ ) and V ( φ ). The specific model is unimportant for what fol-lows.Our starting point for the derivation is the PPN metricfor a single body, which we will refer to as body A withmass M A ˜ g = − G PPN M A r (A4)˜ g ij = (cid:18) γ G PPN M A r (cid:19) δ ij , (A5)where we use tildes to refer to the Jordan frame met-ric, which governs the geodesics of point particles. Theparameter that controls light bending/time delay mea-surements in this metric is ˜ γ . As we will see shortly, fortheories that violate the WEP (such as ours) this is differ-ent from the parameter γ constrained by measurementsof these effects. We will refer to the quantity G PPN asthe PPN gravitational constant because its value controls1the size of effects computed using the PPN metric. It isdistinct from the gravitational constant G appearing inthe action and the gravitational constant measured onEarth. The time delay and gravitational lensing of lightis given by [117]∆ t = 2 (1 + ˜ γ ) G PPN M A c F ( b, x µ ) and (A6) δθ = (cid:18) γ (cid:19) G PPN M A bc F ( b, x µ ) (A7)where b is the impact parameter and F are geometricfactors that depend on the geometry used to perform themeasurement [118]. Their expressions are not necessaryfor what follows.We now wish to calculate the Jordan frame metric inPPN form. we first calculate the Einstein frame metric,defined by g = − g ij = (1 + 2Ψ) δ ij . (A9)One finds [38] ∇ Φ = ∇ Ψ = − πGρ A (A10)so that Φ = Ψ = GM A r (A11)up to irrelevant integration constants set by the boundaryconditions. For the scalar, we are interested in the regimewhere this body has some degree of screening. In thiscase, the equation inside the screening radius is ∇ φ = 0 (A12)while outside the screening radius it is ∇ φ = 8 παGρ A r > r s , (A13)where α = α ( φ ), which is constant for chameleons andgiven by (19) for symmetrons. We have ignored thescalar’s mass m eff = V (cid:48)(cid:48) eff ( φ ) ( ≈ µ for the symmetron)since we expect m eff R (cid:28) e − m eff ( φ ) R (more technical and cumbersome derivationsfind this factor [47]). The solution is then φ = φ − α Q A GM A r , (A14)where the ‘scalar charge’ of body A is Q i = (cid:18) − M A ( r A s ) M A (cid:19) . (A15)Transforming to the Jordan frame using equation (A2)and expanding A ( φ ) to first order in GM A /r one findsequations (A4) and (A5) with G PPN = G (cid:2) α Q A (cid:3) (A16)˜ γ = 1 − α Q A α Q A . (A17)Had we been dealing with a theory with no WEP vio-lations our task would be complete since one could sim-ply apply the constraints on γ to (A17) but WEP viola-tions imply that it is possible that G PPN can differ fromthe value of Newton’s constant measured by local exper-iments. To see this, it is simpler to consider the product GM . 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