Testing gravity on kiloparsec scales with strong gravitational lenses
aa r X i v : . [ a s t r o - ph . C O ] A p r Testing gravity on kiloparsec scales with strong gravitational lenses
Tristan L. Smith
Berkeley Center for Cosmological PhysicsPhysics Department, University of California,Berkeley, CA 94720 (Dated: May 30, 2018)Modifications to GR generically predict time and scale-dependent effects which may be probed byobservations of strong lensing by galaxies. Measurements of the stellar velocity dispersion determinethe dynamical mass whereas measurements of the Einstein radius determine the lensing mass. In GRthese two masses are equal; in alternative gravity theories they may not be. Using measurements ofthe stellar velocity dispersion and strong lensing around galaxies from the Sloan Lens ACS (SLACS)survey we place constraints on lensing in modified gravity theories and extend previous studies byapplying this data to explore its dependence on various properties of the lens such as the lens redshiftor mass and thereby constrain scalar-tensor, f ( R ) gravity theories, and generic parameterizationsof deviations from GR. Besides applying the observations to these specific gravity theories, the dataplaces a constraint on a generic dependence of modifications to GR on the lens mass and redshift.At the 68% confidence level we find that the ratio between the lensing and dynamical masses canonly vary by less then 50% over a mass range for the lens galaxies of 10 . M/M ⊙ . and lessthan 40% over the redshift range 0 . < z < . PACS numbers: 95.30.Sf, 04.50.Kd,04.80.Cc
I. INTRODUCTION
The ability to test our basic understanding of gravityhas been surprisingly limited [1]. Most precision testshave concentrated on the motion of the planets and lightwithin the solar system or the motion of binary pul-sars. Although measurements in the solar system havereached the level of testing deviations from general rela-tivity (GR) to one part in 10 [2, 3], they only constraintheories in the weak gravity limit, on scales of an AU( r ∼ cm), and at a single redshift, z = 0. Binarypulsar systems similarly test theories at z = 0 and on rel-atively small scales but have the added aspect of testinggravity in the limit where it is large due to the compactnature of neutron stars [1].The discovery of an accelerated cosmic expansion [4, 5]has led to a flurry of theoretical activity. Although it ispossible to explain the accelerated expansion within GRby introducing a cosmological constant or new cosmicscalar field or other source of energy density, anotherapproach is to consider these observations as the firstobservational indication of a need for modifications toEinstein’s theory of GR [6].Many groups have attempted to explain the accel-erated expansion within alternative theories of grav-ity. These observations can be explained as the resultof the dynamics of a scalar field within a generalizedscalar tensor theory [7–10]. Another proposal modifiesthe Einstein-Hilbert action by the addition of a generalfunction of the Ricci scalar, f ( R ) [11, 12]. Depend-ing on the functional form of the function f ( R ), sucha term may give rise to late-time accelerated expansion.Refs. [13, 14], proposed a five-dimensional theory of grav-ity which may lead to an epoch of late time accelerated expansion. In addition to studies dedicated to the obser-vational consequences of specific modified gravity theo-ries there has also been interest in parameterizing genericdeviations from GR on cosmological scales [15–18].Many aspects of these modifications can be constrainedor even ruled out by considering tests of gravity made inthe solar system or through observations of binary pul-sars [19–22]. However, given that these theories are nat-urally dynamical and scale dependent, solar system andpulsar tests can be of limited use. Therefore it is impor-tant to test gravity at a variety of scales and redshifts.In particular, since those modified gravity theories whichare able to produce an epoch of late-time accelerated ex-pansion must become dynamically important when theacceleration starts to dominate (around z ∼ . γ PPN and uses the observations to place constraints on how γ PPN may depend on various properties of the lens, suchas its redshift or mass. This paper also extends the anal-ysis to the full SLACS data set of 53 lens systems as wellas uses a more realistic model for the luminosity profileof the lens galaxy (a Hernquist profile, as opposed to apower-law profile used in Ref. [25]). The use of a morerealistic luminosity profile leads to a significant shift inthe best fit γ PPN .This paper is organized as follows. In Sec. II we presenthow scalar modifications to gravity affect the dynamicsof massive test particles (i.e., stars) and photon trajec-tories differently. We discuss how a comparison betweenthe dynamics and lensing signal leads to a test of gravity.In Sec. III we discuss the predictions from general scalartensor theories with a massive scalar field. In Sec. IV wediscuss the predictions from f ( R ) gravity and empha-size its ability to rapidly suppress any deviations fromGR and how this transition presents unique observationalsignatures depending on the mass of the lens galaxy. InSec. VI we discuss how measurements of strong galaxylenses can be used to constrain modified gravity theories.We present our conclusions in Sec. VII. II. LENSING AND DYNAMICS INWEAK-FIELD LIMIT OF MODIFIED GRAVITYTHEORIES
One of the basic ways that we can distinguish betweendifferent theories of gravity is through a comparison be-tween the predicted and observed motion of test particles.Such tests compare the motion of photons (which moveon null geodesics) and the motion of non-relativistic mas-sive particles (which move on time-like geodesics). Thedifferent types of geodesics are sensitive to different com-ponents of the metric and hence their comparison allowsus to measure those components.We start with the line element corresponding to weakgravity, ds = − (1 + 2Ψ) dt + (1 − δ ij dx i dx j , (1)which has been written in the conformal Newtoniangauge and has introduced the two Newtonian potentials,Ψ and Φ. The ‘bare’ Newtonian potential is given by theusual Poisson equation, ∇ Φ N ( ~x ) = 4 πGρ. (2)A general modification to gravity introduces two newequations. One relates the potentials to the underlyingmass density, 12 ∇ (Φ + Ψ) = 4 πµGρ. (3)The other relates the potentials to one anotherΦΨ = γ PPN . (4)Note that GR is regained when µ = 1 and γ PPN = 0.In general both µ and γ PPN can depend on a variety ofquantities that determine the space-time such as the localmass density, position, redshift, and so forth. We will seespecific examples of this in the following sections.In order to distinguish between the two Newtonian po-tentials we compare the dynamics of stars within a galac-tic halo and the deflection of light around the halo. The deflection of the image of a background source throughan angle ˆ α is given byˆ α = Z ~ ∇ ⊥ (Ψ + Φ) dℓ, (5)= 2 µ Z ~ ∇ ⊥ Φ N dℓ, (6)where ~ ∇ ⊥ is the gradient transverse to the photon’s un-perturbed trajectory and dℓ is a length element alongthat trajectory and we have assumed that µ is inde-pendent of position on the relevant scales. In terms ofthe bare potential, Φ N , observations of stellar dynamicsthrough the spherical Jeans equation [28] measure thecombination Ψ = 2 µ γ PPN Φ N . (7)Therefore with a knowledge of Φ N Eqs. (6) and (7) showthat a comparison between lensing and stellar dynamicsprovides a measurement of γ PPN .In Appendix A we derive how a general scalar mod-ification to the GR field equations leads to a modifiedrelationship between the lensing and dynamical masses.The effects of these modifications can be compactly writ-ten in terms of an effective source of stress energy thatwe denote T eff defined in Eq. (A7). III. GRAVITATIONAL LENSING IN GENERALSCALAR-TENSOR THEORIES
Scalar-tensor theories of gravity [19, 29–32] present uswith an example of a class of modified gravity theoriesthat naturally predict a redshift-dependent γ PPN .We will consider a general scalar-tensor theory definedby the action S = 12 κ Z d x √− g (cid:20) ϕR − ω BD ( ϕ ) ϕ ( ∂ α ϕ ) − U ( ϕ ) (cid:21) + S m . (8)The gravitational field equation is given by ϕG µν + g µν (cid:20)
12 ( ∂ α ϕ ) + (cid:3) ϕ + U ( ϕ ) (cid:21) − ω BD ( ϕ ) ϕ ∂ µ ϕ∂ ν ϕ − ∇ µ ∇ ν ϕ = κT µν , (9)with the scalar field equation given by2 ω BD ( ϕ ) ϕ (cid:3) ϕ = − R − (cid:18) ω ′ BD ϕ − ω BD ϕ (cid:19) ( ∂ α ϕ ) +2 U ′ , (10)where a prime denotes differentiation with respect to thefield ϕ .Linearizing the scalar field around its cosmologicalvalue, ϕ = ϕ + δϕ , and specializing to a static casewe have T eff = ∇ δϕ + 2 U ′ δϕ, (11) ∇ δϕ = κT ω BD ( ϕ ) + m δϕ, (12)where the scalar field mass, m , is a function of the back-ground field and derivatives of the potential U and Brans-Dicke function, ω BD , whose exact form is not needed forthis discussion. In general scalar-tensor models whichproduce late-time acceleration have both m ∼ H and U ∼ H so that the mass and potential are negligible onkpc scales. In the absence of a potential and scalar fieldmass we then find the standard result [30]1 + γ PPN ω BD [ ϕ ( z )]4 + 2 ω BD [ ϕ ( z )] , (13)and γ PPN then depends on time through the backgroundevolution of ϕ .Soon after the expansion of the universe was shownto be accelerating many groups proposed scalar-tensormodels as an explanation. In order to produce modelswith expansion histories in agreement with observationsRefs. [7–10] established an algorithm by which observa-tions of both the expansion history as well as the growthof structure would enable a complete determination ofthe scalar-tensor theory. In particular, Refs. [7, 8] real-ized that the specification of the expansion history [inthe form of H ( z )] allows a reconstruction of ϕ ( z ); in theabsence of precise measurements of the growth rate afunctional form for U ( ϕ ) must be specified from whichfollows γ PPN ( z ).An interesting case considers the ability to produceaccelerated expansion by introducing a cosmological con-stant, Ω V , within a scalar-tensor theory which is less thanits value required in the standard ΛCDM cosmology (i.e.,Ω Λ ≃ . z max (which is typically of order unity)after which the theory becomes inconsistent . There-fore, scalar-tensor theories with a cosmological constantΩ V < Ω Λ can only explain the observed accelerated ex-pansion up to z max . This implies that an observation ofthe expansion history at redshifts greater than z max canrule these models out. Because of this Ref. [8] emphasizesthat, as opposed to solar system observations, measure-ments of the luminosity distance, D L ( z ), at larger red-shifts will place the most stringent constraint on thesetheories. Here we shall see that a measurement of γ PPN at a range of redshifts less than z max can also serve todistinguish between these models and ΛCDM.The choice of Ω V in scalar-tensor theories fully speci-fies the potential U which then allows a calculation of γ PPN ( z ). We show the evolution of γ PPN for various As is discussed in detail in Ref. [8] for z > z max the gravitoncarries negative energy. choices of Ω V in Fig. 1. Note that for the scalar-tensormodels considered here the parameters have been chosenso as to be indistinguishable from GR at z = 0 (i.e., inthe solar system). Also note that as Ω V approaches theΛCDM value of 0.7 we regain GR at all redshifts and γ PPN = 1.
FIG. 1: The evolution of γ PPN in scalar tensor theories whichexactly mimic the flat ΛCDM expansion history with Ω M =0 .
3. The various curves correspond to different choices of Ω V going from zero on the bottom to 0.6 at the top in steps of 0.1.As Ω V increases towards the value of Ω Λ = 0 . z = 0(i.e., in the solar system). IV. GRAVITATIONAL LENSING IN f ( R ) GRAVITY
A sub-class of general scalar-tensor theories thatpresent markedly unique predictions are those theoriesfor which the Brans-Dicke parameter identically vanishes.Recently, these theories have been extensively studied inparticular case of f ( R )-theories [11, 12]. These theoriescontain a particularly interesting mechanism, known asthe chameleon mechanism [33, 34], in which the modifi-cations to GR are rapidly suppressed around an objectwith sufficient density. This rapid change in the behaviorof the theory presents a unique scale-dependent lensingsignature.The action for f ( R )-theories takes the form S = 12 κ Z d x √− g [ R + f ( R )] + S m , (14)where f ( R ) is a function of the Ricci scalar R and S m isthe matter action. The gravitational field equation canbe written as[1 + f ′ ( R )] G µν + 12 g µν [ Rf ′ ( R ) − f + 2 (cid:3) f ′ ( R )] −∇ µ ∇ µ f ′ ( R ) = κT µν . (15)In this theory the Ricci scalar becomes a dynamical quan-tity whose equation of motion is determined by the traceof the field equation, (cid:3) f ′ ( R ) = 13 (cid:18) κT + R [1 − f ′ ( R )] + 2 f (cid:19) . (16)In the limit where f →
0, Eq. (16) implies the usualalgebraic relationship R = − κT and GR is regained and T is the trace of the usual stress-energy tensor.Using the trace equation to rewrite the gravitationalfield equation this theory produces T eff = 13 κ [ κT + R ] + 13 κ [2 Rf ′ ( R ) − f ] . (17)Solutions to the trace equation, Eq. (16), determine thelensing predictions for this theory. To understand thesesolutions, we rewrite the trace of the field equation as (cid:3) f ′ ( R ) + d V d f ′ ( R ) = 0 , (18)with d V d f ′ ( R ) ≡ (cid:18) κT + R [1 − f ′ ( R )] + 2 f (cid:19) . (19)We note that for functions f ( R ) which reproduce the ob-served expansion history, the minimum of this potentialyields the general relativistic relationship between R and T , R = − κT [35].Energetics drive the solution of Eq. (16) towards twolimiting cases [35]. If f ′ ( R ) remains close to its asymp-totic, cosmological, value as we move within the galaxythis trades the energy cost of fixing the f ′ ( R ) at a highpoint in its effective potential against the gain in main-taining a nearly homogeneous field. In this case, thesolution to Eq. (16) can be found by linearizing f ′ ( R )around its cosmological value and we have T eff = T / γ PPN = 1 /
2. On the other hand, if the scalar cur-vature is able to reach the minimum of its potential thiswill be at a cost in gradient energy since the scalar cur-vature will have to transition from its asymptotic value, R , to the general relativistic value R = − κT . At theminimum of the potential deviations from GR are highlysuppressed.Within a given object, far away from the center thescalar curvature starts off near its asymptotic value, R ,and evolves with radial distance from the center. If theobject is too ‘small’, in a sense we will make clear in amoment, then R ∼ R throughout the object and devia-tions from relativity will be of order unity. On the other hand, if the object is compact enough then the scalarcurvature is forced to the minimum of its potential and R = − κT within some radius r . Within that radius de-viations from relativity are highly suppressed and we saythat the object is ‘screened’. FIG. 2: The combination κρ ( r ) r for an NFW profile [Eq. 21]for masses incremented by an order of magnitude between10 M ⊙ (bottom curve) to 10 M ⊙ (top curve). The out-ermost point where | f ′ ( R ) | intersects these curves indicatesthe transition radius, r , inside of which deviations from rela-tivity are suppressed. For a given value of | f ′ ( R ) | those haloswith a mass below some threshold will not be screened andwill therefore exhibit order unity modifications to gravity. As shown in Refs. [22, 35] screening occurs within aradius r implicitly given by | f ′ ( R ) | < κρ ( r ) r , (20)where R is the value of the scalar curvature on cos-mological scales and ρ is the local value of the density.Note that if the density of an object is too small thenEq. (20) is not satisfied at any radius and the object iscompletely unscreened with order unity deviations fromGR throughout. One can think of | f ′ ( R ) | as determin-ing a characteristic gravitational potential for the theoryso that when the local gravitational potential ( κρr ) islarger than this characteristic value deviations from GRare highly suppressed. Numerical solutions show that thetransition from R ∼ R to R ≪ R = − κρ occurs over arelatively short length-scale (see, e.g., Fig. 10 in Ref. [35])so we will approximate it by a step function.The most stringent observational constraint on f ( R )gravity theories comes from the requirement that it passsolar system tests. Measurements of the motion of lightin the solar system has placed the constraint γ PPN , ⊙ =1 + (2 . ± . × − [2, 36]. In order for f ( R ) gravitytheories to pass solar system tests the theory must sup-press deviations from GR within our halo leading to theconstraint | f ′ ( R ) | . − [35].Measurements of lensing around other galaxies can alsoserve to constrain this theory. In particular, Eq. (20)shows that f ( R )-gravity predicts a lensing signal aroundgalaxies which depends on halo mass. To see this, con-sider an NFW halo [37] of the form ρ ( r ) = ρ c δ c (cid:18) rr s (cid:19) − (cid:18) rr s (cid:19) − , (21)where r s is the scale radius and ρ c is the critical densityof the universe. The amplitude δ c = (∆ / c / [ln(1 + c ) − c/ (1 + c )] relates the concentration to the virial radiuswith an overdensity, ∆ = 119. We also assume the mass-concentration relation c = 91 + z (cid:18) M . × h − M ⊙ (cid:19) − . , (22)where h is the Hubble parameter in units of 100 km/(sMpc) [38, 39]. This relation reduces the NFW profile toa one-parameter family which we take to be dependenton the virial mass, M . Since ρ NFW r ∝ r for r < r s and ρ NFW r ∝ r − for r > r s it is clear that the inner-most point at which deviations from GR are suppressedin f ( R )-gravity will occur at the scale radius r s .Looking at Eq. (20) and the NFW density profile wecan see that halos with masses which satisfy | f ′ ( R ) | > κρ c δ c ( M ) (23)will not be screened . Since δ c ( M ) decreases with de-creasing M this sets an upper limit to the mass of haloswhich can be screened given a value for | f ′ ( R ) | . This de-pendence is shown as the solid line in Fig. 3. For massesbelow this threshold the theory deviates from GR by fac-tors of order unity and strong lensing around these haloshave γ PPN = 1 /
2. Therefore, a measurement of γ PPN = 1in lower mass galaxies would place a more stringent con-straint on | f ′ ( R ) | .Finally, we must also take into account that | f ′ ( R ) | depends on redshift. For f ( R ) models which producelate-time acceleration we have | f ′ ( R ) | ∝ /H n , where n > | f ′ ( R ) | increases as the universe expands.For a galaxy with ρr ∝ r − m with m > Strictly speaking this condition only applies to halos which areisolated. A strong lens galaxy which sits within a larger halomay be screened by the larger halo even though the mass ofthe lens halo is below the threshold given in Eq. (23) [40]. Thelocal environments of strong lens galaxies can be approximatelydetermined using photometric data and shows that for the dataconsidered in this paper (the SLACS survey) the richness of thelens systems is on average a few with very few close companions[41] indicating that most systems should be sufficiently isolatedfor our purposes here.
FIG. 3: The minimum mass for a halo to be screened givena value for | f ′ ( R ) | . Solar system tests require that theMilky Way’s halo be screened. Since the halo has a mass ∼ M ⊙ this implies that | f ′ ( R ) | . − [35]. Stronglensing measurements around galaxies with smaller masseswill have γ PPN = 1 /
2. The dashed line and upper x -axis show how | f ′ ( R ) | varies as a function of redshift with | f ′ ( R ) | ( z = 0) = 10 − . In general, for f ( R ) models thatproduce late-time acceleration | f ′ ( R ) | ∝ /H ( z ) n . In thisfigure we show the evolution of | f ′ ( R ) | for the model pre-sented in Ref. [35]. The dotted red line indicates the way inwhich this plot should be read: at a given redshift (upper x -axis) a vertical line intersects a given f ( R ) model (dashedcurve); a horizontal line then intersects the solid ‘screening’line; from that point a vertical line drawn to the correspond-ing halo mass (lower x -axis) gives the mass above which thehalo is screened at that redshift. be true in the outer regions of the galaxy to ensure thatthe galaxy has a finite mass) this causes the γ PPN = 1 / f ( R )-gravity predicts a redshift dependent γ PPN as well.The dashed curves in Fig. 3 show the evolution of | f ′ ( R ) | as a function of redshift for the specific f ( R ) model foundin Ref. [35], f ( R ) = − m c ( R/m ) n c ( R/m ) n +1 + 1 , (24)where m ∼ H and n is a free index and the ratio c /c is set by requiring the model have the same expansionhistory as in a ΛCDM model with ˜Ω M and ˜Ω Λ c c ≈ M ˜Ω Λ . (25)Therefore these models have two free parameters whichwe choose to be n and | f ′ ( R ) | ( z = 0). V. STRONG LENSING PARAMETERIZED BYGRAVITATIONAL SLIP
In order to test the predictions of GR we must com-pare predictions and observations with other theories ofgravity. Besides looking at other theories on a case-by-case basis it is more useful to parameterize modificationsto GR and constrain the value of those parameters. Thisapproach has proven very sucessful when interpreting ob-servations of the effects of gravity in the solar systemin the form of the parameterized post-Newtonian (PPN)formalism [1]. In this formalism the relationship betweendifferent parts of the metric are parameterized by con-stant coefficients and observations of the motion of testparticles (both massive and massless) measure the valuesof these coefficients.The PPN formalism relies on the presence of localizedsources of stress-energy and so cannot be applied withoutchange to a cosmological context. Some studies have at-tempted to articulate ways in which to extend the PPNformalism to cosmological observations [15–18]. The ba-sic idea of all of the currently proposed parameterizationsis that current cosmological observations using the cos-mic microwave background, weak lensing, and evolutionof large-scale structure, are only sensitive to the scalarpart of the metric [i.e., the two Newtonian potentials Ψand Φ in the metric given in Eq. (1)]. Furthermore, sinceany modifications to GR that accounts for a phase oflate-time accelerated expansion is significant for z . δ M ) and velocity per-turbations ( θ M ). Considering only those gravity theo-ries where stress-energy is conserved gives two evolutionequations (the continuity and Euler equations). There-fore a generic modified gravity theory is defined whentwo gravitational field equations are specified.The two scalar potentials form a linear combinationwhich is determined by a Poisson-like equation but witha time and space dependent Newton’s ‘constant’ µ ( z, ~x ) ∇ ( A Φ + B Ψ) A + B = 4 πµ ( z, ~x ) Gρ, (26)and their ratio can be parameterized by another time andspace dependent function given in Eq. (4) and repeatedhere, ΦΨ = γ PPN ( z, ~x ) . (27)In GR we have A = 1, B = 0, G ( z, ~x ) = G , and γ PPN ( z, ~x ) = 1. Particular modified gravity theories canthen be parameterized by how the functions G ( z, ~x ) and γ PPN ( z, ~x ) depend on scale and time [15, 17].One particular parameterization, first proposed inRef. [16], supposes γ PPN ( z, ~x ) = [1 + ̟ ′ ρ DE /ρ M ( z )] − =[1 + ̟ / (1 + z ) ] − so that order unity modificationsturn on around the transition from matter dominationto dark energy domination. As described in more de-tail in Ref. [42] this parameterization further chooses to maintain the scalar part of the (0 , i ) component of theEinstein field equations leading to a time dependent µ .In the next Section we show how observations of stronglensing around galaxies over a range of redshifts are ableto constrain the value of ̟ . VI. γ PPN
FROM MEASUREMENTS OF STRONGLENSES
The original idea of measuring γ PPN from observationsof strong lenses was first discussed in Ref. [24] and wasfirst applied to data in Ref. [25]. A qualitative under-standing of how observations of strong lenses can yield ameasurement of γ PPN can be understood by consideringthe following simplified example [24]. In this example thelens density distribution is given by a singular isothermalsphere with the observed line-of-sight velocity dispersion σ obs . The lens then produces an Einstein ring with aradius, R E , given by [43] R E = 4 πσ (cid:18) γ PPN (cid:19) D L D LS D S , (28)where D X is the angular diameter distance to the lens( L ), source ( S ), and between the lens and source ( LS ).The observed spectra give measurements of the sourceand lens redshifts as well as of the line-of-sight velocitydispersion [44]. The angular diameter distances are ob-tained by fixing a fiducial cosmology although the choiceof cosmological parameters does not significantly impactthe final result. The data then yields a measurement of γ PPN = 2 π D S D L D LS R E c σ − . (29)As we will now describe, in practice the problem is morecomplicated since the density profile of the lens cannotbe described by such a simple model.As in the simplified model that was just discussed, inorder to measure γ PPN we must relate the observed stellarvelocity dispersion to the lensing observations. First notethat in an analogy with Gauss’ law the deflection anglein a modified gravity theory parameterized as in Eq. (3)and (4) for a circularly symmetric lens depends on theenclosed mass as ˆ α = 4 D LS D S µGM ( θ ) D L θ ˆ θ, (30)where ˆ θ is a unit vector projected on the sky centeredat the lens, and M ( θ ) is the projected mass enclosedwithin the angle θ . The lens equation [43] then relates theobserved Einstein radius θ E = R E /D L to the enclosedmass and µ R E = 4 D L D LS D S µGM ( R E ) . (31)It is useful to associate an effective ‘lensing’ velocitydispersion with each lens system. The effective velocitydispersion, σ lens , is defined through the measured Ein-stein radius, R E ≡ π σ c D L D LS D S , (32)and the lens equation [Eq. (31)] allows us to relate thisto the projected mass within the Einstein radius M ( R E ) = 4 π σ c µG D L D LS D S . (33)We can therefore write σ = 1 π µGM ( R E ) R E . (34)This ‘lensing’ velocity dispersion, σ , should not beconfused with the observed stellar velocity dispersion, σ obs . In order to measure γ PPN , the equations of hydro-static equilibrium are used to relate the observed stellarvelocity dispersion to M ( R E ).The analysis presented here extends the model due toRef. [45]. The total mass density of the galaxy is modeledas a power law ρ M ( r ) = ρ M, (cid:18) rr ∗ (cid:19) − p . (35)The stellar component is well fit by a Hernquist profile[46] ρ L ( r ) = M ∗ r ∗ πr ( r + r ∗ ) , (36)where M ∗ is the total stellar mass and r ∗ is a scale radiuswhich can be written in terms of the effective radius ofan R / luminosity profile as r ∗ = R eff / . β ≡ − h σ θ ih σ r i . (37)The radial velocity dispersion is found by solving thespherically symmetric Jeans equation [28] for the stel-lar component. Projecting out the line of sight velocitydispersion and performing a weighted average over theluminosity profile within a circular aperture of projectedradius R A the observed stellar velocity dispersion is re-lated to the model parameters through the expression σ = 1 π µGM E ( R E )(1 + γ PPN ) R E (cid:18) R E r ∗ (cid:19) p − g ( p, β, R A /r ∗ , σ see ) , (38)where σ see [47] is the seeing (i.e., blurring due to atmo-spheric distortions), and g ( p, β, R A /r ∗ , σ see ) can be writ-ten in terms of integrals over hypergeometric functions.This FIG. 4: The degeneracy between γ PPN and the slope of thetotal matter density p . In order to generate this curve wehave used the mean values from the SLACS survey: R E = R eff / z L = 0 . β = 0 . The ratio between the observed stellar velocity disper-sion and the lens velocity dispersion, f σ ≡ σ /σ isgiven by f σ = (cid:18)
21 + γ PPN (cid:19) (cid:18) R E r ∗ (cid:19) p − g ( p, β, R A /r ∗ , σ see ) . (39)Observations of the stellar velocity dispersion and theEinstein radius yield measurements of f σ . However, fromFig. 4 it is clear there is a degeneracy between the slopeof the density profile, p , and γ PPN . In order to makeprogress in measuring γ PPN a prior must be placed on p . It is important that any approach taken to place thisprior be independent of the theory of gravity. One ap-proach, discussed in Ref. [48], uses an assumed scalinglaw (related to the fundamental plane [49]) that relatesthe power-law slope, p , to the luminosity, Einstein radius,and effective radius is used to estimate the average valuefor p within a population of lenses. In this approach auniversal value for γ PPN would appear as a constant offsetand does not affect the estimate of h p i = 1 . ± . γ PPN may be non-universal. Furthermore, the analysisin Ref. [48] is done in the limit where the dark matterfraction of in the system is 1. This is a significant sim-plification since the average dark matter fraction within R E of a subset of these systems has been estimated to be25% [50].Another approach is to place a prior on p given by itsdistribution measured in low redshift early-type galaxies FIG. 5: The mean ( left ) and dispersion ( right ) for the distribution of √ f σ / √ γ PPN as a function of the observed effectiveradius, R eff , and the observed Einstein radius, R E . The intrinsic distribution for the slope of the total density profile, p , andthe velocity anisotropy, β , were assumed to be equal to their intrinsic distributions measured in early-type galaxies at lowredshift where more detailed measurements of the stellar kinematics can be measured. where more detailed kinematic data can be used to deter-mine the full density profile. This is the approach takenhere and gives h p i = 1 . σ p = 0 .
08 and h β i = 0 . σ β = 0 .
13 [51, 52]. But as is clear from Fig. 4, theconstraint on γ PPN is very sensitive to the assumed h p i (see Fig. 4). To remove this dependence on h p i in ourfinal constraints we marginalize over the average of value γ PPN within the sample. Therefore, our final constraintscome from the lack of any significant correlation between γ PPN and properties of the lens such as its redshift andmass.Using Eq. (39) the resulting distribution for √ f σ / √ γ PPN was calculated on a grid of val-ues for 0 . ′′ ≤ R eff ≤ ′′ and 0 . ′′ ≤ R E ≤ ′′ usingan aperture radius R A = 1 . ′′ and seeing σ see = 0 . ′′ [50] and is shown in Fig. 5. For a value of γ PPN , anunderlying probability distribution for p and β , andmeasured values for R E and R eff , Eq. (39) gives theunderlying probability distribution of f σ , denoted by( dP/df σ ) (cid:0) γ PPN (cid:1) . The likelihood for γ PPN is then givenby L ( γ PPN ) = (40) Z dPdf σ (cid:0) γ PPN (cid:1) [ f σ | R E , R eff ] G ( f σ ; σ f σ , f σ ) df σ , where G ( f σ ; σ f σ , f σ ) is a Gaussian with a mean equal tothe measured value of f σ and a standard deviation, σ f σ ,equal to the observational error on f σ .Since the resulting probability distribution of √ f σ / √ γ PPN for fixed R eff and R E is well approxi-mated by a Gaussian it is straight-forward to derive thedistribution of f σ for any value of γ PPN , R eff , and R E . We first consider the case where γ PPN is a universalconstant and the results are shown in Fig. 6. In that fig-ure the solid black curve shows the likelihood for the fullSLACS survey (53 systems) [53]. The full joint likelihoodgives the result γ PPN = 0 . ± .
05 at 68% c.l. As we dis-cuss in more detail below, even though this result seemsto be in conflict with GR, it is more of a reflection ofa difference between the mean slope of the mass densityof the galaxies in the local universe as compared to theSLACS galaxies. The other two dashed-dot curves arethe likelihoods when considering a subset of the full sur-vey. The dot-dashed black curve on the right is the like-lihood for the original 15 systems considered in Ref. [25]and the dot-dashed red curve on the left uses 14 of thosesystems but reflects differences in the mass modeling, sur-face brightness measurements, and measurements of thevelocity dispersion. As discussed in Ref. [53] the differ-ence in mass estimates, measured surface brightness andvelocity dispersion, provides a more realistic sense of themeasurement errors for these quantities. The dominanteffect on the measurement of γ PPN is the error in themeasured stellar velocity dispersion. The analysis pre-sented here was slightly more conservative than Ref. [53]and took the statistical errors in σ S quoted in Ref. [53]while enforcing a minimum of 7%. The agreement, at the1 σ level, between the original analysis (dot-dashed black)and the modified analysis (dot-dashed red) indicates theerror budget used in this work appropriately incorporateserrors in the mass modeling, surface brightness measure-ments, and velocity dispersion.Comparing the constraints presented here to the re-sults found in Ref. [25], γ PPN = 0 . ± .
07, we finda similar error but a significantly lower best fit value.
FIG. 6: The joint likelihood for a universal γ PPN . The solidblack curve is the joint likelihood for the full SLACS sample(53 systems) [53] which gives γ PPN = 0 . ± .
05; the blackdashed-dot curve on the right is uses the original 15 systemsconsidered in Ref. [25] which gives γ PPN = 0 . ± .
1; the reddashed-dot curve on the left is a re-analysis of the 14 systemsin common between the original 15 and the full 53 systemswhich gives γ PPN = 0 . ± .
1. As described in Ref. [53] there-analysis of these 14 common systems reflects differencesin the mass modeling, surface brightness measurements, andmeasurements of the velocity dispersion. The effect these dif-ferences have on the inferred constraint to γ PPN is well within1 σ and so is well described by the errors we have included inthe calculation of the likelihood (which are dominated by er-rors on the measured velocity dispersion). All quoted errorsare at 68% c.l. This difference can be explained by noting that an anal-ysis of the SLACS lenses assuming GR (i.e., γ PPN = 1)found an intrinsic distribution for the slope of the mat-ter density, p , of h p i = 2 . σ p = 0 .
12 [48, 50]. Sincethe mean of this distribution is approximately 0.1 awayfrom the mean of the low- z distribution used to constrain γ PPN (i.e., h p i = 1 .
93) and given that the degeneracy be-tween p and γ PPN is nearly linear (see Fig. 4) it followsthat the best-fit value should be about 0.1 away from γ PPN = 1. The main difference between the analysis pre-sented here and the one in Ref. [25] is that this analysisuses a more realistic model for the luminosity profile ofthe lens galaxy (a Hernquist profile), which was also usedto measure p in the original SLACS data [50], as opposedto Ref. [25] which used a power-law profile.The degeneracy shown in Fig. 4 has a significant ef-fect when applying the low redshift measurements of theslope of the total mass density p to the SLACS objectsand leads to an apparent disagreement with GR at the2.4 σ level and it is clear that this constraint on γ PPN is FIG. 7: The measurement of γ PPN in individual lens systemsas a function of the virial mass of the lens halo (top) and lensredshift (bottom). of limited interest. In particular, the lack of knowledgeof the mean value of the slope of the total mass densityin the lens galaxies leads to a bias in any constraint on auniversal value of γ PPN . In order to remove this bias wemarginalize over the mean value of γ PPN for the entiresample of lenses. In this way the data does not constraina universal (constant) value of γ PPN but instead gives ro-bust constraints on how γ PPN may depend on propertiesof the lens such as its redshift and mass. As we saw inprevious sections, various theories of gravity give specificpredictions on how γ PPN depends on both lens redshiftand mass.In order to demonstrate how a variable γ PPN is con-strained we applied the SLACS data to the two modifiedgravity theories described in Sec. III and IV. A variable γ PPN is constrained by calculating the likelihood
L ∝ e − χ / , (41)with χ = X i (cid:8)(cid:2) γ obsPPN , i − Aγ PPN ( z Li , M i ) (cid:3) /σ γ PPN , i (cid:9) , (42)where γ obsPPN , i is the observed value, σ γ PPN , i is the erroron the observation, γ PPN ( z L,i , M i ) is the predicted valuewhich depends on either the lens redshift or its mass, and A is an amplitude which takes values between 0.8 and1.2. Marginalizing over A (which is similar to marginaliz-ing over the bias when using measurements of large-scalegalaxy clustering) allows us remove the degeneracy be-tween h p i and γ PPN (which, as discussed before, is nearlylinear; see Fig. 4) and to explore constraints which arisedue to (the lack of) any correlation or observed relation-ship between γ PPN and lens redshift or halo mass.0The lens redshift is a directly observable quantitywhereas the mass is not. To estimate the total virialmass, M , from observations we use an approximate rela-tion with the observed half-light radius R eff . Using thefact that the stellar to virial mass ratio is approximately0.01 and given the relation between the stellar mass andeffective radius found in Ref. [54] for elliptical galaxieswe can write M = 7 . × (cid:18) R eff kpc (cid:19) . M ⊙ . (43)Fig. 7 shows the measured value of γ PPN as a functionof both halo mass (top panel) and lens redshift (bottompanel). Note that the mean estimated virial mass is 11 × M ⊙ which compares well with the mean virial massdetermined through weak lensing h M i = 14 +6 − × M ⊙ which was made using a subset of the full 53 systemsconsidered here [55].We may approximate a generic non-universal γ PPN asdepending linearly on some property of the lens. In par-ticular, letting x denote a property of the lens (such asits redshift) we model this generic dependence as γ PPN ( x ) = γ ( m x ) + m x x ∆ x , (44)where γ ( m x ) is a constant which depends on the slope m x and ∆ x is the range of x over which we have observa-tions (i.e., the range of lens redshifts for a given survey).In order to remove any sensitivity to the mean value of γ PPN we define γ ( m x ) ≡ ¯ γ PPN − m x ¯ x ∆ x , (45)where ¯ γ PPN is the mean value of γ PPN for all of the lensesin the survey. Here we will only be interested on con-straining how γ PPN may depend on the lens mass andredshift. In these cases, at the 68% confidence level (c.l.),we find that m z = 0 . ± .
24 (0 . < z < .
36) and m M = 0 . ± .
36 (10 . M/M ⊙ . ). A. Constraints to scalar tensor gravity
As described in Sec. III scalar-tensor gravity generi-cally predicts a redshift dependent γ PPN . For the partic-ular case where we account for the observed acceleratedexpansion using a scalar-tensor theory with a cosmolog-ical constant Ω V < Ω Λ the gravitational lensing obser-vations place constraints on the value of Ω V as shown inthe left panel of Fig. 8. As Ω V is made smaller the valueof γ PPN varies more with redshift leading to tension withthe SLACS data (see Fig. 1) The data places the con-straint Ω V > . V > .
06 at 95% c.l.Note that since we have marginalized over the ensembleaverage for γ PPN these constraints rely solely on the lackof any significant correlation between γ PPN and the lensredshift. Since these models are constructed to have negligibledeviations from GR today measurements of either theexpansion history or lensing and dynamics at high z pro-vide the only data which can place meaningful constrainson these theories. B. Constraints to f ( R ) gravity As described in Sec. IV f ( R ) gravity generically pre-dicts a lensing signal which depends on the mass of lenshalo. Using the relationship between the mass thresh-old (below which γ PPN = 1 / | f ′ ( R ) | ( z = 0). Con-straints to the particular model described in Eq. (24)are shown in the center panel in Fig. 8. Placing theprior | f ′ ( R ) | ( z = 0) ≤ − for the particular casewhere n = 1 we find | f ′ ( R ) | ( z = 0) . . × −
68% (2 . × − n = 3 yields | f ′ ( R ) | ( z =0) . × −
68% (2 . × − n = 5 yields | f ′ ( R ) | ( z = 0) . . × −
68% (3 × − | f ′ ( R ) | ( z = 0) is weakened as n gets largersince, as demonstrated in Fig. 3, the larger n is the more | f ′ ( R ) | ( z ) decreases with increasing redshift. Therefore,for the same value of | f ′ ( R ) | ( z = 0) a model with alarger n will have a smaller value of | f ′ ( R ) | ( z ∼ . f ( R ) gravity from the SLACS dataare about a factor of 10 worse than constraints fromsolar system tests. However, they are several ordersof magnitude better than constraints using observationsof anisotropies in the CMB and measurements of thematter power spectrum [56]. A survey which couldmeasure lensing and dynamics around low-mass galax-ies ( M . M ⊙ ) could potentially place constraintson f ( R ) gravity which would improve upon solar systemtests.Note that since we have marginalized over the meanvalue of γ PPN these constraints rely solely on the lackof any significant correlation between γ PPN and the lensmass.
C. Constraints to the gravitational slip
As described in Sec. V a particular way to parameter-ize deviations from GR is to introduce two new time andspace dependent functions: a modified Newton’s ‘con-stant’ µ ( z, ~x ) and γ PPN ( z, ~x ). Assuming that the modifi-cations of GR become important at the same time as theexpansion starts to accelerate inspires the parameteriza-tion [16] γ PPN ( z ) = 11 + ̟ (1 + z ) − , (46)and measurements of the SLACS lenses places a con-straint on ̟ as seen in the right-most panel in1 FIG. 8: Constraints to modified gravity theories and parameterizations using the SLACS data.
Left : The left panel showsconstraints to the energy density in a cosmological constant in units of the critical energy density, Ω V , in scalar tensor theories.As discussed in Sec. III scalar tensor gravity can explain the observed accelerated expansion with a value for the cosmologicalconstant which is less than the value required in GR Ω Λ . As Ω V is made smaller the value of γ PPN varies more with redshiftleading to tension with the SLACS data (see Fig. 1). The SLACS measurements place the constraint Ω V > . V > .
06 at 95% c.l.
Center : The center panel shows constraints to the value of | f ′ ( R ) | using the model presented inEq. (24). The three lines correspond to n = 1 (solid red) | f ′ ( R ) | ( z = 0) . . × −
68% (2 . × − n = 3 (dashedblack) | f ′ ( R ) | ( z = 0) . × −
68% (2 . × − n = 5 (dot-dahsed blue) | f ′ ( R ) | ( z = 0) . . × −
68% (3 × − Right : The right panel shows constraints to the ‘gravitational slip’ ̟ which parameterizes the evolution of γ PPN intime as discussed in Sec. V. The SLACS data places the constraints ̟ = 0 . +0 . − .
27 +0 . − . (68%, 95% c.l.) which is as restrictiveas constraints derived from the cosmic microwave background, weak lensing, and evolution of large-scale structure [42]. Fig. 8. The SLACS data places the constraints ̟ =0 . +0 . − .
27 +0 . − . (68%, 95% c.l.) which is as restrictiveas constraints derived from the cosmic microwave back-ground, weak lensing, and evolution of large-scale struc-ture [42].Note that, as in the case of f ( R ) gravity, since wehave marginalized over the mean value of γ PPN theseconstraints rely solely on the lack of any significant cor-relation between γ PPN and the lens redshift.
VII. CONCLUSIONS
Constraints to modifications of GR from data takenwithin the solar system or from binary neutron star sys-tems within our galaxy are very precise. For instance,radar ranging to the Cassini spacecraft leads to a con-straint γ PPN , ⊙ = 1 + (2 . ± . × − [2]. Althoughthese measurements place important constraints recentinterest in modified gravity theories which can accountfor the observed accelerated expansion have focused at-tention on models with modifications that evolve withboth time and environment.In the case of scalar-tensor theories of gravity timeevolution is a natural consequence of introducing a newscalar degree of freedom. In the case of f ( R ) gravity thechameleon mechanism, where modifications to GR aresuppressed in regions of high mass density, is a naturalconsequence of the fourth order nature of the modifiedfield equations. Other theories, such as DGP gravity [13,14] and the recently proposed Galileon [57], also predict a non-universal γ PPN . It is therefore important to not onlylook for ways to constrain γ PPN in the local universe butto also investigate whether it may change depending ontime, scale, mass, local environment, and so forth.Using measurements of stellar velocity dispersions andstrong lensing around early-type galaxies from the fullSLACS survey we have presented constraints to alterna-tive gravity theories which can account for the observedlate-time acceleration. This analysis updates the resultspresented in Ref. [25] by including more realistic model-ing of the stellar component as well as by using the full53 systems in the SLACS survey. We also extended theanalysis beyond constraining a universal value for γ PPN and applied the data to constrain γ PPN ’s dependence onthe mass and redshift of the lens.Constraints to a universal value for γ PPN must be usedwith caution given the significant degeneracy between theslope of the total matter density, p , and γ PPN . The twomethods discussed in the text which estimate h p i inde-pendently of the theory of gravity both have unquanti-fied systematic errors. In the case where a scaling lawis assumed the scaling law itself may introduce biases;the application of low-redshift observations to the higherredshift SLACS lenses may not be appropriate given thatthe structure of the galaxies may significantly evolve withredshift. To remove this uncertainty the analysis pre-sented here marginalized over the mean of the sampleleading to constraints which only depend on how γ PPN correlates with lens redshift and mass.Attempts to constrain modifications to GR on Mpcscales using observations of galaxy clusters, weak lens-2ing, and galaxy surveys are complementary to the pre-sented here [58–62]. Observations of galaxy clusters allowa measurement of γ PPN though a comparison betweenthe X-ray temperature or virial mass and measurementsof strong lensing. However, these observations have alimited statistical significance leading to constraints on γ PPN to ∼
50% [63]. This is partly due to the lack oflarge homogeneous samples of clusters. It is also relatedto the fact that cluster dynamics are harder to modelleading to larger systematic errors.Although the SLACS survey presents us with an op-portunity to constrain the non-universality of γ PPN thedynamical range of the the SLACS survey are limited:their redshifts range 0 . ≤ z ≤ .
35 and their masses areof order 10 M ⊙ . Future surveys may be able to extendthis to a higher redshift as well as to lower mass galax-ies [64] which would improve the results presented here.For instance, measurements of lensing around galaxieswith masses M < M ⊙ could potentially place amore stringent constraint on f ( R ) gravity theories thansolar system tests. Of course the challenge to measuringstrong lensing around less massive galaxies is that thelensing cross section decreases with decreasing mass. Acknowledgments
After completing this paper, we became aware of simi-lar work by Schwab, Bolton, and Rappaport. The authorthanks Adam Bolton, Robert Caldwell, Daniel Grin, andEric Linder for a careful reading of a previous version ofthe manuscript, and Kevin Bundy, Bhuvnesh Jain andMarc Kamionkowski for useful conversations. This workwas supported by the Berkeley Center for CosmologicalPhysics. The author gratefully acknowledges the hospi-tality of the Aspen Center for Physics where some of thiswork was completed.
Appendix A: Lensing and dynamics in weak-fieldlimit of modified gravity theories
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