Draftversion January3,2017 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 TEST OF PARAMETRIZED POST-NEWTONIAN GRAVITY WITH GALAXY-SCALE STRONG LENSING SYSTEMS Shuo Cao1, Xiaolei Li1, Marek Biesiada1,2, Tengpeng Xu1, Yongzhi Cai1, and Zong-Hong Zhu1* Draft version January 3, 2017 ABSTRACT Basedonamass-selectedsampleofgalaxy-scalestronggravitationallensesfromtheSLACS,BELLS, 7 LSD and SL2S surveys and using a well-motivated fiducial set of lens-galaxy parameters we tested 1 the weak-field metric on kiloparsec scales and found a constraint on the post-Newtonian parameter 0 2 γ = 0.995+−00..003477 under the assumption of a flat ΛCDM universe with parameters taken from Planck observations. General relativity (GR) predicts exactly γ = 1. Uncertainties concerning the total n mass density profile, anisotropy of the velocity dispersion and the shape of the light-profile combine a to systematic uncertainties of 25%. By applying a cosmological model independent method to J ∼ the simulated future LSST data, we found a significant degeneracy between the PPN γ parameter 2 and spatial curvature of the Universe. Setting a prior on the cosmic curvature parameter 0.007 < Ω < 0.006, we obtained the following constraint on the PPN parameter: γ = 1.000+0.−0023. We ] k −0.0025 O conclude that strong-lensing systems with measured stellar velocity dispersions may serve as another important probe to investigate validity of the GR, if the mass-dynamical structure of the lensing C galaxies is accurately constrained in the future lens surveys. . h Subject headings: gravitationallensing: strong - galaxies: structure - cosmology: observations p - o 1. INTRODUCTION Biesiada, Piórkowska,& Malec 2010; Collett & Auger r 2014; Cardone et al. 2016; Bonvin et al. 2016), the dis- t As a successful geometric theory of gravitation, Ein- s tribution of matter in massive galaxies acting as lenses a stein’s theory of general relativity (GR) has been con- (Zhu & Wu1997;Mao & Schneider1998;Jin et al.2000; [ firmed in all observations devoted to its testing to date Keeton 2001; Ofek et al. 2003; Treu et al. 2006a), and (Ashby 2002; Bertotti et al. 2003), in particular in fa- 1 thephotometricpropertiesofbackgroundsourcesatcos- mous experiments (Dyson et al. 1920; Pound & Rebka v mological distances (Cao et al. 2015b). All the above 1960; Shapiro 1964; Taylor et al. 1979). However, the 7 mentionedresultshavebeenobtainedundertheassump- pursuit of testing gravity at much higher precision has 5 tion that GR is valid. Using strong lensing systems ever continued in the past decades, including measure- 3 Grillo et al.(2008)reportedthevalueforthepresent-day mentsoftheEarth-Moonseparationasafunctionoftime 0 matter density Ω rangingfrom0.2 to 0.3 at99% confi- through lunar laser ranging (Williams et al. 2004). On m 0 dence level. This initial result, confirmed in later strong theotherhand,formulatingandquantitativelyinterpret- . 1 ing the test of gravity is another question and an inter- lensing studies (e.g. Cao et al. (2012, 2015a)) is consis- 0 estingproposalinthisrespecthasbeenformulatedinthe tent with most of the current data including precision 7 measurements of Type Ia supernovae (Amanullah et al. frameworksoftheparameterizedpost-Newtonian(PPN) 1 2010)andtheanisotropiesinthecosmicmicrowaveback- framework (Thorne & Will 1971). Different from the : ground radiation (Ade et al. 2014). Currently, the con- v original physical indication (Bertotti et al. 2003), scale cordanceΛCDMmodel isin agreementwith mostofthe i independent post-Newtonian parameter denoted by γ, X availablecosmologicalobservations,in which the cosmo- with γ = 1 representing GR, may serve as a test of the logical constant contributing more than 70% to the to- r theory on large distances. a tal energy of the universe is playing the role of an ex- This paper is focused on the quantitative constraints otic component called dark energy responsible for accel- of the GR as a theory of gravity, using the recently- erated expansion of the Universe. However, there ap- released large sample of galaxy scale strong gravi- pearednoticeabletensionsbetweendifferentcosmological tational lensing systems discovered and observed in probes. For example, regarding the H there is a ten- SLACS, BELLS, LSD, and SL2S surveys (Cao et al. 0 sion between the CMB results from Planck (Ade et al. 2015a). Up to now, most of the progress in 2014) and the most recent Type Ia supernovae data strong gravitational lensing has been made in inves- (Riess et al. 2016). Similarly, the σ parameter derived tigating cosmological parameters (Zhu 2000a,b; Chae 8 from the CMB results from Planck (Ade et al. 2014) 2003; Chae et al. 2004; Mitchell et al. 2005; Grillo et al. turned out to be in tension with the recent tomographic 2008; Oguri et al. 2008; Zhu & Sereno 2008a; Zhu et al. cosmicshearresultsbothfromtheCanadaFranceHawaii 2008b; Cao & Zhu 2012; Cao, Covone & Zhu 2012; Telescope Lensing Survey (CFHTLenS) (Heymans et al. Cao et al. 2012; Cao, Zhu & Zhao 2012; Biesiada 2006; 2012;MacCrann et al.2015)andtheKiloDegreeSurvey 1Department of Astronomy, Beijing Normal University, (KiDS) (Hildebrandt et al. 2016). These tensions partly 100875, Beijing,China;zhuzh@bnu.edu.cn motivate the test of GR performed in the presentpaper. 2Department of Astrophysics and Cosmology, Institute of With reasonable prior assumptions and independent Physics,UniversityofSilesia,Uniwersytecka4,40-007Katowice, measurementsconcerningbackgroundcosmologyandin- Poland 2 Cao et al. ternal structure of lensing galaxies, one can use strong lens center: r2 = R2 + 2. In order to character- Z lening systems as another tool to constrain the PPN ize anisotropic distributionof three-dimensionalvelocity parameters describing the deviations from the GR. dispersion pattern, one introduces (Bolton et al. 2006; This idea was first adopted on 15 SLACS lenses by Koopmans 2006) an anisotropy parameter β Bolton et al. (2006), who found the post-Newtonian pa- β(r)=1 σ2/σ2 (6) rameter to be γ = 0.98 0.07 based on priors on − t r ± galaxy structure from local observations. More recently, whereσ2 andσ2 are,respectively,the tangentialandra- Schwab et al. (2010) re-examined the expanded SLACS t r dialcomponentsofthevelocitydispersion. Inthecurrent sample (Bolton et al. 2008a) and obtained a constraint analysis we will consider anisotropic distribution β = 0 on the PPN parameter γ =1.01 0.05. 6 ± and assume, as it almost always is assumed, that β is Having available reasonable catalogs of strong lenses: independent of r. containing more than 100 lenses, with spectroscopic as Following the well-known spherical Jeans equation well as astrometric data obtained with well defined se- (Binney 1980), the radial velocity dispersion of the lu- lection criteria (Cao et al. 2015a), the purpose of this minous matter σ2(r) in the early-type lens galaxies can work is to use a mass-selected sample of 80 early-type r be expressed as lensescompiledfromSLACS,BELLS,LSD,andSL2Sto provide independent constraints on the post-Newtonian G ∞dr′ ν(r′)M(r′)(r′)2β−2 parameter γ. Throughout this paper we assume a flat σ2(r)= r , (7) r R r2βν(r) ΛCDM cosmology with parameters based on the recent Planck observations (Ade et al. 2014). where β is a constant velocity anisotropy parameter. CombiningthemassdensityprofilesinEq.(4),weobtain 2. METHODANDDATA therelationbetweenthemassenclosedwithinaspherical Our goal will be to constrain deviations from General radius r and M as E Relativity at the level of γ post-Newtonian parameter. 3−α The PPN form of the Schwarzschild metric can be writ- 2 r M(r)= M , (8) ten as √πλ(α)(cid:18)R (cid:19) E E dτ2 =c2dt2(1 2GM/c2r) dr2(1 2γGM/c2r) r2dΩ2 where by λ(x) = Γ x−1 /Γ x we denoted the ratio − − − − (1) of respective Euler’s(cid:0)ga2m(cid:1)ma f(cid:0)u2n(cid:1)ctions. Simplifying the General Relativity corresponds to γ =1. formulae with the notation: ξ = δ +α 2 taken after From the theory of gravitational lensing (Koopmans 2006), we obtain a convenie−nt form for the (Schneider et al. 1992), for a specific strong lensing radial velocity dispersion by scaling the dynamical mass system with the intervening galaxy acting as a lens, to the Einstein radius: multiple images can form with angular separations close 2−α to the so-called Einstein radius θE: σ2(r)= GME 2 r (9) r (cid:20) R (cid:21)√π(ξ 2β)λ(α)(cid:18)R (cid:19) 1/2 E E 1+γ 4GM D − E ls θE =r 2 (cid:18) c2 D D (cid:19) . (2) Inallstronglensingmeasurementsweuse,theobserved s l velocity dispersion is reported, which is a projected, lu- where M is the mass enclosed within a cylinder of ra- minosity weighted average of the radially-dependent ve- E dius equal to the Einstein radius, D is the distance to locity dispersion profile of the lensing galaxy. Its theo- s the source, D is the distance to the lens, and D is the reticalvalue can be calculatedfrom the Eq.(7) with the l ls distance between the lens and the source. All the above assumptionthattherelationshipbetweenstellarnumber mentioneddistancesareangular-diameterdistances. Re- density and stellar luminosity density is spatially con- arranging terms with R = D θ (R is the cylindrical stant. This assumption is unlikely to be violated appre- E l E radius perpendicular to the line of sight – the -axis), ciably within the effective radius of the early-type lens Z we obtain a useful formula: galaxies under consideration. GM 2 c2 D Moreover, the actual observed velocity dispersion is E = sθ , (3) measured over the effective spectrometer aperture θ E ap R (1+γ) 4 D E ls andeffectivelyaveragedbyline-of-sightluminosity. Tak- whichindicates thatonlythe matter withinthe Einstein ingintoaccounttheeffectsofaperturewithatmospheric ring is important according to the Gauss’s law. blurring and luminosity-weighted averaging, the aver- Ontheotherhand,spectroscopicmeasurementsofcen- aged observed velocity dispersion takes the form tral velocity dispersions σ in elliptical galaxies, can pro- c2 D 2 (2σ˜2 /θ2)1−α/2 vide a dynamical estimate of this mass, basedon power- σ¯2= sθ atm E ∗ E law density profiles for the total mass density, ρ, and (cid:20) 4 Dls (cid:21)√π (ξ 2β) − luminosity density, ν (Koopmans 2006): λ(ξ) βλ(ξ+2) Γ(3−ξ) − 2 . (10) r −α ×(cid:20) λ(α)λ(δ) (cid:21)Γ(3−δ) ρ(r)=ρ (4) 2 0 (cid:18)r (cid:19) 0 where σ˜ σ 1+χ2/4+χ4/40 and χ = atm atm r −δ θap/σatm (Sch≈wab et alp. 2010). σatm is the seeing ν(r)=ν0 (5) recorded by the spectroscopic guide cameras during ob- (cid:18)r (cid:19) 0 serving sessions (Cao et al. 2016). The above equation Here r is the spherical radial coordinate from the tells us that we can constrain the PPN parameter γ Test of PPN gravity with galaxy-scale strong lensing systems 3 on a sample of lenses with known redshifts of the lens 300 andofthesource,withmeasuredvelocitydispersionand theEinsteinradius,providedwehavereliableknowledge 290 aboutcosmologicalmodelandaboutparametersdescrib- ing the mass distribution of lensing galaxies (α, β, δ). 280 For the purpose of our analysis, the angular diam- eter distances D (z) between reshifts z and z were 270 A 1 2 calculated using the best-fit matter density parameter 260 Ω given by Planck Collaboration assuming flat FRW m metric (Ade et al. 2014). Moreover, we allow the lu- ap250 σ minosity density profile to be different from the total- mass density profile, i.e., α = δ, and the stellar velocity 240 6 anisotropy exits, i.e., β = 0. Based on a well-studied sample of nearby elliptica6 l galaxies from Gerhard et al. 230 (2001), the anisotropy β is characterized by a Gaus- 220 sian distribution, β = 0.18 0.13, which is also exten- ± sively used in the previous works (Bolton et al. 2006; 210 Schwab et al. 2010). More recently, Xu et al. (2016) measuredthestellarvelocityanisotropyparameterβ and 200 0 0.2 0.4 0.6 0.8 1 its correlationswithredshifts andstellarvelocity disper- z sion, based on the Illustris simulated early-type galax- l ies with spherically symmetric density distributions. It is worth noting from their results that β markedly de- Figure 1. Characteristics of the strong lensing data sample of pends on stellar velocity dispersion and its mean value 80 intermediate mass early-type galaxies. Observed velocity dis- varies from 0.10 to 0.30 for intermediate-mass galaxies persioninside the aperture is plotted against redshiftto the lens. ( 200km/s < σ 300km/s), which is consistent with ap ≤ the values used in our analysis. are characterized by Gaussian distributions: Following our previous analysis (Cao et al. 2016) concerning power-law mass and luminosity density pro- α = 2.00 ; σ = 0.08 files of elliptical galaxies, we used a mass-selected sam- hδi = 2.40 ; σα = 0.11 . (11) δ ple of strong lensing systems, taken from a compre- h i hensive compilation of strong lensing systems observed More importantly, the degeneracy between the two pa- by four surveys: SLACS, BELLS, LSD and SL2S. rameters, γ and δ, is apparently indicated by the results The sample has been defined by restricting the ve- presentedinFig.2,i.e.,asteeperluminosity-densitypro- locity dispersions of lensing galaxies to the interme- fileforthelensingearly-typegalaxieswillleadtoalarger diate range: 200km/s < σ 300km/s. Lenses value for the parameterized post-Newtonian parameter. ap of this sub-sample are located≤at redshifts ranging This tendency could also be seen from the sensitivity from z = 0.08 to z = 0.94. Original data analysis shown below. l l about these strong lenses were derived by Bolton et al. Now the parameters characterizing the total mass- (2008a); Auger et al. (2009); Brownstein et al. (2012); profileshape,velocityanisotropy,andlight-profileshape Koopmans & Treu (2002); Treu & Koopmans (2002, of lenses are set at their best measured values. Perform- 2004);Sonnenfeld et al.(2013a,b),andmorecomprehen- ing fits on γ, we find the resulting posterior probability sivedataconcerningthesesystemscanbefoundinTable density shown in Fig. 3. The result γ = 0.995+−00..003477 (1σ 1 of Cao et al. (2015a). Fig. 1 shows the scatter plot for confidence) is consistent with γ =1 and also with previ- this sample in the plane spanned by the redshift of the ous results of (Bolton et al. 2006) obtained with strong lens and its velocity dispersion. lensing systems. The scatter of galaxy structure parameters is an important source of systematic errors on the 3. MAINRESULTS final result. Taking the best-fitted values of the structure parameters as our fiducial model, we investigated Becauseα,andδcouldnotbeindependentlymeasured how the PPN constraint is altered by introducing the foreachlensinggalaxy,wefirstlytreatedthemasfreepa- uncertaintiesonα, β, andδ aslistedin Eq.(11). There- rameters and inferred α, δ, γ, simultaneously. Perform- fore, firstly, we perform a sensitivity analysis, varying ingfitsonthestronglensingdata-set,the68%confidence the parameter of interest while fixing the other param- level uncertainties on the three model parameters are eters at their best-fit values. In general, one can see from Table 1 and Fig. 4 that constraint on γ is quite α=2.017+−00..009832, sensitive to small systematic shifts in the adopted lens- δ =2.485+0.445, galaxy parameters. By comparing the contribution of −1.393 each of these systematic errors to the systematic error γ =1.010+−10..942552. onγ, we findthat the largestsourcesofsystematic error are the mass density slope α, followedby the anisotropy Fig. 2 shows these constraints in the parameter space parameter of velocity dispersion β and and the luminos- of α, δ, and γ. It is obvious that fits on α and δ are ity density slope δ. Secondly, by considering the intrin- well consistent with the analysis results of Bolton et al. sic scatter of α, β, and δ into consideration, we found (2006); Grillo et al. (2008); Schwab et al. (2010), which γ varying from 0.845 to 1.240 at 1σ confidence level. 4 Cao et al. 3 Table 1 Sensitivityofconstraintsonγ withrespecttothegalaxy 2.5 structureparameters. 2 γ 1.5 Systematics PPNparameter 1 α=2.00;β=0.18;δ=2.40 γ=0.995+−00..003477 α=1.92;β=0.18;δ=2.40 γ=0.860±0.040 0.5 α=2.08;β=0.18;δ=2.40 γ=1.169±0.050 α=2.00;β=0.05;δ=2.40 γ=0.914±0.043 0 1.9 2 2.1 2.2 α=2.00;β=0.31;δ=2.40 γ=1.087±0.043 3 3 α=2.00;β=0.18;δ=2.29 γ=1.111±0.044 α=2.00;β=0.18;δ=2.51 γ=0.883±0.039 2.5 2.5 δ 2 2 cal parameters of the ΛCDM model used in our study. For this purpose, we also considered WMAP9 result of 1.5 1.5 Ω = 0.279 in order to make comparison with Planck m observations. Not surprisingly, the result was that dif- 1 1 ferences were negligible. It could have been expected 1.9 2 α 2.1 2.2 0 1 γ 2 3 because cosmology intervenes here through the distance ratio D /D , which is very weakly dependent on the ls s value of Ω and in flat cosmology does not depend on Figure 2. Constraints onthePPNγ parameter, the total-mass m H at all. and luminosity density parameters obtained from the sample of 0 stronglensingsystems. Bluecrossesdenotethebest-fitvalues. The next generation wide and deep sky surveys with improved depth, area and resolution may, in the near future, increase the current galactic-scale lens sample sizes by orders of magnitude (Kuhlen et al. 2004; 1 Marshall et al. 2005). Such a significant increase of the SL constraint number of strong lensing systems will considerably im- GR prove the constraints on the PPN parameter. Now we 0.8 willillustratewhatkindofresultonecouldgetusingthe future data from the forthcoming Large Synoptic Sur- vey Telescope (LSST) survey, which may detect 120000 d lenses for the most optimistic scenario (Collett 2015). o0.6 o In order to make a good comparison with the results h eli derived with current strong lensing systems (Fig. 2), we k firstlyturntothesimulatedLSSTpopulationcontaining Li 0.4 40000 lensing galaxies with intermediate velocity dis- ∼persions (200km/s< σ 300km/s)3. Performing fits ap ≤ on this simulated strong lensing data-set, we obtain the constraints in the parameter space of α, δ, and γ shown 0.2 in Fig. 5. It is apparent that from the simulated LSST strong lensing data, we may expect the total-mass density parameter α to be estimated with 10−3 precision. 00 0.5 1 1.5 2 However, the degeneracy between the PPN γ parame- γ ter and the luminosity density parameter δ still needs to be investigated with future high-quality integral field unit (IFU) data (Barnabè et al. 2013). In the next sec- Figure 3. NormalizedposteriorlikelihoodofthePPNγ param- tion, we will apply a cosmological-independent method eter obtained with rigid priors on the nuisance parameters (α, β, δ). to study the degeneracy (Ra¨sänen et al. 2015) between cosmiccurvatureandparameterizedpost-Newtonianpa- It means that systematic errors might exceed 25% rameter γ. ∼ of the final result. The large covariances of γ with α and δ seen in Fig. 2 motivate the future use of auxiliary 4. COSMICCURVATUREANDPARAMETERIZED POST-NEWTONIANFORMALISM datato improveconstraintsonα, β andδ. For example, α can be inferred for individual lenses from high reso- In a homogeneous and isotropic Universe, the dimen- lution imaging of arcs (Suyu et al. 2006; Vegetti et al. sionlessdistanced(z1;zs)=(H0/c)(1+zs)DA(z1;zs)can 2010; Collett & Auger 2014; Wong et al. 2015), while be written as constraints on β and δ can be improved with integral 1 z2 dz′ field unit (IFU) data (Barnabè et al. 2013), without the d(z ;z )= sinn Ω , (12) assumption of general relativity (GR). 1 s |Ωk| (cid:20)p| k|Zz1 E(z′)(cid:21) Another issue which should be discussed is by how 3 Our simulatedpLSST sample is obtained with the simulation much is γ affected by the uncertainty of cosmologi- programsavailableonthegithub.com/tcollett/LensPop. Test of PPN gravity with galaxy-scale strong lensing systems 5 α=1.92 β=0.05 δ=2.29 α=2.08 β=0.31 δ=2.51 α=2.00 β=0.18 δ=2.40 GR GR GR d d d o o o o o o h h h eli eli eli Lik Lik Lik 0. 6 0.7 0.8 0.9 1 γ1.1 1.2 1.3 1.4 0. 7 0.8 0.9 1γ 1.1 1.2 1.3 0. 7 0.8 0.9 1γ 1.1 1.2 1.3 Figure 4. Normalizedlikelihoodplotforγ bychoosingdifferentgalaxystructureparameters. SN Ia data covering the redshift range 0 < z 1.414 ≤ (Amanullah et al. 2010). Therefore we bypassed the 2.5 need to assume any specific cosmological model. By 2 using Eq. (13) we were able to calculate the distance ratio d /d depending only on the curvature density ls s γ 1.5 parameter Ω . The reported statistical and system- k atic uncertainties of the distance modulus for individ- 1 ual SNe Ia are considered in the fitting procedure. In 0.5 the Union2.1 SN Ia compilation, light-curve fitting pa- 1.995 2 2.005 2.01 rameterswhich areusedfor distanceestimationare constrained in a global fit. However, compared to the un- 3 3 certaintiesinthemodelingofthestronglensingsystems, the model-dependence of the SNe Ia analysis is likely 2.5 2.5 subdominant (Ra¨sänen et al. 2015). Then we assessed the distance ratios d /d from the strong lensing data δ ls s 2 2 (Einstein radius and velocity dispersion) according to the Eq. 10. For this purpose we used the simulated observations of forthcoming photometric LSST survey 1.5 1.5 (Collett 2015). Using the simulation programs avail- 1.995 2 2.005 2.01 0.5 1.5 2.5 able on the github.com/tcollett/LensPop, we obtained α γ 53000 strong lensing systems meeting the redshift crite- rion 0 < z < z 1.414 in compliance with SNIa data l s ≤ Figure 5. Constraints onthePPNγ parameter, the total-mass usedin parallel. The simulated catalogis derivedon the and luminosity density parameters obtained from the simulated base of realistic population models of elliptical galaxies LSSTstronglensingdata. actingaslenses,withthemassdistributionapproximated by the singular isothermal ellipsoids. where E(z) = H(z)/H is the expansion rate, and Ω 0 k Following the assumptions underlying the simulation, is the spatial curvature density parameter; sinn(x) = we fixed α = δ = 2 and β = 0 in our analysis. We took sinh(x) for Ω > 0, sinn(x) = x for Ω = 0, and k k the fractional uncertainty of the Einstein radius at the sinn(x) = sin(x) for Ω < 0, respectively. For a k level of 1% and the observed velocity dispersion at the stronglensingsystemwiththefollowingnotation: d(z)= level of 10%. Secondary lensing contribution from the d(0;z), d = d(0;z ), d = d(0;z ), and d = d(z ;z ), a l l s s ls l s matteralongtheline-of-sightwasneglectedinouranaly- simple sum rule could be easily obtained as sis4. Fig.6displaysthefittingresultsintheΩ γplane, k − thusillustratingthedependencebetweenthecosmiccur- dls/ds =q1+Ωkd2l −dl/dsp1+Ωkd2s. (13) vflaattuureniavnerdsethteogPePthNerγwpitahratmheetevra.lidItitiysoafppGaRren(Ωt th=at0a, k [The case of Eq. (13) is given in, e.g., Peebles (1993), γ = 1) is strongly supported. More importantly, it is p336.] This fundamental formula provides an model- independent probe to test both the spatial curvature, in combination with weak lensing and baryon acoustic 4 Theassumption of 1% accuracy onthe Einstein radius measurements from future LSST survey is reasonable, although the oscillations (BAO) measurements (Bernstein 2006) and line-of-sighteffect mightintroduce∼3%uncertainties inthe Ein- the FLRW metric, in combination with strong lensing stein radii (Hilbertetal. 2009). However, according to the re- systems and SNe Ia observations (Ra¨sänen et al. 2015). cent analysis by Collett&Cunnington (2016), the lines-of-sight For the purpose of our analysis,we determined the di- for monitorable strong lenses (especially for quadruply imaged quasars) might be biased at the level of 1%. Some attempts to mensionless distances d and d of all “observed” strong l s accountfortheline-of-sightsecondarylensingforquasarscanalso lensing systems (taken from the LSST simulation by befoundinCollettetal.(2013); Greeneetal.(2013);Rusuetal. Collett (2015)) by fitting a polynomial to the Union2.1 (2016). 6 Cao et al. 1 Simulation (∆θ=0.01; ∆σ=0.10) Simulation (∆θ=0.01; ∆σ=0.05) GR 0.8 d o0.6 o −0.02 0 0.02 0.04 0.06 h eli k Li 0.4 1.01 1.005 γ 0.2 1 0.995 0 −0.02 0 Ω0.02 0.04 0.06 0.995 1 1.005 1.01 0.97 0.98 0.99 1γ 1.01 1.02 1.03 k Figure 6. ConstraintsonthePPNparameterandcosmiccurva- Figure 8. Constraints on the PPN parameter from simulated turefromthesimulatedLSSTstronglensingdata. LSST strong lensing data, with a prior on the cosmic curvature −0.007<Ωk <0.006fromPlanck. latest CMB data and baryon acoustic oscillation data (Ade et al. 2014), and get a constraint on the PPN parameter: γ = 1.000+0.0023. When we changed the frac- −0.0025 tionaluncertaintyoftheEinsteinradiustothelevelof1% and the observed velocity dispersion to the level of 5%, the resulting constraint on the PPN parameter became: 0.26 0.28 0.3 0.32 γ =1.000+0.0009. Theposteriorprobabilitydensity forγ −0.0011 −0.8 isshownin Fig.8. Onecanseefromthis plotthatmuch −0.9 more severe constraints would be achieved, and one can w −1 expect γ to be estimated with 10−3 10−4 precision. ∼ −1.1 5. CONCLUSIONS −1.2 0.26 0.28 0.3 0.32 −1.2 −1.1 −1 −0.9 −0.8 Based on a mass-selected galaxy-scale strong gravitational lenes from the SLACS, BELLS, LSD and SL2S 1.02 1.02 surveys and a well-motivated fiducial set of lens-galaxy 1.01 1.01 parameters, we tested the weak-field metric on kilopar- γ 1 1 sec scales andfound a constraintonthe post-Newtonian 0.99 0.99 parameter γ = 0.995+0.037 under the assumption of a −0.047 0.98 0.98 flat universe from Planck observations. Therefore it is 0.26 0.28 Ωm0.3 0.32 −1.2 −1.1 −w1 −0.9 −0.8 0.98 0.99 1 1.01 1.02 in agreement with the General Relativity value of γ =1 with 4% accuracy. Considering systematic uncertainties intotalmass-profileshape,velocityanisotropy,andlight- Figure 7. Constraints onthe PPNparameter andcosmological profileshape,weestimatesystematicerrorstobe 25%. parametersfromthesimulatedLSSTstronglensingdata. ∼ Furthermore, we illustrated what kind of result we interesting to note that there exists a significant degen- could get using the future data from the forthcoming eracy between the spatial curvature of the Universe and LargeSynopticSurveyTelescope(LSST)survey(Collett thePPNparameter,whichcaptureshowmuchspacecur- 2015). We applied a cosmological model independent vature is providedby unit rest mass of the objects along method to study the degeneracy (Ra¨sänen et al. 2015) or near the path of the particles. Similar degeneracy between cosmic curvature and the parameterized post- between γ and the other cosmological parameters (the Newtonian parameter γ. It is apparent that spatially matter density fraction, Ω and the equation of state of flat Universe with the conservation of GR (Ω = 0, m k dark energy, w) can also be seen from Fig. 7. γ = 1) is strongly supported. Moreover, the reduced One can easily check that reduction of the error of uncertainty of Ω leads to more stringent fits of γ. This k Ω would lead to more stringent fits of γ, which en- opensupthepossibilityoftestingPPNwithmuchhigher k courages us to consider the possibility of testing PPN accuracy using strong lensing systems discovered in the at much higher accuracy with future surveys of strong future surveys. By setting a prior on the cosmic cur- lensing systems. We now set a prior on the cosmic curvature with 0.007 < Ω < 0.006, assumed accord- k − vature with 0.007 < Ω < 0.006, according to the ing to the latest CMB plus baryon acoustic oscillation k − Test of PPN gravity with galaxy-scale strong lensing systems 7 data (Ade et al. 2014), the accuracy of γ determination Chae,K.-H.2003, MNRAS,346,746 reached 10−3 10−4 precision. Chae,K.-H.,Chen,G.,Ratra,B.,&Lee,D.-W.2004,ApJ,607, ∼ L71 Therefore, we conclude that samples of strong lens- Collett,T.E.,etal.2013,MNRAS,432,679 ing systems with measured stellar velocity dispersions, Collett,T.E.&Auger,M.W.2014,MNRAS,443,969 much larger than currently available, may serve as an Collett,T.E.2015,arXiv:1507.02657 importantprobe to testthe validity ofthe GR,provided Collett,T.E.&Cunnington, S.D.2016,MNRAS,462,3255 that mass-dynamicalstructure of lensing galaxiesis bet- Dyson,F.W.,Eddington,A.S.,&Davidson,C.1920,Phil. Trans.R.Soc.,220,291 ter characterized and constrained in the future surveys. 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