Table Of Content

Strong lensing probability in TeVeS theory Da-Ming Chen 8 NationalAstronomicalObservatories,ChineseAcademyofSciences, Beijing 0 100012, China 0 E-mail: [email protected] 2 n Abstract. We recalculate the strong lensing probability as a function of the a imageseparationinTeVeS(tensor-vector-scalar)cosmology,whichisarelativistic J version of MOND (MOdified Newtonian Dynamics). The lens is modeled by 0 the Hernquist profile. We assume an open cosmology with Ωb = 0.04 and 1 ΩΛ=0.5andthreedifferentkindsofinterpolatingfunctions. Twodifferentgalaxy stellarmassfunctions(GSMF)areadopted: PHJ(Panter-Heavens-Jimenez,2004) ] determined from SDSS data release one and Fontana (Fontana et al., 2006) h from GOODS-MUSIC catalog. We compare our results with both the predicted p probabilities for lenses by Singular Isothermal Sphere (SIS) galaxy halos in - LCDM (lambda cold dark matter) with Schechter-fit velocity function, and the o observational results of the well defined combined sample of Cosmic Lens All- r Sky Survey (CLASS) and Jodrell Bank/Very Large Array Astrometric Survey t s (JVAS). It turns out that the interpolating function µ(x)=x/(1+x) combined a withFontanaGSMFmatches theresultsfromCLASS/JVASquitewell. [ 2 v PACSnumbers: 98.80.-k,98.62.Sb,98.62.Ve,95.35.+d 3 3 6 1 Submitted to: JCAP . 2 1 7 0 : v i X r a Strong lensing probability in TeVeS theory 2 1. Introduction The standardLCDMcosmologyisverysuccessfulinexplainingthe cosmicmicrowave background (CMB, see, e.g., [78]), baryonic acoustic oscillation (BAO, see, e.g., [27]), gravitational lensing (see, e.g., [38]) and large scale structure (LSS) formation. However, LCDM faces some fundamental difficulties. From the observational point of view, the challenges to LCDM arise from smaller scales. For example, the theory cannot explain Tully-Fisher law and the Freeman law [25, 80]. The most difficult ones are the satellites problem and cusps problem. The most key problems are, of course, the unknown nature of Dark Matter (DM) and Dark Energy (DE). Before CDM particles are detected in the lab, science should remain open to the prospect thatDM(andforthesimilarreasons,DE)phenomenamayhavesomedeepunderlying reason in new physics. There are several proposals for resolving DM and DE problems by modifying Newtonian gravity or general relativity (GR) rather than resorting to some kinds of exotic matter or energy. MOND [48] was originally proposed to explain the observed asymptotically flat rotation curves of galaxies without DM, however, it was noticed thatMONDcanalsoexplainTully-FisherlawandFreemanlaw[46,47]. Itisbelieved thatMONDissuccessfulatgalacticscales[84,88](butsee[37]forsatellitesproblem). The challenges to MOND arise from clusters of galaxies [71], in which, some kind of dark matter, possibly some massive neutrinos with the mass of 2ev, is also needed ∼ to explain the dynamics of galaxies[4]. MOND and its relativistic version, TeVeS [6], are only concerned with DM, remain DE as it is. By adding a f(R) term in Einstein-Hilbert Lagrangian, where R is the Ricci scalar, the so called f(R) gravity theory can account for DE [1, 12, 13, 57, 58, 76, 77, 82]. Another interesting theory is Modified Gravity (MOG) [50], it is a fully relativistic theory of gravitation that is derived from a relativistic action principle involving scalar, tensor and vector fields. MOGhasbeen usedsuccessfully to accountforgalaxycluster masses[9], the rotation curves of galaxies (similar to MOND) [10], velocity dispersions of satellite galaxies [51], globular clusters [52] and Bullet Cluster [11], all without resorting to DM. Most recently, MOG is used to investigate some cosmological observations (CMB, galaxy mass power spectrum and supernova), and it is found that MOG provides good fits to data without DM and DE [53]. Anymodificationstotraditionalgravitytheorymustbetestedwithobservational experiments. Gravitational lensing provides a powerful probe to test gravity theory [75, 85]. It is well known that, in standard cosmology (LCDM), when galaxies are modeled by a SIS and galaxy clusters are modeled by a Navarro-Frenk-White (NFW) profile, the predicted strong lensing probabilities can match the results of CLASS/JVAS quite well [15, 17, 18, 19, 20, 34, 35, 39, 40, 42, 43, 44, 49, 65, 62, 64, 60, 63, 59, 68, 70, 74, 83, 86, 87]. This paperisdevotedtoexplorethestronglensingstatisticsinTeVeStheory. As an alternative to LCDM cosmology,TeVeS cosmologyhas received much attention in the recentliterature,inparticularinthe aspectofgravitationallensing [2,23,89], for reviews see [7, 72]. Before TeVeS, strong gravitational lensing in the MOND regime couldonlybemanipulatedbyextrapolatingnon-relativisticdynamics[69,54],inwhich the deflection angle is only half the value in TeVeS [91]. In TeVeS theory, it is now established that, for galaxy clusters, both weak and strong lensing need extra DM to explain observations [3, 30, 29, 79], possibly neutrinos with the mass of 2ev, like ∼ the dynamics of galaxies. The situation is better for galaxies, as will be shown in Strong lensing probability in TeVeS theory 3 this paper. In our previous paper [22], as a first try to calculate the strong lensing probabilityasafunctionoftheimage-separation∆θ inTeVeScosmology,weassumed a flat cosmology with Ω = 1 Ω = 0.04 and the simplest interpolating function b Λ − µ(x) = min(1,x). In this paper, we assume an open cosmology with Ω = 0.04 and b Ω =0.5andthreedifferentkindsofinterpolatingfunctions. Asformassfunction,in Λ addition to the PHJ GSMF [66] used in our previous paper, we also adopt a redshift- dependent Fontana GSMF [31]. Further more, the amplification bias is calculated based on the total magnification of the outer two brighter images rather than the magnification of the second bright image of the three images as did in our previous work [22]. 2. TeVeS cosmology and deflection angle Gravitational lensing can be used to test TeVeS in two aspects. First, in the non- relativistic and spherical limit, TeVeS reduces to MOND. The deflection angle of the light ray passing through the lensing object can be calculated in MONDian regime (this will be discussed later). Second, the distances between the source, the lens and the observer are cosmological and thus depend on the geometry and evolution propertiesofthebackgrounduniverse. AsarguedbyBekenstein[6,89],thescalarfield φ, which is used to produce a MONDian gravitationalacceleration in non-relativistic limit, contributes negligibly to Hubble expansion. According to the cosmological principle, the physical metric takes the usual Friedmann-Robertson-Walker (FRW) form in TeVeS [5], dτ2 = c2dt2+a(t)2[dχ2+f2(χ)(dθ2+sin2θdψ2)], (1) − K where c is the speed of light, a(t) is the scale factor and K−1/2sin(K1/2χ) (K >0) f (χ)= χ (K =0). (2) K  ( K)−1/2sinh[( K)1/2χ] (K <0) − − As in general relativity (GR), we define the cosmologicalparameters: ρ Λ Kc2 b Ω , Ω , Ω − (3) b ≡ ρ (0) Λ ≡ 3H2 K ≡ H2 crit 0 0 where ρ is the mean baryonic matter density in the universe at present time t b 0 c(rreitdicshalifmt azss=de0n)s,itρyc,riatn(0d)H= =3H1002/0(h8kπmGs)−1=Mp2c.7−81×is1th01e1Hh2uMbb⊙leMcpocn−s3tainst.thWeepcrheoseonset 0 a(t )=1. Sincedχ=cdz/H(z),theproperdistancefromtheobservertoanobjectat 0 redshift z is Dp(z)=c z[(1+z)H(z)]−1dz, where the Hubble parameter at redshift 0 z is (known as Friedmann’s equation) R a˙ H(z) =H Ω (1+z)3+Ω (1+z)2+Ω . (4) 0 b K Λ ≡ a The comoving distance from anpobject at redshift z to an object at redshift z is 1 2 z2 cdz χ(z ,z )= , (5) 1 2 H(z) Zz1 the corresponding angular diameter distance therefore is 1 D(z ,z )= f [χ(z ,z )]. (6) 1 2 K 1 2 1+z 2 Strong lensing probability in TeVeS theory 4 In TeVeS, the lensing equation has the same form as in general relativity (GR), and for a spherically symmetric density profile [89] ∞ D 4b dΦ(r) LS β =θ α, α(b)= dl, (7) − D c2r dr S Z0 where β, θ = b/D and α(θ) are the source position angle, image position angle and L deflection angle, respectively; b is the impact parameter; D , D and D are the L S LS angular diameter distances from the observer to the lens, to the source and from the lens to the source, respectively; g(r) = dΦ(r)/dr is the actual gravitational acceleration [here Φ(r) is the spherical gravitational potential of the lensing galaxy and l is the light path]. It is well known that the stellar component of an elliptical galaxy can be well modeled by a Hernquist profile M r 0 h ρ(r)= , (8) 2πr(r+r )3 h with the mass interior to r as r2M 0 M(r)= , (9) (r+r )2 h where M = ∞4πr2ρ(r)dr is the total mass and r is the scale length. The 0 0 h corresponding Newtonian acceleration is g (r) = GM(r)/r2 = GM /(r + r )2. N 0 h R According to MOND [48, 71,72], the actualaccelerationg(r) is relatedto Newtonian acceleration by g(r)µ(g(r)/a )=g (r), (10) 0 N where µ(x) is the interpolating function and has the properties x, for x 1 µ(x)= ≪ (11) (1, for x 1 ≫ and a = 1.2 10−8cms−2 is the critical acceleration below which gravitational law 0 × transits from Newtonian regime to MONDian regime. The concrete form of a µ(x) function should be determined by observational data (e.g., the rotation curves of spiral galaxies) and expected by a reasonable scalar field theory (e.g., TeVeS). The “standard” function one usually takes is µ(x) = x/√1+x2, which fits well to the rotationcurves of most galaxies. Unfortunately, if the MOND effect is produced by a scalarfield(suchasTeVeS),the“standard”µ(x)functionturnsouttobemultivalued [90]. On the other hand, a “simple” function µ(x) =x/(1+x) suggested by Famaey & Binney [28] fits observational data better than the “standard” function and is consistent with a scalar field relativistic extension of MOND [90, 73]. In order to explore a broad class of modified gravity models, Zhao and Tian [92] proposed a parametrized modification function 1 1 g a kn n 0 = 1+ , (12) µ(g/a0) ≡ gN " (cid:18)gN(cid:19) # in which, MOND gravity corresponds to k = 1/2. Substituting equation (10) into equation (12) with k =1/2, we have a n2 −n1 0 µ(g/a )= 1+ , (13) 0 gµ(g/a ) " (cid:18) 0 (cid:19) # Strong lensing probability in TeVeS theory 5 which can be easily solved to obtain the usual form of the µ function for MOND [92] −2/n 1 1 g µ(x)=x + +xn , x= . (14) 2 4 a " r # 0 It is easy to verify that the “simple” and “standard” µ function are approximated with high accuracy by equation (14) with n = 3/2 and n = 3, respectively [92]. The requirement for a physical and monotonic µ function limits the parameter n to the range of 1.5 n 2.0. In this paper, we consider three cases: n=1.5, 2.0 and 3.0. ≤ ≤ SincetheMONDiangravitationalaccelerationgisexplicitlyexpressedintermsof the Newtonianaccelerationg , it is very convenientto use equation (12) to calculate N the deflection angle 4 ∞ g(r)b α(b)= dl c2 r Z0 ∞ 4GM b 1 a = 0 [1+( 0)n/2]1/ndl (15) c2 r(r+r )2 g Z0 h N By using r =b 1+(l/b)2 and θ =b/D , we have L α(θ)=0′′.207hp−1 c/H0 M ∞ [1+(a0/gN)n/2]1/n dx, (16) (cid:18) DL (cid:19) θ Z0 √1+x2[0.05rh(c/H0)/(DLθ)+√1+x2] whereM =M0/M⋆ andM⋆ =7.64 1010h−2M⊙ isthecharacteristicmassofgalaxies × [66], and a D 2 θ2 c/H r 2 0 =2.38 L 1+x2+0.05 0 h . (17) g c/H M D θ N (cid:18) 0(cid:19) (cid:18) L (cid:19) p In equations (16) and (17), the image position angle θ and the scale length r are in h ′′ units of arcsecond ( ) and Kpc, respectively. We needarelationshipbetweenthe scalelengthr andthe massM,whichcould h be determined by observationaldata. First, the scale length is related to the effective (or half-light) radius R of a luminous galaxy by r = R /1.8 [32]. It has long been e h e recognizedthatthereexistsacorrelationbetweenR andthemeansurfacebrightness e I interior to R [26]: R I −0.83±0.08 . Since the luminosity interiorto R (half- h ei e e ∝h ie e light) is L = L/2 = π I R2, one immediately finds R L1.26. Second, we need e h ie e e ∝ to know the mass-to-light ratio Υ = M/L Lp for elliptical galaxies. The observed ∝ data gives p=0.35 [81]; according to MOND, however, we should find p 0 [72]. In ≈ any case we have L M1/(1+p). (18) ∝ Therefore, the scale length should be related to the stellar mass of a galaxy by r = h AM1.26/(1+p), and the coefficient A should be further determined by observational data. Without a well defined sample at our disposal, we use the galaxy lenses which haveanobservedeffectiveradiusR (andthusr )intheCASTLESsurvey[55],which e h are listed in table 2 of [89]. The fitted formulae for r are h 1.26 M 0.72 Kpc, for p=0.0, M r = (cid:18) ⋆(cid:19) . (19) h 1.24 M 1.26/1.35 Kpc, for p=0.35 M (cid:18) ⋆(cid:19) In later calculations, except indicated, we use the fitted formula of rh for p = 0 as required by MOND. Strong lensing probability in TeVeS theory 6 3. Galaxy stellar mass function In LCDM cosmology, mass function of virialized CDM halos can be obtained in two independent ways. One is via the generalized Press-Schechter (PS) theory, the other is via Schechter luminosity function. In TeVeS, however, the PS-like theory does not exist. Fortunately, the stellar mass function of galaxies is available in the literature, including the one constrained by the most recent data [31, 66]. Before giving the galaxy stellar mass functions (GSMF) appeared in the most recentliterature,itishelpfultoderiveaGSMFdirectlyfromtheSchechterluminosity function and mass-to-light ratio. The Schechter luminosity function is α L L dL φ(L)=φ⋆ exp . (20) L −L L (cid:18) ⋆(cid:19) (cid:18) ⋆(cid:19) ⋆ For L/L =(M/M )1/(1+p) implied by equation (18), we have a GSMF ⋆ ⋆ φ⋆ M α1++p1−1 M 1+1p dM φ(M)= exp . (21) 1+p(cid:18)M⋆(cid:19) "−(cid:18)M⋆(cid:19) # M⋆ While theaveragenumberdensityofgalaxiesφ ,theslopeatlow-massendαandthe ⋆ slope of mass-to-light ratio p may be easily found from the published observational data or assumptions, the characteristic stellar mass of galaxies M can be derived ⋆ from ∞ ρ =Ω ρ (0)= Mφ(M)dM, (22) lum lum crit Z0 where ρ is the luminous baryonic matter density (note that ρ ρ ). The lum lum b ≪ characteristic mass M is ⋆ Ω ρ lum crit(0) M = . (23) ⋆ φ Γ(α+p+2) ⋆ For example, for (φ ,α,Ω ,p)=(0.014h3Mpc−3, 1.1,0.003,0.35)from [41], M = ⋆ lum ⋆ 6.56 1010h−1M⊙; for the same parameters excep−t that p = 0.0 (MOND), M⋆ = 5.56×1010h−1M⊙. × Fortunately, the parameters in equation (21) have been determined by recent observational data. By determining non-parametrically the stellar mass functions of 96545 galaxies from the Sloan Digital Sky Survey Data (SDSS) release one, Panter, Heavens and Jimenez [66] (PHJ, hereafter) give the GSMF [22] M α˜ M dM φ(M)dM =φ exp , (24) ⋆ M −M M (cid:18) ⋆(cid:19) (cid:18) ⋆(cid:19) ⋆ where, we use φ(M) to denote the comoving number density of galaxies with mass between M and M +dM, and φ =(7.8 0.1) 10−3h3Mpc−3, ⋆ ± × α˜ = 1.159 0.008, (25) − ± M⋆ =(7.64 0.09) 1010h−2M⊙. ± × Most recently, in order to study the assembly of massive galaxies in the high redshift Universe, Fontana et al. [31](Fontana, hereafter) used the GOODS-MUSIC catalog Strong lensing probability in TeVeS theory 7 to measure the evolutionof the GSMF and of the resulting stellar mass density up to redshift z =4. The GSMF they obtained is α˜(z) M M dM φ(M,z)dM =φ (z) exp , (26) ⋆ M (z) −M (z) M (z) (cid:20) ⋆ (cid:21) (cid:20) ⋆ (cid:21) ⋆ where φ⋆(z)=n⋆0(1+z)n⋆1, n⋆0 =0.0035, n⋆1 =−2.20±0.18, α˜(z)=α˜ +α˜z, α˜ = 1.18,α˜ = 0.082 0.033, 0 1 0 1 − − ± (27) M⋆(z)=10M0⋆+M1⋆z+M2⋆z2h−2M⊙, M⋆ =11.16,M⋆=0.17 0.05,M⋆ = 0.07 0.01 0 1 ± 0 − ± ItwouldbeinterestingtocomparePHJandFontanaGSMFstothemassfunction of galaxies in LCDM cosmology when the galactic halos are modeled by SIS. The comoving number density of galactic halos with velocity dispersion between v and v+dv [49, 22] is v α˜ v β˜ v φ(v)dv =φ exp β˜ , (28) ⋆ (cid:18)v⋆(cid:19) "−(cid:18)v⋆(cid:19) # v⋆ Figure 1. ComovingnumberdensityforPHJ(solid),Fontana (dotted) andSIS halos (dash). Since Fontana mass function depends on redshift, four cases with redshiftz=0.0,0.5,1.0,2.0aredisplayed. Forcomparison,wenormalizethethree massfunctionstothesamevalueofcharacteristicmassM⋆=7.64×1010h−2M⊙. Strong lensing probability in TeVeS theory 8 For comparison, we need to transform equation (28) from velocity dispersion to halo mass M r200 800π M =4π ρ (r)r2dr = r3 ρ (z), (29) SIS 3 200 crit Z0 wherer isthevirialradiusofagalactichalowithinwhichtheaveragemassdensity 200 is 200 times the critical density of the Universe ρ (z). Substituting the well known crit expression ρ (r)=v2/(2πGr2) into equation (29), it is easy to find SIS v 3 M(z)=6.58×105 kms−1 [Ωm(1+z)3+ΩK(1+z)2+ΩΛ]−1/2h−1M⊙, (30) where Ω is the ma(cid:16)tter den(cid:17)sity parameter(including darkandbaryoniccomponents) m [42]. Equation (30) means that at any redshift z we should have M v3, or for our ∝ purpose, another form 3 M v = , (31) M v ⋆ (cid:18) ⋆(cid:19) we thus have the galaxy mass function for SIS halos φ β˜ M (α˜−2)/3 M β˜/3 dM ⋆ φ(M)= exp . (32) 3 (cid:18)M⋆(cid:19) "−(cid:18)M⋆(cid:19) # M⋆ We plot PHJ and Fontana GSMFs in figure 1 together with the galaxy mass function for SIS halos (comoving number density). For SIS halos, we use (φ ,α˜,β˜) = ⋆ (0.0064h3Mpc−3, 1.0,4.0) [14]. For comparison, we normalize the three mass functions to the s−ame value of characteristic mass M⋆ = 7.64 1010h−2M⊙. Note × that, for Fontana GSMF, the comoving number density of galaxies decreases with increasing redshift, as expected [31]. 4. lensing probability Usually, lensing cross section defined in the lens plane with image separations larger than ∆θ is σ(> ∆θ) = πD2β2Θ[∆θ(M) ∆θ], where Θ(x) is the Heaviside step L cr − function and β is the caustic radius within which sources are multiply imaged. This cr is true only when ∆θ(M) is approximately constant within β , and the effect of the cr fluxdensityratioq betweentheoutertwobrighterandfainterimagescanbeignored. r Generally this is not true, readers are referred to [22] for details. We introduce a source position quantity β determined by qr θ(β)dθ(β) θ(β)dθ(β) =q , (33) r β dβ β dβ (cid:18) (cid:19)θ>0 (cid:12) (cid:12)θ0<θ<θcr (cid:12) (cid:12) where θ0 = θ(0) < 0, the absolute valu(cid:12)e of which(cid:12)is the Einstein radius, and θcr is (cid:12) (cid:12) determined by dβ/dθ = 0 for θ < 0. Equation (33) means that when β < β < β , qr cr thefluxdensityratiowouldbelargerthanq ,whichistheupperlimitofawelldefined r sample. Therefore, the source position should be within β according to the sample qr selection criterion. For example, in the CLASS/JVAS sample, q 10. r ≤ The amplification bias should be considered in lensing probability calculations. For the source QSOs having a power-law flux distribution with slope γ˜ (= 2.1 in the CLASS/JVAS survey), the amplification bias is B(β) = µ˜γ˜−1 [65], where, in this paper, θ dθ θ(β)dθ(β) µ˜(β)= + (34) β dβ β dβ (cid:12) (cid:12)θ0<θ<θcr (cid:18) (cid:19)θ>0 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Strong lensing probability in TeVeS theory 9 is the total magnificationof the outer two brighter images. In our previous work[22], however,the amplificationbias is calculatedbasedon the magnificationofthe second bright image of the three images. Therefore,thelensingcrosssectionwithimage-separationlargerthan∆θandflux densityratiolessthanq andcombinedwiththeamplificationbiasB(β)is[75,19,22] r σ(>∆θ,<q )=2πD2 r L× βqr βµ˜γ˜−1(β)dβ, for ∆θ ∆θ , 0 ≤ Z0  Z0βqr −Z0β∆θ!βµ˜γ˜−1(β)dβ, for ∆θ0 <∆θ ≤∆θqr, (35) where β∆θ is theso0u,rce position at which a lens produfcoers∆thθe>im∆aθgqer,separation ∆θ, ∆θ =∆θ(0) is the separation of the two images which are just on the Einstein ring, 0 and ∆θ =∆θ(β ) is the upper-limit of the separationabove whichthe flux ratio of qr qr the two images will be greater than q . r The lensing probability with image separation larger than ∆θ and flux density ratiolessthanq ,inTeVeScosmology,forthesourceQSOsatmeanredshiftz =1.27 r s lensed by foreground elliptical stellar galaxies is [19, 20, 21, 22] zs dDp(z) ∞ P(>∆θ,<q )= dz φ(M,z)(1+z)3σ(>∆θ,<q )dM, (36) r r dz Z0 Z0 We plot in Figure 2 and Figure 3 the numerical results of the lensing probability according to equation (36). In TeVeS (solid lines), we assume an open cosmology with Ω = 0.04 and Ω = 0.5, as implied by fitting to a high-z Type Ia supernova b Λ luminosity modulus [89]. The lensing galaxy is modeled by Hernquist profile with length scale r = 0.72(M/M )1.26Kpc for constant mass-to-light ratio as required by h ⋆ MOND [see equation (19)]. The interpolating functions with three cases n = 1.5,2.0 and 3.0 are considered(top-down) according to equation (13). In order to investigate the effects of MOND on strong lensing, we also calculated the probabilities (dotted lines)withnomodificationtogravitationtheory(i.e.,inGR)andwithoutdarkmatter (i.e., lensing galaxy is modeled by Hernquist profile). In this case, two types of the fittedformulaeforthelengthscaler withp=0and0.35(top-down)areadopted. In h TeVeSandGR(withnodarkmatter),weadopttheGSMFasthemassfunction(mf), with mf=PHJ in Figure 2 and mf=Fontana in Figure 3. As did in our previous work [22], We recalculatethe lensing probabilitywith imageseparationlargerthan∆θ and flux density ratioless than q , in flat LCDM cosmology(Ω =0.3 andΩ =0.7),for r m Λ the sourceQSOsatmean redshiftz =1.27lensedby foregroundSIS modeledgalaxy s halos [14, 45, 49]: zs dDp(z) ∞ P (>∆θ,<q )= dz dvn¯(v,z)σ (v,z)B, (37) SIS r SIS dz Z0 Zv∆θ where n¯(v,z) = φ(v)(1+z)3, which is related to the comoving number density φ(v) given by equation (28), is the physical number density of galaxy halos at redshift z with velocity dispersion between v and v+dv [49], v 4 D D 2 σ (v,z)=16π3 LS L (38) SIS c D (cid:16) (cid:17) (cid:18) S (cid:19) Strong lensing probability in TeVeS theory 10 Figure 2. Predicted lens probability withan imageseparation angle >∆θ and the flux ratio ≤qr =10. For TeVeS (solid line) and GR (no CDM and without modificationofgravity,dottedline),weassumeanopencosmologywithΩb=0.04 andΩΛ=0.5,modelthelensastheHernquistprofileandadoptPHJGSMF(24); forstandardLCDM(dashedline),weassumeaflatcosmologywithΩm=0.3and ΩΛ =0.7, modelthe lens asthe SISandadopt the massfunction (28). For GR, we consider two different mass-to-light ratio types and thus the expressions of rh,seeequation(19). Forcomparison,thesurveyresultsofCLASS/JVAS(thick histogram)arealsoshown. is the lensing cross section, c D ∆θ′′ v =4.4 10−4 S (39) ∆θ × (cid:18)v⋆(cid:19)s DLS ′′ is the minimum velocity for lenses to produce image separation ∆θ and B is the amplification bias. We adopt (φ ,v ,α˜,β˜) = (0.0064h3Mpc−3,1≥98kms−1, 1.0,4.0) ⋆ ⋆ − for early-type galaxies from [14]. A subset of 8958 sources from the combined JVAS/CLASS survey form a well-defined statistical sample containing 13 multiply imagedsources(lens systems)suitable for analysisofthe lens statistics[56, 8,67, 36]. Theobservedlensingprobabilitiescanbeeasilycalculated[18,19,21]byP (>∆θ)= obs N(>∆θ)/8958,where N(>∆θ) is the number of lenses with separationgreater than ∆θ in 13 lenses. For comparison, the observational probability P (> ∆θ) for the obs surveyresultsofCLASS/JVASisalsoshown(thickhistogram). Itwouldbehelpfulfor us tofigureoutdifferencesamongmodelstosummarizethe valuesofthe probabilities P(>∆θ =0.3′′) in the Table 1.