Strong lensing probability in TeVeS theory

0.38 MB·

by  Da-Ming Chen

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.