Can TeVeS avoid Dark Matter on galactic scales?

0.21 MB·English

by  Nick E. Mavromatos

Can TeVeS avoid Dark Matter on galactic scales?
Nick E. Mavromatos,∗ Mairi Sakellariadou,† and Muhammad Furqaan Yusaf‡
King’s College London, Department of Physics, Strand WC2R 2LS, London, U.K.
A fully relativistic analysis of gravitational lensing in TeVeS is presented. By estimating the
lensing masses for a set of six lenses from the CASTLES database, and then comparing them to
thestellar mass, the deficit between thetwo is obtained and analysed. Considering a parametrised
range for the TeVeS function µ(y), which controls the strength of the modification to gravity, it is
found that on galactic scales TeVeS requiresadditional dark matter with thecommonly used µ(y).
A soft dependence of the results on the cosmological framework and the TeVeS free parameters is
discussed. For one particular form of µ(y), TeVeS is found to require very little dark matter. This
9
choice is however ruled out by rotation curve data. The inability to simultaneously fit lensing and
0
rotation curvesfor a single form of µ(y) is a challenge to a nodark matter TeVeS proposal.
0
2 PACS numbers:

n
a
Thestandard(ΛCDM)cosmologicalparadigmisbased galactic scales.
J
uponColdDarkMatter(CDM),acosmologicalconstant In this letter TeVeS is examined in a similar way,
5
Λ and classical general relativity/Friedman-Robertson- namely by deriving the modified lensing equation and
2
Lemaˆıtre-Walker cosmology. Despite its enormous suc- solving it numerically, to investigate whether it also

] cess and consistency with a plethora of astrophysical shows a dark component from lensing. The (weak) de-
A
data, competing models have been proposed for the pri- pendence of the results on the cosmological models and
G
mary reason of the still unknown nature of the dark en- TeVeS free parameters is then discussed. TeVeS [2] is a
. ergy component and the current undetectability of dark bi-metric model in which matter and radiation does not
h
p matter. To explain the observed flat rotation curves feel the Einstein metric, gαβ, appearing in the canon-
- of galaxies without Dark Matter, Milgrom [1] proposed ical kinetic term in the (effective) action, but a modi-
o
MOdified Newtonian Dynamics (MOND), based upon fied“physical”metric,g˜ ,relatedtotheEinsteinmetric
r αβ
st the relation f(|~a|/a0)~a = −∇~ΦN between the accelera- by g˜αβ = e−2φgαβ −UαUβ(e2φ−e−2φ), where Uµ,φ de-
a tion ~a and the Newtonian gravitational field Φ . The note the TeVeS vector and scalarfield, respectively. The
N
[ constanta ≈1.2×10−10m/s2 ismotivatedbytheaccel- TeVeS action is:
0

1 eration found in the outer regions of galaxies where the
1
v rotationcurveis flat. When f,assumedto be a positive, S = d4x (R−2Λ)
16πG
2 smooth, monotonic function, equals unity, usual Newto- Z (cid:20)
3 1 1
nian dynamics hold, while when it approximately equals − {σ2(gµν −UµUν)φ φ + Gl−2σ4F(kGσ2)}
9 ,α ,β
its argument, the deep MONDian regime applies. 2 2
3
1
1. MONDhasbeensuccessfulinexplainingthedynamics − KFαβFαβ −2λ(UµUµ+1) (−g)1/2
32πG
0 of disk galaxies, however it is less successful for clusters (cid:21)
9 +L(g˜ ,(cid:8)fα,fα,...)(−g˜)1/2, (cid:9) (1)
of galaxies. It was promoted [2] to a classicalrelativistic µν |µ
0
fieldtheory by introducinga TEnsor,VEctor andScalar
: where k,K are the coupling constants for the scalar,
v field(TeVeS).TeVeShasbeencriticisedaslackingafun-
i vector field, respectively; ℓ is a free scale length re-
X damental theoretical motivation. Recently, it has been
lated to a (c.f below, after Eq.(5)); σ is an addi-
argued[3]thatsuchatheorycanemergenaturallywithin 0
r tional non-dynamical scalar field; F ≡ U − U ;
a some string theory models. µν µ,ν ν,µ
λ is a Lagrange multiplier implementing the constraint
In Ref. [4], where the lensing mass in MOND was gαβU U = −1, which is completely fixed by variation
α β
compared to the stellar mass content of the lenses, it of the action; the function F(kGσ) is chosen to give
wasfoundthatcomparableamountsofdarkmatterwere the correct non-relativistic MONDian limit, with G re-
needed in MOND to that required in the standard lens- lated to the Newtonian gravitational constant, G , by
N
ing scenario. This result is in contrast with attempts to G = G /(1+ K+k −2φ ), where φ is the present day
N 2 c c
explain the lensing data on galactic-cluster scales by in- cosmological value of the scalar field. Covariant deriva-
troducing a 2 eV neutrino. In fact, this component of tives denoted by | are taken with respect to g˜ and
µ... µν
dark matter has been shown[5] to cluster on Mpc scales indices are raised/lowered with the metric g . A new
µν
butnotongalacticscaleswherethepreviousanalysiswas function µ(y) is introduced as [2]
conducted. Itwasconcludedthateitherlensingmustop-
1 dF(µ)
erateinaqualitativelydifferentwaywithinthecovariant −µF(µ)− µ2 =y ≡kℓ2(gµν −UµUν)φ φ
,µ ,ν
“parent”theoryofMOND,suchastheTeVeSmodels,or 2 dµ
darkmattershouldbeconsideredwithinMONDevenon kGσ2 =µ kℓ2(gµν −UµUν)φ φ .(2)
,µ ,ν
(cid:0) (cid:1)
2
Possiblechoicesfortheµfunctionwillbediscussedlater. of the galaxy contained within the virial radius, r ; it
vir
The isotropic spherically symmetric Einstein metric also specifies the density profile.
can be generically written as g dxαdxβ = −eνdt2 + Making the approximation m (< r) ≈ M(< r),
αβ s
eζ(dr2+r2dθ2+r2sin2θdϕ2);bothν andζ arefunctions shown [2] to be correct to leading order of r, Eq. (3) is
ofr. Thephysicalmetricg˜ hasthesameform,witheν˜ numerically solved and through a transformation to the
αβ
and eζ˜, related to the Einstein metric functions through physical metric, the deflection angle Eq. (4) is obtained.
ν˜ = ν+2φ, ζ˜= ζ −2φ. Isotropy implies φ = φ(r). As- A specification for the function µ (see, Eq. 2) is re-
suming an ideal pressureless matter fluid, T˜ =ρ˜u˜ u˜ . quired. Previous lensing studies [11, 12] on the non-
αβ α β
Motivated by a homogeneous and isotropic cosmology, relativistic scalar potential approach to TeVeS, adopted
thevectorfieldisconsideredtime-like. Thenormalisation a model for µ(y) given in Ref. [2]. It was however
condition imposed by the Lagrange multiplier in Eq. (1) noted [14, 15] that when this choice is converted into its
gives Uα =(e−ν/2,0,0,0). MONDian equivalent, rotation curve data are not well
Previous attempts [10, 11, 12] at lensing analysis in fitted. Itwashenceproposedtoconsider[16]theMOND
TeVeS have remained non-relativistic, considering only functionwhichbestfitstherotationcurvedataandthen
the effect of adding a scalar potential to the standard convert it into its TeVeS µ(y) analogue. Any intermedi-
Newtonian potential. Hence they are insensitive to any ate choice for µ(y) can be parametrised by α. In what
unique features of TeVeS as a fully relativistic field the- followsthe resultsfromtestingthis parametrisedµ(y)in
ory. Theanalysisgivenhereisusedtosolve(numerically) the fully relativistic treatment of lensing in TeVeS are
the TeVeS equations of motion and obtain explicit ex- presented in order to see if TeVeS can both fit rotation
pressionsfor the physicalmetric quantitiesν˜,ζ˜andthus curve data and lensing data for a single choice of µ(y).
derivethemodifiedBirkhoff’stheoremfortheTeVeSthe- For TeVeS µ(y), F(µ) and MOND f(x) we use:
ory. Thesefunctionsareusedtoobtainthedeflectionan-
y/3
gle andfindthe lensing mass,whichis then comparedto µ(y) =
thestellarmasscontent. Assumingamassdensityprofile 1−p4πkα y/3
m (<r), within a radial distance r, leads to a system of
s 6k3 p 4παµ 1 4παµ
differential equations to determine ζ and ν. A transfor- F(µ) = ln( +1)2+ −
mation to ν˜, ζ˜ then gives the physical metric. The (tt) (4πα)3µ2 " k 1+ 4πkαµ k #
and (θθ) differential equations are: 2x
f(x) = , (5)
1+(2−α)x+ (1−αx)2+4x
(ζ′)2 2ζ′ kG2m2e−(ν+ζ)
ζ′′+ + +eζΛ=− s
4 r 4πµ(y) r4 where the parameter rangpe is 0<α≤1 [16]. The α=0
case for µ(y) gives the weak and intermediate gravity
2πµ2(y) (ν′)2 ν′′ ν′ζ′ ν′
− F(µ)eζ −K + + + limit ofµ(y)andF(µ) is takenfromits explicit form[2].
l2k2 8 2 4 r
(cid:20) (cid:21) The α = 1 gives the function which better fits rotation
−8πGρ˜eζ−2φ , curve data. Since the functions increase monotonically,
(ν′+ζ′) (ν′)2 ζ′′+ν′′ the analysis could be confined to the extremes of the
+ + +eζΛ=
2r 4 2 parameter space, i.e. the α = 0 and α = 1 cases. The
kG2m2e−(ν+ζ) 2πµ2(y) K TeVeS parameters are [10] k = 0.01, K = 0.01, ℓ =
−4πµ(ys) r4 − l2k2 F(µ)eζ + 8 ν′2; (3) k˜b/(4πΞa0), φc =0.001, Ξ=1+K/2−2φc.
The cosmological model used is (Ω ,Ω ,Ω ) =
p m Λ k
where the prime denotes derivatives with re- (0.3,0.7,0),thoughtheeffectofotherchoicesisalsocon-
spect to r and the mass profile is ms(< r) = sidered. The parameter ˜b is fixed from the limit of the
4π rρ˜exp ν + 3ζ −2φr2 dr. The deflection an- y(µ) function as µ becomes << 1, y(µ) ≈ ˜bµ2. For our
0 2 2
parametrisedchoiceofµ(y)˜b=3,specifyingthenl. The
gle reads [2(cid:16)]: (cid:17)
R
deflection angle for a model system is then calculated
∞ 1 r2 −1/2 withthischoiceofparameters. Theresultsforthedeflec-
∆φ=2 eζ˜−ν˜ −1 dr−π ; (4) tionanglein TeVeSarecomparedto the deflectionangle
r b2
Z0 (cid:18) (cid:19) resulting from following the scalarpotential method em-
b2 = eζ˜(r0)−ν˜(r0)r2 is the observable impact parameter ployed in Refs. [10, 11, 12], as well as GR and MOND.
0
The deflection angle results are shown in Fig. 1.
and r is the point of closest approach for the light ray.
0
Tocalculateapossibledarkmattercomponent,asam-
The Navarro-Frenk-White (NFW) profile [9]
ple of double lensing systems from the Castles database
is used, which in Schwarzschild radial coordi-
nates rˆ, with rˆ = eζ˜/2r, reads M(< rˆ) = is analysed and the mass of the lensing galaxy in GR,
−1 MOND,andTeVeSiscalculated. Bycomparingthemass
M ln 1+ Crˆ − Crˆ ln(1+C)− C , where fromlensingtothestellarmasscontentcalculatedfroma
rvir rvir+Crˆ 1+C
thehco(cid:16)ncentrati(cid:17)on C is C ∼ih10 and M is theitotal mass comparisonofphotometryandstellarpopulationsynthe-
3
)
c lar masses (excluding one outlier), it is found that even
e
s in TeVeS on average the dark matter content is 48.5%
c7
r
A TeVeS when α=0, and 34.3% when α=1. The outlier lensing
(
e 6 Scalar system BRI0952-0115 shows a mass overshoot of ≈ 80%
Angl5 GMROND isnteTllaeVremS,asi.se.cotnhteelnetn.sinItgims apsosssinibTleeVtheaStisthleisssstyhsatenmthies

n4 affected by some unknown lens environment effects such
o
cti3 as an unseen cluster mass contribution as has been sug-
e gested of other lenses [11], though there is no data to
l
ef2 =0 =1 concludethis atpresent. Overallthe analysisshowsthat
D 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 TeVeS finds it hard to explain the lensing observed in
these systems usingonly the stellar contentof the galax-
Impact Angle (Arcsec) Impact Angle (Arcsec)
FIG. 1: Deflection angle curves for a NFW profile system ies, a conclusion which stands in contrast to that given
(C =10). The parameters of the system are Dl =1000 Mpc in Refs. [11, 12] despite the two methods predicting sim-
(the luminosity distance of the lens), M = 1011M⊙, k = ilar deflection angles (see, Fig.1). This is a problem for
0.01,K = 0.01,φc = 0.001 and rvir = 15 Kpc. Left panel: TeVeS, which will be made more explicit later on in this
α = 0 case. Right panel: α = 1 case. The deflection angles
letter, when the analysis of varying the free parameters
decrease in MOND and TeVeS as αincreases.
will be performed.

sis using a Chabrier Initial Mass Function (IMF), as in
Ref. [13], the mass deficit which belonged to the “dark” =0
0.5
sector is found. Note that while the authors of Ref. [6]
T
O
claim that the stellar mass estimates are sensitive to the TM
IMF used, it has been argued in Ref. [7] that the main M/ 0.0
competitor of the Chabrier IMF, the Salpeter IMF, ap-
pears to fit the data worse. Other realistic IMF choices -0.5
differ from the Chabrier by an insignificant factor [7].

Theseargumentssupportthe validity ofthe method em-
ployedhere. Finally, inverse raytracing is used to calcu-
=1
latethemass[4]. Themassestimatesweobtainthisway 0.5
T
are shown in Table 1. O
T
M
/
M
GR MOND TeVeS MSTAR 0.0
Lens α=0 α=1 α=0 α=1 Ref. [13] GR
HS0818+1227 34.9 23.9 27.6 19.1 24.4 16.22112..26 M = MTot- MStar MTOeNVDeS
FBQ0951+2635 3.1 2.3 2.6 1.9 2.3 1.120..15 -0.5 10 11 12
BHRE1I0190542-1-0810155 863..16 592..04 628..58 417..19 602..44 232..584251..1207..27 10 Total M1a0s s (Msol)10 0 2RLe ns/R4e 6
FIG. 2: A comparison of the need for dark matter for the
LBQS1009-0252 15.2 10.7 12.2 8.7 10.9 5.574..92 lenses in GR, MOND and TeVeS, for α = 0 (top panels)
HE2149-2745 11.2 7.8 9.0 6.3 8.0 4.663..76 and α = 1 (bottom panels). The right hand panels give the
darkmatterrequirementsasafunctionoftheratioRlens/Re.
AnydarkmatterneededinTeVeSisespecially significantfor
TABLE I: Mass estimates (in 1010M⊙ units) for ΛCDM cos- thosemasseswhichprobefurtherout,wherethemodification
mology: (Ωm,ΩΛ,Ωk) = (0.3,0.7,0), k = K = 0.01, and a to gravity is larger.
NFWhaloprofilewithC =10. Twodifferentcases(α=0,1)
for µ(y) parametrisation are considered.
Theresultsshowthatthecaseα=1,whichisachoice
specifically adopted to fit the rotation curve data, re-
Figure 2 compares the mass estimates between GR, quires an even larger amount of dark matter. Thus, at-
MOND and TeVeS for the two cases of µ(y), with the tempts to fit TeVeS to a no dark matter scenario using
massdifferencebeinggivenasafunctionofthemasscal- the freedominthe parametersofthe class ofµ functions
culated using GR (left panels) and R /R (right pan- used here and in the literature so far, would imply that
lens e
els), where R is the distance out to which our mass thetheoryfitspoorlytherotationcurvesdata,whichwas
lens
estimates and the stellar mass estimates are calculated the original motivation for modifying the gravitational
and R is the half light radius, both given in Ref. [13]. behaviour.
e
Thetoptwopanelsareforα=0,the bottomtwoarefor Theeffectofdifferentcosmologicalparametershasalso
α = 1. Comparing the lensing masses against the stel- been examined to see how they alter the results for the
4
α = 0 case, corresponding to the minimum amount of galaxies.
required dark matter. It is found that the lensing re-
sults are insensitive to the precise cosmological param- Inconclusion,theaboveresultsshowasoftdependence
eters, which is not surprising, since the observational on the free parameters of TeVeS and the cosmological
constraints mostly impose limits on the luminosity and model adopted, but a rather strong one on the form of
angular distance scales [4]. For completeness we state the µ(y) function. Our analysis in this letter points to-
the results, all of which point towards fluctuations in wards the fact that TeVeS, at least within the class of
the amount of dark matter well within the error lim- models considered so far in the literature, cannot sur-
its. In particular, for the case [12] (Ω ,Ω ,Ω ) = vive both gravitational lensing and rotation curve tests.
m Λ k
(0.03,0.46,0.51),averagingover the six galaxies,we find However,we cannot yet exclude completely the possibil-
a 5.7% increase in the amount of dark matter required. ity, admittedly remote, that a class of µ functions, or
For the case (Ω ,Ω ,Ω )=(0.23,0.78,0),considered in more complicatedequivalents thereof, canbe found such
m Λ k
Ref. [8], to fit the Cosmic Microwave Background data that alternative to dark matter scenarios are at play.
with TeVeS, one finds a corresponding average decrease
by 2.2 %, while for φ = 0.01, one finds an average de- It is a pleasure to thank Ignacio Ferreras for discus-
c
crease by 1.1% . sions. The work of N.E.M. and M.S. is partially sup-
Finally,thek, K parametersarevariedindependently ported by the European Union through the Marie Curie
tocheckontherobustnessofourclaims. Inparticular,we Research and Training Network UniverseNet (MRTN-
examine the lensing system HE1104-1805, which within CN-2006-035863), while that of M.F.Y. is supported by
the TeVeS approach (with α = 0) requires the largest an E.P.S.R.C. (UK) studentship.
amountofdarkmatter. Wefindthatthevariationofthe
parameterk hasconsiderablysmallereffectsthanthatof
K. Thissupportsthedominantrˆoleplayedbythevector
fieldinTeVeS.Asimilarresulthasalsobeenpointedout
∗ Electronic address: [email protected]
inRef. [17], but froma differentperspective. Within the
† Electronic address: [email protected]
allowed parameter space 1 ≤ K ≤ 10−5, 1 ≤ k ≤ 10−5,
‡ Electronic address: [email protected]
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