Constraints on the warm dark matter model from gravitational lensing

4 Pages·0.11 MB·English

by  Yan-Jie Xue

DraftversionFebruary5,2008
PreprinttypesetusingLATEXstyleemulateapjv.04/03/99
CONSTRAINTS ON THE WARM DARK MATTER MODEL FROM GRAVITATIONAL LENSING
Yan-Jie Xue and Xiang-Ping Wu
BeijingAstronomicalObservatory,ChineseAcademyofSciences, Beijing100012; andNationalAstronomical
Observatories,ChineseAcademyofSciences,Beijing100012; China
Draft version February 5, 2008
ABSTRACT
Formation of sub-galactic halos is suppressed in warm dark matter (WDM) model due to thermal
motion of WDM particles. This may provide a naturalresolutionto some puzzles in standardcolddark
1 matter (CDM)theorysuchasthecuspeddensityprofilesofvirializeddarkhalosandthe overabundance
0 oflowmasssatellites. OneoftheobservationaltestsoftheWDMmodelistomeasurethegravitationally
0 lensed images of distant quasars below sub-arcsecond scales. In this Letter, we report a comparison of
2
the lensingprobabilitiesofmultipleimagesbetweenCDMandWDMmodelsusingasingularisothermal

n sphere model for the mass density profiles of dark halos and the Press-Schechtermass function for their
a distribution and cosmic evolution. It is shown that the differential probability of multiple images with
J smallangularseparationsdownto 10milliarcsecondsshouldallowoneto setusefulconstraintsonthe
0 WDM particle mass. We discuss b∼riefly the feasibility and uncertainties of this method in future radio
3 surveys (e.g. VLBI) for gravitationallensing.

Subject headings: cosmology: theory — dark matter – galaxies: formation — gravitationallensing
1
v
8 1. INTRODUCTION small mass halos on galactic and sub-galactic scales, than
2 in CDM model. For a massive galaxy like the Milky Way
5 While cosmological models with a mixture of roughly
as a lens at cosmological distance, the Einstein radius of
1 35% cold dark matter (CDM) and 65% vacuum energy
a background quasar is typically 1 arcsecond. So, it is
0 have proved remarkably successful at explaining the ori-
expected that the difference in lensing properties between
1 ginandevolutionofcosmicstructuresonlarge-scales(>1
WDMandCDMmodelswouldoccurbelowsub-arcsecond
0 Mpc), there are a number of independent observations on
/ small scales that seem to be in conflict with the predic- scales. We would like to demonstrate quantitatively to
h what angular separations the two models become to be
p tions ofstandardCDMmodels suchasthe cuspeddensity
distinguishableandhow sensitivelytheir lensing probabil-
- profiles of virialized dark halos and the overabundance of
o satellite structures. Among many other mechanisms sug- ities depend on the WDM particle mass.
r
t gested for resolving these discrepancies, the simplest ap-
s proachis to smear out small scale power by free-steaming
:a due to thermal motion of particles. Warm dark matter 2. LENSINGPROBABILITY
v (WDM) is thus invoked and has recently received a lot of
For simplicity, we use a singular isothermal sphere to
Xi attention in literature (e.g. White & Croft 2000; Col´in, model the mass distribution inside a virialized halo so
r Avila-Reese&Valenzuela2000;Sommer-Larsen&Dolgov that the total mass M within radius r is simply M =
a 2000; Bode, Ostriker & Turok 2000b; references therein). 2σ2r/G. We truncate the mass profile at the virial ra-
v
In the conventional scenario of structure formation by dius r defined by M = 4π∆ ρ r3 /3, where ρ =
vir vir c c vir c
WDM, perturbations on small mass scales Mf 4 3H2(z)/8πG is the critical mass density of the universe,
1011M⊙h2ΩX(Rf/0.1Mpc)3 aredamped,relativeto≈CDM× H(z)=100hE(z)kms−1 Mpc−1 istheHubbleconstant,
models, where ΩX is the WDM density parameter and E(z) = ΩM(1+z)3 +ΩK(1+z)2 +ΩΛ, ΩK denotes the
Rf denotes the characteristic free-streaming length. As curvatureterm, and∆c representsthe overdensityofdark
a result, it suppresses the formation of small halos with matter with respect to the critical density ρ at the col-
c
M <Mf athighredshiftandmeanwhile reducesthe over- lapseredshiftz. Foradarkhaloatzasalens,themultiple
all number of low mass halos in the universe, depending imagesofadistantquasaratz withanangularseparation
s
sensitively onthe WDM particle mass mX. Observational of ∆θ =2θE will occur if it happens to lie in the Einstein
tests of this scenario and hence robust constraints on mX radiusofthedarkhalo: θE =4π(σv/c)2(Dds/Ds),inwhich
are restricted to the studies of less massive systems with D and D are the angular diameter distances from the
ds s
M < Mf. The existing limits in literature are based on lens and observer to the source, respectively. The lensing
the Gunn-Peterson effect and the Ly-α forest, which set cross section for the multiple images is simply πθ2. The
mX > 750 eV (Narayanan et al. 2000). In this Letter, probability that a quasar at zs will have multipleEimages
we would like to examine whether the gravitational lens- with anangular separationgreaterthan ∆θ due to allthe
ing of high-redshift quasars by foreground dark halos can foreground dark halos is thus
be used to distinguish CDM and WDM models or place
any meaningful constraints on m . This arises because
X
the probability of gravitational lensing would be lower in
zs dV ∞ θ2 dN
WDMmodel,asaresultofthesuppressionofformationof P(∆θ)= dz E dM, (1)
Z (cid:18)dz (cid:19) Z (cid:18) 4 (cid:19)(cid:18)dMdV (cid:19)
0 Mmin
1
2
where dV is the comoving volume element, and M is gravitational lensing by Turner, Ostriker & Gott (1984).
min
related to ∆θ through Noticeable departure of the lensing probability in WDM
models from that in CDM model occurs only at small an-
Mmin = 1.50×1012M⊙ h−1 gular separations ∆θ < 1′′ and ∆θ < 0.′′1 for mX > 350
(cid:16)E1(z)(cid:17)(cid:0)∆1′θ′(cid:1)3/2(cid:16)DDdss(cid:17)3/2(cid:0)2∆0c0(cid:1)−1/2. (2) mneVeielldaiasnrdtcosemrceXoancd>hsa1innkeoveVredn,esrrmetsaoplelmecrtaiavkneelgyau.nlaIretffsieespctaliirvkaeetliydoinstthoianftc∼toio1nn0e
We use the Press-Schechter (PS) mass function for the betweenthe WDM andCDM models forarealisticWDM
abundance and evolution of virialized dark halos: particle mass of m 1 keV. We have applied the lens-
X
∼
ing probability P(∆θ) to the quasar luminosity function
dN 2 ρ¯ δ (z) dσ δ (z)2 determined from the 2dF QSO survey over absolute mag-
c c
= exp , (3)
dMdV −rπM σ2 dM (cid:18)− 2σ2 (cid:19) nitudes 26 < MB < 23 and redshifts 0.35 < zs < 2.3
− −
(Boyle et al. 2000). Here we would rather take a less
where ρ¯ is the present-day average mass density of the vigorous approach to the problem by neglecting the mag-
universe, δ (z) is the linear over-density of perturbations nification effect of the double images. It is shown that at
c
that collapsed and virialized at redshift z, and σ is the a limiting magnitude of mB = 20, the fractions of small
variance of the mass density fluctuation in sphere of mass angular separation events with 0.′′001 < ∆θ < 0.′′01 in
M = (4π/3)ρR3. It is convenient to express δ (z) as the total number of doubly-imaged quasars are approxi-
c
δ (z)=δ (1+z)[g(0)/g(z)],whereg(z)isthelineargrowth mately1/100,4/1000and1/1000fortheCDMmodeland
c 0
factor,forwhichwetaketheapproximateformulabyCar- the WDM models with mX = 1 keV and mX = 350 eV,
roll,Press&Turner(1992),andthequantityδ hasaweak respectively. Indeed, observationaltests of these fractions
0
dependence onΩ : δ =1.6866[1+0.01256logΩ (z)]for on milliarcsecond scales turn out to be difficult and chal-
M 0 M
a flatuniversewithnonzerocosmologicalconstant(Math- lenging.
iesen & Evrard 1998). Once a power spectrum, P(k), is
3. DISCUSSION ANDCONCLUSIONS
specified, the mass variance becomes
IntheCDMscenarioofstructureformation,manysmall
σ2(M)= 1 ∞k2P(k)W2(kR)dk, (4) galaxieswouldform,relativeto what arepredictedby the
2π2 Z WDM model. Consequently, the standard CDM model
0
wouldpredictmoremultiple,smallangularseparationim-
inwhichW(x)=3(sinx xcosx)/x3 istheFourierrepre- ages of the gravitationally lensed distant quasars than
−
sentation of the window function. We adopt the following WDM model does. This unique property can be used as
parameterization of WDM power spectrum an effective tool to distinguish these two prevailing mod-
els. It has been shown that the significant difference in
P(k)=Akn T2(k)T2 (k), (5) the distributions of image angular separations between
X CDM
WDM and CDM models occurs at small angular sepa-
where n is the primordialpower spectrum and is assumed rations ∆θ 0.01′′ for the WDM particle mass m 1
X
to be the Harrison-Zeldovich case n = 1, TCDM(k) is keV.Althou∼ghtherearenodataavailableatpresentto∼test
the transfer function of adiabatic CDM model for which our prediction, these angle ranges should be accessible by
we use the fit given by Bardeen et al. (1986), and current radio observations such as VLBI. Recall that sys-
TX(k) acts as the CDM to WDM “ transfer function” tematicradiosurveysforstronggravitationallensinghave
for which we adopt the form provided by Bode et al. proved to be very successful at finding small separation
(2000b): TX(k) = [1 +(αk)2ν]−5/ν, where ν = 1.2 and images in the range 0.′′3-0.′′6 (e.g. Augusto, Wilkinson &
α is related to the WDM particle mass mX through Browne 1998; Augusto & Wilkinson 2001; and references
α = 0.048(ΩX/0.4)0.15(h/0.65)1.3(keV/mX)1.15. The am- therein). It is hoped that future multiple image surveys
plitudeAisdeterminedbyequation(4)usingthermsmass down to 10 milliarcseconds would allow one to place
fluctuation on an 8 h−1 Mpc scale, σ8. useful con∼straints on the WDM particle mass.
We work with a flat cosmological model where ΩM = We now briefly discuss several major uncertainties in
ΩX + Ωb = 0.3, Ωb = 0.03, ΩΛ = 0.7, and h = 0.68, the present predictions: Firstly, we have used a simple
which fixes the shape parameter Γ = ΩMhexp[ Ωb(1+ singular isothermal sphere model for the mass distribu-
−
√2h/Ω )]=0.176. Thenormalizationparameteristaken tionsofdarkhalos. Analternativeistheuniversaldensity
M
from the calibration of cluster abundance, σ = 0.85. We profile suggested by high-resolution N-body simulations
8
compute the lensing probabilities for a quasar at redshift (Navarro, Frenk & White 1995; NFW), which should be
zs = 1, 2 and 3, respectively, and for two choices of the validforthevirializeddarkhalosassmallas109M⊙. Li&
WDMparticlemassm =350eVand1keV.Asacompar- Ostriker (2000) have recently compared the lensing prob-
X
ison, we also give the corresponding lensing probabilities abilities P(∆θ) produced by a singular isothermal sphere
foraCDMmodelwithΩ =0.27. Inordertohighlight model and the NFW profile using roughly the same ap-
CDM
theirdifferences,wedisplayinFigure1thenormalizeddif- proach as ours. It turns out that the NFW profile results
ferential probabilities [1/P(∆θ)][dP(∆θ)/d∆θ] instead of in a significantly small value of P(∆θ) 10−6, insensitive
∼
the total probabilities P(∆θ). For the latter, the typical totheimageseparation∆θ. Wehaverecalculatedthelens-
value for a quasar at z = 2 with ∆θ > 1′′ is nearly the ing probabilities under CDM and WDM models following
s
same for all the three models (two WDM models and the theapproximatetreatmentofLi&Ostriker(2000)forthe
standard CDM model): P(> 1′′) 1 10−3. This re- image separation by NFW profile, ∆θ 2θ where θ is
0 0
≈ × ≈
sult is consistent with the pioneering statistical study of the solution to the lensing equation with the zero align-
3
ment parameter. Unlike Li & Ostriker (2000) who took a reducedascomparedwiththosepredictedbythestandard
constant halo concentration c for massive halos from nu- PS formalism. For example, at z =2 and ∆θ =0.1′′, the
s
merical simulations, we fix the characteristic density δ values of P(∆θ) in terms of the Sheth & Tormen (1999)
c
and halo concentration c for each halo M using the δ -M mass function are 1.2 times smaller than those derived
c
relationviathecollapseredshiftz proposedbyNavarro, from the PS approach,and this ratio is insensitive to cos-
coll
Frenk & White (1997). Nevertheless, we have reached a mologicalmodels. Thedistributionofthedifferentiallens-
conclusionessentially similar to Li& Ostriker(2000). For ing probability normalized by P(∆θ) based on the Sheth
example, the lensing probabilities for a distant quasar at &Tormen(1999)massfunctionremainsroughlythesame
z = 3 with ∆θ 0.001′′ – 1′′ are P(∆θ) 3.7 10−6, as in Figure 1.
s
3.6 10−6 and 2≥.9 10−6 for the CDM m≈odel a×nd the Thirdly, the theoretically predicted distribution of im-
× ×
WDMmodelswithm =1keVandm =350eV,respec- ageseparationsdependscriticallyonthechoiceofthecos-
X X
tively. Thelackofsmallseparationeventsbelowarcsecond mologicalparameters(e.g. Li&Ostriker2000). Therefore,
scales will invalidate our original intention that the study a robust constraint on the WDM particle mass from the
ofdoubly-imagedquasarsmayhelptodistinguishbetween studyoflensingprobabilityformultipleimageswithsmall
CDM andWDM models. However,we note thatthe lens- angular separations needs a priori precise determinations
ing probability produced by NFW profile is unreasonably of many other parameters such as Ω , Ω , σ and H .
M Λ 8 0
small( 10−6). ThisimpliesthattheNFWprofilemaybe Finally, one may worry about the contamination of the
∼
inappropriate to the description of the mass distributions supermassive black holes as lenses in the actual applica-
in the inner regions of dark halos where the multiple im- tionofthegravitationallensingprobabilitytotheobserva-
ages would occur. Indeed, the central mass density of an tional tests of WDM model. Indeed, it is believed that all
NFW profileistooshallowtoactasaneffectivelens. The the AGNs contain supermassive black holes with masses
absence of detectable odd images in the known lens sys- M 108 – 109 M⊙ in their centers. The lensing effect
∼
tems alsolends supportto the existence ofasteeper inner by these supermassive black holes leads to an image sep-
density profile in galaxies, ρ r−2 (Rusin & Ma 2001). aration of typically 0.′′1 for a quasar at cosmological
∝ ∼
In a word, the singular isothermal sphere model may be distance, which is indeed of the same order of magnitude
a reasonable approximation for the inner mass profiles of as the effect produced by small galaxies discussed in the
galactic and sub-galactic halos. present Letter. Yet, the lensing probability of a back-
Secondly, the PS mass function may become inaccurate ground quasar due to the supermassive black holes resid-
for small halos. High resolutionN-Body simulations have ing in AGNs is considerably low ( 10−8) (Augusto &
∼
shownthatthePSapproximationpredictstoomanysmall Wilkinson 2001).
halos at low redshifts (e.g. Bode et al. 2000a; Jenkins et Overall, we feel that gravitational lensing should serve
al. 2001). This will lead to an overestimate of the total as a critical test for WDM model although further work
lensing probability for multiple images, P(∆θ), although willbeneededtoaddressinmoredetailvariousuncertain-
the differential lensing probability normalized by P(∆θ), ties in the issue.
[1/P(∆θ)][dP(∆θ)/d∆θ], is probably less affected. In or-
dertodemonstratetheuncertainty,wehaveusedthemod- We gratefully acknowledge the useful comments by an
ified PS mass function by Sheth & Tormen (1999) and anonymous referee. This work was supported by the
the best fit parameters by Jenkins et al. (2001). It is National Science Foundation of China, under Grant No.
found thatthe totallensingprobabilities forsmallsepara- 19725311 and the Ministry of Science and Technology of
tion events in both CDM and WDM models are slightly China, under Grant No. NKBRSF G19990754.
REFERENCES
Augusto, P.,&Wilkinson,P.N.2001, MNRAS,320,L40 Li,L.-X.,&Ostriker,J.P.2000,ApJ,submitted(astro-ph/0010432)
Augusto,P.,&Wilkinson,P.N.,&Browne,I.W.A.1998,MNRAS, Mathiesen,B.,&Evrard,A.E.1998,MNRAS,295,769
299,1159 Narayanan,V.K.,Spergel,D.N.,Dav´e,R.,&Ma,C.-P.2000,ApJ,
Bardeen, J.M.,Bond, J.R.,Kaiser,N.,&Szalay, A.S.1986, ApJ, 543,L103
304,15 Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1995,MNRAS,275,
Bode,P.,Bahcall,N.A.,Ford,E.B., Ostriker,J.P.2000a,ApJ,in 720
press Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1997,ApJ,490,493
Bode, P., Ostriker, J. P., & Turok, N. 2000b, preprint (astro- Rusin,D.,&Ma,C.-P.,2001,ApJ,inpress
ph/0010389) Sheth,R.K.,&Tormen,G.1999,MNRAS,308,119
Boyle,B.J.,etal.2000, MNRAS,317,1014 Sommer-Larsen,J.,&Dolgov,A.2000,ApJ,inpress
Carroll,S.M.,Press,W.H.,&Turner,E.L.1992,ARA&A,30,499 Turner,E.L.,Ostriker,J.P.,&GottIII,J.R.1984,ApJ,284,1
Col´in,P.,Avila-Reese,V.,&Valenzuela,O.2000,ApJ,542,622 White,M.,&Croft,R.A.C.2000,ApJ,539,497
Jenkins,A.,etal.2001,MNRAS,inpress
Fig. 1.— The differential probabilities of image separations normalized by the total probabilities P(∆θ) for distant quasars at zs =1, 2
and3(frombottom totop)inCDMandWDMmodels.