The SWELLS survey. II. Breaking the disk-halo degeneracy in the spiral galaxy gravitational lens SDSS J2141-0001
A. A. Dutton, B. J. Brewer, P. J. Marshall, M. W. Auger, T. Treu, D. C. Koo, A. S. Bolton, B. P. Holden, L. V. E. Koopmans
aa r X i v : . [ a s t r o - ph . C O ] M a r Mon. Not. R. Astron. Soc. , 1–23 (2011) Printed 8 November 2018 (MN L A TEX style file v2.2)
The SWELLS survey. II.Breaking the disk-halo degeneracy in the spiral galaxygravitational lens SDSS J2141 − ⋆ . Aaron A. Dutton , †‡ , Brendon J. Brewer , Philip J. Marshall , , M. W. Auger ,Tommaso Treu § , David C. Koo , Adam S. Bolton , Bradford P. Holden ,Leon V. E. Koopmans Dept. of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada UCO/Lick Observatory, University of California, Santa Cruz, CA 95064, USA Dept. of Physics, University of California, Santa Barbara, CA 93106, USA Kavli Institue for Particle Astrophysics and Cosmology, P.O. Box 20450, MS29, Stanford, CA 94309, USA Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA Kapteyn Astronomical Institute, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands accepted to MNRAS
ABSTRACT
The degeneracy among the disk, bulge and halo contributions to galaxy rotation curvesprevents an understanding of the distribution of baryons and dark matter in diskgalaxies. In an attempt to break this degeneracy, we present an analysis of the stronggravitational lens SDSS J2141 − ( M ∗ / M ⊙ ) = 10 . +0 . − . . The photometric datacombined with stellar population synthesis models yield log ( M ∗ / M ⊙ ) = 10 . ± .
07, and 11 . ± .
07 for the Chabrier and Salpeter IMFs, respectively. Assumingno cold gas, a Salpeter IMF is marginally disfavored, with a Bayes factor of 2.7.Accounting for the expected gas fraction of ≃
20% reduces the lensing plus kinematicsstellar mass by 0 . ± .
05 dex, resulting in a Bayes factor of 11.9 in favor of aChabrier IMF. The dark matter halo is roughly spherical, with minor to major axisratio q , h = 0 . +0 . − . . The dark matter halo has a maximum circular velocity of V max = 276 +17 − km s − , and a central density parameter of log ∆ V/ = 5 . +0 . − . . Thisis higher than predicted for uncontracted dark matter haloes in ΛCDM cosmologies,log ∆ V/ = 5 .
2, suggesting that either the halo has contracted in response to galaxyformation, or that the halo has a higher than average concentration. Larger samplesof spiral galaxy strong gravitational lenses are needed in order to distinguish betweenthese two possibilities. At 2.2 disk scale lengths the dark matter fraction is f DM =0 . +0 . − . , suggesting that SDSS J2141 − Key words: galaxies: fundamental parameters – galaxies: haloes – galaxies: kine-matics and dynamics – galaxies: spiral – galaxies: structure – gravitational lensing ⋆ Based in part on observations made with the NASA / ESAHubble Space Telescope, obtained at the Space Telescope ScienceInstitute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. These observations are associated with programs10587 and 11978. † [email protected] ‡ CITA National Fellow § Packard Research Fellowc (cid:13)
Dutton et al.
The discovery of extended flat rotation curves in the outerparts of disk galaxies three decades ago (Bosma 1978; Rubinet al. 1978) was decisive in ushering in the paradigm shiftthat led to the now standard cosmological model dominatedby cold dark matter (CDM). The need for dark matter oncosmological scales is also firmly established from observa-tions of the Cosmic Microwave Background, type Ia Super-novae, weak lensing, and galaxy clustering (see, e.g., Spergelet al. 2007). Numerical simulations of structure formationwithin the ΛCDM cosmology make firm predictions for thestructure and mass function of dark matter haloes in theabsence of baryons (e.g., Navarro, Frenk, & White 1997;Bullock et al. 2001; Macci`o et al. 2007; Navarro et al. 2010).It is still unclear, however, whether this standard modelcan reproduce the observed properties of the Universe atgalactic and sub-galactic scales. There are problems relatedto the inner density profiles of dark matter haloes (e.g., deBlok et al. 2001; Swaters et al. 2003; Newman et al. 2009),reproducing the zero point of the Tully-Fisher relation (e.g.,Mo & Mao 2000; Dutton et al. 2007), and the amount ofsmall-scale substructure (e.g., Klypin et al. 1999; Moore etal. 1999; Stewart et al. 2008). There are three classes of solu-tions to these problems: those that invoke galaxy formationprocesses that modify the properties of dark matter haloes;those that change the nature of dark matter itself; and thosein which dark matter does not exist. Thus, measuring thedensity profiles of the dark matter haloes of galaxies of alltypes is a stringent test for galaxy formation theories.From an observational point of view, little is knownabout the detailed distribution of dark matter in the innerregions of disk galaxies, despite the great investment of tele-scope time and high quality measurements of hundreds ofrotation curves (e.g., Carignan & Freeman 1985; Begeman1987; Courteau 1997; de Blok & McGaugh 1997; Verheijen1997; Swaters 1999; de Blok et al. 2001; Swaters et al. 2003;Blais-Ouellette et al. 2004; Simon et al. 2005; Noordermeeret al. 2005; Simon et al. 2005; Chemin et al. 2006; Kuzio deNaray 2006; de Blok et al. 2008; Dicaire et al. 2008; Epinatet al. 2008). The fundamental reason is the so-called disk-halo degeneracy : mass models with either maximal or min-imal baryonic components fit the rotation curves equallywell, leaving the structure of the dark matter halo poorlyconstrained by the kinematic data alone (e.g., van Albada& Sancisi 1986; van den Bosch & Swaters 2001; Duttonet al. 2005). Stellar population models are able to placeconstraints on stellar mass-to-light ratios, allowing inferenceabout the baryonic contribution to the overall mass profile.However, there are a number of uncertainties which limitthe accuracy of this method (e.g., Conroy et al. 2009, 2010).These include systematic uncertainties such as the unknownstellar initial mass function (IMF), and the treatment ofthe various stellar evolutionary phases in stellar populationsynthesis (SPS) models. These result in about a factor of 2uncertainty in the stellar masses estimated from spectral en-ergy distribution (SED) fitting. Moreover, for a given IMFand SPS model, there are uncertainties in the star forma-tion histories, metallicities and extinction which introduce(1 σ ) random errors in measurements of stellar masses forindividual galaxies at the level of 0.15 dex (e.g., Bell & deJong 2001; Auger et al. 2009, Gallazzi & Bell 2009). Nevertheless, galaxy colours and dynamical mass esti-mates have been used in combination by various authors toplace an upper limit on the stellar mass-to-light ratio nor-malisation, favoring IMFs more bottom light than Salpeterfor spiral galaxies and fast rotating low-mass elliptical galax-ies (Bell & de Jong 2001; Cappellari et al. 2006; de Jong &Bell 2007; see, however, Treu et al. 2010, Auger et al. 2010,and van Dokkum & Conroy 2010 for massive ellipticals).However, as theory generally predicts more dark matter inthe inner regions of disk galaxies than is consistent withstandard IMFs (Dutton et al. 2007; Dutton et al. 2010b) alower limit to the stellar mass would provide a more usefulconstraint for ΛCDM.Several other methods have been used to try and mea-sure disk galaxy stellar masses, independent of the uncer-tainties in the IMF. These include: 1) vertical velocity dis-persions of low inclination disks (Bottema 1993; Verheijenet al. 2007; Bershady et al. 2010), 2) bars and spiral struc-ture (Weiner et al. 2001; Kranz et al. 2003), and 3) stronggravitational lensing by inclined disks (Maller et al. 2000;Winn et al. 2003). None of these methods have thus faryielded conclusive results.An approach combining strong gravitational lensingplus kinematics holds great promise, because it takes advan-tage of the different geometries of disks and haloes, whichresults in three effects that enable the disk mass to be mea-sured. 1) An inclined disk will present a much higher pro-jected surface density than a face-on disk, with resultingimage positions and shapes that depend on the disk massfraction. 2) An edge-on disk is highly elliptical in projec-tion, more than expected for any realistic dark matter halo,with resulting total mass ellipticity depending on the diskmass fraction 3) Strong lensing measures mass projectedalong a cylinder (within the Einstein radius), whereas stellarkinematics (rotation and dispersion) measure mass enclosedwithin spheres (see Figure 1). For spherical mass distribu-tions of stars and dark matter, the ratio between the pro-jected mass within a cylinder of radius, r , and the enclosedmass within a sphere of the same radius, r , is independent ofthe relative contribution of the two mass components (leftpanel Figure 2). Therefore, in order to break the degener-acy one has to assume a radial profile shape for both com-ponents (e.g., Treu & Koopmans 2002, 2004; Koopmans &Treu 2003; Koopmans et al. 2006; Treu et al. 2010; Auger etal. 2010). Typically this involves assuming the baryonic massfollows the light, and then assuming a functional form forthe dark matter halo. However, for a disk plus halo system,this ratio is dependent on the relative contribution of thetwo components (right panel Figure 2). Thus if the sphericaland cylindrical masses can be measured accurately enough,the disk halo degeneracy can be broken without assuming aspecific radial profile shape for either component. Further-more, strong lensing plus kinematics can place constraintson the 3D shape of the dark matter halo (e.g., Koopmans, deBruyn, & Jackson 1998; Maller et al. 2000) which is of inter-est because ΛCDM haloes are predicted to be non-spherical(e.g. Allgood et al. 2006; Bett et al. 2007; Macci`o et al. 2008).The power of the strong gravitational lensing methodhas not yet been fully realised, primarily due to the scarcityof known spiral galaxy gravitational lenses. Prior to theSLACS Survey (Bolton et al. 2006, 2008) only a handful ofspiral galaxy lenses with suitable inclinations to enable rota- c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − Figure 1.
Illustration of the different geometries probed by strong lensing and kinematics. Strong lensing measures mass with a cylinder(or more generally an ellipse), whereas stellar and gas kinematics measure mass within spheres (or more generally ellipsoids).
Figure 2.
Differences between projected (cylindrical) mass and enclosed (spherical) mass for a bulge-halo system (left) and a disk-halosystem (right). For each system two models are shown (in red and black). The models have baryonic mass profiles (short-dashed lines,upper panels) with the same shape but normalizations that differ by a factor of two. The dark matter profiles (long-dashed lines, upperpanels) have been chosen so that the total circular velocity curves are close to identical (solid lines, upper panels). For the bulge-halosystem the ratio between projected and enclosed masses (middle panels) is independent of the relative contributions of the bulge andhalo, which differ significantly between the two models (lower panels). However, for the disk-halo system there is a significant differencebetween the projected and enclosed masses, especially at radii smaller than the effective radius. This illustrates the potential of stronglensing plus kinematics to break the disk-halo degeneracy.c (cid:13) , 1–23
Dutton et al. tion curve measurements were known: Q2237+0305 (Huchraet al. 1985; Trott & Webster 2002); B1600+434 (Jacksonet al. 1995; Jaunsen & Hjorth 1997); PMN J2004 − − − − ≃ ◦ ;this set-up approximately maximises the projected disk masswhile allowing an accurate rotation curve to be measured.The original spectroscopic observations ofSDSS J2141 − u, g, r, i, z ) = (20 . , . , . , . , .
48) with errors(0 . , . , . , . , . z d = 0 . ± . ±
14 km s − . The spectrum alsoexhibits nebular emission lines at a background redshift of z s = 0 . .
438 kpc, while in thesource plane it is 1 arcsec = 7 .
196 kpc.This paper is organised as follows. In Section 2 wepresent the imaging observations of SDSS J2141 − − Table 1.
Summary of Imaging ObservationsTelescope Camera Filter Integration Time(s)
HST
WFPC2 F450W 4400
HST
WFPC2 F606W 1600
HST
ACS F814W 420Keck II NIRC2-LGS K’ 2700
Throughout, we assume a flat ΛCDM cosmology withpresent day matter density, Ω m = 0 .
3, and Hubble param-eter, H = 70 km s − Mpc. All magnitudes are given in theAB system. Unless otherwise stated, all parameter estimatesare the median of the marginalised posterior PDF, and theiruncertainties are described by the absolute difference be-tween the median and the 84th and 16th percentiles (suchthat the error bars enclose 68% of the posterior probability).
SDSS J2141 − ≃ . HST ) to the NIR (withKeck Laser Guide Star Adaptive Optics). A summary of theimaging observations is given in Table 1, whilst Figure 3shows the
HST and Keck images of SDSS J2141 − − HST
Hubble Space Telescope (
HST ) observations ofSDSS J2141 − .
05 arcsec. The WFPC2 observations, in the F450W(4400s) and F606W (1600s) filters, were part of the cycle16 supplementary programme GO:11978 (PI:Treu). For theWFPC2 observations four sub-exposures were obtained,and the frames drizzled to a pixel scale of 0 .
05 arcsec.The F814W image confirmed the strong lensing natureof this system by showing that the background object wasmultiply imaged into a three-component arc. It revealed thatthe lens was a disk dominated galaxy, with a high inclinationand dusty disk, and that the bulge was compact and disklike. The F450W and F606W images reveal that the sourceis blue.Models for the point spread functions (PSFs) of theACS and WFPC2 data where obtained by using the programTinyTim (Krist 1995). These PSF models include the effectsof sub-pixel dithering and drizzling and have been foundto provide adequate models for the true PSF (e.g., Boltonet al. 2008; Auger et al. 2009). c (cid:13)000
05 arcsec.The F814W image confirmed the strong lensing natureof this system by showing that the background object wasmultiply imaged into a three-component arc. It revealed thatthe lens was a disk dominated galaxy, with a high inclinationand dusty disk, and that the bulge was compact and disklike. The F450W and F606W images reveal that the sourceis blue.Models for the point spread functions (PSFs) of theACS and WFPC2 data where obtained by using the programTinyTim (Krist 1995). These PSF models include the effectsof sub-pixel dithering and drizzling and have been foundto provide adequate models for the true PSF (e.g., Boltonet al. 2008; Auger et al. 2009). c (cid:13)000 , 1–23 he disk and halo of lens SDSS J2141 − Figure 3.
Optical to near-IR high resolution imaging of SDSS J2141 − ≃ . Table 2.
Summary of bulge plus disk fits together with stellar masses derived from SED fits with a Chabrier (2003) IMF. q R [arcsec] n F450W-K’ F606W-K’ F814W-K’ K’ magnitude log ( M ∗ /M ⊙ )Bulge 0 . ± .
02 0 . ± .
01 1 . ± .
11 3 . ± .
04 2 . ± .
03 1 . ± .
04 17 . ± .
28 10 . ± . . ± .
02 2 . ± . ≡ . . ± .
12 1 . ± .
14 1 . ± .
12 16 . ± .
13 10 . ± . On August 13th 2009, we imaged SDSS J2141 − . × .
04 arcsec pix − . A total of 45 minutes of exposure was ob-tained. Individual exposures were 1 minute in duration (di-vided into two 30-second co-adds). A dither was executed af-ter every set of 5 exposures to improve sky sampling. Ditherswere based on a four point box pattern with sides 8 arcsec.The laser was positioned at the center of each frame, ratherthan fixed on the central galaxy. Observing conditions dur-ing the run were good.The images were processed with the CATS reductionprocedure described by Melbourne et al. (2005). A sky frameand a sky flat were created from the individual science ex-posures after masking out all objects. Frames were thenflat-fielded and sky-subtracted. The images were de-warpedto correct for known camera distortion. The frames werealigned by centroiding on objects in the field, and finallyco-added to produce the final image. A model for the PSF was derived from observations ofa PSF star pair, where the star used for tip-tilt correction isthe same distance from the PSF star as the lens galaxy wasfrom its tip tilt star. The star pair observations were madeimmediately following the lens observations. The PSF starwas found to have FHWM=0 .
10 arcsec (2.5 pixels) and aStrehl ratio of 18%.In the K’-band the extinction of both the lensed imagesand the lens galaxy light due to dust in the lens galaxy isalmost completely absent, revealing a ring like structure, andconfirming the disky nature of the bulge. The backgroundobject appears to have been lensed into a smooth arc inthis filter. The difference between the source structure inthe rest-frame NIR and the rest-frame UV/optical is likelydue to extinction from the lens galaxy artificially creatingthe appearance of three distinct images.
We begin our study of the mass distribution ofSDSS J2141 − c (cid:13) , 1–23 Dutton et al. a S´ersic profile bulge, with each spatial component consist-ing of distinct stellar populations. We first fit the surfacebrightness data to obtain estimates of the shape and profileof the stellar mass density, and then normalise the two pro-files by fitting the bulge and disk fluxes in our 4 filters (thespectral energy distribution, or SED) with stellar populationsynthesis (SPS) models.
In each band, a 2-dimensional model of the lens galaxy sur-face brightness was fitted to the high resolution imagingdata. The model is composed of two elliptically-symmetricS´ersic profile components, representing the disk and thebulge. Σ( x, y ) = Σ exp[ − ( R/R ) /n ] (1)where R = p x + y /q . The S´ersic index n is fixed at 1for the exponential disk, and left free for the bulge. Theremaining parameters for each component are the centroidposition { x c , y c } , scale radius R , the axis ratio q , and theorientation angle φ . The prior probability distributions wereall independent, with uniform priors for φ , x c , y c and q , and“Jeffreys” ( ∝ /x ) priors for R and n , between generousupper and lower bounds.All four bands are fitted simultaneously, with all pa-rameters except for the normalization of the bulge and diskfluxes constrained to be the same in all bands. This approachgives more robust colors of the bulge and disk than is ob-tained when letting the structural parameters float betweenbands.The inferred parameter values for the disk and bulgesurface brightness are given in Table 2. For the bulge com-ponent, we find a S´ersic index of n b = 1 . ± .
11, and bulge(luminosity) fraction which increases from 0 . ± .
03 in theF450W filter to 0 . ± .
05 in the K’ band. These values aretypical for low-redshift late-type spiral galaxies.The bulge has a major-axis half-light radius of R , b =0 . ± . ′′ = 0 . ± .
02 kpc, whilst the disk has a major-axis half-light radius of R , d = 2 . ± . ′′ = 6 . ± .
32 kpccorresponding to a disk scale length R d = 1 . ′′ ± . ′′ =3 . ± .
19 kpc. The ratio between the bulge half-light radiusand the disk scale length is 0 . ± .
02 which is consistentwith those found by MacArthur et al. (2003) in a sample ofmoderately inclined late-type spirals.The bulge has an observed axis ratio of q b = 0 . ± . q d = 0 . ± . q d = cos( i ). However, in general diskshave a finite thickness, which causes the true inclination tobe higher than that inferred from the observed axis ratio.For SDSS J2141 − q = 0 . ± .
02, and thus i = 78 . ± . q , and inclination, i , the 3D minor-to-major axisratio, q is given by q = ( q − cos i ) / (1 − cos i ) (2)Thus for SDSS J2141 − q , b = 0 . ± .
02, and
Figure 4.
Posterior distributions for the inference on the stellarmass based upon SPS models constrained by the high-resolutionphotometry, assuming a Chabrier IMF. q , d = 0 . ± .
02. The disk thickness that we derive forSDSS J2141 − − The two component model for the surface brightness ofSDSS J2141 − M ∗ , the population age A ,the exponential star formation burst timescale τ , the metal-licity Z and the reddening due to dust, τ V . We employ auniform prior requiring 9 log ( M ∗ / M ⊙ )
13, the ageis constrained such that star formation began at some (uni-formly likely) time between 1 z τ has an exponentialprior with characteristic scale 1 Gyr, and we impose uniformpriors on the logarithms of the metallicity and dust extinc-tion such that − log Z − . − log τ V . c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − the bulge and disk components. Assuming that both com-ponents are well described by a Chabrier IMF, we findlog ( M ∗ , b / M ⊙ ) = 10 . ± .
08 and log ( M ∗ , d / M ⊙ ) =10 . ± .
07, for the bulge and disk respectively, justifyingour description of SDSS J2141 − − ( M ∗ / M ⊙ ) = 10 . ± .
07, and thebulge fraction is f bulge = 0 . ± .
04 (the same as in theK’-band light). For a Salpeter IMF the masses are all 0.24dex higher. We will return to this inference in Section 7.2below, where we compare it to the stellar mass implied bythe gravitational analysis.
As indicated in the previous section, it makes sense to con-sider a stellar mass distribution for SDSS J2141 − Based on the results of the previous section, we model thestellar mass distribution of SDSS J2141 − M ∗ , d , co-axial withan oblate bulge of stars of mass M ∗ , b . We then assume thegalaxy to reside in a dark matter halo that is also axisym-metric, and aligned and concentric with the disk and bulge.This assumption that the galaxy and inner dark matterhaloes are aligned is supported by cosmological simulationsof disk galaxy formation (e.g., Deason et al. 2011). We notethat for our strong lensing analysis it is feasible to allow theposition angles of the baryons and dark matter to be offset.However, this would make the model non-axisymmetric andthus make the kinematics considerably harder to model. Thesurface brightness in our four filters constrains the spatialdistribution of stellar mass tightly, under the assumptionthat the stellar mass-to-light ratios are radially constant;we leave the overall normalisation of the stellar mass distri-bution as a free parameter.We do not explicitly include a cold gas disk for tworeasons. Firstly, we do not have direct observations of theatomic and molecular gas in SDSS J2141 − • Exponential Stellar Disk • S´ersic Stellar Bulge • Non-singular Isothermal Ellipsoid (NIE) Dark MatterHaloThis model has 17 parameters in total; they, and their priorPDFs, are given in Table 3. We assign informative priors toall but 4 of these parameters, propagating the uncertaintiesin the surface brightness fits through to the mass model.First, we assume the bulge, disk, and halo inclinationare all the same, and given by the thin disk axis ratio (0.2),as in Section 3 – we assume that this is known with no un-certainty. As we describe below, we use an approximation tothe Exponential profile that allows us to compute predictedobservable quantities efficiently – the size parameters of thebulge and disk in that approximation are determined fromthe results of the previous section, as is (more straightfor-wardly) the bulge axis ratio. We use the derived value of 1.21(see Table 2) for the Bulge S´ersic index, with no uncertainty.We assume that the disk and bulge are different stellarpopulations, and so use the independent stellar mass resultsfrom the previous section to constrain the bulge mass frac-tion, f bulge = M ∗ , b /M ∗ . As already mentioned, we leave thetotal stellar mass M ∗ as a free parameter with uniform prioron its logarithm. This is effectively equivalent to assumingthat the two components have very similar, although un-known, IMF normalization, We do inform the bounds of thisuniform prior using the SPS modelling results, in the follow-ing way. Estimating that the lightest conceivable IMF wouldgive stellar masses systematically a factor of two lower thanChabrier, we take the 3-sigma point of the Chabrier PDFin Figure 4 and subtract 0.3 dex to set a lower limit onlog ( M ∗ / M ⊙ ) of 10.5. Likewise, at the high end we takeSalpeter to be the heaviest IMF and use the 3-sigma pointof the Salpeter PDF in Figure 4 (11.4) as our upper limiton log ( M ∗ / M ⊙ ). We note that none of our results changeif we adopt a higher upper limit to the stellar mass.This leaves 3 parameters that describe the model darkmatter halo: V c , h , r c , h , and q , h . We allow the axis ratio ofthe halo to be greater than unity, corresponding to a pro-late halo, but use a broad lognormal distribution centred onspherical to encode approximately our expectations. For thehalo density profile, the NIE profile has considerable free-dom and can represent a much broader range of behavioursthan those seen in simulation. Therefore, we adopt physi-cally motivated priors to select the cosmologically motivatedsubset of parameters combination. Studies of large sets ofspiral galaxies, using satellite kinematics and weak galaxy-galaxy lensing, in the context of numerical simulations haveshown that the maximum observed circular velocity is typ-ically comparable to the maximum circular velocity of thehalo, even though these two maximums occur at vastly dif-ferent radii (Dutton et al. 2010a). We also know that ro-tation curves do not keep rising indefinitely, but typicallyflatten out within a few scale radii of the disk. To inject thisinformation we require that the asymptotic circular velocityof the halo be comparable to that measured via spectroscopy(see § − with width 50 km s − . The prior is c (cid:13) , 1–23 Dutton et al.
Table 3.
Summary of mass model priors. Bold indicates an uninformative prior,regular text indicates an informative prior (i.e. that the parameter is virtuallyfixed). L N ( a, b ) denotes a lognormal distribution, with a being the central valuefor the variable, and b being the standard deviation for the log of the variable. N ( a, b ) denotes a normal distribution, with a being the central value and b beingthe standard deviation. U ( a, b ) denotes a uniform distribution with lower andupper limits, a and b , respectively. For clarity we have arranged the parametersinto three groups: free parameters first (with uninformative priors), stellar massdistribution parameters (with priors derived from the SPS modelling of Section 3),and finally nuisance parameters.parameter description prior V c , h / km s − dark halo asymptotic circular velocity N (280, 50 ) q , h dark halo 3D axis ratio L N (1, 0.3 ) r c , h /arcsec dark halo core radius U(0.01, 10) log ( M ∗ / M ⊙ ) stellar mass U(10.5,11.4) f bulge bulge stellar mass fraction N (0 . , . ) q bulge bulge 2D axis ratio L N (0 . , . ) R , bulge /arcsec bulge chameleon size L N (0 . , . ) α bulge bulge chameleon index 0.4892 q disk disk 2D axis ratio L N (0 . , . ) R , disk /arcsec disk chameleon size L N (1 . , . ) α disk disk chameleon index 0.63cos( i ) cosine of disk inclination angle 0.2 x c /arcsec spatial offset in x direction N (0 , . ) y c / arcsec spatial offset in y direction N (0 , . ) θ /deg mass-light position angle offset N (1 . , . ) γ lens external shear N (0 , . ) θ γ /deg position angle of external shear U (0 , chosen to be broad enough – the 3-sigma range of this Gaus-sian spans the range 130 to 430 km s − – not to drive thefinal inference and yet tight enough to rule out models wherethe maximum velocity is reached too far out. In addition weimpose a uniform prior PDF for the core radius, allowing itto be at most 24 kpc (10”).In later sections we will introduce the kinematic andlensing data, and then use them to constrain the parametersof this 3-component mass model. However, before getting tothe data, in the rest of this section we give the functionalforms for each mass component, and the predicted observ-ables resulting from them. The axisymmetric ellipsoidal halo is assumed to have a non-singular isothermal (NIE) profile, which we parametrise ina cylindrical coordinate system in the plane of the galaxyfollowing Keeton & Kochanek (1998): ρ NIE ( R, z ; V c , r c , q ) = V πGq e sin − e r + R + z /q . (3)Here, V c is the asymptotic circular velocity, r c is the coreradius, q is the three dimensional axis ratio, and e =(1 − q ) / is the eccentricity. For a zero thickness massdistribution ( q = 0), e/ sin − e = 2 /π . For a spherical massdistribution ( q = 1), e/ sin − e = 1. For a prolate massdistribution ( q > e is imaginary, e/ sin − e is realand greater than 1.This mass profile is often used in gravitational lensanalysis, since its projected mass distribution and deflection angles can be computed analytically (Keeton & Kochanek1998). This model has been used very successfully to modelthe total (dark plus stellar) mass profiles of elliptical galaxylenses (e.g., Bolton et al. 2008). In this work we use the NIEmodel for the halo alone. While the NIE profile has a con-stant central density, it is flexible enough to broadly capturethe change in the density profile in the central regions thatwe expect from numerical simulations of dark matter halos(e.g., Navarro et al. 1997).We would like to model the stellar disk and stellarbulge mass components such that in projection they appearto have exponential and S´ersic profiles respectively. How-ever, we also need 3D distributions for which we can com-pute predicted rotation curves, as well as projected distri-butions convenient for lensing calculations. To achieve thiswe note that the NIE profile can be used to create an ap-proximation to an exponential profile in projection (Malleret al. 2000). This is done by taking the difference of twoNIEs . If ρ NIE ( R, z ; V c , r c , q ) is a softened isothermal ellip-soid, then ρ Chm ( R, z ; V c , r c , q , α ) = ρ NIE ( R, z ; V c , r c , q ) (4) − ρ NIE ( R, z ; V c , r c /α, q )is a “Chameleon” profile with positive density everywhere,and a finite total mass. In Appendix A we derive new formu-lae that provide Chameleon approximations to S´ersic profilesof any index (for 1 ∼ < n ∼ < r c and index α given a S´ersic half light radius R and in-dex n . c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − For our ellipsoidal mass profiles, we can calculate the rota-tion velocity, as a function of radius, of a massless test parti-cle moving on a circular orbit in the plane of the galaxy. Werefer to this velocity as the circular velocity to distinguish itfrom the rotation velocity of the stars and gas, which may belower than the circular velocity due to a velocity dispersioncomponent. The circular velocity profile for the NIE modelis (Keeton & Kochanek 1998) V ( R ; V c , r c , q ) V = 1 − e sin − e r c ( R + e r ) / × tan − (cid:20) ( R + e r ) / q r c (cid:21) , (5)where again e = (1 − q ) / is the eccentricity of the massdistribution and the model is normalised so that, asymptot-ically for R → ∞ , V NIE ( R ) → V c . For the special case of azero thickness mass distribution ( q = 0 , e = 1) Equation 5reduces to V ( R ) V = 1 − r c ( R + r ) / . (6)For the case of a prolate mass distribution ( q > e/ sin − e is real, but since e <
0, for ( R + e r ) < V ( R ) V = 1 − ˜ e sinh − ˜ e r c ( − R − e r ) / × tanh − (cid:20) ( − R − e r ) / q r c (cid:21) , (7)where ˜ e = p | e | .For the chameleon profile the circular velocity is givenby the quadratic difference between the circular velocities ofthe sub-component NIE’s: V ( R ; V c , r c , q , α ) = V ( R ; V c , r c , q ) (8) − V ( R ; V c , r c /α, q ) . Likewise, for the mass model the total circular velocity isgiven by the quadratic sum of the circular velocities of thebulge, disk, and halo components: V ( R ) = V ( R ) + V ( R ) + V ( R ) (9). Projecting the three components onto the sky allows us tocompute deflection angles and predict the observed gravita-tional arc, pixel by pixel.In projection the mass distribution (an oblate or prolateellipsoid with minor to major axis ratio q ) has projectedaxis ratio q given by q = ( q sin i + cos i ) / , (10)where i is the inclination angle (such that i = 0 ◦ correspondsto a face-on disk, and i = 90 ◦ to an edge-on one). In generalthe projected axis ratio, q , will be closer to unity than the3D axis ratio, q . The projected mass density of an NIE model is givenby (Keeton & Kochanek 1998):Σ NIE ( x, y ; b, r c , q ) = V GD d e sin − e q p r c + x + ( y/q ) , = Σ crit b q p r c + x + ( y/q ) . (11)Here, D d is the angular diameter distance to the lens, and e is again the ellipticity, while in the second line b is the minoraxis of the critical curve (and thus b/q is the major axis ofthe critical curve), and Σ crit is the critical surface density ofstrong lensing: Σ crit = c πG D s D ds D d , (12)where D s is the angular diameter distance from the observerto the source, and D ds is the angular diameter distancefrom the lens to the source. For our assumed cosmology,for SDSS J2141 − D d = 497 . D s = 1510 . D ds = 1179 . crit = 4285 . ⊙ pc − .To explain the parts of Equation 11 a little further, theparameter, b , is related to the spherical Einstein radius, b SIS ,via: b = b SIS ( e/ sin − e ) , (13)and the spherical Einstein radius (in radians) is in turn re-lated to the asymptotic circular velocity, V c , via: b SIS = 2 π ( V c /c ) D ds /D s . (14)The deflection angles are given by (Keeton & Kochanek1998) α x = b (1 − q ) / tan − (cid:20) (1 − q ) / x Ψ + r c (cid:21) , (15) α y = b (1 − q ) / tanh − (cid:20) (1 − q ) / y Ψ + q r c (cid:21) , (16)(17)where Ψ = q ( r + x ) + y . The deflection angles fromthe three components of the mass distribution can be sim-ply summed, as they are just the first derivatives of theprojected (lens) potential of each component, and the po-tentials of the three components can be summed themselvesto give the total potential. Likewise, the Chameleon profilein projection is just the difference between two projectedNIE models, and its deflection angles are just the differencebetween those of its NIE components.To predict the positions and structure of the lensed im-ages given a set of mass model parameters, we map eachobserved pixel location back to the source plane using theoverall deflection angle map, and look up the surface bright-ness of a model source at that position. In practice we use asingle, elliptically symmetric source with a S´ersic brightnessprofile, as in e.g., Marshall et al. (2007). We now present the strong gravitational lensing data thatwe will use to constrain our mass model. We first describe c (cid:13) , 1–23 Dutton et al.
Figure 5.
Upper panel: Galaxy subtracted image in the K’-bandobtained using a reflection of the image. The lens galaxy is wellsubtracted near the arc, but there are significant residuals nearthe center. Lower panel: An arc in the right place, obtained bymasking out non-arc features from the image in the upper panel.This was used for the actual fitting, to weaken the fit criterion.Essentially, we want to produce models whose posterior distribu-tion predicts an arc with this morphology, but does not necessarilyneed to match every pixel. the preparation of the arc imaging data, and then show witha simple lens model the information it contains.
Due to the strong effects of dust in the lens system, we focusour lensing analysis primarily on the K’-band NIRC2 image.In the K’-band the lens galaxy appears to be much smootherthan in the optical, but the light distribution is not able to be modelled by a simple surface brightness profile. Thismakes subtracting the lens galaxy light difficult. Our goalis to obtain robust parameter inferences with meaningfuluncertainties, and so we opt for quite a conservative appli-cation of the imaging data. To account for lens subtractionerrors, we create an arc image and a goodness of fit statisticthat rewards a model for having an arc in the right place,with the right shape, and that is all: we do not require thedetailed features of the modified surface brightness profileto be matched.To achieve this, we first subtracted the galaxy lightaround the arc by reflecting the galaxy along the minoraxis. This method provides a better subtraction than mul-tiple S´ersic components, or a radial bspline model (as usedby e.g., Bolton et al. 2006). We then cut out the arc and setthe remaining pixels to zero, before adding noise at the levelof σ = 15% of the peak arc brightness. This 15% value isan initial estimate of the appropriate noise level needed tosuppress models that predict significant lensed features else-where, although faint counter-images are still allowed. Theresulting modified image is shown in Figure 5 (lower panel);we also show the lens subtracted image (upper panel), withits uncertain central region. It is not clear whether or notthere is a counter-image in the centre of the system. We notethat if there is no counter image, then SDSS J2141 − σ on the pixelvalues d ; we can predict these pixel values from a modelsource with parameters θ s (as described in Section 4.4above) given lens model parameters θ m . Denoting the pre-dicted data as d p , we write down the usual chi-squared misfitfunction χ = pixels X i [ d i − d p i ( θ m , θ s )] σ . (18)We allow the data to inform our understanding of the modeluncertainty, by re-scaling the denominator by a factor T .This corresponds to increasing or decreasing the perceivederrors on the pixel values, and provides a mechanism foravoiding over-fitting the arc structure or allowing modelsthat predict undetected flux. (The symbol T stands for“temperature” – increasing the temperature increases thediffusion of the model around its parameter space.) The like-lihood function is then:Pr( d | θ m , θ s ) ∝ exp (cid:18) − T χ (cid:19) (19)The value of T selected was 7.5. This was the highestvalue where the posterior distribution only contained im-ages that resemble the arc morphology. Higher temperaturescaused the posterior models to predict substantial flux thatis not observed, lower temperatures enforced the fit to themodified image to be too strict. The reason for the two-stepprocedure (adding noise and then selecting a temperature)is that Nested Sampling provides the results for all temper-atures in a single run, whereas tuning the noise level itselfwould have required large numbers of trial runs.Using this modified image, and the temperature-raisingscheme, allows us to explore an approximate posterior dis-tribution for the lens model parameters that conditions on c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − Figure 6.
Marginalised posterior PDF for the 2D axis ratio, q ,and circular velocity, V c , for a single SIE component model fittedto the strong lensing data. The dashed lines show the medianvalues of q and V c . the presence of an arc with the observed morphology, andnothing else. To illustrate the unique information that strong gravita-tional lensing provides, we first perform a fit to the lensingdata with a single singular isothermal ellipsoid (SIE) massmodel. The purpose of this exercise is to show that stronglensing places constraints on the axis ratio of the projectedmass, as well as the projected mass within the Einstein ra-dius.Fixing its centroid and orientation to that of the lenssurface brightness, our example SIE lens model has two pa-rameters, minor axis Einstein radius b , and axis ratio q .We assign uniform priors over wide ranges for these pa-rameters, and then explore the posterior PDF using oursampling code (which we introduce in more detail in Sec-tion 7 below). We find the circularised Einstein radius tobe θ Ein = b/ √ q = 0 . +0 . − . arcsec, and the axis ratio tobe q = 0 . +0 . − . . We can transform samples in b and q into the circular velocity, which we expect to be well con-strained. Figure 6 shows the marginalised posterior PDF for q and V c – the circular velocity is indeed well-constrained: V c = 254 +15 − km s − . The shape of the arc also constrains theellipticity of the total mass distribution: since in our three-component model the ellipticity of the disk and bulge arefixed, we expect strong lensing to then provide informationabout the shape of the dark halo. The second set of data that we will use to constrain ourmass model is a galaxy rotation curve, derived from optical emission and absorption line spectroscopy. In this section wedescribe the observations, and the rotation curve extractionprocess, discuss the observed velocity dispersion and our in-terpretation of it, and then derive the likelihood functionthat we will use when fitting our mass model.
Major axis (PA = 87 ◦ ) long-slit spectra were obtained withthe DEep Imaging Multi-Object Spectrograph (DEIMOS),and Low Resolution Imaging Spectrograph (LRIS) on theKeck 10-m telescopes.On October 1st 2008 SDSS J2141 − . A ) with a 1” widthslit resulting in a spectral resolution of ≃ . A . The centralwavelength was 6500 A , resulting in a wavelength range of5200 − A . We took three exposures of 1200s in seeingconditions of 0 . ′′ . The slit was aligned with the major axisof the galaxy, with PA = 87. The spectra were reduced usingroutines developed by D. Kelson (Kelson 2003).On November 27th 2008 we observed SDSS J2141 − ≃ . ′′ ) than that ofour DEIMOS observation. This resulted in increased beam-smearing and reduced sensitivity, and thus we focus ourkinematic analysis on the DEIMOS observations. Cutouts of the DEIMOS long-slit spectrum centered aroundprominent emission and absorption lines are shown in Fig-ure 7. These show clear signs of rotation in both emissionand absorption lines. For the emission lines we measuredrotation curves by locally fitting Gaussian line profiles toone-dimensional spectra extracted along the slit. For the ab-sorption lines we measured the rotation and dispersion pro-file by applying python routines developed by M.W.Augerto one-dimensional spectra extracted along the slit. The up-per right panel also shows the spatially offset [O ii ] emissionlines from the source galaxy.The extracted rotation curve is shown in the upper pan-els of Figure 8. The spatial sampling is ≃ .
59” (5 DEIMOSpixels), corresponding to 1 data point per seeing FWHM.There is good agreement in the rotation curves measured inH α and N ii , except near the very center, where N ii gives ahigher V rot . This is possibly due to H α being contaminatedwith stellar absorption. The Na D and Mg b absorption linesgive lower V rot than the emission lines, especially at largerradii. This is expected due to the increased pressure supportin the stars compared to the gas, so called asymmetric drift.The rotation curve flattens out beyond 3 arcsec (7 kpc),corresponding to 2 disk scale lengths. On the East sidethe rotation curve remains flat out to the last data point(13 kpc). On the West side the rotation curve decreases be-yond 3.5 arcsec (8.5 kpc). We trace this asymmetry to thewarp in the West side of the optical disk, which causes theslit to miss the major axis: beyond 3.5 arcsec we thereforeuse only the East side of the rotation curve.Figure 9 shows the folded rotation curve obtained bycombining the rotation curves from H α and N ii . The data c (cid:13) , 1–23 Dutton et al.
Figure 7.
Cutout images of the DEIMOS optical long slit spectrum of SDSS J2141 − α α emission due to star formation. points shown in this figure are given in Table 4. When com-bining data points we use the error weighted mean. Thenew error is the maximum of the statistical error and halfthe differences between the two data points. From this ro-tation curve the maximum observed rotation velocity is271 ± − (corrected for inclination, but not beamsmearing) at 13 kpc from the galaxy center. For a given set of mass model parameters θ m we can predictthe circular velocity of the stars at each radius, and hencefind parameter vectors that fit the rotation curve data. Todo this we need to fold in the effects of beam-smearing; ourprocedure for this is described in the next section. We com-pare the predicted data v p and the observed data in theusual way. We assume uncorrelated Gaussian errors on theobserved velocities v j σ j = p Kσ ,j + σ (where σ ,j arethe reported error bars, and K and σ extra are free param-eters allowing the true uncertainties to be inferred), andhence construct the familiar likelihood functionPr( v | θ m ) = exp (cid:16) − χ v (cid:17)Q nj =1 σ j √ π (20)where the misfit function χ v = X j (cid:0) v j − v p j (cid:1) σ j . (21)We then take the product of this likelihood and the priorPDFs on the parameters defined in Section 4 to obtain theposterior PDF for the model parameters given the kinemat-ics data. The rotation curve data presented in Figure 9 are the ob-served values, uncorrected for inclination, finite slit widthand seeing effects. We refer to these combined effects asbeam-smearing. Since the disk inclination is high, the sin i correction is small, just a factor of 1 . Figure 8.
Rotation curve (upper panel), velocity dispersion pro-file (middle panel), and line flux profile for SDSS J2141 − (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − Figure 9.
Folded rotation curve from emission lines of H α and[NII]. The location of the bulge half-light radius, R b , Einsteinradius, R Ein , and 2.2 disk scale lengths, R d are indicated withdotted lines. The spatial sampling is one point per seeing FWHM,which is indicated by the yellow circle. This rotation curve isuncorrected for inclination, and beam smearing due to the finite(1 arcsec) slit width and seeing. slit width covers a large fraction of the minor axis of thegalaxy, the effects of finite slit width and seeing are likelyto be significant, especially near the centre of the galaxy.We take this into account when computing the predicteddata v p as the inclined, beam-smeared, model rotation curvewithin a 1 arcsec slit. For computational efficiency we esti-mate the beam-smearing effect using a simplified, rotatingexponential disk model, and then apply this correction tothe model rotation curve. The intrinsic rotation curve ˆ v isgiven by the sum of the bulge, disk and halo components asdescribed above.The beam-smearing calculation is approximate, becausewe don’t know the exact distribution of the H α emission,only the starlight. While we model the H α distribution withan exponential profile, with the scale length of the V-bandlight, the actual distribution is likely to be asymmetric (dueto extinction), and non-exponential (there is a ring of starformation). To minimize the impact of these uncertainties,we have excluded the inner 2 arcsec of data in our massmodels. The spectral line fits described in the previous section alsoyield some information on the velocity dispersion of the sys-tem. The central (within 1”) velocity dispersion from theMg b–[Fe ii ] lines was found to be σ = 180 ± − ,in agreement with the SDSS value (which is integratedover the 3” fibre aperture). Na D gives a lower central ve-locity dispersion, of σ = 119 ± − . Absorption inNa D can come from interstellar gas, as well as stars. SinceSDSS J2141 − D line is not reliably tracing the stellar velocity dis- Table 4.
Observed Rotation Curve from Emission LinesRadius Radius Rotation Velocity Error[arcsec] [kpc] [ km s − ] [ km s − ]0.000 0.00 3.5 5.30.593 1.45 114.1 5.81.185 2.89 153.8 2.91.778 4.33 212.7 2.62.370 5.78 243.8 2.62.963 7.22 259.8 2.33.555 8.67 256.8 2.04.148 10.11 254.9 7.54.740 11.56 263.4 2.35.333 13.00 265.9 3.5 persion. For the emission lines the central (within the inner0 . ′′ ) velocity dispersion of the N ii line is ∼ ±
14 km s − ,similar to that of the stars. However, the velocity disper-sion of the N ii line declines faster with radius than that ofthe stars (middle panel of Figure 8). This is an indicationthat the peak in velocity dispersion in N ii is due to beam-smearing. For the H α line the central velocity dispersion isconsiderably lower than that of N ii , which we ascribe to thepresence of absorption, which we have not corrected for. Inthe outer part of the disk, the observed velocity dispersionof the emission lines is close to that of the instrumental reso-lution of ∼
32 km s − (dotted horizontal line in middle panelof Figure 8), indicating that the intrinsic velocity dispersionof the line-emitting gas disk is too low to be resolved.How could we model the velocity dispersion data? Oursimple dynamical model does not easily allow for this, butwe can make use of the dispersion information as a cross-check in the following simplistic way. The results of Pad-manabhan et al. (2004) and Wolf et al. (2010) show that,for a spherical system, the circular velocity at the half-light radius V circ ( R ) ≃ . σ los , where σ los is the inte-grated line of sight velocity dispersion of the system. Forthe case of SDSS J2141 − bulge half light radius is R , b ≃ . V circ formula above and adopting an un-certainty of 10% results in an estimate of the circular veloc-ity at R of V circ ( R , b ) = 306 ±
31 km s − . In our currentanalysis we do not make use of this constraint. Rather, weuse this as a consistency check to our models which are con-strained by strong lensing and gas rotation curve. We now infer the parameters of our 3-component massmodel using constraints from the strong lensing and kine-matics data presented in the previous two sections. In or-der to sample the posterior distribution for the parameterswe use the Diffusive Nested Sampling code from Breweret al. (2009). Diffusive Nested Sampling is a powerful andefficient alternative to standard MCMC sampling. c (cid:13) , 1–23 Dutton et al.
Figure 10.
Marginalised two-dimensional posterior PDFs for unconstrained mass model parameters (see Table 3) using constraints fromstrong lensing alone. In the histogram panels, the vertical red lines show the median value. The median value together with the offsetsto the 84th and 16th percentiles of the distribution is given in the top right corner. The priors are shown with solid red lines.
We consider inferences from three data sets:(i) strong lensing only(ii) kinematics only(iii) strong lensing plus kinematicsIn Figures 10–12 we plot, for each data set, all possi-ble one-dimensional and two-dimensional marginalised pos-terior PDFs for the four main mass model parameters. Theseparameters are the total (disk+bulge) stellar mass M ∗ , andthe dark matter halo asymptotic circular velocity, V c , h , coreradius r c , h , and 3 dimensional flattening, q , h . The me-dian, 16th and 84th percentiles of the marginalized posteriorPDFs for these parameters individually are given in Table 5.The constraints on stellar mass, dark halo density, anddark halo shape are discussed in more detail below. We firstpoint out some of the main features of these figures. With strong lensing alone (Figure 10), the halo param-eters are poorly constrained: The PDF for the core radiusis almost uniform, while the PDFs for the halo velocity andhalo axis ratio follow the priors. This is expected owing tothe limited range in projected radius probed by the lens-ing constraints. There is, however, a good constraint on thestellar mass: log ( M ∗ / M ⊙ ) = 11 . +0 . − . . This is a result ofthe axis ratio of the projected mass being quite low ( § c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − Figure 11.
Marginalised two-dimensional posterior PDFs for unconstrained mass model parameters (see Table 3) using constraints fromkinematics alone. In the histogram panels the vertical red lines show the median value. The median value together with the offsets tothe 84th and 16th percentiles of the distribution is given in the top right corner. The priors are shown withe solid red lines.
Adding the strong lensing constraints to the kinematicsconstraints breaks some of the degeneracies. Specifically, itremoves the highest stellar mass solutions from the kinemat-ics only analysis. All posteriors are considerably tighter thanthe priors, illustrating the power of the combined analysis:for example, circular velocity is now known to 6% precision,and core radius is well constrained to be smaller than 5 kpc.However, there is still a degeneracy between halo core radiusand stellar mass. There is also a residual degeneracy betweenstellar mass and halo shape — with low stellar mass solu-tions favoring oblate dark matter haloes. This degeneracy isexpected as the total mass needs to be flattened to repro-duce the strong lensing (Section 5.2). The flattening can beachieved with either a significant stellar disk component anda spherical halo, or a less massive disk and a more flattenedhalo. In Figure 13 we show the rotation curves and stronglensing image predicted by two example mass models drawnfrom the posterior PDF given both lensing and kinematicsdata. These models both predict four images of the lensedsource, including a faint counter-image that is consistentwith the noise in the centre of the subtracted image. Bothmodels’ predicted rotation curves fit the outer part of theobserved rotation curve very well; they also match well thecentral part of the observed rotation curve, which was notused in the fit. The two models have either the posteriormedian stellar mass, or much lower stellar mass; they canonly be distinguished in the plot of the intrinsic, pre beam-smeared rotation curves, where the high stellar mass modelhas a significantly higher rotation velocity at radii less thanone arcsec. This region could be probed with higher spatialresolution spectroscopy, or by making use of the velocity c (cid:13) , 1–23 Dutton et al.
Figure 12.
Marginalised two-dimensional posterior PDFs for unconstrained mass model parameters (see Table 3) using constraints fromboth kinematics and strong lensing. In the histogram panels the vertical red lines show the median value. The median value togetherwith the offsets to the 84th and 16th percentiles of the distribution is given in the top right corner. The priors are shown with solid redlines.
Table 5.
Summary of fitted parameters: stellar mass ( M ∗ ); halo asymptotic circularvelocity ( v c , h ); halo core radius ( r c , h ); and 3D halo axis ratio ( q , h ).log ( M ∗ / M ⊙ ) V c , h / [ km s − ] r c , h / [kpc] q , h Lensing 11 . +0 . − . +51 − . +5 . − . . +0 . − . Kinematics 11 . +0 . − . +47 − . +4 . − . . +0 . − . Lensing + Kinematics 10 . +0 . − . +17 − . +2 . − . . +0 . − . c (cid:13)000
Summary of fitted parameters: stellar mass ( M ∗ ); halo asymptotic circularvelocity ( v c , h ); halo core radius ( r c , h ); and 3D halo axis ratio ( q , h ).log ( M ∗ / M ⊙ ) V c , h / [ km s − ] r c , h / [kpc] q , h Lensing 11 . +0 . − . +51 − . +5 . − . . +0 . − . Kinematics 11 . +0 . − . +47 − . +4 . − . . +0 . − . Lensing + Kinematics 10 . +0 . − . +17 − . +2 . − . . +0 . − . c (cid:13)000 , 1–23 he disk and halo of lens SDSS J2141 − Figure 13.
Example mass model that fits the lensing (left panel) and kinematics (right panel) data. In the right panel two models areshown: median stellar mass (solid lines); low stellar mass (dashed lines). The red lines show the intrinsic model circular velocity, whilethe black lines show the model circular velocity after beam-smearing, finite slit width, and inclination effects are taken into account.Only the black points beyond 2 arcsec are included in the fit. The red point at small radii is the constraint from the stellar velocitydispersion, and disfavors the low stellar mass solution. dispersion information. Indeed, our cross-check point fromSection 6.5 would favour the high stellar mass model.In Figure 14 we show the inferred circular velocity pro-file, decomposed into baryonic and dark matter components.These estimates are based on the posterior samples usingthe joint lensing plus kinematics analysis. The solid linesshow the median model from the posterior PDF, while theshaded regions enclose 68% of the models. In the radialregion where we have observational constraints (i.e., fromthe Einstein radius to the last rotation curve point) thetotal circular velocity is well constrained. For example, at2.2 disk scale lengths (8.1 kpc), the total circular velocityis V . = 289 ± − . The circular velocity profiles ofthe baryons and dark matter are not as tightly constrained.Nevertheless, we can still infer interesting constraints onthe dark matter fraction as a function of radius, and thusdetermine whether or not SDSS J2141 − V disk (2 . R d ) /V tot (2 . R d ) = 0 . ± .
10 (Sackett 1997). Here V disk (2 . R d ) is the circular velocity of the disk at 2.2 diskscale lengths, and V tot (2 . R d ) ≡ V . is the total circularvelocity at 2.2 disk scale lengths.A galaxy may have a sub-maximal disk, but still have amaximal baryonic component due to the bulge. Thus we con-sider the contribution of the baryons (i.e., bulge plus disk)to V . to be of more relevance than just the contribution ofthe disk to V . . We find that V bar (2 . R d ) /V . = 0 . +0 . − . ,which suggests that SDSS J2141 − V bar ( R b ) /V tot = 0 . +0 . − . , andthus SDSS J2141 − f DM = 0 . +0 . − . within2.2 disk scale lengths, and f DM = 0 . +0 . − . within the bulgehalf-light radius. Figure 15 shows the posterior PDFs from our joint lens-ing and kinematics analysis together with those from SPSmodels for both Chabrier and Salpeter IMFs. From our lens-ing and kinematics analysis the stellar mass of the galaxyis found to be log M ∗ = 10 . +0 . − . . This is in excellentagreement with the stellar mass derived from SED fittingassuming a Chabrier IMF, which is log M ∗ = 10 . ± . ≃ c (cid:13) , 1–23 Dutton et al.
Figure 14.
Circular velocity profiles (upper panel) and spherical baryon fractions (lower panel) from our joint lensing plus kinematicsanalysis. The solid lines show the median, while the shaded regions enclose 68% of the posterior PDF. The total circular velocity (blackline and grey shaded region) is well constrained outside of the Einstein radius, R Ein , and up to the last rotation curve point at 13 kpc.The contributions of the baryons (red lines and shaded regions) and the dark matter (blue lines and shaded regions) are more uncertain.However, at the bulge half-light radius, R b , the galaxy is baryon dominated (and thus is “maximal”), while at 2.2 disk scale lengths thebaryons fraction is roughly 50% (and thus is “sub-maximal”). Figure 15.
Inference on stellar mass from lensing and kinematics(histogram) compared with SPS models (solid lines) assuming aChabrier IMF (black) and Salpeter IMF (red). The Bayes factorin favor of a Chabrier IMF, compared to a Salpeter IMF is 2.5. ±
10% (assuming a Chabrier IMF), split roughly equallybetween atomic and molecular gas (e.g., Dutton & van denBosch 2009). If the cold gas is distributed like the stars, thenthe lensing+kinematics stellar mass is actually a baryonicmass, greater than or equal to the actual stellar mass. Ifthe cold gas is more extended than the stars, as is often thecase, then we will still be over-estimating the stellar mass,but by a smaller amount. To estimate an upper limit to theimpact of cold gas on our derived stellar masses we assumethat the gas mass for SDSS J2141 − M gas =1 . × M ⊙ , with a standard deviation of 0.3 dex. We thensubtract off the gas mass from the gas free stellar mass toderive the “true” stellar mass. The results of this exercise areshown in Fig. 16. The resulting median and 68% confidenceinterval on the stellar mass is log ( M ∗ / M ⊙ ) = 10 . +0 . − . ,i.e., 0.1 dex lower than when ignoring the cold gas. TheBayes factor in favor of a Chabrier IMF over a SalpeterIMF has increased from 2.7 to 11.9, which corresponds tostrong evidence. Thus by ignoring the cold gas we could be c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − Figure 16.
Effect of gas mass on the inferred stellar mass fromlensing and kinematics. For each model galaxy in the posteriorPDF we draw a gas mass from a log-normal distribution withmean and standard deviation typical for massive spiral galaxies.The resulting PDFs for the stellar and gas mass are shown asblack hatched histograms. For comparison, the blue shaded his-togram shows the posterior PDF on the stellar mass assuming nogas mass. Thus accounting for cold gas mass reduces the stellarmass derived from lensing and kinematics by ≃ . over estimating the stellar mass by ≃ . ± .
05 dex, whichstrengthens the case for an IMF lighter than Salpeter.How does this result compare to previous work? Us-ing maximal disk fits to spiral galaxy rotation curves in theUrsa Major cluster, Bell & de Jong (2001) placed an upperlimit on the stellar mass-to-light ratio normalisation, favor-ing IMFs with stellar masses 0.15 dex lower than Salpeter,the so-called diet-Salpeter IMF. We note that a SalpeterIMF is also disfavored for fast rotating elliptical galaxies(Cappellari et al. 2006; Treu et al. 2010; Auger et al. 2010;Barnab´e et al. 2010), but is favored for massive ellipticalgalaxies (Treu et al. 2010; Auger et al. 2010; van Dokkum& Conroy 2010). Thus comparing our result with those formassive ellipticals, supports the idea that the IMF is not uni-versal, but dependent on galaxy mass and/or Hubble type.By shifting our Salpeter stellar mass PDF by − .
15 dex,we find that for SDSS J2141 − ( M ∗ / M ⊙ ) = 11 . ± . ( M gas / M ⊙ ) = 10 . ± . N-body simulations have shown that in ΛCDM cosmologiesdark matter haloes are expected (in the absence of baryoniceffects) to have very specific structure. The mass densityprofiles can be well approximated by the so called NFWprofile (Navarro et al. 1997). This has a density profile that varies from ρ ( r ) ∝ r − at small radii, to ρ ∝ r − at largeradii. The radius where the logarithmic slope of the densityprofile is d ln ρ/ d ln r = − r s .The halo scale radii are tightly correlated with the virialmasses of dark matter haloes, M vir . This correlation is usu-ally expressed in terms of the halo concentration, c = r vir /r s ,where r vir is the virial radius. Halo concentrations are onlyweakly dependent on halo mass, with a relation of the form c ∝ M − . (Macci´o et al. 2007). The scatter in halo concen-tration, at fixed halo mass, for relaxed haloes is small ≃ . V/ , whichdepends on the maximum halo circular velocity, V max , andthe radius where circular velocity of the halo is half of themaximum, r V/ (Alam, Bullock & Weinberg 2002):∆ V/ = 5 × (cid:18) V max / [100 km s − ] r V/ / [ h − kpc] (cid:19) . (22)For NFW haloes there is a one-to-one mapping between∆ V/ − V max and c − M vir , and thus one can compare theobserved ∆ V/ with predictions for ΛCDM haloes. Figure 17shows the predictions for ∆ V/ − V max in a WMAP 5th yearcosmology (Dunkley et al. 2009) from Macci`o, Dutton, & vanden Bosch (2008). The shaded regions show the 1 and 2 σ in-trinsic scatter. The large symbols show measurements fromdwarf and low surface brightness galaxies after subtractingof the baryons (de Blok et al. 2001; de Blok & Bosma 2002;Swaters et al. 2003). These are in excellent agreement withthe predictions from ΛCDM.In our mass model of SDSS J2141 − V max = V c , h , and r V/ = 1 . r c , h . We can compute ∆ V/ for this model and compare it with the NFW profile ha-los of the simulations. For our model the median and un-certainty (corresponding to 16th and 84th percentiles) islog ∆ V/ = 5 . +0 . − . . The median is 2 . σ higher (in termsof intrinsic scatter) than that predicted for pristine ΛCDMhaloes, although the full posterior PDF overlaps the ΛCDMpredictions, as shown in Figure 17. The 16th percentile ofthe PDF for ∆ V/ only corresponds to a 1 . σ deviation fromthe ΛCDM distribution. Thus there is a suggestion that theSDSS J2141 − V/ for higher M ∗ . There are two interpretations of the higher than ex-pected halo density. 1) The halo has undergone contrac-tion in response to galaxy formation (e.g., Blumenthalet al. 1986). 2) The halo of SDSS J2141 − − M ∗ ), ro-tation velocity at 2.2 disk scale lengths ( V . ), and disk scale c (cid:13) , 1–23 Dutton et al.
Figure 17.
Central density of the dark matter halo, ∆ V/ vsmaximum halo circular velocity, V max . The solid black line showsthe prediction for pristine dark matter haloes in the concordanceΛCDM cosmology (WMAP5) from Macci`o et al. (2008). Theshaded regions show the 1 and 2 σ intrinsic scatter. The coloredsymbols show measurements from dwarf and low surface bright-ness galaxies, after subtraction of the baryons (de Blok et al. 2001,dB01; de Blok & Bosma 2002, dB02; Swaters et al. 2003, S03).The small black dots show samples from the posterior PDF forSDSS J2141 − length ( R d ). For SDSS J2141 − V . ≃
289 km s − , log ( M ∗ , Chab / M ⊙ ) ≃ . R d ≃ . M ⊙ we ex-pect V . = 229 ±
25 km s − (Dutton et al. 2010b). Alterna-tively, for a rotation velocity of V . = 289 km s − we expecta stellar mass of log ( M ∗ / M ⊙ ) = 11 . +0 . − . , assuming aChabrier IMF. For a rotation velocity of V . = 289 km s − we expect R d = 6 . +2 . − . kpc (Courteau et al. 2007; Duttonet al. 2007). Thus SDSS J2141 − ≃ σ and small size (by ≃ . σ ) at fixed rotationvelocity, which means that it has a higher baryonic and totalmass density than typical massive spiral galaxies.The high central density of SDSS J2141 − − Figure 18.
Central density of the dark matter halo, ∆ V/ vsstellar mass, M ∗ . The solid black line shows the prediction forpristine dark matter haloes in the concordance ΛCDM cosmol-ogy (WMAP5). The shaded regions show the 1 and 2 σ intrinsicscatter. The black dots show samples from the posterior PDF forSDSS J2141 − N-body simulations have shown that in ΛCDM cosmologiesdark matter haloes are triaxial, with a preference towardsprolate shapes (Jing & Suto 2002; Bailin & Steinmetz 2005;Kasun & Evrard 2005; Allgood et al. 2006; Bett et al. 2007;Macci`o et al. 2007, 2008). For a halo mass of M vir = 10 M ⊙ ,which is typical for massive spiral galaxies, the minor to ma-jor axis ratio ratio c/a ≃ . ± .
1, and the intermediate tomajor axis ratio b/a ≃ . ± . b/a ≃ .
95 and c/a ≃ .
85, and the axial ratios become ap-proximately independent of radius. Similar results have beenobtained from other cosmological simulations (e.g., Tisseraet al. 2010). It should be noted that these simulations over-predict the baryon to dark halo mass ratios, and thus theeffect of galaxy assembly on the halo shapes could be over-estimated.In our mass models we assume the halo is axisymmet-ric, with 3D axis ratio q , h . A spherical halo has q , h = 1,an oblate halo has q , h <
1, and a prolate halo has q , h > q , h = 1. For our fits to lensing only or kinematicsonly the posterior PDF for q , h is identical to the prior, butfor the joint analysis slightly oblate haloes are preferred: q , h = 0 . +0 . − . , and prolate haloes with q , h > . c (cid:13) , 1–23 trongly disfavored. Thus our results for SDSS J2141 − We have presented an analysis of the strong gravitationallens SDSS J2141 − HST and the Keck telescopes. Thelens galaxy is a high inclination disk dominated galaxy with K ′ -band bulge fraction of 0.2, showing stellar rotation inmultiple spectral lines. A singular isothermal ellipsoid lensmodel provides a circular velocity of V c = 254 +15 − km s − and an axis ratio of q = 0 . +0 . − . .We perform a joint fit to the multi-filter surface bright-ness, lensing and kinematics data using a self-consistent 3-component mass model, and from it draw the following con-clusions: • The lensing and kinematics constraints yield a stellarmass of log ( M ∗ / M ⊙ ) = 10 . +0 . − . (68% confidence inter-val), independent of the IMF. • This value is in excellent agreement with the stellarmass derived from the SED using SPS models and assum-ing a Chabrier (2003) IMF: log ( M ∗ / M ⊙ ) = 10 . +0 . − . .A Salpeter (1955) IMF results in stellar masses 0.24 dexhigher: our analysis marginally favors a Chabrier IMF overa Salpeter IMF, by a Bayes factor of 2.7. • Accounting for the expected gas mass reduces the lens-ing and kinematics stellar mass by 0 . ± .
05 dex, and in-creases the Bayes factor in favor of a Chabrier IMF to 11.9. • At 2.2 disk scale lengths the spherical dark matter frac-tion is f DM = 0 . +0 . − . , suggesting that the baryons aresub-maximal. • The dark matter halo has a maximum circular veloc-ity of V c , h = 276 +17 − km s − , and a core radius of r c , h =2 . +2 . − . kpc. The corresponding central density parameterlog ∆ V/ = 5 . +0 . − . is higher than expected for uncon-tracted NFW haloes in the concordance ΛCDM cosmology,which have log ∆ V/ = 5 . • This high density could either be evidence for halo con-traction in response to galaxy formation (e.g., Blumenthalet al. 1986), or the result of a selection bias towards high con-centration haloes. A larger sample with well-characterisedselection function is required to make further progress. • The dark matter halo is oblate, q , h = 0 . +0 . − . , witha probability of 69%. This finding provides support for thenotion that galaxy assembly turns strongly prolate triaxialdark matter haloes into roughly oblate axisymmetric haloes(e.g., Abadi et al. 2010). ACKNOWLEDGEMENTS
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G., et al. 2000, AJ, 120, 1579c (cid:13) , 1–23 he disk and halo of lens SDSS J2141 − APPENDIX A: THE CHAMELEONAPPROXIMATION TO A S´ERSIC PROFILE
In this appendix we derive an approximation to a S´ersic pro-file as the difference of two non-singular isothermal ellipsoids(NIE’s).The S´ersic profile is specified by three parameters: anormalization, a radial scale, and a shape parameter com-monly known as the S´ersic index, n . In its simplest form itis given by: Σ( R ) = Σ exp " − (cid:18) RR (cid:19) /n (A1)where Σ is the central surface density, and R is the radialscale. The S´ersic profile is commonly expressed in terms ofthe effective radius, R e , which encloses half of the projectedmass, and the effective density, Σ e ≡ Σ( R e ):Σ( R ) = Σ e exp ( − b n "(cid:18) RR e (cid:19) /n − . (A2)Where the term b n ≈ n − .
32. Here we use the asymptoticexpansion of Ciotti & Bertin (1999) to O ( n − ) valid to onepart in ∼ for n > . b n = 2 n −
13 + 4405 n + 4625515 n + 1311148175 n − n + O ( n − ) . (A3)Equation A1 and Equation A2 are related to each other via R e = ( b n ) n R (A4)and Σ e = exp( − b n )Σ . (A5)The Chameleon profile is the difference between twoNIEs with different core radii, but the same normalization(which insures the total mass is finite):Σ chm ( R ; Σ , R , R ) = Σ p R + R − Σ p R + R ! . (A6)For the purpose of comparing to a S´ersic profile, we use thefollowing parametrization.Σ chm ( R ; Σ , R , α ) =Σ − α R p R + R − R p R + ( R /α ) ! . (A7)Where Σ = Σ (1 − α ) /R is the central surface density ofthe chameleon profile, R = R , and α is the ratio betweenthe core radii of the two NIE’s: α = R /R , so that 0 < α <
1. The total mass of the Chameleon profile is M Chm = π Σ R /α. (A8)We wish to find the correspondence between the S´ersicparameters, Σ , Ser , R , Ser , n Ser , and the Chameleon pa-rameters, Σ , Chm , R , Chm , α Chm . We do this by fitting aChameleon profile to a S´ersic profile. We are interested inmass profiles, so we fit R Σ( R ) to give appropriate weight tothe density profile at large radii. Figure A1 shows the bestfit parameters as a function of S´ersic index. We fit these Figure A1.
Relation between Chameleon and S´ersic parametersas a function of S´ersic index.
Table A1.
Parameters of cubic fitting formula (Equation A9) tothe relations in Figure A1. x y x y y y y n log (cid:16) Σ , Chm Σ , Ser (cid:17) n log (cid:16) R , Chm R , Ser (cid:17) n log ( α Chm ) 2.03 -0.739 -0.527 -0.012 -0.008 relations between chameleon and S´ersic parameters with acubic function: y = y + y ( x − x ) + y ( x − x ) + y ( x − x ) (A9)The parameters of these fits are given in Table A1. Ourfitting function is valid for 1 n Ser n = 1 , , , & 4 using the fitting function inTable A1. The left panels show log Σ( R ) vs ( R/R e ) /n . Inthese units S´ersic profiles are straight lines. The Chameleonprofiles deviate most significantly from S´ersic profiles atsmall radii, where the Chameleon profile has a constant den-sity core. The middle panels show R Σ( R ) vs R/R e , whichare the curves that were fitted against. The right panels showthe cumulative mass profile. For radii between 0 . ∼ < R ∼ < . A TEX file preparedby the author. c (cid:13) , 1–23 Dutton et al.
Figure A2.
Chameleon fits (red solid lines) to S´ersic profiles (dotted lines) with n = 1 , , ,
4. The Chameleon profile reproduces thecumulative mass profile of a S´ersic profile to a few percent over most radii interesting for strong lensing and kinematics.c (cid:13)000