Statistics of mass substructure from strong gravitational lensing: quantifying the mass fraction and mass function

A Bayesian statistical formalism is developed to quantify the level at which the mass function slope (alpha) and the projected cumulative mass fraction (f) of (CDM) substructure in strong gravitational-lens galaxies, with arcs or Einstein rings, can be recovered as function of the lens-survey parameters and the detection threshold of the substructure mass. The method is applied to different sets of mock data to explore a range of observational limits: (i) the number of lens galaxies in the survey, (ii) the mass threshold, Mlow, for the detection of substructures and (iii) the uncertainty of the measured substructure masses. We explore two different priors on the mass function slope: a uniform prior and a Gaussian prior with alpha = 1.90+-0.1. With a substructure detection threshold Mlow=3x10^8 Msun, the number of lenses available now (n_l=30), a true dark-matter mass fraction in (CDM) substructure <=1.0% and a prior of alpha = 1.90+-0.1, we find that the upper limit of f can be constrained down to a level <=1.0% (95% CL). In the case of a Gaussian prior on alpha, it is always possible to set stringent constraints on both parameters. We also find that lowering the detection threshold has the largest impact on the ability to recover alpha, because of the (expected) steep mass-function slope. In the future, thanks to new surveys with telescopes, such as SKA, LSST and JDEM and follow-up telescopes with high-fidelity data, a significant increase in the number of known lenses will allow us to recover the satellite population in its completeness. For example, a sample of 200 lenses, equivalent in data-quality to the Sloan Lens ACS Survey and a detection threshold of 10^8 Msun, allows one to determine f=0.5+-0.1% (68% CL) and alpha=1.90+-0.2 (68% CL).Comment: MNRAS (in press

The Massive End of the Stellar Mass Function

We derive average flux corrections to the \texttt{Model} magnitudes of the Sloan Digital Sky Survey (SDSS) galaxies by stacking together mosaics of similar galaxies in bins of stellar mass and concentration. Extra flux is detected in the outer low surface brightness part of the galaxies, leading to corrections ranging from 0.05 to 0.32 mag for the highest stellar mass galaxies. We apply these corrections to the MPA-JHU (Max-Planck Institute for Astrophysics - John Hopkins University) stellar masses for a complete sample of half a million galaxies from the SDSS survey to derive a corrected galaxy stellar mass function at $z=0.1$ in the stellar mass range $9.5<\log(M_\ast/M_\odot)<12.0$. We find that the flux corrections and the use of the MPA-JHU stellar masses have a significant impact on the massive end of the stellar mass function, making the slope significantly shallower than that estimated by Li \& White (2009), but steeper than derived by Bernardi et al. (2013). This corresponds to a mean comoving stellar mass density of galaxies with stellar masses $\log(M_\ast/M_\odot) \ge 11.0$ that is a factor of 3.36 larger than the estimate by Li \& White (2009), but is 43\% smaller than reported by Bernardi et al. (2013).Comment: 11 pages, 8 figures, Accepted to MNRA

Parametrizing the Stellar Haloes of Galaxies

We study the stellar haloes of galaxies out to 70-100 kpc as a function of stellar mass and galaxy type by stacking aligned $r$ and $g$ band images from a sample of 45508 galaxies from SDSS DR9 in the redshift range $0.06\,\le\,z\,\le\,0.1$ and in the mass range $10^{10.0} M_{\odot} < M_{*} < 10^{11.4} M_{\odot}$r. We derive surface brightness profiles to a depth of almost $\mu_r \sim 32 \,\mathrm{mag\,arcsec}^{-2}$. We find that the ellipticity of the stellar halo is a function of galaxy stellar mass and that the haloes of high concentration ($C > 2.6$) galaxies are more elliptical than those of low concentration ($C < 2.6$) galaxies. The $g$-$r$ colour profile of high concentration galaxies reveals that the $g$-$r$ colour of the stellar population in the stellar halo is bluer than in the main galaxy, and the colour of the stellar halo is redder for higher mass galaxies. We further demonstrate that the full two-dimensional surface intensity distribution of our galaxy stacks can only be fit through multi-component S\'{e}rsic models. Using the fraction of light in the outer component of the models as a proxy for the fraction of accreted stellar light, we show that this fraction is a function of stellar mass and galaxy type. For high concentration galaxies, the fraction of accreted stellar light rises from $30\%$ to $70\%$ for galaxies in the stellar mass range from $10^{10.0} M_{\odot}$ to $10^{11.4} M_{\odot}$. The fraction of accreted light is much smaller in low concentration systems, increasing from $2\%$ to $25\%$ over the same mass range. This work provides important constraints for the theoretical understanding of the formation of stellar haloes of galaxies.Comment: Submitted to MNRAS, 18 pages, 19 figure

Resolving on 100 pc scales the UV-continuum in Lyman-α\alpha emitters between redshift 2 to 3 with gravitational lensing

We present a study of seventeen LAEs at redshift 2$<z<$3 gravitationally lensed by massive early-type galaxies (ETGs) at a mean redshift of approximately 0.5. Using a fully Bayesian grid-based technique, we model the gravitational lens mass distributions with elliptical power-law profiles and reconstruct the UV-continuum surface brightness distributions of the background sources using pixellated source models. We find that the deflectors are close to, but not consistent with isothermal models in almost all cases, at the $2\sigma$-level. We take advantage of the lensing magnification (typically $\mu\simeq$ 20) to characterise the physical and morphological properties of these LAE galaxies. From reconstructing the ultra-violet continuum emission, we find that the star-formation rates range from 0.3 to 8.5 M$_{\odot}$ yr$^{-1}$ and that the galaxies are typically composed of several compact and diffuse components, separated by 0.4 to 4 kpc. Moreover, they have peak star-formation rate intensities that range from 2.1 to 54.1 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$. These galaxies tend to be extended with major axis ranging from 0.2 to 1.8 kpc (median 561 pc), and with a median ellipticity of 0.49. This morphology is consistent with disk-like structures of star-formation for more than half of the sample. However, for at least two sources, we also find off-axis components that may be associated with mergers. Resolved kinematical information will be needed to confirm the disk-like nature and possible merger scenario for the LAEs in the sample.Comment: 19 pages, 7 figures, accepted for publication on MNRA

ALMA imaging of SDP.81 - I. A pixelated reconstruction of the far-infrared continuum emission

We present a sub-50 pc-scale analysis of the gravitational lens system SDP.81 at redshift 3.042 using Atacama Large Millimetre/submillimetre Array (ALMA) science verification data. We model both the mass distribution of the gravitational lensing galaxy and the pixelated surface brightness distribution of the background source using a novel Bayesian technique that fits the data directly in visibility space. We find the 1 and 1.3 mm dust emission to be magnified by a factor of u_tot = 17.6+/-0.4, giving an intrinsic total star-formation rate of 315+/-60 M_sol/yr and a dust mass of 6.4+/-1.5*10^8 M_sol. The reconstructed dust emission is found to be non-uniform, but composed of multiple regions that are heated by both diffuse and strongly clumped star-formation. The highest surface brightness region is a ~1.9*0.7 kpc disk-like structure, whose small extent is consistent with a potential size-bias in gravitationally lensed starbursts. Although surrounded by extended star formation, with a density of 20-30+/-10 M_sol/yr/kpc^2, the disk contains three compact regions with densities that peak between 120-190+/-20 M_sol/yr/kpc^2. Such star-formation rate densities are below what is expected for Eddington-limited star-formation by a radiation pressure supported starburst. There is also a tentative variation in the spectral slope of the different star-forming regions, which is likely due to a change in the dust temperature and/or opacity across the source.Comment: MNRAS accepted 2015 April 1

ALMA imaging of SDP.81 - II. A pixelated reconstruction of the CO emission lines

We present a sub-100 pc-scale analysis of the CO molecular gas emission and kinematics of the gravitational lens system SDP.81 at redshift 3.042 using Atacama Large Millimetre/submillimetre Array (ALMA) science verification data and a visibility-plane lens reconstruction technique. We find clear evidence for an excitation dependent structure in the unlensed molecular gas distribution, with emission in CO (5-4) being significantly more diffuse and structured than in CO (8-7). The intrinsic line luminosity ratio is r_8-7/5-4 = 0.30 +/- 0.04, which is consistent with other low-excitation starbursts at z ~ 3. An analysis of the velocity fields shows evidence for a star-forming disk with multiple velocity components that is consistent with a merger/post-coalescence merger scenario, and a dynamical mass of M(< 1.56 kpc) = 1.6 +/- 0.6 x 10^10 M_sol . Source reconstructions from ALMA and the Hubble Space Telescope show that the stellar component is offset from the molecular gas and dust components. Together with Karl G. Jansky Very Large Array CO (1-0) data, they provide corroborative evidence for a complex ~2 kpc-scale starburst that is embedded within a larger ~15 kpc structure.Comment: MNRAS accepted, 6th July 201

SHARP - I. A high-resolution multi-band view of the infra-red Einstein ring of JVAS B1938+666

We present new mass models for the gravitational lens system B1938+666, using multi-wavelength data acquired from Keck adaptive optics (AO) and Hubble Space Telescope (HST) observations. These models are the first results from the Strong-lensing at High Angular Resolution Program (SHARP), a project designed to study known quadruple-image and Einstein ring lenses using high-resolution imaging, in order to probe their mass distributions in unprecedented detail. Here, we specifically highlight differences between AO- and HST-derived lens models, finding that -- at least when the lens and source galaxies are both bright and red, and the system has a high degree of circular symmetry -- AO-derived models place significantly tighter constraints on model parameters. Using this improved precision, we infer important physical properties about the B1938+666 system, including the mass density slope of the lensing galaxy (gamma = 2.045), the projected dark matter mass fraction within the Einstein radius (M_dark/M_lens = 0.55), and the total magnification factor of the source galaxy (~ 13). Additionally, we measure an upper-limit constraint on luminous substructure (M_V > 16.2), based on the non-detection of bright satellite galaxies in all data sets. Finally, we utilize the improved image resolution of the AO data to reveal the presence of faint arcs outside of the primary Einstein ring. The positions and orientations of these arcs raise the intriguing possibility that B1938+666 has a second source galaxy, located at a more distant redshift. However, future work is needed to verify this hypothesis.Comment: 12 pages, 7 figures; Accepted for publication in MNRA

Inference of the Cold Dark Matter substructure mass function at z=0.2 using strong gravitational lenses

We present the results of a search for galaxy substructures in a sample of 11 gravitational lens galaxies from the Sloan Lens ACS Survey. We find no significant detection of mass clumps, except for a luminous satellite in the system SDSS J0956+5110. We use these non-detections, in combination with a previous detection in the system SDSS J0946+1006, to derive constraints on the substructure mass function in massive early-type host galaxies with an average redshift z ~ 0.2 and an average velocity dispersion of 270 km/s. We perform a Bayesian inference on the substructure mass function, within a median region of about 32 kpc squared around the Einstein radius (~4.2 kpc). We infer a mean projected substructure mass fraction $f = 0.0076^{+0.0208}_{-0.0052}$ at the 68 percent confidence level and a substructure mass function slope $\alpha$ < 2.93 at the 95 percent confidence level for a uniform prior probability density on alpha. For a Gaussian prior based on Cold Dark Matter (CDM) simulations, we infer $f = 0 .0064^{+0.0080}_{-0.0042}$ and a slope of $\alpha$ = 1.90$^{+0.098}_{-0.098}$ at the 68 percent confidence level. Since only one substructure was detected in the full sample, we have little information on the mass function slope, which is therefore poorly constrained (i.e. the Bayes factor shows no positive preference for any of the two models).The inferred fraction is consistent with the expectations from CDM simulations and with inference from flux ratio anomalies at the 68 percent confidence level.Comment: Accepted for publication on MNRAS, some typos corrected and some important references adde

Gravitational detection of a low-mass dark satellite at cosmological distance

The mass-function of dwarf satellite galaxies that are observed around Local Group galaxies substantially differs from simulations based on cold dark matter: the simulations predict many more dwarf galaxies than are seen. The Local Group, however, may be anomalous in this regard. A massive dark satellite in an early-type lens galaxy at z = 0.222 was recently found using a new method based on gravitational lensing, suggesting that the mass fraction contained in substructure could be higher than is predicted from simulations. The lack of very low mass detections, however, prohibited any constraint on their mass function. Here we report the presence of a 1.9 +/- 0.1 x 10^8 M_sun dark satellite in the Einstein-ring system JVAS B1938+666 at z = 0.881, where M_sun denotes solar mass. This satellite galaxy has a mass similar to the Sagittarius galaxy, which is a satellite of the Milky Way. We determine the logarithmic slope of the mass function for substructure beyond the local Universe to be alpha = 1.1^+0.6_-0.4, with an average mass-fraction of f = 3.3^+3.6_-1.8 %, by combining data on both of these recently discovered galaxies. Our results are consistent with the predictions from cold dark matter simulations at the 95 per cent confidence level, and therefore agree with the view that galaxies formed hierarchically in a Universe composed of cold dark matter.Comment: 25 pages, 7 figures, accepted for publication in Nature (19 January 2012

The X-shooter Lens Survey - II. Sample presentation and spatially resolved kinematics

We present the X-shooter Lens Survey (XLENS) data. The main goal of XLENS is to disentangle the stellar and dark matter content of massive early-type galaxies (ETGs), through combined strong gravitational lensing, dynamics and spectroscopic stellar population studies. The sample consists of 11 lens galaxies covering the redshift range from $0.1$ to $0.45$ and having stellar velocity dispersions between $250$ and $380\,\mathrm{km}\,\mathrm{s}^{-1}$. All galaxies have multi-band, high-quality HST imaging. We have obtained long-slit spectra of the lens galaxies with X-shooter on the VLT. We are able to disentangle the dark and luminous mass components by combining lensing and extended kinematics data-sets, and we are also able to precisely constrain stellar mass-to-light ratios and infer the value of the low-mass cut-off of the IMF, by adding spectroscopic stellar population information. Our goal is to correlate these IMF parameters with ETG masses and investigate the relation between baryonic and non-baryonic matter during the mass assembly and structure formation processes. In this paper we provide an overview of the survey, highlighting its scientific motivations, main goals and techniques. We present the current sample, briefly describing the data reduction and analysis process, and we present the first results on spatially resolved kinematics.Comment: Accepted for publication in MNRA