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

Gravitational Lens Time Delays in CDM

In standard CDM halo models, the time delay of a gravitational lens is determined by the cold baryon mass fraction, f, of the visible galaxy relative to the overall halo. The observed time delays in PG1115+080, SBS1520+530, B1600+434 and HE2149-2745 give Hubble constants consistent with the HST Key Project value of H0=72+/-8 km/s Mpc only if f>0.2 (1-sided 68% confidence), which is larger than the upper bound of fmax=Omega_b/Omega_0=0.15+/-0.05 estimated from the CMB. If all available baryons cool and f=fmax then the time delays imply H0=65+/-6 km/s Mpc (95% confidence). If local inventories of cold baryons, f=0.013/h70, are correct, then H0=52+/-6 km/s Mpc and the halo parameters closely match isothermal mass models. Isothermal models are also consistent with strong and weak lens studies, stellar dynamics and X-ray observations on these scales, while significantly more centrally concentrated models are not. There is a a conflict between gravitational lens time delays, the local distance scale and standard CDM halo models.Comment: Submitted to ApJ. 22 pages, 7 figure

Galaxy-Scale Strong Lensing Tests of Gravity and Geometric Cosmology: Constraints and Systematic Limitations

Galaxy-scale strong gravitational lenses with measured stellar velocity dispersions allow a test of the weak-field metric on kiloparsec scales and a geometric measurement of the cosmological distance-redshift relation, provided that the mass-dynamical structure of the lensing galaxies can be independently constrained to a sufficient degree. We combine data on 53 galaxy-scale strong lenses from the Sloan Lens ACS Survey with a well-motivated fiducial set of lens-galaxy parameters to find (1) a constraint on the post-Newtonian parameter gamma = 1.01 +/- 0.05 and (2) a determination of Omega_Lambda = 0.75 +/- 0.17 under the assumption of a flat universe. These constraints assume that the underlying observations and priors are free of systematic error. We evaluate the sensitivity of these results to systematic uncertainties in (1) total mass-profile shape, (2) velocity anisotropy, (3) light-profile shape, and (4) stellar velocity dispersion. Based on these sensitivities, we conclude that while such strong-lens samples can in principle provide an important tool for testing general relativity and cosmology, they are unlikely to yield precision measurements of gamma and Omega_Lambda unless the properties of the lensing galaxies are independently constrained with substantially greater accuracy than at present.Comment: 8 pages, 5 figures; Accepted to Ap

Hierarchical Bayesian inference of the Initial Mass Function in Composite Stellar Populations

The initial mass function (IMF) is a key ingredient in many studies of galaxy formation and evolution. Although the IMF is often assumed to be universal, there is continuing evidence that it is not universal. Spectroscopic studies that derive the IMF of the unresolved stellar populations of a galaxy often assume that this spectrum can be described by a single stellar population (SSP). To alleviate these limitations, in this paper we have developed a unique hierarchical Bayesian framework for modelling composite stellar populations (CSPs). Within this framework we use a parameterized IMF prior to regulate a direct inference of the IMF. We use this new framework to determine the number of SSPs that is required to fit a set of realistic CSP mock spectra. The CSP mock spectra that we use are based on semi-analytic models and have an IMF that varies as a function of stellar velocity dispersion of the galaxy. Our results suggest that using a single SSP biases the determination of the IMF slope to a higher value than the true slope, although the trend with stellar velocity dispersion is overall recovered. If we include more SSPs in the fit, the Bayesian evidence increases significantly and the inferred IMF slopes of our mock spectra converge, within the errors, to their true values. Most of the bias is already removed by using two SSPs instead of one. We show that we can reconstruct the variable IMF of our mock spectra for signal-to-noise ratios exceeding $\sim$75.Comment: Accepted for publication in MNRAS, 16 pages, 8 figure

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

The Sloan Lens ACS Survey. I. A Large Spectroscopically Selected Sample of Massive Early-Type Lens Galaxies

The Sloan Lens ACS (SLACS) Survey is an efficient Hubble Space Telescope Snapshot imaging survey for new galaxy-scale strong gravitational lenses. The targeted lens candidates are selected spectroscopically from within the Sloan Digital Sky Survey (SDSS) database of galaxy spectra for having multiple nebular emission lines at a redshift significantly higher than that of the SDSS target galaxy. In this paper, we present a catalog of 19 newly discovered gravitational lenses, along with 9 other observed candidate systems that are either possible lenses, non-lenses, or non-detections. The survey efficiency is thus >=68%. We also present Gemini and Magellan IFU data for 9 of the SLACS targets, which further support the lensing interpretation. A new method for the effective subtraction of foreground galaxy images to reveal faint background features is presented. We show that the SLACS lens galaxies have colors and ellipticities typical of the spectroscopic parent sample from which they are drawn (SDSS luminous red galaxies and quiescent main-sample galaxies), but are somewhat brighter and more centrally concentrated. Several explanations for the latter bias are suggested. The SLACS survey provides the first statistically significant and homogeneously selected sample of bright early-type lens galaxies, furnishing a powerful probe of the structure of early-type galaxies within the half-light radius. The high confirmation rate of lenses in the SLACS survey suggests consideration of spectroscopic lens discovery as an explicit science goal of future spectroscopic galaxy surveys (abridged).Comment: ApJ, in press. 20 pages, numerous figures, uses emulateapj. Replaced to include full-resolution spectro figures. Version with full-resolution imaging figures available at http://www.cfa.harvard.edu/~abolton/slacs1_hires.pdf (PDF) or at http://www.cfa.harvard.edu/~abolton/slacs1_hires.ps.gz (PS). Additional SLACS survey info at http://www.slacs.or

The initial mass function of early-type galaxies

We determine an absolute calibration of the initial mass function (IMF) of early-type galaxies, by studying a sample of 56 gravitational lenses identified by the SLACS Survey. Under the assumption of standard Navarro, Frenk & White dark matter halos, a combination of lensing, dynamical, and stellar population synthesis models is used to disentangle the stellar and dark matter contribution for each lens. We define an "IMF mismatch" parameter \alpha=M*(L+D)/M*(SPS) as the ratio of stellar mass inferred by a joint lensing and dynamical models (M*(L+D)) to the current stellar mass inferred from stellar populations synthesis models (M*(SPS)). We find that a Salpeter IMF provides stellar masses in agreement with those inferred by lensing and dynamical models (=0.00+-0.03+-0.02), while a Chabrier IMF underestimates them (=0.25+-0.03+-0.02). A tentative trend is found, in the sense that \alpha appears to increase with galaxy velocity dispersion. Taken at face value, this result would imply a non universal IMF, perhaps dependent on metallicity, age, or abundance ratios of the stellar populations. Alternatively, the observed trend may imply non-universal dark matter halos with inner density slope increasing with velocity dispersion. While the degeneracy between the two interpretations cannot be broken without additional information, the data imply that massive early-type galaxies cannot have both a universal IMF and universal dark matter halos.Comment: 10 pages 4 figures. Resubmitted to ApJ taking into account referee's comment

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