Weak Gravitational Lensing Measurements Research Articles

Weighing the giants – IV. Cosmology and neutrino mass

We employ robust weak gravitational lensing measurements to improve cosmological constraints from measurements of the galaxy cluster mass function and its evolution, using X-ray selected clusters detected in the ROSAT All-Sky Survey. Our lensing analysis constrains the absolute mass scale of such clusters at the 8 per cent level, including both statistical and systematic uncertainties. Combining it with the survey data and X-ray follow-up observations, we find a tight constraint on a combination of the mean matter density and late-time normalization of the matter power spectrum, σ8(Ωm/0.3)0.17 = 0.81 ± 0.03, with marginalized, one-dimensional constraints of Ωm = 0.26 ± 0.03 and σ8 = 0.83 ± 0.04. For these two parameters, this represents a factor of 2 improvement in precision with respect to previous work, primarily due to the reduced systematic uncertainty in the absolute mass calibration provided by the lensing analysis. Our new results are in good agreement with constraints from cosmic microwave background (CMB) data, both Wilkinson Microwave Anisotropy Probe (WMAP) and Planck (plus WMAP polarization), under the assumption of a flat ΛCDM cosmology with minimal neutrino mass. Consequently, we find no evidence for non-minimal neutrino mass from the combination of cluster data with CMB, supernova and baryon acoustic oscillation measurements, regardless of which all-sky CMB data set is used (and independent of the recent claimed detection of B modes on degree scales). We also present improved constraints on models of dark energy (both constant and evolving), modifications of gravity, and primordial non-Gaussianity. Assuming flatness, the constraints for a constant dark energy equation of state from the cluster data alone are at the 15 per cent level, improving to ∼6 per cent when the cluster data are combined with other leading probes.

The dark matter haloes of moderate luminosity X-ray AGN as determined from weak gravitational lensing and host stellar masses

Understanding the relationship between galaxies hosting active galactic nuclei (AGN) and the dark matter halos in which they reside is key to constraining how black-hole fueling is triggered and regulated. Previous efforts have relied on simple halo mass estimates inferred from clustering, weak gravitational lensing, or halo occupation distribution modeling. In practice, these approaches remain uncertain because AGN, no matter how they are identified, potentially live a wide range of halo masses with an occupation function whose general shape and normalization are poorly known. In this work, we show that better constraints can be achieved through a rigorous comparison of the clustering, lensing, and cross-correlation signals of AGN hosts to a fiducial stellar-to-halo mass relation (SHMR) derived for all galaxies. Our technique exploits the fact that the global SHMR can be measured with much higher accuracy than any statistic derived from AGN samples alone. Using 382 moderate luminosity X-ray AGN at z<1 from the COSMOS field, we report the first measurements of weak gravitational lensing from an X-ray selected sample. Comparing this signal to predictions from the global SHMR, we find that, contrary to previous results, most X-ray AGN do not live in medium size groups ---nearly half reside in relatively low mass halos with Mh~10^12.5 Msun. The AGN occupation function is well described by the same form derived for all galaxies but with a lower normalization---the fraction of halos with AGN in our sample is a few percent. By highlighting the relatively "normal" way in which moderate luminosity X-ray AGN hosts occupy halos, our results suggest that the environmental signature of distinct fueling modes for luminous QSOs compared to moderate luminosity X-ray AGN is less obvious than previously claimed.

Optics ellipticity performance of an unobscured off-axis space telescope.

With the development of astronomy, more and more attention is paid to the survey of dark matter. Dark matter cannot be seen directly but can be detected by weak gravitational lensing measurement. Ellipticity is an important parameter used to define the shape of a galaxy. Galaxy ellipticity changes with weak gravitational lensing and nonideal optics. With our design of an unobscured off-axis telescope, we implement the simulation and calculation of optics ellipticity. With an accurate model of optics PSF, the characteristic of ellipticity is modeled and analyzed. It is shown that with good optical design, the full field ellipticity can be quite small. The spatial ellipticity change can be modeled by cubic interpolation with very high accuracy. We also modeled the ellipticity variance with time and analyzed the tolerance. It is shown that the unobscured off-axis telescope has good ellipticity performance and fulfills the requirement of dark matter survey.

Galaxy Halo Masses from Weak Gravitational Lensing

AbstractIn this review, I discuss the use of galaxy-galaxy weak lensing measurements to study the masses of dark matter halos in which galaxies reside. After summarizing how weak gravitational lensing measurements can be interpreted in terms of halo mass, I review measurements that were used to derive the relationship between optical galaxy mass tracers, such as stellar mass or luminosity, and dark matter halo mass. Measurements of galaxy-galaxy lensing from the past decade have led to increasingly tight constraints on the connection between dark matter halo mass and optical mass tracers, including both the mean relationships between these quantities and the intrinsic scatter between them. I also review some of the factors that can complicate analysis, such as the choice of modeling procedure, and choices made when dividing up samples of lens galaxies.

How the cosmic web induces intrinsic alignments of galaxies

AbstractIntrinsic alignments are believed to be a major source of systematics for future generation of weak gravitational lensing surveys like Euclid or LSST. Direct measurements of the alignment of the projected light distribution of galaxies in wide field imaging data seem to agree on a contamination at a level of a few per cent of the shear correlation functions, although the amplitude of the effect depends on the population of galaxies considered. Given this dependency, it is difficult to use dark matter-only simulations as the sole resource to predict and control intrinsic alignments. We report here estimates on the level of intrinsic alignment in the cosmological hydrodynamical simulation Horizon-AGN that could be a major source of systematic errors in weak gravitational lensing measurements. In particular, assuming that the spin of galaxies is a good proxy for their ellipticity, we show how those spins are spatially correlated and how they couple to the tidal field in which they are embedded. We will also present theoretical calculations that illustrate and qualitatively explain the observed signals.

On the probability distributions of ellipticity

In this paper we derive an exact full expression for the 2D probability distribution of the ellipticity of an object measured from data, only assuming Gaussian noise in pixel values. This is a generalisation of the probability distribution for the ratio of single random variables, that is well-known, to the multivariate case. This expression is derived within the context of the measurement of weak gravitational lensing from noisy galaxy images. We find that the third flattening, or epsilon-ellipticity, has a biased maximum likelihood but an unbiased mean; and that the third eccentricity, or normalised polarisation chi, has both a biased maximum likelihood and a biased mean. The very fact that the bias in the ellipticity is itself a function of the ellipticity requires an accurate knowledge of the intrinsic ellipticity distribution of the galaxies in order to properly calibrate shear measurements. We use this expression to explore strategies for calibration of biases caused by measurement processes in weak gravitational lensing. We find that upcoming weak lensing surveys like KiDS or DES require calibration fields of order of several square degrees and 1.2 magnitude deeper than the wide survey in order to correct for the noise bias. Future surveys like Euclid will require calibration fields of order 40 square degree and several magnitude deeper than the wide survey. We also investigate the use of the Stokes parameters to estimate the shear as an alternative to the ellipticity. We find that they can provide unbiased shear estimates at the cost of a very large variance in the measurement. The python code used to compute the distributions presented in the paper and to perform the numerical calculations are available on request.

Constraining cosmic string parameters with curl mode of CMB lensing

We present constraints on a cosmic string network with a measurement of weak gravitational lensing from CMB temperature map. The cosmic string network between observer and last scattering surface of CMB photons generates vector and/or tensor metric perturbations, and the deflection of CMB photons by these gravitational fields has curl mode which is not produced by the scalar metric perturbations. In this paper, we use the power spectrum of curl mode obtained from Planck to constrain the string tension, G\mu, and the reconnection probability, P. In demonstrating the parameter constraints with Planck curl mode, we also measure the lensing power spectrum from the Atacama Cosmology Telescope (ACT) 2008 season data, which have better angular resolution with lower instrumental noise on a much smaller chunk of the sky. Assuming P=1, the upper bound on tension is G\mu = 6.6\times 10^{-5} with 2\sigma (95% C.L.), using curl mode from Planck, which is weaker than that from the small-scale temperature power spectrum. For small values of P, however, the constraint from curl mode becomes tighter compared to that from temperature power spectrum. For P\lsim 10^{-2}, we obtain the constraint on the combination of the string parameters as G\mu P^{-1}\leq 3.4\times 10^{-5} at more than 2\sigma (95% C.L.).

Initial Results from a Laboratory Emulation of Weak Gravitational Lensing Measurements

Weak gravitational lensing observations are a key science driver for the NASA Wide Field Infrared Survey Telescope (WFIRST). To validate the performance of the WFIRST infrared detectors, we have performed a laboratory emulation of weak gravitational lensing measurements. Our experiments used a custom precision projector system to image a target mask composed of a grid of pinholes, emulating stellar point sources, onto a 1.7 μm cut-off Teledyne HgCdTe/H2RG detector. We used a 0.88 μm LED illumination source and f/22 pupil stop to produce undersampled point spread functions similar to those expected from WFIRST. We also emulated the WFIRST image reconstruction strategy, using the image combination (IMCOM) algorithm to derive oversampled images from dithered, undersampled input images. We created shear maps for this data and computed shear correlation functions to mimic a real weak lensing analysis. After removing only second-order polynomial fits to the shear maps, we found that the correlation functions could be reduced to O(10-6). This places a conservative upper limit on the detector-induced bias to the correlation function (under our test conditions). This bias is two orders of magnitude lower than the expected weak lensing signal. Restricted to scales relevant to dark energy analyses (sky separations >0.5′), the bias is O(10-7)—comparable to the requirement for future weak lensing missions to avoid biasing cosmological parameter estimates. Our experiment will need to be upgraded and repeated under different configurations to fully characterize the shape measurement performance of WFIRST IR detectors.

THE BURST CLUSTER: DARK MATTER IN A CLUSTER MERGER ASSOCIATED WITH THE SHORT GAMMA-RAY BURST, GRB 050509B

We have identified a merging galaxy cluster with evidence of two distinct sub-clusters. The X-ray and optical data suggest that the subclusters are moving away from each other after closest approach. This cluster merger was discovered from observations of the well localized short-duration gamma-ray burst (GRB), GRB 050509B. The Swift/Burst Alert Telescope (BAT) source position is coincident with a cluster of galaxies ZwCl 1234.0+02916. The subsequent Swift/X-Ray Telescope (XRT) localization of the X-ray afterglow found the GRB coincident with 2MASX J12361286+2858580, a giant red elliptical galaxy in the cluster. Deep multi-epoch optical images were obtained to constrain the evolution of the GRB afterglow, including a 27480s exposure in the F814W band with Hubble Space Telescope Advanced Camera for Surveys (ACS), among the deepest imaging ever obtained towards a known galaxy cluster in a single passband. We perform a weak gravitational lensing analysis, including mapping the total mass distribution of the merger system. Combined with Chandra X-ray Observatory and Swift/XRT observations, we investigate the dynamical state of the merger to probe the nature of the dark matter component. Our weak gravitational lensing measurements reveal a separation of the X-ray centroid of the western subcluster from the center of the mass and galaxy light distributions, similar to that of the famous "Bullet cluster". We conclude that the "Burst cluster" is another candidate merger system for determining the nature of dark matter and for studying the environment of short GRBs. We discuss connections between the cluster dynamical state and/or matter composition and compact object mergers, the leading model for the origin of short GRBs. Finally, we present results from a weak lensing survey based on archival Very Large Telescope (VLT) images in the areas of 5 other short GRBs.

Origins of weak lensing systematics, and requirements on future instrumentation (or knowledge of instrumentation)

The first half of this paper explores the origin of systematic biases in the measurement of weak gravitational lensing. Compared to previous work, we expand the investigation of PSF instability and fold in for the first time the effects of non-idealities in electronic imaging detectors and imperfect galaxy shape measurement algorithms. Together, these now explain the additive A(l) and multiplicative M(l) systematics typically reported in current lensing measurements. We find that overall performance is driven by a product of a telescope/camera's *absolute performance*, and our *knowledge about its performance*. The second half of this paper propagates any residual shear measurement biases through to their effect on cosmological parameter constraints. Fully exploiting the statistical power of Stage IV weak lensing surveys will require additive biases A<1.8e-12 and multiplicative biases M<4.0e-3. These can be allocated between individual budgets in hardware, calibration data and software, using results from the first half of the paper. If instrumentation is stable and well-calibrated, we find extant shear measurement software from GREAT10 already meet requirements on galaxies detected at S/N=40. Averaging over a population of galaxies with a realistic distribution of sizes, it also meets requirements for a 2D cosmic shear analysis from space. If used on fainter galaxies or for 3D cosmic shear tomography, existing algorithms would need calibration on simulations to avoid introducing bias at a level similar to the statistical error. Requirements on hardware and calibration data are discussed in more detail in a companion paper. Our analysis is intentionally general, but is specifically being used to drive the hardware and ground segment performance budget for the design of the European Space Agency's recently-selected Euclid mission.

Weak gravitational lensing with deimos

We introduce a novel method for weak-lensing measurements, which is based on a mathematically exact deconvolution of the moments of the apparent brightness distribution of galaxies from the telescope's point spread function (PSF). No assumptions on the shape of the galaxy or the PSF are made. The (de)convolution equations are exact for unweighted moments only, while in practice a compact weight function needs to be applied to the noisy images to ensure that the moment measurement yields significant results. We employ a Gaussian weight function, whose centroid and ellipticity are iteratively adjusted to match the corresponding quantities of the source. The change of the moments caused by the application of the weight function can then be corrected by considering higher order weighted moments of the same source. Because of the form of the deconvolution equations, even an incomplete weighting correction leads to an excellent shear estimation if galaxies and PSF are measured with a weight function of identical size. We demonstrate the accuracy and capabilities of this new method in the context of weak gravitational lensing measurements with a set of specialized tests and show its competitive performance on the GREAT08 Challenge data. A complete c++ implementation of the method can be requested from the authors.

Bispectrum covariance in the flat-sky limit

<i>Aims. <i/>To probe cosmological fields beyond the Gaussian level, three-point statistics can be used, all of which are related to the bispectrum. Hence, measurements of CMB anisotropies, galaxy clustering, and weak gravitational lensing alike have to rely upon an accurate theoretical background concerning the bispectrum and its noise properties. If only small portions of the sky are considered, it is often desirable to perform the analysis in the flat-sky limit. We aim at a formal, detailed derivation of the bispectrum covariance in the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane approach.<i>Methods. <i/>We define an unbiased estimator of the bispectrum, which takes the average over the overlap of annuli in Fourier space, and compute its full covariance. The outcome of our formalism is compared to the flat-sky spherical harmonic approximation in terms of the covariance, the behavior under parity transformations, and the information content. We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding.<i>Results. <i/>Contrary to foregoing work, we find a difference by a factor of two between the covariances of the Fourier-plane and the spherical harmonic approach. We argue that this discrepancy can be explained by the differing behavior with respect to parity. However, in an exemplary analysis it is demonstrated that the Fisher information of both formalisms agrees to high accuracy. Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol, which leads to convenient approximation formulae for the covariances of both approaches.

CARS: The CFHTLS-Archive-Research Survey

Weak gravitational lensing (WL) has been established as one of the most promising probes of cosmology. So far, most studies have exploited the shear effect of WL leading to coherent distortions of galaxy shapes. But WL also introduces coherent magnifications. We want to detect this cosmic magnification effect in large samples of high-redshift galaxies selected from the Deep part of the CFHTLS. Lyman-break galaxies (LBGs) at z=2.5-5, are used as a background sample and are cross-correlated to foreground lens galaxies selected by photo-z's. The signals of LBGs in different magnitude bins are compared to predictions from WL theory. An optimally weighted correlation function is estimated by taking into account the slope of external LBG luminosity functions. For the first time, we detect cosmic magnification in a sample of normal galaxies. These background sources are also the ones with the highest redshifts so far used for WL measurements. The amplitude and angular dependence of the cross-correlation functions agree well with theoretical expectations and the lensing signal is detected with high significance. In particular, we detect an anti-correlation between faint LBGs and foreground galaxies which cannot be caused by redshift overlap. (abridged)

Dark-energy constraints and correlations with systematics from CFHTLS weak lensing, SNLS supernovae Ia and WMAP5

<i>Aims. <i/>We combine measurements of weak gravitational lensing from the CFHTLS-Wide survey, supernovae Ia from CFHT SNLS and CMB anisotropies from WMAP5 to obtain joint constraints on cosmological parameters, in particular, the dark-energy equation-of-state parameter <i>w<i/>. We assess the influence of systematics in the data on the results and look for possible correlations with cosmological parameters.<i>Methods. <i/>We implemented an MCMC algorithm to sample the parameter space of a flat CDM model with a dark-energy component of constant <i>w<i/>. Systematics in the data are parametrised and included in the analysis. We determine the influence of photometric calibration of SNIa data on cosmological results by calculating the response of the distance modulus to photometric zero-point variations. The weak lensing data set is tested for anomalous field-to-field variations and a systematic shape measurement bias for high-redshift galaxies.<i>Results. <i/>Ignoring photometric uncertainties for SNLS biases cosmological parameters by at most 20% of the statistical errors, using supernovae alone; the parameter uncertainties are underestimated by 10%. The weak-lensing field-to-field variance between 1 deg<sup>2<sup/>-MegaCam pointings is 5-15% higher than predicted from <i>N<i/>-body simulations. We find no bias in the lensing signal at high redshift, within the framework of a simple model, and marginalising over cosmological parameters. Assuming a systematic underestimation of the lensing signal, the normalisation increases by up to 8%. Combining all three probes we obtain at 68% confidence ( at 95%), including systematic errors. Our results are therefore consistent with the cosmological constant <i>Λ<i/>. Systematics in the data increase the error bars by up to 35%; the best-fit values change by less than 0.15<i>σ<i/>.

Diagnosing Space Telescope Misalignment and Jitter Using Stellar Images

ABSTRACTAccurate knowledge of the telescope’s point-spread function (PSF) is essential for the weak gravitational lensing measurements that hold great promise for cosmological constraints. For space telescopes, the PSF may vary with time due to thermal drifts in the telescope structure, and/or due to jitter in the spacecraft pointing (ground-based telescopes have additional sources of variation). We describe and simulate a procedure for using the images of the stars in each exposure to determine the misalignment and jitter parameters, and reconstruct the PSF at any point in that exposure’s field of view. As a case study, the simulation uses the design of the Supernova Acceleration Probe (SNAP) telescope. Stellar-image data in a typical exposure determines secondary-mirror positions as precisely as 20 nm. The PSF ellipticities and size, which are the quantities of interest for weak lensing are determined to 4.0 × 10-4 and 2.2 × 10-4 accuracies, respectively, in each exposure, sufficient to meet weak-lensing requirements. We show that, for the case of a space telescope, the PSF estimation errors scale inversely with the square root of the total number of photons collected from all the usable stars in the exposure.

Improved optical mass tracer for galaxy clusters calibrated using weak lensing measurements

We develop an improved mass tracer for clusters of galaxies from optically observed parameters, and calibrate the mass relation using weak gravitational lensing measurements. We employ a sample of ~ 13,000 optically-selected clusters from the Sloan Digital Sky Survey (SDSS) maxBCG catalog, with photometric redshifts in the range 0.1-0.3. The optical tracers we consider are cluster richness, cluster luminosity, luminosity of the brightest cluster galaxy (BCG), and combinations of these parameters. We measure the weak lensing signal around stacked clusters as a function of the various tracers, and use it to determine the tracer with the least amount of scatter. We further use the weak lensing data to calibrate the mass normalization. We find that the best mass estimator for massive clusters is a combination of cluster richness, N_{200}, and the luminosity of the brightest cluster galaxy, L_{BCG}: M_{200\bar{\rho}} = (1.27 +/- 0.08) (N_{200}/20)^{1.20 +/- 0.09} (L_{BCG}/\bar{L}_{BCG}(N_{200}))^{0.71 +/- 0.14} \times 10^{14} h^{-1} M_sun, where $\bar{L}_{BCG}(N_{200})$ is the observed mean BCG luminosity at a given richness. This improved mass tracer will enable the use of galaxy clusters as a more powerful tool for constraining cosmological parameters.

The Dark UNiverse Explorer (DUNE): proposal to ESA’s cosmic vision

The Dark UNiverse Explorer (DUNE) is a wide-field space imager whose primary goal is the study of dark energy and dark matter with unprecedented precision. For this purpose, DUNE is optimised for the measurement of weak gravitational lensing but will also provide complementary measurements of baryonic accoustic oscillations, cluster counts and the Integrated Sachs Wolfe effect. Immediate auxiliary goals concern the evolution of galaxies, to be studied with unequalled statistical power, the detailed structure of the Milky Way and nearby galaxies, and the demographics of Earth-mass planets. DUNE is an Medium-class mission which makes use of readily available components, heritage from other missions, and synergy with ground based facilities to minimise cost and risks. The payload consists of a 1.2m telescope with a combined visible/NIR field-of-view of 1 deg^2. DUNE will carry out an all-sky survey, ranging from 550 to 1600nm, in one visible and three NIR bands which will form a unique legacy for astronomy. DUNE will yield major advances in a broad range of fields in astrophysics including fundamental cosmology, galaxy evolution, and extrasolar planet search. DUNE was recently selected by ESA as one of the mission concepts to be studied in its Cosmic Vision programme.

Shapelets “Multiple Multipole” Shear Measurement

The measurement of weak gravitational lensing is currently limited to a precision of 10% by instabilities in galaxy shape measurement techniques and uncertainties in their calibration. The potential of large, on-going and future cosmic shear surveys will only be realised with the development of more accurate image analysis methods. We present a description of several possible shear measurement methods using the linear “shapelets” decomposition. Shapelets provides a complete reconstruction of any galaxy image, including higher-order shape moments that can be used to generalise the KSB method to arbitrary order. Many independent shear estimators can then be formed for each object, using linear combinations of shapelet coefficients. These estimators can be treated separately, to improve their overall calibration; or combined in more sophisticated ways, to eliminate various instabilities and a calibration bias. We apply several methods to simulated astronomical images containing a known input shear, and demonstrate the dramatic improvement in shear recovery using shapelets. A complete IDL software package to perform image analysis and manipulation in shapelet space can be downloaded from www.astro.caltech.edu/~rjm/shapelets/.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Weak lensing with COMBO-17: estimation and removal of intrinsic alignments

We estimate and remove the contamination of weak gravitational lensing measurements by the intrinsic alignment of close pairs of galaxies. We do this by investigating both the aperture mass B mode statistic, and the shear correlations of close and distant pairs of galaxies. These can be used to quantify non-lensing effects in weak lensing surveys. We reanalyse the {COMBO-17} survey, and study published results from the Red-sequence Cluster Survey and the {VIRMOS-DESCART} survey, concluding that the intrinsic alignment effect is at the lower end of the range of theoretical predictions. We also revisit this theoretical issue, and show that misalignment of baryon and halo angular momenta may be an important effect which can reduce the intrinsic ellipticity correlations estimated from numerical simulations to the level that we and the {SuperCOSMOS} survey observe. We re-examine the cosmological parameter estimation from the {COMBO-17} survey, using the shear correlation function, and now marginalizing over the Hubble constant. Assuming no evolution in galaxy clustering, and marginalizing over the intrinsic alignment signal, we find the mass clustering amplitude is reduced by 0.03 to ??8(??m/0.27)0.6= 0.71 +/- 0.11, where ??m is the matter density parameter. We consider the forthcoming {SuperNova/Acceleration} Probe {(SNAP)} wide weak lensing survey, and the {Canada-France-Hawaii} Telescope Legacy wide synoptic survey; we expect them to be contaminated on scales {\textgreater}1 arcmin by intrinsic alignments at the level of {\textasciitilde}1 and {\textasciitilde}2 per cent, respectively. Division of the {SNAP} survey for lensing tomography significantly increases the contamination in the lowest redshift bin to {\textasciitilde}7 per cent and possibly higher. Removal of the intrinsic alignment effect by the downweighting of nearby galaxy pairs will therefore be vital for {SNAP.}

ChandraObservations of the Lensing Cluster EMSS 1358+6245: Implications for Self‐interacting Dark Matter

We present Chandra observations of EMSS 1358+6245, a relaxed cooling flow cluster of galaxies at z = 0.328. We employ a new deprojection technique to construct temperature, gas, and dark matter profiles. We confirm the presence of cool gas in the cluster core, and our deprojected temperature profile for the hot component is isothermal over 30 kpc < r < 0.8 Mpc. Fitting the mass profile to a Navarro-Frenk-White model yields rs = 153 kpc and c = 8.4. We find good agreement between our dark matter profile and weak gravitational lensing measurements. We place an upper limit of 42 kpc (90% confidence limit) on the size of any constant-density core. We compare this result to recent simulations and place a conservative upper limit on the dark matter particle-scattering cross section of 0.1 cm2 g-1. This limit implies that the cross section must be velocity-dependent if the relatively shallow core mass profiles of dwarf galaxies are a direct result of dark matter self-interaction.