Least Squares Two-Point Function Estimation

The standard estimator for the two-point function of a homogeneous and isotropic random field is a special case of a larger class of least squares estimators that interpolate the function values. Using a different interpolation scheme, two-point function values can be estimated at specific distances, instead of the binned averages.Comment: 3 pages, 1 figur

An unbiased estimator for the ellipticity from image moments

An unbiased estimator for the ellipticity of an object in a noisy image is given in terms of the image moments. Three assumptions are made: i) the pixel noise is normally distributed, although with arbitrary covariance matrix, ii) the image moments are taken about a fixed centre, and iii) the point-spread function is known. The relevant combinations of image moments are then jointly normal and their covariance matrix can be computed. A particular estimator for the ratio of the means of jointly normal variates is constructed and used to provide the unbiased estimator for the ellipticity. Furthermore, an unbiased estimate of the covariance of the new estimator is also given.Comment: 4 pages, accepted by MNRASL; v2 contains explicit covariance matrix for moment

Generalised model-independent characterisation of strong gravitational lenses II: Transformation matrix between multiple images

(shortened) We determine the transformation matrix T that maps multiple images with resolved features onto one another and that is based on a Taylor-expanded lensing potential close to a point on the critical curve within our model-independent lens characterisation approach. From T, the same information about the critical curve at fold and cusp points is derived as determined by the quadrupole moment of the individual images as observables. In addition, we read off the relative parities between the images, so that the parity of all images is determined, when one is known. We compare all retrievable ratios of potential derivatives to the actual ones and to those obtained by using the quadrupole moment as observable for two and three image configurations generated by a galaxy-cluster scale singular isothermal ellipse. We conclude that using the quadrupole moments as observables, the properties of the critical curve at the cusp points are retrieved to higher accuracy, at the fold points to lower accuracy, and the ratios of second order potential derivatives to comparable accuracy. We show that the approach using ratios of convergences and reduced shear is equivalent to ours close to the critical curve but yields more accurate results and is more robust because it does not require a special coordinate system like the approach using potential derivatives. T is determined by mapping manually assigned reference points in the images onto each other. If the assignment of reference points is subject to measurement uncertainties under noise, we find that the confidence intervals of the lens parameters can be as large as the values, when the uncertainties are larger than one pixel. Observed multiple images with resolved features are more extended than unresolved ones, so that higher order moments should be taken into account to improve the reconstruction.Comment: 13 pages, 12 figures, submitted to Astronomy & Astrophysics, comments welcom

Moment-Based Ellipticity Measurement as a Statistical Parameter Estimation Problem

We show that galaxy ellipticity estimation for weak gravitational lensing with unweighted image moments reduces to the problem of measuring a combination of the means of three independent normal random variables. Under very general assumptions, the intrinsic image moments of sources can be recovered from observations including effects such as the point-spread function and pixellation. Gaussian pixel noise turns these into three jointly normal random variables, the means of which are algebraically related to the ellipticity. We show that the random variables are approximately independent with known variances, and provide an algorithm for making them exactly independent. Once the framework is developed, we derive general properties of the ellipticity estimation problem, such as the signal-to-noise ratio, a generic form of an ellipticity estimator, and Cram\'er-Rao lower bounds for an unbiased estimator. We then derive the unbiased ellipticity estimator using unweighted image moments. We find that this unbiased estimator has a poorly behaved distribution and does not converge in practical applications, but demonstrates how to derive and understand the behaviour of new moment-based ellipticity estimators.Comment: 11 pages, 7 figures; v2 matches accepted version with minor change

Model-independent and model-based local lensing properties of CL0024+1654 from multiply-imaged galaxies

We investigate to which precision local magnification ratios, $\mathcal{J}$, ratios of convergences, $f$, and reduced shears, $g = (g_{1}, g_{2})$, can be determined model-independently for the five resolved multiple images of the source at $z_\mathrm{s}=1.675$ in CL0024. We also determine if a comparison to the respective results obtained by the parametric modelling program Lenstool and by the non-parametric modelling program Grale can detect biases in the lens models. For these model-based approaches we additionally analyse the influence of the number and location of the constraints from multiple images on the local lens properties determined at the positions of the five multiple images of the source at $z_\mathrm{s}=1.675$. All approaches show high agreement on the local values of $\mathcal{J}$, $f$, and $g$. We find that Lenstool obtains the tightest confidence bounds even for convergences around one using constraints from six multiple image systems, while the best Grale model is generated only using constraints from all multiple images with resolved brightness features and adding limited small-scale mass corrections. Yet, confidence bounds as large as the values themselves can occur for convergences close to one in all approaches. Our results are in agreement with previous findings, supporting the light-traces-mass assumption and the merger hypothesis for CL0024. Comparing the three different approaches allows to detect modelling biases. Given that the lens properties remain approximately constant over the extension of the image areas covered by the resolvable brightness features, the model-independent approach determines the local lens properties to a comparable precision but within less than a second. (shortened)Comment: 22 pages, published in A&A 612 A17, comments welcom

Lensed: a code for the forward reconstruction of lenses and sources from strong lensing observations

Robust modelling of strong lensing systems is fundamental to exploit the information they contain about the distribution of matter in galaxies and clusters. In this work, we present Lensed, a new code which performs forward parametric modelling of strong lenses. Lensed takes advantage of a massively parallel ray-tracing kernel to perform the necessary calculations on a modern graphics processing unit (GPU). This makes the precise rendering of the background lensed sources much faster, and allows the simultaneous optimisation of tens of parameters for the selected model. With a single run, the code is able to obtain the full posterior probability distribution for the lens light, the mass distribution and the background source at the same time. Lensed is first tested on mock images which reproduce realistic space-based observations of lensing systems. In this way, we show that it is able to recover unbiased estimates of the lens parameters, even when the sources do not follow exactly the assumed model. Then, we apply it to a subsample of the SLACS lenses, in order to demonstrate its use on real data. The results generally agree with the literature, and highlight the flexibility and robustness of the algorithm.Comment: v2: major revision; accepted by MNRAS; lens reconstruction code available at http://glenco.github.io/lensed

Zooming into the Cosmic Horseshoe: new insights on the lens profile and the source shape

The gravitational lens SDSS J1148+1930, also known as the Cosmic Horseshoe, is one of the biggest and of the most detailed Einstein rings ever observed. We use the forward reconstruction method implemented in the lens fitting code Lensed to investigate with great detail the properties of the lens and of the background source. We model the lens with different mass distributions, focusing in particular on the determination of the slope of the dark matter component. The inherent degeneracy between the lens slope and the source size can be broken when we can isolate separate components of each lensed image, as in this case. For an elliptical power law model, $\kappa(r) \sim r^{-t}$, the results favour a flatter-than-isothermal slope with a maximum-likelihood value t = 0.08. Instead, when we consider the contribution of the baryonic matter separately, the maximum-likelihood value of the slope of the dark matter component is t = 0.31 or t = 0.44, depending on the assumed Initial Mass Function. We discuss the origin of this result by analysing in detail how the images and the sources change when the slope t changes. We also demonstrate that these slope values at the Einstein radius are not inconsistent with recent forecast from the theory of structure formation in the LambdaCDM model.Comment: 13 pages, 9 figures, accepted for publication in MNRA

Weak lensing of large scale structure in the presence of screening

A number of alternatives to general relativity exhibit gravitational screening in the non-linear regime of structure formation. We describe a set of algorithms that can produce weak lensing maps of large scale structure in such theories and can be used to generate mock surveys for cosmological analysis. By analysing a few basic statistics we indicate how these alternatives can be distinguished from general relativity with future weak lensing surveys.Comment: 25 pages, 7 figures, accepted by JCAP. v2: references updat