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Register a new user3D weak lensing with spin wavelets on the ball
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We construct the spin flaglet transform, a wavelet transform to analyze spin signals in three dimensions. Spin flaglets can probe signal content localized simultaneously in space and frequency and, moreover, are separable so that their angular and radial properties can be controlled independently. They are particularly suited to analyzing of cosmological observations such as the weak gravitational lensing of galaxies. Such observations have a unique 3D geometrical setting since they are natively made on the sky, have spin angular symmetries, and are extended in the radial direction by additional distance or redshift information. Flaglets are constructed in the harmonic space defined by the Fourier-Laguerre transform, previously defined for scalar functions and extended here to signals with spin symmetries. Thanks to various sampling theorems, both the Fourier-Laguerre and flaglet transforms are theoretically exact when applied to bandlimited signals. In other words, in numerical computations the only loss of information is due to the finite representation of floating point numbers. We develop a 3D framework relating the weak lensing power spectrum to covariances of flaglet coefficients. We suggest that the resulting novel flaglet weak lensing estimator offers a powerful alternative to common 2D and 3D approaches to accurately capture cosmological information. While standard weak lensing analyses focus on either real or harmonic space representations (i.e., correlation functions or Fourier-Bessel power spectra, respectively), a wavelet approach inherits the advantages of both techniques, where both complicated sky coverage and uncertainties associated with the physical modeling of small scales can be handled effectively. Our codes to compute the Fourier-Laguerre and flaglet transforms are made publicly available.
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We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial half-line using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel three-dimensional decomposition which we call the Fourier-Laguerre transform. We relate this new transform to the well-known Fourier-Bessel decomposition and show that band-limitedness in the Fourier-Laguerre basis is a sufficient condition to compute the Fourier-Bessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the Fourier-Laguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and Fourier-Laguerre spaces and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floating-point precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example.
One of the most powerful techniques to study the dark sector of the Universe is weak gravitational lensing. In practice, to infer the reduced shear, weak lensing measures galaxy shapes, which are the consequence of both the intrinsic ellipticity of the sources and of the integrated gravitational lensing effect along the line of sight. Hence, a very large number of galaxies is required in order to average over their individual properties and to isolate the weak lensing cosmic shear signal. If this `shape noise can be reduced, significant advances in the power of a weak lensing surveys can be expected. This paper describes a general method for extracting the probability distributions of parameters from catalogues of data using Voronoi cells, which has several applications, and has synergies with Bayesian hierarchical modelling approaches. This allows us to construct a probability distribution for the variance of the intrinsic ellipticity as a function of galaxy property using only photometric data, allowing a reduction of shape noise. As a proof of concept the method is applied to the CFHTLenS survey data. We use this approach to investigate trends of galaxy properties in the data and apply this to the case of weak lensing power spectra.
Convergence maps of the integrated matter distribution are a key science result from weak gravitational lensing surveys. To date, recovering convergence maps has been performed using a planar approximation of the celestial sphere. However, with the increasing area of sky covered by dark energy experiments, such as Euclid, the Large Synoptic Survey Telescope (LSST), and the Wide Field Infrared Survey Telescope (WFIRST), this assumption will no longer be valid. We recover convergence fields on the celestial sphere using an extension of the Kaiser-Squires estimator to the spherical setting. Through simulations we study the error introduced by planar approximations. Moreover, we examine how best to recover convergence maps in the planar setting, considering a variety of different projections and defining the local rotations that are required when projecting spin fields such as cosmic shear. For the sky coverages typical of future surveys, errors introduced by projection effects can be of order tens of percent, exceeding 50% in some cases. The stereographic projection, which is conformal and so preserves local angles, is the most effective planar projection. In any case, these errors can be avoided entirely by recovering convergence fields directly on the celestial sphere. We apply the spherical Kaiser-Squires mass-mapping method presented to the public Dark Energy Survey (DES) science verification data to recover convergence maps directly on the celestial sphere.
A new spin wavelet transform on the sphere is proposed to analyse the polarisation of the cosmic microwave background (CMB), a spin $pm 2$ signal observed on the celestial sphere. The scalar directional scale-discretised wavelet transform on the sphere is extended to analyse signals of arbitrary spin. The resulting spin scale-discretised wavelet transform probes the directional intensity of spin signals. A procedure is presented using this new spin wavelet transform to recover E- and B-mode signals from partial-sky observations of CMB polarisation.
Weak gravitational lensing measurements are traditionally made at optical wavelengths where many highly resolved galaxy images are readily available. However, the Square Kilometre Array (SKA) holds great promise for this type of measurement at radio wavelengths owing to its greatly increased sensitivity and resolution over typical radio surveys. The key to successful weak lensing experiments is in measuring the shapes of detected sources to high accuracy. In this document we describe a simulation pipeline designed to simulate radio images of the quality required for weak lensing, and will be typical of SKA observations. We provide as input, images with realistic galaxy shapes which are then simulated to produce images as they would have been observed with a given radio interferometer. We exploit this pipeline to investigate various stages of a weak lensing experiment in order to better understand the effects that may impact shape measurement. We first show how the proposed SKA1-Mid array configurations perform when we compare the (known) input and output ellipticities. We then investigate how making small changes to these array configurations impact on this input-outut ellipticity comparison. We also demonstrate how alternative configurations for SKA1-Mid that are smaller in extent, and with a faster survey speeds produce similar performance to those originally proposed. We then show how a notional SKA configuration performs in the same shape measurement challenge. Finally, we describe ongoing efforts to utilise our simulation pipeline to address questions relating to how applicable current (mostly originating from optical data) shape measurement techniques are to future radio surveys. As an alternative to such image plane techniques, we lastly discuss a shape measurement technique based on the shapelets formalism that reconstructs the source shapes directly from the visibility data.
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