S2LET: A code to perform fast wavelet analysis on the sphere

We describe S2LET, a fast and robust implementation of the scale-discretised wavelet transform on the sphere. Wavelets are constructed through a tiling of the harmonic line and can be used to probe spatially localised, scale-depended features of signals on the sphere. The scale-discretised wavelet transform was developed previously and reduces to the needlet transform in the axisymmetric case. The reconstruction of a signal from its wavelets coefficients is made exact here through the use of a sampling theorem on the sphere. Moreover, a multiresolution algorithm is presented to capture all information of each wavelet scale in the minimal number of samples on the sphere. In addition S2LET supports the HEALPix pixelisation scheme, in which case the transform is not exact but nevertheless achieves good numerical accuracy. The core routines of S2LET are written in C and have interfaces in Matlab, IDL and Java. Real signals can be written to and read from FITS files and plotted as Mollweide projections. The S2LET code is made publicly available, is extensively documented, and ships with several examples in the four languages supported. At present the code is restricted to axisymmetric wavelets but will be extended to directional, steerable wavelets in a future release.Comment: 8 pages, 6 figures, version accepted for publication in A&A. Code is publicly available from http://www.s2let.or

Unbiased methods for removing systematics from galaxy clustering measurements

Measuring the angular clustering of galaxies as a function of redshift is a powerful method for extracting information from the three-dimensional galaxy distribution. The precision of such measurements will dramatically increase with ongoing and future wide-field galaxy surveys. However, these are also increasingly sensitive to observational and astrophysical contaminants. Here, we study the statistical properties of three methods proposed for controlling such systematics – template subtraction, basic mode projection, and extended mode projection – all of which make use of externally supplied template maps, designed to characterize and capture the spatial variations of potential systematic effects. Based on a detailed mathematical analysis, and in agreement with simulations, we find that the template subtraction method in its original formulation returns biased estimates of the galaxy angular clustering. We derive closed-form expressions that should be used to correct results for this shortcoming. Turning to the basic mode projection algorithm, we prove it to be free of any bias, whereas we conclude that results computed with extended mode projection are biased. Within a simplified setup, we derive analytical expressions for the bias and discuss the options for correcting it in more realistic configurations. Common to all three methods is an increased estimator variance induced by the cleaning process, albeit at different levels. These results enable unbiased high-precision clustering measurements in the presence of spatially varying systematics, an essential step towards realizing the full potential of current and planned galaxy surveys

A novel sampling theorem on the rotation group

We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by connecting the rotation group to the three-torus through a periodic extension. Our sampling theorem requires $4L^3$ samples to capture all of the information content of a signal band-limited at $L$, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as $O(L^4)$, compared to the naive scaling of $O(L^6)$. For the common case of a low directional band-limit $N$, complexity is reduced to $O(N L^3)$. Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for publication. Code available at http://www.sothree.or

No new cosmological concordance with massive sterile neutrinos

It has been claimed recently that massive sterile neutrinos could bring about a new concordance between observations of the cosmic microwave background, the large-scale structure of the Universe, and local measurements of the Hubble constant, H0. We demonstrate that this apparent concordance results from combining data sets which are in significant tension, even within this extended model, possibly indicating remaining systematic biases in the measurements. We further show that this tension remains when the cosmological model is further extended to include significant tensor modes, as suggested by the recent BICEP2 results. Using the Bayesian evidence, we show that the cold dark matter model with a cosmological constant is strongly favored over its neutrino extensions by various combinations of data sets. Robust data combinations yield stringent limits of ∑mν≲0.3  eV and meffν,sterile≲0.3  eV at 95% C.L. for the sum of active and sterile neutrinos, respectively

Accurate cosmology with galaxy and quasar surveys

Observations of the cosmic microwave background have led to a golden age of cosmology, where precise measurements can be confronted with predictions from cosmological models. Ongoing and future surveys of the distribution of galaxies will continue this revolution: they will enable us to test the laws of gravity, uncover the properties of dark energy and dark matter, and reinforce the connection to high-energy physics. However, current galaxy survey analyses are already limited by our ability to identify and treat observational systematics, and this problem will be even more pronounced in future experiments. Therefore, it is essential to develop novel methods to deal with these complications when testing cosmological models and searching for new physics. This is the focus of this thesis. Firstly, I will present measurements of primordial non-Gaussianity obtained from the clustering of quasars from the Sloan Digital Sky Survey. Primordial non-Gaussianity is a powerful probe of inflation, the leading theory of the initial conditions of the universe, but its effects on the distribution of quasars are mimicked by observational systematics. I will describe a framework to deal with these systematics and robustly measure primordial non-Gaussianity from the clustering of quasars. Secondly, I will present a new set of wavelet transforms on the sphere and the ball. These approaches are highly promising for analysing cosmological and geophysical data and dealing with their systematics in novel ways. Finally, I will examine the recent claims that extra massive neutrinos can resolve the tensions between cosmic microwave background, galaxy survey and supernova observations. I will demon- strate that this conclusion is premature since it is driven by the least robust data sets. Given the growing number of cosmological observables and their varied levels of robustness, combining data sets and dealing with such tensions will become critical in the near future

Second-Generation Curvelets on the Sphere

Curvelets are efficient to represent highly anisotropic signal content, such as a local linear and curvilinear structure. First-generation curvelets on the sphere, however, suffered from blocking artefacts. We present a new second-generation curvelet transform, where scale-discretized curvelets are constructed directly on the sphere. Scale-discretized curvelets exhibit a parabolic scaling relation, are well localized in both spatial and harmonic domains, support the exact analysis and synthesis of both scalar and spin signals, and are free of blocking artefacts. We present fast algorithms to compute the exact curvelet transform, reducing computational complexity from O(L5) to O(L3 log2 L) for signals band limited at L. The implementation of these algorithms is made publicly available. Finally, we present an illustrative application demonstrating the effectiveness of curvelets for representing directional curve-like features in natural spherical images

Wavelet reconstruction of E and B modes for CMB polarisation and cosmic shear analyses

We present new methods for mapping the curl-free (E-mode) and divergence-free (B-mode) components of spin 2 signals using spin directional wavelets. Our methods are equally applicable to measurements of the polarisation of the cosmic microwave background (CMB) and the shear of galaxy shapes due to weak gravitational lensing. We derive pseudo and pure wavelet estimators, where E-B mixing arising due to incomplete sky coverage is suppressed in wavelet space using scale- and orientation-dependent masking and weighting schemes. In the case of the pure estimator, ambiguous modes (which have vanishing curl and divergence simultaneously on the incomplete sky) are also cancelled. On simulations, we demonstrate the improvement (i.e., reduction in leakage) provided by our wavelet space estimators over standard harmonic space approaches. Our new methods can be directly interfaced in a coherent and computationally-efficient manner with component separation or feature extraction techniques that also exploit wavelets

Exact wavelets on the ball

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

Mapping dark matter on the celestial sphere with weak gravitational lensing

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 Vera Rubin Observatory’s Legacy Survey of Space and Time (LSST), and the Nancy Grace Roman Space Telescope, 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 the order of tens of percent, exceeding 50 per cent 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 science verification data to recover convergence maps directly on the celestial sphere

Red clump stars and Gaia: calibration of the standard candle using a hierarchical probabilistic model

Distances to individual stars in our own Galaxy are critical in order to piece together the nature of its velocity and spatial structure. Core helium burning red clump (RC) stars have similar luminosities, are abundant throughout the Galaxy and thus constitute good standard candles. We build a hierarchical probabilistic model to quantify the quality of RC stars as standard candles using parallax measurements from the first Gaia data release. A unique aspect of our methodology is to fully account for (and marginalize over) parallax, photometry and dust correction uncertainties, which lead to more robust results than standard approaches. We determine the absolute magnitude and intrinsic dispersion of the RC in 2MASS bands J, H, Ks, GaiaG band and WISE bands W1, W2, W3 and W4. We find that the absolute magnitude of the RC is −1.61 ± 0.01 (in Ks), +0.44 ± 0.01 (in G), −0.93 ± 0.01 (in J), −1.46 ± 0.01 (in H), −1.68 ± 0.02 (in W1), −1.69 ± 0.02 (in W2), −1.67 ± 0.02 (in W3) and −1.76 ± 0.01 mag (in W4). The mean intrinsic dispersion is ∼0.17 ± 0.03 mag across all bands (yielding a typical distance precision of ∼8 per cent). Thus RC stars are reliable and precise standard candles. In addition, we have also re-calibrated the zero-point of the absolute magnitude of the RC in each band, which provides a benchmark for future studies to estimate distances to RC stars. Finally, the parallax error shrinkage in the hierarchical model outlined in this work can be used to obtain more precise parallaxes than Gaia for the most distant RC stars across the Galaxy