Precision matrix expansion - efficient use of numerical simulations in estimating errors on cosmological parameters

Computing the inverse covariance matrix (or precision matrix) of large data vectors is crucial in weak lensing (and multi-probe) analyses of the large scale structure of the universe. Analytically computed covariances are noise-free and hence straightforward to invert, however the model approximations might be insufficient for the statistical precision of future cosmological data. Estimating covariances from numerical simulations improves on these approximations, but the sample covariance estimator is inherently noisy, which introduces uncertainties in the error bars on cosmological parameters and also additional scatter in their best fit values. For future surveys, reducing both effects to an acceptable level requires an unfeasibly large number of simulations. In this paper we describe a way to expand the true precision matrix around a covariance model and show how to estimate the leading order terms of this expansion from simulations. This is especially powerful if the covariance matrix is the sum of two contributions, $\smash{\mathbf{C} = \mathbf{A}+\mathbf{B}}$, where $\smash{\mathbf{A}}$ is well understood analytically and can be turned off in simulations (e.g. shape-noise for cosmic shear) to yield a direct estimate of $\smash{\mathbf{B}}$. We test our method in mock experiments resembling tomographic weak lensing data vectors from the Dark Energy Survey (DES) and the Large Synoptic Survey Telecope (LSST). For DES we find that $400$ N-body simulations are sufficient to achive negligible statistical uncertainties on parameter constraints. For LSST this is achieved with $2400$ simulations. The standard covariance estimator would require >$10^5$ simulations to reach a similar precision. We extend our analysis to a DES multi-probe case finding a similar performance.Comment: 14 pages, submitted to mnra

Theoretical Aspects of Cosmic Shear and its Ability to constrain Cosmological Parameters

In the last decade weak gravitational lensing by the large-scale structure of the Universe, also called cosmic shear, has become an important tool to constrain cosmological parameters. Despite this success there remain observational and theoretical issues which must be solved to fully investigate the cosmological information of future cosmic shear data. In this PhD thesis I address several of these issues. As a first project I use ray-tracing simulations to compare and optimize cosmic shear data vectors. More precisely, I develop a new data vector by combining two cosmic shear measures, namely the aperture mass dispersion (2ap>) and the two-point correlation function (2PCF). The new data vector has higher information content than a 2ap> data vector and is more robust against contamination compared to a 2PCF data vector. In my second project on cosmic shear measures I examine the ring statistics, which is the most recently developed cosmic shear measure. The ring statistics improves on deficits in the E- and B-mode decomposition of commonly used cosmic shear measures, e.g. the aperture mass dispersion. I optimize the signal strength of the ring statistics, develop an expression for its covariance, and compare its information content to that of the aperture mass dispersion. I find that the ring statistics' data points are less correlated and that a ring statistics' data vector contains more information on cosmological parameters. Finally, I employ the ring statistics to measure a cosmic shear signal from data of the Canada-France-Hawaii Telescope Legacy Survey and constrain cosmological parameters. As a result I obtain s8 (Om / 0.25)= 0.82+0.02-0.04. As a third project I examine cosmic shear covariances and their impact on cosmological parameter constraints. Using simulated data I investigate the cosmology-dependence of these covariances and develop improved methods for a likelihood analysis, which take the cosmology-dependence into account. In addition to the cosmology-dependence, I examine how the shear fields' non-Gaussianity affects cosmic shear covariances and the parameter constraints (in particular for dark energy parameters). I quantify the impact of non-Gaussianity as a function of angular scale and derive a fit-formula for the calculation of non-Gaussian covariances from Gaussian ones

Testing dark energy paradigms with weak gravitational lensing

Any theory invoked to explain cosmic acceleration predicts consistency relations between the expansion history, structure growth, and all related observables. Currently there exist high-quality measurements of the expansion history from Type Ia supernovae, the cosmic microwave background temperature and polarization spectra, and baryon acoustic oscillations. We can use constraints from these datasets to predict what future probes of structure growth should observe. We apply this method to predict what range of cosmic shear power spectra would be expected if we lived in a LambdaCDM universe, with or without spatial curvature, and what results would be inconsistent and therefore falsify the model. Though predictions are relaxed if one allows for an arbitrary quintessence equation of state $-1\le w(z)\le 1$, we find that any observation that rules out LambdaCDM due to excess lensing will also rule out all quintessence models, with or without early dark energy. We further explore how uncertainties in the nonlinear matter power spectrum, e.g. from approximate fitting formulas such as Halofit, warm dark matter, or baryons, impact these limits.Comment: 12 pages, 11 figures, submitted to PR

Core or Cusps: The Central Dark Matter Profile of a Strong Lensing Cluster with a Bright Central Image at Redshift 1

We report on SPT-CLJ2011-5228, a giant system of arcs created by a cluster at z = 1.06. The arc system is notable for the presence of a bright central image. The source is a Lyman break galaxy at z_s= 2.39 and the mass enclosed within the Einstein ring of radius 14 arcsec is ~10^(14.2) M⊙. We perform a full reconstruction of the light profile of the lensed images to precisely infer the parameters of the mass distribution. The brightness of the central image demands that the central total density profile of the lens be shallow. By fitting the dark matter as a generalized Navarro–Frenk–White profile—with a free parameter for the inner density slope—we find that the break radius is 270^(+48)_(-76) kpc, and that the inner density falls with radius to the power −0.38 ± 0.04 at 68% confidence. Such a shallow profile is in strong tension with our understanding of relaxed cold dark matter halos; dark matter-only simulations predict that the inner density should fall as r^(-1). The tension can be alleviated if this cluster is in fact a merger; a two-halo model can also reconstruct the data, with both clumps (density varying as r^(-0.8) and r^(-1.0)) much more consistent with predictions from dark matter-only simulations. At the resolution of our Dark Energy Survey imaging, we are unable to choose between these two models, but we make predictions for forthcoming Hubble Space Telescope imaging that will decisively distinguish between them

2D-FFTLog: Efficient computation of real space covariance matrices for galaxy clustering and weak lensing

Accurate covariance matrices for two-point functions are critical for inferring cosmological parameters in likelihood analyses of large-scale structure surveys. Among various approaches to obtaining the covariance, analytic computation is much faster and less noisy than estimation from data or simulations. However, the transform of covariances from Fourier space to real space involves integrals with two Bessel integrals, which are numerically slow and easily affected by numerical uncertainties. Inaccurate covariances may lead to significant errors in the inference of the cosmological parameters. In this paper, we introduce a 2D-FFTLog algorithm for efficient, accurate and numerically stable computation of non-Gaussian real space covariances for both 3D and projected statistics. The 2D-FFTLog algorithm is easily extended to perform real space bin-averaging. We apply the algorithm to the covariances for galaxy clustering and weak lensing for a Dark Energy Survey Year 3-like and a Rubin Observatory's Legacy Survey of Space and Time Year 1-like survey, and demonstrate that for both surveys, our algorithm can produce numerically stable angular bin-averaged covariances with the flat sky approximation, which are sufficiently accurate for inferring cosmological parameters. The code CosmoCov for computing the real space covariances with or without the flat sky approximation is released along with this paper.Comment: MNRAS accepted; 13 pages, 3 figures, 2 tables; fixed a typo in Eq.41; 2DFFTLog code available at https://github.com/xfangcosmo/2DFFTLog ; 3x2pt covariance code CosmoCov at https://github.com/CosmoLike/CosmoCo