Karhunen‐Loeve Eigenvalue Problems in Cosmology: How Should We Tackle Large Data Sets?

Tegmark, Max; Taylor, Andy; Heavens, Alan

doi:10.1086/303939

Cited by 1,004 publications

(1,163 citation statements)

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“…We closely follow the derivation of the Fisher matrix presented in Tegmark et al (1997). A comma notation is used to indicate derivatives with respect to parameters.…”

Section: Discussionmentioning

confidence: 99%

“…Since T is invertible, the data in x and y contains the same amount of information about the parameters. Accordingly, the Fisher matrix is also invariant under this transformation (Tegmark et al 1997), which is easily demonstrated by inserting (26) into (25). However, in the case of nulling the transformation (19) to the new data vector Π (i) ( ) depends on the cosmological parameters one aims at determining because the elements of T are composed of comoving distances.…”

Section: Fisher Matrix Formalismmentioning

confidence: 99%

“…In the following analysis we will make use of the Fisher matrix formalism (see Tegmark et al 1997 for details) to obtain parameter constraints. Probing the likelihood locally around its maximum, it is computationally much cheaper than a full likelihood analysis and thus useful for error estimates for a large set of models.…”

Section: Fisher Matrix Formalismmentioning

confidence: 99%

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The removal of shear-ellipticity correlations from the cosmic shear signal

Joachimi¹,

Schneider²

2009

A&A

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Aims. Cosmic shear, the gravitational lensing on cosmological scales, is regarded as one of the most powerful probes for revealing the properties of dark matter and dark energy. To fully utilize its potential, one has to be able to control systematic effects down to below the level of the statistical parameter errors. Particularly worrisome in this respect is the intrinsic alignment of galaxies, causing considerable parameter biases via correlations between the intrinsic ellipticities of galaxies and the gravitational shear, which mimic lensing. Since our understanding of the underlying processes of intrinsic alignment is still poor, purely geometrical methods are required to control this systematic. In an earlier work we proposed a nulling technique that downweights this systematic, only making use of its well-known redshift dependence. We assess the practicability of nulling, given realistic conditions on photometric redshift information. Methods. For several simplified intrinsic alignment models and a wide range of photometric redshift characteristics, we calculate an average bias before and after nulling. Modifications of the technique are introduced to optimize the bias removal and minimize the information loss by nulling. We demonstrate that one of the presented versions of nulling is close to optimal in terms of bias removal, given the high quality of photometric redshifts. Although the nulling weights depend on cosmology, being composed of comoving distances, we show that the technique is robust against an incorrect choice of cosmological parameters when calculating the weights. Moreover, general aspects such as the behavior of the Fisher matrix under parameter-dependent transformations and the range of validity of the bias formalism are discussed in an appendix. Results. Given excellent photometric redshift information, i.e. at least 10 bins with a dispersion σ ph 0.03, a negligible fraction of catastrophic outliers, and precise knowledge about the bin-wise redshift distributions as characterized by a scatter of 0.001 or less on the median redshifts, one version of nulling is capable of reducing the shear-intrinsic ellipticity contamination by at least a factor of 100. Alternatively, we describe a robust nulling variant which suppresses the systematic signal by about 10 for a very broad range of photometric redshift configurations, provided basic information about σ ph in each of 10 photometric redshift bins is available. Irrespective of the photometric redshift quality, a loss of statistical power is inherent to nulling, which amounts to a decrease of the order 50% in terms of our figure of merit under conservative assumptions.

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“…We closely follow the derivation of the Fisher matrix presented in Tegmark et al (1997). A comma notation is used to indicate derivatives with respect to parameters.…”

Section: Discussionmentioning

confidence: 99%

Section: Fisher Matrix Formalismmentioning

confidence: 99%

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The removal of shear-ellipticity correlations from the cosmic shear signal

Joachimi¹,

Schneider²

2009

A&A

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“…(16), (18) and the finite difference derivatives for Eq. (17) from the simulated WL maps -one can use the Fisher matrix formalism ( [67], and see [68] and [69] for comprehensive reviews of this application) to compute parameter constraints. In fact, at least four groups have followed this approach recently for weak lensing simulations, some with redshift tomography [53][54][55]70].…”

Section: B Parameter Estimation and Constraintsmentioning

confidence: 99%

Probing cosmology with weak lensing Minkowski functionals

et al. 2012

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In this paper, we show that Minkowski Functionals (MFs) of weak gravitational lensing (WL) convergence maps contain significant non-Gaussian, cosmology-dependent information. To do this, we run a large suite of cosmological ray-tracing N-body simulations to create mock weak WL convergence maps, and study the cosmological information content of MFs derived from these maps. Our suite consists of 80 independent 512 3 N-body runs, covering seven different cosmologies, varying three cosmological parameters Ωm, w, and σ8 one at a time, around a fiducial ΛCDM model. In each cosmology, we use ray-tracing to create a thousand pseudo-independent 12 deg 2 convergence maps, and use these in a Monte Carlo procedure to estimate the joint confidence contours on the above three parameters. We include redshift tomography at three different source redshifts zs = 1, 1.5, 2, explore five different smoothing scales θG = 1, 2, 3, 5, 10 arcmin, and explicitly compare and combine the MFs with the WL power spectrum. We find that the MFs capture a substantial amount of information from non-Gaussian features of convergence maps, i.e. beyond the power spectrum. The MFs are particularly well suited to break degeneracies and to constrain the dark energy equation of state parameter w (by a factor of ≈ three better than from the power spectrum alone). The non-Gaussian information derives partly from the one-point function of the convergence (through V0, the "area" MF), and partly through non-linear spatial information (through combining different smoothing scales for V0, and through V1 and V2, the boundary length and genus MFs, respectively). In contrast to the power spectrum, the best constraints from the MFs are obtained only when multiple smoothing scales are combined.

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“…Our original motivation for applying a PCA (following Vogeley & Szalay 1996 ;Tegmark, Taylor, & Heavens 1997) was to allow optimal compression of the data into the modes that are most important for estimating the parameters we wish to evaluate, with the aim to reduce the computational cost associated with inverting huge correlation matrices and to improve the results given an inaccurate correlation matrix. In the current paper we use a PCA for two other purposes.…”

Section: Principal Component Analysismentioning

confidence: 99%

Cosmological Density and Power Spectrum from Peculiar Velocities: Nonlinear Corrections and Principal Component Analysis

Silberman¹,

Dekel²,

Eldar³

et al. 2001

ApJ

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We allow for nonlinear e †ects in the likelihood analysis of galaxy peculiar velocities and obtain D35% lower values for the cosmological density parameter and for the amplitude of mass density ) m Ñuctuations This result is obtained under the assumption that the power spectrum in the linear p 8 ) m 0.6. regime is of the Ñat "CDM model (h \ 0.65, n \ 1, COBE normalized) with only as a free parameter. ) m Since the likelihood is driven by the nonlinear regime, we "" break ÏÏ the power spectrum at (h~1 k b D 0.2 Mpc)~1 and Ðt a power law at This allows for independent matching of the nonlinear behavior k [ k b . and an unbiased Ðt in the linear regime. The analysis assumes Gaussian Ñuctuations and errors and a linear relation between velocity and density. Tests using mock catalogs that properly simulate nonlinear e †ects demonstrate that this procedure results in a reduced bias and a better Ðt. We Ðnd for the Mark III and SFI data and 0.37^0.09, respectively, with and ) m \ 0.32^0.06 p 8 ) m 0.6 \ 0.49^0.06 0.63^0.08, in agreement with constraints from other data. The quoted 90% errors include distance errors and cosmic variance, for Ðxed values of the other parameters. The improvement in the likelihood due to the nonlinear correction is very signiÐcant for Mark III and moderately signiÐcant for SFI.When allowing deviations from "CDM, we Ðnd an indication for a wiggle in the power spectrum : an excess near k D 0.05 (h~1 Mpc)~1 and a deÐciency at k D 0.1 (h~1 Mpc)~1, or a "" cold Ñow.ÏÏ This may be related to the wiggle seen in the power spectrum from redshift surveys and the second peak in the cosmic microwave background (CMB) anisotropy.A s2 test applied to modes of a principal component analysis (PCA) shows that the nonlinear procedure improves the goodness of Ðt and reduces a spatial gradient that was of concern in the purely linear analysis. The PCA allows us to address spatial features of the data and to evaluate and Ðne-tune the theoretical and error models. It demonstrates in particular that the models used are appropriate for the cosmological parameter estimation performed. We address the potential for optimal data compression using PCA.

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Karhunen‐Loeve Eigenvalue Problems in Cosmology: How Should We Tackle Large Data Sets?

Cited by 1,004 publications

References 36 publications

The removal of shear-ellipticity correlations from the cosmic shear signal

The removal of shear-ellipticity correlations from the cosmic shear signal

Probing cosmology with weak lensing Minkowski functionals

Cosmological Density and Power Spectrum from Peculiar Velocities: Nonlinear Corrections and Principal Component Analysis

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