Wiener Reconstruction of the Large-Scale Structure

Abstract

The formalism of Wiener ltering is developed here for the purpose of reconstructing the large scale structure of the universe from noisy, sparse and incomplete data. The method is based on a linear minimum variance solution, given data and an assumed prior model which speci es the covariance matrix of the eld to be reconstructed. While earlier applications of the Wiener ler have focused on estimation, namely suppressing the noise in the measured quantities, we extend the method here to perform both prediction and dynamical reconstruction. The Wiener lter is used to predict the values of unmeasured quantities, such as the density eld in un-sampled regions of space, or to deconvolve blurred data. The method is developed, within the context of linear gravitational instability theory, to perform dynamical reconstruction of one eld which is dynamically related to some other observed eld. This is the case, for example, in the reconstruction of the real space galaxy distribution from its redshift distribution or the prediction of the radial velocity eld from the observed density eld.When the eld to be reconstructed is a Gaussian random eld, such as the primordial perturbation eld predicted by the canonical model of cosmology, the Wiener lter can 1 be pushed to its fullest potential. In such a case the Wiener estimator coincides with the Bayesian estimator designed to maximize the posterior probability. The Wiener lter can be also derived by assuming a quadratic regularization function, in analogy with thè Maximum Entropy' method. The mean eld obtained by the minimal variance solution can be supplemented with constrained realizations of the Gaussian eld to create random realizations of the residual from the mean. Subject Headings: cosmology: large scale structure of universe; analytical{ methods: data analysis

Bayesian 3D velocity field reconstruction with virbius

“…Several attempts have been made at reconstructing the density field directly from distance data. For example one can note the POTENT method (Bertschinger & Dekel 1989;Dekel et al 1990Dekel et al , 1999, the Wiener filter approach (Zaroubi et al 1995), or the Unbiased Minimum Variance algorithm (Zaroubi 2002). Additionally, the procedure to derive the power spectrum of the velocity field is relatively complex and prone to the same aforementioned systematics, though there have been some early attempts at measuring it (Jaffe & Kaiser 1995;Zaroubi et al 1997;Macaulay et al 2011Macaulay et al , 2012.…”

Goodness-of-fit analysis of the Cosmicflows-2 data base of velocities

“…The scope and implications of the present section exceed and reach beyond the specific aim of the paper but it should be considered as one of our key results. The methodology of Zaroubi et al (1995) is the basis for extracting information from large scale structure observational data, where the underlying fields are assumed to be Gaussian. This methodology assumes a cosmological prior in terms of a power spectrum which allows an inference of velocity and density fields.…”

“…This enables a powerful and rigorous approach to the problem of the reconstruction of the LSS -both the density and velocity fields -from a given database of peculiar velocities. Given a database of peculiar velocities and assuming a prior cosmological model which postulates that the underlying primordial perturbation field is Gaussian of a given power spectrum, the LSS is readily reconstructed by means of the Wiener filter (WF) and constrained simulations (CRs) that sample the scatter around the mean (WF) field (Hoffman & Ribak 1991;Zaroubi et al 1995;Hoffman 2001). This WF/CRs Bayesian framework provides an appealing framework for the reconstruction of the LSS from velocities data within the realm of the standard model of cosmology (Zaroubi et al 1999(Zaroubi et al , 2001(Zaroubi et al , 2002Courtois et al 2012;Tully et al 2014;Pomarède et al 2015).…”

“…The relevant methodology is straightforward and well established. One needs to calculate the likelihood function of the data given the model -namely the probability of the occurrence of the data within the framework of the assumed model (Zaroubi et al 1995;Jaffe & Kaiser 1995;Hoffman 2001;Press et al 2007). The likelihood function establishes the goodness-of-fit (GoF) of the data by the model.…”

MAPCUMBA: A fast iterative multi-grid map-making algorithm for CMB experiments

“…In that case the particular solution turns out to be the Wiener filtering solution (Zaroubi et al 1995;Bouchet & Gispert 1996;Tegmark & Efstathiou 1996;Bouchet & Gispert 1998):…”