Bayesian reconstruction of the ... - University of...

04/12/2007 KICP, Chicago 1

Francisco-Shu Kitaura

Kitaura & Ensslin (2007): arXiv:0705.0429v2Kitaura (2007): LMU

Bayesian reconstruction of the cosmological large-scale structure

KICP, Chicago 04/12/2007

04/12/2007 KICP, Chicago 2

Francisco-Shu Kitaura

Torsten Ensslin, Simon White Jens Jasche, Benjamin Wandelt, Jeremy Blaizot

Bayesian reconstruction of the cosmological large-scale structure

Chicago 04/12/2007

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ARGO: Algorithm for the Reconstruction of the Galaxy-traced Overdensities

Johannes Hevelius, 1690

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Motivation

Goal: cosmography -> obtain 3-Dimensional reconstruction of the dark matter distribution

Input data: galaxy positions from galaxy redshift surveys(for the moment)

joint reconstruction of: + velocity field + gravitational potential

+ power spectrum

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Motivation

Goal: cosmography -> obtain 3-Dimensional reconstruction of the dark matter distribution

Input data: galaxy positions from galaxy redshift surveys(for the moment)

joint reconstruction of: + velocity field

+ gravitational potential + power spectrum

-> + redshift distortion corrections + reconstruction of initial conditions + constrained simulations

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Motivation

Goal: cosmography -> obtain 3-Dimensional reconstruction of the dark matter distribution

Input data: galaxy positions from galaxy redshift surveys(for the moment)

joint reconstruction of: + velocity field + gravitational potential

+ power spectrum

-> + weak signal detections (ISW,SZ) + dark energy studies

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Motivation

Goal: cosmography -> obtain 3-Dimensional reconstruction ofthe dark matter distribution

Input data: galaxy positions from galaxy redshift surveys(for the moment)

joint reconstruction of: + velocity field + gravitational potential

+ power spectrum-> + BAOs -> dark energy studies + cosmological parameter estimation

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Motivation

Goal: cosmography -> obtain 3-Dimensional reconstruction of the dark matter distribution

Input data: galaxy positions from galaxy redshift surveys

continuous DM-density field?

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

Instrumental/physical responseBlurring of the atmosphere/telescopeBlurring of the pixel windowRedshift distortions operatorGalaxy-DM bias

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

Selection function

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

Mask, Window function

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

Structure function in the noise terme.g.: shot noise

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model:

Random noise termwhite noise/ colored noise

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Inverse problem

Observed data d (m-dim vector) Signal s (n-dim vector)

Signal degradation data model: -> + n>>m: rank defficient system -> infinite solutions + direct inversion R^-1 d amplifies the statistical noise

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Bayesian framework

Bayes theorem:

posterior = Likelihood*prior/evidence

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Bayesian framework

1) Define the likelihood

2) Define the prior

3) Maximize

4) Sample the joint PDF: MCMC

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Non-informative priors Informative priors Flat Entropic Gaussian Poissonian

Gaussian COBE-filter (Maisinger et al '97) Wiener-filter DOD (Janssen&Gulkis '92) (Hobson '98) (Rybicki&Press '02) (Hobson&McLachlan '03)

(Sutton & Wandelt '06) (Zaroubi '95) MAGIC (Wandelt '03)

ARGO: MEMG ARGO: WIENER

Poissonian Richardson-Lucy ARGO: MEMP ARGO: GAPMAP ('72-'74)

Inverse Gamma MAGIC (Wandelt '03)

ARGO: PS

Modified Gaussian CMB (Verde '03) (log-normal PDF) SZ (Pierpaoli '05)

PS (Percival '05)

LIK

ELI

HO

OD

PRIOR

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Bayesian framework

2) Define the prior

+ informative prior: Gaussian prior with

+ Gaussian likelihood: WIENER-filter

IRAS Zaroubi et al '952DF Erdogdu et al '04,'06

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Bayesian framework

2) Define the prior

+ informative prior: Gaussian prior with

+ Gaussian likelihood: WIENER-filter

IRAS Zaroubi et al '952DF Erdogdu et al '04,'06

-> requires inversion of huge matrices !

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Bayesian framework

2) Define the prior

+ informative prior: Gaussian prior with

+ Gaussian likelihood: WIENER-filter

IRAS Zaroubi et al '952DF Erdogdu et al '04,'06

-> requires inversion of huge matrices ! Can we do this step efficiently?

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Krylov space method

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Operator formalism

How to apply matrix A ?

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Operator formalism

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Numerical tests

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Structured noise

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Preconditioning

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Deblurring

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Poissonian noise

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Selection function

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Window effect

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Window effect

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Single Bayesian reconstruction step

Linear reconstructionEffective redshift distortions treatment

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Linear power-spectrum

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Bayesian framework

4) Sample the joint PDF: MCMC

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Bayesian framework

4) Sample the joint PDF: MCMC

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

-> Is this feasible ?

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Bayesian framework

4) Sample the joint PDF: MCMC -> Is this feasible ?

-> Gibbs sampling methods MAGIC (Wandelt '03)

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Bayesian framework

4) Sample the joint PDF: MCMC

velocity field reconstruction:

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Linear velocity field: bulk flow

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Non-linear component: virialized structures

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Non-linear power-spectrum

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Bayesian framework

4) Sample the joint PDF: MCMC

Power-spectrum reconstruction:

1) Define the likelihood2) Define the prior3) Maximize4) Sample the joint PDF:

MCMC

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Appendix

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Redshift distortions

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Mask, window function

SDSS observed patches of sky in galactic coordinates

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Numerical method: Iterative inversion schemes

-> Use difference schemes with the non-stationary equation:

Jacobi iteration schemeSteepest DescentKrylov schemes: Conjugate GradientsNewton-Raphson methodLandweber-Fridman scheme

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Numerical method: Iterative inversion schemes

Quadratic form:

Steepest Descent convergence behaviour:

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Numerical method: Iterative inversion schemes

Quadratic form:

Steepest Descent convergence behaviour:

-> the searching directions are constantly repeated !

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Krylov space method

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Krylov space method

Krylov vs Steepest Descent:

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Redshift distortions: correlation function

contours of redshift-space two-point correlation function (2PCF) for galaxies in the SDSS (colour lines)

black lines: isotropic distribution of galaxies.

The rp and π directions are respectively perpendicular and parallel to the line of sight.

Cheng Li, Y.P. Jing, Guinevere Kauffmann, Gerhard Boerner, Simon D.M. White, F.Z. Cheng 2006

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2DF Erdogdu et al '04

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