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

04/12/2007 KICP, Chicago 3

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

Johannes Hevelius, 1690

04/12/2007 KICP, Chicago 4

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

04/12/2007 KICP, Chicago 5

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

04/12/2007 KICP, Chicago 6

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

04/12/2007 KICP, Chicago 7

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

04/12/2007 KICP, Chicago 8

Motivation

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

Input data: galaxy positions from galaxy redshift surveys

continuous DM-density field?

04/12/2007 KICP, Chicago 9

Inverse problem

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

04/12/2007 KICP, Chicago 10

Inverse problem

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

Signal degradation data model:

04/12/2007 KICP, Chicago 11

Inverse problem

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

Signal degradation data model:

04/12/2007 KICP, Chicago 12

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

04/12/2007 KICP, Chicago 13

Inverse problem

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

Signal degradation data model:

Selection function

04/12/2007 KICP, Chicago 14

Inverse problem

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

Signal degradation data model:

Mask, Window function

04/12/2007 KICP, Chicago 15

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

04/12/2007 KICP, Chicago 16

Inverse problem

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

Signal degradation data model:

Random noise termwhite noise/ colored noise

04/12/2007 KICP, Chicago 17

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

04/12/2007 KICP, Chicago 18

Bayesian framework

Bayes theorem:

posterior = Likelihood*prior/evidence

04/12/2007 KICP, Chicago 19

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

04/12/2007 KICP, Chicago 20

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

04/12/2007 KICP, Chicago 21

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

04/12/2007 KICP, Chicago 22

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

04/12/2007 KICP, Chicago 23

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

04/12/2007 KICP, Chicago 24

Krylov space method

04/12/2007 KICP, Chicago 25

Operator formalism

How to apply matrix A ?

04/12/2007 KICP, Chicago 26

Operator formalism

04/12/2007 KICP, Chicago 27

Numerical tests

04/12/2007 KICP, Chicago 28

Structured noise

04/12/2007 KICP, Chicago 29

Preconditioning

04/12/2007 KICP, Chicago 30

Deblurring

04/12/2007 KICP, Chicago 31

Poissonian noise

04/12/2007 KICP, Chicago 32

Selection function

04/12/2007 KICP, Chicago 33

Window effect

04/12/2007 KICP, Chicago 34

Window effect

04/12/2007 KICP, Chicago 35

04/12/2007 KICP, Chicago 36

04/12/2007 KICP, Chicago 37

Single Bayesian reconstruction step

Linear reconstructionEffective redshift distortions treatment

04/12/2007 KICP, Chicago 38

04/12/2007 KICP, Chicago 39

Linear power-spectrum

04/12/2007 KICP, Chicago 40

Bayesian framework

4) Sample the joint PDF: MCMC

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

MCMC

04/12/2007 KICP, Chicago 41

Bayesian framework

4) Sample the joint PDF: MCMC

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

MCMC

-> Is this feasible ?

04/12/2007 KICP, Chicago 42

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

04/12/2007 KICP, Chicago 43

Bayesian framework

4) Sample the joint PDF: MCMC

velocity field reconstruction:

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

MCMC

04/12/2007 KICP, Chicago 44

Linear velocity field: bulk flow

04/12/2007 KICP, Chicago 45

04/12/2007 KICP, Chicago 46

04/12/2007 KICP, Chicago 47

04/12/2007 KICP, Chicago 48

Non-linear component: virialized structures

04/12/2007 KICP, Chicago 49

04/12/2007 KICP, Chicago 50

04/12/2007 KICP, Chicago 51

04/12/2007 KICP, Chicago 52

04/12/2007 KICP, Chicago 53

Non-linear power-spectrum

04/12/2007 KICP, Chicago 54

Bayesian framework

4) Sample the joint PDF: MCMC

Power-spectrum reconstruction:

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

MCMC

04/12/2007 KICP, Chicago 55

04/12/2007 KICP, Chicago 56

04/12/2007 KICP, Chicago 57

Appendix

04/12/2007 KICP, Chicago 58

Redshift distortions

04/12/2007 KICP, Chicago 59

Mask, window function

SDSS observed patches of sky in galactic coordinates

04/12/2007 KICP, Chicago 60

Numerical method: Iterative inversion schemes

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

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

04/12/2007 KICP, Chicago 61

Numerical method: Iterative inversion schemes

Quadratic form:

Steepest Descent convergence behaviour:

04/12/2007 KICP, Chicago 62

Numerical method: Iterative inversion schemes

Quadratic form:

Steepest Descent convergence behaviour:

-> the searching directions are constantly repeated !

04/12/2007 KICP, Chicago 63

Krylov space method

04/12/2007 KICP, Chicago 64

Krylov space method

Krylov vs Steepest Descent:

04/12/2007 KICP, Chicago 65

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

04/12/2007 KICP, Chicago 66

2DF Erdogdu et al '04

04/12/2007 KICP, Chicago 67