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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
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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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04/12/2007 KICP, Chicago 3
ARGO: Algorithm for the Reconstruction of the Galaxy-traced Overdensities
Johannes Hevelius, 1690
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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
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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
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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
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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
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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?
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04/12/2007 KICP, Chicago 9
Inverse problem
Observed data d (m-dim vector) Signal s (n-dim vector)
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04/12/2007 KICP, Chicago 10
Inverse problem
Observed data d (m-dim vector) Signal s (n-dim vector)
Signal degradation data model:
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04/12/2007 KICP, Chicago 11
Inverse problem
Observed data d (m-dim vector) Signal s (n-dim vector)
Signal degradation data model:
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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
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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
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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
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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
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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
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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
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04/12/2007 KICP, Chicago 18
Bayesian framework
Bayes theorem:
posterior = Likelihood*prior/evidence
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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
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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
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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
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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
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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
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04/12/2007 KICP, Chicago 24
Krylov space method
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04/12/2007 KICP, Chicago 25
Operator formalism
How to apply matrix A ?
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04/12/2007 KICP, Chicago 26
Operator formalism
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04/12/2007 KICP, Chicago 27
Numerical tests
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04/12/2007 KICP, Chicago 28
Structured noise
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04/12/2007 KICP, Chicago 29
Preconditioning
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04/12/2007 KICP, Chicago 30
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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04/12/2007 KICP, Chicago 37
Single Bayesian reconstruction step
Linear reconstructionEffective redshift distortions treatment
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04/12/2007 KICP, Chicago 39
Linear power-spectrum
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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
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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 ?
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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
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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
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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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