Estimating The Weak Lensing Power Spectrum

Estimating the weak lensing power spectrum Michael Schneider UC Davis In collaboration with Lloyd Knox

Goal: constrain cosmological parameters with weak lensing data Question: How should we compare the data with theory? Observe: - Shear field is stochastic and non-Gaussian - Theory predictions can be given as N-point correlation functions Frequent solution: compute 2-point correlation function - doesn’t contain all information about non-Gaussian field, but it’s a place to start - we want to adapt CMB methods to compute the weak lensing power spectrum

Recent analyses Survey

Analysis tools

COSMOS

correlation function, tomography, “3D cosmic shear*”

CFHTLS (deep)

correlation function

GaBoDS

correlation function

COMBO-17

power spectrum (2003), “3D cosmic shear*”

The aperture mass and top-hat variance are often computed analytically from the correlation function and shown as well *Heavens et al., MNRAS 373 (2006)

Requirements: Fourier vs. real space

E/B separation simple error structure computational complexity

Real

Fourier

?

?

(ring statistics1)

(ML estimator2)

no

maybe?

moderate

low?

(for Monte Carlo) 1. Schneider & Kilbinger, A&A 462, 841 (2007) 2. Hu & White, ApJ 554,67 (2001), Brown et al., MNRAS, 341,1 (2003)

Pseudo PS estimator (a la CMB (Hivon et al., ApJ 567, 2002)) We need to apply a window to the data in order to apply masks and apodize for the Fourier transform

χ(x) ˜ ≡ W (x)χ(x)

The binned pseudo-power spectrum estimator is:

C˜α ≡

1 2πBα

!

d" w"

Bα

!

dφ" χ ˜∗ (%")χ( ˜ %")

This has expectation value:

!

"

C˜α = Kαα! (Sα! + Nα! )

Pseudo PS estimator has large E/B mixing 10-2 10-3

((+1)/(2))C 

10-4

input E - noise bias B - noise bias noise noise/sqrt(Nmodes)

10-5 10-6 10-7 10-8 10-9 10-10

102

103



Separating E and B (Smith & Zaldarriaga (2006)) Naive computation of C E , C!B on a finite ! patch of sky introduces mixing between true E and B modes from ambiguous modes. Instead, compute power spectra of (spin-0) “potentials”:

" 1 !¯ ¯ χE ≡ "∂ "∂ (γ1 + iγ2 ) + "∂ "∂ (γ1 − iγ2 ) 2 " i !¯ ¯ χB ≡ "∂ "∂ (γ1 + iγ2 ) − "∂ "∂ (γ1 − iγ2 ) 2

“Pure” modes from counterterms Expressed another way:

1 χE (") ≡ − 2 2"

!

" # ∗ d x γ #∂¯ #∂¯ + γ #∂ #∂ W (x)e−i!·x 2

This is the “naive” result with counter terms added to remove the ambiguous modes.

Note that all the extra terms arise because of the weight function In zero noise, unity window limit this is the FT of the convergence

“pure” B modes 10-2 10-3

((+1)/(2))C 

10-4

input B - noise bias pure B - noise bias noise noise/sqrt(Nmodes)

10-5 10-6 10-7 10-8 10-9 10-10

102

103



Stellar masks

Choosing a window function Want to find window function that minimizes error bars while preserving E/B separation Window will depend on models of the signal and noise covariance Counterterms for “pure” E/B separation depend on first and second derivatives of window function these terms can be complicated for the stellar masks

We have not yet solved this problem!

Summary Advantages of shear power spectrum for constraining cosmological parameters: possibly simpler error structure E/B separation possible over finite dynamic range Easy to compute from data and compare to theory Faster for Monte Carlo to learn about errors

Status Implemented flat-sky version of Smith & Zaldarriaga method Pure pseudo-PS successfully applied to simulated data with simple mask structures Stellar masks are more challenging - success requires an effective method to optimize window function