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Estimating the weak lensing

power spectrumMichael Schneider

UC Davis

In collaboration with Lloyd Knox

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Goal: constrain cosmological parameters withweak lensing data

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

- doesnt contain all information about non-Gaussianfield, but its a place to start

- we want to adapt CMB methods to compute the weaklensing power spectrum

Question:How should we compare the data with theory?

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Recent analyses

The aperture mass and top-hat variance are often computedanalytically from the correlation function and shown as well

*Heavens et al., MNRAS 373 (2006)

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Requirements: Fourier vs. real space

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

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Pseudo PS estimator

We need to apply a window to the data in order toapply masks and apodize for the Fourier transform

The binned pseudo-power spectrum estimator is:

This has expectation value:

(x) W(x)(x)

C

= K (S +N)

(a la CMB (Hivon et al., ApJ 567, 2002))

C 1

2B

B

dw

d

()()

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102 103

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

C))2(/)1+((

input

E - noise biasB - noise biasnoisenoise/sqrt(Nmodes)

Pseudo PS estimator has large E/B mixing

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Separating E and B

Naive computation of , on a finitepatch of sky introduces mixing between true

E and B modes from ambiguous modes.Instead, compute power spectra of (spin-0)potentials:

CE

CB

(Smith & Zaldarriaga (2006))

E 1

2(1 + i2) + (1 i2)

B i

2

(1 + i2) (1 i2)

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Pure modes from counterterms

Expressed another way:

This is the naive result with counter terms addedto remove the ambiguous modes.

Note that all the extra terms arise because ofthe weight function

In zero noise, unity window limit this is the FTof the convergence

E() 1

22d2x +

W(x)eix

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102 103

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

C))2(/)1+((

input

B - noise biaspure B - noise bias

noisenoise/sqrt(Nmodes)

pure B modes

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Stellar masks

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Choosing a window function

Want to find window function that minimizes error barswhile preserving E/B separation

Window will depend on models of the signal and noisecovariance

Counterterms for pure E/B separation depend on first andsecond derivatives of window function

these terms can be complicated for the stellar masks

We have not yet solved this problem!

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SummaryAdvantages of shear power spectrum for constrainingcosmological 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 withsimple mask structures

Stellar masks are more challenging - success requires aneffective method to optimize window function