Post on 30-May-2018
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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
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10-8
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10-6
10-5
10-4
10-3
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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