Shear Power of Weak Lensing 10 3 N-body Shear 300 Sampling errors l(l+1)c l /2π εε 10 4 10 5 Error estimate Shot Noise θ y (arcmin) 200 100 10 6 100 1000 l 100 200 300 θ x (arcmin) Wayne Hu U. Chicago
Collaborators Asantha Cooray Dragan Huterer Mike Joffre Jordi Miralda-Escude Max Tegmark Martin White
Collaborators Asantha Cooray Dragan Huterer Mike Joffre Jordi Miralda-Escude Max Tegmark Martin White Microsoft
Why Power Spectrum? Pros: Direct observable: shear-based Statistical properties simple on large-scales (cf. variance, higher order) Complete statistical information on large-scales
Why Power Spectrum? Pros: Direct observable: shear-based Statistical properties simple on large-scales (cf. variance, higher order) Complete statistical information on large-scales Statistical tools developed for CMB and LSS power spectrum analysis directly applicable Information easily combined with CMB As (more with redshifts) precise as CMB per area of sky
Pros: Why Power Spectrum? Direct observable: shear-based Statistical properties simple on large-scales (cf. variance, higher order) Complete statistical information on large-scales Statistical tools developed for CMB and LSS power spectrum analysis directly applicable Information easily combined with CMB As (more with redshifts) precise as CMB per area of sky Cons: Relatively featureless Non-linear below degree scale Degeneracies: initial spectrum, transfer function shape, density growth, and angular diameter distance + source redshift distribution Computationally expensive: extraction by likelihood analysis N pix 3
Statistics and Modelling
Shear Power Modes Alignment of shear and wavevector defines modes ε β
Power Spectrum Roadmap Lensing weighted Limber projection of density power spectrum ε-shear power = κ-power εε l(l+1)cl /2π 10 4 10 5 10 6 k=0.02 h/mpc zs Linear Non-linear 10 7 Limber approximation 10 100 1000 l zs=1 Kaiser (1992) Jain & Seljak (1997) Hu (2000)
Statistics & Simulations Measurement of power in each multipole is independent if the field is Gaussian Non-linearities make the power spectrum non-gaussian Need many simulations to test statistical properties of the field in particular: sample variance White & Hu (1999)
Statistics & Simulations Measurement of power in each multipole is independent if the field is Gaussian Non-linearities make the power spectrum non-gaussian Need many simulations to test statistical properties of the field in particular: sample variance PM simulations ideal if sufficient angular resolution can be achieved lenses sources obs. tiling White & Hu (1999)
PM Simulations Hundreds of independent simulations Convergence Shear 6 6 FOV; 2' Res.; 245 75 h 1 Mpc box; 480 145 h 1 kpc mesh; 2 70 10 9 M White & Hu (1999)
Mean & Sampling Errors Mean agrees well with PD96 + Limber (Jain, Seljak & White 1999) Sampling errors per 6 x 6 field: Ratio Power 10 4 10 5 1.4 1 0.6 256 3 512 3 PD96 linear shot noise vs PD96 White & Hu (1999)
Mean & Sampling Errors Sampling errors ~Gaussian at l<1000 Non-Gaussianity increases sampling errors on binned power spectrum At current survey depths, shot noise dominates in non-gaussian regime Ratio Sampling Errors 0.3 0.1 1.4 1.2 1 vs shot noise 100 l 1000 256 3 512 3 Gaussian White & Hu (1999)
Mean & Sampling Errors Correlation in Band Powers: Correlation 1000 Gaussian 100 l 1000 100 l White & Hu (1999)
Mean & Sampling Errors Correlation in Band Powers: Non-Gaussian Correlation 1000 Gaussian 100 l 1000 100 l White & Hu (1999)
Halo Model Model density field as (linearly) clustered NFW halos of PS abundance: Simulation Shear 6 6 FOV; 2' Res.; 245 75 h 1 Mpc box; 480 145 h 1 kpc mesh; 2 70 10 9 M White & Hu (1999)
Halo Model Model density field as (linearly) clustered NFW halos of PS abundance: Halo Model Shear Peebles (1974); Scherrer & Bertschinger (1991) Komatsu & Kitiyama (1999); Seljak (2001)
Power Spectrum Statistics in Halo Model Power spectrum as a function of largest halo mass included Non-linear regime dominated by halo profile / individual halos increased power spectrum variance and covariance 10 4 εε l(l+1)cl /2π 10 5 10 6 10 7 14 13 12 log(m/m )<11 10 8 profile 10 100 1000 l Cooray, Hu, Miralda-Escude (2000)
Halo Model vs. Simulations Halo model for the trispectrum: power spectrum correlation Simulation Correlation 1000 Halo 100 l 1000 100 l Cooray & Hu (2001)
Halo Model vs. Simulations Halo model for the bispectrum: S 3 dominated by massive halos 10 3 Hu & White (1999) 10 2 16 15 S3 10 1 14 10 0 12 13 11 5 15 25 35 45 55 65 75 85 σ (arcmin) Cooray & Hu (2001)
Halo Model vs. Simulations Explains large sampling errors: 150 100 S 3 50 0 6 6 0 10 20 30 σ (arcmin) White & Hu (1999)
Power Spectrum Estimation
Likelihood Analysis Pros: Optimal power spectrum estimator for Gaussian field Automatically accounts for irregular (sparse sampled) geometries & varying sampling densities Arbitrary noise correlations Marginalize over systematic error templates (ε, β, cross) checks for non-gravitational effects Rigorous error analysis in linear regime
Pros: Likelihood Analysis Optimal power spectrum estimator for Gaussian field Automatically accounts for irregular (sparse sampled) geometries & varying sampling densities Arbitrary noise correlations Marginalize over systematic error templates (ε, β, cross) checks for non-gravitational effects Rigorous error analysis in linear regime Cons: Computationally expensive search in multiple dimensions Not guaranteed to be optimal or unbiased in non-linear regime
Pros: Likelihood Analysis Optimal power spectrum estimator for Gaussian field Automatically accounts for irregular (sparse sampled) geometries & varying sampling densities Arbitrary noise correlations Marginalize over systematic error templates (ε, β, cross) checks for non-gravitational effects Rigorous error analysis in linear regime Cons: Computationally expensive search in multiple dimensions Not guaranteed to be optimal or unbiased in non-linear regime Test iterated quadratic estimator of maximum likelihood solution and error matrix
Testing the Likelihood Method Input: pixelized shear data γ 1 (n i ), γ 2 (n i ); pixel-pixel noise correlation Iterate in band power parameter space to maximum likelihood... Output: maximum likelihood band powers and local curvature for error estimate including covariance l(l+1)c l /2π εε 10 3 10 4 10 5 N-body Shear Sampling errors Error estimate Shot Noise θ y (arcmin) 200 150 100 50 10 6 100 1000 50 Hu & White (2001) l 100 150 200 θ x (arcmin)
Testing the Likelihood Method Sparse sampling test Mean and errors correctly recovered 10 3 N-body Shear 300 Sampling errors l(l+1)c l /2π εε 10 4 10 5 Error estimate Shot Noise θ y (arcmin) 200 100 10 6 100 1000 Hu & White (2001) l 100 200 300 θ x (arcmin)
Testing the Likelihood Method Systematic error monitoring: example underestimated shot noise β-channel appearance in power Systematic error template: jointly estimate or marginalize 10 3 Noise Underestimated Signal+Noise Noise Removed l(l+1)c l /2π 10 4 εε εε 10 5 ββ p noise Shot Noise Shot Noise 100 1000 l Hu & White (2001) 100 1000 l
Cosmological Parameter Forecasts
Degeneracies All parameters of ICs, transfer function, growth, angular diam. distance Power spectrum lacks strong features: degeneracies εε l(l+1)cl /2π 10 4 10 5 10 6 k=0.02 h/mpc zs Linear Non-linear 10 7 Limber approximation 10 100 1000 l zs=1
Degeneracies All parameters of ICs, transfer function, growth, angular diam. distance Power spectrum lacks strong features: degeneracies T 100 80 60 40 MAP Planck CMB: provides high redshift side IC's, transfer fn., ang. diameter to z=1000 Lensing: dark energy, dark matter Hu & Tegmark (1999) 20 CMB 10 100 l (multipole) Redshifts: source (and lens) Breaks degeneracies by tomography Hu (1999)
Tomography Divide sample by photometric redshifts Cross correlate samples g i (D) ni(d) 3 2 1 0.3 0.2 0.1 Order of magnitude increase in precision even after CMB breaks degeneracies Hu (1999) (a) Galaxy Distribution 1 1 2 0 0.5 2 (b) Lensing Efficiency 1 1.5 2.0 D Power 10 4 10 5 25 deg 2, z med =1 22 12 11 100 1000 10 4 l
Error Improvement 100 30 10 3 Error Improvement: 25deg 2 MAP + 3-z no-z 1 MAP 9.6% 14% 0.60 1.94 0.18 0.075 19% 0.28 Ω Ω m h 2 b h 2 w τ n S T/S Ω Λ Cosmological Parameters δ ζ Hu (1999; 2001)
Error Improvement 100 30 10 3 Error Improvement: 1000deg 2 MAP + 3-z no-z 1 MAP 9.6% 14% 0.60 1.94 0.18 0.075 19% 0.28 Ω Ω m h 2 b h 2 w τ n S T/S Ω Λ Cosmological Parameters δ ζ Hu (1999; 2001)
Dark Energy & Tomography Both CMB and tomography help lensing provide interesting constraints on dark energy 2 1 MAP no z 3 z 0.6 0.7 0.8 w 0 0.9 1000deg2 0 0.5 1.0 Ω Λ 1 0.55 0.6 0.65 0.7 l<3000; 56 gal/deg2 Hu (2001)
Direct Detection of Dark Energy? In the presence of dark energy, shear is correlated with CMB temperature via ISW effect z>1.5 Θε l(l+1)cl /2π 10 9 10 10 10 9 10 10 1000deg2 65% sky 1<z<1.5 z<1 10 9 10 10 10 100 1000 l Hu (2001)
CMB Lensing: Tomography Anchor CMB acoustic waves are the ultimate high redshift source! Hu (2001)
Mass Reconstruction CMB acoustic waves are the ultimate high redshift source! original mass (deflection) map reconstructed 1.5' beam; 27µK-arcmin noise Hu (2001)
Power Spectra Power spectrum of deflection and cross correlation with CMB ISW 10 9 dd l(l+1)cl /2π 10 7 Θd l(l+1)cl /2π 10 10 Planck Ideal 10 100 1000 l 10 100 1000 l Extend tomography to larger scales and higher redshift Constrain clustering properties of dark energy through ISW correlation Hu (2001)
Lens Redshifts Shear selected halos: clustered according to well-understood halo bias Shear Halo Identification Mo & White (1996) e.g. aperture mass: Schneider (1996)
Lens Redshifts Shifting of feature(s) in the angular power spectrum: d A (z) 10 0 hh l(l+1)cl /2π 10 1 z=0.3 0.4 z=1.0 1.2 10 2 100 1000 l 4000deg 2, M>10 14 Cooray, Hu, Huterer, Joffre (2001)
Lens Redshifts If baryon features not detected, 10% distances to z~0.7 possible 10 0 hh l(l+1)cl /2π 10 1 z=0.3 0.4 z=1.0 1.2 10 2 100 1000 l 4000deg 2, M>10 14 Cooray, Hu, Huterer, Joffre (2001)
Lens Redshifts If baryon features are detected, ultimate standard ruler! 10 0 hh l(l+1)cl /2π 10 1 z=0.3 0.4 z=1.0 1.2 10 2 100 1000 l 4000deg 2, M>10 14 Cooray, Hu, Huterer, Joffre (2001)
Summary N-body sims and halo model for understanding power spectrum stats. and higher order correlations Likelihood analysis is feasible and near optimal for current generation of surveys
Summary N-body sims and halo model for understanding power spectrum stats. and higher order correlations Likelihood analysis is feasible and near optimal for current generation of surveys properly accounts for sparse or uneven sampling and correlated noise built in monitors for systematic errors techniques for marginalizing known systematics w. templates
Summary N-body sims and halo model for understanding power spectrum stats. and higher order correlations Likelihood analysis is feasible and near optimal for current generation of surveys properly accounts for sparse or uneven sampling and correlated noise built in monitors for systematic errors techniques for marginalizing known systematics w. templates Power spectrum complements CMB information order of magnitude increase in precision Tomography (including CMB lensing) assists in identifying dark energy (including possible clustering) Lens redshifts yield halo number counts and halo power spectra