Shear Power of Weak Lensing
300
10–3
N-body Shear
Error estimate
200
10–4
θy (arcmin)
l(l+1)Clεε/2π
Sampling errors
100
10–5
Shot Noise
10–6
100
1000
l
Wayne Hu
U. Chicago
100
200
θx (arcmin)
300
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
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
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 Npix3
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
10–4
Non-linear
k=0.02 h/Mpc
zs
l(l+1)Clεε /2π
•
•
Linear
10–5
10–6
Limber approximation
10–7
zs=1
10
100
1000
l
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–1Mpc box; 480–145 h–1kpc mesh; 2–70 109 M
White & Hu (1999)
Mean & Sampling Errors
Mean agrees well with PD96 + Limber (Jain, Seljak & White 1999)
Sampling errors per 6° x 6° field:
Power
10–4
2563
5123
PD96
linear
10–5
Ratio
•
•
1.4
shot noise
vs PD96
1
0.6
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
•
•
•
2563
5123
Gaussian
0.3
0.1
1.4
vs shot noise
1.2
1
100
l
1000
White & Hu (1999)
Mean & Sampling Errors
Correlation in Band Powers:
Correlation
•
Gaussian
1000
l
100
l
1000
100
White & Hu (1999)
Mean & Sampling Errors
Correlation in Band Powers:
Non-Gaussian
Correlation
•
Gaussian
1000
l
100
l
1000
100
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–1Mpc box; 480–145 h–1kpc mesh; 2–70 109 M
White & Hu (1999)
Halo Model
•
Model density field as (linearly) clustered NFW halos of PS abundance:
Halo Model
Peebles (1974); Scherrer & Bertschinger (1991)
Shear
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
14
10–5
l(l+1)Clεε /2π
•
•
13
12
log(M/M )<11
10–6
10–7
10–8
profile
10
100
l
1000
Cooray, Hu, Miralda-Escude (2000)
Halo Model vs. Simulations
Halo model for the trispectrum: power spectrum correlation
Simulation
Correlation
•
Halo
1000
l
100
l
1000
100
Cooray & Hu (2001)
Halo Model vs. Simulations
Halo model for the bispectrum: S3 dominated by massive halos
10
3
Hu & White (1999)
10
16
2
15
S3
•
10
10
1
14
13
0
12
11
5
15
25
35
45
55
σ (arcmin)
65
75
85
Cooray & Hu (2001)
Halo Model vs. Simulations
Explains large sampling errors:
150
100
S3
•
50
0
0
6° × 6°
10
20
σ (arcmin)
30
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
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
Cons:
•
•
Computationally expensive search in multiple dimensions
Not guaranteed to be optimal or unbiased in non-linear regime
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
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 γ1(ni), γ2(ni); 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
10–3
N-body Shear
200
Error estimate
150
10–4
θy (arcmin)
l(l+1)Clεε/2π
Sampling errors
100
10–5
Shot Noise
10–6
100
1000
l
Hu & White (2001)
50
50
100
150
θx (arcmin)
200
Testing the Likelihood Method
•
•
Sparse sampling test
Mean and errors correctly recovered
300
10–3
N-body Shear
Error estimate
200
10–4
θy (arcmin)
l(l+1)Clεε/2π
Sampling errors
100
10–5
Shot Noise
10–6
100
1000
l
Hu & White (2001)
100
200
θx (arcmin)
300
Testing the Likelihood Method
•
•
•
Systematic error monitoring: example underestimated shot noise
β-channel appearance in power
Systematic error template: jointly estimate or marginalize
l(l+1)Cl /2π
10–3
10–4
Noise Underestimated
Signal+Noise
εε
10–5
Noise Removed
εε
ββ
pnoise
Shot Noise
100
Shot Noise
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
10–4
Non-linear
k=0.02 h/Mpc
zs
l(l+1)Clεε /2π
•
•
Linear
10–5
10–6
Limber approximation
10–7
zs=1
10
100
1000
l
Degeneracies
•
•
All parameters of ICs, transfer function, growth, angular diam. distance
Power spectrum lacks strong features: degeneracies
MAP
Planck
100
•
∆T
80
60
CMB: provides high redshift side
IC's, transfer fn., ang. diameter to z=1000
Lensing: dark energy, dark matter
Hu & Tegmark (1999)
40
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
3
2
10–4
1
2
0.3 (b) Lensing Efficiency
gi(D)
25 deg2, zmed=1
22
1
Power
ni(D)
(a) Galaxy Distribution
12
10–5
11
0.2
2
0.1
1
0
0.5
1
1.5
2.0
100
1000
l
D
• Order of magnitude increase in precision even after CMB breaks
degeneracies
Hu (1999)
104
Error Improvement: 25deg2
100
MAP +
3-z
no-z
Error Improvement
30
10
3
1
MAP
9.6%
14%
Ωbh2 Ωmh2
0.60
1.94
0.18
0.075
19%
0.28
ΩΛ
w
τ
nS
δζ
T/S
Cosmological Parameters
Hu (1999; 2001)
Error Improvement: 1000deg2
100
MAP +
3-z
no-z
Error Improvement
30
10
3
1
MAP
9.6%
14%
Ωbh2 Ωmh2
0.60
1.94
0.18
0.075
19%
0.28
ΩΛ
w
τ
nS
δζ
T/S
Cosmological Parameters
Hu (1999; 2001)
Dark Energy & Tomography
•
Both CMB and tomography help lensing provide interesting
constraints on dark energy
2
–0.6
MAP
no–z
3–z
–0.7
1
w
–0.8
–0.9
0
–1
0.55
0.6
0.65
0.7
1000deg2
0
ΩΛ
0.5
1.0
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
10–9
1000deg2
65% sky
10–10
l(l+1)ClΘε /2π
•
1<z<1.5
10–9
10–10
z<1
10–9
10–10
10
100
l
1000
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
reconstructed
mass (deflection) map
1.5' beam; 27µK-arcmin noise
Hu (2001)
Power Spectra
•
Power spectrum of deflection and cross correlation with CMB ISW
l(l+1)ClΘd /2π
l(l+1)Cldd /2π
10–9
10–7
10–10
Planck
Ideal
10
100
l
•
•
1000
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
Mo & White (1996)
Halo Identification
e.g. aperture mass: Schneider (1996)
Lens Redshifts
Shifting of feature(s) in the angular power spectrum: dA(z)
10–0
l(l+1)Clhh /2π
•
10–1
z=0.3–0.4
z=1.0–1.2
10–2
4000deg2, M>1014
100
l
1000
Cooray, Hu, Huterer, Joffre (2001)
Lens Redshifts
If baryon features not detected, 10% distances to z~0.7 possible
10–0
l(l+1)Clhh /2π
•
10–1
z=0.3–0.4
z=1.0–1.2
10–2
4000deg2, M>1014
100
l
1000
Cooray, Hu, Huterer, Joffre (2001)
Lens Redshifts
If baryon features are detected, ultimate standard ruler!
10–0
l(l+1)Clhh /2π
•
10–1
z=0.3–0.4
z=1.0–1.2
10–2
4000deg2, M>1014
100
l
1000
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