Weak Lensing Tomography - Wayne Hu's...
Transcript of Weak Lensing Tomography - Wayne Hu's...
Weak Lensing Tomography
Wayne HuFermilab, October 2002
z0.5 1 1.5 2
windows
1.4
1.2
1.0
0.8
0.6
0.60.40.2
mtot=0.2eV
Collaborators
• Chuck Keeton• Takemi Okamoto• Max Tegmark• Martin White
Microsoft
http://background.uchicago.edu/~whu/Presentations/ferminu.pdf
Massive Neutrinos• Relativisticstressesof a light neutrinoslow thegrowthof structure
• Neutrino species withcosmological abundancecontribute to matterasΩνh
2 = mν/94eV, suppressing power as∆P/P ≈ −8Ων/Ωm
∆P
P≈ −0.6
(mtot
eV
)
Massive NeutrinosCurrent data from 2dF galaxy survey indicates mν<1.8eVassuming a ΛCDM model with parameters constrained by theCMB.
•
k (h Mpc-1)
P(k)
(h-
1 M
pc)3
0.1 10.01
103
104
105
2dF
Lensing Observables• Image distortiondescribed byJacobian matrixof the remapping
A =
1− κ− γ1 −γ2
−γ2 1− κ+ γ1
,
whereκ is theconvergence, γ1, γ2 are theshearcomponents
Lensing Observables• Image distortiondescribed byJacobian matrixof the remapping
A =
1− κ− γ1 −γ2
−γ2 1− κ+ γ1
,
whereκ is theconvergence, γ1, γ2 are theshearcomponents
• related to thegravitational potentialΦ by spatial derivatives
ψij(zs) = 2∫ zs
0dzdD
dz
D(Ds −D)
Ds
Φ,ij ,
ψij = δij − Aij, i.e. viaPoisson equation
κ(zs) =3
2H2
0Ωm
∫ zs
0dz
dD
dz
D(Ds −D)
Ds
δ/a ,
Gravitational Lensing by LSS
• Shearing of galaxy images reliably detected in clusters
• Main systematic effects are instrumental rather than astrophysical
Colley, Turner, & Tyson (1996)
Cluster (Strong) Lensing: 0024+1654
Cosmic Shear Data• Shear variance as a function of smoothing scale
compilation from Bacon et al (2002)
Shear Power Spectrum
• Lensing weighted Limber projection of density power spectrum
• ε−shear power = κ power
10 100 1000l
10–4
10–5
10–7
10–6
l(l+
1)C
l /2
πεε
k=0.02 h/Mpc
Limber approximation
Linear
Non-linear
Kaiser (1992)Jain & Seljak (1997)
Hu (2000)
zs=1
zs
PM Simulations
• Simulating mass distribution is a routine exercise
6∞ × 6∞ FOV; 2' Res.; 245–75 h–1Mpc box; 480–145 h–1kpc mesh; 2–70 109 M
Convergence Shear
White & Hu (1999)
Degeneracies• All parameters of initial condition, growth and distance redshift relation D(z) enter• Nearly featureless power spectrum results in degeneracies
10 100 1000l
10–4
10–5
10–7
10–6l(l+1
)Cl
/2
πεε
zs=1
Degeneracies• All parameters of initial condition, growth and distance redshift relation D(z) enter• Nearly featureless power spectrum results in degeneracies
• Combine with information from the CMB: complementarity (Hu & Tegmark 1999)
• Crude tomography with source divisions (Hu 1999; Hu 2001)
• Fine tomography with source redshifts (Hu & Keeton 2002; Hu 2002)
10 100
20
40
60
80
100
l (multipole)
∆T
CMB
MAPPlanck
Error Improvement• Error improvements over MAP/CMB with a 1000 deg2 survey
Hu & Tegmark (1999)
0.029 0.0026 0.76 1.0 0.29 0.64 0.11 0.45 1.2
Ωmh2 Ωbh2 mν nS T/S AΩΛ ΩK τ
MAP
Erro
r Im
prov
emen
t
1
3
10
30
100
Crude Tomography
• Divide sample by photometric redshifts
Hu (1999)
1 2
2
D0 0.5
0.1
1
2
3
0.2
0.3
1 1.5 2.0
g i(D
) n i
(D)
(a) Galaxy Distribution
(b) Lensing Efficiency
22
100 1000 104
10–5
10–4
l
Pow
er
25 deg2, zmed=1
Crude Tomography
• Divide sample by photometric redshifts
• Cross correlate samples
• Order of magnitude increase in precision even after CMB breaks degeneracies
Hu (1999)
1
1
2
2
D0 0.5
0.1
1
2
3
0.2
0.3
1 1.5 2.0
g i(D
) n
i(D
)
(a) Galaxy Distribution
(b) Lensing Efficiency
22
11
12
100 1000 104
10–5
10–4
l
Pow
er
25 deg2, zmed=1
Efficacy of Crude Tomography• Error improvements over MAP/CMB with a 1000 deg2 survey
Hu & Tegmark (1999); Hu (2000)
0.029 0.0026 0.76 1.0 0.29 0.64 0.11 0.45 1.2
Ωmh2 Ωbh2 mν nS T/S AΩΛ ΩK τ
MAP
MAP +Er
ror I
mpr
ovem
ent
1
3
10
30
100
photo-zno-z
Dark Energy & Tomography
• Both CMB and tomography help lensing provide interesting constraints on dark energy
0 0.5 1.0
0
1
2
ΩDE
w
l<3000; 56 gal/deg2
MAPno–z
1000deg2
Hu (2001)
Dark Energy & Tomography
• Both CMB and tomography help lensing provide interesting constraints on dark energy
0 0.5 1.0
0
1
2
ΩDE
w
l<3000; 56 gal/deg2
MAPno–z3–z
1000deg2
Hu (2001)
Dark Energy & Tomography
• Both CMB and tomography help lensing provide interesting constraints on dark energy
0 0.5 1.0
0.55–1
–0.9
–0.8
–0.7
–0.6
0.6 0.65 0.7
0
1
2
ΩDE
w
l<3000; 56 gal/deg2
MAPno–z3–z
1000deg2
Hu (2001)
Dark Sector and Radial Information• Much of the information on the dark sector is hidden in the
temporalor radialdimension
• Evolution ofgrowth rate(dark energy pressure slows growth)
• Evolution ofdistance-redshiftrelation
• Lensing is inherentlytwo dimensional: all mass along the line ofsight lenses
• Tomography implicitly or explicitlyreconstructs radial dimensionwith source redshifts
• Photometric redshift errors currently∆z < 0.1 out toz ~ 1 andallow for ”fine” tomography
Fine Tomography• Convergence– projection of∆ = δ/a for eachzs
κ(zs) =3
2H2
0Ωm
∫ zs
0
dzdD
dz
D(Ds −D)
Ds
∆ ,
Fine Tomography• Convergence– projection of∆ = δ/a for eachzs
κ(zs) =3
2H2
0Ωm
∫ zs
0
dzdD
dz
D(Ds −D)
Ds
∆ ,
• Data islinear combinationof signal + noise
dκ = Pκ∆s∆ + nκ ,
[Pκ∆]ij =
32H2
0ΩmδDj(Di+1−Dj)Dj
Di+1Di+1 > Dj ,
0 Di+1 ≤ Dj ,
Fine Tomography• Convergence– projection of∆ = δ/a for eachzs
κ(zs) =3
2H2
0Ωm
∫ zs
0
dzdD
dz
D(Ds −D)
Ds
∆ ,
• Data islinear combinationof signal + noise
dκ = Pκ∆s∆ + nκ ,
[Pκ∆]ij =
32H2
0ΩmδDj(Di+1−Dj)Dj
Di+1Di+1 > Dj ,
0 Di+1 ≤ Dj ,
• Well-posed (Taylor 2002) but noisy inversion (Hu & Keeton 2002)
• Noise properties differ from signal properties→ optimal filters
Tomography in Practice• Localization and selection of clusters
Wittman et al. (2001; 2002)
0 1 2 3-0.02
0
0.02
0.04
0 0.5 1
0
0.05
0.1
zphot zphot
γ t
Hidden in Noise• Derivatives of noisy convergence isolate radial structures
Hu & Keeton (2001)
400
0 0.5 1 1.5
200
z
∆
0.6
0.4
0.2
κ
(a) Density
(b) Convergence
Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002)
Radial density field
1
0.5
0
–0.5
–1
0 0.5 1 1.5z
Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Taylor 2002; Hu & Keeton 2002)
Radial density fieldWiener reconstruction
1
0.5
0
–0.5
–1
0 0.5 1 1.5z
Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Taylor 2002; Hu & Keeton 2002)
Radial density fieldWiener reconstruction
1
0.5
0
–0.5
–1
0 0.5 1 1.5z
Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002)
Radial density fieldWiener reconstruction
1
0.5
0
–0.5
–1
0 0.5 1 1.5z
Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002)
Radial density fieldWiener reconstruction
1
0.5
0
–0.5
–1
0 0.5 1 1.5z
Growth Function• Localized constraints (fixed distance-redshift relation)
4000 sq. deg
z0.5 1 1.5 2
windows
1.4
1.2
1.0
0.8
0.6
0.60.40.2
mtot=0.2eV
Hu (2002) 3 degenerate mass ν's assumed e.g. Beacom & Bell (2002)
z0.5 1 1.5 2
1.5
1.0
0.5
0.50
-0.5
ρ DE/
ρ cr0
windows
w=–0.8
Dark Energy Density• Localized constraints (with cold dark matter)
Hu (2002)
4000 sq. deg
Dark Energy Parameters• Three parameter dark energy model (ΩDE, w, dw/dz=w')
Hu (2002)
1
0.5-0.9
-1
-1.1
0
-0.5
0.6 0.65 0.7-1
ΩDE0.6 0.65 0.7
ΩDE
w'
w
σ(ΩDE)=0.01
σ(w')=0
0.03
nonenone
4000 sq. deg.
Lensing of a Gaussian Random Field• CMB temperature and polarization anisotropies are Gaussian
random fields – unlike galaxy weak lensing
• Average over many noisy images – like galaxy weak lensing
Lensing by a Gaussian Random Field• Mass distribution at large angles and high redshift in
in the linear regime
• Projected mass distribution (low pass filtered reflectingdeflection angles): 1000 sq. deg
rms deflection2.6'
deflection coherence10°
Lensing in the Power Spectrum• Lensing smooths the power spectrum with a width ∆l~60
• Convolution with specific kernel: higher order correlations between multipole moments – not apparent in power
Pow
er
10–9
10–10
10–11
10–12
10–13
lensedunlensed
∆power
l10 100 1000
Seljak (1996); Hu (2000)
Reconstruction from the CMB• Correlation betweenFourier momentsreflectlensing potentialκ = ∇2φ
〈x(l)x′(l′)〉CMB = fα(l, l′)φ(l + l′) ,
wherex ∈ temperature, polarization fieldsandfα is a fixed weightthat reflects geometry
• Each pair forms anoisy estimateof the potential or projected mass- just like a pair of galaxy shears
• Minimum variance weightall pairs to form an estimator of thelensing mass
Quadratic Reconstruction• Matched filter (minimum variance) averaging over pairs of
multipole moments
• Real space: divergence of a temperature-weighted gradient
Hu (2001)
originalpotential map (1000sq. deg)
reconstructed1.5' beam; 27µK-arcmin noise
Ultimate (Cosmic Variance) Limit• Cosmic variance of CMB fields sets ultimate limit
• Polarization allows mapping to finer scales (~10')
Hu & Okamoto (2001)
100 sq. deg; 4' beam; 1µK-arcmin
mass temp. reconstruction EB pol. reconstruction
Matter Power Spectrum• Measuring projected matter power spectrum to cosmic vari-
ance limit across whole linear regime 0.002< k < 0.2 h/Mpc
Hu & Okamoto (2001)
10 100 1000
10–7
10–8
defle
ctio
n po
wer
"Perfect"
Linear
Lσ(w)∼0.06
∆P
P≈ −0.6
(mtot
eV
)
Matter Power Spectrum• Measuring projected matter power spectrum to cosmic vari-
ance limit across whole linear regime 0.002< k < 0.2 h/Mpc
Hu & Okamoto (2001)
10 100 1000
10–7
10–8
defl
ectio
n po
wer
"Perfect"Planck
Linear
Lσ(w)∼0.06; 0.14
Summary• Gravitational lensing is the only direct probe of the dark sector: composition of dark matter: massive neutrinos nature of the dark energy: scalar field? Λ?
• With sources distributed in redshift, tomography possible• Coarse radial resolution sufficient for recoving linear growth rate dark energy density evolution• Requires good photometric redshifts, elimination of systematics, avoidance of intrinsic alignment contamination
• CMB provides ultimate high-z source for tomography; precision neutrino constraints in principle possible