Patrick Simon Argelander-Institute for Astronomy, Bonn, Germany

3-D mass mapping in
STAGES
using 3-D lensing
Patrick Simon
Argelander-Institute for Astronomy, Bonn, Germany
Collaborators:
C. Heymans, T. Schrabback, A. Taylor, M. Gray (PI STAGES), L. van
Waerbeke, C. Wolf, D. Bacon, M. Balogh, F.D. Barazza, M. Barden, E.F.
Bell, A. Boehm, J.A.R. Caldwell, B. Haeussler, K. Jahnke, S. Jogee, E. van
Kampen K. Lane, D.H. McIntosh, K. Meisenheimer, C.Y. Peng, S.F.
Sanchez, L. Wisotzki, X. Zheng

Outline
Modified Wiener filtering of 3-D lensing data
Radial debiasing and cleaning of map
Data sample: STAGES
A mass map vs. light map comparison
Conclusions

Weak lensing on a grid
Lensing convergence = projected matter fluctuations
∫ ∞
κ(θ) = 3H2 0 Ω m
2c 2 dw W (w)f k(w)
0 a(w)
〈 〉
fk (w s − w l )
W (w l )=
f k (w s )
w s
δ m (f k (w)θ, w)
δ m
(6)
δ m
(7)
θ
δ m
(5)
δ m
(3)
δ m
(4)
δ m
(1)
δ m
(2)
distance w
us

Weak lensing on a grid
Lensing convergence = projected matter fluctuations
∫ ∞
κ(θ) = 3H2 0 Ω m
2c 2 dw W (w)f k(w)
0 a(w)
〈 〉
fk (w s − w l )
W (w l )=
f k (w s )
w s
δ m (f k (w)θ, w)
Chop 3-D volume into slices of constant matter density
κ(θ) ≈
N lp
∑
i=1
Q i δ (i)
m (θ) ,Q i = 3H2 0 Ω m
2c 2
∫ wi+1
w i
dw W (w)f k(w)
a(w)

Weak lensing on a grid
Lensing convergence = projected matter fluctuations
∫ ∞
κ(θ) = 3H2 0 Ω m
2c 2 dw W (w)f k(w)
0 a(w)
〈 〉
fk (w s − w l )
W (w l )=
f k (w s )
w s
δ m (f k (w)θ, w)
Chop 3-D volume into slices of constant matter density
κ(θ) ≈
N lp
∑
i=1
Q i δ (i)
m (θ) ,Q i = 3H2 0 Ω m
2c 2
∫ wi+1
w i
dw W (w)f k(w)
a(w)
On a grid we may therefore write
γ =P γκ κ =P γκ Q δ m

Wiener filter of 3-D lensing data
Combine all “source redshift bins” into one vector
(
) (
)
γ ≡ γ (1) ,γ (2) , . . . =P γκ Q (1) , Q (2) , . . . δ m ≡ P γκ Qδ m
different z-bins
0.3 0.5 0.7 0.9 1.1 1.3 1.5

Wiener filter of 3-D lensing data
Combine all “source redshift bins” into one vector
(
) (
)
γ ≡ γ (1) ,γ (2) , . . . =P γκ Q (1) , Q (2) , . . . δ m ≡ P γκ Qδ m
Need to find solution for δ m for given cosmic shear:
ɛ =P γκ Q δ m + n
shape noise
Source ellipticities binned on grid

Wiener filter of 3-D lensing data
Combine all “source redshift bins” into one vector
(
) (
)
γ ≡ γ (1) ,γ (2) , . . . =P γκ Q (1) , Q (2) , . . . δ m ≡ P γκ Qδ m
Need to find solution for δ m for given cosmic shear:
ɛ =P γκ Q δ m + n
Minimum variance solution for δ
δ mv
m
= SQ t P † γκ
[
N −1 P γκ QSQ t P † γκ + α1 ] −1
N −1 ɛ
S ≡〈δ m δ t m〉
signal covariance
N ≡〈nn † 〉 noise covariance
Simon, P., Taylor, A.N., Hartlap, J., 2009, MNRAS, 399, 48

Wiener filter of 3-D lensing data
Combine all “source redshift bins” into one vector
(
) (
)
γ ≡ γ (1) ,γ (2) , . . . =P γκ Q (1) , Q (2) , . . . δ m ≡ P γκ Qδ m
Need to find solution for δ m for given cosmic shear:
ɛ =P γκ Q δ m + n
Minimum variance solution for δ
δ mv
m
= SQ t P † γκ
[
N −1 P γκ QSQ t P † γκ + α1 ] −1
N −1 ɛ
S ≡〈δ m δ t m〉
signal covariance
N ≡〈nn † 〉 noise covariance
α=1: full Wiener filtering (less noise, biased)
α=0: no prior (plenty noise, unbiased)

Limitations of technique
Mock data
0.5
0.4
0.3
0.2
0.1
on sky
radially
z

Limitations of technique
STAGES
Mock data
re c o nstru c t e d re dshift
0.3
0.5
0.2
0.4
0.1
tru e re dshift
on sky
radially
z

Data sample: STAGES
A901a
about 30 arcmin
α
A901b
21 photo-z slices, z=0...1.3,
with 13 sources/arcmin 2
(matched with COMBO-17 survey)
Three deep mag-bins, m R =[23,27.45]
mag, with 52 sources/arcmin 2
22 lens planes, z=0...1.2
A902
SW
mag bins (GOODS-MUSIC)
STAGES: Gray et al. (2009)
photo-z’s
COMBO-17: Wolf et al. (2004)

Significance of map
about 30 arcmin
3-D mass map (projected onto sky)
A901a
α
A901b
A902
SW
Heymans, C. et al. (2008)
projected S/N surfaces: S/N=2.5,3,3.5...6.0
based on 1000 noise realisations of data

Cleaned and debiased map
A902/SW group region
re dshift
A901 a
A901b
On-sky projection
!
DE C
R.A.
A901a/A901b region
re dshift
C BI
SW
Simon, P., Heymans, C., Schrabback,T.,
2010, submitted
A902
SW /1
SW /2
DE C
R.A.

Mass vs. light comparison
Bin galaxy M star -masses (mass in stars) on grid and crosscorrelate
with lensing mass map:
∑
δ (i)
lum (θ) = j∈M i (θ) M ∗,j
M ∗
− 1 ; CCM(θ) = δ lum(θ)δ m (θ)
σ lum σ m
M star light map
Correlation map
A901a
A901b
A902
SW group
z=0.0-0.36

Mass vs. light comparison
Bin galaxy Mass S/N M star -masses (mass in stars) on grid and crosscorrelate
cleaned with lensing mass map:
map redshift
δ (i)
lum (θ) = ∑
j∈M i (θ) M ∗,j
M ∗
M star light map
− 1 ; CCM(θ) = δ lum(θ)δ m (θ)
σ lum σ m
A901a
red galaxies
A901b
CCM
A902
M star mass
SW group
blue galaxies
z=0.0-0.36

Mass vs. light comparison by eye
A901a
A901b
A902
SW group
Radial spread of A901a noise artefact?

Mass vs. light comparison by eye
A901a
A901b
A902
SW group
Radial spread of A901a noise artefact?
Light confirms structure behind A901b (z=0.70-0.80)

Mass vs. light comparison by eye
A901a
A901b
A902
SW group
Radial spread of A901a noise artefact?
Light confirms structure behind A901b (z=0.70-0.80)
Light confirms structure behind SW group (z=0.43?,0.80)

Mass vs. light comparison by eye
A901a
CB1
A901b
A902
A902
SW group
Radial spread of A901a noise artefact?
Light confirms structure behind A901b (z=0.70-0.80)
Light confirms structure behind SW group (z=0.43?,0.80)
CB1 reconfirmed (COMBO-17; Taylor et al. 2004: z=0.46)

Light-mass cross-correlation coefficient
10//
:;

Conclusions
This work is an application of a new lensing 3-D mass mapping
algorithm to real data
The modified Wiener filter produces a biased image of the
true distribution; z-shift bias can be corrected for
Annoyance of radial cigar-effect inside maps can be partly
removed with heuristic cleaning algorithm presented
3-D map reveals structure behind A901b and SW-group;
may be important for lensing mass modelling
Good light-mass match at z

Quality control

Quality control
Frequency statistics of noise peaks
noise
B-mode
E-mode
Peaks: pixel larger than all neighbour
pixels in transverse&radial direction
Troughs: negative peaks
Noise realisations vs. E/B-mode maps

Quality control
Frequency statistics of noise peaks
noise
B-mode
E-mode
Peaks: pixel larger than all neighbour
pixels in transverse&radial direction
Troughs: negative peaks
Noise realisations vs. E/B-mode maps
tan. shear
Z-scaling of tan. shear inside
apertures (≈1-2 arcmin radius)
mag
rad. shear
γ t (θ c ,z s )=− ∑
i∈M s (z s )
θi ∗ − θ∗ c
ɛ i
θ i − θ c
ε i
θ i
θ c
targets: A901a/b/α, SW and A902

A902 region/less smoothing
CBI
A902
re dshift
C BI
SW
A902
SW /1
SW /2
R.A.

Degradation of correlations
Cross-correlation coefficient between maps:
r = 〈CCM(θ)〉 = 〈δ (θ)δ (θ)〉
σ σ
; − 1 ≤ r ≤ +1

Degradation of correlations
Cross-correlation coefficient between maps:
r = 〈CCM(θ)〉 = 〈δ (θ)δ (θ)〉
σ σ
; − 1 ≤ r ≤ +1
Noise degrades the correlation but can be corrected if
signal and noise power are known:
r true = r ×
√ (
1+ 〈noise2 lum〉
〈signal 2 lum〉
)(
)
1+ 〈noise2 m〉
〈signal 2 m〉
Noise and signal variances are estimated from noise
realisations of mass and light maps;
Map pixel variance = noise variance + signal variance!