Transcript Ｗｅａｋ Ｌｅｎｓｉｎｇ Ｆｌｅｘｉｏｎ Ａｌａｌｙｓｉｓ ｂｙ ＨＯＬＩＣｓ T. Futamase,

Ｗｅａｋ Ｌｅｎｓｉｎｇ
Ｆｌｅｘｉｏｎ
Ａｌａｌｙｓｉｓ ｂｙ ＨＯＬＩＣｓ
T. Futamase,
Astronomical Institute, Tohoku
University, JAPAN
Collaborators:Y. Okura, K. Umetsu
Weak Lensing Flexion
 Flexion Observable- HOLICs
 PSF correction
 Flexion Analysis of Subaru A1689
Data
 Applications
 HSC(Hyper Suprime Camera) for
Subaru

1. Higher-order weak lensing
Linear order
 i   i   i ( )
2 
1     1

Aij  

1




2
1

d i  Aij ( 0 )d j
  * ,   
(   1  i 2 )
Second order
d i  Aij (0) d j 
D  F  G,
1
Dijk (0)d j d k
2
F  * , G  
(（（Goldberg & Natarajan 2002, Bacon et al. 2005)
Complex representation
*
1
Y11  (1   )[ X 11  g X 11  ]  (2 FX 02  F * X 22  GX 22* )
4
Y11  d1  i d 2 ,
X
N
M
X 11  d1  i d 2
 (d1  id 2 )
N M
2
g

1 
(d1  id 2 )
N M
2
Effects of Convergence, Shear, Flexion
Spin-0
Spin-1
Spin-2
Spin-3
Where is Flexion useful?
An intermediate regime between WEAK and STRONG lensing can be well
described by shearing and flexing effects:
Arclets = lensed images with slight curvatures
quadrupole
1
  
2
Spin-2
skewness
1 *
F   
2
Spin-1
3-fold
1
G  
2
Spin-3
Goldberg & Natarajan 02, Bacon et al. 2005
Flexion to Surface Mass Density S/Scrit
Observable Flexion measures the 1st derivative of shear
Non-parametric inversion from flexion to mass density
1
F  *  
2
1
G    
2
   ( F )
1
*
   (   G )
1
* * *
Bacon et al. 2005
Shear vs. Flexion
Resolution limit in ordinal (=Spin-2) weak lensing with
ground based telescopes (Subaru, CFHT, etc.):
1
1
S/N
ng

       

 
FWHM  0.'9
 
5  0.2   0.4   30arcmin

Ordinal WL is sensitive to structures of 1’-10’, which is
1/ 2
2



dominated by clusters of galaxies
Flexion measures the gradient of shear; so is relatively sensitive
to small-scale structures (e.g., galaxies, groups of galaxies)
S (F )
L / r 3
L
~
~
2
S ( )
/r
r
L: image size
r: distance from the lens
Even though the higher-order effect is small, at small
scales (r), for large images (L), Flexion signal might
dominate over Shear signal
2. How to measure Flexion-HOLICs
(Higher Order Lensing Image Characteristics)
Qij   d 2 q[ I ( )]d i d j
spin  2 combination
Q11  Q22  2iQ12

Q11  Q22
 (s)    2g
g    
Qijk   d 2 q[ I ( )]d i d j d k
spin  1
(Q  Q )  i (Q112  Q222 )
  111 122
Q1111  2Q1122  Q2222
spin  3

(Q111  3Q122 )  i (3Q112  Q222 )
Q1111  2Q1122  Q2222

(s)
 (s)
9
 
F
4
3
  G
4
F  * , G  
Okura, Umetsu, Futamase, ApJ, 660, 995,(2007)
HOLICs Moment Method
KSB formalism (Kaiser, Squares, Broadhurst 1995):
Adds in Lensing and PSF induced response to intrinsic ellipticity
that is randomly oriented
Spin-2
Higher-order generalization of KSB moment method by
Okura, Umetsu, Futamase 2007b
Spin-1
  
(s)
Spin-3
  
(s)
 (C ) F  (C ) ( q ) 
 ( D ) G  ( D ) ( q ) 
F
Unbiased estimator
1
F  CF
for Flexion
G
 C q
q
q
q
G  DF
1
  D q
q
Application to A1689 (Subaru)
15 arcmin (2Mpc/h)
Mass + Light contours
from Shear+Magbias data
(Umetsu & Broadhurst 07)
Mass map from Fleixon in a 4’x4’
region using ng=8 gal/arcmin^2 !!!
(Okura, Umetsu, & Futamase 2007b)
Mass Map of A1689 from Spin-1 Flexion
Mass reconstruction in the 4’x4’ core region of A1689 (z=0.18)
E-mode (lensing)
B-mode (noise)
5
6
ng=8 arcmin^-2
0.’3FWHM
Gaussian
530kpc/h
Okura, Umetsu, Futamase 2007b
Main Halo and Subhalos
Results



First successful cluster mass
reconstruction only by Flexion
incorporating full higher-order PSF
anisotropy corrections
Significant detections of cD/group-scale
substructures at 5-6 sigma levels
Main density peak of ~2.5+0.45-0.45 is
consistent with the ACS/Subaru mass model
(Broadhurst, Takada, Umetsu et al. 05).
5. Applications
1. Aperture mass using flexion
M ap ( 0 )   d 2 U (|    0 |) ( )


0
dU ( )  0
Apperture mass can be expresed by
tangential shear

M ap  2  d Q( )~t ( )
0
1
~
 t ( ) 
2

2
0
d  t ( ) ,
 t ( )  Re[ e  2i  ( )]
Similoarly aperture mass can be exoressed
by Flexions
M ap

1
 2  dX ( )
0
2
1 2
~
Fr ( ) 
d

0
2
1 2
~
Gr ( ) 
d

0
2

~
~
Fr ( )  Gr ( )

Fr ( ) ,
Fr ( )  Re[ e  2i F ( )]
Gr ( ) ,
Gr ( )  Re[ e  2i G ( )]
Aperture mass using 2pt. Correlation
functions
shear
flexions
2. Spin-2 HOLICs
Degree N and spin number M complex coordinates
X MN  (d1  id 2 )


2
2
d

I
(

)
X
2

tr Q
d


 
N M
2
 I ( ) X
2
3
1

2
3
d

I
(

)
X
3


(d1  id 2 )
N M
2
tr Q   d 2 I ( ) X 02
   d 2 I ( ) X 04



d
 I ( ) X
2
4
2

2
6
d

I
(

)
X
2


d
d
 I ( ) X
8
2
 I ( ) X
8
0
2
2

(s)
   3g

(s)
   4g

(s)
   5g
Mass reconstruction using various Spin2HOLICs
4
2
X 2 
X2  
A1689
X 
6
2
X 
8
2
Blanck region：N1
X 22
X 24
 (24 moments)
 (26 moments)
Angular
difference
｜｜
 (28 momoments)
｜｜
｜｜
New possibility to do weak lensing
analysis using HOLICs
Not only Flexion(Clusters) but
also (Cosmic) Shear as well
How much we can improve accuracy of
mass reconstruction and power
spectrum measurement(dark energy
parameter) are now in progress
 Hyper
Suprime-Cam
High-z (z<3) galaxy survey ~2000 squire degree
Dark energy observation by Cosmic Shear
•Wide Field of View
(10 times of SC)
•High Sensitivity( 3 times sensitive
than present CCD at 1 micron)
超高感度CCD（8千万
画素16億画素）
Field of View of HSC
Cosmic shear
BAO
現状
2PCF-Power spectrum –Neutrino mass
and so on
2000平方度、銀河3億個を観測。
Z～3までの情報を最良画像で取得
Pde   de