The estimation of the SZ effects with
unbiased multifilters
Diego Herranz, J.L. Sanz, R.B. Barreiro & M. López-Caniego
Instituto de Física de Cantabria
Workshop on the SZ effect & ALMA – Orsay – April 8th 2005
Overview
• Linear multifilters for the detection of
the SZ effects: motivation & review
• Joint study of two signals with the
same spatial profile and different
frequency dependence: bias.
• The unbiased matched multifilter.
• Conclusions.
1
1. Linear multifilters2
PROS CONS
Easy to understand and to implement Adequate for compact sources Robust Fast
• Not as powerful as more sophisticated techniques (higher order statistics, etc) • Not yet optimised for extended/irregular objects
The matched multifilter (Herranz et al, 2002, MNRAS, 336, 1057) is a useful tool to enhance the SZE signal
Blind surveys with low angular resolution (Planck)
1. Linear multifilters (II):3
• How do the clusters look like?
• How do they appear at different wavelengths?
• How does the background behave at the different wavelengths?
1. Linear multifilters (III): data model
Nv
n
sf
d
xxAx
,,1
)()()(
n
F
d
nFd
4
Data (N maps at different frequencies)
Frequency dependence
X
Source profile (beam included)
)()()()( 2D2121
qqqPqnqn
“Noise” (CMB + foregrounds + instrumental noise)
1. Linear multifilters (IV): matched multifilter
ΘPΘ
Θd
1t2
t )()()(
qd
qqeqdbw
w
bqi
5
a) Make so that w(0)=A (unbiased estimator of the amplitude)
b) Make so that w is as small as possible (efficient estimator)
FPF
FPΘ1t
11MMF
qd
MATCHED
MULTIFILTER
1. Linear multifilters (V): two sources of bias
6
Identical shape
Identical spectral behaviour
THE SAME THING
Different shape
Identical spectral behaviour
BIAS
Identical shape
Different spectral behaviour
BIAS
Different shape
Different spectral behaviour
IDEAL SEPARATION
2. Joint study of the thermal and the kinematic SZ effects
)()()()()(kSZt
xxVxyxT
Tx c
nsFd
7
FPs
FPsFPF
ΘsF
1t
1t11t1
MMFtherm
,
0therm
qd
Vyy
qdVyqdy
Vyqdw
cc
cc
ttc
8
cc
eB
er
c
c
yy
Tk
cmvV
y
r
99.0ˆ
1.0
10
5.14
MMF: Bias in the determination of the thermal SZ effect in presence of the kinematic SZ effect
MMF: Bias in the determination of the kinematic SZ effect in presence of the thermal SZ effect
9
sPs
ΘsF
1t
MMFkin
,
0kin
qd
Vyy
Vyqdw
cc
ttc
95.005.1ˆ
1.0
10
5.14
VV
Tk
cmvV
y
r
eB
er
c
c
3. Canceling the bias10
a) Make so that w1(0)=A
b) Make so that w2(0)=0
c) Make so that 1+2 is as small as possible (efficient estimator)
)()( 21 xxxAx
nFFd
3. Canceling the bias of the thermal effect: UMMFt
11
minimum is
0
1
2
w
t
t
qd
qd
s
F
2
1 ,1
sFP
sPs
FPs
FPF
1t
1t
1t
qd
qd
qd
3. Canceling the bias of the kinematic effect: UMMFk
12
minimum is
1
0
2
w
t
t
qd
qd
Φs
ΦF
2
1 ,1
sFPΦ
sPs
FPs
FPF
1t
1t
1t
qd
qd
qd
3. Filter comparison: thermal effect
13
023.0ˆ
104.5
1077.9ˆ6
5
c
cc
y
c
y
yy
y
c
017.0ˆ
107.5
1083.9ˆ6
5
c
cc
y
c
y
yy
y
c
5.11
ˆ
24.0
05.1ˆ
V
VV
V
V
3. Filter comparison: kinematic effect
14
2.0ˆ
26.0
08.0ˆ
V
VV
V
V
3. Filter comparison: kinematic effect (II)
15
4. Conclusions:16
• SZ thermal effect can introduce dramatic systematic effects in the estimation of the kinematic effect
• It is possible to cancel this systematic effect introducing a new constraint in the formulation of the filters:
o It is not necessary to know a priori the thermal effect
o The variance of the estimator increases a little bit
• The errors in the determination of the peculiar velocities of individual clusters remain very large.
• However, once the estimator is unbiased it can be used for statistical analysis of large numbers of clusters (bulk flows, etc)
