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)