Relativistic Corrections to the Sunyaev-Zeldovich

Effect for Clusters of Galaxies

Satoshi NozawaJosai Junior College for Women

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Collaborators: N. Itoh, Y. Kohyama

Carolina Neutrino Workshop April 16-17th, 2004

Contents

(1) Introduction(a) Clusters of Galaxies(b) Thermal SZ Effect(c) Kinematical SZ Effect

(2) Relativistic Corrections to SZ Effects(a) Thermal SZ Effect(b) Kinematical SZ Effect(c) Polarization SZ Effect

(3) Summary

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(1) Introduction

Optical Image by Subaru

(a) Cluster of Galaxies (CG)　 RX J1347 (z=0.45, 5×109 light years)

X-ray Image by Chandra

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0.9 Mpc

Structure of CG Galaxie

svisibleG MM of%10

Hot plasmas

visibleHP MM

pe

of%90

... He,,,

33

8

/cm10:densitynumber

K1010keV:etemperatur-

e

e

N

T ）（

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(b) Thermal Sunyaev-Zeldovich (SZ) Effect Distortion of the Cosmic Microwave Background

(CMB)photon spectrum due to electrons in the CG.(Inverse Compton scattering)

CMB photons

ν

I

ν=218GHz

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Sunyaev & Zeldovich, CommentsAstrophys. Space Phys. 4, 173 (1972)

Low frequency region --- temperature decrement

High frequency region --- temperature increment

Thermal SZ effect determines ΔTCMB ．

l

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erayXCMB

eerayX

eeCMB

TSTl

TlNS

TlNT

Determination of the Hubble constant with CG

--- Thermal SZ effect of CG--- X-ray intensity of CG

: diameter of CG along the line of sight

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l

l

Assumption: CG is spherical.

AD

)distancediameterangular(l

DA

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With the redshift observation, the Hubble constant H0 is determined (H0=v/DA).

yearsbillion5.135.10km/s/Mpc75814730 ～～ H

WMAP data

yearsbillion3.04.13km/s/Mpc5720 H

Advantages of the SZ effects:

・ Total radiation intensity is redshift independent! ECMB ∝ (1+z)4 (energy density of CMB) DA ∝ (1+z)2 (angular diameter distance) Itotal ∝ ECMB/D2 　 --- 　 z-dependences cancel out! ・ Determination of H0, cosmological constants, peculiar velocity field, etc.Disadvantages of the SZ effects: ・ Assumption of the spherical symmetry for CG ・ Small signals ΔTCMB ～ 10-3

Chandra and XMM-Newton have opened a new era for precise observation of the SZ effect.

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Determination of cosmological constants with SZ effect

9/33Temperature Map for RXJ1347

150 GHz (Nobeyama, Japan)

350 GHz (Maxwell, USA)

ν

I

ν=218GHz

Contours: X-ray intensity observed by Chandra

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Recent observations:High temperature CG Te = 10 ～ 20keV.

Relativisitic corrections become significant!

Typical order of magnitude

02.0511/10/ eee mT

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(c) Kinematical Sunyaev-Zeldovich (SZ) Effect

Peculiar motion of CG (Bulk Motion of CG relative to CMB by the local gravitational field) produces ΔTCMB. Doppler Effect

zeCMB vlNT

vz

cv

mT

zz

eee

/

/

Relativisitic corrections become significant!

Sunyaev & Zeldovich,Mon. Not. R. Astr. Soc. 190, 413 (1980)

12/33Typical magnitudes of SZ

effects

km/s000,1v

Dis

tort

ion

of

spec

tral

inte

nsit

y

13/33(2) Relativistic Corrections to SZ Effects

(a)Thermal SZ Effect Itoh, Kohyama & Nozawa, ApJ 502, 7 (1998)

Boltzman equation for the photon distribution function n(ω).

momentaphoton:,momenta,electron:,

functionsondistributiFermi:)(),(

amplitudescatteringComptonInverse:

,)(2

)4(

)},()](1)[()()](1)[({)2(

2)(

422

333

3

kkpp

EfEf

R

kpkpREE

eW

EfnnEfnnWkdpdpd

t

n

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eBeB Tkx

Tkx

,

'

Fokker-Planck expansion )1( x

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2cm

Tk

e

eBe

eBTkx

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Distortion of the CMB spectrum

etemperaturCMB:

oncrosssectiThomson:,,

,][1)(

)(

0

0

44

33

22

100

T

dlNeyTk

X

YYYYYe

Xey

Xn

Xn

TTB

eeeeX

Xe

Expansion in terms of

2cm

Tk

e

eBe

--- Non-relativistic term

Relativistic corrections

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Relativistic corrections

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Distortion of the spectral intensity )(

)(

1 0

3

Xn

Xn

e

XI

X

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・ Relativistic corrections are very important for high-temperature CG Te> 10keV.

・ Relativistic corrections are significant for large X region (sub-millimeter region)For X>10, factor 4 effect!

・ Fokker-Planck expansion approximation is OK for Te < 15keV.

Summary for thermal SZ effect

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(b) Kinematical SZ Effect Nozawa, Itoh & Kohyama ApJ, 508, 17 (1998) Lorentz boosted Boltzman equation

c

vpEE

pEE

EfEfEfEf

CGCG

CGCGCGCG

,1

,1

)()(,)()(

22

Peculiar velocity of CGrelative to CMB.

v

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Distortion of the CMB spectrum

Kinematical SZ effect

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Relativistic corrections

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Distortion of the spectral intensity )(

)(

1 0

3

Xn

Xn

e

XI

X

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・ Relativistic corrections are very important for high-temperature CG Te> 10keV.

Summary for kinematical SZ effect

%3.1)(

%2.8)(

km/s000,1,keV10For

2

e

e

eB

O

O

vTk

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Magnitude of the polarization of CMB

v

(c) Polarization SZ Effect

2

10

1

vlNP e

determinedbecanv

Sunyaev & Zeldovich,Mon. Not. R. Astr. Soc. 190, 413 (1980)

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Itoh, Nozawa & Kohyama ApJ, 533, 588 (2000)

Polarized photons with Lorentz boosted Boltzman equation

ncrossectioonpolarizati:),,(

parameterStokes:),,(

oncrosssectidunpolarize:

)},()](1)[()()](1)[({

1)2(

2)(

321

321

333

3

W

EfnnEfnn

Wkdpdpd

t

n

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42354224

23222

23221

0

2

22

102

~~2

17~~13

~35

1~~~2

11~7

4

~~2

~280

867~2

1~5

21~4

3

,~~

2~

20

1~2

1~5

2~10

3

,~

20

1

,

,1

SXSXXSSXX

SXXSXXF

SXXSXXF

XF

cm

TekdlNey

FFFe

XeyP

e

BeT

eeX

Xe

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Relativistic corrections

Distortion of the spectral intensity

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・ Relativistic corrections are very important for high-temperature CG Te> 10keV.

・ Distortion of the spectral intensity is extremely small.

Summary for Polarization SZ effect

5101

km/s000,1,keV10For

yI

vTk

pol

eB

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(3) Summary

・ We have presented a relativistically covariant formalism for the thermal SZ effect.

・ With the formalism one can treat kinematical SZ effect, polarization SZ effect, and multiple scattering effect in an unified manner.

・ Relativistic corrections are very important for high-temperature CG Te> 10keV for all SZ effects.

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