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
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22
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~~2
17~~13
~35
1~~~2
11~7
4
~~2
~280
867~2
1~5
21~4
3
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2~
20
1~2
1~5
2~10
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,~
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SXSXXSSXX
SXXSXXF
SXXSXXF
XF
cm
TekdlNey
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XeyP
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eeX
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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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