Relativistic Corrections to the Sunyaev-Zeldovich Effect for Clusters of Galaxies. V. Effect of the Motion of the Observer

Abstract

We extend the formalism of the relativistic thermal and kinematical Sunyaev-Zeldovich effects to the observer's system (the Solar System) moving with a velocity $\beta_{S}$ with respect to the cosmic microwave background radiation. We confirm the results recently obtained by Chluba, Huetsi, and Sunyaev in the lowest order of the observer's velocity $\beta_{S}. We give a more general analytic expression for the thermal and kinematical Sunyaev-Zeldovich effects corresponding to the observer's system with the power series expansion approximation in terms of $\theta_{e}$. It is found that the effect of the motion of the observer on the Sunyaev-Zeldovich effect will become important in the future high precision observation projects of the Sunyaev-Zeldovich effect. Comment: 17 pages, 2 figures, in press in A&A

Full text

Page 1

arXiv:astro-ph/0501114v2 6 Jul 2005
Astronomy & Astrophysics manuscript no. manuscript
(DOI: will be inserted by hand later)
February 2, 2008
Relativistic corrections to the Sunyaev-Zeldovich effect for
clusters of galaxies: Effect of the motion of the observer
Satoshi Nozawa1, Naoki Itoh2, and Yasuharu Kohyama3
1Josai Junior College for Women, 1-1 Keyakidai, Sakado-shi, Saitama, 350-0290, Japan
e-mail: snozawa@josai.ac.jp
2Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo, 102-8554, Japan
email: n itoh@sophia.ac.jp
3Mizuho Information and Research Institute, 2-3 Kanda-Nishiki-cho, Chiyoda-ku, Tokyo, 101-8443, Japan
yasuharu.kohyama@gene.mizuho-ir.co.jp
Received / Accepted
Abstract. We extend the formalism of the relativistic thermal and kinematical Sunyaev-Zeldovich effects to the
observer’s system (the Solar System) moving with a velocity βS ≡ vS/c with respect to the cosmic microwave
background radiation. The present formulation makes full use of the Lorentz covariance properties of the problem
and gives solutions with high precision. We confirm the results recently obtained by Chluba, Huetsi, and Sunyaev
in the lowest order of the observer’s velocity βS. We give a more general analytic expression for the thermal and
kinematical Sunyaev-Zeldovich effects corresponding to the observer’s system (the Solar System) with the power
series expansion approximation in terms of θe ≡ kBTe/mc2, where Te and m are the electron temperature and
the electron mass, respectively.
Key words. cosmology: cosmic microwave background — cosmology: theory — galaxies: clusters: general —
radiation mechanisms: thermal — relativity
1. Introduction
Compton scattering of the cosmic microwave background
radiation (CMBR) by hot intracluster gas — the Sunyaev-
Zeldovich effect (Zeldovich & Sunyaev 1969; Sunyaev
& Zeldovich 1972, 1980a,b, 1981) — provides a useful
method for studies of cosmology (see recent excellent
reviews: Birkinshaw 1999; Carlstrom, Holder, & Reese
2002). The original Sunyaev-Zeldovich formula has been
derived from a kinetic equation for the photon distribution
function taking the Compton scattering by electrons into
account: the Kompaneets equation (Kompaneets 1957;
Weymann 1965). The original Kompaneets equation has
been derived with a nonrelativistic approximation for the
electron; however, recent X-ray observations have revealed
the existence of many high-temperature galaxy clusters
(Tucker et al. 1998; Markevitch 1998; Allen et al. 2001;
Schmidt et al. 2001; Allen et al. 2002). Some galaxy clus-
ters have been found to possess the electron temperature
kBTe≃20keV. Wright (1979) and Rephaeli and his collab-
orator (Rephaeli 1995; Rephaeli & Yankovitch 1997) have
Send offprint requests to: Naoki Itoh
done pioneering work taking the relativistic corrections to
the Sunyaev-Zeldovich effect into account for clusters of
galaxies.
In recent years remarkable progress has been made
in theoretical studies of the relativistic corrections to
the Sunyaev-Zeldovich effects for clusters of galaxies.
Stebbins (1997) generalized the Kompaneets equation.
Itoh, Kohyama & Nozawa (1998) have adopted a relativis-
tically covariant formalism to describe the Compton scat-
tering process (Berestetskii, Lifshitz, & Pitaevskii 1982;
Buchler & Yueh 1976), thereby obtaining higher-order rel-
ativistic corrections to the thermal Sunyaev-Zeldovich ef-
fect in the form of the Fokker-Planck expansion. In their
derivation, the scheme to conserve the photon number
at every stage of the expansion proposed by Challinor &
Lasenby (1998) played an essential role.
Nozawa, Itoh & Kohyama (1998b) have extended their
method to the case where the galaxy cluster is moving
with a peculiar velocity with respect to CMBR. They have
thereby obtained the relativistic corrections to the kine-
matical Sunyaev-Zeldovich effect. Challinor & Lasenby
(1999b) then confirmed the correctness of the result ob-

Page 2

2Nozawa et al.: Relativistic Corrections to the Sunyaev-Zeldovich Effect
tained by Nozawa et al. (1998b). Sazonov & Sunyaev
(1998a,b) calculated the kinematical Sunyaev-Zeldovich
effect by a different method.
Itoh, Nozawa & Kohyama (2000) have also applied
their method to calculation of the relativistic corrections
to the polarization Sunyaev-Zeldovich effect (Sunyaev &
Zeldovich 1980b, 1981). They thereby confirmed the re-
sult of Challinor, Ford & Lasenby (1999a) which was ob-
tained with a completely different method. Recent work
on the polarization Sunyaev-Zeldovicheffect include Audit
& Simons (1999), Hansen & Lilje (1999), and Sazonov &
Sunyaev (1999).
Sazonov & Sunyaev (1998a,b) reported the results of
the Monte Carlo calculations on the relativistic corrections
to the Sunyaev-Zeldovich effect. In Sazonov & Sunyaev
(1998b), a numerical table that summarizes the results of
the Monte Carlo calculations was presented. Nozawa et al.
(2000) presented an accurate analyic fitting formula that
has a high accuracy for the ranges 0.00 ≤ θe≤ 0.05 and
0 ≤ X ≤ 20, where θe ≡ kBTe/mec2, X ≡ ¯ hω/kBT0.
Another fitting formula with a still higher precision that
is valid for the more limited ranges 0.00 ≤ θe ≤ 0.035,
0 ≤ X ≤ 15 was developed by Itoh et al. (2000b).
Relativistic corrections to the double scattering effect on
the Sunyaev-Zeldovich effect was calculated by Itoh et al.
(2001). Dolgov et al. (2001) carried out an independent nu-
merical calculation, and their result shows excellent agree-
ment with that of Itoh et al. (2001) for the case of small
optical depth.
Very recently Kitayama et al. (2004) measured the
Sunyaev-Zeldovich effect in the galaxy cluster RX J1347-
1145. They have discovered that the south-east excess
component of this galaxy cluster has a temperature
kBTex= 28.5 ± 7.3keV. They attribute the high temper-
ature of this excess component to a recent major merger
as discussed by Sarazin (2003). Therefore it is important
to present accurate numerical data for kBTe ≥ 25keV.
Itoh & Nozawa (2004) have accordingly presented an ac-
curate numerical table for the relativistic corrections to
the Sunyaev-Zeldovich effect for clusters of galaxies in the
ranges 0.002 ≤ θe ≤ 0.100. For analyses of the galaxy
clusters with extremely high temperatures, the results of
the calculation of the relativistic thermal bremsstrahlung
Gaunt factor (Nozawa, Itoh & Kohyama 1998a) and their
accurate analytic fitting formulae (Itoh et al. 2000b) will
be useful. The nonrelativistic electron-electron thermal
bremsstrahlung Gaunt factor has been also calculated by
Itoh, Kawana & Nozawa (2002a).
It now appears that all the necessary theoretical tools
are in place for accurate analysis of the observational
data of the Sunyaev-Zeldovich effect for galaxy clusters.
Recently, however, Chluba, Huetsi, & Sunyaev (2005)
point out the importance of the influence of the Solar
System’s motion with respect to the CMBR rest frame.
Assuming that the CMBR dipole is fully motion-induced,
we deduce that the Solar System is moving with a ve-
locity βS ≡ vS/c = 1.241×10−3towards the direction
(ℓ,b) = (264.14◦± 0.15◦, 48.26◦± 0.15◦) (Smoot et al.
1977; Fixsen et al. 1996; Fixsen & Mather 2002). Chluba
et al. (2005) have calculated corrections to the Sunyaev-
Zeldovich effect arising from the motion of the Solar
System. In this paper we wish to calculate those correc-
tions to the Sunyaev-Zeldovich effect which are due to the
motion of the Solar System in a more general way. As a
matter of fact, it turns out that one can make full use
of the Lorentz covariance properties of the problem and
obtain solutions with high precision by extending the re-
sults obtained by Itoh, Kohyama & Nozawa (1998) and
Nozawa, Itoh & Kohyama (1998b).
The present paper is organized as follows. In § 2, the
general formalism is presented in deriving the thermal and
kinematical Sunyaev-Zeldovich effects by taking the mo-
tion of the Solar System into account. With the power se-
ries expansion approximation, an analytic expression in-
cluding the relativistic effects is derived for the thermal
and kinematical Sunyaev-Zeldovich effects also by taking
the motion of the Solar System into account. Numerical re-
sults are presented in § 3, and finally, concluding remarks
are given in § 4.
2. Lorentz-boosted Kompaneets equation
In the present section we extend the work of Nozawa, Itoh,
& Kohyama (1998b) by first extending the Kompaneets
equation to a system (the cluster of galaxies) moving
with a peculiar velocity with respect to the CMBR. To
formulate the kinetic equation for the photon distribu-
tion function we use a relativistically covariant formalism
(Berestetskii, Lifshitz, & Pitaevskii 1982; Buchler & Yueh
1976). As a reference system, we have chosen the system
which is fixed to the cosmic microwave background radia-
tion (CMBR). The z-axis is fixed to a line connecting the
observer and the center of mass of the cluster of galax-
ies (CG), first assuming that the observer is fixed to the
CMBR frame. We fix the positive direction of the z-axis
as the direction of the propagation of a photon from the
cluster to the observer. In this reference system, the cen-
ter of mass of the CG is moving with a peculiar velocity
β(≡ v/c) with respect to the CMBR. For simplicity, we
choose the direction of the velocity in the x-z plane, i.e.
β = (βx,0,βz).
In the CMBR frame the four-momenta of the initial
electron and photon are p = (E,p) and k = (ω,0,0,k),
respectively, while the four-momenta of the final electron
and photon are p′= (E′,p′) and k′= (ω′,k′), respec-
tively. The electron distribution functions in the initial
and final states are Fermi–like in the CG frame. They are
related to the electron distribution functions in the CMBR
frame as follows (Landau & Lifshitz 1975):
f(E) = fC(EC),
f(E′) = fC(E′
EC = γ (E − β · p) ,
E′
(1)
C), (2)
(3)
C= γ (E′− β · p′) ,
1
?1 − β2,
(4)
γ ≡
(5)

Page 3

Nozawa et al.: Relativistic Corrections to the Sunyaev-Zeldovich Effect3
where the suffix C denotes the CG frame.
In addition to θe, there is another parameter β. For
most of the CG, β ≪ 1 is realized. For example, β ≈ 1/300
for a typical value of the peculiar velocity v=1,000km/s.
Therefore it should be sufficient to expand in powers of β
and to retain up to O(β2) contributions. We assume the
initial photon distribution of the CMBR to be Planckian
with a temperature T0:
n0(X) =
1
eX− 1,(6)
where
X ≡
ω
kBT0. (7)
Nozawa, Itoh & Kohyama (1998b) have obtained the
following expression for the fractional distortion of the
photon spectrum:
∆n(X)
n0(X)
=
τ XeX
eX− 1θe
+θ3
τ XeX
eX− 1β2
τ XeX
eX− 1β P1(ˆβz)?1 + θeC1+ θ2
τ XeX
eX− 1β2P2(ˆβz) [D0+ θeD1] ,
βz
β
P1(ˆβz) =ˆβz,
1
2
?Y0 + θeY1+ θ2
eY3 + θ4
?
?1
eY2
eY4
+
3Y0 + θe
?5
6Y0 +2
3Y1
??
?
+
eC2
+
(8)
ˆβz ≡
= cosθγ, (9)
(10)
P2(ˆβz) =
?
3ˆβ2
z− 1
?
,(11)
where θγis the angle between the directions of the pecu-
liar velocity of the cluster (β) and the initial photon mo-
mentum (k), which is chosen as the positive z-direction.
The reader should notice that this sign convention for the
positive z-direction is the opposite of the ordinary one.
Thus a cluster moving away from the observer hasˆβz< 0.
However, because of the positive sign in front of the P1(ˆβz)
term, one obtains ∆n(X) < 0 in this case, as one should.
The coefficients are defined as follows:
Y0 = −4 +˜ X ,
Y1 = −10 +47
(12)
2
˜ X −42
5
?
˜ X2+7
10
˜ X3
+˜S2
?
−21
5
+7
5
˜ X −868
˜ X,(13)
Y2 = −15
2
+1023
85
˜ X2+329
5
˜ X3
−44
5
?
?
˜ X4+11
30
+658
˜ X5
+˜S2
−434
55
˜ X −242
5
˜ X2+143
30
˜ X3
?
+˜S4
−44
5
+187
60
˜ X
?
, (14)
Y3 =
15
2
−18594
35
?
+156767
140
?
+1024
35
?
+2505
8
˜ X4+12059
˜ X −7098
5
˜ X2+14253
10
˜ X6+
˜ X3
140
˜ X5−128
21
16
105
˜ X7
+˜S2
−7098
10
+14253
5
˜ X −102267
35
˜ X2
˜ X3−1216
7
˜ X4+64
7
˜ X5
?
˜ X2
+˜S4
−18594
35
+205003
280
˜ X −1920
7
˜ X3
?
+992
105
+˜S6
−544
21
+30375
˜ X
?
, (15)
Y4 = −135
32128
˜ X4+355703
˜ X −62391
10
˜ X5−16568
˜ X2+614727
40
˜ X6
˜ X3
−124389
10
+7516
105
?
+4624139
80
−2717
7
?
+481024
35
?
+19778
105
?
8021
˜ X7−22
7
+614727
˜ X8+11
210
˜ X −1368279
˜ X9
+˜S2
−62391
20 2020
˜ X2
˜ X3−157396
7
˜ X4+30064
7
˜ X5
˜ X6+2761
210
+6046951
160
˜ X3−15972
˜ X7
?
+˜S4
−124389
10
˜ X −248520
7
˜ X2
7
˜ X4+18689
140
˜ X5
?
+˜S6
−70414
21
+465992
105
˜ X −11792
7
˜ X2
˜ X3
?
+7601
210
+˜S8
−682
7
˜ X
?
, (16)
C1 = 10 −47
5
˜ X +7
5
˜ X2+
7
10
˜S2, (17)
C2 = 25 −1117
10
˜ X +847
10
˜ X +121
˜ X2−183
10
˜ X3+11
10
˜ X4
+˜S2
?847
3+11
20
−183
520
˜ X2
?
+11
10
˜S4,(18)
D0 = −2
30
˜ X ,(19)
D1 = −4 + 12˜ X − 6˜ X2+19
30
˜ X3
+˜S2
?
−3 +19
15
˜ X
?
,(20)
and
τ ≡ σT
?
dℓNe,(21)

Page 4

4 Nozawa et al.: Relativistic Corrections to the Sunyaev-Zeldovich Effect
˜ X ≡ X coth
?X
? ,
2
?
, (22)
˜S ≡
X
?X
sinh
2
(23)
where Neis the electron number density in the CG frame,
the integral in Eq. (21) is over the photon path length in
the cluster, and σT is the Thomson cross section.
In the next step of the calculation, we suppose that
the observer’s system (the Solar System) is moving with
a velocity βS(≡ vS/c) with respect to the CMBR. The
photon distribution function in the CMBR system n(ω,k)
and the photon distribution function in the Solar System
nS(ωS,kS) are related by (Landau & Lifshitz 1975)
n(ω,k) = nS(ωS,kS),
ω = γS(ωS+ βS· kS),
(24)
(25)
γS ≡
1
?1 − β2
S
.(26)
The photon unit wave vector in the Solar SystemˆkSand
the photon unit wave vector in the CMBR systemˆk are
related by (Møller 1962)
ˆk =
?ˆkS·ˆβS+ βS
1 +ˆkS· βS
?
ˆβS+
ˆkS− (ˆkS·ˆβS)ˆβS
γS(1 +ˆkS· βS)
,(27)
whereˆβSis a unit vector in the direction of βS.
Therefore, our task is to rewrite Eq. (8) for the ex-
pression corresponding to the Solar System. The isotropic
Planck distribution in the CMBR system, Eq. (6), corre-
sponds to the distribution including the dipolar distortion
in the Solar System
n0
S(XS,ˆkS) =
1
exp{γSXS(1 + βSµS)} − 1,(28)
where
XS ≡
ωS
kBT0,(29)
µS ≡ cosθS,(30)
θSbeing the angle between βSand kS. Here we have used
the relationship
X = γSXS(1 + βSµS).(31)
The velocity vecor β of the galaxy cluster in the CMBR
system is transformed to the velocity vector β′of the
galaxy cluster in the Solar System by the relationship
(Møller 1962)
β =
?
β′·ˆβS+ βS
1 + β′· βS
?
ˆβS+β′− (β′·ˆβS)ˆβS
γS(1 + β′· βS).(32)
Therefore, βz in the CMBR system in Eq. (9) is trans-
formed to
βz ≡ β ·ˆk
=
1
(1 + β′· βS)(1 +ˆkS· βS)
?
+(1 − β2
S)
β′·ˆkS− (β′·ˆβS)(ˆβS·ˆkS)
×
(β′·ˆβS+ βS)(ˆkS·ˆβS+ βS)
???
.(33)
We write the photon distribution function in the Solar
System frame that has a distortion caused by the thermal
and kinematical Sunyaev-Zeldovich effects as
nS(XS,ˆkS) = n0
S(XS,ˆkS) + ∆nS(XS,ˆkS).(34)
Here ∆nS(XS,ˆkS) is given by
∆nS(XS,ˆkS) =
τ XeX
(eX− 1)2
+θ3
?θe
eY4
?Y0+ θeY1 + θ2
?
?5
eY2
eY3 + θ4
?1
+ β2
3Y0+ θe
6Y0 +2
3Y1
??
?
+ β P1(ˆβz)?1 + θeC1 + θ2
+ β2P2(ˆβz) [D0+ θeD1]
eC2
?
,(35)
where one should rewrite X by using Eq. (31) and also
βP1(ˆβz) and β2P2(ˆβz) by using Eqs. (32), (33). The
present result can be used in order to analyze the observa-
tional data of the Sunyaev-Zeldovich effect. We note that
βS= 1.241 × 10−3is a known quantity and µS= cosθSis
also a known quantity for the galaxy cluster under con-
sideration. For present and near-future observations of the
Sunyaev-Zeldovich effect, β2terms in Eq. (35) are beyond
observational accuracy, such that we only consider the first
and third terms in the braces of Eq. (35). When the opti-
cal depth of the galaxy cluster τ is known, we obtain the
best-fit values of θe and βz by fitting the observational
data with the first and third terms of Eq. (35) written for
the Solar System frame. In this way we can obtain the
cluster temperature, as well as the cluster velocity in the
direction of the photon propagation vector observed by
the observer fixed to the CMBR system.
In the somewhat more distant future accuracy in the
observation of the Sunyaev-Zeldovich effect will be hope-
fully improved dramatically. In that case one should also
include the β2terms in Eq. (35) and use the full expres-
sion for analysis of the observational data. In doing so it
should be possible to also determine the cluster velocity
perpendicular to the photon propagation vector. We de-
fine the distortion of the spectral intensity in the Solar
System frame as follows:
∆IS ≡ ∆nS(XS,ˆkS)X3
S.(36)
3. Numerical results
Equations (31) and (35) present our main results for the
effect of the observer’s the motion (the Solar System). It
should be emphasized here that because no expansions
have been made so far in deriving Eqs. (31) and (35) for

Page 5

Nozawa et al.: Relativistic Corrections to the Sunyaev-Zeldovich Effect5
Fig.1. Spectral intensity of the thermal Sunyavev-
Zeldovich effect for kBTe = 10keV and µS = −1 as a
function of XS. The solid curve is the full contribution.
The dash-dotted curve is the lowest-order (Y0) contribu-
tion. The dashed curve is the extraction of the full βS
corrections (multiplied by 100). The dotted curve shows
results of Chluba et al. (2005) (multiplied by 100). Refer
to the main text for detailed explanations.
βS, they are general expressions for arbitrary values of
βS. In practical cases, however, βS ≪ 1 is achieved. By
expanding X around XS and keeping only lowest order
terms for both βS and θe, we have reproduced the Eqs.
(12a) and (12b) of Chluba, Huetsi, and Sunyaev (2005).
In Fig. 1, we plot the spectral intensity of the thermal
Sunyavev-Zeldovich effect for kBTe= 10keV and µS= −1
as a function of XS. Here the solid curve is the full con-
tribution (Y0to Y4terms) of the first line of Eq. (35), and
the dash-dotted curve is the lowest-order (Y0 term) con-
tribution of the first line of Eq. (35). The dashed curve
is the extraction of the full βScorrections of the first line
of Eq. (35), which is multiplied by 100 in order to be
visible in the same figure. This curve contains full βScor-
rections. The dotted curve shows the results of Chluba,
Huetsi, and Sunyaev (2005) (Eq. (12a) in their paper),
where only lowest-order corrections are included for both
βS and θe. This correction is also multiplied by 100 in
order to be visible in the same figure. It should be noted
that the β2
Scorrections are totally negligible. The essential
difference in two curves are due to higher-order relativistic
corrections of θe, which is present in Eq. (35) but is ab-
sent in Eq. (12a) of Chluba, Huetsi, and Sunyaev (2005).
Therefore, higher order relativistic corrections of θe are
important in discussing the effect of the observer’s motion
in the thermal Sunyaev-Zeldovich effect with high preci-
sion.
In Fig. 2, we have also plotted the spectral intensity
of the kinematical Sunyavev-Zeldovich effect for kBTe=
10keV, β = βz= 1/300, and µS= −1 as a function of XS.
Note that the velocity of the cluster relative to the CMBR
and the velocity of the observer relative to the CMBR are
parallel in this case. The solid curve is the full contribu-
Fig.2. Spectral intensity of the kinematical Sunyavev-
Zeldovich effect for kBTe= 10keV, β = βz= 1/300, and
µS= −1 as a function of XS. The solid curve is the full
contribution. The dash-dotted curve is the lowest-order
contribution βP1(ˆβz). The dashed curve is the extraction
of the full βScorrections (multiplied by 100). The dotted
curve shows the results of Chluba et al. (2005) (multiplied
by 100). Refer to the main text for a detailed explanations.
tion (1 to C2terms) of the third line of Eq. (35), and the
dash-dotted curve is the lowest-order contribution of the
third line of Eq. (35). The dashed curve is the extraction of
the full βScorrections of the third line of Eq. (35), which
is multiplied by 100 in order to be visible in the same fig-
ure. This curve contains full βS corrections. The dotted
curve shows the results of Chluba, Huetsi, and Sunyaev
(2005) (Eq. (12b) in their paper), where only lowest-order
corrections are included for both βS and θe. This correc-
tion is also multiplied by 100 in order to be visible in the
same figure. It should also be noted that the β2
tions are totally negligible. The essential difference in two
curves are due to higher-order relativistic corrections of
θe, which is present in Eq. (35) but is absent in Eq. (12b)
of Chluba, Huetsi, and Sunyaev (2005). Therefore, higher
order relativistic corrections of θeare again important in
discussing the effect of the motion of the observer in the
kinematical Sunaev-Zeldovich effect with high precision.
Scorrec-
4. Concluding remarks
We have calculated the effect of the motion of the ob-
server (the Solar System) on the relativistic thermal and
kinematical Sunyaev-Zeldovich effects with the power se-
ries approximation in terms of θe≡ kBTe/mc2. We made
full use of the Lorentz covariance properties of the prob-
lem to obtain solutions with high precision, and con-
firmed the correctness of the results recently obtained
by Chluba, Huetsi, and Sunyaev in the lowest-order of
the observer’s velocity βS. We have given a more gen-
eral analytic expression for the thermal and kinematical
Sunyaev-Zeldovich effects corresponding to the observer’s
system (the Solar System). We also found that the ef-
fect of the motion of the observer (the Solar System) on

Page 6

6Nozawa et al.: Relativistic Corrections to the Sunyaev-Zeldovich Effect
the thermal and kinematical Sunyaev-Zeldovich effects is
marginally important for present and near-future observa-
tions. However, for the next-generation high-precision ob-
servations, we find the effect of the motion of the observer
(the Solar System) could be important and should be
taken into account for analysis of the observational data.
We also note that the next-generation high-precision kine-
matic Sunyaev-Zeldovich effect observations will hopefully
allow us to determine the cluster velocity perpendicular to
the photon propagation vector, as well as the cluster ve-
locity in the direction of the photon propagation vector.
Acknowledgements. We wish to thank Dr. Chluba for sending
their manuscript to us prior to publication of their paper. We
also wish to thank our referee for the helpful suggestions for
improving the original manuscript. This work is financially sup-
ported in part by the Grants-in-Aid of the Japanese Ministry
of Education, Culture, Sports, Science, and Technology under
contracts #13640245, #15540293, and #16540220.
References
Allen, S. W., Ettori, S., & Fabian, A. C. 2001, MNRAS, 324,
877
Allen, S. W., Schmidt, R. W., & Fabian, A. C. 2002, MNRAS,
335, 256
Audit, E., & Simmons, J. F. L. 1999, MNRAS, 305, L27
Berestetskii, V. B., Lifshitz, E. M., & Pitaevskii, L. P. 1982,
Quantum Electrodynamics (Oxford: Pergamon)
Birkinshaw, M. 1999, Physics Reports, 310, 97
Buchler, J. R., & Yueh, W. R. 1976, ApJ, 210, 440
Carlstrom, J. E., Holder, G. P., & Reese, E. D. 2002, ARA&A,
40, 643
Challinor, A., Ford, M., & Lasenby, A., 1999, MNRAS, 312,
159.
Challinor, A., & Lasenby, A., 1998, ApJ, 499, 1
Challinor, A., & Lasenby, A., 1999, ApJ, 510, 930
Chluba, J., Huetsi, G., & Sunyaev, R. A. 2005, A&A, 434, 811
Dolgov, A. D., Hansen, S. H., Pastor, S., & Semikoz, D. V.
2001, ApJ, 554, 74
Fixsen, D. J., Cheng, E. S., Gales, J. M., Mather, J. C., Shafer,
R. A., & Wright, E. L. 1996, ApJ, 473, 576
Fixsen, D. J., & Mather, J. C. 2002, ApJ, 581, 817
Hansen, E., & Lilje, P. B. 1999, MNRAS, 306, 153
Itoh, N., Kohyama, Y., & Nozawa, S. 1998, ApJ, 502, 7
Itoh, N., & Nozawa, S. 2004, A & A, 417, 827
Itoh, N., Nozawa, S., & Kohyama, Y. 2000a, ApJ, 533, 588
Itoh, N., Sakamoto, T., Kusano, S., Nozawa, S., & Kohyama,
Y. 2000b, ApJS, 128, 125
Itoh, N., Kawana, Y., Nozawa, S., & Kohyama, Y. 2001,
MNRAS, 327, 567
Itoh, N., Kawana, Y., & Nozawa, S. 2002a, Nuovo Cimento,
117B, 359
Itoh, N., Sakamoto, T., Kusano, S., Kawana, Y., & Nozawa, S.
2002b, A&A, 382, 722
Kitayama, T., Komatsu, E., Ota, N., Kuwabara, T., Suto, Y.,
Yoshikawa, K., Hattori, M., & Matsuo, H. 2004, PASJ, 56,
17
Kompaneets, A. S. 1957, Soviet Physics JETP, 4, 730
Landau, L. D., & Lifshitz, E. M. 1975, The Classical Theory
of Fields (Oxford: Pergamon)
Markevitch, M. 1998, ApJ, 504, 27
Møller, C. 1962, The Theory of Relativity (Oxford: Oxford
University Press)
Nozawa, S., Itoh, N., Kawana, Y., & Kohyama, Y. 2000, ApJ,
536, 31
Nozawa, S., Itoh, N., & Kohyama, Y. 1998a, ApJ, 507, 530
Nozawa, S., Itoh, N., & Kohyama, Y. 1998b, ApJ, 508, 17
Rephaeli, Y. 1995, ApJ, 445, 33
Rephaeli. Y., & Yankovitch, D. 1997, ApJ, 481, L55
Sarazin, C. L. 2003, preprint [astro-ph/0310157]
Sazonov, S. Y., & Sunyaev, R. A. 1998a, ApJ, 508, 1
Sazonov, S. Y., & Sunyaev, R. A. 1998b, Astronomy Letters
24, 553
Sazonov, S. Y., & Sunyaev, R. A. 1999, MNRAS, 310, 765
Schmidt, R. W., Allen, S. W., & Fabian, A. C. 2001, MNRAS,
327, 1057
Smoot, G. E., Gorenstein, M. V., & Muller, R. A. 1977, Phys.
Rev. Lett., 39, 898
Stebbins, A., 1997, preprint [astro-ph/9705178]
Sunyaev, R. A., & Zeldovich, Ya. B. 1972, Comments
Astrophys. Space Sci., 4, 173
Sunyaev, R. A., & Zeldovich, Ya. B. 1980a, ARA&A, 18, 537
Sunyaev, R. A., & Zeldovich, Ya. B. 1980b, MNRAS, 190, 413
Sunyaev, R. A., & Zeldovich, Ya. B. 1981, Astrophys. Space
Phys. Rev., 1, 1
Tucker, W., Blanco, P., Rappoport, S., David, L., Fabricant,
D., Falco, E. E., Forman, W., Dressler, A., & Ramella, M.
1998, ApJ, 496, L5
Weymann, R. 1965, Phys. Fluid, 8, 2112
Wright, E. L. 1979, ApJ, 232, 348
Zeldovich, Ya. B., & Sunyaev, R. A. 1969, Ap&SS, 4, 301