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arXiv:0902.2595v2 [astro-ph.CO] 28 May 2009

Analytical Study on the Sunyaev-Zeldovich Effect for Clusters of Galaxies

Satoshi Nozawa∗

Josai Junior College, 1-1 Keyakidai, Sakado-shi, Saitama, 350-0295, Japan

Yasuharu Kohyama

Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo, 102-8554, Japan

(Dated: May 28, 2009)

Starting from a covariant formalism of the Sunyaev-Zeldovich effect for the thermal and non-

thermal distributions, we derive the frequency redistribution function identical to Wright’s method

assuming the smallness of the photon energy (in the Thomson limit). We also derive the redistri-

bution function in the covariant formalism in the Thomson limit. We show that two redistribution

functions are mathematically equivalent in the Thomson limit which is fully valid for the cosmic

microwave background photon energies.We will also extend the formalism to the kinematical

Sunyaev-Zeldovich effect. With the present formalism we will clarify the situation for the discrep-

ancy existed in the higher order terms of the kinematical Sunyaev-Zeldovich effect.

PACS numbers: 95.30.Cq,95.30.Jx,98.65.Cw,98.70.Vc

Keywords: cosmology: cosmic microwave background — cosmology: theory — galaxies: clusters: general —

radiation mechanisms: thermal — relativity

I.INTRODUCTION

The Sunyaev-Zeldovich (SZ) effect[1, 2, 3, 4], which arises from the Compton scattering of the cosmic microwave

background (CMB) photons by hot electrons in clusters of galaxies (CG), provides a useful method for studies of

cosmology. For the reviews, for example, see Birkinshaw[5] and Carlstrom, Holder and Reese[6]. The original SZ

formula has been derived from the Kompaneets equation[7] in the non-relativistic approximation. However, recent

X-ray observations (for example, Schmidt et al.[8] and Allen et al.[9]) have revealed the existence of high-temperature

CG such as kBTe≃20keV. Wright[10] and Rephaeli and his collaborator[11, 12] have done pioneering work including

the relativistic corrections to the SZ effect for the CG.

In the last ten years remarkable progress has been made in theoretical studies of the relativistic corrections to

the SZ effects for the CG. Stebbins[13] generalized the Kompaneets equation. Challinor and Lasenby[14] and Itoh,

Kohyama and Nozawa[15] have adopted a relativistically covariant formalism to describe the Compton scattering

process and have obtained higher-order relativistic corrections to the thermal SZ effect in the form of the Fokker-

Planck approximation. Nozawa, Itoh and Kohyama[16] have extended their method to the case where the CG is moving

with a peculiar velocity with respect to the CMB and have obtained the relativistic corrections to the kinematical

SZ effect. Their results were confirmed by Challinor and Lasenby[17] and also by Sazonov and Sunyaev[18, 19]. Itoh,

Nozawa and Kohyama[20] have also applied the covariant formalism to the polarization SZ effect[3, 4]. Itoh and his

collaborators (including the present authors) have done extensive studies on the SZ effects, which include the double

scattering effect[21], the effect of the motion of the observer[22], high precision analytic fitting formulae to the direct

numerical integrations[23, 24] and high precision calculations[25, 26]. The importance of the relativistic corrections is

also exemplified through the possibility of directly measuring the cluster temperature using purely the SZ effect[27].

On the other hand, the SZ effect in the CG has been studied also for the non-thermal distributions by several

groups[28, 29, 30]. The non-thermal distribution functions, for example, the power-law distributions, have a long tail

in high electron energy regions. Therefore the relativistic corrections for the SZ effect could be more important than

the thermal distribution.

Shimon and Rephaeli[31] have discussed on the equivalence of different formalisms to the SZ effect. The relativistic

SZ effect has been studied analytically so far in three different approaches. The first method is the calculation of the fre-

quency redistribution function in the electron rest frame using the scattering probability derived by Chandrasekhar[32].

This method was used by Wright[10] and extended by Rephaeli[11]. We call it as Wright’s method in the present

paper. The second approach solves the photon transfer equation in the electron rest frame. This approach was

used by Sazonov and Sunyaev[18]. We call it the radiative transfer method. The third approach is the relativistic

∗Electronic address: snozawa@josai.ac.jp

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generalization of the Kompaneets equation[7], where the relativistically covariant Boltzmann collisional equation is

solved for the photon distribution function. This approach was used by Challinor and Lasenby[14] and Itoh, Kohyama

and Nozawa[15]. We call it as the covariant formalism in the present paper. In Shimon and Rephaeli[31] they have

shown the equivalence between Wright’s method and the radiative transfer method. They also have claimed the

equivalence between Wright’s method and the covariant formalism. However, no mathematical relations are shown

between the redistribution function in Wright’s method and the expression of the scattering probability in the covari-

ant formalism. Therefore their claim is incomplete. In the present paper we will show explicitly that two approaches

are mathematically equivalent.

On the other hand, recently Boehm and J. Lavalle[30] also have discussed the equivalence of the different approaches

for the SZ effect in the non-thermal distribution. They have shown that the radiative transfer method is equivalent

to the covariant formalism. However, they have concluded that Wright’s method is incorrect. In the present paper

we will show that their conclusion is incorrect. We will show that Wright’s method, which has been widely used in

the literature, is still fully valid.

The fourth method for the study of the SZ effect is the direct numerical integration of the rate equation of the

photon spectral distortion function. The first-order calculation in terms of the optical depth τ was done by Itoh,

Kohyama and Nozawa[15] for τ ≪ 1. The full-order calculation was done by Dolgov et al.[33] for τ ≫ 1. The rate

equation in the present formalism has a simple form. Therefore it is more suitable for the direct numerical application.

We will present the numerical calculation elsewhere[34].

The present paper is organized as follows. In § II, we show the equivalence between Wright’s method and the

covariant formalism of the SZ effect for both thermal and non-thermal distributions. We also derive the rate equations

and their formal solutions for the photon distribution function and for the spectral intensity function. In § III we extend

the formalism to the kinematical SZ effect, and derive the rate equations in Wright’s method. Finally, concluding

remarks are given in § IV.

II. SUNYAEV-ZELDOVICH EFFECT

A. Equivalence between Covariant Formalism and Wright’s Method

Let us consider that both of the CG and the observer are fixed to the CMB frame. As a reference system, we choose

the system which is fixed to the CMB. (Three frames are identical in the present case.) In the CMB frame, the time

evolution of the photon distribution function n(ω) is written as follows[15]:

∂n(ω)

∂t

= −2

?

d3p

(2π)3d3p′d3k′W {n(ω)[1 + n(ω′)]f(E) − n(ω′)[1 + n(ω)]f(E′)} ,(1)

W =(e2/4π)2 ¯ X δ4(p + k − p′− k′)

2ωω′EE′

?κ

κ = −2(p · k) = −2ωE (1 − βµ) ,

κ′= 2(p · k′) = 2ω′E (1 − βµ′) ,

, (2)

¯ X = −

κ′+κ′

κ

?

+ 4m4

?1

κ+1

κ′

?2

− 4m2

?1

κ+1

κ′

?

,(3)

(4)

(5)

where e is the electric charge, m is the electron rest mass, W is the transition probability of the Compton scattering,

and f(E) is the electron distribution function. The four-momenta of the initial electron and photon are p = (E,? p)

and k = (ω,?k), respectively. The four-momenta of the final electron and photon are p′= (E′,? p′) and k′= (ω′,?k′),

respectively. In Eqs. (4) and (5), β = |? p|/E, µ = cosθ is the cosine between ? p and?k, and µ′= cosθ′is the cosine

between ? p and?k′. Throughout this paper, we use the natural unit ? = c = 1, unless otherwise stated explicitly. For

later convenience we rewrite Eq. (3) as follows:

¯ X =¯ XA+¯ XB, (6)

¯ XA= 2 + 4m4

?1

κ+1

κ′

?2

− 4m2

?1

κ+1

κ′

?

, (7)

¯ XB= −4(k · k′)2

κκ′

. (8)

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By eliminating the δ-function, Eq. (1) is rewritten as follows:

∂n(ω)

∂τ

= −

3

64π2

×{n(ω)[1 + n(ω′)]pe(E) − n(ω′)[1 + n(ω)]pe(E′)} ,

?

d3p

?

dΩk′1

γ2

1

1 − βµ

?ω′

ω

?2

¯ X

(9)

dτ = neσTdt,(10)

γ =

1

?1 − β2,

f(E) = neπ2pe(E),

(11)

(12)

where ne is the electron number density, σT is the Thomson scattering cross section, and pe(E) is normalized by

?∞

0dpp2pe(E) = 1. By choosing the direction of the initial electron momentum (? p) along z-axis, the photon momenta

?k and?k′are expressed by

?k = ω

??

1 − µ2cosφk,

?

1 − µ2sinφk,µ

?

,(13)

?k′= ω′??

1 − µ′2cosφk′,

?

1 − µ′2sinφk′,µ′?

, (14)

where φk and φk′ are the azimuthal angles of?k and?k′, respectively. Inserting Eqs. (13) and (14) into Eqs. (7) and

(8), one obtains

¯ XA= 2 +

(1 − cosΘ)2

γ4(1 − βµ)2(1 − βµ′)2− 2

?2ω′

1 − βµ

1 − βµ′+ (ω/γm)(1 − cosΘ),

cosΘ ≡ µµ′+

?

1 − cosΘ

γ2(1 − βµ)(1 − βµ′),(15)

¯ XB=

?ω

γmω

(1 − cosΘ)2

(1 − βµ)(1 − βµ′), (16)

ω′

ω= (17)

1 − µ2?

1 − µ′2cos(φk− φk′), (18)

where cosΘ is the cosine between?k and?k′. It should be noted that¯ XAand¯ XBwill not be mixed each other under

an arbitrary Lorentz transformation, because¯ XAdepends only on µ, µ′, β and γ, whereas¯ XBdepends also on ω and

ω′.

Now let us introduce the transformations for µ and µ′which will play a key role in the present paper.

µ =−µ0+ β

1 − βµ0

µ′=−µ′

1 − βµ′

,(19)

0+ β

0

,(20)

where µ0 = cosθ0 and µ′

frame throughout this paper, unless otherwise stated explicitly. Equations (19) and (20) are the composition of the

Lorentz transformation for the photon angles from the CMB frame to the electron rest frame and the transformation

θ0 → π − θ0, θ′

Eqs. (15) – (18), one obtains as follows:

0= cosθ′

0are cosines in the electron rest frame. The suffix 0 denotes the electron rest

0→ π − θ′

0. Note that the latter transformation is not essential. Applying Eqs. (19) and (20) to

¯ XA= 1 + cos2Θ0, (21)

¯ XB=

?ω

1 − βµ0+ (ω/γm)(1 − cosΘ0),

?

γm

?2ω′

ω

(1 − cosΘ0)2

(1 − βµ0)(1 − βµ′

1 − βµ′

0

0), (22)

ω′

ω

= (23)

cosΘ0≡ µ0µ′

0+

1 − µ2

0

?

1 − µ′2

0cos(φk− φk′), (24)

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where cosΘ0is the cosine between?k and?k′in the electron rest frame. It can be seen that Eq. (21) was surprisingly

simplified compared with Eq. (15). On the other hand Eq. (22) did not change its form compared with Eq. (16). As

will see later in this section, Eqs. (21) and (22) are the key points for connecting the covariant formalism with Wright’s

method. The terms¯ XAand¯ XB did not mix each other by the above reason. Furthermore¯ XAis the expression in

the electron rest frame, whereas¯ XBis not, because it contains β and γm.

The phase space volumes are transformed as follows:

1

γ2(1 − βµ0)2d3p0,

1

γ2(1 − βµ′

d3p =

(25)

dΩk′ =

0)2dΩk′0,(26)

where d3p0= p2dpdµ0dφp, dΩk′0= dµ′

With these variables Eq. (9) is re-expressed by

0dφk′. Note that the z-axis was chosen along?k direction for the d3p integration.

∂n(ω)

∂τ

= −

3

64π2

×{n(ω)[1 + n(ω′)]pe(E) − n(ω′)[1 + n(ω)]pe(E′)} .

?

d3p0

?

dΩk′0

1

γ4

1

1 − βµ0

1

(1 − βµ′

0)2

?ω′

ω

?2

¯ X

(27)

In deriving Eq. (27) we used the relation γ2(1 − βµ) = (1 − βµ0)−1.

Before proceed to the next step, some explanations might be necessary for Eq. (27). In Eq. (27) photon zenith

angles (µ0 and µ′

other hand, energies (ω, ω′and p) and azimuthal angles (φk, φk′ and φp) are left in the CMB frame. As seen later in

this section, this peculiar hybrid coordinate system makes the connection from the covariant formalism to Wright’s

method in a straightforward manner. It is needless to say that the familiar Klein-Nishina formula in the electron rest

frame will be obtained by the Lorentz transformations ω = ω0γ (1 − βµ0) and ω′= ω′

Eqs. (21) and (22).

Now let us introduce an assumption which was also used in Boehm and Lavalle[30].

γω

m≪ 1.

For the CMB (kBTCMB = 2.348 × 10−4eV) photons ω < 5 × 10−3eV is well satisfied. Then ω/m < 1 × 10−8,

which implies γ ≪ 108. Therefore as far as the CMB photon energies are concerned, Eq. (28) is fully valid from

the non-relativistic region to the extreme-relativistic region for the electron energies. With Eq. (28) the following

approximations are valid.

ω′

ω≈1 − βµ′

1 − βµ0

??

E′= E

?

pe(E′) = pe(E)

[1 + O(γω/m)] for power law distribution

0) are described in the electron rest frame with the transformations of Eq. (19) and (20). On the

0γ (1 − βµ′

0) and inserting into

(28)

0

, (29)

¯ XB= Oγω

m

?2?

?

?[1 + O(TCMB/Te)]

, (30)

1 + Oβγω

m

??

, (31)

for thermal distribution

. (32)

As seen from Eqs. (29)–(32), the Thomson limit is realized in the scattering kinematics by the assumption of Eq. (28).

With these approximations Eq. (27) is reduced to

∂n(ω)

∂τ

=

3

64π2

?

d3p0pe(E)

?

dΩk′0

1

γ4

1

(1 − βµ0)3(1 + cos2Θ0)[n(ω′) − n(ω)] .(33)

Furthermore the φk′-integral can be performed and one obtains

?2π

Inserting Eq. (34) into Eq. (33) and assuming the spherical symmetry for pe(E), one obtains as follows:

?∞

?1

1

2π

0

dφk′?1 + cos2Θ0

?= 1 + µ2

0µ′2

0+1

2(1 − µ2

0)(1 − µ′2

0). (34)

∂n(ω)

∂τ

=

0

dpp2pe(E)

×

−1

dµ0

?1

−1

dµ′

0

1

2γ4

1

(1 − βµ0)3f(µ0,µ′

0)[n(ω′) − n(ω)] , (35)

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f(µ0,µ′

0) =3

8

?

1 + µ2

0µ′2

0+1

2(1 − µ2

0)(1 − µ′2

0)

?

.(36)

According to Wright[10] we introduce a new variable s by

es=ω′

ω

=1 − βµ′

1 − βµ0

0

,(37)

which implies dµ′

0= −(1/β)(1 − βµ0)esds. Then Eq. (35) is finally rewritten by

∂n(ω)

∂τ

=

?∞

0

dpp2pe(E)

?smax

−smax

dsP(s,β)[n(esω) − n(ω)] ,(38)

P(s,β) =

es

2βγ4

?µ2(s)

µ1(s)

dµ0(1 − βµ0)

1

(1 − βµ0)3f (µ0,µ′

0) ,(39)

where

smax= ln[(1 + β)/(1 − β)],

µ′

?−1

µ2(s) =

1

(40)

(41)

0= [1 − es(1 − βµ0)]/β ,

µ1(s) =

for s ≤ 0

for s > 0[1 − e−s(1 + β)]/β

,(42)

?[1 − e−s(1 − β)]/β for s < 0

for s ≥ 0

.(43)

Equation (39) is the probability for a single scattering of a photon of a frequency shift s by an electron with a velocity

β, which is described in the electron rest frame. By using the identity relation 1 − βµ′

identical to P(s;β) (Eq. (7)) in Wright[10]. Thus Wright’s redistribution function has been derived from the covariant

formalism.

Now we will derive the redistribution function in the covariant formalism under the assumption of Eq. (28) (the

Thomson limit). The derivation is straightforward but lengthy. We will give the derivation in Appendix A and will

quote the result here. The expressions which correspond to Eqs. (38) and (39) in the covariant formalism (in the

CMB frame) are

0= es(1 − βµ0), Eq. (39) is

∂n(ω)

∂τ

=

?∞

0

dpp2pe(E)

?smax

e2s

2βγ2

−smax

ds˜P(s,β)[n(esω) − n(ω)] ,(44)

˜P(s,β) =

?µ2(s)

µ1(s)

dµ′˜f (µ,µ′) , (45)

µ = [1 − es(1 − βµ′)]/β ,

(46)

˜f(µ,µ′) =3

8

2 +

(1 − µµ′)2+1

2(1 − µ2)(1 − µ′2)

γ4(1 − βµ)2(1 − βµ′)2

− 2

1 − µµ′

γ2(1 − βµ)(1 − βµ′)

, (47)

where smax, µ1(s) and µ2(s) are defined in Eqs. (40), (42) and (43), respectively. In the present paragraph we show

that˜P(s,β) is identical to P(s,β). In order to show the equivalence, we apply the transformations of Eqs. (19) and

(20) to Eq. (45). First, inserting Eqs. (19) and (20) into Eq. (47), one obtains

˜f(µ,µ′) = f(µ0,µ′

0). (48)

The variables µ′and µ0have the relation

µ′=1

β

?

1 −

e−s

γ2(1 − βµ0)

?

, (49)

which implies

dµ′= −

e−s

γ2(1 − βµ0)2dµ0

(50)

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and boudary values

µ0=

?µ2(s) at µ′= µ1(s)

at µ′= µ2(s)µ1(s)

.(51)

Inserting Eqs. (48)–(51) into Eq. (45), one finally obtains

˜P(s,β) =

es

2βγ4

?µ2(s)

µ1(s)

dµ0

1

(1 − βµ0)2f (µ0,µ′

0) ,(52)

which is identical to Eq. (39). Therefore one obtains

˜P(s,β) = P(s,β).(53)

Thus the equivalence between the covariant formalism of the Boltzmann collisional equation[15] and Wright’s

method[10, 11] has been shown mathematically under an assumption γω/m ≪ 1, where the assumption is fully

valid for the CMB photon energies. It should be emphasized that no non-relativistic approximations are made for the

electron energies in deriving Eqs. (39) and (45). This is the reason why the calculations by two different formalisms

produced same results for the SZ effect even in the relativistic electron energies. In Appendix B, we have also shown

the derivation of Eq. (27) in terms of the Klein-Nishina cross section formula.

Boehm and Lavalle[30] also discussed the equivalence between the radiative transfer approach and the covariant

formalism. However, they concluded that Wright’s method was incorrect. We conclude that their conclusion is

incorrect. The reason why they lead the erroneous conclusion is as follows. They start with the covariant form for

the squared Compton amplitude (their Eq. (43)). They derived the familiar Chandrasekhar’s form (their Eq. (50)) by

taking the non-relativistic limit (β → 0) in their Eq. (49). Because of the non-relativistic approximation they used,

they concluded that Wright’s method (Eq. (50)) should not be used for the relativistic calculation. On the other

hand, we have also started with the same covariant form for the squared Compton amplitude. We have derived the

same expression (Eq. (34)) without taking the non-relativistic limit. We have shown that Eq. (34) is connected to

its covariant form by the Lorentz transformations of Eqs. (19) and (20). Therefore Wright’s method is equivalent

to the covariant formalism. We conclude that their criticism is incorrect. Shimon and Rephaeli[31] also claimed the

equivalence between the covariant formalism and Wright’s method. Their Eq. (19) looks similar to Eq. (38), however,

no mathematical relations are shown explicitly in their paper between W in their Eq. (19) and P(s;β) of Wright[10].

B. Rate Equations and Formal Solutions

We now proceed to derive the rate equations and their formal solutions. Since two formalisms are equivalent, one

can use either P(s,β) or˜P(s,β). We start with Eq. (38) and rewrite as follows:

∂n(ω)

∂τ

=

?∞

?1

−∞

dsP1(s)[n(esω) − n(ω)] ,(54)

P1(s) =

βmin

dββ2γ5˜ pe(β)P(s,β),(55)

βmin= (1 − e−|s|)/(1 + e−|s|),(56)

where ˜ pe(β) ≡ m3pe(E). As seen from Eq. (55), P1(s) is the probability for a single scattering of a photon of a

frequency shift s averaged over the electron distribution function, which is so called the redistribution function of a

shift s. The total probability is?∞

∂I(ω)

∂τ

−∞

−∞dsP1(s) = 1. Multiplying ω3to Eq. (54), one obtains the rate equation for the

spectral intensity function.

=

?∞

dsP1(s)?e−3sI(esω) − I(ω)?, (57)

where I(ω) = ω3n(ω)/2π2is the spectral intensity function for ω. Now let us introduce the following key identity

relations:

P(s,β)e−3s= P(−s,β), P1(s)e−3s= P1(−s).(58)

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The derivation is straightforward. Inserting Eq. (58) in Eq. (57) and replacing s by −s, one obtains the rate equation

for the spectral intensity function.

∂I(ω)

∂τ

=

?∞

−∞

dsP1(s)?I(e−sω) − I(ω)?.(59)

It should be remarked that n(esω) appears in RHS of Eq. (54), whereas I(e−sω) appears in RHS of Eq. (59). It is

also straightforward to show that Eq. (54) satisfies the photon number conservation.

d

dτ

?∞

0

dωω2n(ω) = 0. (60)

Let us now derive formal solutions for the rate equations Eq. (54) and Eq. (59). We consider an ideal condition

that the CG is infinitely large. We introduce a new function ˜ n(ω,τ) by

n(ω) ≡ e−τ˜ n(ω,τ).(61)

By inserting Eq. (61) into Eq. (54), one obtains the equation for ˜ n(ω,τ).

∂˜ n(ω,τ)

∂τ

=

?∞

−∞

dsP1(s)˜ n(esω,τ),(62)

where?∞

−∞dsP1(s) = 1 was used. Equation (62) can be integrated and one has

˜ n(ω,τ) = n0(ω) +

?τ

0

dλ

?∞

−∞

dsP1(s)˜ n(esω,λ). (63)

In deriving Eq. (63) an initial condition ˜ n(ω,τ = 0) = n0(ω) was used, where n0(ω) is the initial photon distribution

function. We solve Eq. (63) with a successive approximation method. The first-order term is obtained by inserting

n0(ω) into RHS of Eq. (63).

˜ n1(ω,τ) = n0(ω) + τ

?∞

−∞

dsP1(s)n0(esω). (64)

The second-order term is also obtained by inserting ˜ n1(ω,τ) into RHS of Eq. (63).

˜ n2(ω,τ) = n0(ω) + τ

?∞

+τ2

2!

?∞

−∞

dsP1(s)n0(esω)

?∞

ds1P1(s1)P1(s − s1),

−∞

dsP2(s)n0(esω), (65)

P2(s) ≡

−∞

(66)

where P2(s) is the probability (redistribution function) of a shift s for the double scattering. By repeating the above

procedure N + 1 times, one obtains the (N + 1)-th order term.

˜ nN+1(ω,τ) = n0(ω) +

N

?

j=1

τj

j!

?∞

−∞

dsPj(s)n0(esω), (67)

Pj(s) =

?∞

−∞

ds1P1(s1)···

?∞

−∞

dsj−1P1(sj−1)P1(s −

j−1

?

i=1

si), (68)

where Pj(s) is the probability (redistribution function) of a shift s for the multiple scattering of the j-th order. By

taking the limit N → ∞ in Eq. (67) and replacing limN→∞˜ nN(ω,τ) = ˜ n(ω,τ), one finally obtains the formal solution

for n(ω).

n(ω) = e−τn0(ω) +

?∞

−∞

dsP(s,τ)n0(esω), (69)

P(s,τ) =

∞

?

j=1

τje−τ

j!

Pj(s). (70)

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Multiplying ω3to Eq. (69) and using P(s,τ)e−3s= P(−s,τ), and also replacing s by −s, one obtains the formal

solution for I(ω).

I(ω) = e−τI0(ω) +

?∞

−∞

dsP(s,τ)I0(e−sω),(71)

where I0(ω) = ω3/(2π2)n0(ω).

that Eq. (70) is the Poisson distribution function. The distribution function is commonly used, for example, in

Birkinshaw[5]. In the present paper, however, Eq. (70) is derived as a natural consequence of the present formalism.

In practical cases, the CG has a finite size and the optical depth is small (τ ≪ 1), therefore the first order

approximation is sufficiently accurate for the study of the SZ effect. From Eqs. (69)–(71) one obtains the following

familiar forms for the distortion functions.

Note that this solution can be also derived directly from Eq. (59).Note also

∆n(ω) ≡ n(ω) − n0(ω)

≈ τ

?∞

−∞

dsP1(s)[n0(esω) − n0(ω)] ,(72)

∆I(ω) ≡ I(ω) − I0(ω)

≈ τ

?∞

?

−∞

dsP1(s)?I0(e−sω) − I0(ω)?,

dℓne.

(73)

τ = σT

(74)

The integral in Eq. (74) is done over the photon path length in the CG.

III. KINEMATICAL SUNYAEV-ZELDOVICH EFFECT

Let us now consider the case that the CG is moving with a peculiar velocity?βc(=? vc/c) with respect to the CMB.

As a reference system, we choose the system which is fixed to the CMB. The z-axis is fixed to a line connecting the

observer and the center of mass of the CG. (We assume that the observer is fixed to the CMB frame.) In the present

paper we choose the positive direction of the z-axis as the conventional one, i.e. the direction of the propagation of

a photon from the observer to the cluster, which is opposite to that of Nozawa, Itoh and Kohyama[16]. In the CMB

frame, the time evolution of the photon distribution function n(ω) is same as for the thermal SZ effect as shown in

Nozawa, Itoh and Kohyama[16]. They are given by Eqs. (1)–(5). The electron distribution functions are Lorentz

invariant and are related as follows:

f(E) = fc(Ec),

f(E′) = fc(E′

(75)

(76)

c),

Ec= Eγc

?

?

1

1 +?βc·?β

?

, (77)

E′

c= E′γc

1 +?βc·?β′?

,

, (78)

γc=

?1 − β2

c

(79)

where the suffix c denotes the CG frame. Therefore the formalism of § II will be directly applicable to the present

case. A modification should be made to the electron distribution function pe(E) by

pe(E) = pe,c

?

Eγc

?

1 +?βc·?β

??

, (80)

where pe,c(Ec) is normalized by?∞

0dpcp2

cpe,c(Ec) = 1. To proceed the calculation, one expresses the product?βc·?β

in the coordinate system where?k is parallel to the z-axis. Then one obtains

?βc·?β = βcβ

?

µcµ +

?

1 − µ2

c

?

1 − µ2cos(φc− φp)

?

, (81)

Page 9

9

where µc and φc are the cosine of the zenith angle and the azimuthal angle of?βc, respectively. By applying the

transformation of Eq. (19) to Eq. (81), one obtains

?βc·?β =

βcβ

1 − βµ0

?

µc(−µ0+ β) +1

γ

?

1 − µ2

c

?

1 − µ2

0cos(φc− φp)

?

.(82)

Inserting Eqs. (80) and (82) into Eq. (33), one obtains the expression for the CG with non-zero peculiar velocity in

Wright’s method.

∂n(ω)

∂τ

=

?∞

×1

0

dpp2

?1

−1

dµ0

?1

?

−1

dµ′

0

1

2γ4

1

(1 − βµ0)3f(µ0,µ′

0)

2π

?2π

0

dφppe,c

Eγc

?

1 +?βc·?β

??

[n(ω′) − n(ω)] . (83)

Shimon and Rephaeli[31] also obtained the expression for the kinematical SZ effect based upon Wright’s method,

which is similar to Eq. (83). For the expression of?βc·?β, Eq. (82) agrees with their Eq. (39). As discussed in their

paper, however, they have an extra factor γc

?

factor, see their Eq. (37). As discussed also in Nozawa, Itoh, Suda and Ohhata[26], the reason of the discrepancy

is because they used the phase space in the CG frame instead of the CMB frame. As far as the present formalism

is concerned, we have used the CMB frame as a reference system. Therefore there are no extra factors needed in

Eq. (83). We conclude that the result of Shimon and Rephaeli is in error by the extra factor.

Let us now proceed with Eq. (83). For most of the CG, βc≪ 1 is realized. For example, βc≈ 1/300 for a typical

value of the peculiar velocity vc=1000 km/s. In Nozawa, Itoh and Kohyama[16] they made an expansion in terms of

βcin the Fokker-Planck approximation. They found that O(β2

we will keep O(βc) terms and neglect higher-order terms in the present paper. In this approximation the electron

distribution function is approximated as follows:

1 +?βc·?β

?

in Eq. (83) which comes from Ec/E in their phase space

c) terms are negligible for most of the CG. Therefore

pe,c(Ec) ≈ pe(E)

?

?

?

1 −

a

β2?βc·?β

?

for pe(E) ∝ p−a

1 − a?βc·?β

?

?βc·?β

for pe(E) ∝ E−a

1 −

E

kBTe

?

for pe(E) ∝ exp(−E/kBTe)

. (84)

For simplicity, we consider the thermal distribution function. (Only a minor modification will be needed for the

power-law distributions.) Inserting Eq. (82) into Eq. (84) the integral for the azimuthal angle is performed.

1

2π

?2π

0

dφppe,c

?

Eγc

?

1 +?βc·?β

??

≈ pe(E)

?

1 + βcµc

?γ

θe

??βµ0− β2

1 − βµ0

??

, (85)

where θe≡ kBTe/m. Repeating the same procedure done in § II, one obtains the rate equations for the case of the

CG with nonzero peculiar velocity.

∂n(ω)

∂τ

∂I(ω)

∂τ

=

?∞

?∞

−∞

dsP1(s,βc,z)[n(esω) − n(ω)] , (86)

=

−∞

dsP1(s,βc,z)?e−3sI(esω) − I(ω)?,

P1(s,βc,z) = P1(s) + βc,zP1,K(s),

(87)

(88)

where P1(s) is Eq. (55) and βc,z= βcµcis the peculiar velocity parallel to the observer, because the photon direction

is along z-axis. In Eq. (88), P1,K(s) is the redistribution function due to the peculiar velocity of the CG. It is given

as

P1,K(s) =

?1

βmin

es

2βγ4

dββ2γ5˜ pe(β)PK(s,β), (89)

PK(s,β) =

?γ

θe

??µ2(s)

µ1(s)

dµ0(βµ0− β2)

1

(1 − βµ0)3f (µ0,µ′

0) ,(90)

Page 10

10

where µ′

that Eq. (87) is expressed by e−3sI(esω) instead of I(e−sω) in Eq. (59). This is because P(s,β)e−3s= P(−s,β) as

shown in Eq. (58), however, PK(s,β)e−3s?= PK(−s,β). For the power-law distributions, (γ/θe) should be replaced

by a/β2and a in Eq. (90) for the p-power distribution and the E-power distribution, respectively.

Finally, one obtains the distortions of the photon spectrum and the spectral intensity in the first order approxima-

tion.

0, µ1(s), µ2(s) and βmin are defined in Eqs. (41), (42), (43) and (56), respectively. It should be remarked

∆n(ω) ≈ τ

?∞

?∞

−∞

dsP1(s,βc,z)[n0(esω) − n0(ω)] ,(91)

∆I(ω) ≈ τ

−∞

dsP1(s,βc,z)?e−3sI0(esω) − I0(ω)?. (92)

IV. CONCLUDING REMARKS

We started with a covariant Boltzmann collisional equation of the SZ effect shown in Itoh, Kohyama and Nozawa[15]

for thermal and non-thermal distributions. First we have applied a rational transformation (Eqs. (19) and (20)) to the

photon angles, which is essentially a Lorentz transformation for photon angles from the CMB frame to the electron

rest frame. The transformation has made the expression for the transition probability surprisingly concise form. Then

we have introduced an assumption used by Boehm and Lavalle[30], namely γω/m ≪ 1 (the Thomson limit). The

assumption is fully valid for the CMB photon energies. Under the assumption, we have derived the redistribution

function P(s,β), which is the probability for a single scattering of a photon of a frequency shift s by a electron with

a velocity β. The obtained redistribution function is identical to that of derived with Wright’s method[10, 11].

Similarly, starting from the covariant Boltzmann collisional equation of the SZ effect for thermal and non-thermal

distributions, we have derived the redistribution function˜P(s,β) in the covariant formalism under the assumption

γω/m ≪ 1. We have shown that˜P(s,β) is identical to P(s,β). They are connected by the Lorentz transformation

of Eqs. (19) and (20). Thus we have shown mathematically that Wright’s method is equivalent to the covariant

formalism under the assumption γω/m ≪ 1. This result guarantees that existing works which used Wright’s method,

for example, Birkinshaw[5], Enßlin and Kaiser[28] and Colafrancesco et al.[29], are still fully valid. This result also

explains the reason why two different calculations for the thermal SZ effect agree extremely well even for the relativistic

electron energies.

We have also extended the present formalism to the kinematical SZ effect. Starting from the covariant Boltzmann

collisional equation for the kinematical SZ effect, we have repeated the same procedure. We have derived the redistri-

bution function for the CG with nonzero peculiar velocity in Wright’s method. We have compared the present result

with that of Shimon and Rephaeli[31]. The obtained redistribution function is differ by a factor γc

have clarified the discrepancy between their result and others[16, 17, 18]. Their result is in error by the factor.

?

1 +?βc·?β

?

. We

Acknowledgments

We wish to acknowledge Professor N. Itoh for enlightening us on this subject and also for giving us many useful

suggestions. We would also like to thank our referee for valuable suggestions.

APPENDIX A: REDISTRIBUTION FUNCTION IN COVARIANT FORMALISM

In this appendix we will derive the redistribution function in the covariant formalism. The starting equation is

Eq. (9).

∂n(ω)

∂τ

= −

3

64π2

×{n(ω)[1 + n(ω′)]pe(E) − n(ω′)[1 + n(ω)]pe(E′)} .

?

d3p

?

dΩk′1

γ2

1

1 − βµ

?ω′

ω

?2

¯ X

(A1)

Then we assume the Thomson limit γω/m ≪ 1, which implies the approximations

ω′

ω

≈1 − βµ

1 − βµ′

(A2)

Page 11

11

and¯ XB≪ 1, E′≈ E and pe(E′) ≈ pe(E). Under the assumption, Eq. (A1) is approximated as

∂n(ω)

∂τ

=

3

64π2

?

d3ppe(E)

?

dΩk′1

γ2

1 − βµ

(1 − βµ′)2¯ XA[n(ω′) − n(ω)], (A3)

where¯ XAis given by Eq. (15). In Eq. (A3) the φk′-integration can be done as

1

2π

?2π

0

¯ XAdφk′ = 2 +

(1 − µµ′)2+1

2(1 − µ2)(1 − µ′2)

γ4(1 − βµ)2(1 − βµ′)2

1 − µµ′

γ2(1 − βµ)(1 − βµ′).−2 (A4)

Assuming the spherical symmetry for pe(E), Eq. (A3) is further simplified.

∂n(ω)

∂τ

=

?∞

0

dpp2pe(E)

?1

−1

dµ

?1

2(1 − µ2)(1 − µ′2)

γ4(1 − βµ)2(1 − βµ′)2

−1

dµ′1

2γ2

1 − βµ

(1 − βµ′)2˜f(µ,µ′)[n(ω′) − n(ω)], (A5)

˜f(µ,µ′) =3

8

2 +

(1 − µµ′)2+1

− 2

1 − µµ′

γ2(1 − βµ)(1 − βµ′)

.(A6)

Now let us introduce a new variable s by

es=ω′

ω

=1 − βµ

1 − βµ′, (A7)

which implies dµ = −(1/β)(1 − βµ′)esds. Then Eq. (A5) is finally rewritten by

∂n(ω)

∂τ

=

?∞

0

dpp2pe(E)

?smax

e2s

2βγ4

−smax

ds˜P(s,β)[n(esω) − n(ω)] , (A8)

˜P(s,β) =

?µ2(s)

µ1(s)

dµ′˜f (µ,µ′) ,(A9)

µ = [1 − es(1 − βµ′)]/β , (A10)

where smax, µ1(s) and µ2(s) are given by Eqs. (40), (42) and (43), respectively. Equation (A9) is the redistribution

function in the covariant formalism, which is described in the CMB frame.

APPENDIX B: KLEIN-NISHINA CROSS SECTION

In this appendix we will derive Eq. (27) in terms of familiar Klein-Nishina cross section formula. Notations are

same as those in the main text, unless otherwise stated explicitly. As a reference frame we choose the electron rest

frame. The energy-momentum conservation gives the relation for the photon energies as follows:

ω′

ω0

0

=

1

1 + (ω0/m)(1 − cosΘ0),

?

(B1)

cosΘ0≡ µ0µ′

0+

1 − µ2

0

?

1 − µ′2

0cos(φk0− φk′

0),(B2)

where Θ0is the scattering angle. The Klein-Nishina cross section formula in the electron rest frame is expressed by

dσ

dΩk′

0

=1

2r2

e

?ω′

0

ω0

?2?ω′

0

ω0

+ω0

ω′

0

− sin2Θ0

?

,(B3)

where reis the classical electron radius. With Eq. (B1) one obtains the following usefull relation.

ω′

ω0

0

+ω0

ω′

0

= 2 +

?ω0

m

?2ω′

0

ω0(1 − cosΘ0)2. (B4)

Page 12

12

Inserting Eq. (B4) into Eq. (B3) one can rewrite the Klein-Nishina formula as follows:

dσ

dΩk′

0

=1

2r2

e

?ω′

0

ω0

?2?

1 + cos2Θ0+

?ω0

m

?2ω′

0

ω0(1 − cosΘ0)2

?

, (B5)

It is needless to say that one obtains the Thomson cross section by taking the limit ω0/m ≪ 1 and ω′

Eq. (B5).

Now let us introduce the transformation from the electron rest frame to the CMB frame, where the electron is

moving with a velocity β. The photon energies ω and ω′in the CMB frame are related to ω0and ω′

transformation

0/ω0→ 1 in

0by the Lorentz

ω = ω0γ (1 − βµ0) ,

ω′= ω′

(B6)

(B7)

0γ (1 − βµ′

0) ,

where µ0= cosθ0and µ′

0= cosθ′

0. With the variables ω and ω′one obtains

dσ

dΩk′

0

=1

2r2

e

?1 − βµ0

1 − βµ′

0

?2?ω′

ω

?2?

1 + cos2Θ0+

?ω

γm

?2ω′

ω

(1 − cosΘ0)2

(1 − βµ0)(1 − βµ′

0)

?

. (B8)

As seen from Eq. (B8) the square bracket in the RHS is identical to¯ XA+¯ XB, where they are defined by Eqs. (21)

and (22). Note that Eq. (B8) is the expression in the hybrid coordinate system, where the energies are described in

the CMB system, whereas the zenith angles are described in the electron rest frame.

The cross section is defined by the transition rate divided by the flux of the incident particles. The flux in the CMB

frame is

jinc≡p · k

Eω= 1 − βµ.(B9)

Therefore, one can write Eq. (1) in terms of the cross section in the CMB frame as follows:

∂n(ω)

∂t

= −2

?

d3p

(2π)3(1 − βµ)

{n(ω)[1 + n(ω′)]f(E) − n(ω′)[1 + n(ω)]f(E′)} .

?

dσ

dΩk′

?

dΩk′

(B10)

Since the cross section is Lorentz invariant, one can rewrite Eq. (B10) with the Klein-Nishina cross section in the

hybrid system of Eq. (B8) as follows:

∂n(ω)

∂t

= −2

?

d3p

(2π)3(1 − βµ)

{n(ω)[1 + n(ω′)]f(E) − n(ω′)[1 + n(ω)]f(E′)} .

?

dσ

dΩk′

0

?

dΩk′

0

(B11)

Rewriting the phase space volume d3p by

d3p =

1

γ2(1 − βµ0)2d3p0

(B12)

and inserting Eqs. (B8) and (B12), one finally obtains

∂n(ω)

∂τ

= −

3

64π2

×{n(ω)[1 + n(ω′)]pe(E) − n(ω′)[1 + n(ω)]pe(E′)} .

?

d3p0

?

dΩk′0

1

γ4

1

1 − βµ0

1

(1 − βµ′

0)2

?ω′

ω

?2

¯ X

(B13)

In deriving Eq. (B13) we used the relations γ2(1 − βµ) = (1 − βµ0)−1, f(E) = π2nepe(E), dτ = neσTdt and

σT = 8π/3r2

e. One finds that Eq. (B13) is identical to Eq. (27).

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