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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= O γω

m

?2?

?

?[1 + O(TCMB/Te)]

, (30)

1 + O βγω

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)