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arXiv:0807.4043v1 [astro-ph] 25 Jul 2008

Scaling Relation between Sunyaev-Zel’dovich

Effect and X-ray Luminosity and Scale-Free

Evolution of Cosmic Baryon Field

Qiang Yuana,bHao-Yi WanaTong-Jie Zhanga,c,d,∗Ji-Ren Liud

Long-Long Fenge,fLi-Zhi Fangd

aDepartment of Astronomy, Beijing Normal University, Beijing, 100875,

P.R.China

bKey Laboratory of Particle Astrophysics, Institute of High Energy Physics,

Chinese Academy of Sciences, Beijing 100049, P.R.China

cKavli Institute for Theoretical Physics China, Institute of Theoretical Physics,

Chinese Academy of Sciences (KITPC/ITP-CAS), P.O.Box 2735, Beijing 100080,

P.R. China

dDepartment of Physics, University of Arizona, Tucson, AZ 85721

ePurple Mountain Observatory, Nanjing 210008, P.R. China

fNational Astronomical Observatories, Chinese Academy of Science, Chao-Yang

District, Beijing, 100012, P.R. China

Abstract

It has been revealed recently that, in the scale free range, i.e. from the scale of

the onset of nonlinear evolution to the scale of dissipation, the velocity and mass

density fields of cosmic baryon fluid are extremely well described by the self-similar

log-Poisson hierarchy. As a consequence of this evolution, the relations among var-

ious physical quantities of cosmic baryon fluid should be scale invariant, if the

physical quantities are measured in cells on scales larger than the dissipation scale,

regardless the baryon fluid is in virialized dark halo, or in pre-virialized state. We

examine this property with the relation between the Compton parameter of the ther-

mal Sunyaev-Zel’dovich effect, y(r), and X-ray luminosity, Lx(r), where r being the

scale of regions in which y and Lxare measured. According to the self-similar hier-

archical scenario of nonlinear evolution, one should expect that 1.) in the y(r)-Lx(r)

relation, y(r) = 10A(r)[Lx(r)]α(r), the coefficients A(r) and α(r) are scale-invariant;

2.) The relation y(r) = 10A(r)[Lx(r)]α(r)given by cells containing collapsed objects

is also available for cells without collapsed objects, only if r is larger than the dissi-

pation scale. These two predictions are well established with a scale decomposition

analysis of observed data, and a comparison of observed y(r)-Lx(r) relation with hy-

drodynamic simulation samples. The implication of this result on the characteristic

scales of non-gravitational heating is also addressed.

Preprint submitted to Elsevier25 July 2008

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Key words: cosmology: theory, large-scale structure of universe, X-rays: galaxies:

clusters, hydrodynamics, methods: numerical

PACS: 95.30.Jx, 07.05.Tp, 98.80.-k

1 Introduction

Scaling relation of dimensional quantities is very powerful to reveal the dy-

namical feature of various physical systems. There has been a considerable

effort devoting to study the correlations and scaling laws of various observable

quantities of galaxy clusters. Since virialized self-gravitational system is char-

acterized by one parameter, mass or virial temperature, one can find a set of

scaling relations among mass, size, X-ray luminosity, temperature, and Comp-

ton parameter of Sunyaev-Zel’dovich (SZ) effect if the velocity and mass den-

sity fields of baryon fluid in clusters are assumed to be similar to the virialized

dark matter halos (Kaiser, 1986). Observed data of galaxy clusters did yield

scaling relations (Edge & Stewart, 1991; David et al., 1993; Wu et al., 1999;

Helsdon & Ponman, 2000; Xue & Wu, 2000; Croston et al., 2005). However,

observed scaling relations generally do not support the predictions given by

the baryon-dark matter similarity of virialized dark halos (Helsdon & Ponman,

2000; Lloyd-Davies et al., 2000).

Since Newtonian gravity is scale-free, the self gravitational system of collision-

less dark matter shows scaling behavior if the power spectrum of initial density

perturbations is scale-free. These scaling is regardless of whether the under-

lying gravitational field is virialized (Peebles, 1980). Thus, if the velocity and

mass density fields of cosmic baryon matter are given by a similar mapping

of the fields of dark matter, one may expect the scaling relations of clus-

ters. However, the similar mapping assumption is correct only in linear regime

(Bi et al., 1992), but is baseless in nonlinear regime (Shandarin & Zeldovich,

1989). The nonlinear evolution of cosmic baryon fluid leads to statistically de-

couple of the fluid from dark matter. The statistical properties of the velocity

and mass density fields of baryon fluid do show deviation from the underlying

dark matter field (Pando et al., 2004; He et al., 2005; Kim et al., 2005).

Nevertheless, it has been pointed out by Shandarin & Zeldovich (1989): the

dynamics of cosmic baryon fluid in the expanding universe is scale-free, i.e. no

preferred special scales can be identified in the range from the onset of nonlin-

ear evolution down to the length scale of dissipation. It likes fully developed

∗Corresponding author.

Email addresses: yuanq@mail.ihep.ac.cn (Qiang Yuan), tjzhang@bnu.edu.cn

(Tong-Jie Zhang), fanglz@physics.arizona.edu (Li-Zhi Fang).

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turbulence in inertial range. This idea recently received substantial develop-

ments. With the hydrodynamic simulation sample of the concordance ΛCDM

model, the velocity field of cosmic baryon fluid is found to be extremely well

described by She-Leveque’s (SL) scaling formula (She & Leveque, 1994) in the

“inertial range” (He et al., 2006). The SL formula is considered to be the ba-

sic statistical features of the scale-free evolution of fully developed turbulence.

Moreover, the SL formula comes from self-similar log-Poisson hierarchy, which

is related to the hidden symmetry of the Navier-Stokes equations (Dubrulle,

1994; She & Waymire, 1995). Very recently, it has been shown that the clus-

tering of the mass density field of the cosmic baryon fluid can indeed be well

described by a log-Poisson hierarchical cascade (Liu & Fang, 2008). All the

scaling relations and non-Gaussian features predicted from the log-Poisson

hierarchy are in very good agreement with the hydrodynamic simulation sam-

ples.

These results indicate that, in the scale-free range, the nonlinear evolution

of cosmic baryon fluid reaches a statistically quasi-steady state similar to a

fully developed turbulence. For turbulence of incompressible fluid, the fluid

undergoes a self-similar hierarchical evolution from largest to the smallest

eddies and finally dissipates into thermal motion. For cosmic baryon fluid, the

clustering on different scales can also be described by a self-similar hierarchy,

and the fluid finally falls and dissipates into thermal motion.

This scenario motives us to investigate the scaling relations of clusters from

the self-similar hierarchy of cosmic baryon fluid. If the observed scaling rela-

tions come from the self-similar hierarchy, one can expect that 1.) the relations

of dimensional quantities should be scale-free, i.e. all the scale-dependent co-

efficients of the scaling relations are scale-invariant; 2.) the relations should

be held only if the scales of considered regions are larger than Jeans length,

regardless of whether the underlying gravitational field is virialized, i.e. the

relations given by cells containing collapsed objects is also available for cells

without collapsed objects, only if the scale of cells is larger than the dissipation

scale.

Other relevant motivation comes from the non-gravitational heating of baryon

gas of clusters. In order to solve the deviation from the similarity of virialized

dark halos, various models of non-gravitational heating and cooling of baryonic

gas have been proposed (e.g. Valageas & Silk, 1999; Tozzi & Norman, 2001;

Voit et al., 2002; Zhang & Pen, 2003; Xue & Wu, 2003; Nagai et al., 2007).

Since these cooling and heating may introduce characteristic scales, the self-

similar hierarchy will no longer work on these characteristic scales. Therefore,

it would be worth to detect the scale on which the above-mentioned two

predictions to be broken.

We study these properties with the relation between the Compton parameter

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y of SZ effect and X-ray luminosity Lx. The thermal SZ effect is due to the

inverse Compton scattering of cosmic microwave background (CMB) photons

by hot electrons of baryon fluid. The Compton parameter y depends on the

pressure of electron gas (Zeldovich & Sunyaev, 1969; Sunyaev & Zeldovich,

1980). There are many works on the y-Lxrelation (e.g. da Silva et al., 2004;

Maughan, 2007; Bonamente et al., 2007). We will, however, focus on the above-

mentioned two points, which have not yet been addressed right now.

The outline of this paper is as follows. §2 presents the scaling relations y =

10A(r)Lα(r)

x

with observed samples, and shows that A(r) and α(r) are scale-

invariant. §3 describes the hydrodynamic cosmological simulation samples.

The comparison of the scaling relations of simulation samples with observed

results is presented in §4. The conclusions and discussion are given in §5.

2

y-LxScaling Relations from Observed Samples

2.1Data

To study the scale free properties, we should find the y(r) - Lx(r) relations,

where y(r) and Lx(r) are, respectively, the Compton parameter of SZ effect

and X-ray luminosity measured from regions with spatial scale r. The data

of X-ray luminosity of these clusters are taken from McCarthy et al. (2003)

(Xray1) and Morandi et al. (2007) (Xray2). The X-ray luminosity from area

on comoving scale r is calculated by

Lx(r) =

θr

?

0

Lx(θ)θdθ,(1)

where θr= r/[(1+z)dA(z)] is the angular radius corresponding to the comoving

scale r, and dA(z) is angular diameter distance. Lx(θ) is proportional to the

X-ray surface brightness Sx(θ), which can be well fitted by β-model Sx(θ) =

SX0[1 + (θ/θc)2](1−6β)/2up to θ ∼ 10 arcmin.

Similarly, The mean of y within a region on comoving scale r is given by

y(r) =2

θ2

r

θr

?

0

y(θ)θdθ. (2)

We will use the SZ effect data from Reese et al. (2002) (SZ1) and Bonamente et al.

(2006) (SZ2). The former compiled SZ effects of 18 clusters of galaxies span-

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−5

−4

−3

log[y]

r=0.10 h−1Mpc r=0.20 h−1Mpc

−5

−4

−3

log[y]

r=0.39 h−1Mpc r=0.78 h−1Mpc

101001000

−5

−4

−3

Lx(1043 ergs/s)

log[y]

r=1.56 h−1Mpc

101001000

Lx(1043 ergs/s)

r=3.12 h−1Mpc

Fig. 1. y-Lxrelation of observational samples SZ1+Xray1 (see Table 1). The Comp-

ton parameter y are given by average over areas on the comving sizes 0.10, 0.20, 0.39,

0.78, 1.56, and 3.12 h−1Mpc, respectively. The solid lines indicate the best-fitting

for all observational samples.

ning the redshift range of 0.14 < z < 0.78, and the later includes 38 clusters in

the same redshift range. These data are on angular scales up to ∼ 2 arcmin,

of which the corresponded r is on about the same scale as, or larger than,

the Jeans length on redshift ∼ 0.5. Moreover, the θ-dependencies of y(θ) are

well fitted by β-model y(θ) = y0[1 + (θ/θc)2](1−3β)/2. Therefore, it would be

reasonable to use the β model fitted y(r) to study the y(r) - Lx(r) relation.

We will check this point below.

2.2Result

Figure 1 plots the relation of y(r) vs. Lx(r) on scales r=0.1, 0.2, 0.39, 0.78, 1.56

and 3.12h−1Mpc respectively. In this figure the SZ and X-ray data are taken

from SZ1 and Xray1, respectively. The cluster A370 is excluded as it shows a

3-σ discrepancy with the distance-redshift relation (Reese et al., 2002). Three

clusters, Cl0016, A611 and A697, are also excluded due to lacking the data of

X-ray luminosity, and after all, there are totally 14 clusters used in Figure 1.

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