2.2 The intracluster medium
2.2.6 The Sunyaev-Zel’dovich e ff ect
In the previous chapters we focused mostly on X-ray studies of the ICM. The thermal electron population of the ICM, however, also interacts with the photons of the cosmic microwave back- ground (CMB) leading to the so-called thermal Sunyaev-Zel’dovich effect (SZE), first described
by Sunyaev and Zel’dovich (1970, 1972). We give here a brief overview of the physical processes behind the SZE following the approach of Birkinshaw (1999).
Elements of the SZE
Electrons in the ICM can scatter low energy CMB photons via inverse Compton scattering. Al- though the scatterings are usually still referred to as inverse-Compton processes, they might better be described in this limit as Thomson scattering, since the thermal electron population is almost completely non-relativistic14 and the CMB photons have low energy. For these low- energy interactions the scattering optical depth isτe ≈ neσTReff ∼ 10−2, whereσT is the Thom-
son scattering cross-section, ne the electron density and Reff the effective geometrical light path
length (e.g. roughly the cluster’s diameter). On average a single scattering produces only a small change of the mean photon energy (∆ν/ν) ≈ (kT/mec2) ∼ 10−2, where me is the electron
mass. The overall change in brightness of the CMB is on the 10−4level, i.e. roughly an order of
magnitude larger than the cosmological signal from the primordial anisotropies.
The change in the photon energy (frequency) is described by the Compton scattering equa- tion:
ǫ′= ǫ
1+ ǫ
mec2 (1−cosφ)
, (2.25)
under the assumption that the electron is at rest before the interaction (ǫ ≪ mec2). The photon
energies before and after the interaction areǫandǫ′, respectively, andφis the angle by which the photon is deflected in the encounter (the angle between the post-scattering paths of the electron and deflected photon).
Given that both the CMB and the Maxwellian velocity distribution of the electrons are almost isotropic, photons scattered away from our line of sight are replaced by photons from other directions scattered into our line of sight. This means that there is no observable change in the number of detected photons independently from whether there is an intervening cluster or not!
14ICM temperatures are.10 keV, except for a few cases where shock heating occurs, but this is always only in
clusters?
The reason for this is, that although the number of photons is conserved in these processes their energy spectrum is modified. The CMB photons (a low temperature system) and the ICM gas (hot system) are in interaction and thus energy flows from the ICM to the CMB photons as required by the second law of thermodynamics. This flow is mediated on the particle level by the fact, that the up-scattering of photons (ǫ′ > ǫ in Eq. 2.25) is slightly more likely than the
converse.15 We will provide here a brief simplified derivation which gives insight into the basic principles of this effect.
Let us denoteβ = v/c, where v is the electron velocity and c the speed of light (for a 4 keV
plasmaβ ≈ 0.14) and further in the electron rest frame we denote the photon impact angle to be
θand the angle after scatteringθ′. Eq. 2.25 then can be rewritten as
ν′= ν(1+βµ′) (1−βµ)−1 , (2.26) whereνandν′are the pre- and post-scattering photon frequencies and we denotedµ= cosθ. It is convenient to express the resulting scattering in terms of the logarithmic frequency shift defined as
s=log(ν′/ν). (2.27)
Finally, the probability that the photon experiences a frequency shift s after a single scattering on an electron with velocityβis:
P(s, β) ds= Z p(µ) dµ φ(µ′, µ) dµ ′ ds ! ds, (2.28)
where p(µ) dµis the probability of the photon having the impact angle θ before the scattering (just from the simple Thomson scattering geometry) andφ(µ′, µ) dµ′the probability of scattering from this angle to angle θ′. The φ(µ′, µ) distribution function was derived by Chandrasekhar (1950) and we display only the final P(s, β) function for several values ofβin Fig. 2.6 (left).
As can be seen, P(s, β) is slightly asymmetric, with up-scatterings (positive s) being slightly more likely. The asymmetry and broadening increases with increasing mean electron velocity (i.e. increasing ICM temperature). Since the velocity distribution of the electrons is Maxwellian (we denote it pe(β)), the probability of a frequency shift s for a single photon and single scattering
is given by the convolution:
P1(s)=
Z 1
βmin
pe(β) dβP(s, β). (2.29)
Electrons with velocities smaller thanβmincan not cause a frequency shift s. The effect of single
scattering on the CMB spectrum then is
I(ν′)= ∞
Z
−∞
P1(s) I0(ν) ds, (2.30)
2.2 The intracluster medium 19
Figure 2.6: Left: The inverse Compton scattering probability function P(s, β) (Eq. 2.28), forβ=
0.01,0.02,0.05,0.10,0.20 and 0.50, whereβ = v/c. The probability distribution is increasingly
asymmetric and broadened asβincreases. Right: The spectrum of the CMB (black body, black line) and its distortion after a passage through the ICM of an unrealistically massive cluster with a Compton parameter of y = 0.15. The red line shows the curve often displayed in literature (e.g. Sunyaev and Zel’dovich 1980; Carlstrom et al. 2002), but which was obtained by a first order approximation not applicable to this high values of y. The blue curve shows the exact non- relativistic solution for y = 0.15. See text for more discussion. Image courtesy of E. L. Wright,
http://www.astro.ucla.edu/∼wright/SZ-spectrum.html. where I0(ν) is the incident CMB spectrum (black-body):
I0(ν)= 2 hν3 c2 ehν/kTCMB −1−1. (2.31)
The purpose of this simplified treatment we provided here was to highlight the basic mech- anisms at work. We made several important simplifications - we allowed only a single scat- tering and assumed the Thomson scattering to hold for all values of β. The proper description of the non-relativistic scattering process in this case is provided by the Kompaneets equation (Kompaneets 1956) and the full derivation of the impact of the electrons on the CMB spectrum was first given by Sunyaev and Zel’dovich (1970, 1972). In the following we will provide only the final results and their implications for cluster observations.
Observational signatures of the SZE
The SZE causes an increase in the CMB intensity in the high frequency (Wien) part of the spectrum and a decrement in the Rayleigh-Jeans tail. The transition occurs at a frequency of
of the order of∼mK in the temperature surface brightness of the CMB.
The shape of the SZE spectrum is depicted in Fig. 2.6 (right). The original black body spec- trum is shown in black. The red line shows the distorted spectrum for an unrealistically massive cluster with y = 0.15 (roughly 1000 times more massive than real clusters). As was pointed out by E. L. Wright16 this curve (to be found in e.g. Sunyaev and Zel’dovich 1980; Carlstrom et al.
2002), was obtained using the first order approximation in Sunyaev and Zel’dovich (1980) (their Eq. A7) to the Sunyaev-Zel’dovich effect, and is not applicable for y= 0.15. The exact solution is shown in blue (computed using Eq. A8 of Sunyaev and Zel’dovich 1980) and is significantly wider than the approximation. Note that this still does not include relativistic corrections. The first order approximation is still appropriate for real clusters.
The decrement in the CMB is equal to
∆I(ν)=−2 y I(ν), (2.32) where y is the so-called Compton parameter defined as
y≡ σTkB mec2
Z
Tenedl, (2.33)
The integration runs along the line-of-sight. The decrement∆I(ν) is defined as the difference in the CMB intensity between the distorted spectrum in the direction of the cluster and the black body spectrum of the unobstructed CMB.
Eq. 2.33 shows that the SZE is completely redshift independent.17 This is very different from the X-ray observations, where the cosmological redshift dimming causes a fast decline of the surface brightness∝ (1+z)−4 (Sect. 2.2.2). In the case of SZE the redshift dimming is exactly
compensated by the increase of the CMB intensity∝ (1+z)4(at higher redshift we are probing a younger Universe where the CMB temperature is higher). This is illustrated in Fig. 2.7 on the example of three clusters at redshifts between 0.2−0.8. The dimming of the X-rays is evident, while the SZE decrement is comparable even for the most distant object. This gives SZE surveys the ability to have a nearly redshift-independent selection function and thus allow to detect many distant clusters.
This advantage started to be utilized by large area surveys carried out by large, single dish telescopes: the South Pole Telescope (SPT, Staniszewski et al. 2009; Vanderlinde et al. 2010; Williamson et al. 2011; Foley et al. 2011), the Atacama Cosmology Telescope (ACT, Marriage et al. 2010; Hincks et al. 2010) and by the Planck space mission (Planck Collaboration et al. 2011a,b). The delivered samples have significantly higher median redshifts compared to X-ray selected cluster catalogs. We provide a brief overview of the advances in the field in Sect. 5.1.
An additional feature of the SZE signal is that the Compton parameter is proportional to the integrated pressure along the line-of-sight (compare Eq. 2.33 with the ideal gas pressure
16http://www.astro.ucla.edu/
∼wright/SZ-spectrum.html
17The integrated Compton parameter within a solid angle would depend on the aperture size and thus the angular