2.4 The Sunyaev-Zel’dovich Effect 32
galaxies and clusters. As photons stream through the centre of massive clusters, the probability of their path intersecting with hot ICM electrons is ≈ 1%. The resulting interaction boosts the energy of CMB photons resulting in a≤1mK distortion in the CMB spectrum.
The dynamics of the interaction between non-relativistic electrons and a radiation field is well described by the Kompaneets equation (Kompaneets, 1956). Assuming the electron temperature, Te, is larger than the radiation field temperature,T, the Kompaneets equation can be written as
∂hni
∂t = kBTe mec
σTne x2
∂
∂x µ
x4∂hni
∂x
¶
, (2.4.38)
where, hniis the average photon occupation number, σT is the Thompson scattering cross sec- tion, ne is the electron number density and finallyx =hν/kBT is the dimensionless frequency component. Rephaeli (1995) suggests an alternative form for the scattering kernel that includes relativistic corrections. However, since we are concerned with weak scattering and small elec- tron temperatures (in the case of small to medium-sized clusters), the two kernels are equivalent, we consequently follow the simpler Kompaneets approximation from here on. In light of the low energy scattering environment, a Bose-Einstein distribution forhnican be assumed. In this notation, the spectral intensity is given by
I = 2 (kBTCM B)3
(hc)2 x3hni. (2.4.39)
The specific intensity (a convention common in microwave SZE observations) can be derived by integrating along the line of sight
∆ItSZ = 2 (kBTCM B)3
(hc)2 ycg(x), (2.4.40)
whereg(x)is the spectral distortion term which can be expressed as g(x) = x4ex
(ex−1)2 µ
xex+ 1 ex−1−4
¶
(1 +δtSZ(x, Te)), (2.4.41) whereδtSZ(x, Te)is the correction to the frequency dependence due to relativistic effects. The Compton parameter yc in Eq. (2.4.40) is measure of the gas pressure integrated along the line of sight. Fig. 2.3 demonstrates the effect of the tSZ on the CMB spectrum for a cluster 1000
2.4 The Sunyaev-Zel’dovich Effect 33
Figure 2.3: Distortion of the CMB spectrum caused by the tSZ for a fictional cluster1000times more massive than a typical cluster. The undistorted CMB spectrum is shown as the dashed line, while the distorted one is shown as the solid line. This figure is taken from Carlstrom et al.
(2002).
times more massive than a typical cluster. The distortion is seen as a decrease in photon energy at frequencies below≈218GHz and an increase above this frequency.
The spectral distortion of the CMB can equivalently be expressed as a temperature change
∆TtSZ/TCM B expressed by
∆TtSZ
TCM B =f(x)y=f(x) Z
nekBTe
mec2σTdl, (2.4.42)
where mec2 is the election rest mass energy and the integration is performed along the line of sight. The frequency dependence of the tSZ in this formalism is contained withinf(x)as
f(x) = µ
xex+ 1 ex−1 −4
¶
(1 +δtSZ(x, Te)). (2.4.43)
2.4 The Sunyaev-Zel’dovich Effect 34
The derivative of the blackbody with respect to temperature,|dBν/dT|is the coupling factor betweenTtSZ and∆ItSZ. Fig. 2.4 depicts the distortion of the CMB spectrum due to the SZE, measured in intensity units. The tSZ distortion is easily distinguishable from a CMB temperature fluctuation unlike the kSZ in the non-relativistic regime.
Figure 2.4: Distortion of the CMB spectrum caused by the SZE for a realistic cluster comprising a Compton y parameter of 10−4, electron temperature of 10keV and peculiar velocity of 500 kms−1. The tSZ is depicted by the solid line, while the kinetic SZE is shown by the dashed line.
For comparative purposes, the CMB intensity, scaled by0.0005, is shown as the dotted line. This figure is taken from Carlstrom et al. (2002).
An observable of particular importance for SZE cluster surveys is the integrated tSZ signal.
The total tSZ signal, derived by integrating over the entire cluster, is proportional to the integrated Compton parameterYSZ where
YSZDA2 = µ σT
mec2
¶ Z
P dV, (2.4.44)
2.4 The Sunyaev-Zel’dovich Effect 35
whereDA is the angular diameter distance to the cluster. The gas pressure within clusters is an excellent proxy for the gravitational potential, and thusYSZDA2is expected to be closely related to the cluster mass.
The salient features of the tSZ are: (1) it is a weak distortion of the CMB spectrum which is proportional to the cluster gas pressure along the line of sight; (2) the effect is independent of redshift; (3) it comprises a unique frequency dependence ( with a decrease of the CMB intensity for frequencies < 218 GHz, zero effect at the null ≈ 218 GHz, and an increase of intensity for frequencies higher than the null); (4) the total tSZ signal is proportional the inverse of the squared angular diameter distance implying that SZE surveys will comprise redshift independant thresholds particularly at high redshifts.
2.4.2 Kinetic Sunyaev-Zel’dovich
The kinetic Sunyaev-Zel’dovich (kSZ) effect arises when the gas medium within a cluster is moving with respect to the CMB or Hubble flow. This effect is manifested as a distortion of the CMB spectrum caused by the Doppler effect of the bulk velocity on the scattered CMB photons.
In the non-relativistic regime the kSZ is a pure thermal distortion given by
∆TkSZ TCM B
=−τe
³vpec c
´
, (2.4.45)
where vpec is the cluster peculiar velocity along the line of sight and τe is the optical depth of the cluster. Fig. 2.4 depicts the kSZ effect compared to a scaled version of the CMB as a function of frequency. The distorted CMB spectrum is still Planckian but at a marginally different temperature.
Perhaps the most pertinent feature of the kSZ effect is that it provides a measure of the line of sight peculiar velocity of a cluster at high redshift. Moreover, if the bulk flows can be measured over a range of redshifts, they can be used to place constraints on the process of structure formation (Davis et al., 1992). Unfortunately, the sensitivity of today’s experiments cannot probe this feature, but rapid progress is being made in this regard.