2.1. Thermal Sunyaev-Zel'dovich Effect
The SZE is a small spectral distortion of the CMB spectrum caused by the scattering of the CMB photons off a distribution of high-energy electrons. We only consider the SZE caused by the hot thermal distribution of electrons provided by the ICM of galaxy clusters. CMB photons passing through the center of a massive cluster have a _{e} 0.01 probability of interacting with an energetic ICM electron. The resulting inverse-Compton scattering preferentially boosts the energy of the CMB photon, causing a small ( 1 mK) distortion in the CMB spectrum. To illustrate the small effect, Figure 1 shows the SZE spectral distortion for a fictional cluster that is over 1000 times more massive than a typical cluster. The SZE appears as a decrease in the intensity of the CMB at frequencies below 218 GHz and as an increase at higher frequencies.
Figure 1. The CMB spectrum, undistorted (dashed line) and distorted by the SZE (solid line). Following Sunyaev & Zel'dovich (1980), to illustrate the effect, the SZE distortion shown is for a fictional cluster 1000 times more massive than a typical massive galaxy cluster. The SZE causes a decrease in the CMB intensity at frequencies 218 GHz (~ 1.4 mm) and an increase at higher frequencies. |
The SZE spectral distortion of the CMB, expressed as a temperature change T_{SZE} at dimensionless frequency x (h ) / (k_{B} T_{CMB}), is given by
(1) |
where y is the Compton y-parameter, which for an isothermal cluster equals the optical depth times the fractional energy gain per scattering. Here, _{T} is the Thomson cross-section, n_{e} is the electron number density, T_{e} is the electron temperature, k_{B} is the Boltzmann's constant, m_{e} c^{2} is the electron rest-mass energy, and the integration is along the line of sight. The frequency dependence of the SZE is
(2) |
where _{SZE}(x, T_{e}) is the relativistic correction to the frequency dependence. Note that f (x) -2 in the nonrelativistic and Rayleigh-Jeans (RJ) limits.
It is worth noting that T_{SZE} / T_{CMB} is independent of redshift, as shown in Equation 1. This unique feature of the SZE makes it a potentially powerful tool for investigating the high-redshift Universe.
Expressed in units of specific intensity, common in millimeter SZE observations, the thermal SZE is
(3) |
where I_{0} = 2(k_{B} T_{CMB})^{3} / (hc)^{2} and the frequency dependence is given by
(4) |
T_{SZE} and I_{SZE} are simply related by the derivative of the blackbody with respect to temperature, |dB_{} / dT|.
The spectral distortion of the CMB spectrum by the thermal SZE is shown in Figure 2 (solid line) for a realistic massive cluster (y = 10^{-4}), in units of intensity (left panel) and RJ brightness temperature (right panel). The RJ brightness is shown because the sensitivity of a radio telescope is calibrated in these units. It is defined simply by I_{} = (2k_{B} ^{2} / c^{2})T_{RJ}, where I_{} is the intensity at frequency , k_{B} is Boltzmann's constant, and c is the speed of light. The CMB blackbody spectrum, B_{}(T_{CMB}), multiplied by 0.0005 (dotted line), is also shown for comparison. Note that the spectral signature of the thermal effect is distinguished readily from a simple temperature fluctuation of the CMB. The kinetic SZE distortion is shown by the dashed curve (Section 2.2). In the nonrelativistic regime, it is indistinguishable from a CMB temperature fluctuation.
The gas temperatures measured in massive galaxy clusters are around k_{B} T_{e} 10 keV (Mushotzky & Scharf, 1997; Allen & Fabian, 1998) and are measured to be as high as ~ 17 keV in the galaxy cluster 1E 0657 - 56 (Tucker et al., 1998). At these temperatures, electron velocities are becoming relativistic, and small corrections are required for accurate interpretation of the SZE. There has been considerable theoretical work to include relativistic corrections to the SZE (Rephaeli & Yankovitch, 1997; Molnar & Birkinshaw, 1999; Challinor & Lasenby, 1998; Challinor & Lasenby, 1999; Itoh et al., 1998; Fabbri, 1981; Wright, 1979; Dolgov et al., 2001; Nozawa et al., 1998b; Rephaeli, 1995; Sazonov & Sunyaev, 1998a; Stebbins, 1997; Sazonov & Sunyaev, 1998b). All of these derivations agree for k_{B} T_{e} 15 keV, appropriate for galaxy clusters. For a massive cluster with k_{B}T_{e} 10 keV (k_{B} T_{e} / m_{e} c^{2} 0.02), the relativistic corrections to the SZE are of order a few percent in the RJ portion of the spectrum, but can be substantial near the null of the thermal effect. Convenient analytical approximations to fifth order in k_{B} T_{e} / m_{e} c^{2} are presented in Itoh et al. (1998).
The measured SZE spectrum of Abell 2163, spanning the decrement and increment with data obtained from different telescopes and techniques, is shown in Figure 3 (LaRoque et al., 2003; Holzapfel et al., 1997a; Désert et al., 1998). Also plotted is the best-fit model (solid) consisting of thermal (dashed) and kinetic (dotted) SZE components. The SZE spectrum is a good fit to the data, demonstrating the consistency and robustness of modern SZE measurements.
Figure 3. The measured SZE spectrum of Abell 2163. The data point at 30 GHz is from BIMA (LaRoque et al., 2003), at 140 GHz is the weighted average of Diabolo and SuZIE measurements (filled square; Holzapfel et al. 1997a; Désert et al. 1998), and at 218 GHz and 270 GHz from SuZIE (filled triangles; Holzapfel et al. 1997a). Uncertainties are at 68% confidence with the FWHM of the observing bands shown. The best-fit thermal and kinetic SZE spectra are shown by the dashed line and the dotted lines, respectively, with the spectra of the combined effect shown by the solid line. The limits on the Compton y-parameter and the peculiar velocity are y_{0} = 3.71^{+0.36+0.33}_{-0.36-0.16} × 10^{-4} and _{p} = 320^{+880+480}_{-740-440} km s^{-1}, respectively, with statistical followed by systematic uncertainties at 68% confidence (LaRoque et al., 2003; Holzapfel et al., 1997a). |
The most important features of the thermal SZE are: (1) it is a small spectral distortion of the CMB, of order ~ 1 mK, which is proportional to the cluster pressure integrated along the line of sight (Eq. 1); (2) it is independent of redshift; and (3) it has a unique spectral signature with a decrease in the CMB intensity at frequencies 218 GHz and an increase at higher frequencies.