![]() | Annu. Rev. Astron. Astrophys. 2002. 40:
643-680 Copyright © 2002 by Annual Reviews. All rights reserved |
2.1. Thermal Sunyaev-Zel'dovich Effect
The Sunyaev-Zel'dovich effect (SZE) is a small spectral distortion of
the cosmic microwave background (CMB) spectrum caused by the
scattering of the CMB photons off a distribution of high energy
electrons. Here we focus only on the SZE caused by the hot thermal
distribution of electrons provided by the intra-cluster medium (ICM)
of galaxy clusters. CMB photons passing through the center of a
massive cluster have only a
1% probability of
interacting with an energetic ICM electron. The resulting inverse
Compton scattering preferentially boosts the energy of the CMB photon
by roughly kB Te
/ me c2 causing a small
(
1 mK)
distortion in the CMB spectrum. Figure 1 shows the
SZE spectral distortion for a fictional cluster that is over 1000
times more massive than a typical cluster to illustrate the small
effect. The SZE appears as a decrease in the intensity of the CMB at
frequencies
218 GHz and as an
increase at higher frequencies.
![]() |
Figure 1. The Cosmic Microwave Background
(CMB) spectrum, undistorted (dashed line) and distorted by the
Sunyaev-Zel'dovich effect (SZE) (solid line). Following
Sunyaev &
Zel'dovich (1980a) 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
|
The derivation of the SZE can be found in the original papers of Sunyaev and Zel'dovich (Sunyaev & Zel'dovich, 1970, 1972), in several reviews (Sunyaev & Zel'dovich, 1980a, Rephaeli, 1995, Birkinshaw, 1999), and in a number of more recent contributions which include relativistic corrections (see below for references). This review discusses the basic features of the SZE that make it a useful cosmological tool.
The SZE spectral distortion of the CMB expressed as a temperature
change
TSZE at dimensionless frequency
x
(h
) / (kB
TCMB) is given by
![]() |
(1) |
where y is the Compton y-parameter, which for an isothermal
cluster equals the optical depth,
e, times the
fractional energy gain per scattering,
T is the
Thomson cross-section, ne is the
electron number density, Te is the electron
temperature, kB is
the Boltzmann constant, me c2 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,
Te) is the relativistic correction to the
frequency dependence. Note that
f (x)
-
2 in the non-relativistic and Rayleigh-Jeans (RJ) limits.
It is worth noting that
TSZE
/ TCMB is independent of
redshift as shown in Eq. 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 I0 = 2(kB TCMB)3 / (hc)2 and the frequency dependence is given by
![]() |
(4) |
TSZE
and
ISZE
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
Rayleigh-Jeans (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
= (2kB
2 /
c2) TRJ where
I
is the intensity at frequency
,
kB is Boltzmann's constant, and c is the speed
of light. The CMB blackbody spectrum,
B
(TCMB), 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
non-relativistic regime, it is indistinguishable from a CMB
temperature fluctuation.
The gas temperatures measured in massive galaxy clusters are around
kB Te ~ 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).
The mass is expected to scale with temperature roughly as
Te
M2/3. 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
(Wright, 1979,
Fabbri, 1981,
Rephaeli, 1995,
Rephaeli &
Yankovitch, 1997,
Stebbins, 1997,
Itoh et al, 1998,
Challinor
& Lasenby, 1998,
Sazonov &
Sunyaev, 1998a,
Sazonov &
Sunyaev, 1998b,
Nozawa et al,
1998b,
Challinor
& Lasenby, 1999,
Molnar &
Birkinshaw, 1999,
Dolgov et al,
2001).
All of these derivations agree for kB Te
15 keV,
appropriate for galaxy clusters. For a massive cluster
with kB Te ~ 10 keV
(kB Te / me
c2 ~ 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
kB Te / me
c2 are presented in
Itoh et al (1998).
Particularly relevant for finding clusters with an SZE survey is the
integrated SZE signal. Because the SZE signal is the integrated
pressure, integrating over the solid angle of the cluster provides a
sum of all of the electrons in the cluster weighted by temperature.
This provides a relatively clean measure of the total thermal energy
of the cluster. Integrating the SZE over the solid angle of the
cluster, d =
dA / DA2, gives
![]() |
(5) |
where Ne is the total number of electrons in the clusters, <Te> is the mean electron temperature, DA is the angular diameter distance, and M is the mass of the cluster (either gas or total mass as Mgas = Mtotal fg, where fg is the gas mass fraction). The integrated SZE flux is simply the temperature weighted mass of the cluster divided by DA2. The angular diameter distance DA(z) is fairly flat at high redshift. Also, a cluster of a given mass will be denser and therefore hotter at high redshift because the universal matter density increases as (1 + z)3. Therefore, one expects an SZE survey to detect all clusters above some mass threshold with little dependence on redshift (see Section 4.2).
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; 3) it has a
unique spectral signature with a decrease in the CMB intensity at
frequencies
218 GHz and an
increase at higher frequencies; and
4) the integrated SZE flux is proportional to the temperature
weighted mass of the cluster (total thermal energy of the cluster)
implying that SZE surveys will have a mass threshold nearly
independent of redshift.
2.2. Kinetic Sunyaev-Zel'dovich Effect
If the cluster is moving with respect to the CMB rest frame there will be an additional spectral distortion due to the Doppler effect of the cluster bulk velocity on the scattered CMB photons. If a component of the cluster velocity, vpec, is projected along the line of sight to the cluster, then the Doppler effect will lead to an observed distortion of the CMB spectrum referred to as the kinetic SZE. In the non-relativistic limit, the spectral signature of the kinetic SZE is a pure thermal distortion of magnitude
![]() |
(6) |
where vpec is along the line of sight; i.e., the emergent spectrum is still described completely by a Planck spectrum, but at a slightly different temperature, lower (higher) for positive (negative) peculiar velocities (Sunyaev & Zel'dovich, 1972, Phillips, 1995, Birkinshaw, 1999, see Figure 2).
Relativistic perturbations to the kinetic SZE are due to the Lorentz boost to the electrons provided by the bulk velocity (Nozawa et al, 1998a, Sazonov & Sunyaev, 1998a). The leading term is of order (kB Te / me c2)(vpec / c) and for a 10 keV cluster moving at 1000 km s-1 the effect is about an 8% correction to the non-relativistic term. The (kB Te / me c2)2(vpec / c) term is only about 1% of the non-relativistic kinetic SZE and the (vpec / c)2 term is only 0.2%.
2.3. Polarization of the Sunyaev-Zel'dovich Effect
The scattering of the CMB photons by the hot ICM electrons can result
in polarization at levels proportional to powers of
(vpec / c) and
e. The largest
polarization is expected from the anisotropic
optical depth to a given location in the cluster. For example, toward
the outskirts of a cluster one expects to see a concentric (radial)
pattern of the linear polarization at frequencies where the thermal
SZE is positive (negative). Plots of the polarization pattern are
presented in
Sazonov &
Sunyaev (1999).
Nonspherical morphology for the
electron distributions will lead to considerably complicated
polarization patterns. The peak polarization of this signal will be
order
e times the
SZE signal, i.e., of order
0.025(kB Te / me
c2)
e2
times the CMB intensity. For a massive cluster with
e = 0.01, the
effect would be at the 0.1 µK level
toward the edge of the cluster. In principle, this effect could be
used to measure the optical depth of the cluster and therefore
separate Te and
e from a
measurement of the thermal SZE (see Eq. 1).
It can be shown that polarization of the SZE comes entirely from the
quadrupole component of the local radiation field experienced by the
scattering electron. In the case above, the quadrupole component at
the outskirts of the cluster is due to the anisotropy in the radiation
field in the direction of the cluster center due to the SZE. Sunyaev
and Zel'dovich discussed polarization due to the motion of the cluster
with respect to the CMB and transverse to our line of sight
(Sunyaev &
Zel'dovich, 1980b, see also
Sazonov &
Sunyaev, 1999).
In this case,
the quadrupole comes from the Doppler shift. They found the largest
terms to be of order 0.1e(vpec / c)2 and
0.025
e2(vpec / c) of
the CMB intensity. The latter term, second order in
e, can be thought
of as imperfect cancellation
of the dipole term due to the anisotropic optical depth. Using
e = 0.01 and a
bulk motion of 500 km s-1, results in
polarization levels of order 10 nK, far beyond the sensitivity of
current instrumentation.
The CMB as seen by the cluster electrons will have a quadrupole
component and therefore the electron scattering will lead to linear
polarization. This mechanism could possibly be used to trace the
evolution of the CMB quadrupole if polarization measurements could be
obtained for a large number of clusters binned in direction and
redshift
(Kamionkowski & Loeb, 1997,
Sazonov &
Sunyaev, 1999).
Sazonov and Sunyaev
calculated the expected polarization level and found the maximum CMB
quadrupole induced polarization is
50(e/0.01) nK,
somewhat higher than the expected velocity induced terms discussed
above. The effect is again too small to expect detection in the near future.
However, by averaging over many clusters, detecting this polarization
might be possible with future satellite missions.