## 1. IntroductionOver the last few years, there has been a tremendous increase in
the study of galaxy clusters as cosmological probes, initially through
the use of X-ray emission observations, and in recent years, through
the use of Sunyaev-Zel'dovich (SZ) effect. Briefly, the SZ effect is a
distortion of the cosmic microwave background (CMB) radiation by
inverse-Compton scattering of thermal electrons within the hot
intracluster medium (Sunyaev & Zel'dovich 1980; see Birkinshaw
1998 for a recent review). By combining the SZ intensity change and
the X-ray emission observations, the angular diameter distance,
, to a cluster can be derived (e.g., Cavaliere et
al. 1977). Combining the distance measurement with redshift allows a
determination of the Hubble constant, H The accuracy of the Hubble constant determined from a SZ and X-ray analysis depends on the assumptions. Using numerical simulations, Inagaki et al. (1995) and Roettiger et al. (1997) showed that the Hubble constant measured through the SZ effect can seriously be affected by systematic effects, which include the assumption of isothermality, cluster gas clumping, and asphericity. The Hubble constant can also be affected by statistical effects, including cluster peculiar velocities and astrophysical confusions, such as radio sources & CMB primary anisotropies. The latter statistical effects are expected to produce a broad distribution in the Hubble constant measured for a sample of galaxy clusters, while the former systematic effects are expected to offset the Hubble constant from the true value. In recent years, several other effects have also been suggested to explain the difference between the SZ and X-ray Hubble constant and the ones derived from other techniques. These include the preferential removal of the lensed background radio sources in SZ surveys (Loeb & Refregier 1997), which would systematically lower the Hubble constant by as much as 13% for SZ observations at 15 GHz, and gravitational lensing of the arcminute scale CMB anisotropy (Cen 1998), which would broaden the Hubble constant distribution for a sample of galaxy clusters. The first effect is in opposite direction to the radio source contamination in SZ observations due to galaxy cluster member radio sources, which dominate the radio source number counts towards galaxy clusters. As discussed in Cooray et al. (1998a), the two radio source effects are likely to cancel out. The Loeb & Refregier (1997) effect is also not expected to occur for SZ observations at high frequencies. The second effect, due to gravitational lensing of CMB anisotropy through galaxy cluster potential, is not expected to be a dominant source of error in the Hubble constant, given that Cen (1998) considered the largest upper limits to arcminute scale anisotropies, which have not yet been detected. Apart from the SZ and X-ray Hubble constant, the gas mass fraction,
, measurements from X-ray (also using SZ,
gravitational lensing and optical velocity dispersion measurements),
can also be used to constrain the cosmological parameters. The primary
assumption in such an analysis is that the gas mass fraction, when
measured out to a standard (hydrostatic) radius is constant. Evrard
(1997) applied these arguments to a sample of galaxy clusters using
X-ray data, and put constraints on the cosmological mass density of
the universe, , with some dependence on the
Hubble constant. Under the assumption that the cluster gas mass
fraction is constant in a sample of galaxy clusters, the apparent
redshift evolution of the baryonic fraction can also be used to
constrain the cosmological parameters (e.g., Pen 1997). Cooray (1998)
and Danos & Pen (1998) used the present X-ray gas mass fraction
data to derive in a flat universe
() and in an open universe
(; 90% C.I.). In Shimasaku (1997), the assumption
of constant gas mass fraction in galaxy clusters was used to put
constrains on , the rms linear fluctuations on
scales of 8 h Given the importance of SZ and X-ray emission observations in cosmological studies, we initiated a program to study the systematic effects in the present SZ and X-ray Hubble constant measurements and gas mass fraction measurements. As part of this study, we found a negative correlation between the broad distribution of the Hubble constant and the gas mass fraction measurements. We explain this observation as due to a projection effect of aspherical clusters modeled with a spherical geometry. In Sect. 2, we present the effects of projection on the Hubble constant and the gas mass fraction by projecting triaxial ellipsoidal clusters and extending the work of Fabricant et al. (1984). The observational evidence for projection effects in the present Hubble constant values based on SZ and X-ray route are presented in Sect. 3. In Sect. 4, we outline an alternative method to calculate the Hubble constant, by combining SZ, X-ray, gravitational lensing, and velocity dispersion measurements of clusters, and which is subjected to less projection effects than current method involving only the SZ and X-ray observations. We apply this technique to A2163 based on the published observational data, and derive a new Hubble constant. A summary and conclusions are presented in Sect. 5. © European Southern Observatory (ESO) 1998 Online publication: October 21, 1998 |