## 4. Hubble constant without projection effectsHere, we consider the possibility of deriving the Hubble constant
in a meaningful manner without any biases due to cluster projections.
It has been suggested in literature that observations of a large
sample of galaxy clusters can be used to average out the dependence on
the scale factor if all clusters are prolate, and if all clusters are oblate. Here . When all clusters are prolate and that the semi-major axis used to calculate the Hubble constant, then the distribution has a mean of . However, if the semi-minor axis is used, then the average Hubble constant is underestimated from the true value by about 10%, assuming that the mean is 0.7 for prolate clusters. If all clusters are oblate, and the semi-major axis is used to derive the Hubble constant, then the mean of the distribution overestimates the true value of the Hubble constant by as much as 20%, if the mean is 1.5 for oblate clusters. For oblate clusters, the true value of the Hubble constant can be obtained when the semi-minor axis is used. However, in both oblate and prolate cases, the distribution has a large scatter requiring a large sample of galaxy clusters to derive a reliable value of the Hubble constant. A similar calculation can also be performed for the gas mass fraction to estimate the nature of the value derived by averaging out a gas mass fraction measurements for a large sample of clusters. Here again, a similar offset as in the Hubble constant is present, and measurements of gas mass fraction in a large sample of clusters are needed to put reliable limits on the cosmological parameters, especially the mass density of the universe based on cosmological baryon density (e.g., Evrard 1997). So far, we have only considered the SZ and X-ray observations of
galaxy clusters. By combining weak lensing observations towards galaxy
clusters, we show that it may be possible to derive a reliable value
of the Hubble constant based on observations of a single cluster. The
gravitational lensing observations of galaxy clusters measure the
total mass along the line of sight through the cluster. The SZ effect
measures the gas mass along the line of sight, and thus, the ratio of
SZ gas mass to gravitational lensing total mass should yield a
measurement of the gas mass fraction independent of cluster shape
assumptions and asphericity. Here, we assume that the cluster gas
distribution exactly traces the cluster gravitational potential due to
dark matter, and that these two measurements are affected equally by
cluster shape. This is a reasonable assumption, but however, it is
likely that gas distribution does not follow the dark matter
potential, and that there may be some dependence on the cluster shape
between the two quantities. For now, assuming that the gas mass
fraction from SZ and gravitational lensing is not affected by cluster
projection, we outline a method to estimate the Hubble constant
independent of the scale factor We applied this to SZ, X-ray and weak lensing observations of
galaxy cluster A2163. The SZ observations of A2163 are presented in
Holzapfel et al. (1997), while weak lensing and X-ray observations are
presented in Squires et al. (1996). The SZ effect towards A2163 can be
described with a © European Southern Observatory (ESO) 1998 Online publication: October 21, 1998 |