4. Hubble constant without projection effects
Here, 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 Z and to produce the true value of the Hubble constant, which we define as from individual Hubble constant measurements, , in a large sample of clusters. We investigate the possibility of such an averaging by considering the different projections of clusters at different inclination angles. Assuming the previously described ellipsoidal shape and the effect of the scale factor Z in the Hubble constant, we can over the all possible inclination angles and the ratio to derive the expected average value of the Hubble constant :
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 Z. The gas mass fraction based on SZ and lensing is , while the gas mass fraction based on X-ray emission gas mass and the total mass based on X-ray temperature is . Since the two gas mass fraction measurements are expected to be the same, then one can solve for a combination of h and Z. However to break the degeneracy between h and Z an additional observation or an assumption is needed. In general, there are large number of clusters with X-ray measurements and X-ray based gas mass fraction measurements. By averaging out the gas mass fraction for such a large sample of clusters, we can estimate the universal gas mass fraction value for clusters, e.g. (Evrard 1997; Cooray 1998). If assumed that this gas fraction is valid for the cluster for which SZ and weak lensing observations are available, we can then calculate the Hubble constant.
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 y () parameter of , which includes various uncertainties described in Holzapfel et al. (1997). The weak lensing observations of A2163 has been used to derive the total cluster mass in Squires et al. (1996), and the lensing observations are most sensitive out to a radius of 200" (0.423 h-1 Mpc) from the cluster center, where the total mass is . Using the cluster model ( and ) in Holzapfel et al. (1997), we integrated the SZ temperature change to this radius from cluster center along the line of sight to derive a gas mass of . This represents the gas mass within the cylindrical cut across the cluster, and effectively probes the same region as the weak lensing observations. The gas mass fraction based on the SZ effect and the weak lensing total mass is . When this gas mass is compared to the effective gas mass fraction of clusters, (Evrard 1997; Cooray 1998), we obtain . We have slightly overestimated the error in the average gas mass fraction to take into account the fact that this fraction is measured at the outer hydrostatic radius ( 1 Mpc), and may not correspond to the value at the observed radius of A2163. In Squires et al. (1997), the gas mass fraction was measured to be for A2163, which is in agreement with our universal value, but the value in Squires et al. (1997) may be subjected to a scaling factor. The combined SZ/lensing gas mass fraction and the average gas mass fraction for clusters result in a Hubble constant of km s-1 Mpc-1. Given that we used data from 2 different papers in deriving this Hubble constant, it is likely that this value may be subjected to unknown systematic effects between the two studies. We strongly recommend that a careful analysis of cluster data be carried out to derive the Hubble constant based on SZ, X-ray and weak lensing observations. In addition, total virial masses from velocity dispersion analysis should also be considered in such an analysis to constrain the cluster shape. It is likely that much stronger and reliable result may be obtained through this method, instead of just SZ and X-ray observations. In Holzapfel et al. (1997), the Hubble constant was derived to be 60 km s-1 Mpc-1 for an isothermal temperature model and 78 km s-1 Mpc-1 for a hybrid temperature model. Our value is lower than these two values, but is in good agreement with the average value of based on SZ and X-ray as tabulated in Table 1, which is in agreement with the average gas mass fraction value.
© European Southern Observatory (ESO) 1998
Online publication: October 21, 1998