Probing Dark Matter with Gravitational Lensing

The technique of weak gravitational lensing has proven to be an important tool to study mass distributions in the universe. The projected mass distribution of foreground gravitational structures distorts the images of the faint background galaxies. As a result, gravitational lensing provides a direct measurement of the projected mass density (e.g. Kaiser & Squires 1993).

Until recently, massive structures in the universe were studied through dynamical analysis of their luminous components. These studies have shown that large amounts of dark matter exist in the universe. For clusters of galaxies a popular method uses the motions of the galaxies to estimate the mass using the virial theorem. One can also estimate the cluster mass profile from X-ray observations when one assumes hydrostatic equilibrium and spherical symmetry (e.g. Allen & Fabian 1994).

Both methods assume some dynamical state or geometry in order to obtain the mass or a mass profile. The advantage of gravitational lensing is the fact that no such assumptions are needed. In the regime of weak gravitational lensing one can calculate the projected mass surface density up to some additive constant from the observed distortion pattern (Kaiser & Squires 1993; Kaiser et al. 1995; Schneider & Seitz 1995; Schneider 1995; Squires & Kaiser 1996).

Since the first successful measurements of the weak gravitational distortions (Tyson, Valdes & Wenk 1990), many massive clusters of galaxies have been studied (e.g. Bonnet, Mellier, & Fort 1994; Fahlman et al. 1994; Squires et al. 1996b; Luppino & Kaiser 1997). In principle one can measure the gravitational distortion out to large radii from the cluster centre, beyond the radii where X-ray observations or cluster kinematics can be used to determine the mass distribution.

So far, most weak lensing studies of clusters of galaxies have been undertaken using data from ground based optical telescopes. These are affected by atmospheric seeing, which causes the images of the faint background galaxies to be enlarged and more circular. Recently these efforts have been extended with the weak lensing analysis of a cluster of galaxies, CL 1358+62, using HST observations with a large field of view (Hoekstra et al. 1998). Up to now, other weak lensing studies of clusters of galaxies with HST have been limited to cluster cores (C. Seitz et al. 1996; Smail et al. 1997). Those observations consisted of single pointings, thus suffering from the limited field of view of the HST (about 2 arcmin).

By using a mosaic of 12 pointings, a total field of view of approximately 8 by 8 arcmin could be obtained. This combination of space based observations and a large field of view provided an opportunity to study the cluster CL 1358+62 in great detail.

An advantage of HST observations is the high number density of galaxies one can reach. Previous HST studies have achieved $\simeq 100$ galaxies arcmin^-2 routinely (e.g. C. Seitz et al, 1996; Smail et al. 1997). With a one hour exposure per pointing, a number density of $\sim 50$ useful background galaxies arcmin^-2 can be obtained.

Another important advantage of HST over ground based observations is the size of the point spread function (PSF). Most of the faint objects are small. To recover the lensing signal one needs to correct for the effect of seeing (Bonnet & Mellier 1995, Kaiser, Squires & Broadhurst 1995, Luppino & Kaiser 1997, Fischer & Tyson 1997). For objects with sizes comparable to the PSF, these corrections become very large, amplifying the uncertainty in the ellipticity due to photon noise. As a result, the scatter in the derived ellipticities of the galaxies is larger than the expected scatter due to their intrinsic shapes. Consequently, for a given number density of background objects, the accuracy of weak lensing studies based on HST observations will be higher than the results from ground based data.

Even with the high angular resolution of the HST, a major limitation to the detection of gravitational shear is introduced by the distortions of the PSF over the area of each detector array. This is illustrated in Fig. 1.10 where HST archival data for the globular cluster M4 have been analyzed for the width and orientation of the stellar profiles.

  

Figure 1.10: The upper panel shows the ``polarization'' field of stars taken from observations of the globular cluster M 4. The orientation of the sticks shows the direction of the major axis of the HST PSF whereas the length is proportional to the size of the anisotropy. (From Hoekstra et al. 1998)

After careful calibration and correction it is in fact possible to extract the weak lensing signature as illustrated in Fig. 1.11 out to a distance of $\sim$1.5 Mpc from the cluster center. The observed distortion is consistent with a singular isothermal sphere model with a dispersion of $780\pm50$ km/s. The total projected mass within a radius of 1 Mpc, corresponding to this model is $(4.4\pm 0.6)\times 10^{14}~{\rm M}_\odot$. The errors given here represent the random error due to the ellipticities of the background galaxies. The uncertainty in the redshift distribution introduces an additional, systematic error of $\sim 10\%$ in the weak lensing mass. The weak lensing mass is slightly lower than dynamical estimates and agrees well with X-ray mass estimates. The mass distribution is elongated similar to the light. The axis ratio of $0.30\pm0.15$ and position angle of $-21^\circ\pm7^\circ$ were measured directly from the observations and agree very well with the previous strong lensing determination (Franx et al. 1997). A two-dimensional reconstruction of the cluster mass surface density shows that the peak of the mass distribution coincides with the peak of the light distribution. A mass-to-light ratio of $(90\pm 13)h_{50}{\rm M}_\odot/{\rm L}_{V\odot}$ is indicated, and this appears to be constant with radius.

  

Figure 1.11: Smoothed distortion field g from the blue and red background galaxies toward CL 1358+62. The length of the sticks denote the size of the distortion. The distortion field is calculated by smoothing the observed field from the individual galaxies with a Gaussian of width 0.4 arcmin(FWHM is indicated by the shaded circle). The characteristic pattern due to gravitational lensing is clearly visible. (From Hoekstra et al. 1998)

The impact of the SKA on the field of weak lensing will be particularly profound. With the sensitivities noted in the Introduction, we expect detected source densities of between 100 and 400 arcmin^-2 in an 8 hour integration. Some 98% of these sources will correspond to the same normal galaxies visible to the HST, while the remaining 2% will be active galactic nuclei and radio galaxies. While these source densities are comparable, but superior to those of the HST, there are two important differences. Firstly, the SKA PSF is both compact (smaller than about 0.1 arcsec) and extremely well-defined (to about one part in a million) over the entire field of view. And secondly, the enormous instantaneous field of view (of about one square degree) is sufficient to probe scales of some 20 Mpc on a side (at z = 0.3) per pointing. This should enable clean measurement of cluster mass surface densities as well as routine detection of the weak lensing signature due to large-scale structure, for which the first tentative detections have recently been made (Schneider et al. 1998).