8.3. Interferometric methods

The two techniques discussed provided most of the existing data on the Sunyaev-Zel'dovich effect until very recently. Both techniques are excellent for large-scale surveys of clusters of galaxies which are well matched to the beam-switching technique being used, but provide only modest angular resolutions on the sky (although higher-resolution and two-dimensional bolometer arrays are now becoming available) and hence are suitable only for simple mapping (as in Figs. 17 and 19). Radio interferometry is a powerful method for making detailed images of Sunyaev-Zel'dovich effects. Such images are valuable for making detailed comparisons with X-ray images, and can also measure accurate Sunyaev-Zel'dovich effects while avoiding some of the systematic difficulties of the other techniques. Perhaps for these reasons, interferometry is the most rapidly-growing area for observation of the Sunyaev-Zel'dovich effects (Table 3).

Table 3. Interferometric measurements of the Sunyaev-Zel'dovich effects

Paper		p	B	Telescope
	(GHz)	(arcmin)	(m)
Partridge et al. 1987	4.9	9	35-1030	VLA
Jones et al. 1993	15	6	18- 108	RT
Grainge et al. 1993	15	6	18- 108	RT
Jones 1995	15	6	18- 108	RT
Saunders 1995	15	6	18- 108	RT
Liang 1995	8.8	5	31- 153	ATCA
Carlstrom et al. 1996	29	4	20- 75	OVMMA
Grainge 1996	15	6	18- 288	RT
Grainge et al. 1996	15	6	18- 288	RT
Jones et al. 1997	15	6	18- 108	RT
Matsuura et al. 1996	15	6	18- 108	RT
Note. - is the central frequency of observation. p is the corresponding FWHM of the beam provided by the primary antennas of the interferometer. B is the range of baselines that were used in the work. The telescopes are the Very Large Array (VLA), Ryle Telescope (RT), Australia Telescope Compact Array (ATCA), and Owens Valley Radio Observatory Millimeter Array (OVMMA).

where B(, ) is the brightness distribution of the sky, G(, ) represents the polar diagram of the antennas of the interferometer, (u,v) are the separations of the antennas, measured in wavelengths, (, ) are direction cosines relative to the center of the field of view, and the constant of proportionality depends on the detailed properties of the interferometer (see Thompson et al. (1986) for a detailed explanation of the meaning of this expression and the assumptions that go into it). An image of the sky brightness distribution, B(, ), can be recovered from the measurements V(u,v), by a back Fourier transform and division by the polar diagram function: alternatively, estimation techniques can be used to measure B(, ) directly from the V(u,v).

Most interferometers were originally designed to achieve high angular resolution. The finiteness of interferometer measurements means that not all (u,v) values are sampled: in particular, the design for high resolution means that the antennas are usually placed so that their minimum separation is many wavelengths (and always exceeds the antenna diameter by a significant factor). The Fourier relationship (92) means that the short baselines contain information about the large angular scale structure of the source, and so there is some maximum angular scale of structure that is sampled and imaged by interferometers. The Sunyaev-Zel'dovich effects of clusters of galaxies have angular sizes of several arcminutes - most interferometers lose (``resolve out'') signals on these or larger angular scales, and hence would find extreme difficulty in detecting Sunyaev-Zel'dovich effects.

Figure 21. The interferometer response that would be expected from a 6-cm VLA observation of the Sunyaev-Zel'dovich effect from cluster CL 0016+16, normalized to the effect with zero baseline. The minimum separation of VLA antennas is 650 wavelengths, but projection effects mean that the minimum observable baseline is roughly equal to the antenna diameter, or about 420 wavelengths. Thus on the shortest one or two baselines, for a brief interval, the VLA can observe about 25 per cent of the total available Sunyaev-Zel'dovich effect (which corresponds to about -0.9 mJy if the central Sunyaev-Zel'dovich effect is about -1 mK). Consequently, the VLA is a poor instrument for observing the Sunyaev-Zel'dovich effect in this cluster.

Figure 21 illustrates this effect for model Very Large Array (VLA) observations of cluster CL 0016+16 at = 6 cm. Since the VLA antennas shadow one another at baselines less than the antenna diameter (of 25 m), no information about the amplitude or shape of the visibility curve can be recovered at baselines less than 420. Most of the VLA baselines are much larger than the minimum baseline, even in the most compact configuration (D array). Hence the VLA's effective sensitivity to the Sunyaev-Zel'dovich effect in CL 0016+16 is low. But CL 0016+16 is a cluster at redshift 0.5455, has a small angular size, and so represents one of the best candidate clusters for observation with the VLA - the VLA is therefore not a useful instrument for measuring the Sunyaev-Zel'dovich effects of any clusters unless those clusters contain significant small-scale substructure in the Sunyaev-Zel'dovich effect, or the clusters can have significantly smaller angular sizes and substantial Sunyaev-Zel'dovich effects. Thus, for example, the VLA observations of Partridge et al. (1987) suffered from this effect: in their data the Sunyaev-Zel'dovich effect signal from Abell 2218 was strongly suppressed because of the excessive size of the array.

Smaller interferometers would allow the Sunyaev-Zel'dovich effects to be measured. What is needed is an array of antennas whose individual beam-sizes are significantly larger than the angular sizes of the cluster Sunyaev-Zel'dovich effects, so that many antenna-antenna baselines can be arranged to be sensitive to the effects. A first attempt to customize a telescope for this experiment was the upgrade of the 5-km telescope at Cambridge, UK into the Ryle telescope (Birkinshaw & Gull 1983b; Saunders 1995). In its new configuration, the five central 12.8-m diameter antennas can occupy a number of parking points which provide baselines from 18 m to 288 m. At the prime operating wavelength of 2 cm, the maximum detectable Sunyaev-Zel'dovich effect signal is about -1.3 mJy, and several baselines should see effects in excess of -0.1 mJy.

The choice of operating wavelength for mapping the Sunyaev-Zel'dovich effect is constrained to some extent by confusion, in the same way that the radiometric observations are affected. Some clusters of galaxies (particularly clusters of galaxies with strong Sunyaev-Zel'dovich effects; Moffet & Birkinshaw 1989) contain cluster halo sources, with similar angular size to the cluster as a whole and whose non-thermal radio emission can swamp the Sunyaev-Zel'dovich effects at low frequencies (although their non-thermal Sunyaev-Zel'dovich effects are probably small; Sec. 5). Such sources have steep spectra, and so are avoided by working at higher frequencies. Clusters of galaxies also contain a population of radio sources, many of which are extended (the wide angle tail sources, narrow angle tail sources, etc.). These extended sources are also avoided by working at high frequency, where their extended emission is minimized and where the small-scale emission can be recognized by its different range of Fourier components. Background, flat-spectrum, radio sources can also affect the data, but can be recognized by their small angular size.

Interferometers with a wide range of baselines are useful in this respect: the longer baselines are sensitive to the small-angular scale radio sources which dominate the radio confusion signal (and which affect the radiometric data: see Fig. 15), while the shorter baselines contain both the radio source signal and the Sunyaev-Zel'dovich effect signal. Thus the longer-baseline data can be analysed first to locate the confusing radio sources, and then these sources can be subtracted from the short-baseline data, so that a source-free map of the sky can be constructed and searched for the Sunyaev-Zel'dovich effect. Furthermore, by tuning the range of baselines that are included in the final map, or by appropriately weighting these baselines, a range of image resolutions can be produced to emphasize any of a range of angular structures.

Of course, this technique depends on there being a good separation of angular scales between the radio sources and the Sunyaev-Zel'dovich effects in the clusters: extended, cluster-based, radio sources cannot be removed reliably using this technique, and there are a number of clusters in which no good measurements of (or limits to) the Sunyaev-Zel'dovich effects can be obtained without working at a higher frequency with a smaller interferometer (to avoid resolving out the Sunyaev-Zel'dovich effect). A good choice of operating frequency might be 90 GHz, with antenna baselines of a few metres: a design which also commends itself for imaging primordial fluctuations in the background radiation.

Since many of the brightest radio sources at the frequencies for which interferometers are used are variable (with timescales of months being typical), the subtraction technique must sometimes be applied to individual observing runs on a cluster, rather than to all the data taken together. The brightest sources may also subtract imperfectly because of dynamic range problems in the mapping and analysis of the data: generally interferometric or radiometric observations of clusters are only attempted if the radio source environment is relatively benign. Any source contamination at a level 10 mJy is likely to be excessive, and to cause difficulties in detecting the Sunyaev-Zel'dovich effects, let alone mapping them reliably. Nevertheless, interferometric work has the advantage over radiometric work that the sources (in particular the variable, and hence small angular size sources) are monitored simultaneously with the Sunyaev-Zel'dovich effect, and so interferometer maps should show much better source subtraction.

Although the interferometric technique is extremely powerful, in taking account of much of the radio source confusion, and in allowing a map of the Sunyaev-Zel'dovich effect to be constructed, it does suffer from some new difficulties of its own. First, the range of baselines over which the Sunyaev-Zel'dovich effect is detected may be highly restricted, so that the ``map'' is little more than an indication of the location of the most compact component of the Sunyaev-Zel'dovich effects. This problem can only be solved by obtaining more short baselines, which may not be possible because of excessive antenna size (as with the VLA, for example).

The source subtraction may also cause problems, since strong sources outside the target clusters often lie towards the edges of the primary beam of the antennas of the interferometer. Small pointing errors in the antennas can then cause the amplitude of these sources to modulate significantly, adding to the noise in the map and reducing the accuracy with which the contaminating source signal can be removed from the Sunyaev-Zel'dovich effect. The problem is worst for sources lying near the half-power point of the primary beam, but significant difficulties can be caused by sources lying even in distant sidelobes, although this extra noise does not usually add to produce a coherent contaminating signal at the map center, where the Sunyaev-Zel'dovich effect is normally expected.

Careful attention must also be paid to the question of correlator errors, which can produce large and spurious signals near the phase-stopping center (see Partridge et al. 1987). In order to avoid excessive bandwidth smearing for contaminating sources which must be identified and removed successfully, it is also normal to observe using bandwidth synthesis methods (which split the continuum bandpass of the interferometer into a number of channels). The combination of these individual channel datasets back into a continuum map of the Sunyaev-Zel'dovich effect may sometimes be complicated by steep (or strongly-inverted) sources on the image which have different fluxes in the different channels.

One major advantage of using an interferometer is that the effects of structures in the atmosphere are significantly reduced. Emission from the atmosphere is important only in its contribution to the total noise power entering the antennas, since this emission is uncorrelated over baselines longer than a few metres and does not enter into the (correlated) visibility data. Furthermore, there are no background level problems: an interferometer does not respond to a constant background level, and so a well-designed interferometer will not respond to constant atmospheric signals, the uniform component of the microwave background radiation, large-scale gradients in galactic continuum emission, or ground emission entering through the telescope sidelobes.

The first cluster for which interferometric techniques were used successfully is Abell 2218, which had been shown to have a strong Sunyaev-Zel'dovich effect with a small angular size using single-dish measurements (Birkinshaw, Gull & Hardebeck 1984). Jones et al. (1993) used the Ryle interferometer at 15 GHz, with baselines from 18 to 108 m, to locate sources and to map the diffuse Sunyaev-Zel'dovich effect. The images that they obtained are shown in Fig. 22. Using baselines from 36 to 108 m, and 27 12-hour runs, a high signal/noise map of the cluster radio sources was made (Fig. 22, left). Using only the 18-m baseline, and subtracting the signals from these sources, a map with effective angular resolution about 2 arcmin was then made (Fig. 22, right). This clearly shows a significant negative signal, of -580 ± 110 µJy, centered at 16^h 35^m 47^s +66° 12' 50" (J2000). The corresponding value for the central Sunyaev-Zel'dovich effect in the cluster cannot be determined without knowing the shape of the efficiency curve (e.g., Fig. 21, which is effectively a visibility curve) on baselines less than those that were observed. The Ryle interferometer data could be fitted with models of the form (66), with a parameter space extending from 0.6, _c 0.9 arcmin, T₀ -1.1 mK, to 1.5, _c 2.0 arcmin, T₀ -0.6 mK. The random error on the detection of an Sunyaev-Zel'dovich effect is, therefore, much smaller than the systematic error in the central measurement of the effect - a better range of baselines, and a detection of the Sunyaev-Zel'dovich effect on more than a single baseline, would be needed to improve this situation.

Figure 22. Interferometric maps of Abell 2218, made with the Ryle telescope of the Mullard Radio Astronomy Observatory (Jones et al. 1993). Left: an image made with the longer-baseline data, which is sensitive chiefly to small angular scales. Three faint radio sources dominate the image. Right: an image made with the short-baseline data, after subtraction of the sources detected on the long-baseline image. Here the image is dominated by the Sunyaev-Zel'dovich effect from the cluster.

Much analysis of the Sunyaev-Zel'dovich effect can usefully be carried out in the data, rather than the map, plane - by fitting the model V(u,v) to the measured visibilities. Indeed, the most reliable indication of the reality of a Sunyaev-Zel'dovich effect may be its presence first in visibility plots (like Fig. 21), and such plots are invaluable for assessing the extent of the missing visibility data in (u,v), and hence the fraction of the full Sunyaev-Zel'dovich effect of a cluster that is being detected by the interferometer. Of course, similar calculations are needed for radiometric and bolometric observations of the Sunyaev-Zel'dovich effects, but the efficiency factors (b) are often lower in interferometric work, and so the sampling of the full Sunyaev-Zel'dovich effect is more critical to its interpretation.

More recently, excellent imaging data on the clusters CL 0016+16 and Abell 773 has been published by Carlstrom et al. (1996). These authors used the Owens Valley Millimeter Array (OVMMA) at 1 cm: by equipping an array designed for operation at 3 mm and shorter wavelengths with cm-wave receivers, they were assured of accurate pointing and a relatively large primary beam, so that the interferometer should not over-resolve the Sunyaev-Zel'dovich effects on short baselines. The total negative flux density of CL 0016+16 in this operating configuration is near -13 mJy if the cluster has a central decrement of -1 mK, so that the cluster should be relatively strong (negative) source. With the OVMMA, Carlstrom et al. (1996) detected a total negative flux density of -3.0 mJy after 13 days of observation: their map of the cluster is shown in Fig. 23.

Figure 23. An interferometric map of CL 0016+16, from Carlstrom et al. (1996), superimposed on a grey-scale representation of the X-ray emission of the cluster (from the ROSAT PSPC). The radio data on which this image is based were taken with the Owens Valley Radio Observatory Millimeter Array operated at 1 cm, contain antenna baselines from 20 to 75 m, and have a synthesized beam of about 55 arcsec (as shown in the lower left corner).

The power of radio interferometric mapping of a cluster is apparent in Carlstrom et al.'s map of the Sunyaev-Zel'dovich effect from CL 0016+16. The Sunyaev-Zel'dovich decrement is extended in the same position angle as the X-ray emission (Fig. 2) and the distribution of optical galaxies (and close to the position angle from the cluster to a companion cluster; Hughes et al. 1995). The small-scale structure seen in this image is close to that predicted from the X-ray image, and corresponds closely with the predicted amplitude based on earlier radiometric detections of the Sunyaev-Zel'dovich effect of the cluster (Uson 1986; Birkinshaw 1991).

Figure 24. An image of the Sunyaev-Zel'dovich effect from the cluster MS 0451.6-0305 at z = 0.55, as measured by Joy et al. (in preparation) using the OVMMA. The beamshape of the image is shown in the lower left corner. This cluster was first detected in the Einstein Medium-Sensitivity Survey (Gioia et al. 1990b), and therefore can be regarded as a part of an X-ray complete sample.

The success of recent interferometric mapping campaigns, which have produced results such as Fig. 24 has amply justified demonstrated the potential of this technique to improve on single-dish observations of the Sunyaev-Zel'dovich effect. The critical elements of this breakthrough have been the development of small interferometers dedicated to Sunyaev-Zel'dovich effect mapping over long intervals, and the existence of stable, low-noise receivers with exceptionally wide passbands.