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Astron. Astrophys. 322, 709-718 (1997)

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3. VLM star clusters

3.1. Rationale

The rationale for invoking clustered DM comes from a number of considerations. Firstly, visible globular clusters are observed out to great distances from the Galactic centre suggesting they may have an important role in halo formation, though they are identified with the spheroid population. Secondly, some of the most promising theories for the formation of large amounts of compact halo baryonic DM (Ashman 1990 ; De Paolis et al. 1995 ) predict that the DM should be clustered into dark globular clusters with a typical mass of around [FORMULA] (Ashman 1990 ). These theories predict that the clusters should comprise either brown dwarfs or VLM stars, due to high gas pressures suppressing the minimum fragmentation mass. They also explain the typical mass and spatial distribution of the visible cluster population since the theories are extensions of the Fall-Rees theory of globular-cluster formation (Fall & Rees 1985 ). The baryon Jeans mass at the cosmological epoch of recombination is [FORMULA], which is also close to [FORMULA] for a cosmological density [FORMULA] of order unity and a baryon density [FORMULA] satisfying cosmological nucleosynthetic constraints: [FORMULA], where h is the Hubble constant in units of 100 km s-1  Mpc-1 (Krauss 1995 ). Lastly, limits on the space density of clusters provide firm upper limits on the allowed space density of VLM stars, provided one assumes that the VLM star distribution follows the halo density on the large-scale average (i.e. scales larger than the typical cluster separation).

The effect that clustering can have on number-count statistics is well illustrated by considering the expectation distances both to the nearest unclustered VLM star and to the nearest cluster. Let us assume that VLM stars contribute some fraction [FORMULA] to the halo DM and that their density is everywhere [FORMULA], where [FORMULA] is given by Eq.  1. Let us also take the fraction of VLM stars residing in clusters to be everywhere [FORMULA], so that the fraction of unclustered VLM stars is correspondingly [FORMULA]. Then, for VLM stars with mass m, the local unclustered number density is

[EQUATION]

whilst the local number density of VLM star clusters of mass M is

[EQUATION]

The expectation value for the distance d to the nearest object appearing within a survey of solid angle [FORMULA], assuming a local number density [FORMULA], is (c.f. Kerins & Carr 1994 )

[EQUATION]

where [FORMULA] is the gamma function. So, using Eq.  4 and Eq.  5, the expectation distances to the nearest unclustered VLM star and star cluster are, respectively,

[EQUATION]

where [FORMULA] is normalised to a solid angle 20 times that of a single HST field.

The large value for [FORMULA] indicates the need to consider the variation of [FORMULA] along the line of sight in order to obtain a more accurate answer, though the estimate of Eq.  8 is sufficient to show that one might not see any clusters at all within the HST fields, even if they comprise all the halo DM. Additionally, the expected fluctuation in source counts from field to field will be governed by the number density of clusters and not the spatially-averaged density of the stars themselves. This problem is addressed in the next subsection where I calculate 95% CL lower limits on the expected cluster number density, which in turn provide 95% CL upper limits on the contribution of VLM stars to the halo DM.

Another important consideration is whether one expects HST to be able to resolve individual stars within clusters. Since BFGK filter their observations to search for only stellar-like sources it is conceivable that a significant fraction of the total number of possible VLM star sources have been discarded because they reside in the cores of unresolvable clusters. Since the number of field galaxies far exceeds the number of stars at the typical limiting magnitudes of the HST fields, it may prove extremely difficult to analyse these fields for the presence of unresolved cluster cores.

Using Eq.  8 as a lower limit on the cluster distance one finds a typical angular separation between neighbouring stars (as projected along the line of sight) for a cluster with radius R of

[EQUATION]

compared to a HST pixel resolution of 0.1 arcsec. Since one expects clusters to be centrally concentrated one anticipates Eq.  9 overestimating the angular separation of stars near the core and underestimating the separation near the cluster edge. Therefore it seems that a significant fraction of cluster sources may appear unresolved. This fraction will be assessed more rigorously in Sect.  3.3.

For the cluster scenario to be viable it must be consistent with existing dynamical constraints on such objects. These constraints, which are reviewed in Sect.  3.4, provide upper and lower bounds to both the permitted cluster mass and radius. The maximum contribution of such clusters to the halo is therefore also potentially bounded by these constraints.

Finally, in Sect.  3.5, I show that this scenario is compatible with MACHO microlensing observations, and I use the MACHO limit on the contribution of VLM stars to compute the allowed ratio of clustered to unclustered stars. Potentially, this ratio provides the strongest constraint on the cluster scenario.

3.2. Space density of VLM clusters

In Sect.  2 limits were placed on the space density of unclustered VLM stars from HST number counts, assuming the stellar distribution obeys Poisson statistics. However, since clusters are extended objects, one can not directly apply the same assumption to them, since one may have a cluster which is only partially within the field of view (in fact this will be the case generally for the HST fields due to their small solid angle).

However one can use Poisson statistics to assess the likely number of cluster centres within the field of view. One can therefore extend this idea by imagining a volume centred on the actual field volume, but whose radius of cross section along the length is always larger by an amount R (the cluster radius) and whose length is also larger by an amount R at either end (i.e. by an amount [FORMULA] overall). Limits on the number of cluster centres appearing within this larger volume necessarily correspond to limits on the number of clusters which appear either wholly or partially within the actual survey field volume.

The description above applies to a cylindrical survey volume (i.e. a circular field of view). The size of the larger volume and number-count limits derived from it are sensitive to the geometry of the field volume on which it is centred (i.e. the shape of the field of view), which for the HST is 'L'-shaped rather than circular. However, for most of the cluster radii considered in this paper their angular sizes are always much larger than the angular size of the field of view itself, so the sensitivity to the shape of the field of view is generally very small, and so for simplicity I assume a cylindrical field volume geometry throughout.

Another important consideration when placing limits on cluster number densities is the statistical dependency of neighbouring fields. In principle, sufficiently large clusters could appear in more than one field and so any limits would need to take such correlations into account. Even if one is placing limits based on the absence of clusters within the observed fields one still needs to consider this problem. One can see why by considering two neighbouring fields which are separated by less than a cluster diameter. The prescription above would oversample the actual volume of space required to place limits on the cluster density (i.e. there would be a volume of space which would be common to the periphery of both fields and would therefore be counted twice), resulting in an overestimate of the expected number of clusters. As mentioned in Sect.  2, fields 1 and 2, whilst not overlapping, are nonetheless too close for most of the cluster radii considered here and so field 1, which samples a smaller amount of halo mass than field 2 (using Eq.  2), is discarded. Similarly, field 19, which partially overlaps with field 20, is also discarded leaving [FORMULA] fields. For these remaining fields the spatial separation between nearest-neighbours ranges from [FORMULA]  pc for clusters comprising 0.2- [FORMULA] VLM stars (with a median separation of 340 pc), and from [FORMULA]  pc for clusters comprising 0.092- [FORMULA] stars (with a median value of 110 pc). The statistical dependency of neighbouring fields therefore only becomes important for the very largest cluster radii considered in this section. I therefore ignore such correlations and assume the fields to be statistically independent.

A lower limit on the expectation number of clusters with mass M and radius R, comprising VLM stars of mass [FORMULA], which either wholly or partially appear within the [FORMULA] HST fields is (c.f. Eq.  3)

[EQUATION]

where

[EQUATION]

Note that for [FORMULA], [FORMULA] in Eq.  2, as required.

If one assumes that the HST fields contain no cluster stars at all then the 95% CL upper limit on the allowed halo fraction is [FORMULA] ; 3 being the 95% CL upper limit on the true average number of clusters when none are seen, assuming statistical independency of neighbouring fields and Poisson statistics. Note from Eq.  10 that the upper limit [FORMULA] is independent of [FORMULA] since it represents a limit on the absolute number of clusters. If [FORMULA] is significantly less than unity stronger constraints on the VLM density will come from the limits obtained in Sect.  2 for unclustered VLM stars. This issue is considered in more detail in Sect.  3.5 where I consider the efficiency with which VLM stars need to be clustered.

The limits on [FORMULA] for a range of cluster masses and radii are shown in Fig. 1, assuming [FORMULA]. The solid grid depicts the limits assuming the clusters comprise VLM stars of mass [FORMULA], where as the dotted line shows the limits for clusters comprising hydrogen-burning limit stars ([FORMULA]). The choice of cluster parameters is guided by the dynamical considerations discussed in Sect.  3.4. The plane at [FORMULA] is shown to guide the eye in seeing which part of the parameter space is constrained and which is not. The most strongly constrained region occurs for low cluster masses and large radii, though combinations of cluster parameters for which [FORMULA] (where [FORMULA] is the average halo density within the survey volumes) are unphysical since they denote clusters whose average density is less than that of the local halo background. (Such 'clusters' might instead best be regarded as voids.) For interesting (i.e. significant) values of [FORMULA] and [FORMULA] this unphysical region includes the case where the average angular separation between neighbouring stars, given by Eq.  9, exceeds the HST field of view. Whilst HST provides strong limits on clusters of low mass and large radius, it is clear from the figure that a significant fraction of the parameter space is not constrained by HST observations.

[FIGURE] Fig. 1. 95% CL upper limits on the allowed halo fraction [FORMULA], as a function of cluster mass M and radius R, assuming a clustering fraction [FORMULA]. The solid grid shows the limits for clusters comprising VLM stars of mass [FORMULA] and the dotted grid for stars at the hydrogen-burning limit, [FORMULA]. [FORMULA] is normalised to a local halo DM density [FORMULA], and [FORMULA] is demarcated by the plane on which the skirting around the perimeter of the grid is centred. The region of the grid at low cluster masses and large radii, which dips below this plane, is constrained by HST number counts, whereas the region rising above it denotes cluster parameters which can provide all of the halo DM whilst remaining compatible with HST observations.

It is apparent from Fig. 1 that the relationship between [FORMULA] and M is linear, which is simply a consequence of the fact that [FORMULA] from Eq.  10. Whilst the relationship between [FORMULA] and R is not a simple power law one can crudely approximate the condition for which [FORMULA] can exceed 1 as

[EQUATION]

where for 0.2- [FORMULA] stars [FORMULA]  pc and [FORMULA], and for 0.092- [FORMULA] stars [FORMULA]  pc and [FORMULA]. If instead of providing all of the halo DM one merely requires that clusters provide at least 40% of the DM, the halo fraction indicated by the MACHO experiment for a local halo normalisation [FORMULA] (Alcock et al. 1997), then one requires [FORMULA]  pc and [FORMULA] for 0.2- [FORMULA] stars, or [FORMULA]  pc and [FORMULA] for hydrogen-burning limit stars. These inequalities should be compared to the requirement noted above that clusters must represent local density enhancements rather than decrements, which requires

[EQUATION]

The limit should also be compared to the dynamical constraints to be discussed in Sect.  3.4.

It is important to note that these limits are fairly insensitive to the actual number of VLM stars observed in the HST fields, provided the observed number is relatively small (e.g. [FORMULA]). This is because if even just a portion of one cluster enters one of the fields of view one would generally expect HST to detect hundreds or even thousands of stars in that field due to the high cluster surface density. For the same reason these limits are also relatively insensitive to the surface-density profile of the cluster, at least within the range of reasonable surface-density profiles (i.e. similar to those inferred from the surface-brightness profiles of visible clusters).

3.3. The question of resolvability

The previous subsection dealt with limits imposed from the absence of clusters within the HST fields. However, compact clusters can escape detection even if they do appear in the fields, provided their surface densities are sufficiently high as not to allow them to be resolved, or provided the total surface density through all clusters along the line of sight is sufficiently large. The rigorous point-source selection criteria of BFGK means that any unresolved portions of clusters are discarded, in which case only some fraction [FORMULA] is detectable. Thus the effective number of detectable clusters is [FORMULA], which implies that the effective allowed halo fraction is [FORMULA], where [FORMULA] denotes averaging over fields. Thus if [FORMULA] for a particular set of cluster parameters [FORMULA] then HST point-source counts do not place any limits on the allowed halo fraction in such clusters.

The effect on resolvability due to several clusters aligned along the line of sight can be estimated from the ratio of the average cluster surface density to that between [FORMULA] and [FORMULA]:

[EQUATION]

Assuming the distribution of clusters on the sky is Poissonian, their sky-covering factor is [FORMULA]. Therefore the surface density measured along a line of sight through a cluster will be enhanced over that expected for the single cluster alone by a factor which is on average

[EQUATION]

The median line-of-sight surface density between [FORMULA] and [FORMULA] for the 20 HST fields is [FORMULA] for 0.092- [FORMULA] stars and [FORMULA] for 0.2- [FORMULA] stars, corresponding to values for [FORMULA] of 1.45 and 2.34, respectively, assuming [FORMULA] - [FORMULA] clusters with a 10-pc radius and [FORMULA].

The resolvable fraction of an individual cluster is sensitive to its assumed surface-density profile, as well as its mass and radius and the distance at which it is expected to be observed. I assume here that the cluster surface density follows the surface-brightness profile of many observed globular clusters, which are well described by the King (1962 ) surface-brightness law. Thus the surface number density [FORMULA] as a function of cluster-centric radius r is assumed to be given by

[EQUATION]

where [FORMULA] denotes the central surface density and [FORMULA] is the projected cluster core radius. Since the integrated mass of the cluster must be M the central surface density is

[EQUATION]

where [FORMULA]. For the purpose of simplification I adopt [FORMULA] for all clusters, a value which is typical for visible clusters, though in reality there is a large dispersion about this value.

Clusters appear resolved in the HST fields provided [FORMULA] is less than

[EQUATION]

where [FORMULA]  arcsec for the HST. If [FORMULA] then one expects that at least some of the cluster is unresolvable. From Eq.  16 to 18, the mass fraction of a cluster at distance x which can be resolved is

[EQUATION]

if [FORMULA] or unity otherwise. Note the factor [FORMULA] from Eq.  15, which enters in the upper integral limit. This is because, as argued above, one expects the observed line-of-sight density to be [FORMULA], rather than just [FORMULA].

Thus, for HST field i, the expectation value for the resolvable fraction of clusters is

[EQUATION]

where [FORMULA] is given by Eq.  1. Hence, the effective number of detectable clusters within the [FORMULA] fields is given by Eq.  10 with [FORMULA].

Fig. 2 shows the upper limits on the effective halo fraction [FORMULA], assuming [FORMULA]. Unlike the plot in Fig. 1, the surface does not strictly represent a 95% CL constraint, since for that one would need to evaluate the joint 95% CL variation in the cluster number [FORMULA] and in the resolvable fraction [FORMULA]. Such an evaluation would require detailed Monte-Carlo simulations of the cluster distribution for each parameter set [FORMULA], and for each field i.

[FIGURE] Fig. 2. Upper limits on the effective allowed halo fraction [FORMULA] as a result of taking into account the expectation fraction of cluster stars which can be resolved [FORMULA]. The assumed parameters are the same as for Fig. 1. Note that for low cluster masses and large radii the allowed fraction is the same as for Fig. 1, whilst for large masses and small radii [FORMULA] becomes asymptotically large as [FORMULA].

Fig. 2 shows that the problem of cluster resolvability only becomes important for high-mass, compact clusters, but that it can have a very strong effect on detection in this regime. For the highest mass and most compact clusters considered the resolvable fraction [FORMULA] rapidly approaches zero, so that no point-source observation of a VLM star is expected, even if clusters appear within the field of view. The density enhancement [FORMULA] due to the superposition of clusters is found to have little effect on [FORMULA], since [FORMULA] only becomes significantly larger than unity for clusters with large radius (in which case [FORMULA] in Eq.  15), and these clusters anyway have such low surface densities that the enhancement has little consequence.

As for Fig. 1, one can crudely parameterise the portion of the plot which permits [FORMULA] to be unity by using Eq.  12 with [FORMULA]  pc and [FORMULA] for 0.2- [FORMULA] stars, or [FORMULA]  pc and [FORMULA] for 0.092- [FORMULA] stars. If one instead simply demands consistency with MACHO ([FORMULA]) then the allowed range is slightly larger with [FORMULA]  pc and [FORMULA], or [FORMULA]  pc and [FORMULA] for 0.2- [FORMULA] or 0.092- [FORMULA] stars, respectively.

3.4. Dynamical considerations

One can provide strong constraints on the permissible mass and radius of clusters which contribute significantly to halo DM by considering their dynamical effects on the visible stellar population (e.g. Lacey & Ostriker 1985 ; Carr & Lacey 1987 ; Moore 1993 ; Moore & Silk 1995 ). Such considerations lead to the conclusion that viable cluster parameters [FORMULA] are bounded above and below, forming an 'island' in parameter space (Kerins & Carr 1994 ; Moore & Silk 1995 ). This is important because it means that one cannot invoke arbitrarily massive and compact clusters in order to force agreement between various observations. Furthermore, in principle cluster parameters which are compatible with dynamical limits may be incompatible with source-count and MACHO limits, thereby ruling out the entire scenario.

The issue of dynamical constraints is somewhat complicated by the fact that the limits depend (sensitively in some instances) on Galactic as well as cluster parameters. Whilst I do not explicitly present the dependencies of the constraints on these other parameters (these can be found in the references cited above), I assume the following values for the relevant parameters: a typical Galactocentric distance for clusters in the vicinity of the HST fields of 8 kpc (in fact most of the fields are further from the Galactic centre, where their dynamical effects would be weaker than assumed here); a cluster halo fraction [FORMULA] (required to provide consistency with MACHO observations); an average halo mass density [FORMULA], giving a cluster number density [FORMULA] ; a halo core radius [FORMULA]  kpc; a one-dimensional halo velocity dispersion of 156 km s-1 ; and a Galaxy age of 15 Gyr. The dependency on Galactic parameters means that the dynamical limits presented here do not represent firm limits in the strictest sense, but should nonetheless help in assessing the viability of the cluster scenario. (Note that some of the authors above express constraints in terms of the cluster half-mass radius [FORMULA], assuming the surface-density profile of Eq.  16 with [FORMULA].)

An upper limit on the radius of clusters comes from the requirement that they do not disrupt one another due to collisions occurring within the lifetime of the Galaxy. This implies [FORMULA]  pc for the parameters adopted above. Clusters also need to avoid tidal disruption due to the differential force from the Galactic potential acting across the cluster diameter; a constraint which requires [FORMULA]  pc. Since at any time there is always a finite fraction of cluster members whose velocities exceed the cluster escape velocity, clusters slowly evaporate over time. That they do not evaporate by the present day requires [FORMULA].

The final constraint comes from considering the effect of close or direct collisions between dark clusters and observed diffuse globular clusters (Carr 1978 ; Moore 1993 ; Moore & Silk 1995 ). Moore (1993 ) showed that the requirement that observed diffuse clusters show no sign of disruption today leads to the limit [FORMULA] for dark clusters with small radius, though this value is derived from a rather unusually diffuse cluster that may have atypical properties which result in too strong a constraint. A more robust limit from other less diffuse and more representative clusters demands [FORMULA] for [FORMULA] (B. Moore, private communication). For clusters with larger radius Moore & Silk (1995 ) derive a limit corresponding to [FORMULA]  pc.

These limits, for an assumed star mass [FORMULA], are projected onto the plane [FORMULA] in Fig. 3 (bold lines), together with the corresponding HST limits from Fig. 2, which assume a clustering fraction [FORMULA]. Regions of parameter space for which HST counts permit [FORMULA] are projected onto the plane [FORMULA], corresponding to the halo fraction preferred by MACHO microlensing observations towards the LMC (Alcock et al. 1997). Thus the flat region of the [FORMULA] plane at small cluster radii and large masses simultaneously satisfies MACHO observations and HST counts. It is clear that this region and the region bounded by the dynamical limits are almost mutually exclusive, with only a small area of the [FORMULA] plane satisfying all three requirements. This is due to the strong limits inferred from the absence of dynamical effects on presently-observed diffuse globular clusters, and from evaporation considerations. The surviving parameter region is characterised by a cluster mass [FORMULA] and radius [FORMULA]  pc and is relatively insensitive to the VLM star mass within the range considered in this paper. Clearly, for the cluster scenario to be viable M and R have to be rather finely tuned, though it is interesting that the allowed parameter space coincides well with the baryon Jeans mass at the recombination epoch, as well as to the predictions of some baryonic DM formation theories (e.g. Ashman 1990 ).

[FIGURE] Fig. 3. Comparison of HST limits, MACHO observations and dynamical constraints, assuming [FORMULA] and a VLM star mass of [FORMULA]. The upper limits permissible from HST counts (shown in Fig. 2) for which [FORMULA] are projected onto the plane [FORMULA] ; the central value inferred from MACHO LMC observations for a local halo normalisation of [FORMULA]. Also projected onto the same plane are the dynamical limits discussed in the main text (bold lines), which are sensitive to Galactic as well as cluster parameters. Note that there exists a small region of parameter space with [FORMULA] and [FORMULA]  pc which is simultaneously compatible with all of the limits. The overlap is marginally larger for lower mass VLM stars.

3.5. Microlensing signatures

The effect of clustering halo objects can be to increase or decrease the observed microlensing rate, depending on whether or not the observer's line of sight intersects with more or less than the expected average number of clusters. One can therefore derive firm upper and lower limits on the halo lens fraction by assuming it to be clustered.

Maoz (1994 ) has shown that there are two key features that should help to discriminate between microlensing events in the clustered and unclustered cases. The first is that the microlensing optical depth through a cluster is [FORMULA], where [FORMULA] is the Einstein radius and L is the distance to the lensed source. Since this can be a factor [FORMULA] higher than in the unclustered case for very massive ([FORMULA]) compact ([FORMULA]  pc) clusters, Maoz notes that one expects to observe angular correlations between events on a scale [FORMULA]. Maoz claims that for clusters of this mass a sample as small as 10 events would be sufficient to rule out such clusters from providing all the halo DM at the 95% CL, if such correlations are not observed. Assuming that angular correlations are observed, one should also expect nearby events to have similar timescales, assuming the lenses have similar masses. This is because the events would be due to objects residing in the same cluster, so they would be at the same distance from the observer and would have similar velocities. These conclusions are however based on cluster mass scales which are now ruled out by the dynamical considerations discussed in the previous subsection. The prospects for detecting such signatures from clusters with [FORMULA] are much less optimistic (Metcalf & Silk 1996 ), especially if they comprise only 40% of the DM.

Whether clustering could significantly affect the statistics for the halo DM fraction [FORMULA] depends upon how many clusters one expects to see within the solid angle of observation. The MACHO LMC search covers a solid angle of 11 deg2 (Alcock et al. 1996), so the expected number of clusters between the observer and the LMC ([FORMULA], [FORMULA], [FORMULA]  kpc) is

[EQUATION]

where

[EQUATION]

for a halo with a density profile given by Eq.  1. Thus, for clusters which are simultaneously compatible with dynamical limits and HST source counts (implying [FORMULA] from Fig. 3) one expects [FORMULA] to be within the MACHO field of view if [FORMULA] and [FORMULA].

If the clusters all have the same mass and comprise objects with the same MF, then the observed rate will depend only on [FORMULA]. Assuming the cluster distribution on the sky obeys Poisson statistics, the probability of there being N clusters within the field of view when one expects [FORMULA] (in which case one would infer a halo fraction [FORMULA] when the true fraction is [FORMULA]) is simply [FORMULA].

The MACHO data yields a likely halo fraction [FORMULA] if the lenses are assumed to be unclustered. The inferred value for [FORMULA] would be the same for the cluster scenario provided [FORMULA], otherwise one should take [FORMULA]. The 1- [FORMULA] variation on [FORMULA] is less than 3% for [FORMULA], so the inferred halo fractions for the clustered and unclustered regimes are virtually identical in this case. Only for [FORMULA] do Poisson fluctuations in the number of clusters become important, though in this case the microlensing signal is anyway dominated by the smoothly distributed VLM stars. The conclusion therefore is that for cluster mass scales compatible with dynamical limits their microlensing signatures are indistinguishable from the unclustered case.

It is interesting to consider what the inferred halo fraction from microlensing, together with the HST source-count limits, imply for the likely value for the clustering efficiency [FORMULA]. The halo considered in this paper has a microlensing optical depth [FORMULA] towards the LMC, assuming it is completely comprised of lenses. The 95% CL lower limit on the measured optical depth is [FORMULA] (Alcock et al. 1997), compared to the optical depth contribution expected from all non-halo components of [FORMULA]. Subtracting the contribution from these components gives a lower limit on the halo fraction of [FORMULA]. From Sect.  2 the maximum contribution from unclustered zero-metallicity VLM stars to the halo is 1.4% at the 95% CL. Therefore, consistency between MACHO and HST observations requires that [FORMULA]. That is, one requires a present-day clustering efficiency of 92% or better. This is certainly a very strong demand for the cluster scenario to meet. If one adopts the central value for the measured optical depth ([FORMULA]) then the required clustering efficiency must be at least 97%.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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