Clusters of galaxies from cosmic strings

PHYSICAL REVIEW D VOLUME 38, NUMBER 12 15 DECEMBER 1988

Clusters of galaxies from cosmic strings

E. P. S. Shellard Center for Theoretical Physics, Laboratory for lVuclear Science, and Department of Physics,

Massachusetts Institute of Technology, Cam bridge, Massachusetts 02139

R. H. Brandenberger Department of Physics, Brown University, Providence, Rhode Island 02912

(Received 8 February 1988)

Early discussion of the cosmic-string model of galaxy and cluster formation assumed a one-to-one correspondence between string loops of a given radius and cosmological objects of a given mass. In this paper the self-consistency of this simple picture is investigated and shown to be far from com- plete. We use Abell clusters as a benchmark to test this assumption, analytically estimating rms fluctuations in the R = 1 cold-dark-matter model using random and correlated loop distributions. These results are compared with Monte Carlo simulations which are supplemented with systematic searches for large and rare overdensities. It is found that the majority of Abell clusters are not seeded by individual large loops, but by large mass concentrations of smaller loops. The implications for the cluster correlation function and for cluster morphology are discussed. The breakdown of this one-to-one correspondence is also apparent on galactic scales and mandates a revision of the standard cosmic-string scenario.

I. INTRODUCTION

The cosmic-string s c e n a r i ~ ' ? ~ for the formation of large-scale structure3ps has attracted considerable interest recently. The radical notion that galaxies formed by the accretion of matter about loops of cosmic string has distinct advantages over current models based on adi- abatic density perturbations. The differences are most notable on larger scales where increasingly firm observa- tional evidence of cluster-cluster correlations6 and large peculiar velocities7 has revealed deficiencies in the standard cold-dark-matter (CDM) picture.8 The cosmic- string scenario, on the other hand, allows such large peculiar velocities9 and may predict a universal (scale- invariant) correlation function which agrees reasonably well, both in amplitude and shape, with the observed cluster correlation function."

The latter remarkable observation was based upon the first numerical experiments by Albrecht and ~ u r o k , " and has since been confirmed in the substantially improved simulations of Bennett and ~ o u c h e t . " These results, however, represent an initial loop-loop correlation function, and recently some doubt has been cast on whether this would be sufficiently preserved to appear today for ~ l u s t e r s . ' ~ ' " The two may differ for several reasons. For example, gravitational attraction may distort a primordial cluster correlation function, as indeed it would for galaxies. This possibility, however, has been discounted in a recent paper'3 because there is insufficient time relative to the intercluster length scale. A more serious ob- jection is that the large initial peculiar velocities which are observed for loops at formation actually "wash out" these correlations. l '

It becomes even more problematic given the further possibility that a substantial proportion of clusters may

be seeded, not as assumed by large loops, but by collections of smaller loops. If such a one-to-one correspondence between clusters and large loops were inaccurate then their respective correlation functions could be very different. More importantly, however, this breakdown would entail a significant modification of the standard cosmic-string scenario, and the basis upon which the model has been studied and criticized.

The purpose of this paper is to address these and related statistical questions about cosmic strings. We make an analytic estimate of the fraction of clusters that form from large fluctuations in the mass distribution due to smaller loops as opposed to those due to individual large loops. This is compared with numerical estimates from Monte Carlo simulations of a loop distribution. We conclude that the accepted maxim that "loops provide peaks with which similarly discrete objects can be identified" is open to considerable dispute. Even with the superseded parameters of Ref. 10, we find a very marginal relation- ship between large individual loops and Abell clusters which depends sensitively on the strength of loop correlations, though large loops are still predominant and would seed some 45-60 % of clusters. In this case a relic correlation function of loops might still be in evidence. How- ever, current indications are that the density of strings is considerably higher with loops forming on scales much smaller than the horizon.",14 Loops expected to seed clusters would appear after matter domination, with the net result of increasing the strength of rms fluctuations relative to those from large loops. Small loops, it appears, become dominant on cluster scales, though wakes are likely to play a larger role than previously anticipat- ed.

This analysis is made possible because of strong indica- t i o n ~ ' ~ ~ " " ~ " ~ that a network of cosmic strings, formed in the very early Universe, will quickly evolve toward a

3 8 - CLUSTERS OF GALAXIES FROM COSMIC STRINGS 361 1

"scaling" solution with a constant number of string seg- ments stretching across each horizon volume. The spectrum of loops, whose formation is allowed by string inter- commuting,I6 will also exhibit scale invariance. If n (R,z)dR is the number density of loops in the interval [R,R + d R ] at a red-shift z, then if loops are formed at time t with an average size R -6t,

where vr,v, are dimensionless constants. The original string network simulations" pointed to a scenario with an unexpectedly low density of long strings in the radiation era, with loops forming on the scale of the horizon 6- 1 (v-0.01). Subsequent s imula t ions , l ' "~owever , give higher densities v-0.01-0.1 and smaller loops 6 < 0 . 1 (the two are not unrelated). The exact values of these parameters are still the subject of lively debate so they are included explicitly throughout the discussion.

Now if we assume that there is a one-to-one correspondence between loops of a given radius R and cosmological objects of a given mass and mean separation d, then we have

where z,, is the red-shift at the time t,, of equal matter and radiation.

In this discussion we take Abell clusters of richness class I as the benchmark for a high-density mass fluctuation. These are regions which contain more than 50 bright galaxies within a radius r, =3h Mpc (h,' is the Hubble parameter in units of 50 km S - ' / M ~ C ) . The luminosity of these galaxies must fall within a two-magnitude range of the third-brightest member of the cluster." Abell clusters have a mean separation of d, = 1 lOh Mpc, and so in the old picture (v-0.01, C- 1 ) where cluster loops appeared before t,,,

whereas in the new picture (v-0.1, 6-0.1) after matter domination,

For definiteness we will adopt the "standard" cosmic- string model, incorporating cold dark matter ( R = 1 ) and the string mass scale Gp= l op6 . Since Abell clusters typ- ically'' have an overdensity of - 130, the comoving precollapse radius r, is then

ri --( 130)I /~r , -- 15h g1 Mpc , (4)

derived from the spherical-collapse model, which corre-

sponds to a radius of about 0.3h5,,t,, at t,,. The mass of a loop of radius R is

where p is the mass per unit length and P- 10 is a convo- lution factor." (These considerations determine the normalization for Gp, but neither p nor /3 are of importance to this discussion.)

In the usual cosmic-string scenario it is assumed that all clusters of galaxies satisfying the Abell criterion are created about a cluster loop of radius R,. In Sec. I1 we test this assumption by calculating the rms and most likely mass perturbations 6 M for random, as well as correlated, distributions of loops. In Secs. I11 and IV we summa- rize the numerical results which give a non-Gaussian probability distribution for 6M, and we demonstrate the need to use special techniques when considering the likelihood of rare events in this model. Finally, we determine the total probability of an Abell cluster being seeded by relatively small loops and we explore some further implications of this analysis.

11. MASS FLUCTUATIONS FROM LOOPS

In this section we shall calculate the rms and most likely mass fluctuations due to loops smaller than a given cluster loop of radius R,. Our approximate calculations will include loop correlations, described by a simple model which is easy to implement in numerical analyses and which gives a two-point correlation function similar to that found by ~ u r o k . ' ' We shall calculate the fluctuations at te, on the precollapse comoving scale r, [see Eq. (411 and corripare them with the mass perturbation (6M/M), , ( r j induced on this scale by a single cluster loop. Later we shall consider the alternative that cluster loops come inside the horizon after t,,, by incorporating an appropriate suppression factor. rms mass perturbations from strings have been discussed previously in Refs. 19-21. We shall, therefore, only briefly derive the final equations, though in a more transparent manner from a somewhat different perspective-volume averaging, as opposed to averaging over realizations.

The mass perturbation induced by a cluster loop R, in a sphere of radius ro is

Since the primary emphasis is on the relative magnitude of fluctuations, we will not be overly concerned with the exact value of the background energy density.

The energy-density distribution due to a configuration of cosmic-string loops with radii R , centered at x, is

where 6p, depends on R , . In order to satisfy the integral constraints of general relativity 6 p , ( x ) must have vanish-

3612 E. P. S. SHELLARD AND R. H. BRANDENBERGER - 3 8

ing monopole and dipole moment^.'^ At the time of G p I ( x ) = f + ( X I + f - ( x ) , ( 8 ) string formation, the positive energy-density perturbation f + in the string loop is compensated by a negative where f - predominantly spreads to the horizon while

energy-density f - in radiation. So, formal- f + remains confined. The rms mass perturbation in a ly, we have sphere S," of radius ro averaged over space (volume V ) is

The first term is the random contribution; the second loops above a particular radius (it could, in principle, be term i i # j ) is the correlation, since it vanishes if the sites evaluated from the original numerical data). The two x, are randomly distributed. quantities can be related in the following manner:

The random contribution can be simplified significantly if r, << t,,, which is true for our case of interest ( r , = r, ) 6 = n ; ; < J R dR n ( R ) J d R ' n ( R ' ) C ( R , R f ; r ) since the amplitude o f f , is suppressed relative to f i f by l r l R c

at least a factor of the order of ( r o / t e , ) 3 . Consequently, 2

6pi can be replaced by f +, with an error of only d i R , ) 4 , r < d i R c ) , A - 0 . 2 1 5 ) O ( ( r o / t e q 1'). In turn, f + can be approximated by a 6 function corresponding to a point mass &R,: where n ,,- is the number density of loops of radius

f , + ( x - x , ) = & L R ~ ~ ~ ( x - x , ) . (10) greater than R,, and d ( R , is their average separation. The amplitude A determines the mean number of neigh-

we note that for a large and loop boring loops in excess of random for a given region about size, the volume average can be replaced by a loop. The proposed origin1' for the r - 2 falloff is a frag-

1 mentation process by which a parent loop breaks into N o - V I d ' X Z R : ~ ' ~ ~ - ~ ~ ) = J d~ ln ' R ) l 1 ' ) daughter loops spread over a region of approximate

i volume d? R , ). Correspondingly, we have

Hence, the random contribution to (GM2(ro i ) is

477- N O - ~ = ~ . R ~ ~ ~ ~ ' ~ " i ( a r ' d ~ - =477-A , (16) ( ~ ~ ~ ( r ~ ) ) ~ ~ ~ ~ ~ ~ = - - r ~ i ~ p ) ~ J d R ~ ~ n i ~ ) . 3 (12) d

which yields an average clump size of some 2-3 loops, as Introducing the upper cutoff radius Rc, the opposed to the 10 of Ref. 10 or the 8 within a distance of tive mass perturbation becomes about 0.8d i R ) of Ref. 12. This remains sufficiently un-

1 /4 known, however, for us to include No explicitly. The requirement that C i R , R '; r ) satisfy (15) leads us,

random as a first approximation, to adopt a correlation function (13) of the fo rm

2 which is dominated by R,. C i R , R 1 ; r ) = t A --

The correlation term can be analyzed in a similar I d ( f ) I R 6 i R R , r < _ d i R I

manner. It is dominated by the term obtained by replac- (17) ing 6p1 with f ,+, allowing the volume average to be rewritten using Intuitively, loop sizes would be expected to be correlated

1 over some fraction of the Hubble radius at their time of

- 1 d 3 x 2 R i ~ , f i 3 ( x - x , ) s3 (x+r-x , formation - R . The function in (17) can be regarded as a v 1 - 1 limiting case of more sophisticated examples such as

= J ~ R d R ' R R 1 n ( R ) n ( R ' ) C i R , R ' ; r ) , (14) ~ [ d ( ~ ) / r ] ~ e i ~ - R ' ) where O i R - R ' ! = l only if 1 R - R ' < AR or, perhaps,

where C ( R , R '; r ) is the dimensionless correlation function of l o o ~ s of radius R and R '.

What little is known about C ( R , R '; r ) can be gleaned The net result is similar, though less concise. from the numerical simulations of Ref. 10 where the The nonrandom contribution to ( 6 ~ ~ i r , 1 ) then be- two-point correlation function i ( r ) is evaluated for all comes

CLUSTERS OF GALAXIES FROM COSMIC STRINGS 3613

where the radial integral is split at d ( R =r, because beyond d ( R ) a compensating anticorrelation will begin to come into effect. The relative mass perturbation for the correlated piece can then be evaluated as

which exhibits a flat logarithmic dependence on the upper cutoff R, because of the correlation length scale, and varies as the square root of the strength of the correlations.

We can now write both the random (13) and the correlated (19) terms as fractions of (GM/M),, using the precollapse Abell radius (41, ro =r, ( 2 0 . 3h,,teq 1. With the original string parameters of Sec. I we obtain

and

with A =0.2, r, =O.3h5,te,, and the cluster loop radius R, given in Eq. (3a). This yields a total fraction f t o t = ( f : + f )'/2--0.26, a result independent of the string loop density parameter v . Recall from Eq. (2a) that R, scales as v2l3 if the cluster loop is to emerge before t,,, and so the v-dependent factors cancel.

The non-Gaussian nature of the mass fluctuations means that the rms value is not a true reflection of the most likely mass fluctuation (GM/M),, expected for a region of this size. We can, however, estimate (GM/M),, by restricting the integration over R in (14) and (17) to those radii for which one loop of radius R is likely to be contained in a sphere of radius r , , i.e., for which R 2 R m l with d (R,, ) = r i . The results for the two contributions are

where the numerical values substituted are those used above in (20) and (21). This yields f,,, -0. 15, and again indicates the significance of the strength of the correlations.

I t is apparent from these results that both the random and nonrandom contributions to the rms mass perturba-

tions are significantly smaller than the mass perturbation due to a cluster loop. It should be remembered, however, that we are dealing with a non-Gaussian distribution and so these rms values can only be regarded as indicators. Given their magnitude, they are not sufficient to give a definitive answer to rather subtle questions regarding the probability of rare events, such as voids and rich clusters. Indeed, the results are close enough to necessitate a numerical analysis to probe such small probabilities in greater detail and, as we shall see, collections of small loops are almost of equal importance (contrary to the claims of Ref. 4).

In the case where cluster loops come inside the horizon after t,,, the influence of rms fluctuations is stronger because small loops can accrete matter in the intervening period, S M -z,, / z ( f -'R, 1. The contribution to ( SM2 ) then has two components: one enhanced from loops ap- pearing before t , , and the other suppressed from those afterwards. In the light of the above derivations, it is straightforward to show that at the time t , - f - ' ~ , when the cluster loop emerges we have the approximate ratio

where we have replaced R, using Eq. (3b) and assumed the first term in parentheses to be dominant. Note the

3614 E. P. S. SHELLARD AND R. H. BRANDENBERGER

red-shift growth factor and again observe the very weak dependence on that much sought-after parameter v. Present indications" are that the average value loop of < may be considerably smaller than 0.1, bringing the ratio up towards unity. Regardless of the size of correlations, if these trends prove well founded, it is clear that collections of loops must play a significant role in large-scale structure formation.

111. NUMERICAL RESULTS

In order to obtain the spectrum of mass perturbations and to check the previous analytical estimates (20) and (211, we have performed numerical simulations similar to those used to determine the distribution and scale dependence of peculiar ve lo~ i t i e s .~ A box of length L =ateq was filled with nonoverlapping spheres of radius ri. A distribution of loops with a number density spectrum n ( R ) satisfying (1) was laid down, in one set of simulations at random, in a second set with nontrivial correlations. The total mass in loops in each sphere S,, was cal-

culated. After subtraction of the mean, this gave the distribution of mass fluctuations 6M. We assume throughout Secs. 111 and IV that cluster loops emerge before r e g , leaving the alternative for discussion in Sec. V.

There are several additional free parameters in the simulations: the size of the box, given by a, and the lower R,,, and upper R,,, cutoff radii for the loop distribution. R,,, was chosen to be slightly less than cluster size RC1. Rmi, was varied between R ,I and R ,, /30, in order to establish the independence of the rms fluctuation on its value, within statistical limits. The high-mass end of the distribution was also observed to be unaffected by the value of the lower cutoff (for this range).

We chose two alternative models to implement the correlations between loops given by (1 5) - ( 17). The first (I) corresponded to choosing the position of a loop (radius R ) at random. Following this another No - 1 loops of nearly equal radius were distributed around the first, the distribution of radii being uniform in the interval [O,d ( R ) ] , and the phases random. The position of the next loop was once again chosen at random, and the process was iterated. The radii of the loops were continuous- ly decreased to give a distribution satisfying (1).

The second implementation (11) was that adopted in Ref. 12 in which a parent loop of radius Rp fragmented into a number of daughter loops, in their binary case No=& with a correlation radius of approximately 0.8d ( R ) . The former more randomized method has the advantage that the interloop distances are not unnatural- ly rigidly fixed at a minimum value, and so the correlation function is not just cut off at small distances. At large N o , however, it becomes a decreasingly accurate approximation to an r -' correlation function because of its central bias, unlike the fragmentation scheme. In any case neither model should be taken too seriously because both implicitly produce higher point correlations which may not agree with those of an actual string network (as yet unmeasured).

In Fig. 1 we plot the numerical results for the relative rms mass perturbation as a function of the number of

FIG. 1. rms mass perturbations as a function of the strength of the correlations between string loops. The parameter f is the mass perturbation in a sphere of radius r, as a fraction of the cluster loop mass. The closed circles are the numerical results, the open circles are the theoretical values obtained by scaling the contribution from correlations as v2 / ' . No is the number of loops correlated with a given loop.

correlated points No (for the first implementation I). It can be observed that the results are a good fit with the function

which is of the form predicted in (20) and (211. The magnitude of the first term, the random contribution, is in excellent agreement with (20). The second, the correlated piece in (24), is in reasonable agreement, though some 40% below the prediction of (21) (for No = 3 1. Such a discrepancy is to be expected because the anticorrela- tions, built into the numerical model, have only been approximately addressed analytically. This is related to the earlier comment about n-point correlation functions, and until progress is made on their numerical determination these estimates cannot be greatly improved.

In Fig. 2 we plot the numerical results as a function of v (in the runs of Fig. 1 v was always taken to be v=0.01).

FIG. 2. rms mass perturbations as a function of v with No= 10. The closed circles are the numerical values; the open circles give the theoretical values obtained by scaling the random contribution to f and the correlated piece o f f with the powers of v given in the text: f (v )=O. 1 7 ~ ' / ~ + 0 . lv2".

3 8 - CLUSTERS OF GALAXIES FROM COSMIC STRINGS

Note that varying v is equivalent to a volume scaling, an important consideration if r, is to be altered. Again there is good agreement with the theoretical predictions of the dependence in (20) and (21). This excellent correspondence with the analytic results provides some justification for the extension of the numerical analysis to the estima- tion of the probability of rare events such as the occurrence of Abell clusters.

IV. THE DISTRIBUTION OF MASS PERTURBATIONS

As we have already noted, in the cosmic-string model the distribution of mass fluctuations is non-Gaussian. Hence, knowledge of the rms fluctuation (in this marginal case) is insufficient to determine whether clusters will be seeded by fluctuations of small loops, rather than by single cluster loops R,. We, therefore, define a probability P to be the likelihood of such an occurrence, i.e., SM,,, > SM,, in a given sphere of radius ri (4). We estimate P numerically.

In Fig. 3 we plot the distribution of mass perturbations in simulations with v=0.01 and R,,, = R ,, /30, which gives us a measure of P (after subtracting the mean and integrating above 6M,, ). Figure 3(a) shows the results for a run in which the loops are uncorrelated, and Fig. 3(b) shows those in which the loops are correlated ( N o = 10 distributed uniformly in the radius interval [O,d ( R ) ] ) . Loop correlations lead to a much broader distribution of mass perturbations and a significantly larger tail at the high-mass end.

We determined P for a range of values of v and No. In Fig. 4 we plot the dependence on v for fixed No= 10. This effective volume scaling is of interest if ri is to be varied. The error bars give the maximal spread of P in a

FIG. 3. The distribution of mass perturbations obtained from Monte Carlo simulations (before subtracting the mean perturbation). The top distribution (a) is with correlated loops (No= 101, the lower one (b) with uncorrelated loops. Above a certain threshold M the mass fluctuation could seed an Abell cluster.

FIG. 4. The probability P of an Abell cluster forming in a sphere of radius r,, as a function of v (see Sec. IV). The error bars denote the maximal range of values obtained by repeated simulations. In these runs No = 10.

series of simulations with the same parameters. Figure 5 illustrates the effect of changing the amplitude A of the correlations by varying N o , the typical number of loops in a group ( 1 5 No 5 20). Similar results were obtained by altering the correlation radius interval [0, r,,,], over the r a n g e d ( R ) / 2 < r < 2 d ( R ) f o r v a n d N o fixed.

We were able to conclude that for a wide range of real- istic loop correlations P remained smaller than 1.8%. For the original model values (Sec. I ) with our implementation we obtained P,,,d,,=0.11(i0.03)%, PNo=4

=0.34(+0.10)%, and PN n=,, =0.65(+0.10)%. These

estimates, however, are not of significance until compared with the cluster loop density. Naively, one would merely multiply P by the number of spheres of radius ri contained in a cube of side length d ( R , ) to obtain the actual relative probability

with K = I. Since ( 3 /4a)(d , /r, 13= 100 we might conclude that correlation effects are subdominant. However, this may seriously undercount the regions in which 6 M > 6Mc, occurs because the nonoverlapping spheres employed for sampling may have sliced some of the clus-

FIG. 5. The probability P as a function of No for v=0.01.


ters in, for example, half. This counting problem has been pointed out in Ref. 23 in the context of sampling a Poisson distribution. The amplification factor K requires further investigation.24

To evaluate K we have followed Politzer and res skill^^ and we count the number of independent mass perturbations (in a test sphere of radius r, ) in excess of M,, /M ( r , ) when moving the center of the test sphere on a lattice of cell length E . In order not to overcount, we only include a perturbation if it is a "first encounter," that is, a sphere which defines the beginning of a cluster by being the first with 6M > 6M, in all positive lattice directions. Any other sphere with 6M > 6Mc, with a similar neighbor behind it, is rejected to ensure that each cluster is only counted once.

Analytic approaches to estimating the undercounting factor K in terms of the volume derivative a P / a V proved to be too inaccurate, though not inconsistent. (Condi- tional probability considerations led us as a first approximation to K = [ - ( v / P ) ( ~ P / ~ v ) ] ~ . ) The alternative was to implement the algorithm numerically.

We chose a step size of c = r , , implying that each test sphere overlapped with 26 others. It would have been desirable to have 6 << 2ri , the separation of nonoverlapping spheres, so these estimates provide only a lower bound on the undercounting factor K (though we believe it to be fairly accurate). In our case, we found that K

varied somewhat as a function of N o , increasing slightly with the rarity of the event.

The results of this analysis are again summarized in Table I, in the fifth column, and they reveal that mass fluctuations from individual loops are comparable to those induced by correlated collections of loops. The values given in Table I can be interpreted in the following: A random distribution of loops produces some 30-35 % of Abell clusters through rms fluctuations, as opposed to large cluster loops. A correlated distribution seeds substantially more, with No = 2,3 ( A ~ 0 . 2 ) we have about 40-55 %. These results demonstrate the marginality of the formation mechanisms usually assumed for cosmic strings and would alter the normalizations chosen in Sec. I ( R , and Gp).

It only remains to comment on the second fragmentation implementation (11) adopted for the loop correlations. The results for the fragmentation length equal to the parent loop radius (R, = 8R ) are also shown in Table

I. The number of Abell clusters seeded by rms fluctuations in this case is some 70-75 %. Clearly, this result would seriously affect the normalization adopted in Ref. 12 by which the authors identify Abell clusters. The fact that the slope of the correlation function is shown to steepen to r - 3 may well be an artifact of an initial choice of amplitude which is far higher than current string simulations predict.

V. DISCUSSION AND FURTHER IMPLICATIONS

By analytically and numerically calculating the distribution of mass perturbations on the comoving scale r, corresponding to the Abell cluster radius before collapse we estimated the fraction of clusters which are formed from "high density peaks" rather than by accretion about cluster loops. Assuming cluster loops appear before teq and including loop correlations consistent with ~ u r o k , " we found that a fraction f -0.47( kO. 1 ) of all Abell clusters would form by large fluctuations in 6 M produced by the combined effect of loops smaller than cluster size.

Already this is a significant result because it challenges naive expectations about cosmic strings. This fraction f is most probably much larger, however, because of mounting evidence that "cluster loops" would be formed well after matter domination-compare Eqs. (21) and (24). The one-to-one correspondence of loops and cosmological objects appears, therefore, to have largely broken down by cluster scales and, as we shall see, is by no means clear even on galactic scales. Such results necessitate the reappraisal of the cosmic-string scenario, a model which appears to be considerably more complex than previously assumed.

Our results also have important implications for the correlation function of clusters. Unless the high-density peaks have large amplitude correlations themselves, the correlation function of clusters will differ only slightly from the (universal) correlation function of loops. To make this statement more quantitative, let c l ( r ) be the correlation function of loops and g2( r ) be that of high density peaks. The cluster correlation function ( ( r ) then will be

Assume now that g 2 ( r ) = 0 and that g , ( r ) = A(d, , / r l2 , where d,, is the mean separation of cluster loops. The

TABLE I. The fraction P of regions with a sufficient density fluctuation to seed an Abell cluster (compare with P-- 1% from cluster loops-the standard normalization). The column headed Actual, however, corresponds to P when the undercounting of rare events is taken into account. I and I1 refer to the different implementations employed to simulate the observed correlation function of l o o ~ s (refer . - to Sec. IV).

Implementation Correlation length No./group P (%) Actual (%)

random I I I I I I

3 8 - CLUSTERS OF GALAXIES FROM COSMIC STRINGS 3617

mean separation d, of clusters is then d, = ( 1 - f )-1'2d,l (changing our normalizations appropriately) and, consequently,

This would entail only a relatively meager reduction in the amplitude of the correlation function and we might conclude that the primordial loop-loop correlations of the numerical simulations should survive to be seen in the present Universe.

Such a conclusion, however, depends on the assumption that high-density peaks will not be correlated, which is by no means clear since they are formed by correlated collections of loops. Indeed, since the loops of which the peaks are composed have smaller correlation lengths than cluster loops they would be expected to contribute to {(r) at small radii. This may indeed somewhat steepen the slope of the correlation function on small scales, perhaps as was seen in the simulations of Ref. 12, where the amplitude of {(r) was taken such that Abell clusters formed by cluster loops were in a small minority. The effect, however, would not be so severe for the standard parameters, and large-scale correlations of Abell clusters would be expected to be observed today. The pressing need is for more accurate numerical simulations of the evolution of a network of strings to establish the data on a firmer footing.

Pursuing the discussion on clusters, we can also draw speculative connections between our results and cluster morphology (see Ref. 25 for a review). According to wick^,^^ clusters can be divided into regular and irregular types. Regular clusters have a high central concentration and are roughly spherical. Irregular clusters, on the other hand, have a lower central concentration and no special symmetry. Regular clusters are often dominated by a cD galaxy (a central very massive elliptical galaxy) or by a binary system, while irregulars do not contain similarly prominent galaxies. It is tempting to conjecture that regular clusters may originate by accretion onto large loops, whereas irregulars may be high-density peaks in the initial energy distribution. The central galaxy or binary may be a remnant of the seed loop. However, were such formation mechanisms to be real, we might ex- - pect somewhat different correlation functions for different cluster types.

The loosening of the correspondence between loops of a given radius and cosmological objects of a given mass holds also on smaller scales. We consider seed loops R, with a mean separation equal to that between galaxies (d , = 10h ,' Mpc). In the original cosmic-string model wch c o l d ~ d a r k matter all matter has accreted around some loop by a red-shift of about z, = 10. Consider now a comoving sphere with radius ri equal to the radius which is going nonlinear at z , for a galaxy loop perturbation. Then, following our previous analysis, it can be shown that the rms mass perturbation on this scale is a fraction

of a galaxy seed loop perturbation on the same scale (using the parameters of Sec. I and h50 = 1 ). At z = 10 this is very similar to the fraction obtained in Eq. (20). Conse- quently, we expect the probability P of getting f > 1 to be between 0.1% and 0.7%, as for clusters. As in Sec. IV, the probability P' to have an rms perturbation with f > 1 somewhere in a box of sidelength dg will be given by Eq. (26), replacing d,. The ratio of volumes is about 50, and taking the undercounting factor again to be about K = 5 we obtain 0.25 < P' < 1.75. Thus, a considerable fraction of the galaxies in the field must originate from high- density peaks.

There are further implications on smaller scales. This is evident, for example, when we consider the formation of galaxies near a cluster loop. The following simple counting argument reveals why. Suppose we find the mean number of galaxy loops inside a sphere corresponding to the precollapse comoving Abell radius ri -- 15h 50' Mpc. The Schechter luminosity function tells us that galaxies that lie within two magnitudes of the third brightest in the cluster are relatively luminous. Such galaxies provide about 5070 of the integrated total luminosity, and so have an average separation of about d, = 14h ,' Mpc.

Given this mean, and naively assuming that loop size and luminosity are directly related, we can estimate that a marginal Abell cluster should have only

bright galaxies within a radius r A . This would leave an Abell cluster today with a large deficit of these galaxies, the problem being even more severe for richer clusters such as Coma where it could be counted in the hundreds.

There are several options for the solution of this bias- ing problem. If a one-to-one correspondence between loops and galaxies were to be maintained, then it may be that the original cluster loop cleaves off galactic scale loops during its o~cillations.~' However, the criterion that a loop remain intact for a sufficient length of time to accrete a large amount of matter after which it breaks up, requires considerable fine-tuning of the loop dynamics.28 Moreover, nothing particular favors the formation of loops of the required scale and, in any case, most would have a sufficient initial velocity to escape from the cluster. Only more detailed numerical simulations can establish the feasibility of this speculation.

Alternatively, it seems much more reasonable to suppose that processes other than pure gravitational instabil- ity play a significant role in the early stages of the formation of galaxies in the vicinity of a cluster loop. These could include dissipative-gas dynamical effectsz9 (which depend on cooling time scales) leading to fragmentation and condensation, as well as collisions resulting in merg- ing. Such mechanisms would establish two separate pathways (with some degree of overlap) for the formation of galaxies in the cosmic-string scenario. On the one hand, in the low-density field we have the usual picture of accretion about a loop and, on the other, the occurrence of dominant dynamical effects in high-density regions.


This holds out the hope not only of solving some of the conundrums facing cosmic strings, but also of providing tangible reasons for morphology differences and segrega- tion. Such ideas have proved fruitful in explorations by t h e present auth01-s.~'

ACKNOWLEDGMENTS

W e a r e much indebted t o Wayne Boucher for perform- ing many of the computer runs necessary for Sec. IV. W e

have also benefited from conversations with Andy Al- brecht, Dave Bennett, E d Bertschinger, Adrian Melott, Robert Scherrer, Dick Bond, Neil Turok, and Mar t in Rees. Finally, we a re grateful t o Stephen Hawking a n d the Cambridge Relativity G r o u p for providing the stimu- lating atmosphere in which this work was begun. This work was supported in part by funds provided by the U.S. Department of Energy under Contracts Nos. D E - AC02-76ER03069 and DE-AC02-76ER03 130A023-Task A.

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Clusters of galaxies from cosmic strings