175

RECENT DEVELOPMENTS IN THE COSMIC STRING THEORY OF GALAXY FORMATION

Neil Turok

Department of Theoretical Physics Imperial College London SW7 2BZ, England

ABSTRACT

Recent progress in the cosmic string theory of galaxy formation is reviewed, and a remarkable agreement between the correlation function for cosmic string loops and that for ~bell clusters is reported.

Lecture presented at the ESO Symposium on Cosmology and Particle Physics, Munich, 1986. 176 N. Turok

In this talk I will review some of the recent

developments in the cosmic string theory of galaxy 1 formation and report on an apparently remarkable agreement

between the theory and the observed correlation function of 2 rich clusters of galaxies Of course I should begin by

acknowledging our debt to the Ancient Greeks who were the

first to suggest that galaxies lay on strings (see any

picture of the constellations) although theirs had ends and

branches!

Over the past year there has been considerable

progress in three main areas. First, many of the most

interesting grand unified theories and in particular those

based on superstring theories are now known to predict 3 4 cosmic strings ' • Second, as a result of numerical and

analytical work, the evolution of a network of cosmic 5 6 16 strings is now significantly better understood ' ! Lastly,

the numerical simulations have allowed a series of detailed

calculations of the specific predictions of the string

theory to be made, from which it has so far emerged very 7 8 9 successfully ' '

In spontaneously broken gauge theories, strings may

form if the space of degenerate vacua v (the space of 0 1 minima of the higgs potential) is nonsimply connected • In

the simplest case, the group U(l) is broken completely and

is just a circle. Here we get the simplest, directional v0 strings (the higgs field can wind round the circle in

either direction). Strings can also form if a simply

connected group is broken to a subgroup with one or more

disconnected components. For example, if the subgroup has

one disconnected piece one gets 11 Z 11 strings, where two 2 Recent Developments in the Cosmic String Theory of Galaxy Formation 177

identical strings can annihilate. This case occurs in many 3 candidate grand unified theories •

In the recently discovered superstring theories

examples of both types occur. In the E 8x E 8 theory , most of the proposed symmetry breaking patterns involve breaking

extra U(l)'s, giving directional strings. The S0(32)

theory yields strings for slightly more subtle reasons.

The universal covering group of S0(32) has centre z 2x z 2 which remains unbroken since all fields are in the adjoint representation, but is generally not part of a continuous symmetry in the unbroken subgroup. Hence we again get 4 nondirectional strings •

In these theories, strings would form at a symmetry breaking phase transition in the very early universe. The 1 basic idea, due to Kibble , is that as the universe cools below some temperature T ~ m, the symmetry breaking scale, c the Higgs field ~ falls into the minimum of the potential 1 Vo. ~ is only correlated over some scale ~ ~T c < t and the universe is broken up into domains of size ~ ~. with ~ choosing a random position on v 0 in each. To minimise gradient terms in the energy ~ smooths itself out by choosing the shortest path along v 0 in going from one domain to the next. Edges where ~ winds right around Vo then become string segments. In these theories strings have no ends, so the segments are all part of closed loops or infinite lengths. One then has a tangled network of loops and lengths with a fixed number of segments of length

~ per volume ~3. Simulations of this process were first performed by Vachaspati and Vilenkin, who found that about 178 N. Throk

80% of the string is in_ the form of infinite lengths and the remainder in closed loops with a scale invariant size

distribution i.e. a fixed number of loops of size ~r in a 3 volume r • Both infinite lengths and closed loops are in 11 the form of Brownian trajectories

A crucial parameter in the cosmic string theory is the 2 ~ where µ is the mass per dimensionless number Gµ n(m/mp1 ) 1 6 unit length of the string. As we will see, m ~ 10 GeV is

required for strings to form galaxies and clusters of

galaxies.

It has been argued that such heavy gauge strings are

difficult to form after a period of inflation. Strings

formed before inflation are obviously diluted to

inconsequential levels if the inflation has 'solved' the

flatness and horizon problems - there would be less than

one string per present horizon volume. Gravitational

radiation constraints force the reheating temperature after 17 inflation to be less than ~10 GeV. The reheating

temperature usually calculated is much less than this, but

is actually not the relevant one for forming strings. If 4 the energy density in the inflation field is M with ' 1 7 M<10 GeV, and the decay rate into radiation r then it is easy to see that in an expansion time or so after inflation

2 ends the universe reheats to a temperature TRH (rm M )~. p 1 2 The mass per unit length of the string is given by ~nm , 2 2 2 where the potential V(~) = A(~ - m ) whilst the

temperature of the string-forming phase transition Tc~ A~,which can be much less than m. Requiring TRH > Tc 16 _2 11 and m ~ 10 GeV we find I'A >10 GeV which is easily 5 satisfied in several models for A~10 Recent Developments in the Cosmic String Theory of Galaxy Formation 179

This requires a small amount of "fine tuning" but to no greatewr an extent than is necessary in these models l 0 anyway

Between the time when they are formed, at a l I 2 temperature T ~ (Gµ) mpl and the time when T - Gµmpl' the

strings are heavily damped by collisions with relativistic l particles • This results in the string straightening out -

the scale ~ on which the string is straightened out grows 5/4 1. as t and ~ ~ (G µ) T't at the time when damping ceases to be important.

The subsequent evolution of this network is a complicated problem, and the early literature on the subject contained several errors. It is obviously crucial to know whether the string density comes to dominate over 2 the radiation, whose density Prad- 1/Gt • The earliest discussions claimed that the network would evolve by straightening out so that ~ ~ t, and the density in strings _2 p ~ µt , whatever the string-string interactions s l l 2 were ' This behaviour is desirable in the sense that the fractional density perturbation provided by the strings ps/ptot- Gµ is constant, so the strings neither come to dominate nor disappear as the universe expands. The perturbation they provide is also of the right order of 16 magnitude (for m ~ 10 GeV, a typical GUT scale) to seed the formation of galaxies and clusters of galaxies later on. However simulations by Albrecht and myself have shown that the evolution of the string network does depend crucially on how the strings interact. In particular if the strings simply pass through each other without 180 N. Turok

interacting, the energy in infinite strings remains 3/2 constant, p a: t , and strings rapidly come to s 5 dominate • Luckily for the cosmic string theory it seems more

likely that when two strings cross they almost always

reconnect the other way. The first indications of this

were in numerical simulations by Shellard of the crossing

of "global" strings. He found that for all velocities of

intersection less than ~ 0.9 the strings always reconnect

the other way. Recently Copeland and myself have given an

analytical treatment of the process which also applies to

gauge strings and indicates similar behaviour. Crossing

and reconnection of strings makes it possible for the

infinite lengths of string to slowly chop themselves up

into closed loops. Albrecht and myself also simulated the

case where strings always reconnect the other way and found

strong evidence that with ~ ~ t initially, as one would 1 expect for "grand unif iert" strings , and for directional

strings the energy density in the string network does

indeed fairly rapidly evolve towards the "scaling" 2 behaviour p ~ µt

All our results are consistent with the following

simple picture. The network of strings longer than the

horizon straightens out on scales of the order of the

horizon scale, with a fixed number of lengths crossing each

horizon. The self-intersection of these long strings

continually produces loops with radii r of order the

horizon scale, and more or less str~ightened out on that

scale i.e. with length R. ~ 2nr. These "parent" loops are Recent Developments in the Cosmic String Theory of Galaxy Formation 181

very likely to intersect themselves, breaking up into several smaller loops, but this process rapidly terminates,

producing a clump of non-self-intersecting loops. These

occupy a small and rapidly decreasing fraction of space and

the probability of them reconnecting to the network of long

strings or colliding with another loop rapidly becomes

negligible. Whilst they are produced with considerable

peculiar velocities v ~ 0.2 or so this rapidly redshifts

away and the total distance loops of a given size move

relative to a comoving frame is a small fraction of their

mean separation. The loops just sit at almost fixed

comoving location, oscillating in a periodic way and slowly

radiating gravitational radiation at a calculable 14 1 5 rate ' , until they eventually disappear. Their mass

being almost constant, their energy density scales like

matter and the smallest loops (those just about to

disappear) dominate the total energy density in strings.

The positions of the loops are correlated, due to the way in

which they are produced - "parent" loops producing a clump

of "child" loops.

This "scaling" solution may be understood in a fairly

simple way. Let us assume that there is always a scale ~

below which strings are straight and above which they are

more or less random. For ~ >> t the strings are

conformally stretched by the expansion of the universe and - 2 1 ps « R « t in a radiation dominated universe, where R

is the scale factor. For ~ ~ t long strings self-intersect 3 2 rarely so few loops are chopped off and p ~ t- / as discussed above. This means that the number of string 182 !V. 1Urok

segments F, per horizon grows. When this number becomes

much larger than one, strings frequently collide and

reconnect, reducing F, to a value less than t.

On the other hand, for ~ << t, long strings frequently

self-intersect and chop off loops. With typical velocities

- 1, and neglecting reconnection of loops, the time scale

for a long string to chop off some fraction of it's length

is - F, so as we reduce F, the rate of loss of string from

the network increases indefinitely. This must result in ~

growing, so for ~ - t a balance is struck and the string 2 density ps - µt

This picture contains two assumptions. First, strings

are a straight on a scale ~ but not above ~. This is

reasonable and in fact because of the way the strings were

formed, F, cannot grow faster than t (recalling the way the

strings were formed, the higgs field cannot align itself on

scales larger than t). Secondly, we have ignored

reconnection of loops to long strings. This is justified

provided that the probability of a loop reconnecting is

much smaller than that of one being produced. This is

particularly plausible in an expanding universe when r, - t

- loops produced of size < t remain constant in size whilst

the network of long strings expands with the universe - the

probability for a loop to reconnect rapidly becomes

negligible. However when ~ << t or equivalently for a

string network in a non-expanding universe, the situation

is not so clear.

As he has discussed at this meeting, Kibble has

developed an analytic approach to this problem on the Recent Developments in the Cosmic String Theory of Galaxy Formation 183

assumption that for a string network in a non-expanding

universe an equilibrium distribution of lengths and loops

would be reached resembling the initial conditions of the

network described above. With this assumption it is

possible that for initial conditions ~<

universe that reconnecting of loops compensates at least to

some extent for chopping off of loops, and the density in

strings does indeed come to dominate. Kibble has explored

the consequences of this scenario for lighter strings, with 13 6 16 Gµ ~ 10 or so ' Whether such an equilibrium exists

is not clear at this stage - it may be that the processes

of a loop being chopped off from or joining onto a length

are not at all symmetrical, and no equilibrium exists at

all. Albrecht and myself simulated this case too and it is

clear from our simulations that if such an equilibrium

exists, it is very different from the initial conditions, 5 with a far higher fraction of the string in loops • In my opinion we will have to do more numerical work to build a clearer picture of the details of the processes of chopping off, reconnection of and breaking up of loops before we will be able to construct analytical models with any level of precision.

Let me turn now to the most interesting part of the cosmic string story, the idea that loops chopped off the network as the universe expanded later seeded the formation of galaxies and clusters of galaxies. For simplicity I will only deal with the case of an Q = 1 cold dark matter 7 dominated universe • For several reasons, however, strings in a neutrino dominated Q = 1 universe may be a more attractive scenario. 184 N. Turok

Perturbations start to grow around loops at the time t

of equal matter and radiation density. At this time the

distribution of loops is as follows. The mean separation

of loops of size r is ~ r when they are produced but has 1 I 2 now grown to r(t/r) Loops of size r have number 3 I 2 density ~ (rt) and contribute a density 112 312 µr- t- so the smallest loops dominate. Larger loops

are more massive but rarer.

Density perturbations grow around loops in a fairly

simple way, well approximated by a spheri~al model on

scales much larger than the loop radius and smaller than

the mean separation of loops. Larger loops produce larger

potential wells, accreting larger masses containing smaller

loops as well as matter about them. Thus a hierarchy of

objects, smaller ones in bigger ones, is produced which we 2 shall identify with galaxies and clusters of galaxies •

Now imagine drawing a sphere of radius R

it through space. Neglecting correlations between loops,

r.m.s. fluctuations in the mass inside R due to loops of

radius r are given by

1/ 2 3 _3/2 1/4 5/4 oM/M (N) ( µr) / ( 4 n pR / 3) GµR r t (1)

with p the background density and N the mean number of

loops in the sphere, for N >>1. These fluctuations are

dominated by the largest loops for which N ~ 1 i.e. 2 r ~ R /t. Now imagine looking for regions containing much Recent Developments in the Cosmic String Theory of Galaxy Formation 185

greater mass excess than this typical fluctuation. These

are places where the sphere contains a much larger loop of 2 radius r with R > r >> R /t. In fact this selection procedure will 'find' all loops of a mean separation equal 2 to the mean separation of the regions so selected • This selection procedure is in fact precisely the one

employed observationally by George Abell to define rich 1 7 clusters of galaxies over 25 years ago He defined them as regions containing more than 50 bright galaxies within a 1 radius 1.5h Mpc. For comparison, the mean separation of l 8 l bright galaxies is 5h Mpc so Abell clusters are

exceptionally dense clusters of galaxies. Their mean l 9 l separation de is known to be 55h Mpc. Furthermore, in

theories where small scale structure forms first, such as the cosmic string theory, 6M/M decreases with M and on as

large scales as d , there has been hardly any c gravitationally induced motion of matter since

perturbations started to grow. The positions of Abell

clusters should thus accurately trace the positions of the

loops that produced them.

Now Abell himself noticed that the two dimensional

distribution of Abell clusters on the sky is in fact highly 17 non random , but it was not until quite recently that enough redshift data became available for a precise determination to be made of a good statistic for measuring

this, the two point correlation function ~(r) (the excess 186 N. Throk

probability over random of finding two objects at a separation r). This was done in 1983 by Bahcall and 19 20 Soneira, and Klypin and Kopylov, who found that the two

point correlation function for clusters was similar in

form, but 18 times larger than that for galaxies. It was

also observed to be positive out to distances of more than 1 1 9 lOOh Mpc

This suggested a very clean test of the cosmic string 2 theory • The correlations of loops may be measured in

simulations such as those described above. The two point

correlation function ~(r) of loops of mean separation d

must in fact be dimensionless function of r/d. This is

because all loops are produced in the same way, and at the

time loops of a given size are formed, there is only one

relevant scale, the horizon scale. At formation therefore,

and measuring distances in terms of the horizon scale, all

loops show identical correlations. Subsequent expansion of

the universe simply stretches all separations by the same

amount, so ~(r/d) does not change. Furthermore, there are

no free parameters in ~(r/d) at all, the dynamics of string being independent of µ, the mass per unit length, and

independent of the cosmological parameters h or Q. ~(r/d)

thus provides a very good test of the cosmic string

theory.

Figure 1 shows ~(r/d) calculated from numerical

simulations compared to ~(r/d) as calculated for Abell clusters by Bahcall and Soneira. As you can see, the

agreement is remarkably good. Some simple models that

explain the form of ~(r/d) in the cosmic string theory were presented in ref. 2. Recent Developments in the Cosmic String Theory of Galaxy Formation 187

In the simulations 'U(l)' type strings were used and

assumed to always reconnect the other way when they

crossed. Loops with the separation of Abell clusters today

were produced just at the end of the radiation dominated

era, so a flat Robertson-Walker background with scale

factor a(t) a t~ was used. The effect of peculiar velocities of loops was shown to be minimal (producing the

slight flattening in ~ at small r/d) and negligible in 7 8 terms of accretion of matter ' Having identified loops with the separation of Abell clusters with Abell clusters,

one can then demand that they be massive enough to have

accreted objects with the mass of an Abell cluster by

today. This determination µ, the mass per unit length of

the string, in a much cleaner and more direct way than 7 previous calculations •

!(rid)

Figure 1.

rid 188 N. Turok

Dynamical mass measurements of the masses of Abell

clusters suggest the overdensity inside an Abell radius is oM ~170, slightly larger than that in the spherical model M

after virialisation (~150). This suggests that Abell 7 clusters formed very recently, and allows one to calculate '

with Q=l and h=0.5

6 G µ = 2xl0 (2)

from requiring that loops with the separation of Abell

clusters virialised an Abell overdensity inside an Abell

radius by the present day.

A completely independent calculation of Gµ can be

performed by requiring that loops with the mean separation

of galaxies gave rise to objects as massive as galaxies and

with the observed galaxy-galaxy correlation function. In d , fact, since dg ~ c/ , the 'bare' correlation function of 10 'galaxy' loops is roughly one hundredth that for clusters,

so a gravitational enhancement of the 'bare' galaxy loop

correlation function by a factor of about 5 is needed to

fit observations.

In ref. 7 a simple model for the gravitational

enhancement of correlations is used to calculate Gµ. One 6 finds G µ ~ 4xl0 which is quite close to (2), the

difference being well within the accuracy of the

calculation. Similarly, using the overdensity in a typical 6 galaxy derived from rotation curves one finds Gµ ~ 4x10- Recent Developments in the Cosmic String Theory of Galaxy Formation 189

as well. These values are far more imprecise than (2)

however, so the most one can legitimately claim is that the

cosmic string theory seems at present consistent with 6 galaxy and cluster masses and correlations for Gµ ~ 2xl0

Q = 1, h = 0.5. These are however considerable successes,

which no other theory has so far achieved. More precise

calculations will clearly require large N-body

simulations.

Finally, what are the prospects of detecting cosmic

strings? Thanks to the numerical simulations, fairly

precise statements can now be made. The most direct effect

of strings is their gravitational lensing effect - a

straight length of string produces double images at a 2 1 typical angular separation

5 A0 = 41tGµ 2xl0 "" 5"

using the value for Gµ in (2). Larger or smaller values are obtained as the geometry is varied.

All five of the observed cases of gravitational lensing are within 3" of this value. For only one of them has a candidate object for producing the lensing so far been observed. However one would expect that only ~ Gµ of the area of the sky has been lensed by string so it is 1+ unlikely that so many of the 10 or so observed quesars would be lensed by strings.

A second effect is the anisotropy induced in the microwave background. A single string moving at velocity v perpendicular to the line of sight produces a 1 190 N. Turok

22 discontinuity

&T 8nGµv 1 T

5 which is always less than 4x10 and has rms value 5 ~ 2x10 , below present observational limits. These

discontinuities are rare : with a beam separation of 5' one 8 sees a discontinuity on average 1/20 of the time • The Sachs-Wolfe effect due to accreting loops has also

been calculated and produces rms fluctuations of similar 8 magnitude •

The best limit on the cosmic string theory will soon

come from the effects of the gravitational radiation from

loops. Despite recent claims, this is not in contradiction 9 with the standard nucleosynthesis scenario . However it would lead to variations in the observed frequency of the

millisecond pulsar. The limit on Gµ obtained if the

predicted variations are not observed gets rapidly better 23 as the period T (in years) of observation increases

5 8 G µ < 10 ( a/T)

with a ~ 2~. This limit may become very stringent in a

decade or so.

In conclusion the cosmic string theory has proved to

be an eminently testable theory of galaxy formation. It

has so far yielded several important predictions in good

agreement with observation, and we can look forward to

several new tests in the near future. Recent Developments in the Cosmic String Theory of Galaxy Formation 191

ACKNOWLEDGEMENT

I would like to thank A. Albrecht, R. Brandenberger,

E. Copeland, T.W.B. Kibble and J. Traschen for helpful discussions.

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