Volume 116B, number 2.3 PHYSICS LETTERS 7 October 1982

SELF-INTERSECTION OF COSMIC STRINGS

T.W.B. KIBBLE and Neff TUROK Blackett Laboratory, Imperial College, London SW7 2BZ, England

Received 16 June 1982

We discuss some aspects of the theory of Zel'dovich and Vilenkin attributing the density perturbations needed to initiate the formation of galaxies to cosmic strings, in particular the question of whether closed loops can have the long lifetimes re- quited in Vflenkin's scenario. We show that any initially static string will collapse after half an oscillation period. On the other hand we also show that there exist solutions, differing infinitesimally from these, in which the strings never self-inter- sect. This result makes it plausible that loops can have the requited long lifetimes.

There is as yet no fully satisfactory theory of galaxy formation [ l ] . Zel'dovich [2] first suggested that strings formed at a phase transition very early in the history of the universe might provide the density per- turbations needed to start the condensation process. Such strings are expected in certain grand unified theo- ries [ 3 - 5 ] . If there is a grand unification phase transi- tion at a temperature Tc, then the tension or mass per unit length o f the strings would be ~ ~ T2c ~ M 2 / a G ,

where M x is the corresponding superheavy vector boson mass and a G the grand unified coupling con- stant. If one assumes that the strings take on a random (brownian) configuration with a persistence length of the order of the horizon distance, l ~ t, then one finds for the ratio of the contribution of strings to the total mass density

Ps]P ~ 30 G~ ~ 3 0 ( r c / M p ) 2 ,

where Mp is the Planck mass. To achieve the required density contrast, Ps]O ~ 10-3 , Zel'dovich was forced to assume a very large value for Tc, of order 1017 GeV.

More recently, Vilenkin [6,7] has presented an al- ternative version of this scenario which allows a more modest value of T c. This difference stems from differ- ing assumptions about the behaviour of closed loops. Zel'dovich assumes that the closed loops which appear in addition to the long strings with persistence length t make a negligible contribution. Vilenkin suggests that loops have very long lifetimes and so contribute

much more than do the longer strings to the total mass density. He finds p loops/,O proportional to Tc[M p rather than (Tc[Mp) 2. Hence a value T c ~ 1015 (3eV is adequate.

In deciding whether this alternative theory is viable it is very important to study the dynamics of closed loops, to determine whether they can in fact have the long lifetimes assumed by Vilenkin. This is the problem we address in this letter.

In the very earliest stages o f the evolutionary history of the universe the motion of the strings is heavily damped by friction of the surrounding medium [3], but this situation does not persist for long. Everett [8] showed that the damping is rather longer than original- ly supposed, because the effective cross-sectional width of a string for low-energy particles is larger than its ap- parent width. However, this only slightly postpones the disappearance of damping. Long before the epoch of relevance to galaxy formation the damping has be- come utterly negligible. The strings can then only lose energy by radiation o f some kind.

It seems clear that closed loops play an important part in the energy-loss process. Once formed, a closed loop is doomed to extinction unless perhaps it reinter- sects a longer string and becomes reconnected. It will oscillate, gradually losing energy, until it disappears. It is implicit in the Zel'dovich scenario that there is some fairly efficient energy-loss mechanism, so that loops formed at time t with a typical length scale also of

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Volume 116B, number 2,3 PItYSICS LETTERS 7 October 1982

order t do not survive for more than a few oscillation periods. By contrast, Vilenkin assumes that the only significant energy loss mechanism is gravitational radia- tion. A string with length scale l and mass M ~ p/will oscillate with frequency 60 ~ 1/l and hence lose ener- gy by gravitational radiation at the rate

ff¢ ~ _ G M 2 oo 2 ~ ~r la 2 .

(Here and below we ignore factors of 27r.) Thus its lifetime is

"r ~ I/Gla ~ 108l.

Because of this very long lifetime, there will at any in- stant be large numbers o f loops surviving from much earlier epochs.

One possible criticism of this estimate is that other types of radiation may in fact be more important. How- ever, strings do not carry any net charges, so that any such process must be rather indirect. It must also be remembered that because o f their large mass, the cou- pling of strings to gravity is really quite strong. This point certainly deserves further study, but here we shall assume that gravitational radiation is indeed do- minant.

There is however another problem. A closed loop may from time to time intersect itself and break into two smaller loops, whose lifetime would be shorter. At first sight it seems plausible to suppose that for any random initial configuration there is some probability p per oscillation cycle of self-intersection, and that p should be independent o f the actual size o f the loop. If that were true, one should expect that a loop of length I would break into two loops of length l/2 in a typical time 1/13. These in turn would break into yet smaller loops o f length 1/4 in a further time l /2p, and so on. Obviously this yields a geometric series, giving a total lifetime of order 2liP. Unless p were astonish- ingly small, this would reduce the number of loops to a level much less than required by Vilenkin.

This is the argument we propose to test. It is by no means conclusive, for it is not clear that it is reasonable to approximate the behaviour of a large number of loops by that o f an "average" loop. It may happen that there are some initial configurations that lead to rapid self-intersection, but others that never intersect at all. Indeed, this is precisely what we shall show does happen. The remaining unanswered question is what proportion of strings fall into this category.

Let us first recall the dynamical equations of strings (which may be derived from an action integral pro- portional to the invariant area of the world sheet swept out by the string). As shown by Goddard et al. [9], it is possible to choose the parameterization of the string so that the equation of motion takes a particularly sim- ple form. If l is the time and s the length parameter along the string proportional to the total mass o f the string from a fixed point, the equation for the position r(s, t ) reduces to

I : ' - r" = 0 , (1)

where ~ = dr/dt and r ' = ~r/Os. These quantities also satisfy the constraint equations

~: ' r ' = 0 , ~:2 +r ,2 .__ 1 .

Of course the general solution of (1)is

I r = ~a(s - t ) + ~b(s + t ) ,

where a and b are subject to the constraints

a '2 = b '2 = 1 .

For a closed loop of invariant length L both r and (in the centre of mass frame)~: must be periodic with period L and therefore we must have

a ( s + L ) = a ( s ) , b ( s + L ) = b ( s ) .

In fact the period of the motion is really L/2 rather than L, since it is easily seen that

r(s + L /2 , t + L / 2 ) = r ( s , t ) .

It remains only to examine these solutions for self- intersection. One may suppose that loops are often formed in an initially almost static configuration. First, therefore, let us consider the case of an initially static string, s:(s, 0) = 0. Then a'(s) = b'(s), and by suitable choice o f an arbitrary constant vector we may take a(s) = b(s) . A half-period later we have

r(s, L / 4 ) : ' ½a(s + L / a ) . ~a(s - L / 4 ) +

However,

r(s + L /2 , L / 4 ) = ~a(s + L / 4 ) + ½a(s + 3L/4)

By the periodicity of a these are equal. Thus we have the remarkable result that any initially static string not merely self-intersects but actually collapses to a doubled loop after a half period. Presumably a Z 2 string would then annihilate into particles. More com-

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Volume 116B, number 2,3 PHYSICS LETrERS 7 October 1982

plex process might occur for other strings. At first sight this result might suggest that it is diffi-

cult to find non-self-intersecting strings, but in fact this is not the case. Indeed there exist initial configura- tions that differ infinitesimally from a static one but which do not lead to self-intersection. Here we merely exhibit some simple examples.

For simplicity let us take the length L of the string to be 2n. The simplest type of solution for a (or b) is given by

a'(s) = e 1 cos s + e 2 sin s , (2)

where e 1 and e 2 are orthogonal unit vectors. If both a ' and b ' have this form, we can choose the axis e I in the direction of the intersection o f the circles traced out by a ' and b ' and by suitable choice of the zero of s write a ' as in (2), and b ' as

b'(s) = e 1 cos s + (e 2 cos ¢ + e 3 sin ¢) sin s .

The corresponding solution is no longer initially static, but it is easy to check that

r o t - s, 7 r / 2 ) = r(s, 1 r / 2 ) .

Hence the string collapses to a line after a half period. It is easy to check that adding to (2) terms in cos 2s

and sin 2s does not yield any extra solutions. The next simplest solutions involve cos 3s and sin 3s. (The only other possible single frequency addit ion is one contain- ing cos 5s and sin 5s terms.) By suitable choice o f the zero point they may be written

a'(s) = e 1 [(1 - a) cos s + tv cos 3s]

+ e 2 [(1 - a ) sin s + a sin 3s]

+e32 [a (1 _ a)] 1/2 sin s ,

where a is an arbitrary parameter between 0 and 1. Taking a ' of this form and b ' as in (2) we find a pertur- bation of the initially-static string solution, namely

I r(s, t ) = ~ e 1 [(1 - a) sin s_ + § a sin 3s_ + sin s+]

with s_. = s :t t. It is then straightforward to check that this string does not intersect itself for 0 < c~ < 1, i.e. there is no non-trivial solution o f the equation

r(s, t) = r(s', t). What is particularly noteworthy about this solution

is that only a small perturbat ion away from one of the collapsing solutions is sufficient to make it non- self-intersecting. This suggests that there is in fact a large class of stably-oscillating solutions, so that the loops required by Vilenkin's scenario may indeed have long lifetimes.

There are o f course several questions still unanswer- ed. It has to be shown that other modes o f radiation of strings do not lead to rapid decay of loops. Also it is not clear how special are the solutions we have found. It will be necessary to examine other pertur- bations o f initially-static strings to determine whether the avoidance of self-intersection is a general feature or a fortuitous result of our special choice. Ideally one would wish to estimate what proport ion of loops form- ed by self-intersections of long strings would be in a self-avoiding configuration. This is at present hard to so, but we have at least shown that the class is non- empty.

References

[ 1 ] For a review see: P.I.A. Peebles, The large scale structure of space-time (Princeton UP, Princeton, NJ, 1980).

[2] Ya.B. Zel'dovich, Mon. Not. R. Astron. Soe. 192 (1980) 663.

[3] T.W.B. Kibble, J. Phys. A9 (1976) 1387. [4] T.W.B. Kibble, G. Lazarides and Q. Shaft, Phys. Lett.

113B (1982) 237. [5] T.W.B. Kibble, G. Lazarides and Q. Shaft, Phys. Rev. D.,

to be published (1982). [6] A. Vilenkin, Phys. Rev. Lett. 46 (1981) 17. [7] A. Vilcnkin, Phys. Rev. D24 (1981) 2082. [8] A.E. Everett, Phys. Rev. D24 (1981) 858. [9] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorne,

Nucl. Phys. B56 (1973) 109.

1 - -~e 2[(1 --ot) coss + ] o t c o s 3 s _ + c o s s + ]

- e 3 [ot(l -- a)] 1/2 cos s_ ,

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