Volume 119B, number 1,2,3 PHYSICS LETTERS 16 December 1982

BARYON NUMBER FROM COLLAPSING COSMIC STRINGS

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

Received 14 July 1982

In those grand unified theories that predict the existence of stable strings, closed loops may be formed and collapse, releasing their energy in the form of particles. We point out that this process may be an important contributor to the net baryon number of the universe.

The occurrence of topologically stable strings in some grand unified theories may be a positive feature. As pointed out by Zel'dovich [1] , strings formed at the grand unification phase transition could later provide the density perturbations necessary to "seed" the pro- cess of galaxy formation. Vilenkin [2] has proposed a more realistic scenario relying on stable loops formed by self-intersection of strings rather than on very long strings. This idea has been strengthened by the dis- covery by two of us [3] of a class o f stable spinning loop solutions. We also showed that any initially static loop collapses to a point or, more generally, a doubled loop * 1 (a configuration in which the string winds twice round the same loop) after a time L/4, where L is its initial length (and c = 1). Z 2 strings would then presumably annihilate into particles. Z 3 strings might "add" to form a string in the opposite direction with some release of particles. For other strings yet more complex processes could occur. We also showed that the lowest frequency mode non-static loop collapsed to a line at t <<,L/4. Presumably such loops would sim- ply annihilate into particles.

The purpose of this letter is to show that these col- lapsing loops may also be a positive feature. The su- perheavy bosons released as the String annihilates would decay with some CP and baryon-number viola- tion, as in the standard mechanism for generating baryon number in GUTs. This would occur well after

.1 This point was stated incorrectly in the preprint version of ref. [3].

the grand unification phase transition at T c ~ 1015 GeV. It is obviously an irreversible process, so that the requirement that the system be out of thermal equilibrium is automatically satisfied.

Strings formed at this transition are initially heavily damped by the surround matter [4]. A segment of radius of curvature r experiences an accelerating force I~[r per unit length, where/~ "~ T 2 is the string tension. This is opposed by the damping force opo, where p is the matter density, u the velocity of the string and o the cross-sectional width for string-particle scattering which as Everett [5] has shown is very roughly (neglect. ing a logarithmic factor) of order lIT. Thus the string reaches a terminal velocity Ote r ~ la/opr. Any kinks in a loop of string will tend to straighten out as it col- lapses (since ote r o: l/r) and it will become roughly circular before disappearing, giving rise to very few superheavy bosons.

However this period of strong damping is quite brief. The largest relevant loops at time t are those of radius r ~ t. For these to acquire relativistic speeds, we re- quire t~/apt ~ 1, whish occurs at

t =t e ~ o m 3 P / I ~ 2 ~ 3 X 10 - 2 9 s ,

corresponding to a temperature o f around 1011 GeV. A more careful relativistic analysis defining t e to

be time at which energy loss through damping be- comes small compared to the initial mass energy yields essentially the same result [5].

After this time, damping is negligible and strings move more or less freely. Collapsing loops may be

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Volume 119B, number 1,2,3 PHYSICS LETTERS 16 December 1982

formed in two ways - either when already existing loops enter the horizon, or by the self-intersection of longer strings. It is not at all clear what fraction of these may be formed in initially static or near-static configurations or in other collapsing configurations, but here we shall make what seems to be a not unrea- sonable assumption that the fraction is significant. These loops will then collapse to a doubled loop, a line or in special cases a point. These last will form black holes, while the others (at least in the case of Z 2 strings) annihilate and release their constituent boson quanta, both gauge bosons and Higgs particles. A rough estimate of the number of superheavy bosons released per unit invariant length of string is simply la/m X.

As argued by Vilenkin [2] , a consistent picture re- quires that the increase in string energy due to confor- mal stretching of long strings (i.e. strings extending be- yond the horizon) must be compensated by some mechanism of energy loss. He takes this to be the pro- cess of loop formation and collapse. If so, the rate of creation of loops per unit volume is

dn/dt ~ 1/t 4 .

A typical loop created at time t has a radius of order t [2] and therefore collapses, as explained above, at or before a time t(1 + 3,/4) where ~'t is its length (7 ~ 2rr). If it collapses to a doubled loop it gives rise to a net baryon number of order eTtu /mx , where e is the mean net baryon number produced in the decay of a super- heavy boson.

We can now estimate the total baryon asymmetry produced by all loops collapsing after the damping period. The entropy density is

s = ~ ~r2NT 3 ,'~ 50 T 3 .

During the expansion the ratio nB/s is constant except for the contribution from collapsing loops (or other baryon-number-generating processes). Since the num- ber of bosons decaying is small compared to the num- ber of particles already present, their contribution to the entropy is negligible. Hence

(d/dt) (nB/s) = (I/s) (dnB/dt)loops.

Integrating from t = t e onwards (effectively to ~) we find

(n B/s)fina 1 ~ 300 f e p 2 /m X m3p

300 ( fe/~ 2 ) ( m x /mp) 3 ,

where f i s the fraction of loops produced that collapse in this way. If f is of order 10 -1 andm X ~ 5 X 1014 GeV, we get typically nB/s ~ 10 - 8 e.

If, following Nanopoulos and Weinberg [6], we as- sume that e lies in the range between 10-2 and 1, we obtain a value in good agreement with the present ob- servational bound n B/s ~ 10- 9.8 +- 1.7 [7].

This is of course in addition to any baryon asymme- try created earlier. One interesting feature of the mechanism is that baryon number is not generated uni- formly throughout space but in clumps around the col- lapsing strings. However the scale of these clumps is too small to be of any relevance to galaxy formation.

Great uncertainties remain in this theoretical predic- tion. First, the number of superheavy bosons released per unit length of collapsing string is uncertain. It should be possible to calculate it, but a deeper under- standing of the quantum or at least semiclassical theory of strings is needed. Second, the estimate of the time t e at which the process of baryon-number genera- tion by collapsing strings effectively begins is rather crude. In reality there is no sharp beginning. The pro- cess is a continuous one. Numerical calculations are presently under way to improve this estimate. But since nB/s ~x t e 1/2, we do not expect the result to change very significantly. Lastly, the parameter e is highly model-dependent and cannot at present be cal- culated from first principles.

What we have shown however is that this process of collapsing cosmic strings may be a signficant contri- butor to the total net baryon number of the universe. Certainly in those GUTs that predict the appearance of stable strings it cannot be ignored.

One of us (PB) would like to thank The Royal Com- mission for the Exhibition of 1851, London, England for financial support.

References

[1] Ya. B. Zel'dovich, Mon. Not. Astron. Soc. 192 (1980) 663.

[2] A. Vilenkin, Phys. Rev. Lett. 46 (1981) 17. [3] T.W.B. Kibble and N. Turok, Phys. Lett. l16B (1982) 141. [4] T.W.B. Kibble, J. Phys. A9 (1976) 1387. [5] A.E. Everett, Phys. Rev. D24 (1981) 858. [6] D.V. Nanopoulos and S. Weinberg, Phys. Rev. D20 (1979)

2484. [7] G. Steigman, Ann. Rev. Astron. Astrophys. 14 (1976)

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