Small-scale structure on cosmic strings

PHYSICAL REVIEW D VOLUME 48. NUMBER 8 15 OCTOBER 1993

Small-scale structure on cosmic strings

Daren Austin* School of Mathematical and Physical Sciences, University of Sussex, Brighton, BN1 SQH, Sussex, United Kingdom

(Received 17 March 1993)

We extend a linear-kink model of small-scale structure on cosmic strings and show how it is equivalent to an earlier two-scale model. By reconsidering the connection between the rate of loop production and the linear-kink density we correctly predict the build-up of small-scale structure currently seen in those simulations which neglect the gravitational back reaction. After including the gravitational back reaction we determine the full evolution of the small-scale structure, coupled to loop production and gravitational radiation. We confirm that the inclusion of the gravitational back reaction leads inevitably to the scaling of this structure, and that as many as lo4 kinks may persist on an horizon-sized segment of string.

PACS number(s): 98.80.Cq

In this paper we address the problem of how small-scale structure in the form of kinks is generated on a network of cosmic strings in an expanding Universe [1,2]. Re- cent numerical simulations of cosmic strings have shown a steady build-up of small-scale structure with no sign of it scaling [3,4]. This small-scale structure may play a crucial role in the generation of density and velocity perturbations during the epoch of galaxy formation [5]. In our earlier work we showed by means of a two-scale model, how cosmic string networks can enter into a scaling regime [6,7]. In that model we used a random kink distribution to model the small-scale structure.

More recently models involving a linear kink distribution have been used to analyze this structure 18-101. In this paper we reconsider the role of loop production in these models and compare our results to those of the two-scale model and numerical simulations. Loops are formed from string segments which tend to have a pref- erentially large number of kinks, usually generated as a result of a long-string intercommuting event. By con- sidering intercommuting: we overcome the earlier problems of formulating an accurate description of loop production, yet we are still able to accommodate all loop- producing events. For completeness we retain the earlier loop-production schemes. The results then obtained are in excellent agreement with the earlier two-scale model and correctly predict the build-up of structure on the long strings.

Previous linear-kink models have all suffered from one shortcoming, namely, that the effect of small-scale structure coupling to the long string evolution, through the gravitational back reaction, has been neglected. In this paper we aim to overcome this shortcoming and consider what effects the full coupled evolution has on the scaling of both long strings and the small-scale structure. We shall find that while the long strings can enter scaling early on, the small-scale structure can enter one of two scaling regimes.

For simplicity we shall only consider a spatially flat

'Electronic address: mpfflQuk.ac.sussex.centra1

0556-2821/93/48(8)/3422(5)/$06.00 48

Friedmann-Robertson-Walker universe with the metric

where R is the scale factor for an expanding background. For cosmic strings, the most interesting period of their

evolution is close to the transition from radiation to matter domination, when much of the large-scale structure is believed to have been formed in the early Universe. The corresponding scale factor during these two eras is given by

with m = 1 and m = 2 during the radiation and matter- dominated eras, respectively. The horizon distance is then simply d H = (m + 1) t .

Our aim is to deduce a n evolution equation for N , the total number of kinks in any comoving volume. Kinks are created on strings by the process of intercommuting (intersection of long strings); however, they may be subse- quently removed if a string should self-intersect creating a loop. Kinks decay by stretching, which opens up the kink angles, and gravitational radiation, which rounds off the kinks. Hence there will be a rate equation for N of the form

where the terms represent contributions from stretching, intercommuting, loop production and gravitational radiation. From this we will be able to calculate a corresponding equation for the linear kink density D defined by

where L, is the total length of long strings. Of the processes listed above, intercommuting is the

only one which has no effect on the total length of string. For cosmic strings energy is simply proportional to length implying that E,, the total energy in long strings, evolves according to

R3422 @ 1993 The American Physical Society

SMALL-SCALE STRUCTURE ON COSMIC STRINGS R3423

where we again use comoving volumes throughout. For kinks on an infinite string, Hindmarsh has shown

that the rate of energy loss due to gravitational radiation is simply proportional to the kink density [ll]. Hence if D is the linear kink density, then

E,, = -rGpDE,. (6)

r is a constant related to the overall form of the string and for long strings is about 50 [12,13].

As the Universe expands, cosmic strings are confor- mally stretched [14] such that

where LY = 1 - 2(x2). In the two-scale model a is found to be closely related to correlations between left and right movers, but we shall assume that it is roughly a constant during the two eras. Numerical simulations of cosmic string networks by Bennett and Bouchet [3] suggest values in the region a,,d e 0.14 & 0.04 and ~ ~ , , t t 0.26 & 0.04.

To simplify matters, rather than calculate the coupled evolution of E, and D (which would require us to use a specific model of loop formation), we shall assume that the long string energy density p, scales a t all times. This assumption has recently been confirmed by the work of [15] who find that the long strings appear to enter an "intermediate" scaling regime before small-scale structure forms. For strings to scale would imply that their density should be

during both the radiation and the matter-dominated eras. The factor ym means that in any horizon-sized volume, there will typically be y m ( m + 1)2 strings of length d H . From this assumption we can calculate the loop term in Eq. (5) necessary to maintain scaling, and hence the rate kinks are removed from the long strings.

The approach outlined above would appear to provide an accurate calculation scheme for deriving the evolution of D, but there is a problem; Eq. (5) includes a term proportional to D, whose evolution we are trying to es- tablish. In assuming that E , scales we implicitly assume that D must also reach some scaling density, since it is unlikely that the long-string network can scale while the small-scale structure continues to grow. Indeed, were the number of kinks on an horizon-sized segment ( D = DdH) to become large, say of order (J?Gp)-l, the network's principal energy loss mechanism would be gravitational radiation not loop production. This has been suggested numerically in Ref. [13]. During the matter-dominated era this coupling is crucial because loop production ap- pears to be unimportant during the late stages of evolution, implying that gravitational radiation and stretching provide the only kink-loss mechanisms.

Rather than solve the true coupled evolution of E,

and D (thereby assuming a specific model of loop production), we can instead construct a consistency argument in which we can analyze to what extent scaling kink densities are consistent with the long-string energy scaling. This approach differs &om that of either Allen and Cald- well, or Quashnock and Piran, who ignore the coupling process altogether.

Our expression for the scaling long-string density implies that in any comoving volume R3V, the total energy in long strings will be

The volume Vm can be thought of as the horizon size at the beginning of each era, or indeed the end of the radiation era if we are to consider the intermediate transition.

If we assume that the kink density does reach some scaling value of the form

for some constant value IS,, then the corresponding rate of energy loss required by loop production to maintain scaling is

after substitution for D( t ) in Eq. (5). We have already mentioned that loops are formed in

regions with a great deal of structure. In an attempt to model this the original two-scale model [6] introduced a "kinkiness" parameter such that the number of kinks removed by loops is a factor q times greater than the linear kink density would predict. Of course it is highly unlikely that this factor will be a constant. Kinks are present on the strings with some angular distribution, and as such would imply that q is a function of kink angle. In this paper we shall consider only a constant value of q, although it is our aim to generalize this to include kink-angle effects. The rate of kink removal by loops is thus

Consider now the effect of intercommuting. Since the formation of loops is really just a special case of intercommuting, we can effectively combine the two processes, at least as far as kink production is concerned. As in the two-scale model [6], we adopt a random-segment model of intercommuting.

Let us assume that the string network consists of N randomly moving segments, each of length D-l , with a mean velocity (x2) = (1 - a) /2 . If we select two segments at random of lengths l1 and 12, with respective starting points x l and x2, then there will be an intersection in the short time period St provided that

for some sl, s2, and t lying in the ranges 0 < s l < 11,

R3424 DAREN AUSTIN - 48

0 < s z < 1 2 , and 0 < t < 6t. Fixing x l imposes the condition that xz should lie in the small volume

The probability that the two segments intersect within a given time 6t is just SV/V, which means that the expected number of kinks forming in this time is i n ~ ' 6 V / V , where n takes into account the behavior of left- and right-moving segments and will usually take the value 2. The typical lengths of x i and xl are Jw and 2/-, respectively.

Averaging over all angles and assuming that V is the - -

comoving volume of the system gives

and x is given by

The a dependence is weak allowing us to set xrad C?

xmat cz 0.24. Since we have broken the long string into the small-

est possible segments our expression for the change in N must include intercommutings which give rise to loop production. Indeed only a small kaction O(y;2) of these intercommutings will be between long strings. With this in mind we can easily calculate the expected loop size at formation. We find that

where we have assumed that most intercommutings are self-intersections which form loops.

Typical values of y,, X, and a would suggest that during the early stages of the radiation-dominated era (L loops) /d~ e lo-'. Since m - 2 is negative definite at this time, as small-scale structure builds up r G p D , increases and (Lloops)/dH is reduced to a smaller value in keeping with the current simulations. Of course it is possible that FGpD, may become much larger than m-2 in which case (L1oops) would increase again, though this is forbidden by the scaling values of r G p D , which suggest

During the matter-dominated era (Lloops)/dH remains roughly constant because m - 2 - a m is a small number compared to r G p D , .

A further possibility is that loops carry away a roughly constant number (no) of kinks [9,10]. In that instance we would expect an average loop length of the form

for some value of no perhaps of order 10. Using this model of loop production means that the number of kinks added to the network by the formation of loops will be

AC = n ~ ~ l ~ ~ ~ ~ / p n ~ . loop.

The generation of loops is the least well-understood aspect of the cosmic string scenario, but it is at least conceivable that the true loop production mechanism is some mixture of the two extremes discussed here. Cer- tainly during the early stages of evolution (LlooPs) is a sizable fraction of the horizon size. As small-scale structure begins to form (Lloops) decreases until it becomes comparable with the interkink separation. By tuning the parameters x and no we can combine the two kink producing terms, thereby approximating the true loop formation process. In fact we shall see that the behavior of the eventual scaling solution is insensitive to the precise loop-production mechanism.

Stretching causes the kinks to open up reducing their angles [14], this in turn induces a factor of the form

Gravitational radiation causes kinks to become rounded, giving each kink a lifetime proportional to (XGpD)-l. The factor X is a dimensionless constant of order 1 which serves to disguise our incomplete knowledge of the back- reaction process [ll]. Thus the corresponding kink decay term from gravitational radiation is of the form

Combining our expressions for the rate equation for N , then switching to a new variable

defined as the number of kinks on an horizon-sized segment of string, we obtain the evolution equation

where the parameters a , b, c, and d are defined by

a = i n ( m + 1)xym, b = A m+l [(q - n/no)(m - 2 - a m ) + 3 - 2 a m ] , (25)

1 C = ( 9 - n/no)rGp , d=.Z& m+l '

We begin by solving this equation neglecting gravitational radiation (i.e., we set c = d = 0). The resulting equation for D has the general solution

and we have set D(to) = Do, which will be roughly zero a t the beginning of the radiation-dominated era, but need not be so after the Universe emerges from the transition to matter domination.

Evidently for the kink density to reach a stable scaling solution, we require that b < 0, or equivalently that q be larger than some critical value:

q > (3 - 2am)/(2 - m + a m ) + n/no. ( 2 7 )

This expression is only weakly dependent on the earlier

48 SMALL-SCALE STRUCTURE ON COSMIC STRINGS R3425

loop production term (no). This weak dependence on the small-loop formation process was also observed in the two-scale model [7], where scaling was found to be independent of the small-loop production efficiency c,h.

Numerical values of a imply that for the kink density to scale, the kinkiness parameter q during the two eras must be greater than grad > 2.3 and q,,tt > 3.7, respectively. These critical values of q are also in good agreement with the two-scale model. In our earlier papers we found corresponding critical values of q,,d > 1.8 and qmatt > 2.8. Although the two sets of parameters are defined for different kink distributions (linear and random), it is encouraging that the critical values are so similar, with both providing a lower bound on the kinkiness parameter.

Should bound (27) be satisfied, loops can remove kinks fast enough such that D reaches the scaling value D, = -a/b, i.e.,

1

lim D(t) = D, = ~ n ( m + l)'xym t+Oo ( q - n / n 0 ) [ 2 - m + a m ] - 3 + 2 a 2

from which we note that up to small numerical factors, - D , r~ ym during each era. This reproduces another of the conclusions of the two-scale model, namely, that the interstring correlation length and the persistence length along the strings will be of the same magnitude at scaling.

So, unless the kinkiness parameter for the small-scale structure is greater than some critical value, then the small-scale structure will not scale. This result implies that without gravitational radiation, scaling is possible if q is greater than the critical value, but the kink density remains small. This observation was made clear in our earlier work on the two-scale model which showed that the earliest simulations were indeed scaling [7].

For values of q less than the critical value grows as a power law, and as such can grow arbitrarily large. Since the values of q quoted in the literature are smaller than the critical value [3], and the simulations do not include gravitational radiation, the model presented here correctly predicts the build-up of small-scale structure observed by Bennett and Bouchet and other simulations.

Figure 1 shows a plot of D during the radiation- dominated era, for q = 1 and q = 4. Using the simulation

results mentioned earlier, we have again set a = 0.14 and yl = 15. The scaling behavior for q = 4 is clearly evi- dent, with D reaching a value of about D, r~ 5, as is the power-law growth for q = 1. Unless q is larger than the critical value, without some extra energy-loss mechanism loops cannot remove kinks at sufficient rates to stem their build-up on the long strings, and hence scaling of the small-scale structure is not possible.

Consider now the effect of adding in the gravitational back reaction [Eq. (24)]. Since we have assumed that D scales and c # 0, it is not possible to find a general solution to this equation. Instead, we can investigate the consistency of the scaling solution that we have imposed. We find that for (d - c) << b2 there are two scaling solutions:

The first of these solutions requires that either b > 0 and d > c, or b < 0 and d < c. The second solution simply requires that b < 0 as before.

In effect these inequalities imply that if q is greater than some critical value (b > 0) then loops remove string (and kinks) more effectively than gravitational radiation and to maintain scaling the gravitational radiation from the small-scale structure will be smaller than that from the long strings themselves (d > c). Conversely, if q is smaller than the critical value then gravitational radiation will be the dominant string (and kink) loss mechanism, and the radiation from the small-scale structure will be greater than that from the long strings (d < c ) .

Transforming these conditions into bounds on q we find that either

(3 - 2am)/(2 - m + a m ) > q - n/no > X/I',

(3 - 2am)/(2 - m + a m ) < q - nlno < X/r, (30)

while the third bound remains unchanged. Although we do not fully understand the process of

gravitational radiation, we know that X/r must be a factor much smaller than unity. The physical requirement that q 2 1 implies that for any network, the second of these bounds is certainly ruled out. The first bound is however consistent with D, becoming large. Indeed if q satisfies the first bound the scaling solution is given by

- (q - n/no)(m - 2 - a m ) + 3 - 2am D, 21

- (4 - nlno) r l . (31)

It should be stressed that these results have only shown that, when gravitational radiation is included scaling of the kink density is consistent with the long strings scaling. Since the combined bounds placed on q [bounds (27) and (30)] cover its entire range, we conclude that the consistency argument holds for all values of q. If q is smaller than the critical value, D can become very large

In tho before scaling begins, whereas larger values of q imply FIG. 1. Evolution of 5 as a function of the ratio t / t o during that loop production efficiently removes kinks and scal-

the radiation-dominated era. Current simulations correspond ing is reached early on. Thus we have the important to the power-law growth shown for q = 1. conclusion that scaling of the small-scale structure is an

R3426 DAREN AUSTIN

inevitable consequence of the long strings scaling, and es- timating the kinkiness parameter during a simulation can predict the eventual scaling density. These results imply that without a full numerical treatment of gravitational radiation the simulations are unlikely to show scaling of the small-scale structure.

As an illustration of the early kink build-up we can drop the coupling term by setting c = 0. Both Allen and Caldwell, and Quashnock and Piran have used this idea to show how scaling is reached, though we have shown that to calculate the eventual scaling density, the terms involving c cannot be ignored. Certainly the resulting uncoupled equation will be valid in the region - D << (rGP)-', but this is not really the region of physical interest. The general solution to the decoupled equation is

where A = -(4ad - b 2 ) . Since a is positive definite, - D always tends to the scaling value ( m + b ) / 2 d . Of course we know from our consistency argument that this should include a term involving c in the denominator. The scaling value D, evidently depends on the sign of b. If b < 0 the network scales but D remains small as we have already seen.

Assuming that b > 0 and keeping leading qrder terms, the general solution reduces to the solution derived by Allen and Caldwell (AC) and Quashnock and Piran (QP): namely,

with DL N b l d . To relate our parameters to theirs we

can make use of the substitutions

d - E , b K (AC), d E ~ l / l ? ~ G p , no E -y (QP). (33)

We see from this solution that D tends to the constant value DL. Yet this is of the same order as D,. Using a simple perturbative expansion we deduce that the full coupled evolution equation will reach scaling, with an eventual scaling value given by Eq. (31).

In summary, we have shown how a linear-kink model of small-scale structure can accurately reproduce the results of the two-scale model of cosmic string evolution. Moreover, using consistency arguments the full evolution of the long-string and kink densities has been inferred. From this work we conclude that those simulations which do not include the gravitational back reaction are not showing scaling of the small-scale structure, since the values of q quoted are below the critical values derived here. When the back reaction is included, we found that scaling of the small-scale structure is an inevitable consequence of the long strings scaling. Moreover, the eventual scaling density is related to the value of q during the evolution. If q is below about 2.3 during the radiation-dominated era and 3.7 during the matter-dominated era, then the kink density will become very large before scaling, with perhaps as many as D = lo4 kinks on an horizon-sized segment of string. If on the other hand q is larger than these values, then the back reaction plays no role whatsoever and the kink density scales at small values in agreement with the two-scale model of cosmic string evolution.

The model presented here is only a very simple one. We have yet to discuss the evolution of the kink density during the transition to matter domination and its sub- sequent effect on structure formation. These problems and the question of possible angular effects in the kink distribution will be treated in a forthcoming paper.

I would like to thank Tom Kibble, Ed Copeland, and David Coulson for useful discussions. This work has been funded by the SERC.

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Small-scale structure on cosmic strings