Effects of friction on cosmic strings

We study the evolution of cosmic strings taking into account the frictional force due to the surrounding radiation. We consider small perturbations on straight strings, oscillation of circular loops and small perturbations on circular loops. For straight strings, friction exponentially suppresses perturbations whose co-moving scale crosses the horizon before cosmological time $t_*\sim \mu^{-2}$ (in Planck units), where $\mu$ is the string tension. Loops with size much smaller than $t_*$ will be approximately circular at the time when they start the relativistic collapse. We investigate the possibility that such loops will form black holes. We find that the number of black holes which are formed through this process is well bellow present observational limits, so this does not give any lower or upper bounds on $\mu$. We also consider the case of straight strings attached to walls and circular holes that can spontaneously nucleate on metastable domain walls.


Introduction
Cosmic strings are topological defects that may have formed during phase transitions in the early universe (see e.g. [1]). Their properties and observational consequences, especially in connection with their possible role in the formation of large scale structure in the universe, have been extensively studied during the past decade (see e.g. [2]).
Cosmic strings of mass per unit length µ would have formed at cosmological time of order t 0 ∼ (Gµ) −1 t P l (G is Newton's constant and t pl is the Planck time). It is well known that immediately after the phase transition the dynamics of strings would be dominated by the force of friction [3,4]. This force is due to the scattering of thermal particles off the string. Friction would dominate the dynamics until a time of order t * ∼ (Gµ) −2 t P l . In most of the investigations about cosmic strings, the effects of friction have been neglected. The reason is that if cosmic strings have to play a role in galaxy formation, then they have to form near the grand unification scale. In that case their mass per unit length is of order Gµ ∼ 10 −6 and friction is important only for a very short period of time.
However, if strings have formed at later phase transitions, say, closer to the electroweak scale, their dynamics would be dominated by friction through most of the thermal history of the universe. It is therefore of some interest to study the evolution of cosmic strings with friction in a quantitative way. The relativistic equation of motion for strings with friction was given by Vilenkin [4]. The main purpose of this paper is to solve this equation in a few simple cases, which should be representative of more complicated situations.
The plan of the paper is the following. In Section 2 we review the results of Ref. [4], in order to fix the notation. In Section 3 we study linearized perturbations on an infinite straight string. In Section 4 we consider the dynamics of oscillating circular loops. The evolution of linearized perturbations on circular loops is discussed in Section 5.
It is known that an exactly circular loop would form a black hole when it collapses under its tension [6,7]. Since friction tends to erase perturbations whose co-moving scale crosses the horizon before t * , loops smaller than t * will tend to be approximately circular. In Section 6 we study the possibility that such loops form black holes. Note that these black holes can have masses of order M ∼ µt * , which can be considerably large if Gµ is sufficiently small. We discuss possible observational consequences of these black holes.
Finally, in Section 7, we consider perturbations on strings which are attached to domain walls, and in particular, to circular holes which can spontaneously nucleate in a metastable wall. Our conclusions are summarized in Section 8.

Strings with friction
In this section we summarize the results of Ref. [4], where the equation of motion for a string with friction was found. A cosmic string can be represented as a two dimensional worldsheet in spacetime x µ = x µ (ξ a ), where ξ a , (a = 0, 1) are two arbitrary parameters on the worldsheet and x µ are spacetime coordinates (µ = 0, ..., 4). In flat space, the equation of motion for the strings is µ✷x ν = 0, where ✷ denotes the d'Alembertian on the worldsheet. When one includes the effects of curvature of the spacetime and a force of friction F ν , the equation of motion reads [4] µ −✷x ν + Γ ν στ x σ ,a x τ,a = F ν (u λ ⊥ , T, σ) Here Γ ν στ are the four dimensional Christoffel symbols (greek indices run over spacetime coordinates). The force of friction F ν will depend on the temperature of the surrounding matter, T , the velocity of the fluid transverse to the worldsheet u ν ⊥ ≡ u ν − x ν ,a x σ,a u σ , and the type of interaction between the particles and the string, which we symbolically represent by σ. Vilenkin [4] found the form of F ν for the case when friction is dominated by Aharonov-Bohm scattering of charged particles with the pure gauge field outside the string [5] F ν = βT 3 u ν ⊥ .
The numerical coefficient β is given by where the summation is over all effectively massles degrees of freedom (i.e. mass m << T ) b a = 1 for bosons, b a = 3/4 for fermions and ν a is the phase change experienced by a particle as it is transported around the string. For the case of a Friedman-Robertson-Walker (FRW) universe, with the four-velocity of the fluid given by u ν = (a −1 , 0, 0, 0), and choosing the gauge ξ 0 = τ,˙ x · x ′ = 0, Eq.(1) reduces tö Here˙≡ d/dτ , ′ ≡ d/dξ 1 , H ≡ȧ/a 2 is the Hubble parameter and [Using the definition of ǫ it can be easily seen that Eq. (5) is actually not independent of (4)].
As noted in Ref. [4], the effect of friction is to make the replacement in the equations of motion for a free string in an expanding universe. In what follows we will consider a(τ ) ∼ τ α (α = 1 in the radiation dominated era and α = 2 in the matter era). Since T ∼ a −1 , the friction term h(τ ) dominates at early times, while the expansion term H(τ ) will dominate at late times. It is clear that friction will be unimportant for τ >> τ * , where τ * is the time at which both terms are equal In terms of cosmological time t defined by dt = a(τ )dτ and using Einstein's equations in the radiation era, H 2 = (8π 3 G/90)N T 4 , one finds [4] (for Gµ > ∼(At p /t eq ) 1/2 ). Here t eq is the time of equal matter and radiation densities and A = [90β 4/3 /32π 3 N ] 3/2 , with N the effective number of massless degrees of freedom. The coefficient A can be rather small. Taking sin 2 πν a ∼ 1/2 in the expression for β one has This estimate should actually be taken as an upper bound for A, since some of the particle species may not interact with the string.
If the strings are so light that Gµ < ∼(At p /t eq ) 1/2 then t * > t eq and (7) is not valid. Then we have to take into account that T ∼ t −2/3 in the matter era and we have 3 Linearized perturbations on a straight string Consider a straight string at rest, with trajectory x = (ξ, 0, 0) and introduce a small displacement in the plane transverse to the string, δ x = e ikξ y(τ ) = e ikξ (0, y 2 (τ ), y 3 (τ )).
Substituting in (4) and keeping only terms linear in δ x we havë where, as mentioned before, α = 1 in the radiation era and α = 2 in the matter era. The case without friction corresponds to taking τ * = 0. Then (9) is just the Modified Bessel equation. The corresponding solution that is well behaved as τ → 0 is given by where ν = α − 1/2. At early times, τ << k −1 , the solution tends to a constant, whereas at late times, τ >> k −1 , the solution is oscillatory Modes with wavelength larger than the horizon size (τ << k −1 ) are frozen in, and their physical amplitude y phys ≡ a(τ )y is conformally stretched by the expansion. Once the wavelength of a mode falls within the horizon (τ >> k −1 ), the perturbation starts oscillating with constant physical amplitude [1]. When friction is included (τ * = 0), equation (9) can only be solved in the high frequency approximation (k >> τ −1 * ). This is actually the interesting regime, in which friction plays a role. In the opposite limit (k << τ −1 * ), we just saw that perturbations are frozen in up to a time τ ∼ k −1 >> τ * due to the expansion term 2α/τ alone, and by the time they start oscillating the friction term has become negligible.
To solve for the case k >> τ −1 * we introduce the variable Substituting in (9), one finds that Ψ satisfies the Schrödinger equation with potential Now Eq.(13) can be solved in the WKB approximation. This is done in Appendix A.
With the boundary condition that the perturbations are frozen in with the expansion at early times, the solution is Here where B is Euler's Beta function and τ k is the classical turning point of the corresponding Schrodinger problem V (τ k ) = k 2 . The phase φ k is just a constant. In radiation dominated universe It can be checked that the conditions for the validity of the WKB approximation are satisfied provided that k >> τ −1 * and α > 1/2. Comparing (14) with (11) we can summarize as follows. Without friction perturbations of wave number k are conformally stretched up to a time τ ∼ k −1 , after which they start oscillating with constant physical amplitude With friction included, short wavelength perturbations k >> τ −1 * are conformally stretched up to a time τ k given by (16), hence the factor a(τ k ) in (15). Since τ k > k −1 , friction contributes to increase the amplitude of the perturbation by a factor (τ k k) α , from time τ ∼ k −1 up to time τ k . After τ k , the perturbation starts oscillating and losing energy to friction. This is represented by the two exponential factors in (14). By the time τ ∼ τ * the amplitude of the perturbations has been damped by a factor exp(−γkτ k ). For τ >> τ * the modes do not lose any more energy to friction, the first exponential in (15) has reached its asymptotic value of unity and the string oscillates with constant physical amplitude As expected, friction plays no role after τ * . Also, the exponential suppression in (18) becomes less and less dramatic as k approaches τ * (i.e. k −1 ∼ τ k ∼ τ * ), in agreement with the intuitive expectation that friction does not affect wavelengths of co-moving size comparable or larger than τ * .
As mentioned before, the WKB approximation is only valid for α > 1/2. The case α ≤ 1/2 is markedly different because friction never 'switches off'.
As a result, the amplitude of perturbations is eventually damped to zero for all wavelengths. The simplest example is the flat space case, (H = 0, h = constant.) In this case the equation for y is that of a damped harmonic oscillator and all excitations disappear after a characteristic lifetime which is given by τ ∼ h −1 for k >> h, and by τ ∼ 2hk −2 for k << h (taking a = 1).

Circular loops
A circular loop can be parametrized as where R 0 is a constant and ξ ∈ [0, 2πR 0 ]. The equations of motion (4) and (5) read (see Appendix B) In the case of flat space and no friction, H = h = 0, ǫ is a constant, so Eq.(4) is linear and one finds the well known oscillatory solution Note that, as a consequence of exact circular symmetry, this solution collapses to a point. Of course the same will happen when we include friction and expansion. In reality, when the string shrinks to a point it would form a black hole [6,11]. We will consider this process in more detail in Section 6, but for now we shall ignore it, and continue the solutions beyond the singular points. The reason is that here we are interested in the energy loss of a loop due to friction. Although we consider the circular loop for simplicity, similar results would apply to nearly circular loops which do not shrink to a point. In the generic case H + h = 0 we cannot find exact analytic solutions, so we have to use numerical solutions and analytic approximations.

a-Loops with friction but no expansion
It is instructive to start by considering the case H = 0 and h = const. = 0. Consider a loop initially at rest and whose initial radius R 0 is sufficiently large. It is clear that as long as aR >> h −1 the motion of the loop will be overdamped, with characteristic velocitẏ [i.e. we neglect theR term in (20)]. That this is a good approximation can be checked by computingR from the previous equation and comparing it to the other terms in (21). On the contrary, for aR << h −1 the damping term can be neglected and the string undergoes a relativistic collapse similar to the frictionless one. This suggests that the energy of the string at the moment of its first collapse will be independent of R 0 (for R 0 >> h −1 ), and will be roughly equal to the energy of a loop of physical size h −1 , This estimate can be made rigorous by using a simple scaling argument. Introducing dimensionless variables p ≡Ṙ, u ≡ 2haR and v ≡ 2haR 0 ǫ, the differential equations (20) and (21) can be reduced to where there is no reference to time or to h. The initial conditions R = R 0 where u 0 ≡ 2haR 0 . The energy of the string at the moment of first collapse is The coefficient χ ≡ v(u = 0) can be found by numerical integration. Although v(u = 0) depends on u 0 , the result rapidly saturates to a constant χ ≈ .57 for u 0 larger than 1.
After the first collapse the loop will undergo a series of oscillations, losing energy to friction in each one of them. Let us estimate this energy loss. Since the behaviour of energy is controlled by Eq. (21) we will be interested in the quantity <Ṙ 2 >, where the brackets denote the temporal average between two consecutive collapses of the loop. We have where in the first step we have integrated by parts and in the second we have used the equation of motion (20). Repeating similar steps for the calculation of < RṘ 3 > one can generate a perturbative expansion in powers of ahR. The first terms are From (24) it is clear that after the first collapse ahR < ∼.3, so the second term is a very small correction (at most of order 10 −2 ) to the first and we have From (21) the fraction of the energy lost in between two consecutive collapses is given by ∆E/E ≈ −2 <Ṙ 2 > hT , where T is the time spent in the oscillation. Therefore E ∝ exp(−ht) and the loop will disappear in a characteristic lifetime of order ∼ h −1 after the first collapse.

b-Loops with friction and expansion
The case without friction in expanding universe has been previously studied in the literature [1]. Loops are conformally stretched by the expansion until they cross the horizon. This happens at time t c defined by Here r c is the physical radius of the loop at horizon crossing. After that, the loop oscillates with constant physical amplitude. This behaviour is illustrated in Fig 1a, which corresponds to the numerical solution of (20) with h = 0 and for a radiation dominated universe (α = 1). The result is plotted in terms of dimensionless cosmological time t/t c . The upper line represents E/E c , where E is the energy of the loop and E c = 2πµr c . In the same figure, we plot r/r c , where r = aR is the physical radius, andṘ, the velocity of the string with respect to the cosmological fluid.
With friction included, for loops such that r c >> t * , the evolution is not much different from the one just described. The reason is that by the time these loops cross the horizon and start oscillating, we already have h << H.
The effect will be important for loops that cross the horizon well before t * (i.e. r c << t * ). In Fig. 1b we plot the time evolution of a loop with r c = 10 −3 t * , in radiation dominated era (same conventions as in Fig. 1a). Not surprisingly, friction delays the time at which the loop first collapses. Initially, this increases the energy of the loop, since it is stretched up to a radius much larger than t c . Later, as the loop shrinks, it loses energy to friction. It is interesting to observe that both effects roughly compensate each other, in the sense that at the time of first collapse, t f we have E(t f ) ≈ E c , just like in the frictionless case. After t f the loop keeps losing energy during each oscillation, up to a time t * , when friction switches off. From that time on, the loop oscillates with constant physical amplitude.
It is possible to give an approximate analytical description of this evolution. At early times the motion of the loop is overdamped. Neglecting thë R term in (20) we have Before t * , H can be neglected and we can integrate (25) to find The loop will first collapse at a value of conformal time which is of order Actually, Eq.(26) is only valid as long asṘ << 1. After that, theR term in (20) becomes comparable to the others. Since R 0 << τ f , the effective friction coefficient, ah, will not appreciably change during the relativistic collapse (which occurs at a conformal time close to τ f ). From (24), the energy of the loop at the moment of first collapse is In the radiation era (α = 1), we have E f ≈ E c independently of R 0 , in good agreement with the numerical results. In the matter dominated era, for R 0 << τ * we have E f >> E c , so by the time τ f loops have actually more energy than they would have had in the absence of friction. After t f , the radius of the loop is much smaller than the horizon and the quantity 2a(H + h) is slowly varying compared with the period of oscillation. From the arguments of the previous subsection we have <Ṙ 2 >≈ 1/2, and from (21) the change in ǫ during one oscillation is ∆ǫ/ǫ = −a(H + h)∆τ . Since ∆ǫ << ǫ, one can take the continuous limit: and since E ∝ aǫ, the energy of the loop is given by In the case without friction (τ * = 0) we have E ≈ constant, in agreement with the numerical results (Fig. 1a). With friction included, the energy drops until time τ * , after which loops oscillate with constant energy.

Perturbations on circular loops
Let R(τ ) be a solution of (20), representing the evolution of a circular loop. A perturbed loop can be parametrized as Here (ρ, θ, z) are co-moving cylindrical coordinates, in which the metric takes the form ds 2 = a 2 (τ )(−dτ 2 + dρ 2 + ρ 2 dθ 2 + dz 2 ), y ρ is a radial perturbation and y z is a perturbation transverse to the plane of the loop.
It is straightforward to write (4) in cylindrical coordinates and then substitute (29) to find the linearized equations for the perturbations. This is done in Appendix B. Decomposing y ρ as a sum over modes and similarly for y z , the resulting equations arë where we have omitted the index + or −. The sum in (30) starts at L = 2 rather than at L = 0 because it is easy to see that to linear order the L = 0 and L = 1 modes do not correspond to deformations from the circular shape but rather to small translations and rotations of a circular loop [8].
Since the function R(τ ) is not known analytically, Eqs.(31) and (32) have to be solved numerically. However, before we do that, the behaviour of the perturbations can be guessed from the results of the previous sections.
Let us first consider the case without friction. Up to the time t c the loop is conformally stretched by the expansion and the perturbations will approximately behave as perturbations on a straight string. Ignoring oscillatory cosine factors, the amplitude of the perturbations at this time will be y L (t c ) ≈ y 0,L a(R 0 /L)/a(τ ), where y 0,L is the initial perturbation. Here we have used (11) making the substitution k → L/R 0 . After t c , the loop comes within the cosmological horizon and starts collapsing. A loop on subhorizon scales behaves approximately as it would behave in flat space. The theory of perturbations on a circular loop collapsing in flat space was solved in [8] (see Appendix A of that reference). It was shown that the amplitude of transverse perturbations stays constant during the collapse, whereas the amplitude of radial perturbations shrinks by a factor of L as the loop shrinks to r << r c . Therefore by that time we have To check the validity of these approximations one has to numerically solve Eqs.(31) and (32), whith h = 0. We have done that for different values of L, in the radiation dominated universe and using the boundary condition that perturbations are initially at rest. The result is plotted in Fig. 2 as a function of wave number L. The circles denote the ratio y ρ phys,L divided by the r.h.s. of equation (33) at the time when the loop first collapses. Similarly, the crosses denote the ratio of y z phys,L to the r.h.s. of (34). Ignoring the "valleys" in Fig.  2, these ratios are of order 1, which means that Eqs. (33) and (34) give a very good estimate. The valleys in Fig. 2 are due to oscillatory behaviour of the perturbations, which we have ignored in our argument.
The effects of friction can be introduced along similar lines, and they will only be important for co-moving wavelengths (R 0 /L) < τ * . For such wavelengths the r.h.s. of equations (33) and (34) has to be corrected by a factor of 2 This is obtained comparing (17) with (18), where now k is given by L/R 0 . Again, one can check the accuracy of this approximation by solving the equations of motion for the perturbations, now with h = 0. The results for the case of a radiation dominated universe are plotted in Fig. 3a for R 0 = τ * , and in Fig. 3b for R 0 = 5τ * . The circles and crosses denote the same quantities as in Fig. 2. It is seen that the suppression factor (35), depicted as a solid line, gives the right answer to very good approximation. The above considerations apply only to loops with R 0 > ∼τ * . For R 0 << τ * the perturbations do not have time to be damped before the loop starts shrinking, so the factor (35) actually overestimates the effect of friction. We shall consider this in more detail in the next section.

Black hole formation
Let r c be the physical radius of a circular loop at horizon crossing. The mass of this loop is 2πr c µ and the Schwarzschild radius corresponding to this mass is r s = 4πGµr c .
As the loop shrinks under its tension, its rest mass is converted into kinetic energy so that the total energy of the loop remains constant (neglecting friction and gravitational radiation for the moment). If a loop is exactly circular then it will eventually shrink to a size smaller than r s and form a black hole [6,7]. (This is only true for strings that form as a result of gauge symmetry breaking. Strings which form as a result of global symmetry breaking would radiate all of their energy in the form of Goldstone bososns before they shrink to the size of their Schwarzschild radius [19]). If the loop is not exactly circular it will still form a black hole provided that the size of the perturbations is sufficiently small. Let τ s denote the time when the unperturbed loop would shrink to the size of its Schwarzschild radius. It is clear that if then a black hole will still form.
From the analysis of the previous section one could argue that loops of string with R 0 << τ * could easily form black holes. The argument is that since friction exponentially suppresses wiggles on scales smaller than τ * , these loops would be circular to very good approximation. This would result in the copious production of black holes with masses up to where m pl is the Planck mass. There are observational upper bounds on the density of black holes of masses M > ∼10 14 m pl , which would be evaporating at the time of nucleosynthesis or later (see [9] and references therein). Then, from (37), one would be able to put constraints on topologically stable cosmic strings of very low tension, Gµ < 10 −14 .
However, this argument needs to be refined, since for R 0 << τ * one cannot simply use the exponential suppression factor (35) to estimate the effect of friction. This is because these loops collapse well before τ * and friction has not had enough time to damp the perturbations. Therefore, it is important to study the behaviour of perturbations on loops with R 0 << τ * in more detail.
At very early times, both the loop and the perturbation will be overdamped. Neglecting second derivatives in (32) we havė Using (25) we have the interesting relation so the perturbations shrink faster than the radius of the loop, the relative perturbation decreasing as the coordinate radius shrinks from its initial value R 0 . Following similar steps we also have To find the limit of applicability of the overdamped approximation, one can calculateR andÿ z from (25) and (39) and compare them to the damping term in equations (20) and (32). One readily finds that (25) is valid forṘ << 1, while (40) is valid forṘ << L −1 . The time at whichṘ ∼ L −1 coincides also with the time at which the relevant physical scale comes within the effective horizon h −1 and starts oscillating. Once a perturbation starts oscillating it is damped very efficiently, so the perturbations which will be more difficult to eliminate are those with L = 2, which are overdamped up to the time when the loop becomes relativistic, R ∼ 1. From (25) this happens when R ∼ [ah] −1 , at a time of order τ f , given by (27). That is, the loop becomes relativistic when where in the last step we have restricted attention to loops which are collapsing in the radiation era. After that, during the relativistic collapse, the loop is within the effective horizon and it can be seen that perturbations behave very much like they would on a circular loop in flat space. That is, as the loop shrinks the amplitude of transverse perturbatins stays constant whereas radial perturbations only shrink by a factor of L [8].
As a result, all the supression in the lowest modes L = 2 comes from the overdamped regime. From (41) with (R/R 0 ) given by (42), and using that r s ∼ Gµh −1 (τ f ) we obtain that at the time t s when the unperturbed loop crosses its Schwarzschild radius For strings formed at a phase transition, the initial value of the relative perturbation y ρ 2 (t 0 )/R 0 can be of order one. Then, in order to satisfy (36) we need However, if this condition is met, the energy of the loop at the moment of first collapse is E ∼ µr c < ∼(Gµ) 2 m P l , much smaller than the planck mass, which simply means that a black hole will not form.
Therefore friction by itself is not sufficient to ensure the formation of a black hole if we start from an arbitrarily wiggly loop: the lowest modes L = 2 are not sufficiently damped. This was not obvious a priory and it is in contrast with what would happen if the universe were not expanding. In that case h is a constant and from (40) and (41) all loops whose initial size aR 0 is much larger than (Gµ) −1/2 h −1 would shrink to form black holes of mass M ∼ µh −1 .
Even in an expanding universe, some of the loops produced at the phase transition might just happen to be circular enough initially that they would form black holes, even without friction. With friction included, the number of loops that will form black holes is larger. The question is then what fraction of the ensemble will go into black holes or, in other words, what is the probability of black hole formation P bh .
Before we try to answer this question we should know how P bh is constrained from cosmological observations. Assuming a scale invariant distribution of loops with number density at horizon crossing given by [1] dn(r c ) = νr −4 c dr c , where ν is a parameter of order one or smaller, and given that these loops form black holes of mass M = 2πµr c with probability P bh , the number density of black holes is Here we have included a factor of (r c /t) 3/2 in the distribution to account for the dilution of the strings (or black holes) by the expansion of the universe. Black holes of mass M ∼ 10 10 − 10 11 g would evaporate during cosmological nucleosynthesis, producing high energy particles which would deplete the deuterium and helium when standard nucleosynthesis has almost concluded. This process has been studied in Ref. [10], using a distribution of black holes of the form (44) with the numerical coefficient left as a free parameter. The bound that these authors obtained can be translated into (assuming that the present abundance of deuterium is of cosmological origin). Note that if P bh ∼ 1, then we would be in conflict with this constraint for Gµ > ∼10 −18 . It is then important to estimate P bh . An upper bound can be obtained directly from Eq.(43). The probability that y ρ 2 (t s ) < R s is equal to the probability that Treating y ρ 2 (t 0 ) as a random variable normally distributed, with r.m.s. amplitude of order R 0 , and noting that for L = 2 there are two independent modes [the + and the − modes in Eq.(30)], that probability is bounded by where in the last step we have used M > ∼10 14 m p . As a result, the constraint (45) will always be satisfied for cosmic strings in the low energy range (38) that we were interested in.
What can one say about heavier strings? Note that P bh can actually be much lower than the r.h.s. of (46), and the bound (45) is likely to be satisfied even for large values of the string tension Gµ ∼ 10 −6 . In this case the loops that form black holes in the mass range 10 14 − 10 20 m pl cross the horizon much later than t * , so friction only smoothes out the perturbations of very large L. This does not help very much when we try to form black holes. The smoothness of the strings in this case has to be attributed to other damping mechanisms such as gravitational radiation. The probability of black hole formation in the absence of friction has been estimated by Hawking [11] and others [12,8], In the case of a loop formed by n straight segments, Hawking found p ∼ 2n. This form for P bh results from the fact that in order to form a black hole it is necessary to "fine tune" a set of 2n angles with acuracy given by Gµ.
Typically, p will be of the order of the number of random variables that parametrize the ensemble of loops, since for a black hole to form all the parameters have to be fine tuned with an accuracy given by Gµ. Polnarev and Zembowicz [12] studied P bh for a family of loops containing excitations in the first and third harmonics in a Fourier expansion of the solutions to the Nambu equations of motion. They studied a two parameter family and they found p ≈ 2 (with some uncertainty due to the arbitrariness in the definition of a probability distribution in the space of parameters.) However, p is likely to be larger (and therefore P bh smaller) since one need not be restricted to a two parameter family. In particular, the general solution including the first and third harmonics is a five parameter family (see e.g. [13]), and one may expect p ∼ 5.
Other observational constraints come from black holes of mass M ∼ 10 20 m pl , which would be evaporating at the present time, producing intense bursts of γ-rays (see e.g. [18]). This constraint has been studied in [12], whose authors concluded that if p < ∼2 there is conflict with observations for strings heavier than Gµ

Strings attached to walls
In this section we consider perturbations on strings which are attached to planar domain walls [1]. For simplicity, we shall ignore the expansion of the universe. Also we shall restrict ourselves to perturbations which lie on the plane of the wall.
Then we are effectively left with a 2+1 dimensional problem, in which the string is the boundary between a region of "false vacuum" (the wall) with energy per unit area equal to the surface tension σ, and a region of "true vacuum" where there is no wall. In the absence of friction, the equations of motion are [14] − µ✷x µ = σn µ .
Here ✷ is the covariant d'Alembertian on the worldsheet of the string and n µ is the space-like unit vector normal to the worldsheet, with n µ n µ = 1 and n µ ∂ a x µ = 0. Our sign convention is that n µ points towards the wall. It is easy to see that equation (48) has a solution representing a straight string which is constantly accelerating due to the tension of the wall. The wall gradually disappears as the string moves forward, its energy going into kinetic energy of the string [14]. The string undergoes the so-called hyperbolic motion, asymptotically approaching the speed of light at late times. Eq. (48) has to be modified to include the force of friction F ν . To keep things general we take without specifying the dependence in the transverse velocity of the fluid u ⊥ = u µ n µ . (Note that u ν ⊥ defined in Section 2 is given by u ν ⊥ = u ⊥ n ν .) Then the equations of motion read We can study perturbations on any solution x µ (ξ a ) of (50) using the covariant formalism of Ref. [14], suitably modified to include friction. Since only perturbations which are normal to the string are physically observable, one need only consider perturbations of the form Here φ represents the proper magnitude of the perturbation, i.e. the normal displacement as measured by an observer that is moving with the string. Multiplying (50) by n ν we can write − µK a a = (σ + F ), where K ab = n µ ∇ a ∂ b x µ is the extrinsic curvature of the worldsheet. Latin indices are raised and lowered using the worldsheet metric The linearized equation of motion for the perturbations can be found from variation of Eq.(51) µδK a a = −δF, where δ denotes the variation induced, to linear order in φ, by the small perturbation δx µ . Following [14], we have δK a a = ✷φ + K ab K ab φ. On the other hand δF = F ′ (u ⊥ )δu ⊥ , where F ′ = dF/du ⊥ . Also, δu ⊥ = u µ δn µ , and the change in n µ induced by the perturbation is [14] δn µ = −g ab φ, b ∂ a x µ . Therefore, the resulting equation of motion for φ is where the "mass" is given by In the last step we have used the Gauss-Codazzi relation K ab K ab = (K a a ) 2 −R, where R is the intrinsic curvature scalar on the worldsheet. The r.h.s. of eq. (52) acts like a friction term for the perturbation φ provided that F ′ > 0. But this condition is always met, since it just means that the force of friction increases with the transverse velocity of the fluid.
In the spirit of Refs. [14,15], Eq.(52) can be seen as the equation of motion for a scalar field φ which is "living" in the worldsheet of the defect. Now this equation has a different mass than in the case without friction, and it also has a right hand side in which the field has a derivative coupling to some external sources (which are essentially the 'temporal' components of the tangent vectors to the string.) In the case when the unperturbed string is straight, lying along the y axis and moving with trajectory x = x(t), the metric on the worldsheet is Here t is the time in the rest frame of the fluid and τ is the proper time of an observer who is moving with the string. In this case the metric on the worldsheet is flat, the curvature scalar vanishes and the effective mass M 2 is tachyonic. This is also true in the absence of friction [14], and it essentially means that modes with wavelength larger than |M| −1 are unstable. Now, including friction, the difference is that the tachyonic mass "switches off" as the string approaches its terminal velocity v. Indeed, the terminal velocity of the string is determined by the vanishing of the driving force in (50), that is but this implies that the mass term for the perturbations vanishes. When the straight string has reached the terminal velocity v, equation (52) reduces to where k is the wave number of the perturbation, and γ v = (1 − v 2 ) −1/2 is the relativistic factor corresponding to the terminal velocity. The friction term in the r.h.s. of (55) now causes the perturbations to decay exponentially with a lifetime which is easily calculable. Taking a force of the form (2) we have In cosmological situations the temperature is always lower than the energy scale of the wall (typically of order σ 1/3 ), therefore the terminal velocity will be relativistic v ∼ 1 and the gamma factor will be of order γ v ∼ σT 3 (ignoring the numerical factor β). Then the r.h.s. of (55) reads It is then straightforward to see that the perturbations decay with proper lifetime given by τ ∼ µ/σ for k >> σ/µ and τ ∼ σµ −1 k −2 for k << σ/µ. A more interesting case is that of a string which is at the boundary of a circular hole that has spontaneously nucleated on a metastable domain wall (see e.g. Ref. [16]). This process is the 2+1 dimensional analogue of the formation of true vacuum bubbles in the problem of false vacuum decay.
Without friction, perturbations on these bubbles (or holes) have been studied in ref. [14]. In that case, the unperturbed solution was a circular hole whose radius R expands with constant acceleration, The effective mass for the perturbations was M 2 = −σ/2µ 2 , which is tachyonic. Because of the expansion of the hole, any perturbation eventually reaches a wavelength larger than M −1 , at which point it becomes unstable and starts growing like φ ∝ R ≈ t [14]. From an intrinsic point of view, one can say that the string is unstable, in the sense that φ, (the perturbation measured by a co-moving observer) grows in time. However, an external observer measures a perturbation which is Lorentz contracted [14] Since from (56) γ ∼ R, we have ∆ ∼ const. at late times. The relative perturbation ∆/R decreases and the string becomes more circular as the hole expands [14]. Including friction, the main difference will be that the string reaches a terminal velocity v, and the Lorentz contraction factor will go to a constant at late times. Also, the mass term switches off as the string approaches the terminal velocity. Let us see, then, what is the fate of the perturbations when friction is included in the dynamics. The string worldsheet is given by x µ = (t, R cos θ, Rsinθ, 0), and the metric induced on the worldsheet is The normal vector is n µ = (1 −Ṙ 2 ) −1/2 (−Ṙ, cos θ, sin θ, 0). The extrinsic curvature can be easily calculated and one finds The velocity is bounded by the terminal velocityṘ < v, so at late times R ≈ vt andR decays faster than R −1 . Then, neglecting theR term in (57) the equation of motion for the perturbations takes the form with φ = φ L (τ ) exp(iLθ). Note that as the hole expands, the tachyonic mass squared −γ 2 v R −2 "redshifts" at the same rate as the wave number term L 2 R −2 . One might expect instability for modes with L < γ v , since for these the total effective mass squared is negative. However, at sufficiently late times we can use the overdamped approximation and neglectφ in (58). Then a simple integration shows that, since R −2 decreases faster than τ −1 , the amplitude of the perturbations asymptotically approaches a constant φ → const. From an intrinsic point of view the asymptotic behaviour of the perturbations is very different from the frictionless case, in which φ was growing in time. However, now γ v is constant and we have ∆ = γ −1 v φ → const., so from the point of view of an external observer which is at rest the behaviour is similar to that of the frictionless case. The relative perturbation ∆/R also goes to zero, and the loop becomes more and more circular as the hole expands.

Conclusions
In this paper we have studied the evolution of cosmic strings taking into account the force of friction. The results are conveniently expressed in terms of t * , defined as the time at which friction "switches off" [equations (7) and (8)].
For small perturbations around a straight string, a WKB analysis shows that they are exponentially suppressed if their wavelength crosses the horizon before the time t * . Relative to the frictionless case, the amplitude of the perturbations has to be multiplied by the suppression factor (35), which in the case of a radiation dominated universe (α = 1) reduces to Here λ is the physical wavelength of the perturbation at the time t * and γ ≈ 1.2. The suppression of the amplitude of these perturbations is due to the oscillation of the perturbations before the time t * . After t * the perturbations oscillate with constant physical amplitude. Perturbations whose wavelength crosses the horizon after t * are practically unaffected by friction. Similarly, for circular loops we should distinguish between large loops with r c >> t * , and small loops with r c << t * . Here r c is the radius of the loop at horizon crossing. Large loops are unaffected by friction, after they cross the horizon they start oscillating with constant physical amplitude r c . Small loops do not start collapsing relativistically right after they cross the horizon. During the non-relativistic evolution, the velocity at which the string moves with respect to the fluid is given bẏ where r is the physical radius of the loop, T is the temperature and β is the numerical factor in (2). Due to the frictional force, these loops are dragged by the expansion more than they would in the absence of friction, growing to a size larger than r c . However as they collapse they lose energy to friction. It is interesting to note that in a radiation dominated universe both effects approximately compensate each other, in the sense that at the time when they first collapse to a point their energy is given by E f ≈ 2πµr c , just as in the frictionless case (in matter dominated universe E f is actually larger than 2πµr c ). In subsequent oscillations, the loops will keep losing energy, up to the time t * . After that they oscillate with constant energy. This energy can be obtained from (28) in the limit of large times, and one has We have also studied perturbations on circular loops. If the loops are larger than t * , then the perturbations behave in much the same way as the perturbations on a straight infinite string. Perturbations whose wavelength crosses the horizon before t * are suppressed according to (59), whereas if it crosses later than t * they are unaffected by friction. If the loops are much smaller than t * then it is not correct to use the suppression factor (59), essentially because the loops start their relativistic collapse much before t * . As a result, the perturbations in the lowest modes are only suppressed as a power law in (r c /t * ) [see e.g. (43)].
Because the suppression in the relative amplitude of the lowest modes is only power law, the probability that loops of strings become circular enough to form black holes is very small (to form a black hole one needs that the relative amplitude of the perturbations be of order Gµ, which is very small). As a result, the number of black holes of masses M > ∼10 14 m P l (which would be evaporating at the time of nucleosynthesis or later) formed by strings which cross the horizon before t * is too small to have any observable consequences (for all values of µ). For loops which cross the horizon later than t * , friction only eliminates the perturbations of very large wavenumber, so the probability that they form black holes is not substantially enhanced by friction.
We have also studied perturbations on strings attached to walls. For this we have used the covariant formalism developed in ref. [14], generalizing it to include the force of friction. In particular we have studied the case of a circular hole which spontaneously nucleates on a metastable domain wall. We find that perturbations on the string that is at the boundary of the hole are initially unstable, growing at the same rate as the radius of the expanding hole. However, as the loop reaches its terminal velocity v < 1, the instability switches off and the perturbations freeze out at a constant amplitude. As a result, the relative perturbation decreases in time, and the hole becomes increasingly circular as it expands. This behaviour is similar to the behaviour that one obtains in the absence of friction [14]. However the mechanism by which perturbations freeze out is very different. In the former case it is due to the force of friction, which balances the instability due to the tension of the wall. In the latter case the freezing occurred for purely kinematical reasons, since the unperturbed string asymptotically approached the speed of light.
Since perturbations are frozen in at early times, the coefficients C ± are determined by imposing the boundary condition that the co-moving perturbation y should approach some constant value y 0 as τ → 0. This immediately requires C + = 0, Using the standard connection formulas of the WKB formalism, the solution in the 'classically allowed' region is where p 2 = [k 2 − V ] 1/2 . The simplest case, which is also the most interesting, is the radiation dominated universe α = 1. In this case the potential assumes a particularly simple form V = τ 2 * τ −4 , and the turning point is given by τ k = (τ * k −1 ) 1/2 . Then, the exponent Γ(k, τ k ) can be calculated analytically Integrating by parts we have where B is Euler's Beta function. For α = 1, the potential is more complicated and the turning points cannot be given explicitly in terms of τ * and k. However, for k >> τ −1 * , the potential under the barrier can be approximated by the second term and we have Neglecting the first term in V (τ ) and following the same steps as before we have Γ = kτ k 1 4α − 2 B 1 + 2α 4α , 1 2 .
Let us derive the equations for the perturbations on circular loops. Taking z = y z (τ θ) << R it is immediate from (B3) to obtain the equation for small transverse perturbations (32). The equation for the radial perturbations requires more work. First we write the perturbed solution as ρ = R(τ ) + ∆, and θ = (ξ/R 0 )+δ, where ∆ and δ are small deviations from the unperturbed values. Substituting in (B1) one finds, to linear order in the perturbations Also to linear order where ǫ 0 is the unperturbed value. Substituting in the previous expression and using (20) we find which is the equation of motion for radial perturbations (31).

Figure captions
• Fig.1a Evolution of a circular loop without friction in radiation dominated universe, as a function of cosmological time t. Here t c is the time at which the loop crosses the horizon, r is the physical radius of the loop and r c is the radius at horizon crossing. The energy of the loop E rapidly approaches the value E c ≡ 2πµr c after the loop crosses the horizon, and remains approximately constant. We also plotṘ, the velocity of the string with respect to the cosmological fluid.
• Fig.1b Evolution of a circular loop with friction in radiation dominated universe, with r c = 10 −3 t * . (Same conventions as in Fig. 1a.) • Fig.2 Numerical results for the evolution of perturbations on a circular loop in radiation dominated universe, ignoring the force of friction. The results are plotted as a function of wave number L. The circles denote the ratio y ρ phys,L divided by the r.h.s. of equation (33) at the time when the loop first collapses. Similarly the crosses denote the ratio of y z phys,L to the r.h.s. of (34). Ignoring the oscillatory behaviour, this ratios are of order one, which means that (33) and (34) give a very good estimate.
• Fig.3a Numerical results for the evolution of perturbations on a circular loop in radiation dominated universe, including the force of friction, for R 0 = τ * . Circles and crosses denote the same quantities as in Fig.  3. It is seen that the suppression factor (35), depicted as a solid line, gives the right answer to very good approximation.