- Email: [email protected]
On the decay of cosmic string loops
Nuclear Physics B293 (1987) 812-828 North-Holland, Amsterdam
ON THE DECAY OF COSMIC STRING LOOPS Robert H. BRANDENBERGER Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
Received 6 November 1986 (Revised 4 March 1987)
The power loss from a cosmic string loop by particle-antiparticle emission is determined using an order of magnitude estimate of back reaction effects on the motion of the loop. We consider perturbative decay processes in vacuum and at finite temperature. Nonperturbative decay processes may be more important. We discuss one example which is used in the back reaction analysis. We conclude that all nongravitational decay processes discussed are subdominant compared to gravitational radiation. The analysis applies to nonsuperconducting strings resulting from spontaneous breaking of a local gauge symmetry.
1. Introduction C o s m i c strings [1] are linear topological defects which arise in certain gauge theories in which a local gauge symmetry is spontaneously broken. There is a simple topological criterion for the existence of strings. The v a c u u m manifold M must be n o n - s i m p l y connected, i.e. %(M) 4= 1 where % is the first h o m o t o p y group of M. C o s m i c strings have no ends. Thus they are either infinite or closed loops. Recently cosmic strings have received a lot of attention as a possible explanation for the existence of structures in the universe such as galaxies and clusters of galaxies [2-5]. In the cosmic string theory of galaxy formation closed loops form massive seeds which will attract the surrounding gas to form structures [5]. In order for this m e c h a n i s m to work, string loops must survive for more than one Hubble expansion time. By conservation of topological charge (winding number) infinite strings cannot decay. Closed loops are not protected by such a conservation law. According to " s t a n d a r d lore" [2, 3] and to order of magnitude estimates [6] the d o m i n a n t energy loss m e c h a n i s m for closed loops is gravitational radiation. Srednicki and Theisen [7] have recently reanalyzed this issue more carefully and find the following power loss due to particle-antiparticle emission 1 where # -
o 2 is the mass per unit length in string, o is the scale of symmetry
0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R.H. Brandenberger / Cosmic string loops
813
breaking and R is the radius of the loop. Except for loops created near the critical temperature Tc - o this decay rate is many orders of magnitude smaller than the power loss PG by gravitational radiation PG
-
v(G/~)/t,
(2)
where 3' is a constant whose value is about , / - 1 0 2 according to numerical computations and analytical estimates [8]. The power loss in eq. (11) exceeds the result obtained from order of magnitude estimates [6] by a factor of R o . For cosmological loops this is the ratio between a cosmological scale R - t o (t o is the present Hubble radius) and a microphysical length 0 -1 (the width of the string). The value of this factor is about 1 0 57 for o - 1016 GeV. Most of the contribution to (1) comes from cusps, points on the string x ( l ) which reach the speed of light. Cusps are, however, artifacts which arise only if all back reaction effects on the motion of the string are neglected. In this paper we extend the analysis of Srednicki and Theisen [7] in two respects. First we present an initial attempt to include the effects of the back reaction, second we extend the decay analysis to finite temperature. Srednicki and Theisen [7] consider only perturbative decay processes. We present an order of magnitude estimate of the energy loss due to cusp annihilation, one of many possible nonperturbative decay mechanisms and the mechanism used in order to estimate the back reaction. We conclude that the back reaction will smooth the cusps and thus result in a maximal loop velocity smaller than the speed of light. As a consequence the perturbative decay processes are suppressed. We also conclude that, based on the one mechanism considered here, nonperturbative processes can be much more important. Our estimate for the decay rate by cusp annihilation is (3)
P~osp - ~t( R o ) - ' / 3 •
For cosmologically interesting loops this is still many orders of magnitude smaller than the energy loss by gravitational radiation. The outline of the paper is as follows. In sect. 2 we determine the perturbative decay rate of a string loop as a function of the maximal velocity Vmax 1 - e of any point on the string. In sect. 3 the smoothing of cusps due to the finite thickness of the string is discussed. No point on a loop outside a cusp region reaches the speed of light. We determine the maximal velocity which such points can reach. The power loss due to interaction of the cusp region must be analyzed separately. We estimate the order of magnitude of this nonperturbative process. In sect. 5 the perturbative analysis is extended to finite temperature. A semiclassical argument is employed. The analysis in this paper is valid for local strings for which the fields decay exponentially as a function of the distance from the string, (it has been shown [9] that global strings decay predominantly by Goldstone boson emission). To simplify =
814
R.H. Brandenberger / Cosmic string loops
the analysis we consider a toy model in which the complex scalar field q) which gives rise to strings has a Mexican hat potential = ¼Xo(l
(4)
12- o 2 ) .
Here o denotes the scale of symmetry breaking and 2, o is the self-coupling constant for q). q) is coupled to a gauge singlet scalar field ~p of mass m 0 via the interaction lagrangian t i - - -2~1~12¢ 2 • (The analysis can easily be A word about notation: constant, mp1 the Planck attention to homogeneous with a scale factor a(t)
(5)
extended to other couplings.) we use units in which h = c = k B = 1. G is Newton's mass. In cosmological considerations we restrict our and isotropic Friedmann-Robertson-Walker universes
ds2=
-dtZ + a( t)2 dx 2.
(6)
The Hubble "constant" H is H(t) =
a(t)
.
(7)
2. Perturbative decay of a rotating loop in vacuum We will first consider the perturbative decay of a rotating loop held fixed at two points. There are advantages in choosing a rotating loop rather than an oscillating loop. For a rotating loop no point ever reaches the speed of light, even neglecting back reaction. For oscillating loops most of the power comes from points which reach the speed of light, i.e. from cusps. Since rotating strings have no cusps, back reaction will only insignificantly change the power of particle-antiparticle radiation. In fact as will be shown in sect. 4, the power for a rotating loop can be used to obtain an estimate for the energy loss of oscillating loops with back reaction taken into account. A similar trick was used by Turok [10] to estimate the strength of gravitational radiation from closed loops. The rotating loop configuration is given by (fig. 1) _s(l, t) = (p cos o~lcos tot, p cos oJl sin ~t,
3l~),
(8)
with p2 + 82 = R2 = ~ - 2 . 1 is the string coordinate and runs f r o m
-
(9)
~ -1 to ½¢r¢0-1 as the point on the ~r~o
R.H. Brandenberger / Cosmicstring loops z
815
@y
A 6R X
B
S(f,t)
Fig. 1. Sketch of the rotating loop configuration of eq. (8). Only half of the loop is shown. The loop is held fixed at points A and B and rotates about the z axis.
string moves from B to A. _s(l, t) satisfies the flat space equations of motion for a string except at points A and B. At these points there are artificial boundary conditions which correspond to the string being held fixed. Fig. 1 shows only half the string. The second half is obtained by rotating s(l, t) by ~r about the z axis. Without loss of generality we can restrict our attention to the half loop. In calculating the vacuum decay amplitude for a string to emit a particle-antiparticle pair we follow the analysis of Srednicki and Theisen [7]. After change of variables to define a new real field 14,1 = a + ~
(lO)
the vacuum S-matrix element can be evaluated to linear order in perturbation theory. Let Is)_o~ denote the initial string at asymptotically early times and Is')oo the final string state at asymptotically late times. Then
= (s'~(kl)+(k2)+lSls) = hfd4x(s'lep2(x)lS)b(d,(kl)~p(kz)+l,kZ(x)10)f = Xfd'x(s'l,l,2(x)ls)
e'~,+k2~ ~ + ~(X2),
+ O(X z)
(11)
where subscripts b and f denote matrix elements restricted to the Hilbert space of ch and ~b respectively. In evaluating the final expectation value we make several assumptions which are only justified for local strings. We assume the string is very thin, i.e. that the field
R.H. Brandenberger / Cosmic string loops
816
falls off exponentially as a function of the distance from the string. Then q~Z(x) can be replaced by a delta function along the string s(l, t). We also neglect the change in the string configuration caused by the emission process, i.e.
(s'lq,2(x)l s)
= 0
if Is') =g bs).
(12)
Since the value of ff at the center of the string is a and the width of the string is w e can write
a ~ ~kol/2o-1,
(slqj2(x)ls) = [~o, 1/2 d183(x _ _s(l, t))02a2(1 -- $2) 1/2,
(13)
a - ~roa 1 / 2
where the final factor is required by reparametrization invariance of the string world sheet [11]. Hence the S-matrix element is
~(s+(kl)q~+(k2)ls)_~ --_ 2,),o1[ ~'-1/2 d l f d t e x p ( i ( k O + k O ) t - i ( k _ l a ~rw 1/2 X
+k_2)s(l,t))
,I
(1 - g2)1/2 + 0()~2).
(14)
For an oscillating string and neglecting back reaction there are points which reach the speed of light. These points give stationary points of the integrand and dominate the final answer, as discussed by Srednicki and Theisen [7]. If the back reaction is taken into account the maximal velocity of the string will be slightly less than the speed of light and the value of the integral in (14) will change by orders of magnitude. For a rotating loop there are no points on the string which reach the speed of light. Hence there are no stationary points of the integrand. The integral must be evaluated using the Riemann-Lebesgue lemma, and the answer will be a nonsingular function of the maximal velocity Uma~ (as long as Ureax < 1) and thus will change only infinitesimally if back reaction is included and changes Ureax infinitesimally. Infinite straight strings cannot decay by conservation of winding number. A consistency check on (14) is to apply it to such an infinite string
s(l,t)=(O,O,Z),
-m
(15)
In this case (14) vanishes by the Riemann-Lebesgue lemma. The consistency check is satisfied. For the rotating loop configuration (8) the S-matrix element (14) (henceforth denoted by s) becomes
s = XX o 1
dt
eiEt[ ~rR/2 dl exp - ikp cos oal - - c o s oat + ~- ,~R/2 k
× (1 - ~2cosZ(oal))l/2,
sin oat - ikfioal (16)
817
R.H. Brandenberger / Cosmicstring loops
where k = (k~ + my ~2~1/2 kx = k I + k 2 and E = k ° + k °. By the Riemann-Lebesgue / lemma the integral vanishes if k~6toR > 1 (17) or ko>l. As an order of magnitude estimate for (16) we then get s-
XXolfdt em' ~rRO(1 -
kp)O(1 - k~8)
(18)
and 2 2 At Isl 2 - X )t o -ff (~'R)20(1 - ko)O(1 - k~6),
(19)
where A t is the total time. F r o m (19) the total power Pv due to particle-antiparticle emission from the string is obtained by integrating over the available phase space: 1 1 f d3kl d3k2' 0 P v A t = 8 (2q7)6 J k 0 ~020 (kx +k°)lsl
2
81 1(2~) 6 x2X°2(~'R)2 [ f d k f d k ~ O ( 1 - 8 ( k f + k ~ ) )
x
f d2kl d2_k20(1 -
p[_kI +
_k21)
1
x
] I
(k 2 + k f 2 + mg) 1/2 (k_ 2 + k~2+ mg) 1/2
(20)
The total momenta k = k I + k 2 and k z = kf + k~ are constrained by the theta functions. The relative momenta K = k 1 - k 2 and K ~'= k f - k~ are not yet constrained. However, the particle-antiparticle emission process is a local process and the energy E of the emitted particles must be smaller than the energy which can be released by making the string locally straight, where locally means on a length related to E by the uncertainty principle [12]. Hence K and K z are constrained to be smaller than Em~x Emax ol/2R - 1/2. (21) - -
Now it is easy to estimate the order of magnitude of the phase space integral in (20), denoted here by I. I - 8-1/9- 2Emax - ~- lp- 2~1/2R - 1/2
(22)
Hence the total power from vacuum decay of the string into particle-antiparticle
R.H. Brandenberger / Cosmic string loops
818
pairs is Pv
1 1 ) 2 ) . - 2 ~ 2,~ - 2 R - 1,., 1 / 2 32 ()'2~r"2 . . . . 0 " ' p . . . . 1
o
- 1/2
1
. ,2X2Xo2R-2~-28-1(Ro) 1/2
(23)
32 (z~r)
where in the final step we have introduced dimensionless variables tS and 3 by P = ~3R,
3 = 3R.
(24)
and 3 can vary between 0 and 1 and measure the deformation of the loop. The perturbative decay rate (23) is smaller than the result by Srednicki and Theisen [7] by a factor of (Ra) -1/2. It is, however, larger than the original order of magnitude estimates. The factor ~-2g-1 depends on the maximal velocity of the loop. As we will argue in sect. 4 it can be as big a s ( R o ) 1/3.
3. Nonperturbative decay by cusp annihilation Cusps, points on the cosmic string loop with x_'(l) = 0 (where the prime denotes a derivative with respect to the string length parameter l) and which reach the speed of light, are generic to solutions of the string equations of motion. They are, however, artifacts of neglecting the nonzero width of the string. In this section we shall estimate the smoothing of the cusp when taking the finite thickness of the string into account. At a cusp two branches of the string are sufficiently close for microphysical forces to become important. Such interactions may lead to a substantial energy loss. We shall estimate the order of magnitude of this effect. Cusps arise once per oscillation period. The general expansion of the cosmic string solution about a cusp at ( t , / ) = (0,0) is [13]
[ t - ½(a 2 +/32 + 2/2)(½/3 + 12t)_a/3(t21+ 113) _x(t,l)=lla(t2+12)+/31t+3(1t3+12t)+~(t21+]13)
I
,
(25)
ylt+e(½'3+12')+~(t21+~13)
where a, /3 and y are constant coefficients of the order 1/R and 3, e, ~ and ~ are constants of the order 1/R 2. The width of the string is w - a-1. From (25) we can determine the length l c of the cusp region, defined as the region where Ix(0, 1) - _x(0, -1)1 ~
(26)
R.H. Brandenberger / Cosmic string loops
819
The result is lc - wl/3(a2f12 + ~2 + ~2)-1/6
- -
o_t/3R2/3.
(27)
We shall now assume that a fraction of the order 1 of the string mass in the cusp region evaporates as a consequence of microphysical interactions. This gives an estimate of the nonperturbative energy loss from the loop by this process. Since cusps arise periodically in each oscillation period the power PN of radiation from the loop is PN -- I'tlc/R - ° 2 ( ° R ) -1/3"
(28)
The above represents an upper bound on the energy loss by cusp annihilation. PN cannot increase by matter streaming into the cusp region from neighbouring segments at the string loop. From (25) it follows that the cusp region (measured for l - lc) persists for a time interval A t - o - l ( R o ) 1/3 .
(29)
The maximal amount of energy which can stream into the cusp region is therefore a E - ~ t a - l ( R o ) 1/3 ,
(30)
which is smaller than the mass already in the cusp by a factor ( R a ) -1/3 Outside of the cusp region defined by (27) points on the string never reach the speed of light. The maximal velocity can be obtained by considering (25). To lowest nontrivial order 1 - ½(a 2 + ~(t,l)
=
+ v 2 ) ( t : + t 2) at + fll
(31)
The maximal velocity is taken on at t = 0 at the edge of the cusp region, i.e. for l = l c. The result is Om~x -
1
- ½(alc) 2.
(32)
As in the introduction e shall denote the difference between the speed of light and Vmax. Then by (27) e - (oR) -2/3.
(33)
We shall use this result in the following section to estimate the perturbative power loss of an oscillating loop.
820
R.H. Brandenberger / Cosmic string loops
4. Back reaction effects on the decay of oscillating loops In sect. 2 we determined the perturbative power loss from a rotating loop which never reaches the speed of light. The answer was expressed in (23) as a function of the parameters t3 and ~ which determine the specific geometry of the loop. t~ and are related by t32 + g2 = 1,
(34)
so there is only one free parameter 8 which can vary between 0 and 1 . 8 determines the maximal velocity of the loop. From (8) we have
U m ~ = p = (1 --~2) 1/2.
(35)
For 8 << 1, Um~ approaches the speed of light. The velocity difference e is then approximately 1 ^2
(36)
and the power P from (23) can be written in terms of e p _ ~2~k0 2R -2( Ro )1/2_ 1/2
(37)
The idea of the following analysis is to apply (37) to oscillating loops by using back reaction estimates to determine e. A higher and a lower estimate for e shall be given. The first only takes into account the smoothing of cusps as a way of obtaining Vm~, < 1, the second is an attempt to include the dominant decay mode, gravitational radiation, into the back reaction analysis. We shall first discuss the lower estimate for e. The maximal velocity of a string loop will occur at or near cusps. As discussed in the previous section cusps will be smeared out by finite width effects. The maximal velocity outside the cusp region is given in (32). Hence ~ - (oR) 2/3 and by (37) p_
~.2~.o2R_lo(Ro)
1/6.
(38)
The above gives the upper estimate for the perturbative power loss of an oscillating loop. Several comments should be made. All processes in the cusp region have been included in the estimate for nonperturbative decay rate PN by cusp annihilation (see eq. (28)). We see that the perturbative power P is reduced by taking back reaction into account. However, provided there are no other mechanisms which prevent the formation of cusps, there is the possibility of an energy loss by cusp annihilation which far exceeds that by perturbative processes. Mechanisms which could prevent the formation of cusps even in the zero width limit of cosmic strings include interactions of the string which change its equations
821
R.H. Brandenberger / Cosmic string loops
of motion. Such effects have recently been discussed in the case of superconducting cosmic strings [13,14]. Here we shall consider nonsuperconducting cosmic strings and investigate the possible effect of gravitational radiation from the string on the dynamics of the loop. The power of gravitational radiation has been estimated analytically by Turok [10], using the rotating loop (8). In linear theory (but not using the quadrupole approximation which would be invalid for a rapidly rotating loop) the power Pc is given by
dP~
_k) _
dI2 ~r
(39)
_k)i2),
where T"'(¢o, k_) =/~
S-~- f e i'll
d i e -/-k-~(`.,>
(1L
7, )
~i~j- sis;
.
(40)
T = (2¢r)-lR is the period of rotation. Estimating the integrals for s(l, t) given by (8) we get Po - (4~r)2(G/t)# = 3,(G/z)~t,
(41)
with ~, - 10 2. Note that the result is independent of the maximal velocity and hence will be a good approximation for gravitational radiation from an oscillating loop (with back reaction taken into account). We shall now treat the oscillating loop as a system with one degree of freedom. This is an unrealistic idealization, but we hope it will be useful in an attempt to estimate the maximal effect of the back reaction, i.e. the largest value of e. The one degree of freedom loop is similar to a harmonic oscillator. Gravitational radiation from the loop corresponds to a damping term in the equation of motion of the harmonic oscillator and will lead to a decrease in the velocity of the oscillator after ¼ period. Similarly gravitational radiation will modify the equation of motion of the loop and will lead to a smaller maximal velocity. We shall estimate e in analogy with the harmonic oscillator example. Consider a harmonic oscillator m.~ + fx = O,
(42)
with f / m = ¢o2= R - 2 and with amplitude A = R. The maximal velocity of the undamped oscillation is 1. Now we introduce a damping term + F2 + to2x = 0
(43)
and fix F by demanding that the resulting energy loss per unit time P equal the
822
R.H. Brandenberger / Cosmic string loops
power Po from gravitational radiation for a loop. We get !
P = ½rm - v(GtDt~,
(44)
r= 2~(Gt~)~.
(45)
and with m = f l l ~ R
1 -1 is The velocity of the damped oscillator after a quarter cycle t = uro~ Yc - 1 - ½Ft = 1 - ¼F~rw 1
=
1 - e,
(46)
with 7 e - ~G/~.
(47)
Eq. (47) represents our estimate for the velocity reduction as a consequence of the back reaction effects due to gravitational radiation. Besides the caveats already mentioned there is another difficulty in the above argument. For the harmonic oscillator the frequency remains unchanged as the amplitude decreases, for the loop the frequency w is inversely proportional to R. Since the decrease in amplitude over ¼ oscillation period is negligible, this difficulty should be unimportant. If gravitational radiation indeed has the maximal effect estimated above, then the perturbative power P by particle-antiparticle emission from the string is p _ X2Xo 2R -2( R o )1/2( r _ l~/a~ ) -1
(48)
and thus exceeds the result for a rotating loop only by a factor ( f l - l y G # ) - i l0 s, small compared to the difference between (23) and (38). The main result, however, is that independent of whether the higher or lower estimate for Vm~x holds, the resulting perturbative energy loss from oscillating loops is smaller than what is obtained without including back reaction, and much smaller than the energy loss due to gravitational radiation. -
5. Thermal decay: a classical analysis
So far we estimated the perturbative energy loss by particle-antiparticle emission from a string in a vacuum. In a cosmological context the string is in a thermal bath which includes ~ particles. Thermal effects may change the decay rate. In this section we present a classical analysis of the thermal decay of the string. If the ~k particle is a boson then we expect the decay rate to be amplified by stimulated emission compared to the vacuum decay rate. A quantum analysis of the decay for both bosonic and fermionic qJ fields will be presented elsewhere [15].
R.H. Brandenberger / Cosmicstring loops
823
The analysis is based on a perturbative Green function method. We consider the production of # particles in an external classical ~ field corresponding to the rotating string of eq. (8). In the classical limit, ~ particle production corresponds to an excitation of the # field. The equation of motion for a bosonic ~ field coupled to the complex scalar field via the interaction lagrangian (5) is (using the change of variables (10)) ~ - 17'2++ (m 2 + ~.(o + dp)2)~ = O.
(49)
The ha 2 term should be viewed as a renormalization of the mass of ~. From now on we shall denote by m 2 the combination m 2 + ~0 2. We shall split ~ into an unperturbed solution ~0 and a perturbation #x. = #0 +
(50)
q'o satisfies the homogeneous equation ~'0 - V'2#o + m2~bo= 0
(51)
and ~i the linearized perturbation equation ~ i - W2~bi+ m2~i = -X(2q~a + d#2) ~0-
(52)
q~(_x, t) can be determined using the retarded Green function Gm of the operator 02 -- V 2 "-I-m 2
1 f d4p e -ip(t'x) Gin(x-,t)- (2~r)4 ret ~-~-- ~--2 "
(53)
The subscript on the integral denotes that the contour of the P0 integration is to be taken above the two poles (fig. 2). ~bi(_x, t) is given by •i(x,/)
=
-~fdt'd3x_'G,,(x_-x_',t-t')d/o(X_',t')(2oep+ep2).
(54)
In the classical analysis q'0(-, x ! t') is homogeneous and its value is determined by equating scalar field energy density and thermal energy density 1 . 2R2 ~,n V 0 -_ -_ 1 ~ 2 T 4 ,
or
(55)
824
R.H. Brandenberger / Cosmic string loops
Po
,,g,
_(p2+m2if2
( p2+ m 2 )1'2
Fig. 2. Location of the poles for the P0 integrationin the retarded Green function G,,.
As in the vacuum decay calculation it is a consistency condition to check that for a long straight string there is not net energy loss. The calculation is sketched in the appendix. We find that ~bi(_x,t) is independent of time and exponentially suppressed as the distance from the core of the string increases. Hence there is no net energy flux from the string. We now outline the calculation for the rotating string configuration s(l, t) in (8). For local strings we replace q~(_x,t) by a delta function q,(_x, t) =
fd163(x_-s_(l,t))oa2(1-~2) 1/2,
(2oep+eP2)(x,t)=3oZazfd183(x_-s(l,t))(1-g2) 1/2.
(57) (58)
Hence
q~(x, t) = -p f d3x'dt'd4p e-ip(t-'"~--~-')f
(59)
where we have combined all the constants into
P
3 ~ O 1 (qr2 ]1/2 T2 (2'n")4 ~ - ] 7
(60)
The d3x ' integral is trivial. The first step is to perform the P0 integral. For t - t' > 0 we must close the contour in the lower half plane (fig. 2), and the result is 1
~k~(x-,t)= 27rPf dldrd3_p (p2 + m2)l/2exp{ i_p(x_-s_(l,t)) } ×sin((p2+m2)l/2(t-t')}O(t-t')(1-g2(l)).
(61)
For t - t' < 0 the contour is closed in the upper half plane and the integral vanishes as it must for a retarded Green function solution.
R.H. Brandenberger / Cosmic string loops
825
Next we evaluate the dl integral using the Riemann-Lebesgue lemma. If (compare with (17)) IpA> 8 -1 or p=
( p 2 ..[_py2)l/2 > p - 1
(62)
we have an integral of a rapidly oscillating function which by the above mentioned lemma vanishes. Hence etp-x
t) - 2=e f d3p (p2 +m2)'/2
f~oodt'sin{(p2 +m2)l/2(t- r) }
×wRO(8 - 1 - ]p~l)O(p-l-p).
(63)
The third step is to perform the dp~ and d2_p integrals. Since the radius R of the loop is much larger than the string width a - o-1 and since the renormalized mass m is X1/20 or larger, the approximation R-1 < m will be well justified. Using this approximation we obtain 1
0
m
-oo
¢i(x,t)-Z~rRP-- f
dt'sin(mt')nfin(8-1, lzl-:)min{p-2, r-2},
(64)
with r = (x 2 +y2)1/2. If At is the total time interval, then in the near field limit p - 1 < r - : and 8 -1 < [z[-1
lm2,L2t x A t ) - ½(2~r)2p28- 2p-4m-lAtR 2 '/" I \ - - ,
(65)
The total energy in the ~ field ( - total energy radiated from the string by particle-antiparticle emission) is obtained by multiplying (65) by the near field volume which is of the order R 3
E ( At ) = PTAt
7/.2 T 4
-
9)k2)ko2(2~)-6_ _ __ R_18_ 2p_4 A t 30 m 3
(66)
(see eq. (24)). Thus the power PT of thermal decay is T4
PT -- ~2)k 0 2 m-~R - le- x,
(67)
where like in sects. 3 and 4 e is the deviation of the maximal velocity from the speed of light.
826
R.H. Brandenberger / Cosmic string loops
The power from stimulated emission at finite temperature is to be compared with the zero temperature perturbative power P from (37). Taking m - o and e given by (33) we get
(] 4
5t4( ompl/5J6
(68)
where in the final step we have evaluated the ratio for loops with radius of the order of the horizon. We conclude that at temperatures close to the critical temperature of the phase transition stimulated emission will dominate. For loops which could be important for cosmological questions today, stimulated emission is negligible. 6. Conclusions The nongravitational decay of cosmic string loops has been discussed. The analysis applies to nonsuperconducting strings which result from spontaneous breaking of a local symmetry. Both the perturbative decay rate by particle-antiparticle emission from the loop and the nonperturbative decay rate from cusp annihilation were considered. The perturbative decay rate was calculated at zero and at finite temperature using an order of magnitude estimate of the back reaction. We conclude that back reaction significantly reduces the perturbative decay rates. By how much depends on the maximal velocity of the loop. A low estimate for the suppression factor can be obtained if we assume that back reaction from gravitational radiation does not prevent cusps and that finite width effects are the only effects which smoothen the singularities. If gravitational radiation acts as a damping term in the string equation of motion then the suppression factor is much larger. If back reaction effects do not prevent cusps, then the energy loss due to cusp annihilation is much larger than the perturbative decay rates. This indicates that in general nonperturbative processes are important. However, both perturbative and nonperturbative decay rates are much smaller than the decay rate due to gravitational radiation. Stimulated emission at finite temperature is important only at temperatures close to the critical temperature of the phase transition. The computations of the perturbative decay rates were performed in a simple toy model in which a gauge singlet scalar field q, is coupled to the fields of the cosmic string model. If ~b is a nongauge singlet scalar field the interaction lagrangian contains more terms, in particular terms coupling the gauge fields A~ to ~. The At,A~2 terms in L I can be analyzed along the same lines as discussed in this paper. The overall effect will be to multiply all the decay rates by a constant of the order 1. Fermion pair emission is more complicated and will be discussed in ref. [15]. Our analysis does not apply to superconducting strings. For these, electric currents can flow along the string loop. This can give rise to many electromagnetic decay channels [16].
R.H. Brandenberger/ Cosmicstringloops
827
Many points discussed in this paper were raised in stimulating conversations with A. Matheson, M. Srednicki, S. Theisen and N. Turok. I wish to thank them for their help. I am grateful to S. Hawking, C. T h o m p s o n and E. Witten for useful comments on the first draft of the manuscript. Part of the work was done at the 1986 Aspen C e n t e r for Physics Workshop on Cosmic Strings and Inflation.
Appendix A. CONSISTENCY CHECK A self consistency check shows that the approximation scheme of sect. 5 gives vanishing energy flux from an infinite straight string. F o r an infinite straight string in a local gauge theory q~2(x, y, z, t) =
a2o28(x)6(y).
(A.1)
Inserting into (54) we get
e-ip(x_-x,,t-t,) m2
q~,(x_, t) = - p f d3x'dt'd4p ~(x')~(y') p2 _p2 -
_
=(2~r)2pfdpxdpye-ip~xe-ip'Y(p~+p2+m2)
-1
(A.2)
W i t h o u t loss of generality we can choose x = 0 and y > 0. Then
1
~i(x,t)=(2~r)3Pfo°°dpx[pZ+mZ]l/ze-YtP2+m211/2.
(A.3)
This is time independent and decays exponentially with y. Thus there is no energy flux from the static infinite string.
References [1] T. Kibble, J. Phys. A9 (1976) 1387 [2] Ya Zel'dovich, Mon. Not. R. Astr. Soc. 192 (1980) 663; A. Vilenkin, Phys. Rev. Lett. 46 (1981) 1169 [3] A. Vilenkin, Phys. Reports 121 (1985) 263 [4] N. Turok, Phys. Lett. 123B (1983) 387; A. Vilenkin and Q. Shaft, Phys. Rev. Lett. 51 (1983) 1716; J. Silk and A. Vilenkin, Phys. Rev. Lett. 53 (1984) 1700 [5] N. Turok and R. Brandenberger, Phys. Rev. D33 (1986) 2175; H. Sato, Prog. Theor. Phys. 75 (1986) 1342; A. Stebbins, Ap. J. (Letters) 303 (1986) L21
828
R.H. Brandenberger / Cosmic string loops
[6] T. Vachaspati, A. Everett and A. Vilenkin, Phys. Rev. D30 (1984) 2046 [7] M. Srednicki and S. Theisen, Phys. Lett. 189B (1987) 397 [8] T. Vachaspati and A. Vilenkin, Phys. Rev. D31 (1985) 3052; N. Turok, Nucl. Phys. B242 (1984) 520 [9] R. Davis, Phys. Rev. D32 (1985) 3172 [10] N. Turok, in ref. [8] [11] A. Vilenkin, private communication [12] A. Matheson, private communication; R. Brandenberger and A. Matheson, in ref. [15] [13] D. Spergel, T. Piran and J. Goodman, Nucl. Phys B291 (1987) 847; A. Vilenkin and T. Vachaspati, Tufts Univ. preprint (1986) [14] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910 [15] R. Brandenberger and A. Matheson, Brookhaven preprint (1987) [16] E. Witten, Nucl. Phys. B249 (1985) 557; E. Chudnovsky, G. Field, D. Spergel and A. Vilenkin, Phys. Rev. D34 (1986) 944; J. Ostriker, C. Thompson and E. Witten, Phys. Lett. 181B (1986) 243; C. Hill, D. Schramm and T. Walker, Fermilab preprint (1986)
ON THE DECAY OF COSMIC STRING LOOPS Robert H. BRANDENBERGER Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
Received 6 November 1986 (Revised 4 March 1987)
The power loss from a cosmic string loop by particle-antiparticle emission is determined using an order of magnitude estimate of back reaction effects on the motion of the loop. We consider perturbative decay processes in vacuum and at finite temperature. Nonperturbative decay processes may be more important. We discuss one example which is used in the back reaction analysis. We conclude that all nongravitational decay processes discussed are subdominant compared to gravitational radiation. The analysis applies to nonsuperconducting strings resulting from spontaneous breaking of a local gauge symmetry.
1. Introduction C o s m i c strings [1] are linear topological defects which arise in certain gauge theories in which a local gauge symmetry is spontaneously broken. There is a simple topological criterion for the existence of strings. The v a c u u m manifold M must be n o n - s i m p l y connected, i.e. %(M) 4= 1 where % is the first h o m o t o p y group of M. C o s m i c strings have no ends. Thus they are either infinite or closed loops. Recently cosmic strings have received a lot of attention as a possible explanation for the existence of structures in the universe such as galaxies and clusters of galaxies [2-5]. In the cosmic string theory of galaxy formation closed loops form massive seeds which will attract the surrounding gas to form structures [5]. In order for this m e c h a n i s m to work, string loops must survive for more than one Hubble expansion time. By conservation of topological charge (winding number) infinite strings cannot decay. Closed loops are not protected by such a conservation law. According to " s t a n d a r d lore" [2, 3] and to order of magnitude estimates [6] the d o m i n a n t energy loss m e c h a n i s m for closed loops is gravitational radiation. Srednicki and Theisen [7] have recently reanalyzed this issue more carefully and find the following power loss due to particle-antiparticle emission 1 where # -
o 2 is the mass per unit length in string, o is the scale of symmetry
0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R.H. Brandenberger / Cosmic string loops
813
breaking and R is the radius of the loop. Except for loops created near the critical temperature Tc - o this decay rate is many orders of magnitude smaller than the power loss PG by gravitational radiation PG
-
v(G/~)/t,
(2)
where 3' is a constant whose value is about , / - 1 0 2 according to numerical computations and analytical estimates [8]. The power loss in eq. (11) exceeds the result obtained from order of magnitude estimates [6] by a factor of R o . For cosmological loops this is the ratio between a cosmological scale R - t o (t o is the present Hubble radius) and a microphysical length 0 -1 (the width of the string). The value of this factor is about 1 0 57 for o - 1016 GeV. Most of the contribution to (1) comes from cusps, points on the string x ( l ) which reach the speed of light. Cusps are, however, artifacts which arise only if all back reaction effects on the motion of the string are neglected. In this paper we extend the analysis of Srednicki and Theisen [7] in two respects. First we present an initial attempt to include the effects of the back reaction, second we extend the decay analysis to finite temperature. Srednicki and Theisen [7] consider only perturbative decay processes. We present an order of magnitude estimate of the energy loss due to cusp annihilation, one of many possible nonperturbative decay mechanisms and the mechanism used in order to estimate the back reaction. We conclude that the back reaction will smooth the cusps and thus result in a maximal loop velocity smaller than the speed of light. As a consequence the perturbative decay processes are suppressed. We also conclude that, based on the one mechanism considered here, nonperturbative processes can be much more important. Our estimate for the decay rate by cusp annihilation is (3)
P~osp - ~t( R o ) - ' / 3 •
For cosmologically interesting loops this is still many orders of magnitude smaller than the energy loss by gravitational radiation. The outline of the paper is as follows. In sect. 2 we determine the perturbative decay rate of a string loop as a function of the maximal velocity Vmax 1 - e of any point on the string. In sect. 3 the smoothing of cusps due to the finite thickness of the string is discussed. No point on a loop outside a cusp region reaches the speed of light. We determine the maximal velocity which such points can reach. The power loss due to interaction of the cusp region must be analyzed separately. We estimate the order of magnitude of this nonperturbative process. In sect. 5 the perturbative analysis is extended to finite temperature. A semiclassical argument is employed. The analysis in this paper is valid for local strings for which the fields decay exponentially as a function of the distance from the string, (it has been shown [9] that global strings decay predominantly by Goldstone boson emission). To simplify =
814
R.H. Brandenberger / Cosmic string loops
the analysis we consider a toy model in which the complex scalar field q) which gives rise to strings has a Mexican hat potential = ¼Xo(l
(4)
12- o 2 ) .
Here o denotes the scale of symmetry breaking and 2, o is the self-coupling constant for q). q) is coupled to a gauge singlet scalar field ~p of mass m 0 via the interaction lagrangian t i - - -2~1~12¢ 2 • (The analysis can easily be A word about notation: constant, mp1 the Planck attention to homogeneous with a scale factor a(t)
(5)
extended to other couplings.) we use units in which h = c = k B = 1. G is Newton's mass. In cosmological considerations we restrict our and isotropic Friedmann-Robertson-Walker universes
ds2=
-dtZ + a( t)2 dx 2.
(6)
The Hubble "constant" H is H(t) =
a(t)
.
(7)
2. Perturbative decay of a rotating loop in vacuum We will first consider the perturbative decay of a rotating loop held fixed at two points. There are advantages in choosing a rotating loop rather than an oscillating loop. For a rotating loop no point ever reaches the speed of light, even neglecting back reaction. For oscillating loops most of the power comes from points which reach the speed of light, i.e. from cusps. Since rotating strings have no cusps, back reaction will only insignificantly change the power of particle-antiparticle radiation. In fact as will be shown in sect. 4, the power for a rotating loop can be used to obtain an estimate for the energy loss of oscillating loops with back reaction taken into account. A similar trick was used by Turok [10] to estimate the strength of gravitational radiation from closed loops. The rotating loop configuration is given by (fig. 1) _s(l, t) = (p cos o~lcos tot, p cos oJl sin ~t,
3l~),
(8)
with p2 + 82 = R2 = ~ - 2 . 1 is the string coordinate and runs f r o m
-
(9)
~ -1 to ½¢r¢0-1 as the point on the ~r~o
R.H. Brandenberger / Cosmicstring loops z
815
@y
A 6R X
B
S(f,t)
Fig. 1. Sketch of the rotating loop configuration of eq. (8). Only half of the loop is shown. The loop is held fixed at points A and B and rotates about the z axis.
string moves from B to A. _s(l, t) satisfies the flat space equations of motion for a string except at points A and B. At these points there are artificial boundary conditions which correspond to the string being held fixed. Fig. 1 shows only half the string. The second half is obtained by rotating s(l, t) by ~r about the z axis. Without loss of generality we can restrict our attention to the half loop. In calculating the vacuum decay amplitude for a string to emit a particle-antiparticle pair we follow the analysis of Srednicki and Theisen [7]. After change of variables to define a new real field 14,1 = a + ~
(lO)
the vacuum S-matrix element can be evaluated to linear order in perturbation theory. Let Is)_o~ denote the initial string at asymptotically early times and Is')oo the final string state at asymptotically late times. Then
= (s'~(kl)+(k2)+lSls) = hfd4x(s'lep2(x)lS)b(d,(kl)~p(kz)+l,kZ(x)10)f = Xfd'x(s'l,l,2(x)ls)
e'~,+k2~ ~ + ~(X2),
+ O(X z)
(11)
where subscripts b and f denote matrix elements restricted to the Hilbert space of ch and ~b respectively. In evaluating the final expectation value we make several assumptions which are only justified for local strings. We assume the string is very thin, i.e. that the field
R.H. Brandenberger / Cosmic string loops
816
falls off exponentially as a function of the distance from the string. Then q~Z(x) can be replaced by a delta function along the string s(l, t). We also neglect the change in the string configuration caused by the emission process, i.e.
(s'lq,2(x)l s)
= 0
if Is') =g bs).
(12)
Since the value of ff at the center of the string is a and the width of the string is w e can write
a ~ ~kol/2o-1,
(slqj2(x)ls) = [~o, 1/2 d183(x _ _s(l, t))02a2(1 -- $2) 1/2,
(13)
a - ~roa 1 / 2
where the final factor is required by reparametrization invariance of the string world sheet [11]. Hence the S-matrix element is
~(s+(kl)q~+(k2)ls)_~ --_ 2,),o1[ ~'-1/2 d l f d t e x p ( i ( k O + k O ) t - i ( k _ l a ~rw 1/2 X
+k_2)s(l,t))
,I
(1 - g2)1/2 + 0()~2).
(14)
For an oscillating string and neglecting back reaction there are points which reach the speed of light. These points give stationary points of the integrand and dominate the final answer, as discussed by Srednicki and Theisen [7]. If the back reaction is taken into account the maximal velocity of the string will be slightly less than the speed of light and the value of the integral in (14) will change by orders of magnitude. For a rotating loop there are no points on the string which reach the speed of light. Hence there are no stationary points of the integrand. The integral must be evaluated using the Riemann-Lebesgue lemma, and the answer will be a nonsingular function of the maximal velocity Uma~ (as long as Ureax < 1) and thus will change only infinitesimally if back reaction is included and changes Ureax infinitesimally. Infinite straight strings cannot decay by conservation of winding number. A consistency check on (14) is to apply it to such an infinite string
s(l,t)=(O,O,Z),
-m
(15)
In this case (14) vanishes by the Riemann-Lebesgue lemma. The consistency check is satisfied. For the rotating loop configuration (8) the S-matrix element (14) (henceforth denoted by s) becomes
s = XX o 1
dt
eiEt[ ~rR/2 dl exp - ikp cos oal - - c o s oat + ~- ,~R/2 k
× (1 - ~2cosZ(oal))l/2,
sin oat - ikfioal (16)
817
R.H. Brandenberger / Cosmicstring loops
where k = (k~ + my ~2~1/2 kx = k I + k 2 and E = k ° + k °. By the Riemann-Lebesgue / lemma the integral vanishes if k~6toR > 1 (17) or ko>l. As an order of magnitude estimate for (16) we then get s-
XXolfdt em' ~rRO(1 -
kp)O(1 - k~8)
(18)
and 2 2 At Isl 2 - X )t o -ff (~'R)20(1 - ko)O(1 - k~6),
(19)
where A t is the total time. F r o m (19) the total power Pv due to particle-antiparticle emission from the string is obtained by integrating over the available phase space: 1 1 f d3kl d3k2' 0 P v A t = 8 (2q7)6 J k 0 ~020 (kx +k°)lsl
2
81 1(2~) 6 x2X°2(~'R)2 [ f d k f d k ~ O ( 1 - 8 ( k f + k ~ ) )
x
f d2kl d2_k20(1 -
p[_kI +
_k21)
1
x
] I
(k 2 + k f 2 + mg) 1/2 (k_ 2 + k~2+ mg) 1/2
(20)
The total momenta k = k I + k 2 and k z = kf + k~ are constrained by the theta functions. The relative momenta K = k 1 - k 2 and K ~'= k f - k~ are not yet constrained. However, the particle-antiparticle emission process is a local process and the energy E of the emitted particles must be smaller than the energy which can be released by making the string locally straight, where locally means on a length related to E by the uncertainty principle [12]. Hence K and K z are constrained to be smaller than Em~x Emax ol/2R - 1/2. (21) - -
Now it is easy to estimate the order of magnitude of the phase space integral in (20), denoted here by I. I - 8-1/9- 2Emax - ~- lp- 2~1/2R - 1/2
(22)
Hence the total power from vacuum decay of the string into particle-antiparticle
R.H. Brandenberger / Cosmic string loops
818
pairs is Pv
1 1 ) 2 ) . - 2 ~ 2,~ - 2 R - 1,., 1 / 2 32 ()'2~r"2 . . . . 0 " ' p . . . . 1
o
- 1/2
1
. ,2X2Xo2R-2~-28-1(Ro) 1/2
(23)
32 (z~r)
where in the final step we have introduced dimensionless variables tS and 3 by P = ~3R,
3 = 3R.
(24)
and 3 can vary between 0 and 1 and measure the deformation of the loop. The perturbative decay rate (23) is smaller than the result by Srednicki and Theisen [7] by a factor of (Ra) -1/2. It is, however, larger than the original order of magnitude estimates. The factor ~-2g-1 depends on the maximal velocity of the loop. As we will argue in sect. 4 it can be as big a s ( R o ) 1/3.
3. Nonperturbative decay by cusp annihilation Cusps, points on the cosmic string loop with x_'(l) = 0 (where the prime denotes a derivative with respect to the string length parameter l) and which reach the speed of light, are generic to solutions of the string equations of motion. They are, however, artifacts of neglecting the nonzero width of the string. In this section we shall estimate the smoothing of the cusp when taking the finite thickness of the string into account. At a cusp two branches of the string are sufficiently close for microphysical forces to become important. Such interactions may lead to a substantial energy loss. We shall estimate the order of magnitude of this effect. Cusps arise once per oscillation period. The general expansion of the cosmic string solution about a cusp at ( t , / ) = (0,0) is [13]
[ t - ½(a 2 +/32 + 2/2)(½/3 + 12t)_a/3(t21+ 113) _x(t,l)=lla(t2+12)+/31t+3(1t3+12t)+~(t21+]13)
I
,
(25)
ylt+e(½'3+12')+~(t21+~13)
where a, /3 and y are constant coefficients of the order 1/R and 3, e, ~ and ~ are constants of the order 1/R 2. The width of the string is w - a-1. From (25) we can determine the length l c of the cusp region, defined as the region where Ix(0, 1) - _x(0, -1)1 ~
(26)
R.H. Brandenberger / Cosmic string loops
819
The result is lc - wl/3(a2f12 + ~2 + ~2)-1/6
- -
o_t/3R2/3.
(27)
We shall now assume that a fraction of the order 1 of the string mass in the cusp region evaporates as a consequence of microphysical interactions. This gives an estimate of the nonperturbative energy loss from the loop by this process. Since cusps arise periodically in each oscillation period the power PN of radiation from the loop is PN -- I'tlc/R - ° 2 ( ° R ) -1/3"
(28)
The above represents an upper bound on the energy loss by cusp annihilation. PN cannot increase by matter streaming into the cusp region from neighbouring segments at the string loop. From (25) it follows that the cusp region (measured for l - lc) persists for a time interval A t - o - l ( R o ) 1/3 .
(29)
The maximal amount of energy which can stream into the cusp region is therefore a E - ~ t a - l ( R o ) 1/3 ,
(30)
which is smaller than the mass already in the cusp by a factor ( R a ) -1/3 Outside of the cusp region defined by (27) points on the string never reach the speed of light. The maximal velocity can be obtained by considering (25). To lowest nontrivial order 1 - ½(a 2 + ~(t,l)
=
+ v 2 ) ( t : + t 2) at + fll
(31)
The maximal velocity is taken on at t = 0 at the edge of the cusp region, i.e. for l = l c. The result is Om~x -
1
- ½(alc) 2.
(32)
As in the introduction e shall denote the difference between the speed of light and Vmax. Then by (27) e - (oR) -2/3.
(33)
We shall use this result in the following section to estimate the perturbative power loss of an oscillating loop.
820
R.H. Brandenberger / Cosmic string loops
4. Back reaction effects on the decay of oscillating loops In sect. 2 we determined the perturbative power loss from a rotating loop which never reaches the speed of light. The answer was expressed in (23) as a function of the parameters t3 and ~ which determine the specific geometry of the loop. t~ and are related by t32 + g2 = 1,
(34)
so there is only one free parameter 8 which can vary between 0 and 1 . 8 determines the maximal velocity of the loop. From (8) we have
U m ~ = p = (1 --~2) 1/2.
(35)
For 8 << 1, Um~ approaches the speed of light. The velocity difference e is then approximately 1 ^2
(36)
and the power P from (23) can be written in terms of e p _ ~2~k0 2R -2( Ro )1/2_ 1/2
(37)
The idea of the following analysis is to apply (37) to oscillating loops by using back reaction estimates to determine e. A higher and a lower estimate for e shall be given. The first only takes into account the smoothing of cusps as a way of obtaining Vm~, < 1, the second is an attempt to include the dominant decay mode, gravitational radiation, into the back reaction analysis. We shall first discuss the lower estimate for e. The maximal velocity of a string loop will occur at or near cusps. As discussed in the previous section cusps will be smeared out by finite width effects. The maximal velocity outside the cusp region is given in (32). Hence ~ - (oR) 2/3 and by (37) p_
~.2~.o2R_lo(Ro)
1/6.
(38)
The above gives the upper estimate for the perturbative power loss of an oscillating loop. Several comments should be made. All processes in the cusp region have been included in the estimate for nonperturbative decay rate PN by cusp annihilation (see eq. (28)). We see that the perturbative power P is reduced by taking back reaction into account. However, provided there are no other mechanisms which prevent the formation of cusps, there is the possibility of an energy loss by cusp annihilation which far exceeds that by perturbative processes. Mechanisms which could prevent the formation of cusps even in the zero width limit of cosmic strings include interactions of the string which change its equations
821
R.H. Brandenberger / Cosmic string loops
of motion. Such effects have recently been discussed in the case of superconducting cosmic strings [13,14]. Here we shall consider nonsuperconducting cosmic strings and investigate the possible effect of gravitational radiation from the string on the dynamics of the loop. The power of gravitational radiation has been estimated analytically by Turok [10], using the rotating loop (8). In linear theory (but not using the quadrupole approximation which would be invalid for a rapidly rotating loop) the power Pc is given by
dP~
_k) _
dI2 ~r
(39)
_k)i2),
where T"'(¢o, k_) =/~
S-~- f e i'll
d i e -/-k-~(`.,>
(1L
7, )
~i~j- sis;
.
(40)
T = (2¢r)-lR is the period of rotation. Estimating the integrals for s(l, t) given by (8) we get Po - (4~r)2(G/t)# = 3,(G/z)~t,
(41)
with ~, - 10 2. Note that the result is independent of the maximal velocity and hence will be a good approximation for gravitational radiation from an oscillating loop (with back reaction taken into account). We shall now treat the oscillating loop as a system with one degree of freedom. This is an unrealistic idealization, but we hope it will be useful in an attempt to estimate the maximal effect of the back reaction, i.e. the largest value of e. The one degree of freedom loop is similar to a harmonic oscillator. Gravitational radiation from the loop corresponds to a damping term in the equation of motion of the harmonic oscillator and will lead to a decrease in the velocity of the oscillator after ¼ period. Similarly gravitational radiation will modify the equation of motion of the loop and will lead to a smaller maximal velocity. We shall estimate e in analogy with the harmonic oscillator example. Consider a harmonic oscillator m.~ + fx = O,
(42)
with f / m = ¢o2= R - 2 and with amplitude A = R. The maximal velocity of the undamped oscillation is 1. Now we introduce a damping term + F2 + to2x = 0
(43)
and fix F by demanding that the resulting energy loss per unit time P equal the
822
R.H. Brandenberger / Cosmic string loops
power Po from gravitational radiation for a loop. We get !
P = ½rm - v(GtDt~,
(44)
r= 2~(Gt~)~.
(45)
and with m = f l l ~ R
1 -1 is The velocity of the damped oscillator after a quarter cycle t = uro~ Yc - 1 - ½Ft = 1 - ¼F~rw 1
=
1 - e,
(46)
with 7 e - ~G/~.
(47)
Eq. (47) represents our estimate for the velocity reduction as a consequence of the back reaction effects due to gravitational radiation. Besides the caveats already mentioned there is another difficulty in the above argument. For the harmonic oscillator the frequency remains unchanged as the amplitude decreases, for the loop the frequency w is inversely proportional to R. Since the decrease in amplitude over ¼ oscillation period is negligible, this difficulty should be unimportant. If gravitational radiation indeed has the maximal effect estimated above, then the perturbative power P by particle-antiparticle emission from the string is p _ X2Xo 2R -2( R o )1/2( r _ l~/a~ ) -1
(48)
and thus exceeds the result for a rotating loop only by a factor ( f l - l y G # ) - i l0 s, small compared to the difference between (23) and (38). The main result, however, is that independent of whether the higher or lower estimate for Vm~x holds, the resulting perturbative energy loss from oscillating loops is smaller than what is obtained without including back reaction, and much smaller than the energy loss due to gravitational radiation. -
5. Thermal decay: a classical analysis
So far we estimated the perturbative energy loss by particle-antiparticle emission from a string in a vacuum. In a cosmological context the string is in a thermal bath which includes ~ particles. Thermal effects may change the decay rate. In this section we present a classical analysis of the thermal decay of the string. If the ~k particle is a boson then we expect the decay rate to be amplified by stimulated emission compared to the vacuum decay rate. A quantum analysis of the decay for both bosonic and fermionic qJ fields will be presented elsewhere [15].
R.H. Brandenberger / Cosmicstring loops
823
The analysis is based on a perturbative Green function method. We consider the production of # particles in an external classical ~ field corresponding to the rotating string of eq. (8). In the classical limit, ~ particle production corresponds to an excitation of the # field. The equation of motion for a bosonic ~ field coupled to the complex scalar field via the interaction lagrangian (5) is (using the change of variables (10)) ~ - 17'2++ (m 2 + ~.(o + dp)2)~ = O.
(49)
The ha 2 term should be viewed as a renormalization of the mass of ~. From now on we shall denote by m 2 the combination m 2 + ~0 2. We shall split ~ into an unperturbed solution ~0 and a perturbation #x. = #0 +
(50)
q'o satisfies the homogeneous equation ~'0 - V'2#o + m2~bo= 0
(51)
and ~i the linearized perturbation equation ~ i - W2~bi+ m2~i = -X(2q~a + d#2) ~0-
(52)
q~(_x, t) can be determined using the retarded Green function Gm of the operator 02 -- V 2 "-I-m 2
1 f d4p e -ip(t'x) Gin(x-,t)- (2~r)4 ret ~-~-- ~--2 "
(53)
The subscript on the integral denotes that the contour of the P0 integration is to be taken above the two poles (fig. 2). ~bi(_x, t) is given by •i(x,/)
=
-~fdt'd3x_'G,,(x_-x_',t-t')d/o(X_',t')(2oep+ep2).
(54)
In the classical analysis q'0(-, x ! t') is homogeneous and its value is determined by equating scalar field energy density and thermal energy density 1 . 2R2 ~,n V 0 -_ -_ 1 ~ 2 T 4 ,
or
(55)
824
R.H. Brandenberger / Cosmic string loops
Po
,,g,
_(p2+m2if2
( p2+ m 2 )1'2
Fig. 2. Location of the poles for the P0 integrationin the retarded Green function G,,.
As in the vacuum decay calculation it is a consistency condition to check that for a long straight string there is not net energy loss. The calculation is sketched in the appendix. We find that ~bi(_x,t) is independent of time and exponentially suppressed as the distance from the core of the string increases. Hence there is no net energy flux from the string. We now outline the calculation for the rotating string configuration s(l, t) in (8). For local strings we replace q~(_x,t) by a delta function q,(_x, t) =
fd163(x_-s_(l,t))oa2(1-~2) 1/2,
(2oep+eP2)(x,t)=3oZazfd183(x_-s(l,t))(1-g2) 1/2.
(57) (58)
Hence
q~(x, t) = -p f d3x'dt'd4p e-ip(t-'"~--~-')f
(59)
where we have combined all the constants into
P
3 ~ O 1 (qr2 ]1/2 T2 (2'n")4 ~ - ] 7
(60)
The d3x ' integral is trivial. The first step is to perform the P0 integral. For t - t' > 0 we must close the contour in the lower half plane (fig. 2), and the result is 1
~k~(x-,t)= 27rPf dldrd3_p (p2 + m2)l/2exp{ i_p(x_-s_(l,t)) } ×sin((p2+m2)l/2(t-t')}O(t-t')(1-g2(l)).
(61)
For t - t' < 0 the contour is closed in the upper half plane and the integral vanishes as it must for a retarded Green function solution.
R.H. Brandenberger / Cosmic string loops
825
Next we evaluate the dl integral using the Riemann-Lebesgue lemma. If (compare with (17)) IpA> 8 -1 or p=
( p 2 ..[_py2)l/2 > p - 1
(62)
we have an integral of a rapidly oscillating function which by the above mentioned lemma vanishes. Hence etp-x
t) - 2=e f d3p (p2 +m2)'/2
f~oodt'sin{(p2 +m2)l/2(t- r) }
×wRO(8 - 1 - ]p~l)O(p-l-p).
(63)
The third step is to perform the dp~ and d2_p integrals. Since the radius R of the loop is much larger than the string width a - o-1 and since the renormalized mass m is X1/20 or larger, the approximation R-1 < m will be well justified. Using this approximation we obtain 1
0
m
-oo
¢i(x,t)-Z~rRP-- f
dt'sin(mt')nfin(8-1, lzl-:)min{p-2, r-2},
(64)
with r = (x 2 +y2)1/2. If At is the total time interval, then in the near field limit p - 1 < r - : and 8 -1 < [z[-1
lm2,L2t x A t ) - ½(2~r)2p28- 2p-4m-lAtR 2 '/" I \ - - ,
(65)
The total energy in the ~ field ( - total energy radiated from the string by particle-antiparticle emission) is obtained by multiplying (65) by the near field volume which is of the order R 3
E ( At ) = PTAt
7/.2 T 4
-
9)k2)ko2(2~)-6_ _ __ R_18_ 2p_4 A t 30 m 3
(66)
(see eq. (24)). Thus the power PT of thermal decay is T4
PT -- ~2)k 0 2 m-~R - le- x,
(67)
where like in sects. 3 and 4 e is the deviation of the maximal velocity from the speed of light.
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R.H. Brandenberger / Cosmic string loops
The power from stimulated emission at finite temperature is to be compared with the zero temperature perturbative power P from (37). Taking m - o and e given by (33) we get
(] 4
5t4( ompl/5J6
(68)
where in the final step we have evaluated the ratio for loops with radius of the order of the horizon. We conclude that at temperatures close to the critical temperature of the phase transition stimulated emission will dominate. For loops which could be important for cosmological questions today, stimulated emission is negligible. 6. Conclusions The nongravitational decay of cosmic string loops has been discussed. The analysis applies to nonsuperconducting strings which result from spontaneous breaking of a local symmetry. Both the perturbative decay rate by particle-antiparticle emission from the loop and the nonperturbative decay rate from cusp annihilation were considered. The perturbative decay rate was calculated at zero and at finite temperature using an order of magnitude estimate of the back reaction. We conclude that back reaction significantly reduces the perturbative decay rates. By how much depends on the maximal velocity of the loop. A low estimate for the suppression factor can be obtained if we assume that back reaction from gravitational radiation does not prevent cusps and that finite width effects are the only effects which smoothen the singularities. If gravitational radiation acts as a damping term in the string equation of motion then the suppression factor is much larger. If back reaction effects do not prevent cusps, then the energy loss due to cusp annihilation is much larger than the perturbative decay rates. This indicates that in general nonperturbative processes are important. However, both perturbative and nonperturbative decay rates are much smaller than the decay rate due to gravitational radiation. Stimulated emission at finite temperature is important only at temperatures close to the critical temperature of the phase transition. The computations of the perturbative decay rates were performed in a simple toy model in which a gauge singlet scalar field q, is coupled to the fields of the cosmic string model. If ~b is a nongauge singlet scalar field the interaction lagrangian contains more terms, in particular terms coupling the gauge fields A~ to ~. The At,A~2 terms in L I can be analyzed along the same lines as discussed in this paper. The overall effect will be to multiply all the decay rates by a constant of the order 1. Fermion pair emission is more complicated and will be discussed in ref. [15]. Our analysis does not apply to superconducting strings. For these, electric currents can flow along the string loop. This can give rise to many electromagnetic decay channels [16].
R.H. Brandenberger/ Cosmicstringloops
827
Many points discussed in this paper were raised in stimulating conversations with A. Matheson, M. Srednicki, S. Theisen and N. Turok. I wish to thank them for their help. I am grateful to S. Hawking, C. T h o m p s o n and E. Witten for useful comments on the first draft of the manuscript. Part of the work was done at the 1986 Aspen C e n t e r for Physics Workshop on Cosmic Strings and Inflation.
Appendix A. CONSISTENCY CHECK A self consistency check shows that the approximation scheme of sect. 5 gives vanishing energy flux from an infinite straight string. F o r an infinite straight string in a local gauge theory q~2(x, y, z, t) =
a2o28(x)6(y).
(A.1)
Inserting into (54) we get
e-ip(x_-x,,t-t,) m2
q~,(x_, t) = - p f d3x'dt'd4p ~(x')~(y') p2 _p2 -
_
=(2~r)2pfdpxdpye-ip~xe-ip'Y(p~+p2+m2)
-1
(A.2)
W i t h o u t loss of generality we can choose x = 0 and y > 0. Then
1
~i(x,t)=(2~r)3Pfo°°dpx[pZ+mZ]l/ze-YtP2+m211/2.
(A.3)
This is time independent and decays exponentially with y. Thus there is no energy flux from the static infinite string.
References [1] T. Kibble, J. Phys. A9 (1976) 1387 [2] Ya Zel'dovich, Mon. Not. R. Astr. Soc. 192 (1980) 663; A. Vilenkin, Phys. Rev. Lett. 46 (1981) 1169 [3] A. Vilenkin, Phys. Reports 121 (1985) 263 [4] N. Turok, Phys. Lett. 123B (1983) 387; A. Vilenkin and Q. Shaft, Phys. Rev. Lett. 51 (1983) 1716; J. Silk and A. Vilenkin, Phys. Rev. Lett. 53 (1984) 1700 [5] N. Turok and R. Brandenberger, Phys. Rev. D33 (1986) 2175; H. Sato, Prog. Theor. Phys. 75 (1986) 1342; A. Stebbins, Ap. J. (Letters) 303 (1986) L21
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R.H. Brandenberger / Cosmic string loops
[6] T. Vachaspati, A. Everett and A. Vilenkin, Phys. Rev. D30 (1984) 2046 [7] M. Srednicki and S. Theisen, Phys. Lett. 189B (1987) 397 [8] T. Vachaspati and A. Vilenkin, Phys. Rev. D31 (1985) 3052; N. Turok, Nucl. Phys. B242 (1984) 520 [9] R. Davis, Phys. Rev. D32 (1985) 3172 [10] N. Turok, in ref. [8] [11] A. Vilenkin, private communication [12] A. Matheson, private communication; R. Brandenberger and A. Matheson, in ref. [15] [13] D. Spergel, T. Piran and J. Goodman, Nucl. Phys B291 (1987) 847; A. Vilenkin and T. Vachaspati, Tufts Univ. preprint (1986) [14] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett. 58 (1987) 1910 [15] R. Brandenberger and A. Matheson, Brookhaven preprint (1987) [16] E. Witten, Nucl. Phys. B249 (1985) 557; E. Chudnovsky, G. Field, D. Spergel and A. Vilenkin, Phys. Rev. D34 (1986) 944; J. Ostriker, C. Thompson and E. Witten, Phys. Lett. 181B (1986) 243; C. Hill, D. Schramm and T. Walker, Fermilab preprint (1986)