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V O L U M E 59, N U M B E R 12 PHYSICAL REVIEW LETTERS 21 SEPTEMBER 1987
Gravitational Particle Production in the Formation of Cosmic Strings
Leonard Parker Department of Physics, University of Wisconsin- Milwaukee, Milwaukee, Wisconsin 53201
(Received 1 June 1987)
The formation of cosmic strings causes a change in the metric of space-time. This change in the gravitational field creates particles, including photons, neutrinos, and gravitons. An estimate is made of the power and total energy output per unit length of string formed. For a grand-unification-scale string the total energy output is about 10^^ ergs c m ~ \ with a power output during the time of formation of about 10^^ ergs sec~* cm ~'.
PACS numbers: 98.80.Cq, 04.60.+n
Cosmic strings form through a phase transition in the early universe according to a wide class of grand-unified theories (GUT's) . ^ Cosmic strings characterized by the GUT scale can act as seeds for galaxy formation and can account for various properties of the observed distribution of galaxies. '̂̂ In a subset of these theories, the cosmic strings formed are superconducting/ Oscillations of the strings give rise to gravitational radiation, and in the case of superconducting strings electromagnetic radiation occurs, as well as a spewing out of particles when the superconducting current becomes suflfi-ciently large.
Here a different process will be investigated. This process is the creation of particles and radiation by the changing gravitational field during formation of cosmic strings. To my knowledge this has not been previously discussed. I will show that this mechanism produces a very intense burst of energy. I will treat the problem by calculating the Bogoliubov transformation produced by the changing gravitational field, a method first introduced in order to calculate particle creation by the expansion of the universe.^
The metric in the vicinity of a cosmic string has the form^
ds^=dt^-dz^-dr^-rH\-a/27t]^de\ (1)
where 0^ O^IK and a is the deficit angle of the conical singularity which constitutes the core of the string. The interior solution,^ which corresponds to rounding of the vertex of the cone, is not important for the present discussion. The deficit angle is related to the mass per unit length jj of the string (in units with h=c = \) by^ ailn) ~^ ^AjiG, where G is the gravitational constant.
Strings form as the result of a phase transition occurring at a cosmic time t in the early universe. The mean spacing between strings at time of formation is of the order of the correlation length ^, which can be no larger
than the particle horizon size at time of formation and is probably a factor of 10 ~^ or 10 "^ smaller. Thus <^=/r, where 1 0 ~ ^ < / < 2 (the horizon is 2t in a radiation-dominated universe).
Consider a cylindrical region of space of radius
R-^^-ft. (2)
surrounding along its axis a length L of string just after formation. Within that region the metric is approximately given by Eq. (1). Before string formation begins, the metric in that region can be approximated by the Minkowski metric. In order to estimate the particle production by the gravitational effect of string formation, I approximate the metric in the cylindrical region by
ds^=dt^- •dz^-dr^-r^a^(t)de\ (3)
where before the phase transition, a ( / ) = 1 , and after the phase transition, a(t) = l — alliz. In general, a metric of the above form is not conformally flat, having a nonzero Weyl tensor. Therefore, one would expect particles of all types, including neutrinos, photons, and gravitons, to be created by this metric, with energies and masses limited only by the rate of change of the metric during the stage of string formation.
To determine what is produced, I impose vanishing boundary conditions at r = /? and periodic boundary conditions in z with period L on the quantum fields. Then, the particles created as the string forms are effectively reflected from the walls of the cylinder, remaining inside so that their number and energy can be found after the process has ceased.
To estimate the number and energy produced per field, I consider the simplest case of a massless Hermitian scalar field 0. The coupling to the scalar curvature is not important since the metric is not conformally flat. For simplicity, we take minimal coupling, 0 0 = 0 , or for this metric
a ^ it)dt[a(t)dt(l)] - dh - r ~^ dr(r dr(l)) - a ~^(t)r -^ dU ''^'O. (4)
Although this equation is not separable, we can obtain a rough estimate of the production by means of the sudden approximation. Of course, it actually takes a finite time interval At for the string to form, that is, for the deficit angle
© 1987 The American Physical Society 1369
V O L U M E 59, N U M B E R 12 P H Y S I C A L R E V I E W L E T T E R S 21 SEPTEMBER 1987
to grow from 0 to a. Because the string actually forms in a finite time interval, the production of particles in modes having energies and momenta large with respect to 1/A/ will be suppressed.
In the sudden approximation, we take
aO) = 1, for ^ < /o;
1 —alln, for t > to.
The field 0 is taken to be continuous at time ^o- The correct boundary condition on its time derivative, which then follows from Eq. (4) or from conservation of the Klein-Gordon scalar product, is
( ^ | , - = ( l - a / 2 ; r ) 0 | , + .
In the initial flat space-time, one can write
0 ( x , / ) = X , U y / / ( x , / ) + ^ / / / ( x , / ) ] ,
where j = {n,m,s) with « , m = 0 , ± 1 , ± 2 , = 1,2,3, . . . . Here
and s
can similarly make the expansion
where
with Vm = \m\/(l-a/27r) and N2='V2~^^^(2W,) ~^^' ^[JU(Ws^-k^)'^^R)]-\ V2 = (l-a/27r)7rR'L^ and k=2Kn/L, The values of Ws are determined by the boundary condition
As before, the gj are orthonormal with respect to conserved scalar product and the Bj are annihilation operators.
Imposition of the continuity conditions (with r o = 0 for simplicity) leads to the following relation between final and initial creation and annihilation operators:
c'c " r ^ — „
ff^si ikz imt
• 1 / 2 /
| ( ( c y^2) l /2^) , where
=Z.'K ./?* ) (5)
= KR^L, and k—lnnlL. The values of cOs are determined by the boundary condition
/ |™|((ft)|-A:2)'/2/?)=0.
The fj are orthonormal with respect to the conserved scalar product and the Aj are annihilation operators.
In the final space-time with conical deficit angle a, one
and
Here
W.
w. CO.
1/2
1/2
a 2K
(D^
1 -2;r
Ws
W.
1/2
1/2
(6)
^^)'/V). I,.,-{\-a/2ny^HR^J\^\{{o)}-k^yf^R)Jl^i{W}-k^y^^R)]-'j\
The average number produced in mode n,m,s from the initial vacuum is
^nms ~'Z^s'Ps'sPs's'>
and the average energy produced is
^ nms '^ s^^ nms-
The zero-point energy, which gives rise to a Casimir energy associated with a string, has been omitted, as it has already been calculated,^ and my concern here is with the production of particles.
Up to now the results are exact for the model considered. For a string characterized by the GUT scale, one has fi^=^ \0~^G~^ and a/2;r ==:̂ 4 x 10 ~ ̂ . Therefore, we can work to lowest order in a/2K. Since the part of ^^ -̂ multiplying I^>^ is already of order a/27r, one needs only the leading term in the expansion of 7 -̂̂ , namely l^'s ^^ J ^s's- Working to lowest order and using an approximation for the zero of the Bessel functions, one then finds that
P^,^(n,m)^(a/27r){l{ \ m \ (s-\- { \m \ - ^ )]l4n^RyL^-h (s-h { \ m y] iH^.v (7)
From this it is clear that, for the case of instantaneous formation, the spectrum does not fall off rapidly enough to give finite total number and energy when summed over modes. As noted earlier, the effect of the finite actual time At of formation is to make the spectrum fall rapidly for momenta and energies much larger than 1/A/.
The time interval At for formation depends on how
long it takes for the large fluctuations near the transition temperature to die down so that the string can freeze out. For strings forming at the GUT time, this time interval is thought to be of the order of the cosmic time t (it can be no longer and gives the most conservative estimate). Thus, I take At ^ t.
To make a rough estimate of the energy produced,
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V O L U M E 59, N U M B E R 12 PHYSICAL REVIEW LETTERS 21 SEPTEMBER 1987
consider the case in which L ^ (̂ . This should approximate the plausible situation in which the string forms in a connected set of straight segments, each of the order of the correlation length. In this case, edge effects will cause a significant deviation of the gravitational field in the cylinder from the form in Eqs. (1) and (3). The probable effect of these gravitational inhomogeneities would be to increase the particle creation above our present estimate. With R and L of order ^=ft and At ==== t, the production of particles in modes above the lowest mode is suppressed because the corresponding energies and momenta are large with respect to \/t. When / is much smaller than 1, even production in the lowest mode would be suppressed. I will take f^^ 1 and estimate the energy produced inside the cylinder by retaining only the contribution of the lowest mode, which has the quantum numbers m = 0 , s = \, and n=0,
When m = 0 , one has W^^cosy since both are given by the condition that JQHW,^-k^)^^^R) =0. It follows from Eq. (6) that p^.^ ~ — aiSjr) ~^S^>^. In addition, for A2=0 (and hence k=0) one has W\ =2.40//?. Therefore, taking into account only the lowest mode and assuming that JV types of particles are produced, we have
E^0Ma/2n)^JVR-\ (8)
and the energy per unit length produced by this mechanism as the string forms is
E/L^0.6(a/2K)^JVt -2 (9)
where we have used L^^ R^=^ t. If / had not earlier been set equal to 1, a factor of / ~ ^ would appear in (9). However, because for small / even the energy of the lowest mode becomes large with respect to I/A/, this factor o f /~^ in (9) would be counteracted by the suppression of production caused by the finite time interval for formation of the string. Thus, it is plausible that the value of E/L when / is of order 10 ~^ does not differ much from the value given in Eq. (9).
For a GUT-scale string forming at the GUT time, /— 10 ~^^ sec, with ji and a as given earlier, and with J V ~ 10^, one has
£ : / L ^ 3 x 1 0 ' " ergs cm (10)
This is small with respect to the string's mass per unit length, fj. ~ 10"*̂ ergs c m ~ ^ The power output per unit length during formation is
E/L A/ ==̂ 3 X 10^^ ergs sec (11)
Thus, the formation of cosmic strings would be accompanied by an exceedingly intense burst of particles and radiation. By way of comparison, the power emitted in gravitational radiation due to oscillations of a closed string is independent of its length,^ and is of order 10"*̂ ergs sec ~ ^ This equals the power emitted during forma
tion of 10 ~^^ cm of string. (However, the gravitational radiation by the oscillating loop lasts for a much longer time and thus can significantly deplete the total energy of the loop.) The short time during which this burst occurs limits the magnitude of the resulting average energy density it produces.
The average energy density produced at time / by this mechanism is
p ^ E/nLR 2 — 1 X 10^^ ergs cm - 3 (12)
This is small with respect to the energy density of order 10^^^ ergs cm~^ which is already present in a radiation-dominated universe at the GUT time. However, if, as usually postulated,^ string formation occurs near the end of the inflationary stage in an inflationary model of the universe, when the ambient radiation density is low, the radiation density created by string formation would be significant relative to that already present.
If a plausible mechanism were to be found which causes GUT-scale cosmic strings to form at a later cosmic time, then the possibly observable consequences of the accompanying burst of particles and radiation would have to be considered. Gravitational particle production may also be significant when strings cross and reconnect, a process which occurs as strings evolve.
It is worth noting that once one knows that E/L is of order ia/ln)'^ and R and Ar are of order i, the result for E/L in Eq. (9) is determined to within a few orders of magnitude on dimensional grounds. For example, a similar calculation for a long segment of string with L^t (which involves summation over n) yields a result for E/L only 1 order of magnitude smaller.
Finally, we note that if one extrapolates the results derived here, then as the parameters characterizing a cosmic string approach the Planck scale, the radiation produced during formation would approach the total energy of the string. In such a case, reaction back would be important, and interesting new phenomena involving the topological obstruction to evaporation of the string would be expected.
It is a pleasure to thank J. D. Bekenstein, L. H. Ford, and A. Vilenkin for valuable discussions, and the National Science Foundation for support under Grant No. PHY-86-03173.
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