Gravitational radiation from cosmic strings

Volume 107B, number 1,2 PHYSICS LETTERS 3 December 1981

GRAVITATIONAL RADIATION FROM COSMIC STRINGS

Alexander VILENKIN Physics Department, Tufts University, Medford, MA 02155, USA

Received 23 March 1981 Revised manuscript received 18 May 1981

It has been recently shown that vacuum strings produced at the grand unification phase transition can generate density fluctuations sufficient to explain the galaxy formation. Here I estimate the energy density and the spectrum of the background gravitational radiation produced by the strings.

It is now generally believed that elementary particles and their interactions can be described by gauge theories with spontaneous symmetry breaking. In such theories the symmetry can be restored [ 1 ] at temperatures greater than some critical temperature, T c. In the standard, hot cosmological model, a phase transition will occur when the universe cools below T c. This phase transition can result in the develop- ment of a vacuum structure, which can take the form of vacuum domain walls, strings, or monopoles, de- pending on the topology of the gauge group [2]. I f the symmetry is broken in several steps, then one has a series of phase transitions, each of which can produce its own vacuum structure.

It has been recently argued [3,4] that vacuum strings produced at (or near) the grand unification phase transition can generate cosmological density fluctuations sufficient to explain the galaxy formation. In the scenario of ref. [4], the galaxies condensate around oscillating closed loops of string, while the loops lose their energy by gravitational radiation and gradually disappear. Other possibilities have also been suggested [3,4]. In the present paper we shall find the energy density and the spectrum of the background gravitational radiation produced by the strings.

The evolution of cosmic strings has been discussed in refs. [2 -7 ] . Here, we follow the treatment of ref. [4]. At the time of formation, the strings have the shape of brownian trajectories with a persistence length of order or smaller than the horizon (t). Ten-

sion in convoluted strings results in oscillations on scales smaller than t. These oscillations are damped by various dissipation mechanisms, and the strings gradually straighten out.

The effective damping time due to string-particle scattering is [5] t d "-" p/po, where p is the linear mass density of the strings, a is the string-particle cross- section, and p is the energy density of the universe, p ~ (zr2/15) N T 4 "~ 3/321rGt 2. (Here T is the temperature and N ~ 102 is the equilibrium number of particle species. I use the system of units in which ~ = c = k = 1). Everett [7] has shown that, for particles of energy T, a ~ zr 2 (Till 2 m/T) -1 , where m is the mass scale associated with the strings (m -1 is the width of the strings). The damping time becomes of order t at t "~ t , , where t , can be found from

t , ~ N 1/2 (G/a)- 2 tplln-4 [N 1/4 (Gm 2 t , / tp l ) 1/2 ]

and tpl is the Planck time. According to ref. [4], the strings responsible for the galaxy formation have m "~ 3 X 1015 GeV,/a "- 1023 g/cm, G/a ~ 10 -5 and t ,

10 -35 s. At t > t , the dominant straightening mechanisms are the cosmological redshifting of the oscillations and the loop formation by intersecting strings. Approximately one closed loop per horizon scale is expected to form during the interval At ~ t. This can be written as

dn/d t ~ ~t -4 , (1)

where n is the number density of the loops and ~ ~< 1

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is the efficiency parameter which depends in particu- lar on the probability of "changing partners" by intersecting strings. Here we assume that ~ ~ 1. (The case of ~ ~ 1 will be discussed below.) The size of the loops formed at time t is also of order t.

From time to time the loops can self-intersect and break into smaller pieces. The frequency of such self- intersections is difficult to estimate. If they are fre- quent, then the loops rapidly deteriorate, eventually decaying into relativistic particles. Here we shall assume that self-intersections are rare enough, so that the dominant energy-loss mechanism is the gravitational radiation. (The opposite limiting case is briefly discussed at the end of the paper.) The energy loss by gravitational radiation for an oscillating loop of size R is of the order

d M / d t ~ - G M 2 R 460 6 ~ - G/.t 2 , (2)

where M ~ / sR is the mass of the loop and ~ is the characteristic frequency of oscillations, 60 ~ R - 1 . The lifetime of the loop is

r ~ MI d M / d t I - i ~ ( G # ) - I R . (3)

Let us now estimate the energy of the gravitational radiation produced by the loops formed in the time interval t 1 < t < t i + dt 1 . We shall do a simple-minded calculation omitting all numerical factors and assum- ing that the loops radiate all their energy at the same frequency 60 ~ t i -1 and disappear at t 2 ~ (G#) -1 t 1. (In fact, the frequency increases as the loops lose their mass, but it can be shown that this effect does not change the order-of-magnitude estimations of the present paper.) The density of the loops at formation (t

t l ) is dn 1 ~ t i -4 dt 1, and their density at t' (t i < t ' < t2) in the radiation era is

dn' ~ ( t l / t ' ) 3[2 dn 1 ~ t { 5 [ 2 t ' - 3 / 2 dt i . (4)

The energy density radiated by the loops during the time interval d t ' is

de' ~ Gla2dn' d t ' . (5)

It is convenient to refer all quantities to the end of the radiation era which we take to be the time of equal matter and radiation energy densities, teq. The cosmic scale factor is a ~ t 1/2 at t < teq and a cc t2/3 at t > teq. The redshifted energy density at t ~ teq corresponding to eq. (5) is

deeq ~ ( t ' / t eq) 2 de ' . (6)

The frequency and the frequency range at teq are, respectively,

6Oeq ~ ( t , / teq) 1/2 t~-I , (7)

d60eq ~ ti -1 ( t ' teq) -1 /2 d t ' . (8)

From eqs. ( 4 ) - ( 8 ) we obtain

deeq/dweq ~ G#2cO2q(tl/teq) 1/2 dt I . (9)

The contribution of all loops formed at different times can be found by integrating eq. (9) over t 1 . To find the limits of integration, we note that the loops formed at t ~ t 1 give rise to ~Oe~ in the range (t l teq) -1/2 < 60eq < (Glatlteq~ -1/2" This can be re- written as

2 -1 < (G//teq602q) -1 (10) (teq60eq) < t 1

Integrating (9) over the interval (10), we obtain

(de/d60)e q "~/a(G/a) - 1/2 te 2 60eql. (11)

In the derivation of eq. (1 1) we assumed that both t I and t 2 are smaller than teq and greater than t , (for t < t , , friction is important and eqs. ( 1 ) - ( 3 ) may not be valid).

This gives

(G/ateq) -1 < 60eq < ( t e q t * ) - l / 2 • (12)

Noticing that the electromagnetic radiation energy density, e(~ ) is

e('r)eq ~ (Gt2q) -1 (13)

we can rewrite eq. (1 1) as

e(~,), -1 (14) (de/d60)eq ~ (G/a) 1/2 eq --eq •

Since the quantity (60/e(7)). (de/d60) does not change in the course of expansion, the present-day spectrum of the gravitational radiation is obtained from eq. (14) by dropping the subscripts "eq":

de/d60 ~ (Gla) 1/2 e(~) 60-1 . (1 5)

It is convenient to introduce the quantity

~-~g(60) ~ ~jp~i de/d60 ~ (Gfl) 1/2 ~2 7 , (16)

where ~2. r = e(7)/p c and Pc is the critical density. ~2g(60) is the energy density of the gravitational radiation, in terms of the critical density, in the frequency interval A60 ~ w.

For a numerical estimation we shall use the values

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G / ~ 1 0 -5 ,~2. r ~ 1 0 -4 , t e q ~ 5 X 1 0 1 2 s , t ~ 3 X 1017 s. Then,

~2g(co)~3 X 10 - 7 , 1 0 - 1 1 s - 1 < c o < 1 0 8 s -1 ,

(17)

where the frequency range is determined from eq. (12). Loops formed at t 1 > Glateq contribute to the low frequency part of the spectrum, co < 10 -11 s -1 . The calculation of this contribution is slightly more elabo- rate, since one has to consider separately the cases t I < t ' < teq , teq < t ' < t2, and teq < t 1 < t. The final result is

~2g(co) ~ 10 -12 co -1/2 ,

3X 10 - 1 3 s - 1 < 6 o < 1 0 - 1 1 s -1 ;

~2g(co) ~ 3 × 106co,

10 -17 s < c o < 3 X 10 -13 s -1 . (18)

The total energy density of the gravitational radiation (in terms of Pc) is

~g = f~2g(co)co -1 dco ~ 10 -5 . (19)

For the cosmological scenario of ref. [4] it is important that the efficiency of the loop formation is high (~ ~ 1 in eq. (1)) and that the loops live suffi- ciently long ( r >~ R ). We shall now briefly discuss what happens if these conditions are not satisfied.

I f ~ is so small that the loop formation can be neg- lected, then the only straightening mechanism at t > t , is the cosmological redshift of oscillations. How- ever, this mechanism is very effective [6] and the characteristic scale of the system of strings at time t is ~ t. The density fluctuations due to strings on the scales L ~ t are 8p/p ~ G/~ (the fluctuations on greater scales are smaller). These fluctuations stay there for about one expansion time and then shift to larger scales, but this is probably sufficient to produce matter density fluctuations of a comparable magnitude. For the galaxy formation, we need 8p/p ~ 10 -3 . This corresponds to G/~ ~ 10 -3 , m ~ 3 X 1016 GeV, and t , ~ 4 X 10 -39 s. In this scenario, the energy density radiated by the strings from time t ' to t ' + d r ' is

de' ~ Gla2t'- 3dt '

and the characteristic frequency is co' ~ 1/t'. A calcu-

lation similar to that leading to eq. (16) gives

~g(co) "" (Gbt)2~.), "" 10 -10 (20)

in the frequency range 10 -16 s -1 < co < 109 s -1 , and

~'2g ~ 10 -8 . (21)

Finally, let us consider the case when ~ ~ 1, but the loops rapidly deteriorate because of self-intersections. For definiteness, we shall make an extreme as- sumption that a loop makes, on average, only a few oscillations before it self-intersects and breaks into smaller pieces. When the size of the pieces becomes ~ m - 1 , they decay into relativistic particles. (This can give rise to interesting astrophysics effects which will be discussed elsewhere.) In this case a loop of mass M produces gravitational radiation in the range la/M < co < m with the spectrum dE/dco ~ GlaM/co. The lifetime of the loop is r ~M/la and the total energy radiated in the form of gravitational waves is E

GlaMln(Mm/la) "~ M. Like in the previous case, for the galaxy formation we need G/~ ~ 10-3 . After a simple calculation, we find

~2g(W) ~ (G/.02 ~.rln(co t 2/3 telq/3)

10 -10 ln(1016 co), (22)

with 10 -16 s -1 < co < 109 s -1 , and

~2g ~ 3 × 10 -7 . (23)

All equations ( 2 0 ) - ( 2 3 ) include only the gravitational radiation produced at t . < t ' < teq.

The observational limits on the gravitational radiation density in various parts of the spectrum have been recently discussed in refs. [8,9]. The most strin- gent limit comes from cosmological nucleosynthesis considerations and implies ~2g < ~27, where ~2g in- cludes only the gravitons generated before the nucleosynthesis. All other limits are weaker than ~g < 0.01. It is clear from eq. (19) that all these requirements are satisfied, and thus the string model in consistent with the existing constraints on ~2g(W).

It is interesting to compare eqs. (17), (18) with the gravitational radiation background which can be produced by other sources. Carr [9] has recently discussed the gravitational waves generated in various scenarios of pregalactic and postgalactic black hole formation. He finds that, for the black hole density

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~'2BH ~ 1 and the efficiency of gravitational wave gen- eration e ~ 0.1, the quantity ~'2g(6o) can be as large as 1 0 - 2 - 1 0 - 4 in the range 10 , 7 s ±1 < 6o < 103 s -1 . However, for a less optimistic choice of parameters, ~2g(6o) can well be smaller than the value 3 X 10-7 of eq. (17). The black-hole-generated background in other parts of the spectrum is much smaller than that produced by strings, even for the optimal choice of parameters.

The characteristic shape of the spectrum (17) (I2g(6o) = const, in a wide frequency range) can, in principle, be used for an experimental test of the string scenario. The most promising method of detect- ing the low-frequency gravitational background radiation is the Doppler tracking of the interplanetary spacecraft. Various sources of noise and the accuracy of the method have been discussed by Bertotti and Carr [10]. They fred that at present one cannot hope to detect gZg(6o) smaller than 10 -5 at co "" 10 -7 s. (The accuracy is smaller at higher frequencies. At low-

er frequencies the accuracy can be greater, but the time of measurement becomes very long.) They have also indicated some ways of improving the Doppler tracking technology, and it does not seem impossible that gZg(6o) ~ 10 . 7 will become detectable within a decade or so.

References

[1] D.A. Kirzlmits and A.D. Linde, Phys. Lett. 42B (1972) 471.

[2] T.W.B. K~ble, J. Phys. A9 (1976) 1387. [3] Ya,B. Zel'dovich, Mon. Not. Ro Astx. Soc. 192 (1980)

663. [4] A. Vilenkin, Phys. Rev. Lett. 46 (1981) 1169. [5] T.W.B. Kibble, Phys. Rep. 67 (1980) 183. [6] A. Vilenkin, Cosmic strings, Phys. Rev. D., in press. [7] A.E. Everett, Phys. Rev. D, in press. [8] R.L. Zimmerman and R.W. Hellings, Astrophys. J. 241

(1980) 475. [9] B.J. Cart, Astron. Astrophys., to be published.

[10] B. Bertotti and B.J. Cart, Astrophys. J. 236 (1980) 1000.

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Gravitational radiation from cosmic strings