Evolution of a system of cosmic strings

Evolution of a system of cosmic strings

Nuclear PhysicsB252 (1985)227-244 North-Holland,Amsterdam 227 EVOLUTION OF A SYSTEM OF COSMIC STRINGS T.W.B. Kibble Blackett Laboratory, Imperial C...

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Nuclear PhysicsB252 (1985)227-244 North-Holland,Amsterdam

227

EVOLUTION OF A SYSTEM OF COSMIC STRINGS T.W.B. Kibble Blackett Laboratory,

Imperial College, Prince Consort Road, London SW7 2BZ

The formation and destruction of closed loops in a system of strings is discussed. Coupled equations are set up to describe the evolution of the system and it is shown that it must either evolve to a scaling solution where the persistence length is proportional to time or to a situation where the strings come to dominate the energy density. The first case provides a very attractive cosmological scenario in which the loops generate the density fluctuation needed for galaxy formation. I. INTRODUCTION Strings generated at a phase transition in the early history of the I universe may have interesting cosmological effects. In particular, they may provide a natural origin for the density fluctuations condensation

A vital role in the scenario their subsequent generate important

required

to initiate

the

process leading to galaxy formation. 2'3 is played by the formation of closed loops and

decay by gravitational

radiation.

the most significant density fluctuations, to know their distribution.

They are the seeds that and it is therefore very

The main aim of this paper is to treat

this process in more detail than heretofore. The string tension p (equal to the mass per unit length) scale of the symmetry-breaking

transition.

is related to the

For grand unified strings,

expects 3 ~ = MX2/~ where ~ = 0.02 and M X is of order 10 15 GeV. dimensionless

parameter G~, where G is Newton's constant,

though it is not hard to construct models

one

Thus the

is of order 10-6 ,

in which this value is substantially

smaller or larger. At the symmetry-breaking random configuration length L.

strings are formed in a more or less

which may be described as Brownlan with some persistence

This means roughly speaking that the direction of a string is

correlated over distances one expects

transition

less than L but not beyond.

In such a configuration,

that each volume L 3 will contain on average a length L of string,

so that the mean string density is

Ps = p/L2"

(I)

In the early stages the motion of strings is heavily damped by interaction with the dense surrounding medium. I straighten.

The persistence

Small kinks in the string will tend to

length will increase due to this, and also due to

0550-3213/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

7".W.B. Kibble / A system o f cosmic strings

228

the universal expansion. From time to time strings may cross and exchange partners, kinks that straighten in turn. exchange of partners occurs. through one another.

thus creating new

It is not a priori obvious how readily this It might be that strings could simply pass

In the case of global strings (corresponding to the

breaking of a global rather than a local symmetry) Shellard 4 has shown by numerical simulation that the probability p of exchanging partners is essentially unity.

It could be that local strings would behave differently,

though it seems unlikely. least not very small.

Here I shall assume that p is of order one, or at

Then the configuration should remain essentially

similar, with a persistence length L that grows faster than the overall expansion but remains roughly equal to the mean distance between strings. p were small these two quantities could become quite different. 5)

(If

This will

continue until the damping ceases to be important, which occurs 1'6 when L is roughly of the order of the expansion time t. The important question is what happens after this point.

Once the damping

is weak the strings will be accelerated to relativistic speeds.

If there is no

mechanism for transferring energy from the system of strings to other matter then eventually (as we shall see in detail) the strings will come to dominate the mass density, and (unless this has happened only rather recently) the universe would expand much too fast. There is however at least one energy loss mechanism, namely the formation of closed loops and their subsequent decay through gravitational radiation. 3

I

shall argue that there are only two possible outcomes of the evolutionary process.

Either the system of strings will reach a steady final state in which

the persistence length L scales with the horizon distance (L = t) or, if the rate of loop formation is insufficient,

the strings will come to dominate.

Which actually happens depends on the detailed parameters of the model.

To

show this, it will be necessary to set up equations for the time development of the energy of the system of strings, separating out the loops of various sizes. This discussion puts on a firmer footing estimates of the number density of loops.

The results essentially confirm the conclusions of earlier

calculations,

though with minor differences.

2. ENERGY OF A SYSTEM OF STRINGS The starting-point is the equation of motion of a string.

It is convenient

here to use the 'conformal" time • in terms of which the Robertson-Walker line interval is

229

T, W.B. Kibble / A system o f cosmic strings

ds 2 = R2(~)

(d~ 2 - d~2),

where ~ represents unimportant sections

the comoving spatial coordinates.

during the time of interest,

to be flat.)

(Curvature

effects are

so for simplicity we take the spatial

We also choose the parameter 7

d labelling the position

along the string so that

~Sx 5x - ~-7 " ~--~ = 0.

~-x'

(2)

Then the string action takes the form

S = ~

f

d~ dd R2(~)

[x'2(i-~2)] I/2.

The corresponding Euler-Lagrange

8-7

(R2e~)

5

equation

8

is

(R 2 1 x'),

(3)

7 -

= ~

where x,2

(4)

The physical significance

of E may be seen by writing down the expression

for the total energy within a comoving volume V,

E =

f

d3x (3g)I/2

go0

TOO = ~R

f

do E.

(5)

V Thus BRe is the energy per unit parameter

interval.

It is easy to show from the equation of motion that



~

&2.

(6)

In the absence of expansion, convenient

e would be a constant.

Indeed in that case it is

to choose e = i, or equivalently

~2 + ~ , 2 = i,

(R=O).

(7)

If this condition holds initially it will continue to hold at later times. an expanding universe, however,

e always decreases with time and even if it

were initially uniform along the string it would not remain so.

In

230

72 W.B. Kibble / A system o f cosmic strings

From the equations

(5) and (6) it follows at once that

: ~ f do e (I-2~ 2) As one might expect,

(8)

it is possible,

at least under certain conditions,

write this expression in a form that has contributions the volume V.

Multiplying

the equation of motion

to

only from the surface of

(3) by x and integrating

along the string we obtain

f do x. -~z~ (R2e~) : f do ~x° ~

Integrating

~:~

(R 2 7 x').

by parts and using (4), we find that (8) may be written

{[_~~x.x' ~ ] + ~1

where the square bracket

f

d o ~-~ (R2e

x-~)},

is the integrated

<9)

term coming from the points where

strings cross the boundary of the region V. It is not hard to show that for a random configuration term too has contributions configuration

only from the boundary of V.

the sign of ~-~ is random,

of strings For such a

and therefore

f d o R2E x'~ -~ 0

(i0)

where the approximate average.

equality sign signifies

The integral of the time-derivative

only because more across

that this result holds on yields a non-vanishing

the values of o at the end-points the boundary.

n'x'

the last

result

change with time as the strings

Thus from (9) we obtain

1

e'~

: ~ I {-7~..', ~ x.x.

(n)

,,n.x'-----T ~ ~'~}

where n is the outward normal to V and the sum is over all points where strings cross the boundary.

3. ENERGY OF A CLOSED LOOP It is instructive Provided

to apply equations

that the loop is contained

boundary terms.

However

(8) and (9) to a single closed loop.

entirely within V there are then no

(i0) is then no longer obviously

For a small closed loop, whose circumference fact true.

In that limit,

2~

true.

satisfies

the motion is strictly periodic

~ << t, it is in

(in terms of t

Z W.B, Kibble / A system o f cosmic strings

231

rather then ~) and consequently the time derivative term in (9) averages out to zego over a period.

Thus E vanishes.

Comparing with (8), we see that an

equivalent statement is that

f v 2 5 <~2> E

d o e ~2 ~

I 2'

(~ << t),

(12)

f do E i.e. the r.m.s, velocity of the string is 1//2. What is important here is not really the size of the loop but rather the persistence length.

Consider a long but highly convoluted loop, with a

persistence length small compared to its size, and small compared to the expansion time.

In that case, no short section of the string can 'know' that

it is part of a large closed loop rather than of an infinite Brownian string. Hence x and x N will again be almost uncorrelated, and E will be very nearly constant. As before the r.m.s, velocity will be close to i//2. Only if the persistence length becomes comparable with the expansion time 8 Bhattacharjee and Turok showed by

does one find that E changes significantly.

analytic and numerical studies of waves on a string that when what they call the 'length in wave' exceeds the horizon distance 2t, the string is conformally stretched and E grows proportionally to R, while in the opposite case it is constant on average. v is very small.

When the wave is outside the horizon the r.m.s, velocity

Once it is inside, v rises to 1//2.

An interstlng special case that can be treated in detail is that of a circular loop.

This is a dynamical system of one degree of freedom.

radius is R(~)q(~),

If the

it is described by the Hamiltonian I/2

H = [p2 + ( 2 ~ R 2 q ) 2 ]

(13)

It is convenient to make the canonical transformation to Q, P, where p =

tan Q

p2+(2~R2q) 2 ,

P

2 ~pR2q

In terms of the new variables,

= (~)I12

R - ~ sin 2Q,

4 ~ R2

the equations of motion are

P = 2P E cos 2Q.

(14)

(In a non-expanding universe, P is constant and Q is proportional to the time. )

72W.B. K i b b l e / A system o f cosmic strings

232

The energy of the loop is directly related to P:

E = ( 4 ~ P ) 1/2.

Thus the rate of change of E is given by the second of the equations

(14).

Consider for example a loop that is initially at rest in comoving coordinates. Then E will increase during the first quarter-period, until Q = ~/4, then decrease until Q = 3~/4, and so on.

Because R/R is decreasing, E will

oscillate with decreasing amplitude,

tending asymptotically to a value between

its starting point and its first maximum. earlier conclusions, the horizon.

Essentially,

in agreement with

the loop is conformally stretched so long as it is outside

During that time E grows proportionally to R.

Once it comes

inside, the energy oscillates before settling down to a constant value. We can apply these results also to a system of long strings.

Provided that

the scale size L is small compared to t, it would make little difference to the short-time behaviour if all the strings crossing the horizon were cut and rejoined to form loops within it.

Hence we may reasonably expect that the

boundary terms in (11) are unimportant, constant.

Correspondingly,

so that E would again be approximately

the r.m.s, velocity should be close to 1/,/2.

At

the other extreme, if L is large compared to t, we expect the strings to be conformally stretched and E to grow proportionally to R. an r.m.s, velocity near zero.

This corresponds to

If L is of the same order as t, then clearly v

should lie somewhere between 0 and 1//2.

Thus E would increase, but more

slowly than R.

4. PROBABILITY OF STRING CROSSING The mechanism for energy loss from the system of strings involves the formation of closed loops and their subsequent decay by gravitational radiation.

Since these two processes involve very different time scales it is

possible to consider them separately. gravitational radiation.

For the moment,

It is necessary, however,

therefore, we ignore the

to consider together the

processes of formation of closed loops by self-intersectlon of sections of string and their destruction by reconnection to longer strings. Let us begin by trying to estimate the frequency of string crossings. Consider a small segment of string, of length %. another string?

How often does it intersect

We can obtain at least a crude estimate by replacing the

Brownian string configuration with persistence length L by a collection of independently moving straight segments each of length L. there will be V/L 3 such segments.

In a large volume V

To a first approximation, we may suppose

that they are moving randomly and independently, with an r.m.s,

velocity v.

T. W.B. Kibble / A system o f cosmic strings

233

In reality the motion of segments of string is of course correlated by the fact that they are connected to each other, but this may be regarded as a secondorder effect. It will be convenient now to revert to the ordinary time t rather than the conformal time ~.

The probability that our segment of length ~ will encounter

one of the other segments within a short time interval 6t is of order

%v 6t/L 2.

(15)

1 [More precisely, we should include in (15) a factor of ~, the average of [email protected],

and also perhaps a factor of (l-v2), arising from the fact that p and L should be identified not with the actual length RN6~H but with the adiabatic invariant Rg6o.

Such factors of 2 or less are important when checking some of the

formulae against particular solutions, but not for our present purposes.] Now consider a small loop of size % (i.e. of circumference 2~%).

The

probability that it will encounter another segment of string within 6t is essentially (15). dropped).

(The factor of 2~ roughly cancels the factor we have already

The probability that it will disappear by reconnection to a longer

string is therefore

p%v6t/L 2,

(16)

where as before p is the probability that crossing strings exchange partners. We shall be interested later in the possibility of reaching a scaling solution in which

L = yt

(17)

for some constant y.

If this relation does hold then it is clear that the

probability that a loop of size % will survive without reconnection from time t up to +~ is

exp(-p%v/y2t).

Evidently if % >> yL/pv the loop is very unlikely to survive.

On the other

hand a small loop, with % << yL/pv, once formed, will in all probability survive.

Thus to get a good estimate of the number of loops with £ ~ L it is

necessary to take account of the destruction process as well as of formation.

T.W.B. Kibble / A system o f cosmic strings

234

5. ENERGY IN LOOPS AND LONG STRINGS To discuss

the rate of production of loops it is convenient

E the energy of small loops. contained

to separate

from

Let e(%)d% be the energy in the volume V

in loops of size between

% and %+d%.

Note that the number density of

loops in this size range is

n(%)d~

e(%)d% 2~%V "

(18)

For very large sizes, infinite

strings becomes

% >> L, the distinction increasingly

between loops and segments of

artificial,

because such large loops have

almost no chance of survival as distinct entities. an arbitrary upper cutoff,

We could therefore

perhaps an order of magnitude

include the energy of loops above that size within E. necessary

to choose a specific cutoff value,

larger than L, and

However

especially

impose

it is not

since e(%) in any case

falls off for large %. Now consider configuration~ persistence

the process of loop formation. there is really only one relevant

length L.

For a Brownian string scale in the problem,

the

Thus we may reasonably assume that the probability

formation of a loop of size % within factors as in (16) multiplied

the time interval

6t involves

by some function of the ratio %/L.

of

the same Thus the

fraction of the energy E lost to loops in the size range % to %+d% within

6t

is pv~St a(~) d~

Lv -

L--'

where a is an as yet unknown function.

(This definition

differs slightly from

that given in the talk on which this paper is based.) We have already estimated

the probability

that a loop will be destroyed

by

reattachment.

Thus the fraction of the energy e(%)d~ that will be lost to the

network within

6t is again given by (16).

Taking both processes

into account we fin d that (8) should be replaced by

the coupled equations

pv ~) .£d% pv ~d~ = ~ E(l-2v 2) - m ~-- f a( ~ y - + ~ - f e(£) --~-

(where we have used the definition

pv.~ ~ pv~ ~(%) = E ~ a( ) - L2 e(~).

(19)

(12) of v 2) and

(20)

T.W.B. Kibble

/ A system o f cosmic strings

235

Note that the dots now denote derivatives with respect to t, not ~. Loops can of course intersect each other, and in principle we ought also to include terms representing this and the inverse process where a loop intersects 9 itself and breaks into two. Their inclusion as separate terms would however complicate the argument without adding anything of principle. If the universe were not expanding, the system would reach equilibrium. Then by (18) and (20) the number density of strings would be

a(~)

=

neq(%)d%

L~

dR ~--~%L" 2

(21)

This relation can be used together with numerical simulations of random string configurations to estimate the function a. A scaling argument, confirmed by the simulation of Vachaspati and i0 suggests that for % substantially larger than L (21) should be

Vilenkin,

independent of L, i.e. that a(x) falls off like i/x 3.

Their simulation

suggests that about 20% of the total length of string should be in the form of loops.

This implies

f a(x)dx = 0.2.

(22)

If the lower limit is set at I, this would require a(x) = 0.4/x 3. Because of the lattice cutoff this simulation cannot predict the value of a(x) for x < I.

Presumably a(x) falls off rapidly as x ÷ 0, because it is

difficult to form loops with % << L. < i.

However a(x) is certainly not zero for x

Indeed the peak of a(x) may well lle somewhat below unity.

6. SCALING SOLUTION Now let us consider the solution to the equations (19) and (20) that apply in the expanding universe.

The question we wish to ask is whether it will

approach a 'scaling' solution. solution, in which (17) holds.

Let us begin therefore by looking for such a n We assume that R ~ t , where of course n = l ~

or 2/3 respectively in a radiation-dominated or matter-dominated universe.

E

=

Since the energy of a Brownian system of strings is

(23)

~V/L 2,

it follows that

~= ~ -

2~-=

.

T.W.B. Kibble / A system o/cosmic strings

236

In a scaling solution, the fraction of the energy E in loops should depend on size according to the relation e(i)d% E

f(~) d% L--'

(24)

where f contains no explicit time-dependence. dependence because L varies with time.

~(~) = ~

There is implicit time-

Thus

- ~

Hence substituting into (19) and (20) we obtain

2 - 2n(l+v 2) = pv fxa(x)dx - pv fxf(x)dx T T

(25)

xf'(x) + 3(l-n)f(x) = PV x[f(x) - a(x)]. T

(26)

and

These equations (if they possess a solution) determine the unknown function f and the constant TFor given a(x) the solution of (26) with the boundary condition that f ÷ 0 as

x

~

~

is co

f(x) = pv x3n-3 ePVX/y f dy y3-3n e-pVy/y a(y). Y x

(27)

The value of y must then be chosen to satisfy co

2 - 2n(l+v 2) = pv f dy a(y) {y T 0

_ p V y 3-3n ~ p v y / T ~ dx ~ n - 2 Y 0

ePVX/T}.

(28)

2 The matter-dominated case is particularly simple: for n = 5' (28) becomes

2 2) = ~-pv ! dy ya(y) e-pVy/T, ~(l-2v

2 (n = ~).

(29)

Let us denote the left and right hand sides of (28) by h(y) and g(y)

72W.B. Kibble / A system o f cosmic strings

respectively.

It is important to remember that v is dependent on y.

i we expect v = I/#2, while v = 0 for y >> i. = 0

237

to 2-2n as y + =.

Next let us examine g(y).

pv

Clearly at large y,

(30)

+ ®.

g(y) = ~-- f xa(x)dx,

For y <<

Thus h(y) ranges from 2-3n at y

0

Because v decreases from 1/#2 to 0 in the vicinity of y = I, g(y) falls off more rapidly than I/y.

In the opposite limit it is easy to check that

co

(31)

g(0) : -(2-3n) f a(x)dx. 0

According to (22) the integral here is of order 0.2, or perhaps somewhat larger if we take account of loops with % < L. 2 Clearly for n < ~, g(0) is negative.

The function rises to a maximum,

probably at a value of y of order p or somewhat less (because of the rapid decrease near y = I).

If h(y) intersects g(y) there will be two roots of (28),

say Y1 and Y2 (YI > Y2) (see Figure i).

g?

2

n-!

n=-

h

>

Y[

Y FIGURE 1

Y

FIGURE 2

For n=I/2, h(y) ranges from i/2 at ¥ = 0 to i as y + =.

Thus there will he

solutions if the maximum of g(y) is significantly larger than 1/2.

Given the

asymptotic values above~ this seems quite likely though far from certain. 2 n = T' both g and h vanish at y = 0 (see Figure 2).

For

71 W.B. Kibble / A system o f cosmic strings

238

2 For ~ < n < i there is only one nonzero root, Y1 (which approaches ~ as n + i). Suppose now that the loops have reached equilibrium with the network of strings, i.e. that equation (27) is satisfied.

Suppose also that g(¥) > h(y).

Then by referring back to equation (19) we see that E would be less than required to maintain the scaling solution.

This in turn means that L will be

growing faster than required i.e. that y will be increasing.

Specifically, one

finds

j~ 1 Y = ~[g(y)

(32)

- h(y)].

Thus if g > h, the system will evolve towards larger y, as shown in Figure i. Inversely,

if initially g < h, it will move to the left.

Clearly,

the stable root is YI-

If the initial value of y is larger than

Y2, the system will always evolve towards the stable scaling solution. On the other hand if initially y < Y2 then the system evolves towards y = O. The same thing happens if the maximum of g(y) is too low to intersect the curve of h(y).

This corresponds to the situation where the rate of energy loss to

loops is not sufficient to prevent the system of strings coming to dominate the universe.

When this happens, since y << 1 and v = 1//2, we expect that the 2 energy density E/V will fall like R -3, i.e. n = ~ . The system of strings behaves just like pressureless dust.

This raises the interesting possibility

that strings might be the dark matter that closes our universe.

Grand-

unification strings would be too massive and so come to dominate too early, but it might be possible to identify dark matter with strings produced at some intermediate-energy transition. This question will be discussed further in a 5 (Vilenkin has made a similar suggestion but based on

future publication.

strings with a very small interchange probability p.)

7. DECAY OF LOOPS Let us now turn to the fate of the surviving loops. From (23), (24) and (27), we see that the number density of very small loops, will % << L, is given by

n(~)d~

where

3n = e(~)d~ ~ ~ d~ 2~%V = 2~ (e) ~

(33)

T. W.B. Kibble /A system o f cosrnic strings

pv =f dx Y 0

3-3n e-pVx/y a(x).

239

(34)

The total energy in loops according to this formula is of course divergent because of the infinite number of very small loops.

This is because we have so

far ignored the energy loss by gravitational radiation. Once a string has become isolated from the rest of the network it takes no further part in the universal expansion.

Provided that its size is small

compared with the expansion time, its motion is strictly periodic 9 in t, with period ~i. As the loop oscillates it will radiate gravitational waves.

The order of

magnitude of the power dissipation may be estimated from the quadrupole formula

dE GE 2~06%4. d--~ -=

P =-

( 35 )

Strictly speaking, the conditions for the applicability of this formula do not hold, but Turok II has shown by explicit calculations that it nevertheless gives a reasonable order-of-magnitude estimate. Using E = 2 ~ %

and 0~ = 2/% we find that the lifetime is (up to a factor of

order one) E % t% = ~ =G~--

If G~ = 10-6 the lifetime of the loop is about a million times its initial radius.

If it is created at time t O with size

%0, its size at any later time

is given by

% = i0 - G~(t - to).

(36)

Loops of a given size % at time t may have been created at different times, but in most cases when the size was roughly of order L.

Thus for the purpose

of estimating the extent of shrinkage they have undergone, we may reasonably assign them all the same conceptual creation epoch to, such that %0 = LO, ie. we set

%0 = yto = % + G~t

T.W.B. Kibble / A system o f cosmic strings

240

(assuming that G~ << y). The number density of these loops is given by the previous formula (33) modified for the effects of shrinkage,

n(£)d £

~ d£ 2~ (~+G~t)4-3n(yt)3n

This formula holds for % << yt.

(37)

For £ of order yt, the distribution is

sensitive to the detailed form of a(x), according to (27).

However for most

practical purposes it is a reasonable approximation to suppose that (37) holds for all £ < yt, with n(£) = 0 for £ > yt. The total energy density p% in loops is obtained by multiplying (37) by 2 ~ £ and integrating.

For a radiation-dominated universe, with p = 3/32~Gt 2, we

find that the fraction of the energy in the form of loops is p%

128~

P

9y3/2

1/2 (Gp)

By comparison, Ps

32~

P

3y2

,

(n = I/2).

(38)

if Ps is the energy density in long strings (i.e. E/V), we have

G~,

(n = 1/2).

(39)

We may expect that in general the numerical factors ~ and y are roughly of order unity.

Then clearly p£ is much larger than Ps"

After the time t of equal matter and radiation densities we have to eq distinguish the loops formed before and after teq. The expression for p£ is then more complicated, and contains a logarithmic time-dependence. It is worth noting that the commonest loops at any given time are the smallest still surviving, with size of order £ ~ G~t ~ 10-6t and a mass M related to the mass M t inside the horizon by

M/Mt N (G~)2 N 10-12.

The typical spacing between the loops is roughly =-I/3(?G~)-3t.

8. DENSITY PERTURBATIONS The loops generate density perturbations that constitute an approximately Gaussian random process.

It is convenient to expand in a Fourier series

T.W.B. Kibble / A system o f cosmic strings

6~(x)

241

ik.x (4o)

where k is a comoving wave vector. of size % contribute %.

(The wave length is k = 2~R/~k~.)

to those Fourier amplitudes with wave lengths

If we assume no correlation betwee the positions

of different

Loops

larger than loops, we

find

<6 k 6k,> = 0

< " ~~k2 > ~

for k ~ k"

= V

~ ~p

"

n(~)d ~,

(41)

where V = R3V0 is the total volume used for the Fourier expansion subscript 0 denotes present values.

We normalise R by R 0 = I.)

(40).

(The

To be more

precise, we should use not sharp cutoff at R/k, but a cutoff function related to the mean a density distribution order-of-magnitude

estimates

The corresponding by 12

6M 2

in a loop of given size.

However for

(41) is quite sufficient.

r.m.s, fluctuation

in mass on the scale k = 2~R/k is given

V0 k3 (42)

YI6 k II2 •

(2=)3

To be specific,

N

let us consider a time just afterthe

and radiation densities. R

time t

eq

o f

equal matter

Then for

R

eq < k < eq Yteq G~teq

i.e. 30 pc < k < 30 Mpc we find 18=

kteq 5/2 (43)

whence 6M/M = M -5/12. However

it has been pointed out II that although this result is technically

correct it gives a misleading

picture of the typical mass perturbation.

r.m.s, value is dominated by extremely rare very large loops. indication of the typical fluctuation

The

A better

is given by imposing a cutoff at the size

T.W.B. Kibble / A system o f cosmic strings

242

%(k) of the largest loops likely to be found in the region.

This is given by

X3 f n(%) d% = 1 %(k) which yields %(k) = k2/yt, and hence in place of (43) (=..)2~

18= kt 2 = (Gp)2 (R eq) y2 eq

(44)

and so 6M/M = M -I/3. It is important to note however that there will be non-random phases among the Fourier components

6k. The loops are small objects separated by large N Thus one finds, for example,

distances.

6~k'+k",O V

<6k 6k' 6k">

min(R/k) ~ f ( 9 0

3 )

n(~)d~

(G~) 3 k -3/2 ,

so that the phases of these three components are correlated. is even more strongly dominated by the very rare large loops.

(This expression Imposing the

cutoff as before yields instead k-3.) In addition our assumption is certainly wrong. nearby

'twins',

that the positions of the loops are uncorrelated

In particular,

we may expect a considerable number of

resulting from the breakup of larger loops.

Thus (41) should

be replaced by

<'16ki12> = 7

{

~

Rlk f %2n(%)d% 0

R/k

R/k

f

f

0

0

%%' n(%) n(%') Ng(~--, ki ~-k%'.) d£ d%'}

where g is related to the Fourier transform of the loop-loop correlation function.

Because of this effect the fluctations

Another significant large velocities, 'independent'

are not in fact Gaussian.

point is that the loops are in general created with

typically of order v//N, where N is the number of

segments constituting

the loop.

During the expansion the

T.W.B. Kibble / A system o f cosmic strings

velocity

is reduced according

long straight

to v = I/R by the usual red-shift.

string slows down faster than this.

because it behaves effectively In addition, scattering.

v

=

_

243

like a particle

(Note that a

In that case v = I/R2,

of increasing mass, m ~ R.)

the strings are slowed by drag due to small-angle

gravitational

Silk and Vilenkin 13 showed that

v

t,

G % in ~ ,

t,

=

v3t 2 G--~--"

Thus the slower loops are rapidly retarded below their escape velocity which is (G~) I/2 independent

of size.

baryons after the decoupling

These slow loops will then start to capture

time, and hence generate black holes with masses

in the range

106 to i0 I0 M . Finally, long straight strings moving through the o universe would create a wake whose total mass is comparable to that of the string. 13

Such wakes might generate

simplest scenarios

'pancakes',

though their number in the

is not enough to be the prime source of galaxies.

9. CONCLUSIONS We have seen that strings can generate density perturbations right magnitude and on reasonable on pure adiabatic In effect,

fluctuations,

scales.

we have much more structure on small scales.

strings are a way of generating

particularly

interesting

of about the

As compared with the scenario based

isothermal

that the fluctuations

fluctuations.

exhibit non-Gaussian

It is features

and non-random phases. The main effect is due to the small loops which have survived a long time. Loops can act as seeds for galaxy formation.

ACKNOWLEDGEMENTS I am indebted

to several participants

comments and criticism,

and especially

at the Bielefeld Workshop

to Bernard Carr, George Lazarides,

Qaisar Shafi, Paul Shellard and Alex Vilenkin.

REFERENCES I)

T.W.B. Kibble,

J. Phys. A.9 (1976) 1387.

2)

Ya B. Zel'dovich,

3)

A. Vilenkin, Phys. Rev. Lett. 46 (1981) Phys. Rev. D24 (1981) 2052.

4)

P. Shellard,

Mon. Not. Roy. Astr. Soc. 192 (1980) 663.

private communication.

for useful

1169, 1496(E);

244

T.W.B. Kibble / A system o f cosmic strings

5)

A. Vilenkin, Tufts University preprint.

6)

A.E. Everett, Phys. Rev. D24 (1981) 858; T.W.B. Kibble, Acta. Phys. Polon. BI3 (1982) 723.

7)

P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. 109.

8)

N. Turok and P. Bhattacharjee, Phys. Rev. D29 (1984) 1557.

9)

T.W.B. Kibble and N. Turok, Phys. Lett. BII6 (1982) 141.

Phys. B56 (1973)

10) T. Vachaspati and A. Vilenkin, Tufts University preprint TUTP-84-1. ii) N. Turok, Santa Barbara preprint UCSB-TH-3 1984. 12) P.J.E. Peebles, The Large Scale Structure of the Universe, (Princeton U.P., 1980). 13) J. Silk and A. Vilenkin, Tufts University preprint TUTP-84-4.