UCLEAR PHYSIC',

PROCEEDINGS SUPPLEMENTS

ELSEVIER Nuclear Physics B (Proc. Suppl.) 43 (1995) 50-53

C o s m o l o g i c a l P e r t u r b a t i o n s f r o m C o s m i c S t r i n g s

M. Hindmarsh ~

~School of Mathematical and Physical Sciences,

University of Sussex, Brighton BN1 9QH, UK

Some aspects of the theory of cosmological perturbations from cosmic strings and other topological defects are

outlined, with particular reference to a simple example: a spatially fiat CDM-dominated universe. The conserved

energy-momentum pseudo-tensor is introduced, and the equation for the density perturbation derived from it. It

is shown how the scaling hypothesis for defect evolution results in a Harrison-Zel'dovich spectrum for wavelengths

well inside the horizon.

One of the most exciting problems in cosmology

today is to explain the origin of the primordial

fluctuations, whose imprint we have seen in the

temperature of the cosmic microwave background

[1]. It is generally t hough t that these fluctua-

tions owe their origin to processes happening very

early on in the history of the Universe, perhaps

at about 10 -36 seconds after the Big Bang. In

the last 15 years or so, two classes of theory have

emerged to explain how this came about: those

based on inflation and those on ordering dynamics

after a cosmological phase transition (sometimes

loosely called topological defects).

Inflationary fluctuations (see e.g. [2]) arise from

the amplification of quantum fluctuations in a

weakly-coupled scalar field by the strong gravi-

tational field of an early phase of accelerated ex-

pansion. They have the statistical properties of a

Gaussian random field, for which the calculations

of correlation functions (such as the power spec-

t rum of density fluctuations) are relatively easy.

Cosmic strings [3] and other defects [4], on the

other hand, are certainly not Gaussian. They are

formed in an early phase transition, in an initial

distribution more akin to a Poisson process, al-

though one of the major problems of the string

scenario is to understand the statistical proper-

ties of the evolving network. In this talk I shall

0920-5632/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0920-5632(95)00449-1

show how an understanding of string statistics

can be converted into knowledge of the statistics

of the primordial fluctuations, using the general

relativistic theory of cosmological perturbations

from "stiff" sources developed by Veeraraghavan

and Stebbins [5]. To bring out the essentials I

shall consider a very simple case: an ~ = 1 uni-

verse dominated by cold dark matter (CDM), and

study the density perturbation only. Much of

what I say applies to textures [6] as well.

Firstly, I outline notation and conventions. For

the space-time metric I write

g , , (~ , x) = a2(~/)(~/,, + h,.(77, x)), (1)

so that r/ is conformal time, x are the comov-

ing coordinates, ~,~ = d iag(-1 , 1, 1, 1,) is the

Minkowksi metric, and h,~ is the (scaled) metric

perturbation. I shall use the synchronous gauge

h0, = 0. Greek indices from near the middle of

the alphabet run from 0 to 3, while Roman ones

are purely spatial indices• Natural units are used

throughout. Strings are stiff sources: that is, their energy-

momentum tensor is separately conserved, so that

0oo + a-o+ +O~Ooz = O, (2) a

5 o 0 o , + - o~+OjO~j = o. (3) a

M. Hindmarsh/Nuclear Physics B (Proc. Suppl.) 43 (1995) 50-53 51

These equations follow from the equations of mo-

tion of the sources, which are unaffected by small

variations in the background gravitational field.

For strings, O,~ can be expressed in terms of the

linear mass density it and the coordinates of the

string world-sheet, X ' ( a , rl):

e t tu (T / , x ) = (~(3) ( x - - X ( o ' , / ] ) )

x i t fd~(2"2~-2"2~) , (4)

where X" = OX'/Ocr and )(~' = OX'/O~. The evolution of the perturbations to the met-

ric and to the energy-momentum tensor of the

matter content of the Universe are then de-

scribed by the linearised Einstein equations [7],

together with the covariant energy-momentum

conservation of the matter. This evolution can be

neatly expressed as the ordinary conservation of

an energy-momentum pseudo-tensor %~ [5]. For

CDM plus strings (or other stiff sources) it takes

the form

i hh+®o ° TOO = a2 p6 8ToGa

T Oi ~ - - O O i

" 1 d (]tij -- ~Jt~ij) + Oij, (5) r'3 - 8~C

where p is the background CDM density, 6 is

the fractional perturbation to it, and ttij is the

traceless part of hij. The conservation equation

0~m~ = 0 reduces in this case, after the use of

the equations of motion of the defects (3) and the

constraint equation h = -2(~, to

+ - 5 = 4~rGO+, (6) a

where O+ = O00 + O,~. This is a second order

inhomogeneous linear ODE, and is easily solved

in terms of two Green's functions,

1 + , (7) = g )

= g (s)

given a source function O+(~,x) and initial

conditions 6i and ~i at time ~ . The ini-

tial conditions are not, arbitrary, for they must

have arisen from a causal evolution, satisfy-

ing pseudo-energy-momentum conservation. On

super-horizon scales at r;~, the total pseudo-

energy vanishes, or

a2P~Si(k) + 4zrG a ~ + O00 = 0. (9)

This equation expresses the fact that energy is

locally conserved when making strings (or other

defects). The initial energy in the strings is com- pensated by that in the matter.

The general solution of equation (6) can be split

into two parts, as is usual with inhomogeneous

differentiM equations,

6z07) = Oz(q,~k)6 i + G2(U,zk)~, (10)

// 6s07) = dT/g~(~/ ,r / )e+(r/) . (11) i

These were termed the initial and subsequent compensation by Veeraraghavan and Stebbins

[5]. The initial compensation soon settles into

a purely growing mode, with

~I(~7) = 1---0 (36i + 7h5i)

- a ( ~ ) 2 ° ° ° ( ~ d 5 a~p, ' (12)

where I have used (9) in the second equality. We

see that the initial compensation is entirely de-

termined by the initial defect density.

This raises an important question: how does

the Harrison-Zel'dovich spectrum (]512) ,-- @k

arise? This spectrum is special because it is

scale invariant: the r.m.s, mass fluctuations in a

horizon-sized volume are constant. To demon-

strate how a scale-invariant spectrum emerges

from the assumption of scaling in the network,

we must first show that the initial compensation

is cancelled by a piece of the subsequent com-

pensation. After all, we should not expect the

52 M. Hindmarsh/Nuclear Physics B (Proc. Suppl.) 43 (1995) 50~53

power spectrum to depend on the initial condi-

tions, for they may well produce white noise (k°),

having been generated by a causal process inside

the horizon at ~i. Using the equation of motion

of the defect network (3), we find

T~3/ 000 (Vi)

I fn ~f/'4 )

+ - 7 ] o0 , ( , ' ) . (13)

The first term on the first line cancels the initial

compensation, the second is a transient, and we

are left with the second line as the true source of

the density perturbations in the dark matter com-

ponent. Thus it is extremely important to get the

initial conditions right in numerical simulations,

otherwise one may well find spurious white noise

in the final power spectrum, which will dominate

any Harrison-Zel'dovich component at late times

[6].

To calculate the power spectrum, we need the

density perturbations well inside the horizon, for

which k~/ >> 1. All that survives (provided

O0,(r/ ,k) drops off fast enough) is the growing

mode, which is the piece proportional to 7/2. Thus

(16(V)12)--+ \ 10 ] ~4k+kJ

/: X d , ld ,2 (O0i(~]1, k)O;i(T}2, k)). (14) i

The angle brackets denote an average over an en-

semble of realisations of the defects. We can de-

duce the form of the right hand side, using the

scaling hypothesis for defects.

Firstly, we do not expect the momenta

Ooi(rl, k) to be correlated at two different times

rh, ~/2 if It/2 - r / l l >> k -1. Let us denote the correlation time by Tk -1. Secondly, statisti-

cal isotropy in the defect network means that

{Ooi(r~)O~j0/)} = 15ij{IOoi(rl)12). Thus,

(15(")12) -+ 5 \ lO ] v4Tk

x drh (IO0+(Vl, k)12). (15) i

Thirdly, the scale invariance tells us that the

r.m.s, momentum density fluctuations are con-

stant on the horizon scale:

fo ~ d3 k 2 ~-~-~-ff(l®oi(~/, k)l ) oc r/-4. (16)

Therefore we may write

(IO0i(?],k)l 2) = v-l?7-1~2s(°i)(k?~), (17)

where S(°i)(z) is a dimensionless scaling function

associated with the momentum density, and V is

the normalisation volume. Finally, we have

<1~(77) ]2) -'-+ V-1 3 ( 4~rG# ~2 \ 10 / ~4Tk

~n ~ d~ll x (18)

Thus, provided the momentum scaling function

is well-behaved, and the integral converges, we

recover the Harrison-Zel'dovieh spectrum. There

is perhaps a logarithmic correction, if the high

frequency limit of the source power spectrum is

white noise, as would be supplied by a population

of small string loops. Given the expression for

00~, easily calculable from (4), we should expect

S (°+) to be proportional to the mean square string

velocity v 2 and the average number of Hubble

lengths of string per Hubble volume ~2.

A similar procedure was adopted by Albrecht

and Stebbins [8] to derive an analytic approxima-

tion to the power spectrum for strings. However,

they did not explicitly remove the initial compen-

sation, and simulated it by a window function.

One then finds that the power spectrum depends

on (IO+12), which goes as v 4. Nevertheless, the

functional form of their approximation that they

M. Hindmarsh/Nuclear Physics B (Proc. Suppl.) 43 (1995) 50-53 53

give is reasonable, even if the details are not quite right.

In general terms, we learn from the above that the appropriate objects for the numerical study of the density perturbation power spectrum are the

O* two-time correlation functions (O00(rh) 0o(r/2)) O* and (®0i(~h) 0i(~/2)). Getting good statistics is

rather hard with fully realistic string simulations [9], but simulations in fiat space [10] run very much faster, and are thus ideal for numerical ex- periments on the form of the correlation func- tions, and the calculations of the observable quan- tities [11]. It is to be hoped that they will also aid us in understanding the coarse-grained features of string evolution and in constructing analytic models [12], which attempt to derive dynamical equations for quantities such as ~ and v.

R E F E R E N C E S

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3. M.B. Hindmarsh and T.W.B. Kibble, preprint NI94025, IMPERIAL/TP/94-95/5, SUSX-TP-94-74, hep-ph/9411342 (1994)

4. A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects, Cam- bridge University Press, Cambridge (1994).

5. S. Veeraraghavan and A. Stebbins, Ap. J., 365

(1990) 37 6. U-L. Pen, D. Spergel and N. Turok, Phys.

Rev. Lett., D49 (1994) 692 7. S. Weinberg, Gravitation and Cosmology, Wi-

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(1989) 973; D.P. Bennett and F.R. Bouchet, Phys. Rev., D41 (1990) 2408; E.P.S. Shellard and B. Allen, in The Formation and Evolu- tion of Cosmic Strings, G.W. Gibbons, S.W. Hawking and T. Vachaspati (eds.), Cam- bridge University Press, Cambridge, 1990, pp

421-448. 10. A.G. Smith and A. Vilenkin, Phys. Rev., D36

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11. D. Coulson, P. Ferreira, P. Graham and N. Turok, Nature, 368 (1994) 27

12. D. Austin, E. Copeland and T.W.B. Kibble, preprint IMPERIAL/TP/93-94/45, SUSX- TH-93-3/6, hep-ph/9406379, (1994)