434

PERTURBATION SPECTRUM IN THE INFLATIONARY STAGE WHEN A COSMICSTRING IS PRESENT

A. A. Saharian

We examine the spectrum of perturbations for a scalar field with an arbitrary curvature coupling parameter

in the de Sitter stage of cosmological expansion when a cosmic string is present. These perturbations are

caused by vacuum fluctuations in the field and serve as seed perturbations for the formation of galaxies in

the postinflationary stage. A cosmic string disrupts the homogeneity of a de Sitter space, so that the

spectrum of the perturbations depends on the distance from the string. This dependence is oscillatory in

character with a period on the order of the perturbation wavelength.

Keywords: cosmology: inflation: cosmic string

1. Introduction

The de Sitter space is one of the most popular manifolds in the general theory of relativity. It corresponds to

the maximally symmetric solution of the Einstein equations with a positive cosmological constant, so many physical

problems are exactly integrable against the background of a de Sitter space.

One of the most remarkable achievements of modern observational cosmology is the discovery of the accelerated

expansion of the universe in our epoch [1,2]. This is based on observational data on supernovae with large red shifts.

In terms of the general theory of relativity an accelerated expansion of the universe presupposes the existence of a

nongravitational source of energy with a negative pressure, usually referred to as dark energy. (See, for example, Ref.

3 and the references therein, about models of dark energy.) The simplest model of dark energy involves the existence

of a cosmological constant that is well matched to observational data. If the source controlling the expansion of the

Astrophysics, Vol. 53, No. 3, 2010

0571-7256/10/5303-0434 ©2010 Springer Science+Business Media, Inc.

Original article submitted April 7, 2010; accepted for publication May 26, 2010. Translated from Astrofizika,Vol. 53, No. 3, pp. 479-491 (August 2010).

Academician G. Saakyan Department of Theoretical Physics, Erevan State University, Armenia; e-mail: saharian@ysu.am

435

universe is the cosmological constant, then the de Sitter space turns out to be an attractor for the geometry of the future.

Another important area of cosmological research where the de Sitter space plays an important role is inflationary

models [4,5]. These models assume a quasi-de Sitter phase of expansion in the initial stage of expansion of the

universe. This leads to a natural solution for a variety of problems in standard cosmology: the problem of the horizon

and flatness, large-scale homogeneity and isotropy, the formation of galaxies, etc. However, the most remarkable

achievement of the inflationary models is the generation of initial inhomogeneities in the distribution of matter owing

to quantum fluctuations in the inflationary epoch. According to the currently most popular scenario for the formation

of the large-scale structure of the universe, galaxies and clusters of galaxies are formed on just these kinds of

inhomogeneities as a result of gravitational instability during the epoch of dominant matter [6]. The spectrum of the

perturbations generated during the inflationary period is in good agreement with observational data on anisotropies in

the temperature of the cosmic background radiation. These data indicate that the density perturbations have a spectrum

that is nearly scale invariant [7].

This paper is a study of the variations in the perturbation spectrum for a scalar field in the de Sitter stage of

expansion owing to the presence of a cosmic string. The model is described and the corresponding eigenfunctions for

a scalar field with an arbitrary curvature coupling parameter are written down in Section 2. In Section 3 the spectral

function of the vacuum fluctuations is determined for the case of an arbitrary angular deficit caused by the presence

of a string. A particular case in which the general formula derived here becomes much simpler is discussed in Section

4. The major results are summarized in Section 5.

2. Model and eigenfunctions

Various kinds of topological defects (monopoles, cosmic strings, domain walls) may be created as the result

of phase transitions during the early stages of cosmological expansion [8,9]. Of these, the most important are cosmic

strings. A new mechanism for the formation of cosmic strings in the framework of brane models has recently been

proposed [10-12]. In the simplest theoretical model for an infinite straight cosmic string, space-time is locally flat

outside the string and has a delta-function curvature tensor localized at the string. In quantum field theory the

corresponding nontrivial topology leads to a nonzero vacuum average of the physical observable. (See Refs. 13 and

14 and the references given there). We have examined the vacuum polarization of a cosmic string against the

background of a de Sitter space-time in a previous paper [15].

Scalar fields play a fundamental role in modern theories of elementary particles. This paper examines a

quantum scalar field ( )xϕ against the background of a de Sitter space in the presence of a cosmic string. The

corresponding element of length is given by

( ), 2222222 dzdrdredtds t +φ+−= α (1)

where the parameter α determines the radius of curvature of the space-time and is related to the Hubble constant H

by H1=α . In Eq. (1), 0≥r and 00 φ≤φ≤ determine the coordinates on a two-dimensional cone, while +∞<<−∞ t

436

and +∞<<−∞ z . When no cosmic string is present, π=φ 20 . When a cosmic string is present it leads to an angular

deficit 02 φ−π that is related to the mass of the string m per unit length by μπ=φ−π G82 0 , where G is the

gravitational constant.

For a scalar field with a coupling parameter x the field equation is

( ) , 02 =ϕξ++∇∇ Rmll (2)

where l∇ is the covariant differentiation operator, m is the mass of the field quantum, and 212 α=R is the scalar

curvature of the de Sitter space. In the cases of minimally and conformally coupled scalar fields, for the curvature

coupling parameter we have 0=ξ and 61=ξ , respectively. Besides the accompanying time coordinate, it is convenient

to introduce the conformal time τ given by α−α−=τ te , where 0<τ<−∞ . In terms of the conformal time coordinate

the element of length (1) takes a conformally flat form with a scale factor of ( )2τα .

The perturbation spectrum is one of the basic characteristics of vacuum fluctuations and plays an important role

in models of cosmological inflation. In the following we examine the change in the perturbation spectrum of a scalar

field against the background of a de Sitter space owing to the presence of a cosmic string. A complete set of

eigenfunctions of Eq. (2) is needed in order to determine the perturbation spectrum. The choice of eigenfunctions

determines the vacuum state of the field. In a de Sitter space-time the most natural vacuum state is the Bunch-Davies

vacuum [16], which is invariant with respect to the de Sitter group and transforms in the adiabatic approximation to

the Minkowski vacuum.

Assuming that the field is in a Bunch-Davies vacuum state, in this geometry the eigenfunctions are given by

( ) ( )( ) ( ) , , 23

2123 3

3kpkeprJkHCx inzik

|n|qnpk +=ηη=ϕ φ+ν (3)

..., ,2 ,1 ,0, 0, 3 ±±=+∞<≤+∞<<−∞ npk (4)

where τ=η , k3 is the projection of the momentum onto the string axis, p is the momentum in the plane perpendicular

to the string axis, and ( )xJμ and ( )( )xH 1ν are the Bessel and Hankel functions, respectively. The order of the Hankel

function and the parameter q in Eq. (3) are determined from the equations

. 2, 1249 022 φπ=α−ξ−=ν qm (5)

Note that, depending on the curvature coupling parameter and the mass of the field, ν may be real or purely imaginary.

For a conformally coupled massless scalar field we have 21=ν and the corresponding Hankel function is expressed

in terms of elementary functions as ( ) ( ) ziezH iz 2121 π−= . For a minimally coupled massless field 23=ν and

( ) ( ) ( ) izezizzH 23123 2 −+π−= . The constant C in Eq. (3) is found from the normalization condition for the

437

eigenfunctions,

( ),

16 2

22

πα=

πν−ν i*

qpeC (6)

where the asterisk denotes the operation of complex conjugation.

3. The perturbation spectrum

One of the basic characteristics of the vacuum fluctuations of the scalar field is the two-point function (Wightman

positive-frequency function)

( ) ( ) ( ) , 00 , xxxxW ′ϕϕ=′ (7)

where x and x’ denote space-time points and ( ) ( ){ }xx npknpk∗ϕϕ

33 , corresponds to the Bunch-Davies vacuum state. This

function determines the correlations of the vacuum fluctuations at different space-time points, as well as the response

of detectors for particles in this state of motion [17]. Expanding the field operator in the complete set of functions

....., and noting the commutation relations for the creation and annihilation operators, we can determine the two-point

function:

( ) ( ) ( ) .xxdkdpxxWn

npknpk∑ ∫∫+∞

−∞=

∗+∞

∞−

+∞′ϕϕ=′ ,

3330

(8)

Substituting Eqs. (3) for the eigenfunctions in this equation, we find that

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )( )[ ] .

16 ,

11

032

2233

∗νν

∞′−

∞+

∞−

∞+

−∞=

φ′−φπν−ν

η′η′×

×πα

η′η=′ ∫ ∫∑

kHkHrpJprJ

edkdppeeq

xxW

|n|q|n|q

zzik

n

inqi *

(9)

The vacuum average square of the field is given by this expression in the limit of coincident arguments.

The spectral function ( )tkP ,ϕ is related to the two-point function by

( ) ( ), , ,0

tkPk

dkxxW ϕ

∞

∫= (10)

where k is the modulus of the wave vector. For calculating the two-point function in the coincidence limit we introduce

438

the polar coordinates in the (p, k3) plane. Equations (9) and (10) then yield the following expression for the spectral

function:

( )( )

( ) ( )( ) ( ),

14 ,

1

02

2

0

2132

2

∫∑−

ηηπα

=∞

=ν

πν−ν

ϕ

∗

x

krxxJdx'kHk

qetkP

qn

n

i

(11)

where the prime on the summation sign means that the term with n = 0 should have weight 1/2. In the absence of

a cosmic string, we have q = 1 and the series in Eq. (11) is calculated using the formula ( ) 212

0

=∑∞

=xJ' n

n. As a result,

we obtain the well known expression for the spectral function in a de Sitters space-time:

( )( )( )

( ) ( )( ) . 8

,213

2

20 ηη

πα= ν

πν−ν

ϕ

∗

kHke

tkPi

(12)

Note that this quantity is independent of the observation point; this is a direct consequence of the maximal symmetry

of the de Sitter space. A cosmic string disrupts the homogeneity of the de Sitter space, so that he perturbation spectrum

depends on the distance to the string.

We begin by examining the behavior of the spectral function in a de Sitter space with no cosmic string in the

asymptotic regions of the parameter kη . The limiting case of 1>>η k corresponds to perturbations with wavelengths

much shorter than the Hubble length a (the radius of the horizon in the de Sitter space). In this limit we have

( ) , ,10 α<<λ

λ≈ϕ phys2

phys

P (13)

where ( )kphys ηπα=λ 2 is the physical length of the wave. As is to be expected, in its leading order the spectral

function coincides with the corresponding result for a Minkowski space.

In the opposite limit of wavelengths we have 1<<η k . In this limit the wavelength of the perturbation is much

greater than the Hubble radius and the corresponding behavior of the spectral function is qualitatively different for real

and imaginary ν . Using the asymptotic formulas for the Hankel function[18], for positive n in the leading order we

have

( ) ( ) ( ), 2 223

230 νΓ

απη≈

ν−

ϕk

P (14)

where ( )xΓ is the Euler gamma function. For a minimally coupled massless scalar field we have 23=ν , so that Eq.

(14) yields the scale-invariant spectral function ( ) ( )20 21 πα≈ϕP . Observational data on the temperature anisotropy of

the microwave background radiation indicate that for a minimally coupled massless field the exact formula has the

439

form ( ) ( )[ ] ( )220 21 παη+=ϕ kP . For a massive scalar field with a mass much less than the Hubble energy scale in the

inflation period, i.e., 1<<αm , the nonzero mass leads to small corrections in the scale invariant spectrum. The

following asymptotic expression holds for imaginary ν and in the limit 1<<ηk :

( ) ( ) ( ) ( )[ ] . 2ln2coscoth4

0023

30

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

φ−ην+νπνπ

απη≈ϕ kB

kP (15)

The constants B0 and 0φ in this formula are determined by ( ) 0

02 φ=νΓ ieBi . In this case the spectral function

approaches zero as 31 physλ .

Figure 1 shows the dependence of the spectral function on ηk for minimally and conformally coupled scalar

fields for different values of the parameter αm . These curves show that the spectral function decreases with increasing

mass of the field.

We now examine the perturbation spectrum when a cosmic string is present, as given by Eq. (11). The integral

in this formula is given in terms of the hypergeometric function by [19]

( ) ( )( ) ( )

. 1232

;12 ,23 ;21

112

2212

1

02

2

+Γ+Γ−+++π=

−+∫ qnqn

aqnqnqnFa

y

ayyJdy

qnqnqn

(16)

Thus, in general, for the spectral function we find

( ) ( )( ) ( ), , , , 0 rkqFtkPtkP ϕϕ = (17)

where we have introduced the notation

Fig. 1. The spectral function in a de Sitter space without a cosmic string forminimally (left) and conformally (right) coupled scalar fields. The numbersnext to the curves are values of the parameter αm .

η

α2P

ϕ

����� �� �� �� ��

η

�� �� �� �� ��

���

���

���

���

���

���

���

���

���

���

���

���

���

�

���

�� =α �

���

�� =α

440

( ) ( )( ) .

22

;12 ,23 ;212 , 2

221

0

qn

n

xqn

xqnqnqnF'qxqF

+Γ−+++= ∑

∞

=(18)

The function ( )tkP ,ϕ depends on k and r through the combinations physkk α=η and α=η physrr , where kphys

and

rphys are the physical wave number and proper distance from the string axis. Equation (17) shows that the effect of

a cosmic string on the perturbation spectrum is determined by the function F(q, x). Note that this function is universal

and does not depend on the mass of the field, the curvature coupling parameter, or the energy. We also introduce the

following integral representation of this function:

( ) ( ). 1

2 ,0

2

2

1

0∑∫∞

=−=

nqn xyJ'

y

ydyqxqF (19)

For x << 1 we have the approximation

( ) ( ) . 22

2

31 ,

22

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+Γ+−≈

q

xxqxqF

q

(20)

For 21 <≤ q and ( )121 −>> qx , an asymptotic formula for F(q, x) can be obtained by applying the Abel-Plana

summation formula [20] to the series in Eq. (19). This yields

( ) ( )( )

( ) . 1

1

1

1sin

42cos21 ,

02223 ∫

∞

⎟⎠⎞⎜

⎝⎛

−−

−ππ−−≈

yqy eeyhdy

x

xxqF (21)

In terms of the angle 0φ , this asymptotic formula is correct in the region π≤φ<π 20 . The simplest form for integral

values of q is obtained in the next section. As expected, in this limit the effects of the string are suppressed.

Figure 2 is a plot of the function F(q, x). In particular, on the string axis we have F(q, 0) = q, and also F(1,

x) = 1. As the figure shows, the string causes oscillations in the spectral function that depend on the distance from the

string. The period of these oscillations is on the order of the perturbation wavelength.

We now examine the limiting cases of the general formula (17). For points near the string, kr << 1, we have

( ) ( )( ) ( ) ( )( ) .

22

2

31 , ,

220

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+Γ+−≈ ϕϕ q

krkrtkqPtkP

q

(22)

At large distances from the string, rk >> 1, the asymptotic formula for the spectral function is obtained from Eq. (21).

In inflationary cosmology an important role is played by the magnitude of the perturbation spectrum at wavelengths

much greater than the scale of the horizon, 1<<α=η physkk .

For cosmological applications the most important case is a minimally coupled scalar field with mass much less

441

than the characteristic energy scale in the inflation period, 1<<αm . The characteristic energy scale in inflationary

models is usually on the order of 1014 GeV. In that case, as noted above, when no string is present the perturbation

spectrum for these wavelengths is scale invariant (the Harrison-Zeldovich spectrum).

We now consider the behavior of the perturbation spectrum in the limit 1<<ηk for different distances from

the string axis. For distances less than or on the order of the scale of the horizon, we have 1≤α=η physrr . In that

case the second argument of the function F(q, kr) in Eq. (17) is small, so we can use the approximate equation (20).

This yields

( ) ( ) ( ) ( )( ) .

22

2

31

4 ,

222

22⎥⎥⎦

⎤

⎢⎢⎣

⎡

+Γ+−η+

απ≈ϕ q

rkrkk

qtkP

q

(23)

For distances from the string axis on the order of the perturbation wavelength we have kr ~ 1, so the exact

formula for F(q, kr) should be used. For 1<<ηk the corresponding spectral function coincides with the function F(q,

kr) to within the constant factor ( )221 πα .

5. A special case

A simpler result for the perturbation spectrum when a cosmic string is present can be obtained in the special

case where the parameter q is an integer. Then the series in Eq. (11) can be written in the form

Fig. 2. The function F(q, x) which determinesthe effect of a cosmic string on the perturbationspectrum.

�

�

�

��x

�

�

F(q,

x)

��

��

��

��

��

q

442

( ) ( ) ( ) ( ). cos0

2

0

yJyJqn'yJ' qnqnn

qnn

−

∞

=

∞

=π= ∑∑ (24)

For the sum on the right hand side of this equation we use the formula

( ) ( ) ( ) ( )( ). sin22

1cos

1

00

0∑∑−

=−

∞

=π=π

q

lqnqn

n

qlyJq

yJyJqn' (25)

After substituting Eqs. (24) and (25) with y = krx in Eq. (11) and integrating using the formula given in Ref. 19, we

arrive at

( )( )

( ) ( )( ) ( )[ ]( ) .

sin2

sin2sin

8 ,

1

0

2132

2

∑−

=ν

πν−ν

ϕ ππηη

πα=

q

l

i

qlkr

qlkrkHk

etkP

*

(26)

Note that the part corresponding to the term l = 0 coincides with the spectral function in a de Sitter space with no string

(cf. Eq. (12)). The terms with 0≠l are induced by the string. Equation (26) can also be derived directly by the

method of images.

Figures 3 and 4 show the dependences on kh and ηr of the spectral functions for minimally and conformally

coupled massless scalar fields when a cosmic string with an angular deficit corresponding to q = 3 is present. As noted

above, these values correspond to the physical wave number and proper distance to the string axis, measured in Hubble

units.

Fig. 3. The spectral function in a de Sitter space for aminimally coupled massless scalar field when a cosmicstring with an angular deficit corresponding to q = 3, ispresent.

� r/η

�

α2P

ϕ

��

kη

�

�

�

�

��

��

��

443

4. Conclusion

In this paper we have studied the change in the perturbation spectrum of the quantum fluctuations in a scalar

field owing to the presence of a cosmic string during the de Sitter stage of expansion of the universe. The appropriate

spectral function has been found using the two-point function and is given by Eq. (17) for the general case of an

angular deficit. The string interaction is described by the function F(q, x) specified by Eq. (18). This function is

universal and does not depend on time, the mass of the field, or the curvature coupling parameter. The string disrupts

the homogeneity of the de Sitter space and the perturbation function depends on the observation point. For points on

the axis of the string the simple Eq. (22) holds. At large distances from the string the part of the spectral function

owing to the presence of the string goes to zero. The general formula becomes much simpler in the special case where

the parameter q, which determines the angular deficit in the geometry of a cosmic string, is an integer. The corresponding

formula is given by Eq. (26). The term with l = 0 on the right hand side of this equation is the spectral function in

a de Sitter space when no cosmic string is present, while the terms with l > 0 are generated by the string.

In de Sitter space time the wavelengths of vacuum fluctuations increase exponentially as the universe expands,

while the radius of the horizon remains constant. This implies that at some time the wavelength becomes equal to the

radius of the horizon. With further expansion the spectral function approaches zero, except for the case of a minimally

coupled massless scalar field, where, in the absence of a string, this function is fixed at ( )221 πα . In the postinflationary,

radiation dominated stage, the perturbations in the energy density owing to vacuum fluctuations of the scalar field lead

to perturbations in the density of matter. According to the standard scenario, it is these perturbations which are the

source of the gravitational instability responsible for the formation of the large-scale structure of the universe. In the

postinflationary, radiation dominated stage and afterward, during the period where nonrelativistic matter is dominant,

the horizon expands more rapidly than the accompanying scale of the density perturbation. As a result, a perturbation

Fig. 4. As in Fig. 3 for a conformally coupled masslessfield.

� r/η

�

α2P

ϕ

��

kη �

�

�

�

��

��

444

with the given wavelength again intersects the horizon, but now in the opposite direction. Perturbations in the energy

density lead to an anisotropy in the temperature of the cosmic microwave background radiation. These anisotropies

are an object of intense study in modern observational cosmology. They contain important information on

inhomogeneities in the distribution of matter during the recombination period. In particular, observational data on

anisotropies in the background radiation impose significant restrictions on inflationary models.

The change in the perturbation spectrum owing to the presence of a cosmic string during the inflationary period

leads to corresponding changes in the temperature anisotropies in the background radiation. In particular, the above

results imply that we should expect additional oscillations to show up in the dependence of the spectrum of the

anisotropies in the background radiation on the angular scale. Observational data on these anisotropies will lead to

limits on the string parameters in the inflationary stage, as well as on their densities. A detailed comparison of the

above results with observational data will require further analysis of the evolution of the perturbations during the

postinflationary period. We shall present an analysis of this kind in our next paper.

In this paper we have examined the changes in the perturbation spectrum caused by a cosmic string in the de

Sitter expansion stage. An alternative mechanism for the generation of primordial inhomogeneities by cosmic strings

during the radiation dominated stage of the expansion of the universe has been widely discussed in the literature [9].

Observational data on temperature anisotropies in the cosmic background radiation obtained with the WMAP satellite

(for recent data, see Refs. 21 and 22) exclude that mechanism as the major source of the primordial inhomogeneities

in the distribution of matter in the universe.

This work was supported by grant No. 119 of the Ministry of Education and Science of the Republic of

Armenia.

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