alexey s. koshelev and sergey yu. vernov- analysis of scalar perturbations in cosmological models...

8/3/2019 Alexey S. Koshelev and Sergey Yu. Vernov- Analysis of scalar perturbations in cosmological models with a non-local

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arXiv:1009.0746v1

[hep-th]

3Sep2010

Analysis of scalar perturbations in cosmological

models with a non-local scalar field

Alexey S. Koshelev

Theoretische Natuurkunde, Vrije Universiteit Brussel and

The International Solvay Institutes,

Pleinlaan 2, B-1050 Brussels, Belgium, [email protected]

Sergey Yu. Vernov

Skobeltsyn Institute of Nuclear Physics, Moscow State University,

Vorobyevy Gory, 119991, Moscow, Russia, [email protected]

Abstract

We develop the cosmological perturbations formalism in models with

a single non-local scalar field originating from the string field theory de-

scription of the rolling tachyon dynamics. We construct the equation for

the energy density perturbations of the non-local scalar field in the pres-

ence of the arbitrary potential and consider the most specific example of

perturbations when important quantities in the model become complex.

1 Introduction

Current cosmological observational data [1, 2] strongly support that the presentUniverse exhibits an accelerated expansion providing thereby an evidence fora dominating dark energy (DE) component [3]. Recent results of WMAP [2]together with the data on Ia supernovae and galaxy clusters measurements,give the following bounds for the DE state parameter wDE = 1.02+0.140.16. Thepresent cosmological observations do not exclude a possibility that the DE withw < 1 exists, as well as an evolving DE state parameter w. Moreover, therecent analysis of the observation data indicates that the varying in time darkenergy with the state parameter wDE , which crosses the cosmological constantbarrier, gives a better fit than a cosmological constant (for details see reviews [4]and references therein).

Construction of a stable model with w < 1 is a challenge leading to theconsideration of originally stable theories admitting the NEC violation in some

Postdoctoral researcher of FWO-Vlaanderen.

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limits. Recently a new class of cosmological models obeying this property whichis based on the string field theory (SFT) [5] and the p-adic string theory [6] has

been investigated a lot [7][26]. It is known that both the SFT and the p-adic string theory are UV-complete ones. Thus one can expect that resulting(effective) models should be free of pathologies.

Models originating from the SFT are distinguished by presence of specificnon-local operators. The higher derivative terms in principle may produce thewell known Ostrogradski instability [27] (see also [28, 8])1. However the Ostro-gradski result is related to higher than two but a finite number of derivatives.In the case of infinitely many derivatives it is possible that instabilities do notappear [19].

The SFT inspired cosmological models [7] are considered as models for darkenergy (DE). The way of solving the Friedmann equations with a quadratic po-tential, by reducing them to the Friedmann equations with many free massivelocal scalar fields, has been proposed in [10, 12] (see also [25]). The obtainedlocal fields satisfy the second order linear differential equations. In the repre-sentation of many scalar fields some of them are normal and some of them arephantom (ghost) ones [12, 25]. Cosmological models coming out from the SFTor the p-adic string theory are considered in application to inflation [15][22]to explain in particular appearance of non-gaussianities. For a more generaldiscussion on the string cosmology and coming out of string theory theoreticalexplanations of the cosmological observational data the reader is referred to[31]. Other models obeying non-locality and their cosmological consequencesare considered in [32].

As a simplest model originating from SFT one can consider a theory with onescalar field whose kinetic operator is non-local. Equations for cosmological per-turbations in such kind of model where the scalar field Lagrangian is quadratic

covariantly coupled with Einstein gravity were derived in [23]. In the presentpaper we develop and improve that formalism accounting an arbitrary potentialof the scalar field and also consider the most specific example of perturbationswhen characteristic quantities of the model become complex.

The paper is organized as follows. In Section 2 we describe the non-local non-linear SFT inspired cosmological model. In Section 3 we sketch the constructionof background solutions in the linearized model and perturbation theory formodels with non-local scalar field 2. In Section 4 we consider the perturbationsin the case of two complex conjugate roots. In Section 5 we summarize theobtained results and propose directions for further investigations.

1Additional phantom solutions, obtained by the Ostrogradski method in some modelscan be interpreted as non-physical ones, because the terms with higher-order derivatives areregarded as corrections essential only at small energies below the physical cutoff[29, 30].

2

For applications of other multi-field cosmological models and related technical aspects seefor instance [33].

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2 Model setup

2.1 Actions

We work in (1 + 3) dimensions, the coordinates are denoted by Greek indices , , . . . running from 0 to 3. Spatial indexes are a , b , . . . and they run from 1 to3. The four-dimensional action motivated by the string field theory is as follows[34, 10, 11]:

S =

d4x

g

R

16GN+

1

g2o

1

2TF0()T Vint(T)

0

. (1)

Here GN is the Newtonian constant: 8GN = 1/M2P, where MP is the Planckmass, is the string length squared, go is the open string coupling constant. Weuse the signature (, +, +, +), g is the metric tensor, R is the scalar curvature, = D =

1

g

gg and D being a covariant derivative, T is a

scalar field primarily associated with the open string tachyon. The functionF0() may be not a polynomial manifestly producing thereby the non-locality.Fields are dimensionless while [go] = length. Vint(T) is an open string tachyonself-interaction. It does not have a quadratic in T term. In the sequel weintroduce dimensionless coordinates x/

and after this set = 1.

Then function F0 is assumed to be an analytic function of its argument,such that one can represent it by the convergent series expansion with realcoefficients:

F0 =n=0

fnn and fn R. (2)

Equations of motion are

G = 8GNg2o

T, (3)

F0()T = Vint(T), (4)where G is the Einstein tensor, the energy-momentum (stress) tensor is asfollows

T =1

2

n=1

fn

n1l=0

lT n1lT + lT n1lT

g

glT

n1lT +lTnlT

g g2o0

1

2TF0()T Vint(T) .

It is easy to check that the Bianchi identity is satisfied on-shell and for F0 =f1 + f0 the usual energy-momentum tensor for the massive scalar field isreproduced.

Let us emphasize that the potential of the field T is V = f02 T2 + Vint(T).Let T0 be an extremum of the potential V. One can linearize the theory in its

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neighborhood using T = T0 + . To second order in one gets the followingaction

S2 =

d4xg

R16GN

+ 12g2o

F()

, (5)

where F= F0 V(T0) and = 0 + V(T0)g2o . Equations (3) and (4) are validfor the latter action after the replacement F0 Fand 0 at Vint(T) = 0.Note that all Taylor series coefficients fn, except f0, are the same for F0 andF. Equation (4) now is

F() = 0. (6)Non-local cosmological models of type (5) with

Fsft() = 2+ 1 c e2,

were previously considered in [11, 12, 17]3. Actions (1) and (5) are of the main

concern of the present paper.

2.2 Background solutions construction in the linearized

model

While solution construction in the full non-linear model (1) is not yet knownthe classical solutions to equations coming out the linearized action (5) werestudied and analyzed in [10, 11, 12, 17, 20, 23, 25]. Thus, we just briefly noticethe key points useful for purposes of the present paper.

The main idea of finding solutions to the equations of motion is to start withequation (6) and to solve it, assuming the function is a sum of eigenfunctionsof the dAlembert operator:

=i

i, where i = Jii and F(Ji) = 0 for any i = 1, . . . , N . (7)

Hereafter we use N (which can be infinite as well) denoting the number ofroots and omit it in writing explicit summation limits over i. Without loss ofgenerality we assume that for any i1 and i2 = i1 condition Ji1 = Ji2 is satisfied.In this way of solving all the information is extracted from the roots of thecharacteristic equation F(J) = 0.

In an arbitrary metric the energymomentum tensor in (3) evaluated on sucha solution is [23]

T = iF(Ji)

ii g2

gii + Ji2i

g2og. (8)

3In [17] for example it has been shown that solving the non-local equations using thelocalization technique is fully equivalent to reformulating the problem using the diffusion-likepartial differential equations. One can fix the initial data for the partial differential equation,using the initial data of the special local fields. This specifies initial data for a non-linearmodel, and these initial data can be (numerically) evolved into the full non-linear regimeusing the diffusion-like partial differential equation.

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The last formula is exactly the energy-momentum tensor of many free massivescalar fields. If

F(J) has simple real roots, then positive and negative values of

F(Ji) alternate, so we can obtain phantom fields.We can consider the solution as a general solution if all roots of F are

simple. The analysis is more complicated in the case of double roots [25] and weskip this possibility for simplicity. Using formula (8) we obtain the Ostrogradskirepresentation [27, 28] for action (5):

S3 =

d4x

g

R

16GN

i=1

F(Ji)2g2o

gii + Ji

2i

.

one can see that S3 is a local action if the number of roots N is finite.

2.3 Energy-momentum tensor in the FriedmannRobertson

Walker Universe

We stress that all the above formulae are valid for an arbitrary metric and thegeneral solution. From now on, however, the only metric we will be interestedin is the spatially flat FriedmannRobertsonWalker (FRW) metric with theinterval given by

ds2 = dt2 + a2(t) dx21 + dx22 + dx23 (9)where a(t) is the scale factor, t is the cosmic time.

Background solutions for are taken to be space-homogeneous. The energy-momentum tensor in (3) in this metric can be written in the form of a perfectfluid T = diag( ,p,p,p ), where

=1

2

n=1

fn

n1l=0

t

lT tn1lT +lTnlT

12

TF0()T + Vint(T) + g2o0,

p =1

2

n=1

fn

n1l=0

t

lT tn1lT lTnlT+

+12TF0()T Vint(T) g2o0.

Using the above notations we get equation (3) in the following form:

3H2 = 8G

, H = 4G( + p), (10)

where G

GN/g

2o is a dimensionless analog of the Newtonian constant, H =

a/a and a dot denotes a derivative with respect to the cosmic time t. Theconsequence of (10) is the conservation equation:

+ 3H( + p) = 0. (11)

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3 Cosmological perturbations with single non-

local scalar fieldNow we turn to the main problem of the present paper: derivation of cosmo-logical perturbation equations in models with a non-local scalar field. We arefocused on the scalar perturbations, because both vector and tensor pertur-bations exhibit no instabilities [35]. Scalar metric perturbations are given byfour arbitrary scalar functions [35, 36]. Changing the coordinate system onecan both produce fictitious perturbations and remove real ones. Natural wayto distinct real and fictitious perturbations is introducing gauge-invariant vari-ables, which are free of these complications and are equal to zero for a systemwithout perturbations. There exist two independent gauge-invariant variables(the Bardeen potentials), which fully determine the scalar perturbations of themetric tensor [36, 35, 23]. To construct the perturbation equations one can use

the longitudinal (conformal-Newtonian) gauge, in which the interval (9) withscalar perturbations has the following form (in terms of the Bardeen potentials):

ds2 = a()2(1 + 2)d2 + (3)ab (1 2)dxadxb

(12)

where is the conformal time related to the cosmic one as a()d = dt. Thethe Bardeen potentials and are as usually Fourier transformed with respectto the spatial coordinates xa having thereby the following form: (, xa) =(, k)eikax

a

and similar for . The obtained equations contain only gaugeinvariant variables, so they are valid in an arbitrary gauge.

Although the metric perturbations are defined in the conformal time framein the sequel the cosmic time t will be used as the function argument and allthe equations will be formulated with t as the evolution parameter.

To the background order energy and pressure densities are given by (10). Tothe perturbed order one has

=1

2

n=1

fn

n1l=0

t(

lT)tn1lT + tlT t(n1lT)

2tlT tn1lT + (13)+ (lT)nlT +lT (nlT)

12g2o

(T Vint Vint)T ,

p =1

2

n=1

fn

n1l=0

t(

lT)tn1lT + tlT t(n1lT)

2tlT tn1lT (14) (lT)nlT lT (nlT)+ 1

2g2o(T Vint Vint)T ,

vs =k

a( + p)

n=1

fn

n1l=0

tlT (n1lT), (15)

s = 0. (16)

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where vs gives the perturbed T0a components of the stress-energy tensor and s

is the anisotropic stress. Using the Einstein equations one gets that s = 0 is

equivalent to = . The Bardeen potential is proportional to the gaugeinvariant total energy perturbation

+ 3(1 + )H vsa

k= k

2

4G

a 2. (17)

The function is a solution of the following linear differential equation (seedetails in [23]):

+ H(2 + 3c2s 6w) ++

H(1 3w) 15H2w + 9H2c2s +

k2

a2

+

k2

a2 = 0.

(18)

Here w = p/ is the equation of state parameter, c2s = p/ is the speed of sound,k = kaka is the comoving wavenumber, = + 3(1 + )Hvs ak , and

= p

+ (1 c2s)ak v s = (1c2s)

+p

n=1

fnn1l=0

tlT (n1lT)

n=1

fnn1l=0

(lT)nlT +lT (nlT)

+ 1g2o

(T Vint Vint)T .

The latter quantity is identically zero for a local scalar field. Therefore = 0is the attribute of the non-locality here.

For the linearized model (5) we can consider the background solution asgiven by (7) to obtain in the more convenient form. To do this the followingrelations are useful

(n) = n +n1m=0

m()n1m and

n1m=0

xm = xn 1

x 1 .

Using (7) explicitly, one has

(n) = n +i

n Jni Ji ()i.

Perturbing the equation of motion for , one has

(F) =n=0

fn(n) = 0. (19)

More explicitly this equation can be written as

(F) = Fi

1

Ji ()i + i

= 0 (20)

where we have put =i

i.

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We stress here that such a decomposition of is arbitrary since originallywe have only one scalar field. Consequently no conditions on individual i arise.

Since the decomposition is arbitrary we may require for all but one i that

(( Ji)i) = 0.

For the remaining i, say i = 2, the most general condition would be

(( J2)2) = ( J2) with F()) = 0.

Solution to the above equation on is already known from the investigation ofthe background. It is exactly equation (4) and its solution is of the form (7)

=i

ii, where ( Ji)i = 0

and i are arbitrary constants. Notice, i = i because i can depend onxa while i is space homogeneous. Nevertheless, we see that the most generalfunctions i can be absorbed in i and one can put all i = 0 without any lossof generality.

After some algebra one can get the following expression for

= 2 + p

m,l

F(Jm)F(Jl)Jmmm2l ml. (21)

where ij =ii

jj

satisfy the following set of equations

ij + 3H+ ii

+j

j ij +3H+

k2

a2 ij =

=

Jii

i Jjj

j

m

F(Jm)2m + p

im + jm

+

2

1 + w

(22)

Equation (18) with the above derived and equations (22) in a closed formdescribe the perturbations in the case of linearized model. Comparing theseequation with the equation for perturbations in a system with many fields (seeeqs. (82) and (85) in [37], or [38] for example) we see they do coincide. Thusperturbations become equivalent in the model with one free non-local scalar fieldand in the model with many local scalar fields. In our model, however, the quan-tity which should be considered as energy density perturbation is . Functionsij play auxiliary role and normally should not be given an interpretation.

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4 Complex roots J in the linearized model

4.1 One pair of complex conjugate roots J1 = J and J2 =J. The background

If a complex number J is a root of F, then J is a root of Fas well. System(10) becomes:

H2 =8

3GN

F(J)2go

2 + J2

+

F(J)2go

2+ J2

+

H = 4GN

go

F(J)2 + F(J) 2

.

(23)

In terms of real fields and such that = + i, = i, we get thefollowing kinetic term:

Ek =F(J)

2go2 +

F(J)2go

2

=drgo

2 2

+ 2

digo

, (24)

where dr = e(F(J)) and di = m(F(J)). In the case di = 0 Ek has a nondi-agonal form. To diagonalize kinetic term we make the following transformation:

= + C1, = C1 + , C1 dr+d2r+d

2i

diIn terms of and system (23)

has the following form:

H2 =8

3GN

C

2g2o

2 2 + Jr(2 2) + 2Jm

+

,

H =4GNC

g2o 2 2

,

(25)

where Jr = e(J), Jm = m(J), C =d2i (d

2r+d

2i )dr+d2r+d2i

d2r+d2i+dr

d2r+d

2i

. So, in the case of

two complex conjugated roots we get a quintom model (for details of quintommodels see reviews [4]).

What is interesting (but not surprising, though) one cannot have non-interactingfields passing to the real components. Precisely, fields will be quadratically cou-pled in the Lagrangian. This means that the usual intuition about field prop-erties (like signs of coefficients in front the kinetic term or the mass term) maynot work. In addition to the latter system of equations we use the equations ofmotion for the scalar fields

+ 3H + J = 0, + 3H + J = 0 (26)

System of equations (23) and (26) has the fixed points at = f = 0 and3H2 = 3H2f = 8GN, which is real at > 0. Equations (23) and (26)describe the late time evolution of the model with Lagrangian ( 1). This modelpossesses a solution with the scalar field tending to the minimum of the potential(i.e. 0) and the Hubble parameter going to the constant. Such solution

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was constructed numerically and was proven to be a solution in [ 14]. Also theasymptotic form of this solution was derived in [10].

Let us analyse the Lyapunov stability of the fixed point. Using formulaefrom [39], we come to conclusion the fixed point with is asymptotically stableat

Hf > 0, Jr < 0. (27)

If at the fixed point Hf < 0 or Jr > 0, then this fixed point is unstable. AtJr = 0 or Hf = 0 one can not use the Lyapunov theorem to analyse the stabilityof the fixed point. Note that the conditions (27) suffice for stability in not onlythe FriedmannRobertsonWalker metric but also the Bianchi I metric [39]. Inthis paper we consider the stability of the fixed point with respect to arbitraryperturbations.

The idea is to compute a solution to (26) using the constant H = H0 and thencompute the correction to H using (23). Then the procedure can be iterated to

compute higher corrections. It was proven in [10] such iteration does converge.Solution to (26) with constant H = H0 is obviously

= +e+t + et, = +e

+t + et (28)

where = 3H02

1

1 4J

9H20

. Considering we see that only one term

in the solution converges when t in general (if both converge we select theslowest one). Lets assume it is the first one proportional to + constant. Thenin order to pick the (slowest) convergent solution we put = 04. We define+ 0 and + .

The first correction to the constant Hubble parameter in case only decayingmodes in are present is

H = H0 + h = H0 + h0

2 + 2

. (29)

Constant h0 is not independent and is related with 0. We note that h is oforder 2. The last expression is a good approximation for H in the asymptoticregime when h H0. Further one can find the scale factor to be

a = a0 exp

H0t +

h02

2

+

2

. (30)

4.2 Cosmological perturbations in the neighborhood of

the solution with complex masses

Configurations with a single scalar field were widely studied and those appearingin the non-local models do not have any distinguished properties. Roughlyspeaking configurations with many scalar fields were explored as well but wehave here new models featuring complex masses and complex coefficients in

4Hereafter we adopt the rulez =

z

meaning that the phase of the complex numberruns in the interval [, ) and for z = rei the square root is |r|ei/2.

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front of the kinetic terms. As it was stressed above there is no problem withthis for the physics of our models while properties of such models, in particular

the cosmological perturbations with such scalar fields were not studied in depth.Thus we focus on perturbations in the configuration with complex roots J. Thesimplest case is one pair of complex conjugate roots where the backgroundquantities were derived in previous Subsection.

First we note that the only function ij is 12 which we shall denote . Thus,there are only two equations in the system. We focus on the asymptotic regimeh H0 and after some algebra one arrives to the following system of equations

( + p)

+ (3H0 + +

) +

3H+ k2

a20e2H0t

=

=

J

J

F(J)22 F(J)22

+ 2g2o

(31)

( + p)

+ H0(8 + 3c2s) +

15H20 + 9H

20c

2s +

k2

a20e2H0t

=

=2k2F(J)F(J)2220 0 2

a20g2o

J

J

e2(H0++

)t.

(32)

where we should use

H = 2h0

2 + 2

+ p = F(J)22 + F(J)22cs

2 =F(J)2(2J)+F(J)2(2J)F(J)2(2+J)+F(J)2(2+J)

The latter system of equations is ready to be solved numerically but in order toget some insight in what is going on it is instructive to make some assumptions

about the value J. This makes some analytic progress possible.We recall the SFT origin of the model. Practically this means that values

of J are determined with the string scales while H0 is expected to be muchsmaller. Therefore, it is natural to assume that |J| H0. This implies iJ. Using the explicit expression for 1 = 0et 0ei

Jt , representing

= x/2 + iy/2 and introducing = i Rg2o

ext the equations of interest can be

written as

cos(yt) + 2(

x2 + y2 sin(yt + b) x cos(yt)) ++

cos(yt)

6h0

x2 + y2ex(tt0) sin(y(t t0) b) + k

2

a20+ x2

2xx2 + y2 sin(yt + b) = 2y

(33)

cos(yt) + 2

x2 + y2 sin(yt b) +

+ 3H0

k2

3a20H0 x

2+ y2 cos(yt c) = 2k

2y

a20.

(34)

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out that masses of these local fields may easily become complex and such acase constitutes the above-mentioned example. The characteristic feature of

the present setup is that all the local fields in fact are not physical and play arole of auxiliary functions introduced for the reduction of the complicated non-local problem to a known one. As it was noted in [10, 11], for a very wide classof the SFT inspired models the local counterpart is not yet studied. Lookingstrange such configurations do not produce a problem for the model since theyare not physical quantities.

Perturbation equations for this local model are (18) and (22) where onlyN 1 functions 1j are independent. The discussion on how a cosmologicalconstant can be generated during the tachyon evolution is presented in [7, 10].We note that perturbations in a quintom model very close to our setup witha phantom field without potential and an ordinary scalar field with quadraticpotential were studied in [40]. Perturbations in models with many scalar fieldswere studied in literature considering various cosmological scenarios [41].

In the present paper we have worked the indicative example where two scalarfields with complex conjugate masses are present. We have demonstrated nu-merically that in the case |

J| H0 the gauge invariant energy density pertur-

bation associated with the matter sector does decay in all wavelength regimesin contrary to ordinary scalar field models. The general case of complex massesdeserves deeper investigation and will be considered in the forthcoming publi-cations [42].

Looking further it is interesting to consider perturbations in other non-localmodels coming from the SFT. For instance, models where open and closed stringmodes are non-minimally coupled may be of interest in cosmology. An exampleof the classical solution is presented in [43]. Furthermore it should be possibleto extend the formalism presented in this paper to other models involving non-

localities like modified gravity setups [32, 44].

Acknowledgements

The authors are grateful to I.Ya. Arefeva, B. Craps, B. Dragovich, and V.F. Mu-khanov for useful comments and discussions. This work is supported in partby RFBR grant 08-01-00798 and state contract of Russian Federal Agency forScience and Innovations 02.740.11.5057. A.K. is supported in part by the Bel-gian Federal Science Policy Office through the Interuniversity Attraction PolesIAP VI/11, the European Commission FP6 RTN programme MRTN-CT-2004-005104 and by FWO-Vlaanderen through the project G.0428.06. S.V. is sup-ported in part by the grant of Russian Ministry of Education and Science NSh-4142.2010.2.

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alexey s. koshelev and sergey yu. vernov- analysis of scalar perturbations in cosmological models...

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