EXPANDING AdS, BRANES: THE DEPENDENT EIC3ENVALUE PROBLEM A» PRDUCTION OF PARTICLES Текст научной статьи по специальности «Физика»

Brevik I.,1 Lund M. W., Ru0 G.

EXPANDING AdSs BRANES: TIME DEPENDENT EIGENVALUE PROBLEM AND PRDUCTION OF

PARTICLES

Department of Energy and Process Engineering, Norwegian University of Science and Technology, M-7491 Trondheim,

Noway

11ntroduction

We begin by considering two static three-branes, situated at fixed positions y = 0 and y-R in the transverse y direction. The branes are embedded in an AdSs space, and are subject to fine-tuning. This is the classic scenario of Randall and Sundrum [1]. There are by now several papers on brane cosmology in general [2]. If A<0 is the five-dimensional cosmological constant in the bulk, the Einstein equations are

^AB ~^Sab^- "*■ 8ab-^ = K T-AU, (0

where xA ,x2 ,xf, y), and k2=8eG5 is the gravitational coupling. On the first, y = 0 brane (the Planck brane) the tensile stress x0 is assumed positive. This is physically tantamount to assuming that the brane contains an ideal fluid whose equation of stare is /?(i = -p,,, i.e., a "vacuum fluid”, but being without any mechanical stress j3j. Thus t0 means physically the same as -p{).

We introduce the effective four-dimensional cosmological constant on the two branes,

A» = - A + — K4tI. X„=~ A + — K4Ti. (2)

0 6 36 1 * 6 36 *

If XQ>0 then XR >0 automatically [3]. Thus the system is two dS4 branes embedded in an AdSs bulk.

We assume that the space is spatially flat, k = 0. The metric will be written as [4]

ds2 = A2 (y)(-dt2 + e2^%lkdxidxk) + dy2 (3)

with (i - yj-A/6 , H0 = being the Hubble

constant on the first brane, and

A(y) = ^-sinh[jl(yw-| y |)]. (4)

p.

Here yH >0 is the horizon, determined by the relation sinh(jiyH )-[i/. We assume R<yH , so that the horizon at which gn = 0 does not occur in between the branes. Note that the metric (3) reduces to the RS metric [1] in the limit when X0 = 0.

1 Email: iver.h.brevik@ntnu.no

Consider now a massive scalar field <t> in the bulk. Its action can in view of the Z, symmetry be written as

S =^d*xj*fdyyFG(gAadA<I>dB<S> - M2#)„ (5)

with G = det(g/Uf),

In Ref. [5] we calculated the thermodynamic energy at finite temperatures, in the restrictive case of fine-tuning (k = 0). Then, we could make use of the particle concept for the field without any conceptual problems. Our purpose in the present paper is to discuss certain aspects of the more complicated case when the four-dimensional cosmological constants Xn and Xg are positive. In the next section we review briefly for reference purposes the thermodynamic energy calculation in the fine-tuned case. In Sect. 3 we solve the eigenvalue problem for the scalar fieki, in principle. The calculation is carried out in full for the

limiting case when A,, is small, ■%//u 1

Somewhat surprisingly, i limit the formalism

does not allow any real ar the Kaluza-Kiei.n

masses /a.. Finally, in Sect. n we consider the opposite limit in which a is large, but active during a brief period of time T only. We here restrict ourselves to the case of one single brane. This situation means physically a rapid expansion of the de Sitter space, from an initial static (fine-tuned) case I to another static case II. The situation is tractable analytically when one makes use of the “sudden” approximation in quantum mechanics. The production of particles is estimated from use of the Bogoliubov transformation, relating the states I and II. The method was first made use of in a cosmological context by Parker [6].

2 On the fine-tuned case

We assume the branes to be spatially flat, k = 0, and also to be fine-tuned, X0 = XK = 0. As mentioned avove, we can here make use of the particle picture. We assume a finite temperature, T = 1/(3 , and start by noting that the free energy F for a bosonic scalar field in a three-dimensional volume V is given by [5]

ßf = -lnZ = vj-

d'p

(2k?

In

2sinh[ -ߣp

(6) ßFKK =V [ dx

where Z is the partition function and Ep = ^/p2 +M2

the particle energy.

The metric is now ds2 = e~2llMr\afidxadx* + dy2. (7)

We allow for a boundary mass terra by letting M —> M , where

M2 = M2 + 2i>jiiS(>>) - S(j - R)),

(B)

b being a constant [7]. The field equation (a-M2 )# = 0 in the bulk takes the form

[e^Y^a,,+e4rt>ia>,(e^4pW3y)--M2]<i>(xa,,y)=o, (9)

The Kaluza-Klein decomposition

(10)

yields the following equation for the KK masses mn:

-f„\y) + 4sign(>’)|!/„'(>’) + M2fn(y)}

(11)

xln

2sinh

ilitf 2k

IßvWaWj

(15)

In D(x)

whose solution (here given for ()<>■</?) can be written as

v*itb xn ~Mni(a\i), a~iivX. v = ,/4 + M1 /¡i2 , The

boundary conditions are found by integrating Eqs, (11) across the branes. The field may be either even (untwisted) or odd (twisted) under the 2, symmetry. We define the altered Bessel functions ,/v (z) = (2 - b)Jv (z) + zJ' (z)»

>v (z) = (2 - b)Yv (z) + z Y'(z),

and consider here only the even case, /’,(}') = -

The KK masses are given as roots of the equation Dn (x) s I (xn) yv {axn) - jv (axn) yv (xn) = 0» (13)

and the coefficient bn in Eq. (12) is

K =-iMn)i y\{xn) ■

In the physically reasonable limit of mn « p and

a <s 1 we get

Xn =|n. + '~|jic, n = 1,2»..., (14)

the approximation being better the larger the value of n. The quantities xn are thus for low n of order unity. At finite temperatures the bosonic free KK energy

FKK can be expressed as a contour integral in the following form:

for the even modes, the contour C encompassing the zero points for D(x) [5].

This concludes our brief review of the k = 0, X = 0 theory, and we now proceed to consider the non-static case for which X is different from zero.

3 Two branes, when X>0

We shall still keep the spatial curvature k equal to zero, but assume now that X0 > 0 which, as noted above, implies that also XR > 0. The metric is given by Eqs. (3) and (4). To begin with, we do not restrict X(] to be small.

The presence of curved branes (meaning here the time varying scale factor a = eH°‘), complicates the calculation of the vacuum energy in two ways. First, it results in a non-trivial spectrum of the KK excitations on the brane, Secondly, for each ; • • • excitation the vacuum energy becomes more o the curvature. Actually, the determinant of the relevant di complicated to such an exten becomes rather intractable. Foi conformal fields have been inve papers are listed in Ref. fbj; t

conformal and partly also with i........... >

we we will focus attention on one specific issue, namely how to find the energy eigenvalues of the massive scalar field.

The y equation giving the KK masses now takes the form

-/„Ty) + 4sign(>')jicoth[jj.(y - y„ )]/„'(>’)

K\i2 (16)

~M Jn (} ) . - 2 r / V1 J/i ()')’

A,sinhTli()'->w)j

for which the solutions are seen to be either even or odd. We consider here the solution for y > 0.

Introducing the variable z by z = cosh |J,(yH - y), we can write the solution as 1

/„(z) = -

~(P!(z) + cnQ!,(z)),

(17)

N„(z2- If4

where jP| and Ql are associated Legendre

polynomials, a = 4 + M2 / p2 -1 / 2,

§ = ^9/4 -ml IX, and Nn is a normalization constant. Note that when y increases from 0 to R , z

decreases from z0 to zR, where z0 = -<Jl + pL2 /X0 , zR =cosh[(i(>’fl - /?)]. The Legendre polynomials are usually taken to converge for | z |< 1, but can be analytically continued to \z |> 1, which is the region of interest here. (To make the functions single valued, a cut is made along -«> < z < 1 •)

For odd solutions we have as boundary conditions

fnl z =0, (18) J n 'Zq.Z# ’ /

so that the coefficients cn in Eq. (17) are determined as

c„ =-Pl(z0)/£(z0) = -P!(zK)/Q^zs), (19)

and the KK masses are given by p!(zM(zr)-p!(zM(z0)-o. (20)

Note that the KK masses enter through the upper index in the associated Legendre functions.

For even solutions the boundary conditions are likewise found by integrating Eq. (16) across the branes. Let us introduce the notation

Pi(z) = -

q\(z) = ~y~

112

~z + b\Pl(z) + (fi(z)y,

^z + b\(£(z) + ((£(z)y.

(21)

Then, a brief calculation shows that the boundary conditions yield for the coefficients c„

c= - ¡i(zD}Iql(z0) = -[}i(zM)Jq^(zK), and the corresponding KK masses are given by pUz^jUzr)-li(z,<)qliz(,) = Q. (23)

We outline how the KK masses mtl can be calculated, when we start from the natural assumption that {fi,X0,/?} are known input parameters. The horizon is first determined from the equation sinh(|iyH) = ii/y[k0 , and z0 and zR are found from

the expressions above. If moreover the scalar mass M is known, the value of a follows, and the KK masses finally follow by solving Eqs. (20) and (23) in the odd and even cases, respectively.

Due to the appearance of ma in the upper index (3, the formalism becomes however quite complicated. We shall not here carry on the analysis further, except from pointing out the following and perhaps surprising behavior of the formalism in the limiting case of a small-X0, narrow-gap situation. Let us first assume that X0/|i «1. Then,

z,, = s[l + F / X0 = cosh(|i>’w )»1.

Moreover, if we take the gap width R to be small compared with the horizon, R « yH , then we can also assume that zR = cosh[|i(yw - i?)j»1. It becomes thus natural to make use of the following approximate

(22)

expressions for the analytically continued Legendre functions, valid when | z!»1:

F»(z) =

+ 1 + 0

2“T| a + |

~ Z +•■

F -a-

V7cr(a-p + l) 2a+1Vjîr(-a-P)

z2))

This expression Similarly,

F(a + |3 + 1)

holds when

(24)

2a ±1,±3,....

Ôt!(z) = /ît:

F a + -

1 + 0

1

(25)

which holds when 2a & -3,-5,... [9]. In the case of odd solutions, insertion of Eqs. (24) and (25) in Eq. (20) leads to the condition 1/F(a-p +1) == 0 , which means a~p + l = ~H with n = 0,1,2,.... In other words

m.

1 I M

—1—- n +-----h 4 “I-----

X 2 y

(26)

9

1' I 2 y jl"

This is a condition, however, that catr,r'* K“ satisfied for any a > 0. The case of even sc leads to the same conclusion. Thus in the limit c X0 and narrow gap R there is no physical solution for rn„ .

: . :.orgy production calculated via the Bogolubov transformation

Instead of dealing with the complicated case of a time dependent metric, it is sometimes possible to proceed in another way which is both mathematically simple and physically instructive. The method applies in cases where the change of the system takes place abruptly (corresponding to the “sudden approximation” in quantum mechanics). The method implies use of the Bogoliubov transformation relating two vacua, designated in the following by I (in) and 11 (out). As mentioned in Sect. 1, in a cosmological context the method was introduced by Parker [10], and it has been made use of later, for instance in connection, with the formation of cosmic strings [10,11].

We assume now that there is only one single brane, situated at y = 0 , As before, we take k > 0, X > 0, and we start from the state I in which the metric is static and the particle concept for the field thus directly applicable. This state corresponds to times f <0 . The line element on the brane at these times is thus ds) =~dr +5ikdxldxk. During the time period 0 < t < T the de Sitter metric on the brane is rapidly

expanding, and the line element is ds2 ~dt2 +e2H°'ditdxid.xi. For t>T (state II) the metric is again assumed static, so that dsl = -dt2 +a2Sitdx‘dxk, with a = eH°T . For state II

the particle concept is thus again applicable. The vacua corresponding to states I and II are denoted by |0,1) and |0,//>.

The field equation on the brane reads, in both static cases,

#n-a9ic|>;i+mf#n=0, (27)

and in state 1 the basic modes can be written .. (2tc)"’'2

-exp(/k;x-iO),i),

(28)

- f d>k' / ft • N

UkU J pjçÿ +Pkk'Mk'/)>

implying for the operators au ~ j(2T^~^ak1t<3k,/' +PktA'/f)-

(32)

(33)

The number of particles N(k) in k space in state II, produced by the rapid expansion, is

N(k)^{0JI\4,au 10,II) = J~-C|Pk* f, (34)

J (¿ny

where we calculate

Ptt — (Uk‘ll > Ukl ) — -

pL |§(k + k') (35)

4 O),

We insert this into Eq. (34), and make use of the effective substitutions

J^lK2ii)3§(k + k')]2 -»

!(2 tc)3'

■ (27t)3S(k + k')l

•V,

Ik —>~k

where V is the volume, to get V

N (A) =

4(2%)

V®//

+-Z--2I.

CO,

(36)

(37)

“k; — nr—

■sj2(S),

with k7 = (fcj, , /c,)/. The dispersion relation is

0)2 = k] + m2. In state II the basic modes uh„ are given by the same expression, only with the replacements 0)f—»(%, k,-^k„. Here

k„ = (k\,k2„k3)u and aru =a"2k^ +m2 (note that ki,km ~u~2k’„). In both cases the mode

functions satisfy the normalization condition (uk, nk.) = -/ jiik (x)(i0uk>(x)d3x - 8(k - Id), (29)

the other products vanishing, as usual. We consider the same portions in space before and after the expansion,

i e., xf^i^s.dd Correspondingly, we identify the

cevanant components of the wave vectors,

ie - k:‘ =k,, The dispersion relations then become

0)] --k' +»c, cor, = or’ld + ml (30)

We expand #„ in either I or II modes,

cDa = +

J (2tc)"

and relate the two modes via a Bogoliubov

transformation [12]

Multiplying with co;, to get the energy of each mode, we find by integrating over all k the following simple expression for the produced energy:

: = jîv£©„

CO,

-4~-

V®«

3l

CD,

-2

k2dk.

(38)

Here K is introduced as an upper limit, to prevent

the UV divergence arising from the idealized sudden approximation.

We shall consider only one limit of this expression. As noted in Seel. 2, xa = (m, / 1 usually,

implying that /?*,,/ji«l . Let as simply assume hete that mn = 0, so that m, /a)/f =cx according to Eq. (30). Then, Eq. (38) yields

E = ~kVK

or a,

To make, a rough estimate of the energy produced we may, following Parker [10], take into account that the process actually takes some finite time At. The, consequence is that the spectrum falls rapidly for frequencies much larger than 1 /At. Let us simply assume that K is of the same order of magnitude as 1 / At. Then, Eq. (39) yields in dimensional units

TthV

4?(AÎ04

1 +

1

or

2

aj

(40)

If we take a to be of order unity, and set At ~t ~ 10~33 s (i.e., inflationary times), then this expression yields EIV ~ 10” ergs/cm3. For the sake of comparison, we note that this is about 13 orders of magnitude less than the energy density associated with the formation of a cosmic string [10].

References

1. RandallL, Sundrum R. // Phys. Rev. Lett. 1999. V. 83. P. 3370; 1999, V. 83. P. 4690.

2. See, for example, Binetruy P., Deffayet C., Eilwanger U„ Langlois D. II Phys. Lett. 2000. V. B477. P. 285; Birtetruy P., Deffayet C., langlois D. // Nucl. Phys. 2000. V. 565. P. 269; Langlois D„ Maartens R.'Wands D. II Phys. Lett. 2000. V. B489. P. 259; Langlois D. // Prog. Theor. Phys.

Timoshkin A. V. The asymptotic conformai invariance in Chern-Simons theory with matter in...

V. 148, Suppt. P. 181, [hep-th/0209261]. Quantum aspects are discussed in Nojiri S., Odintsov S.D., Osetrin K.E. Phys. Rev. 2001. V. D63. P.

084016; Nojiri S., Odintsov S.D., Zerbíni S. II Phys. Rev. 2000. V. D82. P. 064008.

3. Brevik i„ Borkje K., Morten J.P. II Gen. Rei. Grav. in press, gr-qc/0310103,

4. Brevik I., Ghoroku K„ Odintsov S.D., Yahiro M. II Phys. Rev. 2002. V. D66. P. 064016.

5. Brevik I., Milton K.A., Nojiri S., Odintsov S.D. II Nuci, Phys. 2001. V. B599. P. 305.

6. Parker LII Phys. Rev. 1969. V.183. P. 1057.

7. Gherghetta T„ Pomarol A. II Nucl. Phys. 2000. V. B586. P. 141.

8. Nojiri S., Odintsov S.D., Zerbini S. II Class. Quant. Grav. 2000. V.17. P. 4855; Garriga J., Pujólas O., Tanaka T. II Nucl, Phys., 2001. V, B605.

P. 192; Naylor W., Sasaki M. II Phys. Lett. 2002. V. B542. P. 289; Elizalde E., Nojiri S„ Odintsov S.D., Ogushi S. II Phys. Rev. 2003. V. D 67.

P. 063515.

9. Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products. Academic Press, Inc., New York, 1980, formula 8,778.

10. Parker L. II Phys. Rev. Lett. 1987. V. 59. P, 1369.

11. Brevik I., Toverud T. // Phys. Rev. 1995. V. D 51. P. 691.

12. Birrell N.D., Davies P.C.W. Quantum Fields in Curved Space. Cambridge University Press, England, 1982,

Nojiri S.1

DARK ENERGY AND MODIFIED GRAVITIES

Department of Applied Physics, National Defence Academy, Hashirimizu Yokosuka 239-8686, JAPAN

PACS numbers; 98.80.-k,04.50.+h,11.10.Kk,11,10.Wx

I Introduction

By the recent observations of the universe, we have found that the universe is undergoing a phase of accelerated expansion at the present epoch from about 5x I0iu years ago and the universe is flat.

rr"....relerated expansion of the universe has been

by the observation of the type la ; (SNIa) [11. This, type of supernovae has an aoosiption spectrum of Silicon, which helps us to distinguish the supernovae from other types. An important thing is that the intrinsic luminosities of SNIa are almost uniform and their absolute maginitude is almost -20, from which we can find the distance between the earth and the supernovae. From the distance, we can find how old the SNe are. We can also the speed of the SNe from the redshift. Combining the distance and the redshift, the accelerated expansion of the universe has been established.

The flatness of the universe can be found from the observation of the anisotropies of Cosmic Microwave Background (CMB) observed by the baloons, the BOOMERANG project [2] and the MAXIMA-1 project [3]. The spectrum of the CMB has a characteristic structure and the CMB was produced in the early universe, that is, when the universe becomes transparent or the mean free path of the light becomes infinite. Then the CMB has propagated very long distance, about 10 billion light years and if the uni vers is curved, the structure of the spectrum should be

deformed. The deformation has, however, not observed, which tells that the universe should be flat.

If the universe (the spacial part) is flat, we may assume the metric of tlte flat PRW universe in the following form:

ds1 =-dt2 + u(t)2 ^ (dx"f. (I)

Here a is called a se< universe is expanding ai-.- , • '• the accelerated expansion, Y¥C \H tCiii w, which is called as “equation of The present data tell

W = L— j, (2)

P

Here p is the pressure and p is the energy density of the universe. We may explain the relation between w and the accelerated expansion later but if w<-\/3, the universe is accelerating, which Eq. (2) tells. As w is negative, if the energy density p is positive as usual, the pressure p should be negative.

In case of the radiation, we have w~— since the

3

trace of energy-momentum tensor vanishes: I' = -p + 3j? = 0. Usual matters, like baryon, can be regarded as dust with >v = () (p--()). The cosmological term can gives w = ~-l. It is, however unnatural if the cosmological term, which may be suggested from (2), is a part of the fundamental

* E-mail: snojifi@yukawa.kyoto-u.ac.jp, nojiri@nda.ac.jp