Assisted inflation in the Randall–Sundrum scenario

Physics Letters B 528 (2002) 188–192 www.elsevier.com/locate/npe

Assisted inflation in the Randall–Sundrum scenario Yun-Song Piao a,c , Winbin Lin a , Xinmin Zhang a , Yuan-Zhong Zhang b,c a Institute of High Energy Physics, Chinese Academy of Sciences, PO Box 918(4), Beijing 100039, China b CCAST (World Laboratory), PO Box 8730, Beijing 100080, China c Institute of Theoretical Physics, Chinese Academy of Sciences, PO Box 2735, Beijing 100080, China

Received 11 October 2001; received in revised form 5 January 2002; accepted 5 January 2002 Editor: J. Frieman

Abstract We extend the Randall–Sundrum (RS) model by adding a fundamental scalar field in the bulk, then study the multi-field assisted inflationary solution on the brane. It is shown that this model not only satisfies observations, but also provides a solution to the hierarchy problem. Furthermore in comparison to the chaotic inflation model with a single field in four space– time dimensions, the parameters in our model required for a successful inflation are natural.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction Over the past two years the Randall–Sundrum (RS) [1] model has received much attention. In the RS scenario two 3-branes with opposite tensions are taken to sit at fixed points of an S 1 /Z2 orbifold with an AdS5 bulk geometry. It is shown that the gravity is localized on the brane of positive tension and the exponential warp factor in the space–time metric generates a scale hierarchy on the brane of negative tension. In this Letter we will study the cosmological implications of the RS model. In the literature the related topics about the cosmological solutions of the brane-world picture [2–4] have been discussed intensively in the recent years. In modern cosmology inflation plays an important role. Recently BooMERANG and MAXIMA show

that the location of the first peak in the Microwave Background Radiation angular power spectrum is consistent with a flat universe, which agrees with the prediction of inflation. In spite of the remarkable success of the inflation models, there are some serious difficulties associated with the fine-tuning of the model parameters. To overcome these difficulties Liddle et al. [5] recently (see also Ref. [6]) have proposed an assisted inflation model in which multi-scalar fields are introduced to drive the inflation cooperatively. In the present Letter we propose a specific assisted inflation model by introducing a fundamental bulk scalar field in the RS scenario, and we will show that due to the large number of the Kaluza–Klein (KK) modes of the bulk scalar field, this model satisfies observations without the fine-tuning of the model parameters. At the same time it keeps the nice features of the RS scenario in solving the hierarchy problem. We have noticed the inflationary solutions

E-mail address: [email protected] (Y.-S. Piao). 0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 2 ) 0 1 2 2 6 - 1

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in the RS scenario in [7], however, the fine-tuning of the model parameters is still required in these models. In Section 2 we describe our model and study the behavior of the inflationary solution. Section 3 is our conclusion.

189

In the equations above σ (y) = ky where k =
κ˜ 2 /6)1/2 is the bulk curvature, and δn, δa, δb ∼ (−Λ O(ρ0 , ρl ) where ρ0 , ρl are the energy densities of matter on the two branes since in the limit of ρ0 , ρl → 0 the RS solution should be recovered. Therefore, without loss of generality, b = 1 is taken in this Letter and the metric in Eq. (4) is reduced to

2. The model Consider a model which generalizes the RS model
with mass [1] by adding a fundamental scalar field Φ m in the bulk. The action of the model is S = Sbulk + Sbrane + Sbulkscalar , where     1
− Λ
, R Sbulk = d 4 x dy g˜ (1) 2κ˜ 2  √ Sbrane = d 4 x −g(Lm,0 − V0 )|y=0  √ + d 4 x −g(Lm,l − Vl )|y=l , (2)   1 1 2 2 A


Sbulkscalar = d x dy g˜ ∂A Φ∂ Φ − m Φ . 2 2 (3) The RS model described by action S = Sbulk + Sbrane in Eqs. (1) and (2) consists of two 3-branes in an
AdS5 space with a negative cosmological constant Λ. The two 3-branes are assigned to have opposite brane tensions and located at the orbifold fixed points y = 0 and y = l. Note that the tilde in Eqs. (1)–(3) represent the five-dimensional variables and x A = (x µ , y). The distance between the 3-branes is assumed to be stabilized by the mechanisms in [8,9]. In general the five-dimensional metric which respects the cosmological principle on the brane can be written as 

4

ds 2 = −n2 (t, y) dt 2 + a 2(t, y) dx 2 + b2 (t, y) dy 2. (4) Following Ref. [3], we consider the case where the energy density of the scalar field in the bulk and the matter in both 3-branes much less than the bulk cosmological constant and the brane tension, so that this metric can be linearized around the RS solution [3]    a(y, t) = a(t) exp −σ (y)b(t) 1 + δa(y, t) ,    n(y, t) = exp −σ (y)b(t) 1 + δn(y, t) ,   b(y, t) = b(t) 1 + δb(y, t) . (5)

ds 2 = e−2σ (y)gµν dx µ dx ν − dy 2,

(6)

where gµν dx µ dx ν = dt 2 − a 2 (t) dx 2 is the flat Friedmann–Robertson–Walker (FRW) metric. One can see that this metric gives rise to a expansion rate identical to the observers on the two 3-branes [3,4]. Assisted inflation model requires multi-scalar fields [5,6]. In the model under consideration, upon com
gives rise to a set of pactification the scalar field Φ effective scalars in the four dimension and the KK modes of the bulk scalar serve as inflaton in the assisted inflation scenario. Now we closely follow the procedure in Ref. [10] and work out the effective potential of the KK scalars. Starting with the action in Eq. (3) and the metric in Eq. (6), and then making use of the integration by parts we can rewrite the action of the bulk scalar field as    1 4
µΦ
d x dy e−2σ (y)∂µ Φ∂ Sbulkscalar = 2  
∂ −4σ (y) ∂ Φ
e +Φ ∂y ∂y 
2 . (7) − m2 e−4σ (y)Φ A set of scalar fields in the fifth dimension is denoted by Ψi (y) which satisfies  dy Ψi (y)Ψj (y) exp (−2σ ) = δij , and   ∂ −4σ (y) ∂Ψi e − m2 e−4σ (y)Ψi = −m2i e−2σ (y)Ψi , ∂y ∂y (8)
y) as a sum over modes, i.e., we expandΦ(x,
Φ(x, y) = i Φi (x)Ψi (y). Then, after integratingthe

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five-dimensional action over the extra dimension y, obtain an effective four-dimensional action which describes the evolution of the scale factor a(t) on the two 3-branes  S=

√ 1 −g 2 R 2κ   √

1 + d 4 x −g ∂µ Φi ∂ µ Φi − m2i Φi2 . (9) 2 d 4x

i

In Eq. (9), 2κ1 2 = 2k1κ˜ 2 (1 − e−2kl ) from which one can see that for ekl 1, M ∼ k ∼ mp with mp being the four-dimensional Planck scale. The solution to Eq. (8) is Ψi (y) = aiν e

2σ (y)

  mi σ (y) e Jν k   mn σ (y) e + biν Nν , k

(10)

where aiν is a normalization factor, Jν and Nν are the Bessel functions. The jump conditions for two 3-branes give rise to two relations which can be used to solve for mi and biν . As usual, the bulk field
y) upon compactifications manifests itself to Φ(x, a observer in the four space–time dimension as an infinite “tower” of scalars with masses mi . And the mass spectrum satisfies the equation:  2Jν

mi ekl k





  mi ekl mi ekl Jν + = 0, k k

(11)

2

where ν = 4 + mk 2 . From Eq. (11), one can obtain a relation between mi and m. In Fig. 1 we plot the KK scalar mass spectrum as a function of the bulk scalar mass m. One can see that for m
0.1k ∼ 0.1mp , the dependence of the mass spectrum mi on the mass scale m of the bulk scalar field is very weak. So the mass formula could be approximately rewritten as

 mi  ξ1 1 + (i − 1)π ke−kl .

(12)

In the usual single-field inflation in four space– time dimension or the assisted inflation in the large

Fig. 1. The KK field masses vs. the bulk scalar mass. The x-axis is m m/k, and the y-axis is ki ekl . The three curves correspond to the three lightest KK modes, respectively.

extra dimension [11], the scalar mass is determined by the COBE observation, so one obtains a strong constrain on the mass, i.e., m ∼ 10−6 mp . We will show below that this is not the case in the model under consideration. For the infinite “tower” of the scalar field the natural cut off of the mass spectrum is the Plank scale mp . With such a cutoff we have the total number of the KK fields exp(kl) Ntot ≈ (13) . πξ1 Now we study the inflationary solution of our scenario. Given the action (9) we have the equations of motion  4π 2 Φ˙ i + m2i Φi2 , H2 = (14) 3m2p i

Φ¨ i + 3H Φ˙ i + m2i Φi = 0.

(15)

In the slow-rolling approximation, the second derivative term in Eq. (15) is neglected, we then obtain that   2 2 Φ1 (t) mi /m1 Φi (t) = , (16) Φi (0) Φ1 (0) where the Hubble parameter H ≡ a/a ˙ represents the expansion rate of two 3-branes and Φi (0) is the initial values of Φi (t). To calculate the amplitude of the adiabatic perturbation generated during inflation and the corresponding index of the spectrum, we use the usual formula

Y.-S. Piao et al. / Physics Letters B 528 (2002) 188–192

191

for the perturbations of the multi-field in the four- dimensional space–time. Following Refs. [12] and [13], we have for our model the e-folding number 2π 2 N= 2 (17) Φi , mp i

and the amplitude of the adiabatic perturbation    ∂N 2 H  δH = . 2π ∂Φi

(18)

i

In general the initial values of the multi scalars in the assisted inflation model could be taken to be different, however for simplicity as what has been done in [11] we assume the same for the initial values of all KK fields. With such an assumption, the discussions would be much simple. Substituting (16) into (17) and (18), one arrives at 2π ri , N = 2 Φ12 (0) (19) mp i

 δH =

    4π m1 Φ12 (0)   µi ri ri , 3 m3p i

(20)

i

where µi ≡ (mi /m1 )2 and ri ≡ (Φ1 (t)/Φ1 (0))2µi . During the inflation, the KK fields with masses less than the Hubble parameter drive the inflation while those with masses larger are diluted quickly. Quantitatively, for m2N
H 2 < m2N +1 , we have N

4πΦ12 (0) µi ri < µN +1 , µN
3m2p

(21)

i=1

where N denotes the number of KK fields which are in the process of slow-rolling during the inflation. Because the difference between mN and mN +1 is the same order as mN (see Eq. (12)), i.e., µN ∼ µN +1 , Eq. (21) can be reduced to N

µN 

4πΦ12 (0) µi ri . 3m2p

(22)

i=1

Making use of the definitions of µi and ri , Eqs. (19) and (22) can be rewritten as

N 2π 2 Φ1 (t) 2[1+(i−1)π] , N = 2 Φ1 (0) (23) Φ1 (0) mp i=1

Fig. 2. N as a function of Φ1 (0). The x-axis is Φ1 (0) in the unit of mp , the y-axis is N . In the numerical calculation we take N = 60.



1 + (N − 1)π

2



N 2 Φ1 (t) 2[1+(i−1)π] 4πΦ12 (0)  1 + (i − 1)π . 3m2p Φ1 (0) i=1 (24) Taking N  60 as required when the COBE scale exits the Hubble radius, and eliminating Φ1 (t) in (23) and (24), we obtain a relation between N and Φ1 (0), which is shown in Fig. 2. One can see that the number of the KK fields participating in the slowrolling process increases as Φ1 (0) decreases. Combining (19), (20) and (22), we get 

1 δH  √ 2π 1 √ 2π

m1  NµN mp √ m1

1 + (N − 1)π N . mp

(25)

Making use of the COBE observation δH  10−5 , and m1  ξ1 ke−kl , we obtain exp(−kl) 

10−6 . N

(26)

Note that ekl ∼ 1016 which is required to solve the hierarchy problem between the Planck scale and the electroweak scale. Eq. (26) shows that the number of the KK fields in the slow-rolling is large: N ∼ 1010 . From Fig. 2, we see that in this case Φ1 (0)  mp . The corresponding index of the spectrum is 2 2 d ln δH d ln δH =− d ln k dN 1 2 dN − − , N N dN

n−1=

(27)

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which is close to zero, i.e., a near scale-invariant spectrum. The effective potential of the total KK fields in our model on the 3-brane can be written as:  N 1 2 2 V  m1 Φ1 (0) (28) µi ri . 2 i

Substituting (22) into (28), and taking into account that µN ≡ (mN /m1 )2 ∼ N 2 (notice that N ∼ 1010 1), we obtain V∼

3 2 2 2 m m N . 8π 1 p

Acknowledgements We thank R. Brandenberger for stimulating discussions. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 19925523, 10047004, 19835040 and 10175070, and also supported by Ministry of Science and Technology of China under Grant No. NKBRSF G19990754.

References (29)

Since m1  ξ1 ke−kl and e−kl ∼ 10−16 , we have V ∼ 10−12 m4p . This shows that although the number of KK fields is quite large (i.e., ∼ 1010 ) the potential energy V is still far below the Planck scale. Therefore, the following assumption is reasonable: the energy density of matter on the 3-brane is much less than both the bulk cosmological constant and the brane tension. So that the quadratic term of the energy density in our model is not important.

3. Conclusion In summary, we have considered a model by including a bulk scalar field in the RS scenario and studied the inflationary solution. We have calculated the amplitude of the scalar perturbation generated during inflation and the index of the spectrum. Our results show that the initial value of the inflaton is not required to be higher than the Planck scale and the scalar boson mass m is not required to be much less than the Plank scale, furthermore this model solves the hierarchy problem. All of these features are attractive compared with the chaotic inflation model of a single scalar field in the four-dimensional space–time in which the initial value of the scalar field must be higher than the Planck scale to make inflation happen and its mass must be far less than the Planck scale to satisfy the COBE observation.

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