Cosmological perturbations from multi-field inflation in generalized Einstein theories

Nuclear Physics B 610 (2001) 383–410 www.elsevier.com/locate/npe

Cosmological perturbations from multi-field inflation in generalized Einstein theories Alexei A. Starobinsky a,b , Shinji Tsujikawa b,c , Jun’ichi Yokoyama d a Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117334 Moscow, Russia b Research Center for the Early Universe, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan c Department of Physics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan d Department of Earth and Space Science, Graduate School of Science, Osaka University,

Toyonaka 560-0043, Japan Received 8 June 2001; accepted 25 June 2001

Abstract We study cosmological perturbations generated from quantum fluctuations in multi-field inflationary scenarios in generalized Einstein theories, taking both adiabatic and isocurvature modes into account. In the slow-roll approximation, explicit closed-form long-wave solutions for field and metric perturbations are obtained by the analysis in the Einstein frame. Since the evolution of fluctuations depends on specific gravity theories, we make detailed investigations based on analytic and numerical approaches in four generalized Einstein theories: the Jordan–Brans–Dicke (JBD) theory, the Einstein gravity with a non-minimally coupled scalar field, the higher-dimensional Kaluza–Klein theory, and the R + R 2 theory with a non-minimally coupled scalar field. We find that solutions obtained in the slow-roll approximation show good agreement with full numerical results except around the end of inflation. Due to the presence of isocurvature perturbations, the gravitational potential Φ and the curvature perturbation ζ do not remain constant on super-horizon scales. In particular, we find that negative non-minimal coupling can lead to strong enhancement of ζ in both the Einstein and higher derivative gravity, in which case it is difficult to unambiguously decompose scalar perturbations into adiabatic and isocurvature modes during the whole stage of inflation.  2001 Elsevier Science B.V. All rights reserved. PACS: 04.62.+v; 04.90.+e

1. Introduction The beauty of the inflationary paradigm is that it both (1) explains why the present-day Universe is approximately homogeneous, isotropic and spatially flat, so that it may be described by the Friedmann–Robertson–Walker (FRW) model in the zero approximation [1], E-mail address: [email protected] (S. Tsujikawa). 0550-3213/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 1 ) 0 0 3 2 2 - 4

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and (2) makes detailed quantitative predictions about small deviations from homogeneity and isotropy including density perturbations which produce gravitationally bound objects (such as galaxies, quasars, etc.) and the large-scale structure of the Universe. It is the latter predictions that make it possible to test and falsify this paradigm (for each its concrete realization) like any other scientific hypothesis. Fortunately, all existing and constantly accumulating data, instead of falsifying, continue to confirm these predictions (within observational errors). Historically, among models of inflation making use of a scalar field (called an inflaton), the original model or the first-order phase transition model [2] failed due to the graceful exit problem, which was taken over by the new [3] and the chaotic [4] inflation scenarios where the inflaton is slowly rolling during the whole de Sitter (inflationary) stage. The latter property was shared by the alternative scenario with higher-derivative quantum gravity corrections [5] (where the role of an inflaton is played by the Ricci scalar R) just from the beginning. Note that a simplified version of this scenario — the R + R 2 model — was even shown to be mathematically equivalent to some specific version of the chaotic scenario [6] (see also a review in [7]). Turning to inhomogeneous perturbations on the FRW background, the inflationary paradigm generically predicts two kinds of them: scalar perturbations and tensor ones (gravitational waves) which are generated from quantum-gravitational fluctuations of the inflaton field and the gravitational field, respectively, during an inflationary (quasi-deSitter) stage in the early Universe. The spectrum of tensor perturbations generated during inflation was first derived in [8], while the correct expression for the spectrum of scalar perturbations after the end of inflation was obtained in [9]. For completeness, one should also cite two papers [10] and [11], where two important intermediate steps on the way to the right final answer for scalar perturbations were made, in particular, in the latter paper the spectrum of scalar perturbations during inflation was calculated for the Starobinsky inflationary model [5]. In order to obtain a small enough amplitude of density perturbations in all the above mentioned slow-roll inflationary models, the inflaton should be extremely weakly coupled to other fields. It is therefore not easy to find sound motivations to have such a scalar field in particle physics (see, however, [12,13]). Reflecting such a situation, the extended inflation [14] scenario was proposed to revive a GUT Higgs field as the inflaton by adopting non-Einstein gravity theories. Although the first version of the extended inflation model, which considers a first-order phase transition in the Jordan–Brans–Dicke (JBD) theory [15], resulted in failure again due to the gracefulexit problem [16], it triggered further study of more generic class of inflation models in non-Einstein theories [17], in particular extended chaotic inflation [18] where both the inflaton and the Brans–Dicke scalar fields are in the slow rolling regime during inflation. Note that the natural source of Brans–Dicke-like theories of gravity is the low-energy limit of the superstring theory [19,20] with the Brans–Dicke scalar being the dilaton. Several analyses have been done on the density perturbations produced in extended new or chaotic inflation models [21–25], all of which made use of the constancy of the gauge-invariant quantity ζ [26] or its equivalent on super-horizon scales and matched it directly to quantum field fluctuations at the moment of horizon crossing which would be the correct procedure in a single component inflationary model. However, in the presence

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of two sources of quantum fluctuations (i.e., the inflaton and the Brans–Dicke scalar field), ζ does not remain constant during inflation due to the appearance of isocurvature perturbations [27]. In such a case, mixing between adiabatic and isocurvature perturbations may occur due to ambiguity in the definition of the latter ones. Note that it is nothing unusual or unexpected about non-conservation of the quantity ζ even in the long-wave, or super-horizon, k  aH limit since ζ is not a conserved integral of motion even in the one-field, k  aH case, if we follow the meaning of this term used in classical mechanics. Here, a(t) is the FRW scale factor, H ≡ a/a, ˙ where a dot denotes the time derivative, and k = |k| is the conserved covariant momentum of a perturbation Fourier mode. Namely, the “conservation” of ζ in the latter case is restricted to a part of initial conditions for perturbations for which the decaying mode is not strongly dominating. In the opposite case, since then ζ = O(k 2 Φ), the k 2 term in equations for perturbations, e.g., in Eq. (3.40) below, may not be neglected even in the k  aH case. Of course, any classical dynamical system with N degrees of freedom has exactly 2N conserved combinations of its coordinates and conjugate momenta irrespective of the fact if it is integrable or not (or even chaotic), but the functional form of these combinations is generically not universal and strongly depends on initial conditions. That is why no special attention is paid to such constant quantities in mechanics. The quantity ζ is just the example of such a conserved combination for the growing mode. It is the specifics of the inflationary scenario where the case of strongly dominated decaying mode is excluded that leads to the illusion of the universal conservation of ζ in the one-field, k  aH case. Of course, when more modes for each k appear in a multi-field case, generic non-conservation of ζ becomes more transparent. This general remark explains numerous findings of non-conservation of this quantity in particular cases. However, this circumstance does not affect the predictive power of the inflationary paradigm at all since metric perturbations, in particular the gravitational potential Φ, may be calculated during and after inflation without any reference to ζ conservation or non-conservation. The only problem is that evolution of isocurvature modes of scalar perturbations (in contrast to adiabatic ones) is not universal, and its knowledge requires some additional assumptions about behaviour of matter after the end of inflation (in particular, isocurvature modes disappear completely if the total thermodynamic equilibrium is reached at some moment of time). Returning to evolution of perturbations during inflation, in Ref. [28], two of the present authors made the first correct analysis of the issue in the case of slow-roll inflation in the original Brans–Dicke gravity, keeping the above point in mind and extending the method used in [29,30] to find spectra of all modes of adiabatic and isocurvature fluctuations. This was extended to constrain the parameters of general scalar–tensor theories (e.g., the Damour–Nordvedt model [31]) from the spectrum of the CMB anisotropy in Refs. [32, 33]. Later it was found that density perturbations in multi-field models can be derived analytically in the scheme of slow-roll approximations with general potentials of nonEinstein theories [34] including the case of the separated potential, U = V1 (ϕ1 )V2 (ϕ2 ) [35]. General analytic formula for evaluating the spectral index in multi-field inflation was developed in Ref. [36], which neglected isocurvature modes. It was also found in

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Ref. [37] that long-wavelength solutions of cosmological perturbations can be derived by differentiating background equations using the Hamilton–Jacobi method. At present there are several generalized Einstein theories which can provide inflationary solutions, e.g., generalized scalar–tensor theory, Einstein gravity with a non-minimally coupled scalar field, higher-dimensional Kaluza–Klein theories, f (R) gravity theories. Making use of the conformal equivalence between these theories, cosmological perturbations can be analyzed in unified manner as studied in Ref. [38]. In the present paper we analyze density perturbations generated in slow-roll inflationary models for a general class of generalized Einstein theories in the presence of two scalar fields. We make use of the conformal transformation [39] which transforms the original, or the Jordan frame to the Einstein frame in which equations are somewhat simpler. If this class of gravity theories is considered as a low-energy limit of the superstring theory, then the Jordan frame is also called the string frame. While the closed form solutions for large scale perturbations can be generally obtained by the slow-roll analysis of Ref. [34] including noncanonical kinetic terms, the evolution of perturbations depends on specific gravity theories. In the JBD theory where the Brans– Dicke parameter is constrained to be ω > 500 from observations [40], the gravitational potential is dominated by adiabatic perturbations [28,33], in which case the variation of ζ is restricted to be small as we will see later. On the other hand, it was recently found that negative non-minimal coupling with a second scalar field other than inflaton can lead to significant growth of ζ by the analysis in Jordan frame [41]. In this case it is not obvious whether the slow-roll analysis provides correct amplitudes of field and metric perturbations, since these exhibit strong enhancement during inflation by negative instability. In this paper we will investigate the validity of slow-roll approximations by numerical simulations in the Einstein frame. We will also work on the evolution of cosmological perturbations in a higher-dimensional theory and the R + R 2 theory with a non-minimally coupled scalar field. The latter corresponds to the case where explicit and closed forms of solutions in the super-horizon limit are obtained by slow-roll analysis in spite of the coupled form of the effective potential. The rest of the present article is organized as follows. In Section 2 we present the Lagrangian in the Einstein frame and introduce several generalized Einstein theories which can be recasted to the Lagrangian by conformal transformations. Then in Section 3 basic equations and closed form solutions for large-scale perturbations are given. In Section 4 we apply the results of Section 3 to specific gravity theories, namely, JBD theory, Einstein gravity with a non-minimally coupled scalar field, higher-dimensional Kaluza– Klein theory, and R 2 + (1/2)ξ Rχ 2 theory. In order to confirm analytic estimates, we also show numerical results by solving full equations of motion. We present conclusions and discussions in Section 5.

2. Inflation in generalized Einstein theories Consider the following two-field model with scalar fields ϕ1 and ϕ2 :

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S=

d 4x

√ −g



 1 1 1 −2F (ϕ1 ) 2 2 (∇ϕ e R − ) − (∇ϕ ) − U (ϕ , ϕ ) 1 2 1 2 , 2κ 2 2 2

387

(2.1)

where κ 2 /8π = G is the Newton’s gravitational constant, F (ϕ1 ) is a function of ϕ1 , and U (ϕ1 , ϕ2 ) is a potential of scalar fields. Many of the generalized Einstein theories are reduced to the Lagrangian (2.1) via conformal transformations [39]. We have the following theories, which may provide inflationary solutions [21]. 1. Theories with a scalar field ψ coupled to gravity whose action is written by 
  2  2 1  4   ˆ − h(ψ) ∇ψ − ∇φ − V (φ) , S = d x −gˆ f (ψ)R(g) 2

(2.2)

where V (φ) is a potential of inflaton, φ. In this work we consider the following theories. (a) Jordan–Brans–Dicke (JBD) theory with a Brans–Dicke field, ψ [15]. In this case f and h are f=

ψ , 16π

h=

ω , 16πψ

(2.3)

where ω is the Brans–Dicke parameter which is restricted as ω > 500 from observations [40]. Making a conformal transformation, gµν = Ω 2 gˆµν ,

(2.4)

where

 κχ κ2 ψ ≡ exp √ , Ω = 8π ω + 3/2 2

(2.5)

we obtain the action in the Einstein frame (2.1) with replacement, ϕ1 → χ,

ϕ2 → φ,

(2.6)

and F = (β/4)κχ, U (χ, φ) = e−βκχ V (φ), (2.7) √ with β = 8/(2ω + 3). (b) Nonminimally coupled massless scalar field, ψ, with an interaction, (1/2)ξ Rψ 2 [41,42]. In this case f and h read f=

1 − ξ κ 2ψ 2 , 2κ 2

1 h= . 2

(2.8)

Applying the conformal transformation (2.4) with Ω 2 = 1 − ξ κ 2 ψ 2 and defining a new field χ in order for the kinetic term to be canonical as


1 − (1 − 6ξ )ξ κ 2 ψ 2 dψ, χ= (2.9) (1 − ξ κ 2 ψ 2 )2

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we obtain the action (2.1) with replacement (2.6) and  1  F = ln1 − ξ κ 2 ψ 2 , 2 V (φ) U (χ, φ) = e−4F (χ)V (φ) = (2.10) . (1 − ξ κ 2 ψ 2 )2 The induced gravity theory [43] is also described by the action (2.2). In this theory the scalar field ψ has its own potential of the form, V (ψ) = (λ/8)(ψ 2 − η2 )2 with f = ((/2)ψ 2 and h = 1/2. ¯ is introduced in N = D + 4 2. The higher-dimensional theories where the inflaton, φ, dimensions  
 R 1  2   N ¯ ¯ S = d x −g¯ (2.11) − ∇φ − V φ , 2κ¯ 2 2

are the N -dimensional gravitational constant and a scalar curvature, where κ¯ 2 and R respectively. We compactify the N -dimensional spacetime into the four-dimensional spacetime and the D-dimensional internal space with length scale b. Then the metric can be expressed as 2 2 dsN = gˆµν dx µ dx ν + b2 dsD ,

(2.12)

where gˆ µν is a four-dimensional metric. Assuming that extra dimensions are compactified on a torus which has zero curvature, 1 one gets the following action after dimensional reduction [21]:
  b D 1 4 S = d x −gˆ b0 2κ 2 

     ∂µ b∂ν b µν 2 1 ˆ 2   ˆ (2.13) ∇φ − V φ × R + d(d − 1) gˆ − κ¯ , 2 b2  is the scalar curvature with respect where b0 is the present value of b, and R to gˆµν . In order to obtain the Einstein–Hilbert action, we make the conformal transformation (2.4) with a conformal factor,  χ 2 Ω = exp D (2.14) , χ0 where a new scalar field, χ , is defined by    b D(D + 2) 1/2 χ = χ0 ln (2.15) , with χ0 = . b0 2κ 2 Then the four-dimensional action in the Einstein frame can be described as (2.1) with replacement (2.6) and  2D F = 0, U (χ, φ) = exp(−βκχ)V (φ), with β = (2.16) . D+2 1 Note that there exist other methods of compactifications. One of them is the compactification on the sphere

[44], in which case stability of extra dimensions and the evolution of cosmological perturbations during inflation are studied in Refs. [45,46].

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Note that when inflaton is introduced in the four-dimensional action after compactifi√ cation, we find F = e−(β/2)κχ and U (χ, φ) = e−βκχ V (φ) with β = 8D/(D + 2). In this case, however, we do not have inflationary solutions since the effective poten√ tial does not satisfy the condition: β < 2, which is required for power-law inflation to occur (see the next section). 3. The f (R) theories where the Lagrangian includes the higher-order curvature terms, i.e., ∂f/∂R depends on the scalar curvature R [38]:
S=

d 4x

   1   2 . −gˆ f (R) − ∇χ 2

(2.17)

In this case the conformal factor     2 2  ∂f  , Ω = 2κ  ∂R 

(2.18)

describes a dynamical freedom in the Einstein–Hilbert action. Introducing a new scalar field       3 2  ∂f  , φ= (2.19) ln 2κ  2 ∂R  2κ the action in the Einstein frame is described as (2.1) with replacement, ϕ1 → φ,

ϕ2 → χ,

(2.20)

and κφ F=√ , 6

 √   √ 2 6 (sign) 6 U (φ, χ) = (sign) exp − κφ κφ − f (φ, χ) , R(φ, χ) exp 3 3 2κ 2 (2.21) where sign = (∂f/∂R)/|∂f/∂R|. For example, in the R 2 theory [5] with a nonminimally coupled massless χ field, i.e., f (R) =

1 1 R + qR 2 − ξ Rχ 2 , 2κ 2 2

(2.22)

the effective two-field potential is described as U (φ, χ) =

m4pl (32π)2q

e−(2

√  √ 6/3)κφ ( 6/3)κφ

e

2 − 1 + ξ κ 2χ 2 ,

(2.23)

where we have chosen a positive sign. In this case the φ field behaves as an inflaton and leads to an inflationary expansion of the Universe.

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3. Cosmological perturbations in two-field inflation and analytic estimates in slow-roll approximations 3.1. Basic equations Let us first derive the background and perturbed equations by the action (2.1). We consider perturbations around the flat Friedmann–Robertson–Walker (FRW) background in the longitudinal gauge ds 2 = −(1 + 2Φ) dt 2 + a 2 (t)(1 − 2Ψ )δij dx i dx j ,

(3.1)

where a(t) is a scale factor, and Φ, Ψ are gauge-invariant potentials [47–49]. Hereafter we use the relation Φ = Ψ,

(3.2)

which comes from the fact that the anisotropic stress vanishes at linear order. Transforming back to the Jordan frame, this relation does not hold generically [27,38]. Variations of the action (2.1) yield the following background equations for the cosmic expansion rate H ≡ a/a ˙ and the homogeneous parts of scalar fields:  κ 2 1 2 1 −2F 2 2 ϕ˙ + e ϕ˙2 + U , H = (3.3) 3 2 1 2  κ2  2 H˙ = − ϕ˙1 + e−2F ϕ˙ 22 , 2

(3.4)

ϕ¨1 + 3H ϕ˙ 1 + U,ϕ1 + F  e−2F ϕ˙ 22 = 0,

(3.5)

ϕ¨2 + 3H ϕ˙ 2 + e2F U,ϕ2 − 2F  ϕ˙1 ϕ˙2 = 0,

(3.6)

where a prime denotes a derivative with respect to ϕ1 . The Fourier-transformed, first order Einstein equations for the metric and field fluctuations are written as Φ¨ + 4H Φ˙ + κ 2 U Φ    κ2  = ϕ˙1 δ ϕ˙1 − U,ϕ1 + F  e−2F (ϕ˙ 2 )2 δϕ1 + e−2F ϕ˙2 δ ϕ˙2 − U,ϕ2 δϕ2 , 2 

2   k2 ˙ Φ = − κ ϕ˙ 1 δ ϕ˙1 + 3H ϕ˙ 1 + U,ϕ1 − F  e−2F ϕ˙22 δϕ1 − H 2 a 2    + e−2F ϕ˙ 2 δ ϕ˙ 2 + U,ϕ2 + 3H ϕ˙ 2e−2F δϕ2 ,

 κ2  ϕ˙ 1 δϕ1 + e−2F ϕ˙2 δϕ2 , Φ˙ + H Φ = 2

(3.7)

(3.8)

(3.9)

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391

  ϕ˙ 2  k2 2 −2F δϕ1 δ ϕ¨1 + 3H δ ϕ˙1 + 2 + U,ϕ1 ϕ1 − e 2 a + 2F  (ϕ1 )e−2F ϕ˙2 δ ϕ˙2 + U,ϕ1 ϕ2 δϕ2 = 4ϕ˙ 1 Φ˙ − 2U,ϕ1 Φ,

(3.10)

 2   k 2F ˙ + e U,ϕ2 ϕ2 δϕ2 δ ϕ¨2 + 3H − 2F δ ϕ˙2 + a2   − 2F˙ δ ϕ˙1 + e2F 2F  Uϕ2 + Uϕ1 ϕ2 δϕ1 = 4ϕ˙2 Φ˙ − 2e2F U,ϕ2 Φ.

(3.11)



The relation (3.8) clearly indicates that metric perturbations are determined when the evolution of scalar fields are known. 3.2. Closed form solutions in slow-roll approximations The use of slow-roll approximations allows us to obtain closed form solutions for perturbations on large scales [28,32,34,35]. Under this approximation, the background equations are simplified as κ2 U, 3

(3.12)

3H ϕ˙ 1 + U,ϕ1 = 0,

(3.13)

3H ϕ˙ 2 + e2F U,ϕ2 = 0.

(3.14)

H2 =

Combining Eqs. (3.12)–(3.14) with Eq. (3.4), we find H e2F U,ϕ2 , κ2 U

(3.15)

  2  1 U,ϕ1 2 H˙ 2F U,ϕ2 = + e . H 2 2κ 2 U U

(3.16)

ϕ˙1 = −

−

H U,ϕ1 , κ2 U

ϕ˙2 = −

In JBD and higher-dimensional theories where the potentials take the form, U (ϕ1 , ϕ2 ) = e−βκϕ1 V (ϕ2 ), it is straightforward to show that the scale factor evolves as power-law. In this case integrating ϕ˙1 = βH /κ over t, we find ϕ1 (t) =

β a(t) β ln + ϕ1f ≡ − z + ϕ1f , κ af κ

(3.17)

where a subscript f denotes the value of each quantity at the end of inflation and z is the number of e-folds of inflationary expansion after the time t. Assuming that V (ϕ2 ) takes a constant value V0 during inflation, one finds 2/β 2  2

1/2 2 β κ −βκϕ1 (0) e (t − t0 ) V0 , a = a0 (3.18) 3 2 where a0 and t0 are constants. Then we have a power-law inflationary solution when β < √ 2. For example, in the JBD case with potential, (2.7), inflation takes place with a large

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√ power exponent because β is constrained to be β = 8/2ω + 3 < 0.09 from observations [40]. In higher-dimensional theories with potential √ (2.16), inflation is realized for arbitrary extra dimensions D, because the condition, β < 2, always holds. Consider large scale perturbations with k  aH . Neglecting Φ˙ and those terms which include second order time derivatives in Eqs. (3.9)–(3.11), one finds Φ=

 κ2  ϕ˙1 δϕ1 + e−2F ϕ˙2 δϕ2 , 2H

(3.19)

3H δ ϕ˙1 + U,ϕ1 ϕ1 δϕ1 + U,ϕ1 ϕ2 δϕ2 + 2U,ϕ1 Φ = 0,

(3.20)

    3H δ ϕ˙2 + e2F U,ϕ2 ,ϕ δϕ1 + e2F U,ϕ2 ,ϕ δϕ2 + 2e2F U,ϕ2 Φ = 0.

(3.21)

1

2

Note that this approximation may not be always valid especially when perturbations exhibit nonadiabatic growth during inflation. We will check its validity in the next section. Introducing new variables, x and y, with δϕ1 = U,ϕ1 x and δϕ2 = e2F U,ϕ2 y, Eqs. (3.20) and (3.21) yield 3H x˙ +

U,ϕ1 ϕ2 U,ϕ2 e2F (y − x) + 2Φ = 0, U,ϕ1

(3.22)

3H y˙ +

(e2F U,ϕ2 ),ϕ1 U,ϕ1 e−2F (x − y) + 2Φ = 0. U,ϕ2

(3.23)

Subtracting Eq. (3.22) from Eq. (3.23), we obtain the following integrated solution  
A y − x = Q3 exp (3.24) dt , 3H where Q3 is a constant, and A≡

U,ϕ1 ϕ2 U,ϕ2 e2F (e2F U,ϕ2 ),ϕ1 U,ϕ1 e−2F + . U,ϕ1 U,ϕ2

(3.25)

Taking notice of the relation   κ2  ˙ κ2  ˙ U x + U,ϕ2 ϕ˙2 (y − x) = U y + U,ϕ1 ϕ˙1 (x − y) , 2H 2H and making use of Eq. (3.24) and background equations (3.12)–(3.14), we find 
 J Q3 H U,ϕ1 ϕ2 U,ϕ2 e2F dt, x =− + U,ϕ2 ϕ˙2 U U,ϕ1 U κ2 
 Q3 J H (e2F U,ϕ2 ),ϕ1 U,ϕ1 y =+ dt, + U ϕ ˙ ,ϕ 1 1 2 U κ U,ϕ2 U Φ=

where


J ≡ U exp

 A dt . 3H

(3.26)

(3.27) (3.28)

(3.29)

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Then the final closed-form solutions for longwave perturbations are expressed as  
t   δϕ1 = (ln U ),ϕ1 Q1 + Q3 ln(ln U ),ϕ1 ,ϕ J dϕ2 , 2

393

(3.30)

t∗

 δϕ2 = e

2F


t

(ln U ),ϕ2 Q2 − Q3

   2F  ln e (ln U ),ϕ2 ,ϕ J dϕ1 , 1

(3.31)

t∗

Φ =−

 1 (ln U ),ϕ1 δϕ1 + (ln U ),ϕ2 δϕ2 , 2

(3.32)

with 
t     2F    J = exp − ln e (ln U ),ϕ2 ,ϕ dϕ1 + ln(ln U ),ϕ1 ,ϕ dϕ2 . 1

2

(3.33)

t∗

Here integration constants, Q1 , Q2 , and Q3 satisfy the relation Q2 = Q1 + Q3 , which comes from Eq. (3.24). The Q3 terms appear due to the presence of isocurvature perturbations. These constants are evaluated by the amplitudes of quantum fluctuations of scalar fields at horizon crossing, t∗ . The fluctuations are generated by small scale perturbations (k > aH ), so that they can be considered as free massless scalar fields which are described by independent random variables [28,30]. Then the field perturbations when they crossed the Hubble radius (k  aH ) are written in the form:   H (t∗ ) H (t∗ ) F (t∗ ) eϕ1 (k), δϕ2 (k)t =t∗ = √ e eϕ2 (k). δϕ1 (k)t =t∗ = √ 2k 3 2k 3 Here eϕ1 and eϕ2 are classical stochastic Gaussian quantities, described by   eϕ1 (k)eϕ∗2 (k ) = δij δ (3)(k − k ), eϕ1 (k) = eϕ2 (k) = 0, where i, j = ϕ1 , ϕ2 . From Eqs. (3.15), (3.30), and (3.31), we find  
t   δϕ1 κ2 ln(ln U ),ϕ1 ,ϕ J dϕ2 , =− Q1 + Q3 2 ϕ˙ 1 H

(3.34)

(3.35)

(3.36)

t∗

 
t   2F  δϕ2 κ2 Q1 + Q3 − Q3 ln e (ln U ),ϕ2 ,ϕ J dϕ1 . =− 1 ϕ˙ 2 H

(3.37)

t∗

Making use of Eqs. (3.34), (3.36), and (3.37), the integration constants are expressed as  H 2 (t∗ ) eϕ1 (k) Q1 = − √ , ϕ˙1 κ 2 2k 3 t∗  H 2 (t∗ ) eϕ1 (k) F eϕ2 (k) −e . Q3 = √ (3.38) ϕ˙ 1 ϕ˙ 2 κ 2 2k 3 t∗

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We shall introduce the so-called Bardeen parameter, defined by  H2 Φ˙ ζ =Φ− , Φ+ H H˙

(3.39)

which is conserved on large scales for adiabatic single-field inflationary scenarios. In the multifield inflationary scenario, however, the conservation of ζ does not necessarily hold even on large scales due to the presence of isocurvature perturbations. This can be easily seen by considering the time derivative of ζ [32,35]:  δϕ1 δϕ2 H k2 ˙ζ = Z, Φ +H − (3.40) ϕ˙1 ϕ˙2 H˙ a 2 where Z≡

2e−2F ϕ˙1 ϕ˙2 (ϕ¨1 ϕ˙2 − ϕ˙ 1 ϕ¨2 + F˙ ϕ1 ϕ2 ) + F,ϕ1 ϕ˙1 e−4F ϕ˙ 24 (ϕ˙ 12 + e−2F ϕ˙ 22 )2

.

(3.41)

Note that this relation was obtained without assuming slow-roll approximations. The quantity,  δϕ1 δϕ2 − , Sϕ1 ϕ2 ≡ H (3.42) ϕ˙1 ϕ˙2 represents a generalized entropy perturbation between ϕ1 and ϕ2 fields[50,51]. Since the second term in Eq. (3.40) is absent in single-field adiabatic inflationary scenarios, the Bardeen parameter is conserved on super-horizon scales. However, in the multifield case with Sϕ1 ϕ2 = 0 and Z = 0, the presence of isocurvature perturbations leads to the variation of ζ . In the next section, we will investigate the evolution of large-scale perturbations in specific gravity theories.

4. Applications to specific gravity theories Among the generalized Einstein theories which we presented in Section 2, most of them take the following separated form of potentials except for the f (R) theories: U (ϕ1 , ϕ2 ) = V1 (ϕ1 )V2 (ϕ2 ). In this case, we have an integration constant by making use of Eq. (3.15):

V1 2F V2 C = −κ 2 e dϕ1 + κ 2 dϕ2 . V1,ϕ1 V2,ϕ2

(4.1)

(4.2)

This characterizes the trajectory in field space in two-field inflation. Since J = e−2(F −F∗ ) in the separated potential, (4.1), Eqs. (3.30) and (3.31) are easily integrated to give δϕ1 =

V1 Q1 , V1

δϕ2 =

 V2  Q1 e2F + Q3 e2F∗ , V2

(4.3)

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together with the gravitational potential,    1 V1 2 1 V2 2  Φ =− Q1 e2F + Q3 e2F∗ . Q1 − 2 V1 2 V2

395

(4.4)

Introducing new integration constants, C1 ≡ −κ 2 (Q1 + Q3 e2F∗ ) and C3 ≡ −κ 2 Q3 e2F∗ , and making use of Eq. (3.16), one finds δϕ1 = −(C1 − C3 )

V1 , κ 2 V1

  V δϕ2 = − C1 e2F − C3 (e2F − 1) 2 2 , κ V2

    Φ = C1 (ϕ1 + e2F (ϕ2 − C3 (ϕ1 + (e2F − 1)(ϕ2   H˙ = −C1 2 − C3 (ϕ1 + (e2F − 1)(ϕ2 , H where (ϕ1 and (ϕ2 are given by   1 V1 2 1 V2 2 (ϕ1 ≡ 2 , (ϕ2 ≡ 2 . 2κ V1 2κ V2 From Eq. (3.38) C1 and C3 are expressed as    H 2 (t∗ )  2F eϕ1 (k) 3F eϕ2 (k) C1 = √ 1−e +e , ϕ˙ 1 ϕ˙2 t∗ 2k 3   eϕ (k) H 2 (t∗ ) 3F eϕ2 (k) e − e2F 1 . C3 = √ ϕ˙2 ϕ˙ 1 t∗ 2k 3

(4.5)

(4.6)

(4.7)

(4.8)

The gravitational potential can also be decomposed in a different way. For example, let us 3 , defined by 1 and C introduce the integration constants C 2 2(F∗ −Ff ) 1 ≡ −κ 2 Q1 − κ e Q3 C 1 + αf   H 2 (t∗ ) e2(F −Ff ) eϕ1 (k) e3F −2Ff eϕ2 (k) = √ + , 1− 1 + αf ϕ˙ 1 1 + αf ϕ˙2 t∗ 2k 3  2 (t )  eϕ (k) eϕ (k) ∗ 3 ≡ −κ 2 e2(F∗ −Ff ) Q3 = H √ − e2(F −Ff ) 1 , e3F −2Ff 2 C ϕ˙ 2 ϕ˙ 1 t∗ 2k 3

(4.9)

where the subscript f denotes the value at the end of inflation, and α≡

(ϕ1 2F e (ϕ2

.

(4.10)

Then Eq. (4.4) reads  2F   H˙ e (ϕ1 2Ff   + −e Φ = −C1 2 − C3 (ϕ2 . H 1 + αf 1 + αf

(4.11)

This decomposition corresponds to the case where the second term in the rhs of Eq. (4.11) vanishes at the end of inflation.

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During slow-roll we have that |H˙ /H 2 |  1 in Eq. (3.16), which yields from Eqs. (3.39), (4.6), and (4.11) as ζ −

(ϕ + (e2F − 1)(ϕ2 H2 Φ = C1 − C3 1 (ϕ1 + e2F (ϕ2 H˙   (ϕ1 + e2F − (1 + αf )e2Ff (ϕ2 = C˜1 − C˜3 . (1 + αf )((ϕ1 + e2F (ϕ2 )

(4.12) (4.13)

Note that both decompositions coincide each other in the limit αf → 0 and Ff → 0. When αf and Ff are nonvanishing, the second term in Eq. (4.6) gives contributions to the gravitational potential, Φ. When this term is negligible relative to −C1 H˙ /H 2 during the whole stage of inflation, the first and second terms in Eq. (4.6) can be identified as adiabatic and isocurvature modes, respectively. However, in some generalized Einstein theories which we discuss in the following subsections, the final Φ is dominated by the second term in Eq. (4.6). In those cases we can no longer regard the second term as the isocurvature mode at the end of inflation. The relation (4.12) or (4.13) indicates that ζ is not generally conserved during inflation when both fields are evolving due to the presence of isocurvature perturbations. The generalized entropy perturbations are written as 3 e2(Ff −F ) . Sϕ1 ϕ2 = −C3 e−2F = −C

(4.14)

Since Sϕ1 ϕ2 and Z do not vanish generally, these work as a source term for the change of ζ . The evolution of ζ depends on specific gravity theories as we will show below. 4.1. JBD theory Let us first apply the results in the previous section to the JBD theory with Eq. (2.7). In this case Eqs. (4.6) and (4.12) are  2   (β/2)κχ  β H˙ Φ = −C1 2 − C3 (4.15) + e − 1 (φ , 2 H  ζ = C1 − C3 1 −

 1 . e(β/2)κχ + β 2 /(2(φ )

(4.16)

In the JBD theory, β is required to be β
0.09 from observational constraints, which yields (χ = β 2 /2  1. In addition to this, the value of χ should be practically vanishing, χf  0, at the end of inflation in order to reproduce the present value of the gravitational constant, because χ generally grows only as logarithm of t after inflation and is even constant during radiation-dominated stage in this theory. These lead to Ff = (β/4)κχf  0 and αf = e−2Ff (χf /(φf  0, which implies that the decomposition of Eq. (4.11) looks almost the same form as that of (4.6). Since the second term in the rhs of Eq. (4.15) is negligible relative to the first term after inflation, the term proportional to C1 represents the growing adiabatic mode [48], while the one proportional to C3 corresponds to the

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isocurvature mode [28,33]. During inflation, ζ evolves due to the change of the term, W ≡ e(β/2)κχ + β 2 /(2(φ ).

(4.17)

As a specific model of inflation, let us first consider chaotic inflation driven by a potential, V (φ) = λ2n φ 2n /(2n). In this model, inflation occurs √ at large φ and it is ˙ terminated when |φ/φ| becomes as large as H at φ = φf = 2n/κ. Making use of Eqs. (3.17) and (4.2) we find
φ κ

2

   κ 2 2 2n 2  2  V (ϕ) 2 dϕ = φ − 2 = 2 1 − eβκχ/2 = 2 1 − e−β z/2  z.  V (ϕ) 4n κ β β

φf

(4.18) The error in the last expression is less than 12% for z  60 and β  0.09. Then Eq. (4.17) is rewritten as  2 2 β2 W = 1 − e(β/2)κχ + + . (4.19) n n 2n Since W is constant for n = 2, ζ is conserved in the quartic potential, V (φ) = λ4 φ 4 /4. In other cases such as the quadratic potential, ζ evolves during inflation due to the presence of isocurvature perturbations, although its change is typically small. At the end of inflation, ζ takes almost constant value, ζ  C1 . One may worry that the above results are obtained by imposing slow-roll conditions, which are only approximations to the full equations of motion. In order to answer such suspicions, we numerically solved full equations (3.7)–(3.11) along with the background equations (3.3)–(3.6). We adopt the quadratic potential V (φ) = m2 φ 2 /2, and start integrating from about 60 e-folds before the end of inflation. We found that the evolution of field and metric perturbations are well described by analytic estimations except around the end of inflation. The evolution of Φ for the adiabatic and isocurvature mode is plotted in Fig. 1 for the case of β = 0.09, which shows that the contribution of the isocurvature mode is small as estimated by Eq. (4.15). The adiabatic growth of the gravitational potential terminates for mt  20, after which the system enters a reheating stage. During reheating no additional growth of super-Hubble metric perturbations occurs in this scenario, unless some interactions between inflaton and other field, σ , such as g 2 φ 2 σ 2 /2 are not introduced. The evolution of ζ for β = 0.09 is plotted in Fig. 1, which shows that the change of ζ is small during inflation. Since its growth is sourced by the e(β/2)κχ term in Eq. (4.19), we find in Fig. 1 that ζ approaches almost constant value around the end of inflation. During reheating ζ is conserved except for the short period when H˙ passes through zero. Thus one can use analytic expressions for adiabatic curvature perturbations based on slow-roll approximations in order to constrain the model parameters of the potential. Note that the number of e-folds, zk , after the comoving wave-number k leaves the Hubble radius during inflation satisfies k 2 = e−(1−β /2)zk (2zk )n/4 , kf

for 1  zk
60.

(4.20)

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ad ≡ k 3/2 Φad , the isocurvature metric Fig. 1. The evolution of the adiabatic metric perturbation Φ iso ≡ k 3/2 Φiso , and the Bardeen parameter ζ˜ ≡ k 3/2 ζ for the case of β = 0.09 in the perturbation Φ JBD theory with quadratic inflaton potential, V (φ) = m2 φ 2 /2. We choose the initial values of scalar fields as φ = 3mpl and χ = −1.2 mpl , in which case χ is nearly zero at the end of inflation.

We can then express the amplitude of curvature perturbation on scale l = 2π/k as [28] −1  3  2 2k |C1 |2  Φ(l) = 1 + 3(1 + w) 2π

−1 2 β 2 zk /2   κ e 2λ2n nzk n/2 2 = 1+ 3(1 + w) 2π 3n κ2     1 2zk 2 2 × e−β zk /4 (4.21) + 1 − e−β zk /2 , n β which is valid until the second horizon crossing after inflation and also for 1  zk
60. Here w is the ratio of the pressure to the energy density. Since the large-angular-scale anisotropy of background radiation due to the Sachs– Wolfe effect is given by δT /T = Φ/3, we can normalize the value of λ2n by the COBE normalization [13]. For β = 0.09, since this gives the relation, 1 δT 12λ2 /mpl ≡ 12m/mpl for n = 1 √ = Φ(zk  60)  (4.22) for n = 2 41 λ4 T 3 = 1.1 × 10−5 ,

(4.23)

one finds m = 1 × 1013 GeV, n = 1, λ4 = 7 × 10

−14

,

n = 2,

(4.24) (4.25)

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which is not much different from the values obtained assuming the Einstein gravity [52]. Since the behavior of the system approaches to that in the Einstein gravity as we increase ω, we can conclude that in Brans–Dicke theory, model parameters of chaotic inflation should take the same value as in the Einstein gravity. Next we consider new inflation with a potential λ V (φ) = V0 − φ 4 , 4 for which we find
φ κ

2 φf

 V (ϕ) 1 ∼ κ2 κ 2 V0 1 ∼ dϕ = −  z. = V  (ϕ) 2λ φ 2 φf2 2λφ 2

(4.26)

(4.27)

We can again express the amplitude of curvature fluctuation as a function of zk , which is now related with k as k 2 = e−(1−β /2)zk , kf

(4.28)

−1 2 3(1 + w)    2  eβ zk /2  β 2 zk /4 λ 3/2 −β 2 zk /2 (2zk ) + κHf 1−e × e (4.29) , 3 β  with Hf ≡ κ 2 V0 /3. Taking β = 0.09 again, it predicts the amplitude of δT /T to be compared with COBE data as  Φ(l) = 1 +

√ Hf δT (zk  60)  27 λ + 0.49 . T mpl

(4.30)

Since Hf should also satisfy Hf
10−5 , mpl

(4.31)

to suppress long-wave gravitational radiation of quantum origin [53], we obtain λ
2 × 10−13,

(4.32)

from Eq. (4.30). Again its amplitude is practically no different from the case of the Einstein gravity. 4.2. Nonminimally coupled scalar field case Let us first briefly review the single-field inflationary scenario with a non-minimally coupled scalar field (ξ Rφ 2 /2). In chaotic inflationary models, Futamase and Maeda [54] found that the non-minimal coupling is constrained as |ξ |
10−3 in the quadratic potential,

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by the requirement of sufficient amount of inflation. 2 In the quartic potential, such a constraint is absent for negative ξ , and as a bonus, the fine tuning problem of the selfcoupling λ in the minimally coupled case can be relaxed by considering large negative values of ξ [56–58]. Several authors evaluated scalar and tensor perturbations generated during inflation [59–62] and preheating [63] in this model. Since the system is reduced to the single-field case with a modified inflaton potential by a conformal transformation, we cannot expect nonadiabatic growth of ζ on large scales. We shall proceed to the case of the non-minimally coupled ψ field with Eq. (2.10) in the presence of inflaton, φ. In this theory the evolution of field and metric perturbations was studied in Ref. [41] in the Jordan frame. It was found that ζ can grow nonadiabatically during inflation on super-horizon scales for negative ξ . Here we will show that similar results are obtained by the analysis in the Einstein frame. From Eqs. (4.5), (4.6), and (4.12), we obtain the following explicit solutions: δχ = −4(C1 − C3 ) 

ξψ

, 1 − (1 − 6ξ )ξ κ 2 ψ 2    V  (φ)   C1 1 − ξ κ 2 ψ 2 + C3 ξ κ 2 ψ 2 , δφ = − 2 κ V (φ)

Φ = −C1

(4.33)

    H˙ − C3 (ψ − ξ κ 2 ψ 2 (φ , 2 H

(4.34)

(ψ − (ξ κ 2 ψ 2 )(φ , (ψ + (1 − ξ κ 2 ψ 2 )(φ

(4.35)

ζ = C1 − C3 with

8(ξ κψ)2 (ψ = , 1 − (1 − 6ξ )ξ κ 2 ψ 2

 1 V  (φ) 2 (φ = 2 . V (φ) 2κ

(4.36)

When ξ is negative, the coefficient of ψ in the rhs of Eq. (3.15) is always positive, which leads to the rapid growth of ψ (and χ ). Eq. (4.33) indicates that long wave δχ fluctuations are amplified with |ψ| being increased. This is due to the fact that the effective mass of δχ becomes negative after horizon crossing [41], whose property is different from the JBD case. In the JBD case, δχ is almost constant during inflation [see Eq. (4.5) with V1 = e−βκχ ], which restricts the nonadiabatic growth of large scale metric perturbations. In contrast, in the present model, Φ and ζ exhibit strong amplification due to the excitation of low momentum field perturbations unless |ψ| is initially very small. The second terms in Eqs. (4.34) and (4.35) appear in the presence of non-minimal coupling, whose contributions are negligible when |ξ |κ 2 ψ 2  1. With the increase of |ψ|, however, isocurvature perturbations are generated during inflation, which can lead to nonadiabatic growth of Φ and ζ . When the second term in Eq. (4.34) grows to of order the first term, the adiabatic mode includes the isocurvature mode partially. In this case one 2 This constraint is loosened by considering topological inflation, see Ref. [55].

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Fig. 2. The evolution of background fields, φ˜ = φ/mpl and χ˜ = χ/mpl , and large-scale field perturbations, δ φ˜ ≡ k 3/2 δφ/mpl and δ χ˜ ≡ k 3/2 δχ/mpl , in the massive chaotic inflationary scenario with a non-minimally coupled χ field for ξ = −0.02. The initial values of scalar fields are chosen as φ∗ = 3mpl and χ∗ = 10−2 mpl . The slow-roll results of Eq. (4.33) are also plotted, where we denote δ φ˜ (s) and δ χ˜ (s) in the figure. We find that slow-roll approximations are valid except in the final stage of inflation (mt > 17). We remove the evolution of δ φ˜ (s) and δ χ˜ (s) after inflation.

cannot completely decompose adiabatic and isocurvature modes in the final results by the expression, Eq. (4.34). Let us consider the massive chaotic inflationary scenario with initial conditions, φ∗ = 3mpl and χ∗ = 10−2 mpl . We plot in Fig. 2 the evolution of field perturbations for ξ = −0.02 in two cases, i.e, (i) solving directly the perturbed equations, (3.7)–(3.11), (ii) using the slow-roll solutions, Eq. (4.33). In spite of the rapid growth of field perturbations, slow-roll analysis agrees reasonably well with full numerical results. The enhancement of δχk fluctuations stimulates the amplification of δφk fluctuations for mt  10. Inflationary period ends around mt  17, after which the slow-roll results begin to fail. In Fig. 3, the evolution of Φ and ζ is depicted. We also plot the first and the second term in the rhs (s) (s) of Eq. (4.34), where we denote Φ1 and Φ2 , respectively. Although Φ is dominated (s) (s) (s) by the Φ1 term in the initial stage, Φ2 catches up Φ1 around mt  7 , after which (s) the Φ2 term completely determines the evolution of Φ. We find in Fig. 3 that slowroll approximations are valid right up until the end of inflation. The growth of metric perturbations stops when the system enters a reheating stage. If we use the decomposition of Eq. (4.11), the second term gives negligible contribution to the gravitational potential around the end of inflation. In the early stage of inflation, however, its contribution is comparable to the first term, in which case the isocurvature mode cannot be completely separated from the adiabatic mode. When fluctuations are sufficiently amplified, it is inevitable that both adiabatic and isocurvature modes mix each

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 ≡ k 3/2 Φ and the curvature perturbation ζ˜ ≡ k 3/2 ζ Fig. 3. The evolution of the metric perturbation Φ with same parameters as in Fig. 2. The amplification of field perturbations leads to the nonadiabatic (s) ≡ k 3/2 Φ (s) , where Φ (s) and Φ (s) denote (s) ≡ k 3/2 Φ (s) and Φ growth of Φ and ζ . We also plot Φ 1 1 2 2 1 2 (s)

the first and the second terms in Eq. (4.34), respectively. Φ is mainly sourced by the Φ2 term, after (s)

(s)

(s)

(s)

Φ2 catches up Φ1 . We remove the evolution of Φ1 and Φ2 after inflation.

other with the growth of fluctuations, which means that complete decomposition is difficult during the whole inflationary stage. In Fig. 3 the Bardeen parameter, ζ , is nonadiabatically amplified sourced by the second term in Eq. (4.35). Whether this occurs or not depends upon the strength of the coupling, ξ , and the initial χ . When both are small and the second term in Eq. (4.35) is negligible relative to the C1 term during inflation, we can regard the first and second terms in Eq. (4.35) as adiabatic and isocurvature modes, respectively. In the simulations of Figs. 2 and 3, we take χ∗ = 10−2mpl , in which case numerical calculations imply that the conservation of ζ is violated for ξ
−0.01. When ξ
−1, strong amplification of ζ is inevitable even for very small values of χ far less than mpl . For positive ξ , conservation of ζ is typically preserved due to an exponential suppression of χ during inflation [41,64,65]. Regarding detailed investigation about the observational constraints of the strength of ξ , see Ref. [41] whose results are similar to those in the analysis of the Einstein frame. 4.3. Higher-dimensional theories In the higher-dimensional theory with Eq. (2.16), the kinetic term takes√a canonical √ form. In this theory the condition, D > 1, gives the constraint, 2/3 < β < 2, which is different form the JBD theory with β  1. Larger values of β correspond to the steep exponential potential of the χ field, which leads to the rapid evolution toward the χ

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direction. In this case inflaton decreases slowly relative to the χ field. Then the expansion of the universe is described by the power-law solution, Eq. (3.18). For example, in the polynomial inflaton potential, V (φ) = λ2n φ 2n /(2n), classical trajectories of scalar fields are given by Eq. (4.2) as C=

κ2 κ χ + φ2. β 4n

Differentiating Eq. (4.37) with respect to t yields √      χ˙  βκ 2πβ  φ   =  φ˙  2n |φ| = n  m . pl

(4.37)

(4.38)

This relation indicates that for the values of φ greater than mpl with β and n being of ˙ in which case χ rapidly evolves along the order unity, |χ| ˙ is typically larger than |φ|, ˙ grows comparable to |χ|, exponential potential. The power-law inflation continues until |φ| ˙ √ corresponding to φ/mpl  n/( 2πβ). After φ falls down this value, φ begins to evolve faster than χ toward the local minimum at φ = 0 in the φ direction. In this stage the system deviates from the power-law expansion (3.18). From Eqs. (4.6), (4.11), (4.12), and (4.13), Φ and ζ evolve during inflation as Φ = −C1

β 2 − 2αf (φ β2 H˙ H˙   = − C , − C − C 3 1 2 3 2 2(1 + αf ) H2 H

(4.39)

β 2 − 2αf (φ β2 1 − C 3 (4.40) , = C β 2 + 2(φ (1 + αf )(β 2 + 2(φ ) √ √ where αf = β 2 /(2(φf ) is of order unity for 2/3 < β < 2. Since the condition, β 2  (φ , holds during power law-inflation, the Bardeen parameter in this stage takes almost a 1 − C 3 /(1 + αf ). As |φ| ˙ grows relative to |χ|, constant value, ζ  C1 − C3 = C ˙ ζ begins to evolve due to the change of (φ in Eq. (4.40). This corresponds to the stage where deviations from power-law inflation become relevant. When (φ in Eq. (4.40) becomes comparable to 1 . β 2 /2 (i.e., αf  1) around the end of inflation, we have ζ  C1 − C3 αf /(1 + αf ) = C After inflation ζ takes this conserved value. ζ = C1 − C3

4.4. The R 2 theory with a non-minimally coupled χ field In the f (R) theories, effective potentials do not generally take separated forms, Eq. (4.1), as found in Eq. (2.21). Nevertheless we have closed form solutions, Eqs. (3.30)– (3.33), by which the evolution of cosmological perturbations can be studied analytically. Let us analyze the R 2 inflationary scenario with a non-minimally coupled χ field as one example of the f (R) theory [see Eqs. (2.22) and (2.23)]. Note that when χ = 0 the system has an effective potential U=

m4pl (32π)2q

2  √ 1 − e−( 6/3)κφ .

(4.41)

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The φ field defined by Eq. (2.19) plays the role of an inflaton and leads to inflationary expansion of the Universe in the region, φ  mpl [5,66]. In the absence of the nonminimally coupled χ field, the resulting spectrum of density perturbations after the end of inflation was found in [67] (using equations for perturbations of the FRW model for the Einstein gravity with one-loop quantum corrections derived in [68]) and then rederived in [69]. Here we study how the effect of non-minimal coupling alters the adiabatic evolution of cosmological perturbations in the single field case. In the presence of non-minimal coupling, Eq. (3.33) is reduced to J=

√ 6/3)κφ ](1 − ξ κ 2 χ 2 ) ∗ √ . [1 − (1 − ξ κ 2 χ∗2 )e−( 6/3)κφ∗ ](1 − ξ κ 2 χ 2 )

[1 − (1 − ξ κ 2 χ 2 )e−(

Then Eqs. (3.30) and (3.31) are integrated to give √ √ 2 6(1 − ξ κ 2 χ 2 )e−( 6/3)κφ √ δφ = − 3κ[1 − (1 − ξ κ 2 χ 2 )e−( 6/3)κφ ]   ξ κ 2 (χ 2 − χ∗2 ) √ × C1 − C3 , [1 − (1 − ξ κ 2 χ∗2 )e−( 6/3)κφ∗ ](1 − ξ κ 2 χ 2 ) √   1 − (1 − ξ κ 2 χ∗2 )e−( 6/3)κφ √ √ C1 + C3 , δχ = − 1 − (1 − ξ κ 2 χ 2 )e−( 6/3)κφ 1 − (1 − ξ κ 2 χ∗2 )e−( 6/3)κφ∗

(4.42)

(4.43)

4ξ χ

where

C1 = −κ 2 Q1

and

C3 = −κ 2 Q3 .

(4.44)

Therefore, Φ and ζ are expressed as

C3 H˙ √ − 2 H 1 − (1 − ξ κ 2 χ∗2 )e−( 6/3)κφ∗    √   ξ κ 2 (χ 2 − χ∗2 ) ( 6/3)κφ 2 2 × (φ − ( − 1 − ξ κ χ e , χ ∗ 1 − ξ κ 2χ 2

Φ = −C1

ζ = C1 −

(4.45)

C3

√ 6/3)κφ∗  √  (φ ξ κ 2 (χ 2 − χ∗2 ) − (χ e( 6/3)κφ − (1 − ξ κ 2 χ∗2 ) √ , (φ + e( 6/3)κφ (χ

1 − (1 − ξ κ 2 χ∗2 )e−( ×

where (φ and (χ are defined by √  2  1 U,φ 2 4 (1 − ξ κ 2 χ 2 )e−( 6/3)κφ √ = , (φ ≡ 2 2κ U 3 1 − (1 − ξ κ 2 χ 2 )e−( 6/3)κφ √   2 1 U,χ 2 ξ κχe−( 6/3)κφ √ (χ ≡ 2 =8 . 2κ U 1 − (1 − ξ κ 2 χ 2 )e−( 6/3)κφ

(4.46)

(4.47) (4.48)

In the absence of non-minimal coupling, one finds adiabatic results, Φ = −C1 H˙ /H 2 and ζ = C1 . In two-field inflation with a non-minimally coupled χ field, the presence of isocurvature perturbations can lead to nonadiabatic growth of Φ and ζ as found in

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the second terms in Eqs. (4.45) and (4.46). Their contributions are negligible when the conditions, |ξ |κ 2 χ 2  1 and (χ  1, holds during inflation. The latter condition is similar to the former one when |ξ |
1. From Eq. (3.15), since χ˙ is approximately written as χ˙ = −

4ξ H 1 − (1 − ξ κ 2 χ 2 )e−(

√ χ, 6/3)κφ

(4.49)

the χ field exhibits exponential decrease for positive values of ξ . For negative ξ , however, χ is exponentially amplified during inflation, which means that the condition, |ξ |κ 2 χ 2  1, can be violated. The longwave δχ fluctuation grows with the increase of χ as found in Eq. (4.44). On the other hand, the growth of δφ begins only when the second term in Eq. (4.43) becomes comparable to the first term. We plot the evolution of δχ , δφ, and the slow-roll results (4.43) and (4.44) for ξ = −0.025 with initial conditions, φ∗ = 1.1 mpl and χ∗ = 10−3 mpl (see Fig. 4). In this case √ inflation ends around t˜ ≡ mpl t/ 96πq ≈ 130 with e-foldings, N ≈ 63. Again the slowroll analysis is quite reliable except around the end of inflation. In Fig. 5 we also depict the (s) (s) evolution of Φ and ζ , and the first (= Φ1 ) and second (= Φ2 ) terms in Eq. (4.45). In the initial stage of inflation where χ and δχ are not sufficiently amplified, the gravitational potential is dominated by the Φ1(s) term, in which case Φ2(s) may be regarded as the (s) (s) isocurvature mode. However, after t˜ ≈ 70 where Φ2 catches up Φ1 , we find in Fig. 5 that (s) Φ2 mainly contributes to the gravitational potential. As is similar to the case of the chaotic inflationary scenario with a non-minimally coupled χ field, adiabatic and isocurvature modes mix each other with the growth of the χ fluctuation. If one defines the isocurvature mode as the one which gives negligible contribution to the gravitational potential, it can not be completely separated from the adiabatic mode during the whole stage of inflation. The conservation of ζ is typically violated when the term proportional to C3 in Eq. (4.46) surpasses the one proportional to C1 . For the initial values, φ∗ = 1.1 mpl and χ∗ = 10−3 mpl , ζ exhibits nonadiabatic growth for ξ
−0.02. Negative large non-minimal coupling such as ξ
−1 leads to strong amplification of ζ unless χ is initially very small. Although we do not make detailed analysis here, the basic property is quite similar to the case of the Subsection 4.2. These results are also expected to hold for other inflationary models with a non-minimally coupled χ field, since the scalar curvature is proportional to the potential energy of inflaton which slowly decreases during inflation.

5. Conclusion We have studied cosmological perturbations from multi-field inflation in generalized Einstein theories. Most of the generalized Einstein theories recasts to the Lagrangian (2.1) in the Einstein frame by conformal transformations. Making use of slow-roll approximations, we have closed form solutions for field and metric perturbations in long wavelength limit. In multifield inflationary scenarios, not only adiabatic but isocurvature perturbations are generated, which leads to the variation of the curvature perturbation, ζ . While slow-

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Fig. 4. The evolution of background fields, φ˜ = φ/mpl and χ˜ = χ/mpl , and long-wave field √ perturbations, δ φ˜ ≡ k 3/2 δφ/mpl and δ χ˜ ≡ k 3/2 δχ/mpl as a function of time, t¯ ≡ mpl t/ 96πq in the R 2 inflationary scenario with a non-minimally coupled χ field for ξ = −0.025. The initial values of scalar fields are chosen as φ∗ = 1.1 mpl and χ∗ = 10−3 mpl . We also plot the slow-roll results, where we denote δ φ˜ (s) and δ χ˜ (s) in the figure. The evolution of δ φ˜ (s) and δ χ˜ (s) is removed after inflation.

 ≡ k 3/2 Φ and the curvature perturbation ζ˜ ≡ k 3/2 ζ Fig. 5. The evolution of the metric perturbation Φ 2 (s) ≡ k 3/2 Φ (s) in the R inflationary scenario with same parameters as in Fig. 4. We also plot Φ 1 1 (s) (s) (s) (s)  ≡ k 3/2 Φ , where Φ and Φ and Φ denote the first and the second terms in Eq. (4.45), 2

2

1

2 (s)

(s)

respectively. We remove the evolution of Φ1 and Φ2 after inflation.

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roll analysis provides general expressions of large-scale perturbations in unified manner, the evolution of cosmological perturbations depends on specific gravity theories. In this work we considered the following four gravity theories. (1) The Jordan–Brans–Dicke theory with a Brans–Dicke field χ and inflaton φ. Using the Brans–Dicke parameter constrained by observations, the isocurvature mode in the gravitational potential Φ is negligible relative to the adiabatic mode. Therefore, the variation of ζ is typically small in this theory. In particular, for the quartic potential, ζ is conserved in the slow-roll analysis. (2) A non-minimally coupled scalar field χ in the presence of inflaton φ. When the coupling ξ is negative, χ and its long-wave fluctuations exhibit exponential increase during inflation, leading to the nonadiabatic amplification of Φ and ζ due to the existence of isocurvature perturbations. Even in this case we find that slow-roll analysis agrees well with full numerical results. When field and metric perturbations are sufficiently amplified, adiabatic and isocurvature modes of the gravitational potential mix each other. (3) Higher-dimensional Kaluza–Klein theory with dilaton χ and inflaton φ. In this theory the inflationary period can be divided into two stages: the first is the power-law inflationary stage where χ evolves along the exponential potential and the second is the deviation from power-law inflation due to the rapid evolution of φ around the end of inflation. In the former stage ζ is nearly constant but its change occurs at the transition between two stages. (4) R 2 theory with a non-minimally coupled scalar field χ . This system has an additional scalar field φ playing the role of inflaton after conformal transformations. Although this theory has a coupled effective potential which is different from the above theories (1)– (3), we have integrated forms of long-wave field and metric perturbations by slow-roll analysis, which is found to be quite reliable right up until the end of inflation. Negative non-minimal coupling again leads to the nonadiabatic growth of Φ and ζ , in which case complete decomposition between adiabatic and isocurvature modes is difficult. While we analyzed generalized Einstein theories involving two scalar fields, there exist other multi-field inflationary scenarios such as hybrid inflation [70] and models of two interacting scalar fields [71]. During preheating after inflation, there has been growing interest about the evolution of cosmological perturbations for the simple two-field model with potential U (φ, χ) = λφ 4 /4 + g 2 φ 2 χ 2 /2 [64,72]. It is certainly of interest to constrain realistic multi-field models based on particle physics by produced density perturbations, together with the constraints by gravitinos [73] and primordial black hole over-production during preheating [74].

Acknowledgements A.S. is thankful to Profs. Katsuhiko Sato and Masahiro Kawasaki for hospitality in RESCEU, the University of Tokyo. He was also partially supported by the Russian Fund for Fundamental Research, grants 99-02-16224 and 00-15-96699. S.T. thanks Bruce A. Bassett, Kei-ichi Maeda, Naoshi Sugiyama, Atsushi Taruya, Takashi Torii, Hiroki

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Yajima, and David Wands for useful discussions. He was supported by the Waseda University Grant for Special Research Projects. The work of J.Y. was partially supported by the Monbukagakusho Grant-in-Aid, Priority Area “Supersymmetry and Unified Theory of Elementary Particles”(#707) and the Monbukagakusho Grant-in-Aid for Scientific Research No. 112440063.

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