Effective field theory approach to quasi-single field inflation and effects of heavy fields

Abstract

We apply the effective field theory approach to quasi-single field inflation, which contains an additional scalar field with Hubble scale mass other than inflaton. Based on the time-dependent spatial diffeomorphism, which is not broken by the time-dependent background evolution, the most generic action of quasi-single field inflation is constructed up to third order fluctuations. Using the obtained action, the effects of the additional massive scalar field on the primordial curvature perturbations are discussed. In particular, we calculate the power spectrum and discuss the momentum-dependence of three point functions in the squeezed limit for general settings of quasi-single field inflation. Our framework can be also applied to inflation models with heavy particles. We make a qualitative discussion on the effects of heavy particles during inflation and that of sudden turning trajectory in our framework.

Published for SISSA by Springer

Received: December 2, 2012 Revised: April 8, 2013 Accepted: May 19, 2013

Published: June 13, 2013

E ective eld theory approach to quasi-single eld ination and e ects of heavy elds

Toshifumi Noumi,a,1 Masahide Yamaguchib and Daisuke Yokoyamab,2

aInstitute of Physics, University of Tokyo,

Komaba, Meguro-ku, Tokyo 153-8902, Japan

bDepartment of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] ,mailto:[email protected]

Web End [email protected]

Abstract: We apply the e ective eld theory approach to quasi-single eld ination, which contains an additional scalar eld with Hubble scale mass other than inaton. Based on the time-dependent spatial di eomorphism, which is not broken by the time-dependent background evolution, the most generic action of quasi-single eld ination is constructed up to third order uctuations. Using the obtained action, the e ects of the additional massive scalar eld on the primordial curvature perturbations are discussed. In particular, we calculate the power spectrum and discuss the momentum-dependence of three point functions in the squeezed limit for general settings of quasi-single eld ination. Our framework can be also applied to ination models with heavy particles. We make a qualitative discussion on the e ects of heavy particles during ination and that of sudden turning trajectory in our framework.

Keywords: Spontaneous Symmetry Breaking, Cosmology of Theories beyond the SM, Space-Time Symmetries

ArXiv ePrint: 1211.1624

JHEP06(2013)051

1Present address: Mathematical Physics Laboratory, RIKEN Nishina Center, Saitama 351-0198, Japan.

2Present address: Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea.

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP06(2013)051

Web End =10.1007/JHEP06(2013)051

c

Contents

1 Introduction 1

2 E ective eld theory approach and quasi-single eld ination 32.1 E ective eld theory approach to ination 32.2 Quasi-single eld ination 6

3 Most generic action of quasi-single eld ination 73.1 Action in the unitary gauge 73.2 Ambiguity of the action in the unitary gauge 93.3 Action for the Goldstone boson and the decoupling regime 113.4 Examples 143.4.1 Original model discussed by Chen and Wang 143.4.2 E ects of heavy particles 143.4.3 A class of two-eld models 16

4 Power spectrum 194.1 Constant turning trajectory 224.2 Qualitative features of sudden turning trajectory 25

5 Three point functions in the squeezed limit 27

6 Summary and discussion 30

A Integrating out heavy elds 31A.1 Role of kinetic term 32A.2 Derivative coupling, partial integral, and Hamiltonian formalism 34A.3 Extension to cosmological perturbation 36

B Numerical calculations of power spectrum 37

C Analytical calculation of power spectrum for c = c 39

C.1 Asymptotic behavior of D(, , x) 44

1 Introduction

Ination gives the most natural solution to the horizon and the atness problems of the big-bang theory as well as generates the primordial perturbations [1, 2], whose properties well coincide with the recent observations of cosmic microwave background anisotropies like the Wilkinson Microwave Anisotropy Probe [3]. Models of ination can be classied into two

1

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categories with respect to relevant degrees of freedom during ination: single-eld models and multiple eld models. Recently, the most general single eld ination model with the second order equations of motion [4] has been invented in the context of Horndeski [5, 6] and Galileon theories [79]. Then, the bispectra of primordial curvature [1015] and tensor perturbations [16, 17] are obtained as well as their powerspectra [4].

E ective scalar elds are ubiquitous in the extensions of the standard model of particle physics such as supergravity and superstring. Then, it is well motivated to consider multiple eld models of ination. Such multiple eld models are roughly divided into three classes:(i) only one eld is light, while the other elds are very heavy compared to the Hubble scale during ination. Generically, this class virtually falls into the single eld category [18]. However, it is recently discussed that heavy modes can a ect the dynamics of light mode in some particular cases [1928]. (ii) there are multiple light elds, in which isocurvature perturbations are generated in addition to curvature perturbations. (iii) only one eld is light, while the masses of other elds are comparable to the Hubble scale during ination. This class is called quasi-single eld ination model [2931].

In supergravity, ination necessarily involves supersymmetry (SUSY) breaking, whose e ects are transmitted into other scalar elds as Hubble induced masses [3235].1 Therefore, quasi-single eld ination is naturally realized in supergravity and it is well motivated by the model building based on supergravity or inspired by superstring. Furthermore, it is known that massive isocurvature modes which couple to the inaton and have Hubble scale masses can give signicant impacts on primordial curvature perturbations. In the original paper [29, 30] by Chen and Wang, it was shown that, for example, scalar three point functions take the intermediate shapes between local and equilateral types. Based on these backgrounds, we would like to discuss quasi-single eld ination model in general settings.

Recently, e ective eld theory approach to ination has been invented in [3840], which is based on the symmetry breaking during ination: time di eomorphism is broken by the time-dependent background evolution during ination. Then, based on the unbroken time-dependent spatial di eomorphism, the e ective action for ination can be constructed systematically in unitary gauge, where inaton is eaten by graviton and there are no perturbations of inaton. By use of the Stckelberg trick, the curvature perturbation can be associated with the Goldstone boson , which non-linearly realizes the time di eomorphism. The key observation is that the Goldstone could decouple from the metric uctuations in some parameter region which we call the decoupling regime. In the decoupling regime, the dynamics of Goldstone is described by a simplied action, which does not contain metric perturbations. As a consequence, calculations of scalar perturbations are also simplied and seeds of non-Gaussianities become clear.

In this paper, we apply this e ective eld theory approach to quasi-single eld ination. First, in unitary gauge, we write down the most general action invariant under the time-dependent spatial di eomorphism and constructed from graviton and the massive isocurvature mode. The obtained action is expanded systematically in uctuations and derivatives around the FRW background. By the Stckelberg trick, we introduce the

1The methods to keep inaton at against such SUSY breaking e ects are reviewed in refs. [36, 37].

2

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action for Goldstone boson and carefully discuss its decoupling regime. Using the action in the decoupling regime, the power spectrum is calculated in the general setting of quasi-single eld ination. The momentum dependence of scalar three-point function is also discussed in the general setting. Our framework can be also applied to ination models with heavy particles. As an application, we make a qualitative discussion on the e ects of heavy particles during ination and that of sudden turning trajectory.

The organization of this paper is as follows. In the next section, we briey review the e ective eld theory approach and quasi-single eld ination. In section 3, the most general action for quasi-single eld ination is constructed via e ective eld theory approach. The decoupling regime of the obtained action is also discussed. In section 4, the power spectrum is calculated rst in the general setting of quasi-single eld ination with constant mixing couplings. Then, the e ects of sudden turning trajectory on the power spectrum is qualitatively discussed. In section 5, the momentum dependence of scalar three-point functions are discussed. Final section is devoted to summary and discussions. Technical details of the calculation of the power spectrum are summarized in appendices.

2 E ective eld theory approach and quasi-single eld ination

In this section we briey review the e ective eld theory approach to ination developed in [38] and the quasi-single eld ination model proposed in [29, 30].

2.1 E ective eld theory approach to ination

Ination is an accelerated cosmic expansion with an approximately constant Hubble parameter:

ds2 = dt2 + a2(t)dxidxi with H(t) =

aa , =

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HH2 1 . (2.1)

It is characterized by the spontaneous breaking of the time-di eomorphism:

h(t, x)i = 0(t) , (2.2)

where (t, x) is a certain scalar operator. Here we chose the frame in which the vacuum expectation value of (t, x) is spatially uniform. Assuming the degrees of freedom relevant to the cosmological perturbation and invariance under the time-dependent spatial di eo-morphism, xi xi = xi + i(t, xj), which is not broken by the condensation 0(t), we can

construct the e ective action for ination.

In the simplest case, relevant degrees of freedom are three physical modes of graviton: two transverse modes and one longitudinal mode related to the inaton. As discussed in [38], any action of graviton invariant under the time-dependent spatial di eomorphism can be written in terms of the Riemann tensor R, the time-like component of the metric g00, the extrinsic curvature K on constant-t surfaces, the covariant derivative , and

the time coordinate t:

S =

Z

d4xg F (R, g00, K, , t) , (2.3)

3

where all the free indices inside the function F must be upper 0s. Note that g00 should be treated as a scalar when considering its covariant derivative, and we can use g00 for

example. The explicit form of the extrinsic curvature K is

K =hn =

0g00 + 0g00

2(g00)3/2

00g0g00

2(g00)5/2

+ g0(g + gg)2(g00)1/2

, (2.4)

pg00is a unit vector perpendicular to constant t surfaces and h = g + nn is the induced spatial metric on constant t surfaces. In [38], it was shown that the action (2.3) can be expanded around a given FRW background as

S =Z

where n =

0

d4xg

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1

2M2PlR + M2Pl

Hg00 M2Pl(3H2 +

H)

+F (2)(g00, K, R; 0, g, g, , t)

, (2.5)

where the function F (2) starts with quadratic terms of the arguments g00, K, and R and all the free indices must be upper 0s. The arguments g00, K, and R are dened by2

g00 = g00 + 1 , (2.6)

K = K Hh , (2.7) R = R 2H2h[h] + (

H + H2)(h00 + (3 permutations)) . (2.8)

They are covariant under time-dependent spatial-di eomorphism and vanish on the FRW background. Notice that the action of single eld ination in the uniform inaton gauge can be reproduced by gauge-xing the time-dependent spatial di eomorphism as

gij(x) = a2(t)e2(x)(e(x))ij with ii = iij = 0 , (2.9)

where (x) is the scalar perturbation.

For the calculation of correlation functions of the scalar perturbation , it is convenient to introduce the action for the Goldstone boson by the Stckelberg method. We perform the following time-di eomorphism on the action (2.5) in the unitary gauge:

t ~t, xi ~xi with ~t+ ~

(~t, ~x) = t , ~xi = xi . (2.10)

In general, the transformation (2.10) is realized by the following replacement:

0 0 + , f(t) f(t + ) , Z

d4xg ,

, g g , g g , R R , (2.11)

where we dropped the tilde for simplicity and g00 transforms, for example, as

g00 g00 + 2g0 + g . (2.12)

2Here and in what follows, we concentrate on the spatially at FRW background.

4

d4xg Z

The transformation rules of K and h also follow from (2.11) straightforwardly. These procedures lead to the following action for the Goldstone boson :

S = Z

d4xg

1

2M2PlR + M2Pl

H(t + ) g00 + 2g0 + g

M2Pl

3H2(t + ) + H(t + )

, (2.13)

where the dots stand for the terms corresponding to F (2). The obtained action enjoys the time-di eomorphism by assigning to the non-linear transformation rule3

(x) ~

(~x) = (x) 0(x) with t ~t= t + 0(x) , xi ~xi = xi , (2.14)

and the action in the unitary gauge can be reproduced by gauge-xing the time-di eomorphism as (x) = 0.

It is important to recognize that, in the action (2.13), terms with graviton uctuations have less derivatives than those without graviton. Because of this property, it is expected that the mixing of the Goldstone boson and graviton becomes irrelevant to the dynamics of the Goldstone boson at a su ciently high energy scale. For example, let us consider the following simplest case:

S = Z

+ . . .

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d4xg

1

2M2PlR + M2Pl

H(t + ) g00 + 2g0 + g

M2Pl

3H2(t + ) + H(t + )

. (2.15)

In the canonical normalization (c MPl(

H)1/2, gc MPlg), the mixing term

M2Pl H g0 can be written as

M2Pl H g0 (

H)1/2 g0c c 1/2H g0c c , (2.16)

and it can be neglected in the energy scale E 1/2H. In other words, when the slow-roll

parameter is small, the mixing becomes irrelevant inside the horizon. The dynamics of inside the horizon are then determined by the following action in the decoupling limit:

S

Z

d4x a3

, (2.17)

where the metric reduces to the FRW background. More generally, the mixing of the Gold-stone boson and graviton becomes irrelevant inside the horizon when the free parameters of the action are in some regime (decoupling regime) as in the slow-roll regime for the above

3Here it should be noted that while the action is invariant under the time-di eomorphism, it does not have the shift symmetry + constant because of the time-dependent free parameters such as H(t + )

or H(t + ). Expanding the parameters in , we nd, for example, that has a mass term M2Pl

H22,

M2Pl H(t+)

12 2 + (i)2 a2

M2Pl(3H2(t+)+ H(t+))

which is sub-leading in the slow-roll approximation.

5

simplest case. In the decoupling regime, the calculation of scalar correlation functions also becomes tractable. Taking the spatially at gauge4

gij(x) = a2(t)(e(x))ij with ii = iij = 0 , (2.18)

the scalar perturbation (x) is given by (x) = H(t)(x) at the linear order, and the

calculation of correlation functions of reduces to those of , which can be obtained using the simplied action in the decoupling limit. This kind of simplication in the decoupling regime is one of the advantages to use the e ective eld theory approach. In the next section we extend this approach to quasi-single eld ination.

2.2 Quasi-single eld ination

The original model [29, 30] of quasi-single eld ination is described by the following matter action:

Smatter =

Z

d4xg

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1

2(

~R + )2g

1

2g Vsr() V ()

, (2.19)

where and are the tangential and radial directions of a circle with radius ~R and the potential Vsr() along the tangential direction is of slow-roll type. The homogeneous backgrounds and their equations of motion are given by

= 0(t) , = 0 (constant) ,

3M2PlH2 = 1

2R2

20 + V (0) + Vsr(0) , 2M2Pl

H = R2

20 , (2.20)

0 + V sr(0) = 0 ,

where R = ~R + 0. Expanding the action around the homogeneous background, it yields the following second order action of the uctuations = 0 and = 0:

S(2)matter =

Z

d4xg

V (0) = R

20 , R2

0 + 3R2H

1

2R2g

1

2g R

0 0

1

2(V (0)

20)2

.

(2.21)

The mixing coupling

converts the 3 coupling, for example, into three point functions of , and hence this model can potentially give a large non-Gaussianities. Furthermore, it is known that the squeezed limit of scalar three point functions is sensitive to the mass of :

lim

k3/k1=k3/k2=0

hk1k2k3i 3/2k61 , (2.22)

where =

r94 m2

H2 and m2 = V (0)R20 < 94H. As this simple model implies, massive

scalar elds of Hubble scale mass can cause a non-trivial behavior of non-Gaussianities. In the following sections we discuss more general setting for quasi-single eld ination using the e ective eld theory approach.

4Strictly speaking, the name of this gauge may be inadequate because there are still tensor uctuations and hence the spatial hypersurface is not exactly at.

6

3 Most generic action of quasi-single eld ination

In this section, we construct the most generic action of quasi-single eld ination using the e ective eld theory approach. After constructing the action in the unitary gauge rst, we derive the action for the Goldstone boson and discuss its decoupling regime. Relations between our approach and models in the literatures are also discussed.

3.1 Action in the unitary gauge

In the unitary gauge, the relevant degrees of freedom in quasi-single eld ination are three physical modes of graviton and an additional scalar eld . The typical mass of is supposed to be of the order of the Hubble scale during the inationary era. In this subsection, we construct the most generic action invariant under the time-dependent spatial di eomorphism from graviton and the scalar eld up to the third order uctuations. Here it should be noticed that the action constructed in this section can be applied to any two eld models because no conditions on are imposed.5

Extending the procedures in [38] to our case, the most general action invariant under the time-dependent spatial di eomorphism is given by

S = Z

d4xg F (R, g00, K, , t, ) , (3.1)

and it is expanded around the given FRW background as

S =Z

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d4xg

1

2M2PlR + M2Pl

Hg00 M2Pl(3H2 +

H)

+F (2)(g00, , K, R; 0, g, g, , t)

, (3.2)

where all the free indices inside the functions F and F (2) must be again upper 0s and F (2) starts with quadratic terms of the arguments g00, , K, and R.6 Then, let us

5For example, we do not require the shift symmetry of , + constant, which is assumed in multi-

eld ination [39]. The mass of is not necessarily of order Hubble scale so that the action constructed in this section can be applied not only for quasi-single eld ination but also ination models with an additional heavy scalar.

6In general, sums of terms linear in the uctuations can be practically second order. For example, let us consider the term

[integraltext]

d4xg [bracketleftbig]f

. Although this kind of action seems to be rst order apparently, it turns out to be second order after taking into account the equation of motion for : f1(t) + f2(t) + 3Hf2(t) = 0. Then, the function F (2) seems to contain such a combination of linear order terms. However, using the relation

[integraldisplay] d4xgf(t)0(. . .) = [integraldisplay] d4xgf(t)[radicalbig]g00n

(. . .)

1(t) + f2(t)0

= [integraldisplay] d4xg[parenleftbigg] g00

f(t)

1

2 f(t)0 ln(g00) + f(t)[radicalbig]g00K

(. . .)

= [integraldisplay] d4xg[parenleftBig]

f(t) + 3Hf(t) + O(g00, K)[parenrightBig]

(. . .) , (3.3)

we can rewrite it into the second and higher order terms in , g00, and K. Similar discussions hold for more general cases and we conclude that F (2) starts with quadratic terms of g00, , K , and R . See also appendix B in [38].

7

write down possible terms in the action up to the third order uctuations. Schematically, we write the action in the following way:

S = Sgrav + S + Smix , (3.4)

where the rst term Sgrav in the right hand side denotes terms constructed from g00, K, and R, the second term S denotes those only from , and the last term Smix denotes those mixing the graviton uctuations and . As discussed in [38], the rst term Sgrav can be expanded as

Sgrav =

Z

d4xg

1

2M2PlR + M2Pl

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H(t)g00 M2Pl

3H2(t) + H(t)

+ M42(t)

2! (g00)2 +

M43(t)

3! (g00)3 + . . .

, (3.5)

where the dots stand for terms of higher order in the uctuations or with more derivatives. When we rewrite the action in the unitary gauge in terms of the Goldstone boson , the terms displayed in (3.5) are described by and its rst order derivatives. In this paper, we consider the action up to the same order in derivatives of and .

Let us rst construct the second order action. The second order action S(2) containing and its rst order derivative can be written generally as

S(2) = Z

d4xg

1(t)

2 g +

2(t)

2 (0)2

3(t)

2 2 + 4(t)0

, (3.6)

where we note that terms such as (0)2 can be absorbed into other terms up to higher order uctuations by integrating by parts. As discussed in [39], the second term leads to a non-trivial sound speed c of given by c2 = 1/(1 + 2). The second order mixing S(2)mix is generally given by

S(2)mix = Z

d4xgh

1(t)g00 + 2(t)g000 + 3(t)K

i

. (3.7)

It is convenient to note the relation

Z

d4xgf(t)K(. . .) = Z

d4xg

f(t)0(. . .)

f(t) + 3Hf(t)

(. . .)

f(t)

+ 2 g00(. . .) +

f(t)

2 g000(. . .) + . . .

, (3.8)

which can be obtained using (3.3) in footnote 6 twice. Here the last dots stand for higher order terms in the uctuations, which can be written using g00, 0g00, and K. Using

this relation (3.8), S(2)mix in (3.7) is rewritten as

S(2)mix =

Z

d4xgh

1(t)g00 + 2(t)g000 + 3(t)0 (

3(t) + 3H3(t))

i

, (3.9)

where 1 = 1 +

3/2, 2 = 2 + 3/2, and we dropped higher order terms constructed from , g00, 0g00, and K. In the following, we employ eq. (3.9) as a denition of the second order mixing action S(2)mix.

8

The third order action S(3) of is generally given by

S(3) = Z

d4xgh

1(t)3 + 2(t)20 + 3(t)(0)2 + 4(t)(0)3

+ 5(t) + 6(t)0

i

1(t)g00 2 + 2(t)g00 0 + 3(t)g00(0)2 + 4(t)0g000

+ 5(t) g00 + 6(t)(g00)2 + 7(t)(g00)20

i

. (3.11)

Here, it may be wondered if the terms such as K 2, K 0, R002, and Kg00 can appear at the same order in derivatives. However, they can be absorbed into other third order terms in (3.11) and the second order terms in (3.6) and (3.9) by integrating by parts as we did to rewrite (3.7) into (3.9). The term proportional to R2 can also appear but such a term vanishes in the decoupling limit, so we do not consider it here for simplicity.

To summarize, the most generic action in the unitary gauge can be written up to the third order uctuations as follows:

S = Sgrav + S(2) + S(2)mix + S(3) + S(3)mix , (3.12)

where Sgrav, S(2), S(2)mix, S(3), and S(3)mix are dened in (3.5), (3.6), (3.9), (3.10), and (3.11).

3.2 Ambiguity of the action in the unitary gauge

In multiple eld ination models, there are some ambiguities of the action in the unitary gauge: there are degrees of freedom of the eld redenition of and time coordinate transformations vanishing on the background trajectory = 0. Using these degrees of freedom, it is possible to drop some terms and simplify the action. Without loss of generalities, the action can be written into the following three normalizations using these ambiguities.7

Normalization 1: 1 + 2 = 1. Let us rst consider the kinetic term of . The second order action S(2) of can be expanded up to the second order uctuations as

Z

derivative of can be written as

, (3.14)

the action still takes the form (3.6) after the redenition.

7Note that it is not possible in general to impose some of the three conditions at the same time.

9

, (3.10)

and the third order mixing S(3)mix is given by

S(3)mix = Z

d4xgh

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2 c2(i)2a2 3 3H4 41 + 2 2 , (3.13)

where the normalization factor is dened as 2 = 1 + 2. Although the factor is time-dependent in general, it can be taken unity by redening as ~

= . Since the

d4x a3

2 2

= 1~

0

Normalization 2: 4 = 3 = 2 = 0. Using the time coordinate transformation vanishing on the background trajectory = 0, the following form of interaction terms can be eliminated:

Z

d4xgh

f(t, )0

i

, (3.15)

where f(t, ) is a function of t and but does not contain derivatives of . As a simple example, let us consider the action

S = Z

d4xg

1

2M2PlR + M2Pl

H(t)g00 M2Pl

3H2(t) + H(t)

+ f(, t)0

. (3.16)

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Under the time coordinate transformation

t ~t with t = ~t (~t, ) , (3.17) the action (3.16) is transformed into

S = Z

d4xg

1

2M2PlR + M2Pl

H(t )

g00(1 t)2 2(1t) 0 + ()2

M2Pl

3H2(t) +

H(t)

+f(, t)(1t)0 f(, t)

. (3.18)

Therefore, if we take such that

= f(, t ) 2M2Pl H(t )

, (t, = 0) = 0 , (3.19)

the action (3.18) reduces to

S = Z

d4xg

1

2M2PlR M2Pl

3H2(t ) +

H(t )

+ M2Pl H(t )

g00(1 t)2 ()2

f(, t )

, (3.20)

which does not contain interaction terms in the form f(t, )0. Note that the conditions (3.19) can be always solved at least as an expansion in . It is straightforward to extend this discussion to general cases and we conclude that interaction terms in the form of (3.15) can be eliminated using the time coordinate transformation vanishing on the background trajectory = 0. In particular, we can set 4 = 3 = 2 = 0 without

loss of generalities.

Normalization 3: 1 = 1, 5 = 0. The action S in eq. (3.12) contains the following term:

Z

d4xg

1

2f2(t, )g

, (3.21)

where f(t, ) is a function of t and , and does not contain derivatives of . The function is expanded in as f2(t, ) = 1(t) 25(t) + O(2). By the eld redenition, it is possible

10

to eliminate this kind of derivative couplings and to rewrite (3.21) into the canonical form of the kinetic term. Let us dene ~

as ~

= F (t, ) such that

F (t, ) = f(t, ) , F (t, = 0) = 0 . (3.22)

Since the derivative of ~

is given by

~

= f + 0tF , (3.23)

the action can be rewritten as

Z

d4xg

1

2g~

~

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+ tF 0~

1

2g00 (tF )2

, (3.24)

which still takes the form (3.12). Here F should be regarded as a function of ~

and t.

It also should be noticed that, though we can set the function f to be unity, the terms proportional to 0 and g00 appear.

3.3 Action for the Goldstone boson and the decoupling regime

In this subsection we construct the action for the Goldstone boson and discuss its decoupling regime. As in the last section, we perform the time di eomorphism (2.10). Practically, it is realized by the following replacements:

g00 g00 + 20 + , 0 0 + ,

, f(t) f(t + ) ,

Z

d4xg Z

d4xg . (3.25)

With these replacements, Sgrav, S(2), and S(2)mix are rewritten as

Sgrav =

Z

d4xg

1

2M2PlR+M2Pl

H(t+) g00+20+

M2Pl 3H2(t+)+ H(t+)

+ M42(t + )

2! g00 + 20 +

2 + M43(t + ) 3!

g00 + 20 +

,

(3.26)

3

S(2) =

Z

d4xg

1(t + )

2 +

2(t + )

2 0 +

2

3(t + )

2 2 + 4(t + )(0 + )

, (3.27)

S(2)mix =

Z

d4xg

1(t + ) g00 + 20 +

+ 2(t + ) g00 + 20 +

+

+ 3(t + )(0 + )

0

3(t + ) + 3H(t + )3(t + )

. (3.28)

The third order actions S(3) and S(3)mix can be obtained in a similar way.

11

In order to discuss the decoupling regime of the action, we rst clarify in which regime graviton uctuations become irrelevant to tree-level three point functions of . For this purpose, let us take the spatially at gauge (2.18) and use the ADM decomposition:

ds2 = (N2 NiNi)dt2 + 2Nidxidt + a2(e)ij dxidxj with ii = iij = 0 . (3.29) Here and in what follows we use the spatial metric hij = a2(e)ij and its inverse hij =

a2(e)ij to raise or lower the indices of Ni. The inverse metric g are written in terms of N, Ni, and hij as

g00 =

1N2 , g0i = gi0 =

NiN2 , gij = hij

NiNj

N2 . (3.30)

In this gauge, there are no second order mixing terms of and ij because ij has two spatial indices and is transverse-traceless. Then, the tensor uctuation ij does not contribute to tree-level three point functions of . Therefore, possible contributions of graviton uctuations come only from the auxiliary elds N = N 1 and Ni. As discussed in [41], it

is su cient for the calculation of three-point functions to solve the constraints up to rst order. Expanding the actions (3.26)(3.28) up to the second order in , N, and Ni,

Sgrav =

Z

d4xa3

M2Pl(3H2+c2H)N22M2PlHNiNi+M2Pl14NiijNj M2Pl1 4Ni2Ni

JHEP06(2013)051

M2Pl H c2

2c2 (i)2 a2

3M2PlH22+M2Pl(2c2 H 6HH)N +2M2Pl HNii

,

(3.31)

S(2) =

Z

d4x a3

2

2 c2

(i)2

a2

3

2 2 4

, (3.32)

S(2)mix =

Z

d4x a3

21 (N ) 22 (N ) N

3

+

3 + 3H3

+ 3Nii + 3

+ ii

a2

+

3

3

H3

, (3.33)

the constraints are solved up to rst order as follows:

N =

HH

3 2M2PlH

, Ni = a2i

with = a22

c2HH2 (H + H ) + 1M2PlH 2M2PlH + 3 2M2PlH

c2HH +

. (3.34)

Here the sound speed c2 of are dened as c2 = HM2Pl/( HM2Pl 2M42). The factors

c2 = 1/(1 + 2) and 2 = 1 + 2 are the sound speed and the normalization factor of , respectively. Using the canonical normalization

c MPl(

H)1/2c1 , c , Nc MPlN , Nic MPlNi , (3.35)

12

3 2M2PlH

and redening the coupling constants 1, 2, and 3 correspondingly as

c1

c MPl(

H)1/2 1 , c2

c MPl(

H)1/2 2 , c3

c MPl(

H)1/2 3 , (3.36)

we rewrite the constraints (3.34) as

Nc ~

1/2

c2c 12c3 c , (3.37)

Nic ~

1/2 i

2

c + 12 ~

Hc + c1c c2

c

1

2c3

c

JHEP06(2013)051

+ 1

2c3c

2

+ (c2 1)~H

~

2H

1

2

!, (3.38)

where we have dened ~

~

~

H in analogy with usual slow-roll parameters and . It is manifest that Nc and Nic are suppressed by the parameter ~

1/2 and contri-

= c2

HH2 and ~

=

butions from Nc and Nic become irrelevant in the limit ~

0.8 In this limit, tree-level

three point functions of are determined by the following action in the decoupling limit:

Sgrav =

Z

d4xa3

M2Pl H c2

2 c2 (i)2 a2

M2PlH(c2 1) 3 (i)2 a2

4M433 3

3M2Pl

H22 t

M2Pl H c2

2 + M2Pl H (i)2a2 3M2plH H3

, (3.39)

S(2) =

Z

d4xa3

2 2

2c2(i)2a2 33H4 42 2 +2(1c2) 2 (ii) a2

4

iia2 +(2)2 2 (2c2)2 (i)2a2 32 2 4 , (3.40)

S(2)mix =

Z

d4x a3

(21 +3) + (22 3) + 3 iia2 3 H3

1

2 (i)2 a2

+ (21 + 3) + 32 2 22 ii a2

2

(i)2

a2

+ (2

2

3)

+

3

ii a2

3 H 3 + 12 H3

2 , (3.41)

S(3) =

Z

d4x a3

1 + H2 + 13

2

3 + (3 5) 2 + (4 + 6) 3

+ 5 (i)2a2 6

(i)2 a2

, (3.42)

S(3)mix =

Z

d4x a3

21

2 + 22

+ 2(3 + 5)

2 24

25

(i)2 a2

. (3.43)

8To be precise, we need to assume that ~

is small enough to vanish when multiplied by other parameters in the calculation. For example, we need to assume that ~

HE 1.

+ 46

2 47

2

13

It should be noticed that non-trivial cubic interactions appear generically when the sound speed c of is small, 4 is non-zero, or mixing couplings 1 and 2 exist as well as the sound speed c of is small.

3.4 Examples

Before closing this section, we clarify the relation between our approach and models in the literatures. For this purpose, we rst discuss the original model of quasi-single eld ination [29, 30], and then, we investigate the e ects of heavy particles during ination. At the end of this subsection, a class of two eld models will be considered.

3.4.1 Original model discussed by Chen and Wang

As was reviewed in section 2, the original model [29, 30] of quasi-single eld ination is described by the following matter action:

Smatter =

Z

d4xg

JHEP06(2013)051

1

2(

~R + )2g

1

2g Vsr() V ()

. (3.44)

The homogeneous backgrounds are given by = 0(t) and = 0 (constant), which leads to the action in the unitary gauge = 0 = 0,

Smatter =

Z

d4xg

1

2(R + )2g00

20

1

2g Vsr(0) V (0 + )

= d4xg

1

2R2

20 g00 (Vsr(0) + V (0))

1

2g

1

2

V (0) 20

2

R

20 g00

V (0)

3! 3

1

2

20 g002 + O(4)

, (3.45)

where R = ~R + 0, = 0 and we used the background equations of motion. Using

the equations of motion (2.20), the action (3.45) can be written in terms of the Hubble parameter H as

Smatter =

Z

d4xg

M2Pl Hg00M2Pl(3H2+

H)

1

2g

1

2

V (0)+ 2M2Pl H R2

2

+ 2M2Pl

HR g00

V (0)

3! 3 +

M2Pl H

R2 g002 + O(4)

, (3.46)

which corresponds to the following parameters in our framework,

1 = 1 , 3 = V (0) + 2M2Pl

HR2 , 1 =

2M2Pl H

R , 1 =

13!V (0) ,

HR2 , ( others ) = 0 . (3.47)

3.4.2 E ects of heavy particles

Recently, it is argued that the existence of heavy particles can cause a non-trivial sound speed of e ective single eld ination [1928]. As was mentioned earlier, our framework is

14

1 = M2Pl

also applicable for such ination models with heavy particles. In the following, we give a simple explanation for the e ects of heavy elds.

Let us start from the following simplest case:

S = Z

d4xg

m2

1

2M2PlR + M2Pl

Hg00 M2Pl(3H2+

.

(3.48)

Here m is the mass of and is the mixing coupling between the adiabatic mode and the massive particle . We assume that the mass of is much larger than the Hubble scale during ination, m H, and the time-dependence of is negligible compared to the

mass m. In such a regime, the kinetic term of becomes irrelevant and the dynamics is determined by

S

Z

H)

1

2g

2 2+ g00

JHEP06(2013)051

d4xg "

1

2M2PlR+M2Pl

Hg00M2Pl(3H2+

H)

m2 2

m2 g00 2+ 22m2 (g00)2#,

(3.49)

which implies that the perturbation quickly responds to the variation of the adiabatic mode g00. Integrating out the massive particle , we obtain the following e ective action for single-eld ination:

Se =

Z

d4xg

1

2M2PlR + M2Pl

Hg00 M2Pl(3H2 +

H) + 2

2m2 (g00)2

. (3.50)

In particular, the last term gives the following non-trivial sound speed:

c2 =

HM2Pl

, (3.51)

which reproduces the result in [1928]. Note that it is obvious in our approach that the e ective action contains the (g00)2 interaction: our result explains not only the e ective sound speed but also non-trivial cubic e ective interactions of the Goldstone boson associated with the (g00)2 term.

The above discussions can be extended to more general settings. Let us consider the following action with more generic mixing couplings:

S = Z

HM2Pl + 22/m2

d4xg

1

2M2PlR + M2Pl

Hg00 M2Pl(3H2 +

H)

m2

2 2 + 1g00 + 2g000 + 30 (

1

2g

3 + 3H3)

. (3.52)

When the mass of is much larger than the Hubble scale during ination, m H, and

the time-dependence of is is negligible compared to the mass m, the low energy e ective action can be obtained via the following procedure (see appendix A for more detailed discussions):

1. Drop the kinetic term of heavy elds.

15

2. Eliminate derivatives of heavy elds by partial integrals.

3. Complete square the Lagrangian and integrate out heavy elds.

To perform the second step, it is convenient to introduce the following relations, which follow from the formulae (3.3) and (3.8):

Z

d4xg f(t)g000

= d4xg h

( f(t) + 3H f(t))g00 f(t)0g00 + . . .i

, (3.53)

JHEP06(2013)051

Z

d4xg h

f(t)0 (

f(t) + 3H f(t))

i

Z

= d4xg "

f(t)K

f(t)

2 g00

f(t)

2 g000 + . . .#

, (3.54)

where dots stand for higher order terms in g00, K, and their derivatives. From these relations, it follows that

Z

d4xg h

1g00 + 2g000 + 30 (

3 + 3H3)

i

(3.55)

Z

= d4xg "

1

g00 +

2 3 2

g000 + 3 K + . . .

#

0g00+3K+. . . .

Note that the rst equality is the same as the relation used to rewrite (3.7) into the form of (3.9) plus higher order terms. Then, it is straightforward to obtain the following e ective action for single-eld ination using the above prescription:

Se =

Z

Z

= d4xg

1+

2+3H2

3

3 2H3

g00 2 3 2

d4xg"

1

2M2PlR + M2Pl

Hg00 M2Pl(3H2 +

H) (3.56)

+ 1

2m2

1+

2+3H2

3

3 2H3

g00 2 3 2

0g00 + 3 K

2+. . .

#,

where dots stand for higher order terms in g00, K, and their derivatives. It turns out that interactions such as (0g00)2 and (K)2 appear in the e ective action as well as the (g00)2 interaction.

3.4.3 A class of two-eld models

Let us then consider a class of two-eld models described by the following matter action:

Smatter =

Z

d4xg

1

2ab(a)gab V (a)

(a = 1, 2) , (3.57)

where ab(a) is the metric on the eld space. This class of models were carefully studied in [42] and recently discussed in [22] to investigate e ects of massive particles during ination. Suppose that the trajectory of the homogeneous background elds a0(t) is on a curve

16

a =

a(), and the background elds are given by a0(t) =

(0(t)). We also assume that

> 0. Dening the coordinates (, ) of the eld space such that the curve = 0 coincides with the trajectory curve, the elds (x) and (x) describe the adiabatic mode and the isocurvature mode, respectively. There are still many choices of the coordinates or degrees of freedom of the eld redenition. In the following, we consider two types of basis of the elds and discuss their properties in our framework.

Orthogonal basis. We rst consider the orthogonal-basis. We can always take the coordinate (, ) such that

= = 0 . (3.58)

By the eld redenition of , we further require (, = 0) = 1. In this basis, the matter

action (3.57) is given by

Smatter =

Z

d4xg

JHEP06(2013)051

1

2(, )g

1

2(, )g V (, )

, (3.59)

and it can be expanded in the unitary gauge = 0 = 0 up to the third order

uctuations as

Smatter =

Z

d4xg

1

2

20 g00

1

2

20()0g00 + 1

4

20()02

1 4

20()0g002 + 1

12

20()03

1

2()0 g

1

2()0 g V0

1

2V 02

16V 03

, (3.60)

where we have used the equation of motion for . In this subsection, we write derivatives of the metric and the potential evaluated at the classical value, for example, as ()0 = (, )|=0,=0 and V 0 = V (, )|=0,=0. Using the equations of motion for graviton,

M2Pl H =

1

2

20 , M2Pl(3H2 + H) = V0 , (3.61)

we can rewrite (3.60) as

Smatter =

Z

d4xg

M2Plg00 H M2Pl(3H2 +

H)

1

2()0 g

1

2

V 0 + M2Pl H()0

2 + M2Pl H()0g00 12()0 g

1 6

V 0 + M2Pl H()0

3 + 1

2M2Pl

H()0g002

, (3.62)

which is described in our framework as

1 = ()0 , 3 = V 0 + M2Pl H()0 , 1 = M2Pl H()0 , (3.63)

1 =

1 6

V 0 + M2Pl H()0

, 5 =

1

2()0 , 1 =

1

2M2Pl

H()0 , (others) = 0 .

We notice that this basis corresponds to the second normalization of in section 3.1.

17

Canonical basis. We then choose the coordinate (, ) such that

(, ) = 1 , (, = 0) = 1 , (3.64)

where the normalization of is always canonical and that of is canonical only on the trajectory curve. In this basis, the matter action (3.57) is given by

Smatter =

Z

d4xg

1

2(, )g (, )g

1

2g V (, )

, (3.65)

and it can be expanded in the unitary gauge = 0 = 0 up to the third order

uctuations as

Smatter =

Z

d4xg

JHEP06(2013)051

1

2

20 g00 + 12

20()0

1

2

20()0g00 + 1

4

20()02

1 4

20()0g002 + 1

12

20()03

0()00

0()00

1

2

0()020

1

2g

V0 V 0

1

2V 02

16V 03

, (3.66)

where the equation of motion for implies V 0 = 12

20()0

3H

0()0 +

0()0 +

0 t()0

. In terms of the Hubble parameter H, we can rewrite it as

Smatter =

Z

d4xg

M2Plg00 H M2Pl(3H2 +

H) 2MPl(

H)1/2()00

(V 0 + M2Pl

H()0)

1

2g

1

2

V 0 + M2Pl H()0

2

2MPl(

H)1/2()00 + M2Pl H()0g00

1 6

V 0 + MPl H()0

3

2

2 MPl(

H)1/2()020 + 12M2Pl

H()0g002

, (3.67)

which is described in our framework as

1 = 1 , 3 = V 0 + M2Pl H()0 , 4 = 2MPl(

H)1/2()0 , 1 = M2Pl H()0 ,

3 = 2MPl(

H)1/2()0 , 1 =

1 6

V 0 + MPl H()0

,

H()0 , (others) = 0 . (3.68)

The above action apparently includes terms linear in . However, it can be easily shown that such terms turn out to be more than second order after integration by parts. We notice that this basis corresponds to the third normalization of in section 3.1.

18

2 =

2

2 MPl(

H)1/2()0 , 1 = 1

2M2Pl

4 Power spectrum

In this section we calculate the power spectrum for a class of quasi-single eld ination models in our framework. In the following, we take the decoupling limit ~

0 and use

the action for the Goldstone boson . It is also assumed that the background trajectory satises 1. Up to the second order uctuations, the action for and takes the

following form in the decoupling limit:

S = Z

d4x a3

2 2

2 c2(i)2a2 + 2 2

2 c2(i)2a2 m22

+ ~

1

JHEP06(2013)051

+ ~

2

+ ~

3c2 ii

a2

, (4.1)

where we have dropped sub-leading terms M2Pl

H2 and

H in the regime 1,

and have dened

2 =

2M2Pl Hc2 , m2 =

3 3H4

42 , (4.2)

~

1 = 21 +

3 , ~

2 = 22 3 ,

~

3 = c23 .

The corresponding Hamiltonian is given by

H = Z

d3x

1

2a3 2P 2 + 2P 2

+ a3

2

2c2 (i)2a2 + 2c2(i)2a2 + 2m22

2 ~1P a322

~

2PP

3a c2ii + 1

2a3 42

~

~

22P 2 + . . .

, (4.3)

where P and P are canonical momentum variables conjugate to and , respectively. In this paper, we assume that the mixing couplings can be treated as perturbations and calculate the power spectrum up to the second order in the mixing couplings ~

is. The

dots in (4.3) stand for term irrelevant to the power spectrum at this order, and therefore, we drop them in the following.

Let us then calculate the power spectrum using the in-in formalism. Before going to concrete models, we rst introduce general expressions for the power spectrum. Choosing the free part Hfree and the interaction part Hint of the Hamiltonian as

H = Hfree + Hint , (4.4)

Hfree =

Z

d3x Hfree (4.5)

=

Z

d3x

1

2a3 2P 2 + 2P 2

+ a3

2

2c2 (i)2a2 + 2c2(i)2a2 + 2m22 ,

Hint =

Z

d3x Hint (4.6)

=

Z

d3x

2~1P a322~2PP ~3a c2ii + 12a3 42~ 22P 2

,

19

the dynamics of canonical variables in the interaction picture are determined by

= HfreeP = a32P ,

= Hfree

P = a32P , (4.7)

P =

= 2(c2a2i a3m2) . (4.8)

The elds and are then expanded in the Fourier space as

k = uk ak + uk ak , k = vk bk + vk bk (4.9)

with the standard commutation relations

[ak, ak] = (2)3(3)(k k) , [bk, bk] = (2)3(3)(k k) . (4.10)

Here the mode functions uk and vk satisfy the equations of motion in the free theory and depend on k = |k|:

k + H(3 2 + ~

) uk + c2 k2a2 uk = 0 ,

vk + H(3 + 2 ) vk +

Hfree

= 2c2a2i ,

P =

Hfree

m2 + c2 k2 a2

vk = 0 with =

H . (4.11)

Their normalization follows from

2 a3 (uk uk uk uk) = 2M2PlH2~

JHEP06(2013)051

a3 (uk uk uk uk) = i , 2 a3 (vk vk vk vk) = i . (4.12)

Using these expressions, the Hamiltonian in the interaction picture can be written as

Hint(t)=

Z

(2)3 a3(t)

d3k

~ k k

~ k k

3c2 k2

~ a2 k k+

1

22

~ k k

(t) . (4.13)

Then, the expectation value of k(t)k (t) is calculated as

hk(t)k (t)i = h0|

T exp

i

Z

t0 dtHint(t)

k(t)k (t)

T exp

i

Z

t0 dtHint(t) |

0i

t

t

t

t0 dt1h0|k(t)k

= h0|k(t)k

(t)|0i 2Re

i

Z

(t)Hint(t1)|0i

+

Z

t

t0 d

~t1

Z

t

t0 dt1h0|Hint(

~t1)k(t)k (t)Hint(t1)|0i

2Re

Z

t0 dt1 Z

t1

t0 dt2h0|k(t)k

(t)Hint(t1)Hint(t2)|0i

t

+ . . . , (4.14)

where the dots stand for the higher order terms in the couplings. In terms of the mode functions and the couplings, the general form of the two point function (4.14) is given up to the leading order corrections from the mixing couplings by

hk(t)k (t)i = (2)3(3)(k + k)uk(t)uk(t)h1 + C1 + C2 + C3

i

, (4.15)

20

where C1, C2, and C3 are dened by

C1 = 2

Z

t0 dt1 a3

~

1 ukvk + ~

2 uk vk + ~

3c2 k2

a2 ukvk

t

2

, (4.16)

(t1)

u2k(t)

|uk(t)|2

Z

t0 dt1 a3

~

1 ukvk + ~

2 uk vk + ~

t

C2 = 4Re

3c2 k2

a2 ukvk

(t1)

Z

" t0 dt2 a3

~

1 ukvk + ~

2 uk vk + ~

3c2 k2

a2 ukvk

t1

(t2)

#

, (4.17)

C3 = 2Re

"

i u2k(t)

|uk(t)|2

Z

#

t

t0 dt1 a3 2

. (4.18)

It is also convenient to rewrite C = C1 + C2 + C3 as follows:

C =4Re

" Z

t

~

22 uk2(t1)

t0 dt1 a3

~

1 ukvk + ~

2 uk vk + ~

3c2 k2

a2 ukvk

u2k(t)

|uk(t)|2

~1 ukvk + ~2 uk vk + ~3c2 k2a2 ukvk (t1)

Z

t0 dt2 a3

~

1 ukvk + ~

2 uk vk + ~

t1

3c2 k2

a2 ukvk

JHEP06(2013)051

(t2)

#

+ 2Re

"

i u2k(t)

|uk(t)|2

Z

t

t0 dt1 a3 2

~

22 uk2(t1)

#

. (4.19)

Since the scalar perturbation is given at the linear order by = H, we obtain the

expectation value of k(t)k (t) as

hk(t)k (t)i = (2)3(3)(k + k)

22k3 P(k) , (4.20)

where the power spectrum P(k) is given by

P(k) = H2(t)k322 uk(t)uk(t) (1 + C) . (4.21)

The factor C can be considered as a deviation factor from single eld ination. Here it

should be noticed that in the derivation of the above general expression we assumed only 1, ~

1, and the perturbativity of the mixing couplings. In principle, we can calculate

the power spectrum for any models satisfying those three conditions. Note that, as pointed out in [29, 30], the perturbativity of mixing couplings is justied even in the case when they are large only in a su ciently short period of time, which is realized for example by the sudden turning background trajectory [2124, 26, 27]. In the rest of this section, we rst perform the calculation concretely for the case when the time-dependence of mixing couplings is irrelevant (constant turning trajectory), and then we discuss the qualitative features of the case when the mixing couplings are large in a su ciently short period of time (sudden turning trajectory).

21

4.1 Constant turning trajectory

In this subsection, the power spectrum is calculated in the case that the time-dependence of mixing couplings is irrelevant. To make the calculation tractable, we assume that the time-dependence of , ~

, , c, and m is negligible and we use the de-Sitter approximation. In this approximation, the equations of motion (4.11) for the mode functions uk and vk can be written as

uk

2 uk + c2 k2uk = 0 , vk

2 vk + c2 k2vk +

m2

H22 vk = 0 , (4.22)

where the conformal time d = a1dt is given by = 1/(aH) in the de-Sitter approxi

mation and the primes denote derivatives with respect to . The equations (4.22) can be solved as follows:

uk = 12MPl~

1/2(ck)3/2 (1 + ick)eick =

1

2MPl~

1/2(ck)3/2 (1 ix)eix , (4.23)

vk = ie

i2 + i4 H

2 ()3/2H(1)(ck)=ie

i2 + i4 H

2(ck)3/2 x3/2H(1)(rsx), (4.24)

where x = ck, rs = c/c, and we chose the Bunch-Davies vacuum for and . The

function H(1) = J + iY is the Hankel function and is dened as

=

r94 m2

H2 for m <32H , = i

JHEP06(2013)051

rm2

H2 94 for m >32H . (4.25)

The time derivatives of uk and vk are given by

uk = Huk =

H 2MPl~

1/2(ck)3/2 x2eix , (4.26)

vk =Hvk = ie

i2 + i4 H2

2(ck)3/2 x3/2

(3/2 )H(1)(rsx) + (rsx)H(1)1(rsx)

, (4.27)

where we used the identity zzH(1) = zH(1)1 H(1). Therefore, the factor C dened

in (4.19) takes the form

C = e

i2 ()

42

c2

M2Pl(

H) Re" Z

0 dx1

~

1

H + ( 3/2)

~

2

~

3

A1(x)

eA1(x1)

~ A2(x)

eA2(x1)

+ ~

3

A3(x)

eA3(x1)

Z

x1 dx2

~

1

H + ( 3/2)

~

2

~ A1(x2) ~2A2(x2) + ~ 3A3(x2) #

c2

2M2Pl(

H)Re

i

Z

0 dx 2

~

22e2ix

, (4.28)

where we set t0 = and t = , and we denedA1(x) = x1/2eixH(1)(rsx) , A2(x) = rsx1/2eixH(1)1(rsx) , A3(x) = ix1/2eixH(1)(rsx) ,

eA1(x) = x1/2eixH(2)(rsx) , eA2(x) = rsx1/2eixH(2)1(rsx) ,

eA3(x) = ix1/2eixH(2)(rsx) .(4.29)

22

When the time-dependence of mixing couplings is irrelevant, mixing couplings ~

s can be

evaluated at the time of horizon-crossing = (ck)1. In such a case, C is given by

C =

c2

2M2Pl(

H)Re"

~

1/H + ( 3/2)

~

2

~

3

2

I11

+ ( ~

1/H + ( 3/2)

~

~ 3)(

~

2 I12 +

~

3 I13)

~ ( ~1/H + ( 3/2)~2 ~3)I21 ~2 I22 +~3 I23

+ ~

3

( ~1/H + ( 3/2)~2 ~3)I31 ~2 I32 +~3 I33 14~ 22

#

, (4.30)

JHEP06(2013)051

where Iijs are integrals dened by

Iij = 4 e

i2 ()

Z

0 dx1

Ai(x1)

x1 dx2 Aj(x2) . (4.31)

The last term in (4.28) was calculated using the i-prescription as

c2

2M2Pl(

eAi(x1)

Z

H)Re

i

Z

0 dx 2

~

22e2ix

= 14

c2

2M2Pl(

e2ix

0

H)

~

22Re

=

1 4

c2

2M2Pl(

H)

~

22 . (4.32)

We then have

C =

c2

2M2Pl(

~

21H2 C11 +

~

22 C22 +

~

23 C33 +

~

~ 2 C12 +

~

~ 3 C13 +

H)

~ 3 C23

!, (4.33)

and Cijs are given by

C11 = Re

h I11 i

, C22 = Re

| 3/2|2 I11 ( 3/2) I12 ( 3/2) I21 + I22 1 4

,

C33 = Re [ I11 + I33 I13 I31 ] , C12 = Re [( + 3) I11 I12 I21] ,

C13 = Re [2 I11 + I13 + I31] , (4.34)

C23 = Re [( + 3) I11 + I12 + I21 + ( 3/2) I13 + ( 3/2) I31 I23 I32] .

Here the explicit form of the power spectrum is

P(k) = H2 82M2Pl c

"1 + c2 2M2Pl( H)

~

21H2 C11 +

~

22 C22 +

~

23 C33 +

~

~ 2 C12

~

+ ~ 3 C13 +

~ 3 C23

#

, (4.35)

where H, , c, , ~

i, and rs are evaluated at the time of horizon-crossing. Then, the calculation of the power spectrum reduces to the evaluation of Re[ Iij + Iji], Im[ I12 I21], and Im[ I13 I31].

23

100

100.0

50.0

10

10.0

5.0

C11

1

C22

1.0

0.5

0.1

0.1

0.01

0 1 2 3 4 5

0 1 2 3 4 5

m!/H m!/H

100

100

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10

10

C33

1

C12

1

0.1

0.1

0.01

0 1 2 3 4 5

0.01 0 1 2 3 4 5

m!/H

m!/H

100

100

10

10

C13

C23

1

1

0.1

0.1

0.01 0 1 2 3 4 5

0.01 0 1 2 3 4 5

m!/H

m!/H

Figure 1. Cijs for xed rs = c/c. The dots are numerical results for rs = 0.1 (red), 0.3 (orange),

1 (yellow), 3 (green), and 10 (blue). The curves are analytic results for rs = 1 obtained in the next subsection.

As is understood from the denition, Iijs and Cijs are functions of m and rs = c/c.

For general value of rs, it is di cult to perform the integrals analytically and we performed numerical calculations by contour deformations (see appendix B for details). For the special case rs = 1, however, it is possible to perform the integrals Iijs analytically by extending

the results in [31], and the results are summarized in appendix C. In such a way, Cijs are calculated and the obtained results are summarized in gure 1 and gure 2. We nd that they monotonically decrease in m for xed rs, but they are not monotonic for xed m. As is discussed in section 3.4.2, the e ects of mixing interactions appear in the form of 2i/m2 in the heavy mass limit and it is implied that Cij 1/m2 for large m,

which is consistent with our results in this subsection. Therefore, the power spectrum is not a ected by heavy particles unless the mixing couplings are comparable to the mass of heavy particles [1928].

24

100

100

10

10

C11

C22

1

1

0.1

0.1

0.01 0.1 0.2 0.5 1.0 2.0 5.0 10.0

0.01 0.1 0.2 0.5 1.0 2.0 5.0 10.0

rs

rs

rs rs

rs

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100

100

10

10

C33

1

C12

1

0.1

0.1

0.01

0.1 0.2 0.5 1.0 2.0 5.0 10.0

0.01 0.1 0.2 0.5 1.0 2.0 5.0 10.0

rs

100

100

10

10

C13

C23

1

1

0.1

0.1

0.01 0.1 0.2 0.5 1.0 2.0 5.0 10.0

0.01 0.1 0.2 0.5 1.0 2.0 5.0 10.0

Figure 2. Cijs for xed m. The dots are numerical results for m/H = 1.2 (red), 1.6 (yellow),

2.0 (green), and 4.0 (blue).

4.2 Qualitative features of sudden turning trajectory

In this subsection, we discuss qualitative features of the case when the mixing couplings are large in a su ciently short period of time (sudden turning trajectory). As a simplest example, let us rst consider the case when the ~

1 coupling is the only relevant mixing coupling and it is proportional to a delta function:

~

1 = (t t) = a1(t) ( ) ,

~

2 = ~

3 = 0 , (4.36)

where t is the time of sudden turning, is the corresponding conformal time, and the mode k crossing the horizon at t = t is given by k = 1/(c). We assume that

is small enough to be treated perturbatively. In this case, the deviation of the power

25

spectrum from that of single eld ination can be written as9

C = 22 Re h

a6 ( uk uk) uk vk vk(t)i

, (4.37)

which is related to the power spectrum by (4.21). For simplicity, suppose that time-dependence of m, , c, and c is negligible at the time of sudden turning and the mode functions uk(t) and vk(t) are given by

uk(t) = 12MPl~

1/2(ck)3/2 (1 ix)eix , (4.38)

vk(t) = ie

i2 + i4 H

2(ck)3/2 x3/2H(1)(rsx) , (4.39)

where x = k/k and parameters such as m are evaluated at the time of sudden turning. Then, (4.37) is given by

C = 2

42

c2

M2Pl(

H)F(x) with F(x) = e

i2 ()x sin2 x|H(1)(rsx)|2 . (4.40)

For k k or x 1, the asymptotic behavior of F(x) is

F(x)

42 ()2 r2sx32 for m(t) <

3 2H ,

JHEP06(2013)051

1 2

i2 2 ()(rsx)

e

i2 2 ()(rsx) +e

2 x3 for m(t) > 3

2H ,

(4.41)

which is consistent to the intuition that modes outside the horizon at t = t are not much a ected by the sudden turning. For k k or x 1, it reduces to

F(x)

2 r1s sin2 x =

1 r1s(1 cos 2x) . (4.42)

This kind of oscillating behavior was also found in [2224, 26]. The turning trajectory generically oscillates around the turning point and the mixing couplings at the turning point become more regular than delta functions. In such a case, it is expected that the oscillating behavior of short modes ck [greaterorsimilar] 1/ begins damping at some scale characterized

by the oscillation of the trajectory. Let us next conrm such a behavior explicitly for a

concrete example with a nite width of turning. We consider the following ~

1 prole:

~

1 =

((H)( )1 for 2 < < + 2 , 0 otherwise ,

(4.43)

9In our calculation we have treated the mixing coupling as an interaction. In such an interaction picture, it is manifest that the deviation from single eld ination starts from the second order in . On the

other hand, it is also possible to treat the mixing coupling as a part of the kinetic and mass terms. In that picture, the commutation relations (4.10) of creation and annihilation operators are a ected by the mixing as well as the mode functions uk and vk are modied. However, in some literatures, these modications are not taken into account adequately and the deviation from single eld ination is calculated to start from the rst order in .

26

0.8

m! = H m!

0.8

= 3H

0.6

0.6

0.4

0.4

F!

F!

0.2

0.2

0.0

0.0

2 4 6 8 10

2 4 6 8 10

[Minus]0.2

[Minus]0.2

k/k! k/k!

Figure 3. F for xed ck and c = c. The left gure is for m = H and the right one is

for m = 3H. The dots are numerical results for ck = 5 (blue), 10 (green), and 15 (red). The curves are analytic results in the limit of ck 0 (delta function limit).

where we normalized so that it reproduces the coupling (4.36) in the limit 0. For

this class of couplings, we dene F by

C = 2

42

c2

M2Pl(

H)F(x) . (4.44)

Since it is di cult to calculate F analytically, we can conrm the expected damping be

havior of F by numerical calculations and the results are given in gure 3. To summarize, C vanishes in the long mode limit ck 1/, oscillates for short modes ck [greaterorsimilar] 1/, and

damps at some scale characterized by the oscillating trajectory around the turning point. It is straightforward to extend the above discussions to the case with non-vanishing ~

2

and ~

3. We generically expect to nd a similar behavior of C: vanishing in the long mode

limit ck 1/, oscillating for short modes ck [greaterorsimilar] 1/, and damping at some scale

characterized by the oscillating trajectory around the turning point.

5 Three point functions in the squeezed limit

In this section, we discuss the momentum dependence of three point functions in the squeezed limit. We take the decoupling limit and assume that time dependence of s, s, s, and H is negligible. Under these assumptions the second order action is given in (4.1) and (4.2), and three point vertices are given by

S(3) =

Z

d4x a3

JHEP06(2013)051

M2Pl H(c2 1) +

4M43 3

3 + M2Pl H(c2 1) (i)2 a2

+ (1 + 46)

2 + (32 47)

2

22

ii

a2 + 1

(i)2

a2

2

(i)2

a2

21

2 + (4 + 22)

+

2(1c2) 23 + 25 2

25

(i)2a2 + 4

iia2 2(1c2)

ii a2

24

+ 13 22

+ (3 5)

2 + 5 (i)2a2 + (4 + 6)

3 6

(i)2 a2

. (5.1)

27

!k1 !k2

!k1 !k2 !k3

!k1 !k2 !k3

!2"

!2"

!"

!"

!k3

!2"

!"

"

"

"

" t = ! t1 t2 "

Figure 4. Feynman diagrams for the rst term of (5.2). The solid lines denote the propagations of and the dotted lines denote those of .

In the following, we discuss what kind of momentum dependence in the squeezed limit appears from the above three point vertices.10

As an example, let us start from the case when the mixing coupling ~

1

and the three

" t t = ! t1

2

t = ! t1

t2

point vertex 46

2 are relevant. In this case, the three point function of takes the form

hk1(t)k2(t)k3(t)i

3/2(ck1)3/2(ck2)3/2(ck3)3/2 Re" Z

dt1 a3 uk1(t1) uk2(t1)

~ 6 (2)3(3)(k1 + k2 + k3)

M3Pl~

JHEP06(2013)051

vk3(t1)

Z

dt2 a3 uk3(t2)vk3(t2) vk3(t1) Z

t1 dt2 a3 uk3(t2)vk3(t2)

t1

vk3(t1)

Z

dt2 a3 uk3(t2)vk3(t2) #

+ (2 permutations) . (5.2)

Here there are three terms in the curly brackets, and they correspond to the Feynman diagrams in gure 4, respectively. In the squeezed limit, k1 = k2 = k and k3/k = 1,

the long mode k3 crosses the horizon much earlier than the short modes k1 and k2. Since the interactions are considered to be relevant around the horizon, it is expected that the relevant contribution arises from (1, 2) (1/k1, 1/k3), and therefore, the middle term

in the curly brackets becomes irrelevant in the squeezed limit. In fact, we can conrm this expectation explicitly from the expression (5.2), and the integrals in the curly brackets can be written at the leading order in as

Z

dt1 a3 uk1(t1) uk2(t1) 2i Im

vk3(t1)

Z

dt2 a3 uk3(t2)vk3(t2)

. (5.3)

10For the calculation of three point functions using the in-in formalism, it is necessary to obtain the Hamiltonian description of the system. Since the

coupling contains the second order derivative of , careful discussions are required when it is relevant in the action (5.1) and calculate using the Hamiltonian formalism in the interaction picture. In this paper, we do not consider such a situation for simplicity and concentrate on other cubic couplings. It should be also noted that the form of interactions in the Hamiltonian formalism in the interaction picture does not coincide with minus that in the Lagrangian formalism because the action contains derivative interactions. Correspondingly, the coe cients of cubic couplings in the Hamiltonian are changed from minus those in the action (5.1).

28

We notice that the t2-integral is k3-independent and k3-dependence of the total integral appears only via vk3(t1) originated from the three-point vertex. Then, the momentum dependence of the rst term in (5.2) is given by

the rst term in (5.2)

3/2k6 for m < 3

2H ,

(5.4)

where is a -dependent phase factor. We note that the only information necessary to derive (5.4) was the fact that the eld of momentum k3 in the three point vertex takes the form . For the other two permutation terms in (5.2), the k3-dependence of the integral is determined by uk3 originated from the three point vertex, and their momentum dependence is given by

the other two permutation terms in (5.2) 1k6 . (5.5)

Then, the rst term dominates for small . Therefore, when the mixing coupling ~

1 and

three point coupling 6 are relevant, the momentum dependence of scalar three point functions in the squeezed limit is given by

lim

k3/k1=k3/k2=0

hk1k2k3i

3/2k6 sin[i log + ] for m > 3

2H ,

JHEP06(2013)051

3/2k6 for m < 3

2H ,

(5.6)

It is straightforward to extend the above discussion for more general cases. First, for general ~

is, we can show that when the mixing couplings convert of momentum k3 = k to , the t2-integral becomes -independent in the limit 1 and the three

point vertex determines the -dependence of the diagram. Second, the only information necessary to obtain the momentum dependence is the form of the eld of momentum k3 in

the vertex. In the previous examples, when it takes the form in the vertex the diagram was proportional to 3/2 or 3/2 sin[i log + ], and it was proportional to 1 for

. More generally, we can obtain the relations in table 1 between momentum dependence of the diagram and the form of the eld of momentum k3 in the three point vertex. Here it should be noted that the -dependent phase factor depends on the details of mixing couplings. Finally, as discussed in the previous example, momentum dependence of the contribution from each vertex is now identied for 1 so that it is straightforward to

obtain momentum dependence of the contribution to scalar three point functions from each three point vertex displayed in (5.1). The results are summarized in table 2. Here note

that although the contribution from the

(i)2a2 vertex seems to be proportional to 2k6

apparently, explicit calculations show that this kind of leading contribution vanishes and the three point functions begin with terms proportional to 1k6.

As we have seen, the momentum dependence of scalar three point functions in the squeezed limit has robust information about mass of and three point vertices.

29

3/2k6 sin[i log + ] for m > 32H .

form of the eld of momentum k3

momentum dependence of the diagram

3k6

1k6

i

a

2k6

,

3/2k6 for m < 32H

3/2k6 sin[i log + ] for m > 3

2H

i

a

1/2k6 for m < 32H

1/2k6 sin[i log + ] for m > 3

2H

Table 1. Momentum dependence of the diagram.

three point vertices

momentum dependence

3,

(i)2 a2

1k6

JHEP06(2013)051

3/2k6 for m < 32H

3/2k6 sin[i log + ] for m > 32H

2,

2

,

2,

,

2,

3, 2

,

2, (i)2a2 ,

3,

(i)2 a2

ii

a2

2k6

a2 ,

(i)2

a2

, ii

a2 ,

ii a2

(i)2

3/2k6 for m < 2H

2k6 for m > 2H

(i)2 a2

1/2k6 for m < 2H

1k6 for m > 2H

Table 2. Three point vertices and momentum dependence.

6 Summary and discussion

In this paper we discussed quasi-single eld ination using the e ective eld theory approach. We rst constructed the most generic action in the unitary gauge based on the unbroken time-dependent spatial di eomorphism, and then constructed the action for the Goldstone boson by the Stckelberg method. Its decoupling regime was also discussed carefully, and the action in the decoupling regime implies that non-trivial cubic interactions

30

generically appear and non-negligible non-Gaussianities can arise when the sound speed c of is small, 4 is non-zero, or mixing couplings 1 and 2 exist as well as the sound speed c of is small. Using the obtained action, two classes of concrete models were discussed: the constant turning trajectory and the sudden turning trajectory.

In the constant turning case, we rst calculated the power spectrum of scalar perturbations numerically for general values of rs = c/c and analytically for the special case rs = 1. We then discussed the momentum dependence of scalar three point functions in the squeezed limit for general settings of quasi-single eld ination. It was shown that the momentum dependence is determined only from the cubic interactions and the cubic interactions were classied into ve classes. The three point functions in the squeezed limit take the intermediate shapes between local and equilateral types when the mixing couplings are relevant, and this kind of momentum dependence characterizes quasi-single eld ination. Recently in [43], the detectability of such a momentum dependence was discussed for some cases. It would be interesting to discuss the detectability of the momentum dependence in the form of 3/2 sin[i log +], which arises in the second class with m > 32H. It is also important to calculate the full bi-spectrum for general settings of quasi-single eld ination.

In the sudden turning case, we made a qualitative discussion of the power spectrum. It was found that the deviation factor C from single eld ination vanishes for long modes

ck 1/, oscillates for short modes ck [greaterorsimilar] 1/, and damps at some scale charac

terized by the oscillating trajectory around the turning point. Since our framework makes the contributions from the mixing couplings clear, it would be useful to discuss more on the sudden turning trajectory.

Our framework can be considered as a starting point for systematic discussions on multiple eld models, and there would be a lot of applications such as those mentioned above. We hope to report our progress in these directions elsewhere.

Acknowledgments

We would like to thank Mitsuhiro Kato for useful discussions. The work of T.N. was supported in part by JSPS Grant-in-Aid for JSPS Fellows. The work of M.Y. was supported in part by the Grant-in-Aid for Scientic Research No. 21740187 and the Grant-in-Aid for Scientic Research on Innovative Areas No. 24111706. D.Y. acknowledges the nancial support from the Global Center of Excellence Program by MEXT, Japan through the Nanoscience and Quantum Physics Project of the Tokyo Institute of Technology.

A Integrating out heavy elds

In section 3.4.2, we discussed e ects of heavy elds by the following procedure:

1. Drop the kinetic term of heavy elds.

2. Eliminate derivatives of heavy elds by partial integrals.

3. Complete square the Lagrangian and integrate out heavy elds.

31

JHEP06(2013)051

In this appendix we make some comments and careful discussions on the rst and second steps of this procedure.11

A.1 Role of kinetic term

We rst discuss the procedure to integrate out heavy elds using the following simple model:

S = Z

dt

12

2 m22

+ f[i(t); t]

, (A.1)

where f[i(t); t] is a function of light elds is and time t. Before discussing the cosmological perturbation, let us recall the case when we calculate correlation functions in the momentum space and the total energy of the system is conserved. Suppose that the typical energy scale E of external states is much smaller than the mass of the heavy eld: E m.

Then, the typical energy scale of internal is and is also of the order E and much smaller than m because of the energy conservation. In such a case, the kinetic term 12

2 becomes

irrelevant compared to the mass term because it can be written as 12

2 12E22 12m22.

We therefore drop the kinetic term of the heavy eld and obtain

S = Z

dt

12m22 + f[i(t); t] =

Z

dt

" m2 2

1m2 f[i(t); t]

2+ 12m2 f[i(t); t]

2

,

(A.2)

#

JHEP06(2013)051

which reduces to the following e ective action after integrating out :

Se = 1 2m2

Z

dt f[i(t); t]

2 . (A.3)

In this way, the rst step of the procedure is justied in the momentum space when the momentum conservation holds.

In the cosmological perturbation, we calculate correlation functions in the real time coordinate. Furthermore, the time translation is broken by the time-dependent background. As an illustrative toy model for the cosmological perturbations, let us next consider to calculate correlation functions of the model (A.1) in the real time coordinate space. In such a case,

may seem to behave as

m because the heavy eld oscillates like

eimt on shell: it may not be so obvious whether the kinetic term of can be neglected.

To clarify this point, let us make a careful discussion without neglecting the kinetic term of . We rst rewrite the kinetic term and the mass term as

1

2

Z

dt

2 m22

=

1

2

Z

dt1

Z

dt2 K(t1, t2)(t1)(t2)

with K(t1, t2) = m2(t1 t2) + (t1 t2) , (A.4)

and introduce the inverse P(t1, t2) of the kinetic operator K(t1, t2) satisfying

Z

dtP(t1, t)K(t, t2) =

Z

dtK(t1, t)P(t, t2) = (t1 t2) . (A.5)

11See also [28] for related discussions.

32

We then rewrite the action as

S =

1

2

Z

dt1dt2 K(t1, t2)

(t1)

Z

dt1P(t1, t1)f[i(t1); t1]

(t2)

Z

dt2P(t2, t2)f[i(t2); t2]

+ 1

2

Z

dt1dt2 P(t1, t2)f[i(t1); t1]f[i(t2); t2] . (A.6)

Integrating out , the following e ective action is obtained:

Se = 1 2

Z

dt1dt2 P(t1, t2)f[i(t1); t1]f[i(t2); t2] . (A.7)

Here note that no approximations are used so far. Let us then consider the property of

P(t1, t2) when is heavy. Since the condition (A.5) can be rephrased as

2t1P(t1, t2) + m2P(t1, t2) = 2t2P(t1, t2) + m2P(t1, t2) = (t1 t2) , (A.8)

P(t1, t2) can be expanded in 1/m2 as

P(t1, t2) = 1m2 (t1 t2)

1m2 2t1P(t1, t2)

= 1

m2 (t1 t2)

1m4 (t1 t2) +

1m4 4t1P(t1, t2)

JHEP06(2013)051

= . . .

= 1

m2 (t1 t2) +

Xn=1(1)nm2n (2n)(t1 t2) . (A.9)

Using this expression, the e ective action (A.7) can be written as

Se = 1

2m2

Z

dt

1 m2

"

f[i(t); t]

2 +

Xn=1f[i(t); t](1)nm2nd2ndt2n f[i(t); t]

#

. (A.10)

Therefore, if the time-dependence of f[i(t); t] is negligible compared to the mass m of , or in other words, if the elds is are light and the explicit time-dependence of f[i(t); t] is irrelevant compared to m, the second term in (A.10) is negligible and the e ective action reduces to

Se = 1

2m2

Z

dt f[i(t); t]

2 , (A.11)

which coincides with the result (A.3) obtained by dropping the kinetic term of .

To summarize, the kinetic term 12

2 can be neglected when the time-dependence of f[i(t); t] is negligible compared to the mass m of the heavy eld . It would be notable that, when we neglect the kinetic term 12

2, the kinetic operator K(t1, t2) takes the form K(t1, t2) = m2(t1 t2) and its inverse P(t1, t2) is given by

P(t1, t2) = 1m2 (t1 t2) , (A.12)

which coincides with the rst term in (A.9). Therefore, it can be considered that the delta-function like behavior of P(t1, t2) originates from the mass term and the kinetic term plays

a role to regularize the singular behavior by reproducing the second term in (A.9).

33

A.2 Derivative coupling, partial integral, and Hamiltonian formalism

Let us next consider the second step of the procedure using the following two actions:

S = Z

dt

12

2 m22

+

g[i(t); t]

, (A.13)

S =

Z

dt

12

2 m22

d dtg[i(t); t]

, (A.14)

where g[i(t); t] is a function of light elds is and time t. Since (A.13) and (A.14) are related to each other by partial integrals, they are expected to describe the same dynamics. In particular, they are expected to reproduce the same e ective theory in the low energy regime. However, it may be wondered that the mixing term in (A.13) becomes relevant when is heavy because

m on shell and that the low energy dynamics can be

di erent from those of (A.14). In this subsection we would like to clarify this point and show that (A.13) and (A.14) describe the same dynamics as is expected from the partial integral perspective.

Similarly to the previous discussions, the action (A.13) can be written in terms of

K(t1, t2) and P(t1, t2) as

S =

1

2

Z

JHEP06(2013)051

dt1dt2 K(t1, t2)

(t1)

Z

dt1

t1 P(t1, t1)

g[i(t1); t1]

(t2)

Z

dt2

t2 P(t2, t2)

g[i(t2); t2]

+ 12 Z

i(t2); t2] , (A.15)

and we obtain the following e ective action after integrating out :

Se = 1 2

Z

dt1dt2

dt1dt2

t1t2P(t1, t2)

g[i(t1); t1]g

t1t2P(t1, t2)

g

i(t1); t1

g

i(t2); t2

. (A.16)

It follows from the expression (A.9) of P(t1, t2) that

t1t2P(t1, t2) = t1t2

"

1m2 (t1 t2) +

1 m2

Xn=1(1)nm2n (2n)(t1 t2)#

=

Xn=1(1)nm2n (2n)(t1 t2) , (A.17)

and therefore, the e ective action (A.16) can be written as

Se = 1

2

Z

dt

Xn=1(1)n m2n g

t, i(t) d2n dt2n g

i(t); t

(A.18)

= 1

2m2

Z

dt

"

d dtg

i(t); t 2+

Xn=1 (1)n m2n

d dtg

i(t); t d2n dt2n

d dtg

i(t); t

#

,

34

which is nothing but the e ective action for (A.14) obtained by applying the previous result (A.10). We therefore conclude that the actions (A.13) and (A.14) describe the same dynamics.

Then, what was wrong in the naive expectation that the mixing term in (A.13) becomes relevant for heavy ? To answer this question, it may be instructive to reconsider the above discussion using the following concrete form of P(t1, t2):

P(t1, t2) = 1 2im

h(t1 t2)eim(t1t2) + (t2 t1)eim(t1t2)i, (A.19)

which is essentially the same as the Feynman propagator. The property of the e ective action (A.16) is determined by t1t2P(t1, t2) and it is given for the choice (A.19) by

t1t2P(t1, t2) = m2P(t1, t2) (t1 t2) . (A.20)

Here the rst term is obtained by taking derivatives of eimt originated from mode functions of and the factor m2 is the expected one from the observation that

m. An important

point is that we also have the second term obtained by taking derivatives of both of step functions and eimt. Because of this second term, the leading order term in the 1/m2 expansions of m2P(t1, t2) is canceled out as

t1t2P(t1, t2) = m2

1 m2

JHEP06(2013)051

"(t1 t2) +

Xn=1(1)nm2n (2n)(t1 t2)# (t1 t2)

=

Xn=1(1)nm2n (2n)(t1 t2) , (A.21)

and the naive expectation that the interaction is enhanced by the mass of turns out to be wrong. The lesson is that it is important to take into account derivatives of step functions in the Feynman propagator appropriately when we discuss derivative interactions: the mass factor naively expected from time derivatives of massive elds can be canceled out.

It would be also notable that similar situations appear in the Hamiltonian formalism in the interaction picture. Let us consider the Hamiltonian system corresponding to (A.13). For simplicity, suppose that g[i(t); t] does not depend on time derivatives of is. Then, the momentum conjugate P of and the Hamiltonian H are given by

P =

+ g[i(t); t] , (A.22)

H =

P

12 2 12m22 + g[i(t); t]

2 . (A.23)

Choosing the free part Hfree and the interaction part Hint of the Hamiltonian as

Hfree = 1

2P 2 +

1

2m22 , (A.24)

Hint = Pg[i(t); t] +

1

2 g[i(t); t]

= 1

2P 2 +

1

2m22 Pg[i(t); t] +

1

2 g[i(t); t]

2 , (A.25)

35

the dynamics of canonical variables in the interaction picture are determined by

= HfreeP = P ,

P =

Hfree

= m , (A.26)

and we have

Hfree = 12

2 + 1

2m22 , (A.27) Hint =

g[i(t); t] + 1

2 g[i(t); t]

2 . (A.28)

Note that the rst term in (A.28) corresponds to the interaction part of the Lagrangian and Hint has the additional second term 1

2 g[i(t); t]

2 as is usual in systems with derivative

interactions. Since m, the rst term in (A.28) is enhanced by the mass m of

and it may be wondered that the interaction of the system becomes relevant when is heavy. However, it turns out that the enhancement is canceled out by the second term

1

2 g[i(t); t]

2 just as the second term in (A.20) cancels out the leading order in the 1/m2 expansions of the rst term in (A.20): the additional second term in (A.28) plays an important role.

A.3 Extension to cosmological perturbation

The above discussions can be extended to the cosmological perturbation straightforwardly. Let us consider the following action:

Z

d4xg

JHEP06(2013)051

1

2g

m2

2 2 + f[i(x); x]

. (A.29)

The kinetic term and the mass term of can be written as

Sfree =

1

2

Z

d4x1g(x1) Z

d4x2g(x2) (x1)K(x1, x2)(x2)

with K(x1, x2) = (m2

)4(x1 x2)

g

, (A.30)

where = g is the covariant dAlembertian operator and is the covariant

derivative. Introducing the inverse P(x1, x2) of K(x1, x2),

Z

d4xg K(x1, x)P(x, x2) = Z

d4xg P(x1, x)K(x, x2) =

4(x1 x2) g

, (A.31)

the following e ective action is obtained after integrating out :12

Se = 1

Z

d4x2gP(x1, x2)f[i(x1); x1]f[i(x2); x2] . (A.32)

12Since the kinetic operator K(x1, x2) contains metric perturbations, non-trivial log det K contributions

arise from the Gaussian path integrals of . However, we do not consider these contributions for simplicity because they do not appear as long as tree-level perturbations are discussed around a given background spacetime.

2 d4x1g Z

36

Just as in the previous toy models, the conditions (A.31) can be rephrased as

(m2

1)P(x1, x2) = (m2

2)P(x1, x2) =

4(x1 x2) g

, (A.33)

and P(x1, x2) can be expanded in 1/m2 as

P(x1, x2) = 1 m2

4(x1 x2) g

+

1m2 P(x1, x2)

= . . .

= 1

m2

4(x1 x2) g

+ 1

m2

Xn=1

JHEP06(2013)051

1 m2

n 4(x1 x2)g. (A.34)

Then, the e ective action can be written as

Se = 1 2m2

Z

d4xg "

f[i(x); x]

2 +

Xn=1f[i(x); x]

m2

nf[i(x); x]

#

, (A.35)

which reduces to the following form when the spacetime-dependence of f[i(x); x] is negligible compared to the mass m of :

Se = 1

2m2

Z

d4xg f[i(x); x]

2 . (A.36)

Therefore, in the cosmological perturbation around FRW backgrounds, the kinetic term of can be neglected when the mass m of is much larger than the Hubble parameter H and the mass mi of other elds i: m H, mi.

B Numerical calculations of power spectrum

To perform the integral (4.31) numerically, there are two technical obstacles [2931]: spurious divergences at x = 0 and oscillating behaviors at x = .

The integral (4.31) contains two integrals,

R

0 dx1 Ai(x1)

x1 dx2 Aj(x2) and

R

0 dx1

eAi(x1)

R

R

x1 dx2 Aj(x2), and each integral diverges for some parameter region. However, we can show that such divergences cancel out in the calculation of Cijs. For example,

let us consider C11 for real 0 < < 3/2. The asymptotic behavior of each integral in I11

around x1 = x2 = 0 is given by

Z0 dx1 x1/21 Zx1dx2 x1/22 012 , (B.1)

which diverges for > 1/2. We rst notice that this kind of leading singularities cancel out between two terms and the asymptotic behavior of the total integral I11 is given by

i

Z0 dx1 x1/21 Zx1dx2 x1/22 i 022 (B.2)

up to a real constant number. Although it is still singular for > 1, such singular contribution is pure-imaginary and does not contribute to Re[ I11]. The higher order terms are

37

nite for < 3/2 and hence we conclude that C11 = Re[ I11] is nite. In a similar way, we

can show that Re[ Iij + Iji], Im[ I12 I21], and Im[ I13 I31] are nite for 0 < < 3/2 and

= pure-imaginary, and therefore all the Cijs are nite. To avoid this kind of spurious

singularities in the numerical calculation, we introduce a IR cut o IR:

Iij = 4 e

i2 ()

Z

x1 dx2 Aj(x2) , (B.3)

dx1

Ai(x1)

eAi(x1)

Z

where we set IR = 1010 in our calculation.

As is usual in the Feynman diagram calculation in the momentum space, the integral

Iij oscillates at x = because of the oscillating behavior of the mode functions uk and

vk, which makes the numerical calculation di cult. Following [2931], we perform contour deformations to avoid this kind of technical di culties. Let us rst consider the integral Re[ Iij + Iji]. It is convenient to rewrite it as follows:

Re[ Iij +Iji] =

4 e

JHEP06(2013)051

Z

Z

x1 dx2 Ai(x2)

i2 ()Re

dx1Ai(x1)

x1 dx2Aj(x2)+Z

dx1Aj(x1)

Z

Z

Z

dx1

eAi(x1)

x1 dx2 Aj(x2) Z

dx1

eAj(x1)

Z

x1 dx2 Ai(x2)

=

4 e

i2 ()Re

Z

dx Ai(x)

Z

dy Aj(y)

Z

Z

dx1

eAi(x1)

x1 dx2 Aj(x2) Z

dx1

eAj(x1)

Z

x1 dx2 Ai(x2)

. (B.4)

The rst term in the bracket can be Wick rotated without crossing any poles as

Z

0 dx Ai(IR + ix) Z

0 dy Aj(IR + iy) , (B.5) and the last two terms are Wick rotated as

Z

Z

0 dy Aj(IR + ix + iy) +Z

0 dy Ai(IR + ix + iy) .(B.6)

eAi(IR + ix)

eAj(IR + ix)

Z

Then, we obtain the following expression of Re[ Iij + Iji]:

Re[ Iij +Iji] =

4 e

0 dy Aj(IR + ix + iy) + (i j) . (B.7)

To avoid the singular behavior around x = 0 discussed above, we further modify the contour as follows:

Re[ Iij + Iji] =

4 e

i2 ()Re

Z

Ai(IR + ix)+

eAi(IR + ix)

Z

Z

0 dy Aj(IR + ix + iy)

i2 ()Re

i

1 Ai(IR + x)+

eAi(IR + x)

Z

0 dy Aj(IR+1+ix + iy) + (i j) . (B.8)

38

+

Z

Ai(IR +1+ ix) +

eAi(IR +1+ ix)

Z

By performing similar contour deformations, Im[ Iij Iji] can be also expressed as follows:

Im[ Iij Iji]=

4 e

i2 ()Re

Z

1 Ai(IR + x)

eAi(IR + x)

Z

0 dy Aj(IR + ix + iy)

+

4 e

i2 ()Im

Z

0 dx

Ai(IR + 1 + ix) +

eAi(IR + 1 + ix)

Z

0 dy Aj(IR + 1 + ix + iy)

(i j) . (B.9)

The expressions (B.8) and (B.9) are used in our numerical calculations of the power spectrum.

C Analytical calculation of power spectrum for c = c

In this appendix we calculate the power spectrum for the case rs = c/c = 1. For this class of models, we can analytically calculate the integrals Iijs by extending the results

in [31]. We rst introduce a function A(, , x) dened by

A(, , x) = x

1

2 +eixH(1)(x) . (C.1)

In terms of A, Ai can be written as

A1(x) = A(0, , x) , A2(x) = A(1, 1, x) , A3(x) = iA(1, , x) , (C.2)

and hence the x-integrals in the rst term and the x2-integral in the second term of (4.31) reduce to that of A. Similarly to the case in [31], the indenite integral D of A can be

expressed using hyper-geometric functions as

D(, , x) = Z

dx A(, x) (C.3)

= 2x

1

2 + ()

JHEP06(2013)051

i(12 + ) 2

F2

12 ,12 + ;32 + , 1 2; +2ix

+ ei 2x

1

2 ++ ()

12 + ,12 + + ;32 + + , 1 + 2; +2ix

.

i(12 + + )

2F2

Then, the integral Iij can be written as

Iij = 8 e

i2 ()(i)n11+n22 D

(n1, + n2, ) D(n1, + n2, 0)

D(1, +2, )

D(1, +2, 0)

D(1, +2, ) D(1, +2, x1) , (C.4)

39

4 i1+2e

i2 ()

Z

0 dx1

eAi(x1)

where (n1, n2), (1,2) = (0, 0), (1, 1), (1, 0) for i, j = 1, 2, 3, respectively. Here note that D(, , x) gives a nite value in the limit x = (see appendix C.1 for the derivation):

D(, , ) =

1 i2

1 sin

(1/2 + )

(1/2 )

(1/2 + + )

(1/2 + )

! ()(2i)ei2 (1/2)

(C.5)

= (1 i)e

i2

cos (2i)

1

( + 1)

(12 + + ) (12 + )

(12 + ) (12 )

, (C.6)

where we used an i-prescription to drop oscillatory terms e2ix at innity x . It

should be also noted that the asymptotic behavior of D(, , x) around x = 0 is given by

D(, , x) = 2 ()

i(12 + )

x

JHEP06(2013)051

1

2 + + O(x

32 +)

+ ei 2 () i(12 + + )

x12 ++ + O(x32 ++)

, (C.7)

which is singular for Re[1/2 + ] < 0. However, as mentioned in appendix B, this kind

of singularities cancel out in the calculation of Cijs and we drop them in the following

calculation. We therefore rewrite (C.4) as follows:

Iij = 8 e

i2 ()(i)n11+n22 D

(n1, + n2, )

D(1, +2, )

4 i1+2e

i2 ()

Z

0 dx1

eAi(x1)

D(1, +2, ) D(1, +2, x1)

= 2

4

1cos2 2n11(1)n11+n22

1

2 + + n2

n1

1

2

n2

1

1

2

n1 (1 + n1)

1

2 ++2

1 (1 +1)

2

4 i1+2e

i2 ()

Z

0 dx1

eAi(x1)

D(1, +2, ) D(1, +2, x1) , (C.8)

with (a)m = (a + m)

(a) .

We next perform the x1-integral in (C.8). To perform this kind of integrals, we use a trick of resummation [31]. Expanding

eAi in x and using the identity for Bessel functions,

xJ(x)eix =

Xm=0amxm with am = 2m+im (m + + 1/2)m! (m + 2 + 1) , (C.9)

eAi is rewritten as

eAi = in1+n2x

12 +n1eix isin ( + n2)

ei(+n2)J+n2(x) J( +n2)(x)

= i1+n1+n2 sin

ei

Xm=0a+n2mx12 +m+n1+n2+

(1)n2

Xm=0a(+n2)mx12 +m+n1n2

,

(C.10)

40

where again (n1, n2) = (0, 0), (1, 1), (1, 0) for i = 1, 2, 3, respectively. Using this expres

sion, the x1-integral in (C.8) can be written as a sum of integrals in the form

Z

0 dx xp (D(, , ) D(, , x)) . (C.11)

Integrating by parts and using xD(, , x) = A(, , x) and x1+pA(, , x) = A(1 + +

p, , x), we rewrite it as

Z

0 dx xp (D(, , ) D(, , x))

x1+p

= 1 + p (D(, , ) D(, , x))

JHEP06(2013)051

0+

Z

0 dx

11 + pA(1 + + p, , x)

= 1

1 + p

D(1 + + p, , ) +

x1+pD(, , x) D(1 + + p, , x)

x0

= 1

1 + pD(1 + + p, , ) . (C.12)

Here we again dropped the contribution from D at x = 0, which vanishes or cancels out in

our calculation as mentioned earlier. Then, the x1-integral can be written as follows:

4 i1+2e

i2 ()

Z

0 dx1

eAi(x1)

D(1, +2, ) D(1, +2, x1) (C.13)

=

4

i1+n1+1+2

sin e

i2 (++n2)

Xm=0a+n2m12 + m + n1 + n2 +

D(1/2 + m + n1 + n2 + +1, +2, )

e

+n2)m12 + m + n1 n2

D(1/2 + m + n1 n2 +1, +2, )!

.

i2 (n2)

Xm=0a(

After some lengthy calculations, we obtain

4

i1+n1+1+2

sin e

i2 (n2) a(

+n2)m12 +m+n1n2 D

(1/2 + m + n1 n2 +1, +2, )

= i ei sin

2n112(1)m+2 (12 + m + n1 ( + n2))

1

2 + m ( + n2)

(C.14)

1+n1+1

((m+1)n1n2+~n1~n2(mn1n2122)n1n2+~n1~n2 for real , (m+1)n1n2+~n1~n2(mn1 n2122)n1n2+~n1~n2 for pure-imaginary ,

where (z) (1z) =

sin z was used. Finally, it is necessary to re-sum (C.14) with respect

to m. In the case of Re[ I11] (ni =i = 0), for example, we perform the resummation

41

as follows:

4 e

i2 ()

Z

0 dx1

eA1(x1)

D(0, , ) D(0, , x1)

= i

4 sin

Xm=0

"

(1)mei(m + 12 + )2

(1)mei (m + 12 )2 #

= i

4 sin

ei (1, 2,12 + ) ei (1, 2,12 )

(1) 14 2 .

Here (z, s, ) and (n)(z) are the Lerch transcendent and the polygamma function, respectively:

(z, s, ) =

=

i ei

16 sin

(1)

34 + 2

(1) 14 +2 + i ei 16 sin

(1)

34 2

Xm=0zm(m + )s , (n)(z) = (1)n+1n!

(n)

z + 1 2

(n)

z 2

#

JHEP06(2013)051

Xm=01(m + z)n+1 , (C.15)

which satisfy the following relation:

(1, n + 1, z) = (1)n n! 2n1"

. (C.16)

We then obtain

Re[ I11] =

24 cos2 + Re

"

i ei

16 sin

(1)

34 +2 (1)

14 +2

14 2 #, (C.17)

which reproduces the result in [31]. In a similar way, we can obtain analytic expression for

Iijs, and the results are summarized as follows:

I22 =

i ei

16 sin

(1)

34 2 (1)

2

16 cos2 +

1 16

i128 sin

12

2

32

2

ei(23)(25)ei(2+1)(21)

+ i

256 sin (2 1)2(2 3)2

ei

1, 2,12 +

ei 1, 2, 52

for real ,

2| 12|2| 32|2

16 cos2 +

1 16

i128 sin

ei 382 + 122+7233 + 2 + ei 15126362+5631 + 2

+ i

8 sin (3 + 42)

ei

1, 1,32 +

ei 1, 1, 12

+ i

256 sin (42 1)(42 9)

ei (1, 2,

52 + ) ei (1, 2,

1

2 )

for pure-imaginary ,

(C.18)

42

I33 = 2

+ 12

16 cos2

116 (C.19)

+ i

128 sin

2

12

2

ei(21)(2 3) ei(2 + 1)(2 + 3)

i256 sin (2 + 1)2(2 1)2

ei

1, 2,32 +

ei 1, 2, 32 ,

I12 =

2 12

8 cos2 i16 sin

32

ei(2 1) + ei(2 + 1)

(C.20)

+ i

16 sin (3 + 42)

ei

1, 1,12 + ei

1, 1,12

i32 sin (3 + 2)(1 + 2)

ei

1, 2,12 +

ei 1, 2, 12 ,

I21 =

2 12

8 cos2 (C.21)

+ i(1+42) 32 sin

ei

32

1, 1, 12 + ei

1, 1,32

i(42 5)

32 sin

JHEP06(2013)051

ei

1, 1,12 + ei

1, 1,52

i(3 + 2)(1 + 2)

32 sin

ei

1, 2,12 + ei

1, 2,52 ,

I13 = 2 + 12

8 cos2 +i16 sin

12

ei(2 1) + ei(2 + 1)

(C.22)

i(1 + 42)

16 sin

ei

1, 1,12 +

ei

1, 1,12

+ i(1 + 42) 32 sin

ei

1, 2,12 +

ei

1, 2,12 ,

I31 = 2 + 12

8 cos2 + i(1+42) 32 sin

ei

12

1, 1,12 +

ei 1, 1, 12

i(3 + 42)

32 sin

ei

1, 1,32 +

ei

1, 1,32

i(1 + 42) 32 sin

ei

1, 2,32 + ei

1, 2,32 , (C.23)

I23 =

2| 12|2 + 12

16 cos2 (C.24)

+ 1

16

32

i128 sin

ei(11 + 122) + ei 1 + 2

3 + 2 (1 20 + 202)

,

+ i(1 + 2)(1 4 + 42)

32 sin

ei

1, 1, 12 + ei

1, 1,32

i(1+2)2(3+2)(1+2)

256 sin

ei

1, 2,12 + ei

1, 2,52 ,

43

I32 =

2| 12|2( 32)( + 12)16 cos2 (C.25)

ei 1 + 21 + 2 (1 20 + 202) + ei(1 24 + 122)

+ i(1 + 2)(1 4 + 42)32 sin

ei

116

i128 sin

1, 1,12 +

ei

1, 1,12

+ i(3 + 2)(1 + 2)2(1 + 2)

256 sin

ei

1, 2,32 +

ei 1, 2, 32 .

Xm=0(1)m =11 + 1 =12. As we displayed in gure 1, the analytic results

obtained in this appendix and the numerical results for rs = 1 well coincide with each other.

C.1 Asymptotic behavior of D(, , x)

In this subsection we derive the asymptotic behavior (C.5) in the limit x of the

function D(, , x):

D(, , x) = 2x

1

2 + ()

Here we used

JHEP06(2013)051

i(12 + ) 2

F2

12 ,12 + ;32 + , 1 2; 2ix

(C.26)

+ ei 2x

1

2 ++ ()

12 + ,12 + + ;32 + + , 1 + 2; 2ix

.

i(12 + + )

2F2

We use the following asymptotic expansion of hypergeometric functions:

2F2(a1, a2; b1, b2; z) = (b1) (b2)

(a1) (a2)ezza1+a2b1b2

Xk=0ckzk (C.27)

+ (b1) (b2) (a2 a1)

(a2) (b1 a1) (b2 a1)

(z)a13F1(a1, a1 b1 + 1, a1 b2 + 1; a1 a2 + 1; 1/z)

(z)a23F1(a2, a2 b1 + 1, a2 b2 + 1; a2 a1 + 1; 1/z) ,

where cks are numerical factors independent of z. The rst term gives an oscillating term

e2ix, which can be dropped by an i-prescription. Then, let us consider the contribution

of the last two terms. They are respectively in the form

1 i2

1sin xe

+ (b1) (b2) (a1 a2)

(a1) (b1 a2) (b2 a2)

i2 (1/2)3F1(1/2 + , 1/2 , ; 1 ; i/(2x))

+ 1

i2

1 sin

(1/2 + )

(1/2 )

()(2i)e

i2 (1/2)

3F1(1/2 + + , 1/2 + , 0; 1 + ; i/(2x)) , (C.28)

and

1 i2

1sin xe

i2 (1/2)3F1(1/2 + , 1/2 , ; 1 ; i/(2x))

1 i2

1 sin

(1/2 + + )

(1/2 + )

()(2i)e

i2 (1/2)

3F1(1/2 + + , 1/2 + , 0; 1 + ; i/(2x)) , (C.29)

44

where note that hypergeometric functions pFq(a1, . . . , ap; b1, . . . , bq; z) are symmetric under

the permutations of ais and those of bis, respectively. The rst term in (C.28) and that in (C.29) cancel out and we obtain

D(, , x) = 1

i2

1 sin

(1/2 + )

(1/2 )

()(2i)e

i2 (1/2)

3F1(1/2 + + , 1/2 + , 0; 1 + ; i/(2x))

1 i2

1 sin

(1/2 + + )

(1/2 + )

()(2i)e

i2 (1/2)

JHEP06(2013)051

3F1(1/2 + + , 1/2 + , 0; 1 + ; i/(2x)) , (C.30) where note that we did not use any approximation so far. Finally, taking the limit x ,

we conclude that

D(, , )=

1 i2

1 sin

(1/2+)

(1/2)

(1/2++)

(1/2+)

! ()(2i)ei2 (1/2), (C.31)

where we used 3F1(a1, a2, a3; b; 0) = 1. Using the identity

(z) (1 z) =

sin z , (C.32)

we can also rewrite (C.31) as follows:

D(, , ) =

(1 i)e

i2

cos (2i)

1

( + 1)

(12 + + ) (12 + )

(12 + ) (12 )

, (C.33)

which reproduces the result in [31] for = 0.

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