Starobinsky-like two-field inflation

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Web End = Eur. Phys. J. C (2016) 76:26DOI 10.1140/epjc/s10052-016-3888-0

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Web End = Starobinsky-like two-eld ination

Sho Kaneda1,a, Sergei V. Ketov1,2,3,b

1 Department of Physics, Tokyo Metropolitan University, Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan

2 Kavli Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Chiba 277-8568, Japan

3 Institute of Physics and Technology, Tomsk Polytechnic University, 30 Lenin Ave., Tomsk 634050, Russian Federation Received: 23 October 2015 / Accepted: 7 January 2016 / Published online: 20 January 2016 The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We consider an extension of the Starobinsky model, whose parameters are functions of an extra scalar eld. Our motivation is to test the robustness (or sensitivity) of the Starobinsky ination against mixing scalaron with another (matter) scalar eld. We nd that the extended Starobinsky model is (classically) equivalent to the two-eld ination, with the scalar potential having a at direction. For the sake of fully explicit calculations, we perform a numerical scan of the parameter space. Our ndings support the viability of the Starobinsky-like two-eld ination for a certain range of its parameters, which is characterized by the scalar index ns = 0.96 0.01, the tensor-to-scalar ratio r < 0.06,

and a small running of the scalar index at |s| < 0.05.

1 Introduction

Cosmological ination in the early Universe is practically well established both theoretically and experimentally. It gives the universal solution to many problems of the Standard Cosmology, because it predicts homogeneity of our Universe at large scales, its spatial atness, its large size and entropy, as well as the almost scale-invariant spectrum of cosmological perturbations, in remarkable agreement with the COBE, WMAP, PLANCK, and BICEP2 measurements of the cosmic microwave background (CMB) radiation spectrum. Ination is also thought of as the amplier of microscopic quantum eld uctuations in vacuum, and it is the only known mechanism for seeds of the macroscopic structure formation.

The standard mechanism of ination in eld theory uses a scalar eld (called inaton), whose potential energy drives ination. The inaton scalar potential should be at enough to meet the slow-roll conditions during the inationary stage. Physical nature and fundamental origin of inaton and its

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

interactions to the standard model (SM) elementary particles are unknown.

Starobinsky ination [15] offers the gravitational origin of inaton by identifying it with the spin-0 part of spacetime metric. In the Higgs ination [68] inaton is identied with the Higgs eld of the SM. Both those single-eld inationary models offer the very economic and viable descriptions of chaotic ination together with the clear origin of the inaton eld either from gravitational theory or from particle theory, respectively. As regards slow-roll ination, the predictions of the Starobinsky and Higgs inationary models are essentially the same (see below in this section).

The simplest Starobinsky model of ination is based on the modied gravity action [1]

S[g] = d4xg

2 R +

112M2 R2 (1.1)

in terms of 4D spacetime metric g(x) with the scalar curvature R, where we have used the natural units with the reduced Planck mass MPl = 1 and the spacetime signature

(+, , , ). Slow-roll ination takes place in the high-

curvature regime (M H 1 and | H| H2), where

the Hubble function H(t) has been introduced. Then the Starobinsky inationary solution (attractor!) takes the simple form

6 (texit t), 0 < t texit. (1.2)

The inationary model (1.1) has a single mass parameter M whose value is xed by the observational cosmic microwave background (CMB) data as M = (3.0106)( 50Ne ) where Ne

is the e-foldings number. The predictions of the Starobinsky model (1.1) for the spectral indices ns 1 2/Ne 0.964,

r 12/N2e 0.004 and low non-Gaussianity are in agree

ment with the WMAP and PLANCK 2013 data (r < 0.13 and r < 0.11, respectively, at 95 % CL) [9], though they

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26 Page 2 of 14 Eur. Phys. J. C (2016) 76 :26

are in disagreement with the BICEP2 measurements (r =

0.2 + 0.07, 0.05) [10]. The enhancement of the tensor-to-

scalar-ratio r of the Starobinsky model to the higher values can be achieved via modication of the simplest Ansatz (1.1) by (matter) quantum corrections (beyond one loop) [11,12].However, the Planck 2015 data [13] excludes a signicant enhancement of r beyond r = 0.08. Therefore, the Starobin

sky model (1.1) still perfectly ts the current observational data.

It raises the natural question on the theory side about the robustness of the simplest Starobinsky model (1.1) against mixing scalaron with other (matter) scalars. Though the current observational data favors a single-eld ination, it is very unlikely that any single-eld inationary model is capable to provide the ultimate description of ination. As regards a more fundamental description of ination in super-gravity and string theories, multi-eld ination is a must; see e.g., Refs. [1416]. The direct observational evidence for multi-eld ination would be a detection of primordial isocurvature perturbations beyond the adiabatic spectrum (see Appendix A for details).

In this paper, we study the two-eld extensions of the Starobinsky model by non-minimal couplings, motivated by generic supergravity extensions of Eq. (1.1) in Ref. [17].

The action (1.1) can be dualized by the LegendreWeyl transform [18,19] to the standard (quintessence) action of the Einstein gravity coupled to a single physical scalar (canonically normalized inaton) having the scalar potential

V () =

4 M2 1 e

LJ = gJ

2 (1 +

2J )RJ +

2 g

J JJ VH(J) ,

(1.4)

having the real scalar eld J(x) non-minimally coupled to gravity with the coupling constant , and the Higgs scalar potential

VH(J) =

4 2J v2 2. (1.5)

The action (1.4) can be rewritten to the Einstein frame by the Weyl transformation

g =

(1 + 2J)

. (1.6)

It gives rise to the standard EinsteinHilbert term (12 R) for

gravity in the Lagrangian. However, it also leads to a non-minimal (or non-canonical) kinetic term of the scalar eld J.

To get the canonical kinetic term, a scalar eld redenition is needed, J (J), subject to the condition

d dJ =

1 + (1 + 6)2J 1 + 2J

. (1.7)

As a result, the non-minimal theory (1.4) is classically equivalent to the standard (canonical) theory of the scalar eld (x) minimally coupled to gravity,

= g

2 R +

2 g V () , (1.8)

and having the scalar potential [6]

V () =

VH(J())

[1 + 2J()]2

. (1.9)

Given a large positive 1, one easily nds in the large-

eld limit

23 1 that

32 log(1 + 2J) (1.10) and

V ()

2. (1.3)

During slow-roll ination the R2 term dominates in the action (1.1), whereas the coupling constant in front of it is dimensionless. It implies the (approximate) rigid scaling invariance of the Starobinsky ination in the high-curvature (high-R; or in the large-eld +) limit [8]. The scaling

invariance is not exact for nite values of R, and its violation is exactly measured by the slow-roll parameters, in full correspondence to the observed (nearly conformal) spectrum of the CMB perturbations. The (approximate) atness of the inaton scalar potential implies the (approximate) shift symmetry of the inaton eld. It also implies the alternative physical interpretation of the inaton eld as the pseudo-Nambu Goldstone boson associated with spontaneous breaking of the scale invariance [2022].

Similar observations apply to the Higgs ination in the presence of a non-minimal coupling of the Higgs eld to the spacetime scalar curvature [6]. It also has the approximate (rigid) scale invariance and, actually, the same scalar potential (1.3) during slow-roll ination [8]. The Higgs ination is based on the Lagrangian (in the Jordan frame) [6]

1 exp 23 2 (1.11)

indeed. Comparing Eqs. (1.3) and (1.11) gives rise to the identication [8]

M =

3 1. (1.12)

The LHC and TEVATRON measurements of the masses of Higgs and t-quark, however, imply (via the renormalization

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Eur. Phys. J. C (2016) 76 :26 Page 3 of 14 26

group and the Standard Model particle content) that the effective Higgs potential coupling constant becomes negative at around 1011 GeV [23], which is lower than the expected scale of ination. It means that the SM has to be extended by new particles and new physics.

It is still possible that inaton is neither Starobinsky scalaron nor Higgs eld, but a mixture of them. This possibility leads to a two-eld ination also. Another motivation to study the Starobinsky-like two-eld ination comes from 4D, N = 1 supergravity with chiral matter superelds,

where inaton is automatically extended to a complex eld as the leading bosonic eld component of an N = 1 scalar

supermultiplet. For example, as demonstrated in Ref. [17], a generic N = 1 supergravity extension of the simplest

Starobinsky model (1.1) leads to the non-minimal couplings of the Higgs eld to both R and R2 gravity terms. It is commonplace in string cosmology that the inaton is mixed with other scalars (moduli), so that a stabilization of the latter is required for ination.

Our paper is organized as follows. In Sect. 2 we dene the new class of two-eld inationary models as a combination (and a generalization) of Eqs. (1.1) and (1.4), and rewrite them to the more standard (dual) form. Those inationary models interpolate between the Starobinsky and Higgs (single-eld) inationary models, and they can accommodate a broader range of values for the tensor-to-scalar ratio.In Sect. 3 we set up the equations of motion, and classify our model against the other two-eld inationary models studied in the literature. In Sect. 4 we focus on the particular case by dropping the Higgs part of the scalar potential. In Sect. 5 we summarize our numerical ndings in the special two-eld model of the Starobinsky-like ination. Section 6 is our Conclusion. The technical details as regards linear perturbations, their spectra, and their evolution are collected in Appendix A.

2 Starobinsky- and Higgs-inspired two-eld ination models

Our new inationary model (in Jordan frame) of a real scalar eld non-minimally coupled to the Starobinsky (R + R2)

gravity is given by

(g)1/2L =

2 f 2()R +

1 12M2() R2

Both functions enter the Lagrangian (2.1) via their squares, in order to avoid ghosts. It is worth mentioning that both non-minimal couplings are required by renormalization of the (R + R2) gravity coupled to matter. In other words,

we just replaced the parameters of the Starobinsky (R + R2)

gravity by functions of a (Higgs) scalar eld .

Should the scalar eld be stabilized by its scalar potential V () to some vacuum expectation value 0, our model reduces to the standard Starobinsky model (Sect. 1). Should the M2() be sent to innity, the Higgs inationary setup is recovered.

In the case of the truly Higgs eld , its scalar potential takes the form

VH() =

4 2 v2 2 (2.2)

in terms of the coupling constants > 0 and v = 0 0.

Thus, the model (2.1) describes all the quintessence models with a non-minimal coupling to R (like the Higgs ination) and the R + R2 gravity model of Starobinsky (1.1) as

the particular cases. A non-minimal coupling to the R2 term is our new feature when M() is truly eld-dependent.

In order to understand the physical signicance of our model, and put it under the standard treatment in theoretical cosmology (in Einstein frame), let us replace the R2 by the 2 R 2 in Eq. (2.1), where the new scalar has been

introduced as follows:

(g)1/2L =

f 21

13M2 R

1 12M2 2

2 g VH(). (2.3)

It is easy to check that it is classically equivalent to the original model (2.1) because the equation of motion of the eld is algebraic, and its solution reads = R.

Introducing the notation

A(, ) = f 21()

13M2() (2.4)

allows us to rewrite Eq. (2.3) to the BransDicke-type form

(g)1/2L =

2 AR

112M2 2 +

2 g V (). (2.1)

Its non-minimal couplings are described by two generic functions f () and M() in place of the constant parameters MPl and M of the original Starobinsky model (1.1).1

1 More general couplings, including an arbitrary function of and R, were considered in Ref. [24].

2 g VH().

(2.5)

When assuming positivity of A (to avoid ghosts), i.e.

f 21() >

13M2 , (2.6)

the Weyl transformation of metric, g A(, )g,

gives rise to the standard (EinsteinHilbert) term for gravity in the classically equivalent (dual) Lagrangian,

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26 Page 4 of 14 Eur. Phys. J. C (2016) 76 :26

(g)1/2L =

2 R +

The rst term of the scalar potential has a at direction along

4A2 g A A +

2A g

112M2 2 A2VH(), (2.7)

where we have used the Einstein-frame metric and have ignored an additive total derivative. Hence, our model takes the form of the non-linear sigma model (NLSM) [25]

(g)1/2L =

2 R+

32 ln f 2(), (2.15)

while the second term in the case (2.2) has the absolute minimum at = v. Therefore, along the at direction (2.15),

the full scalar potential (2.14) has the absolute minimum at = v in the Minkowski vacuum. Moreover, along the at

direction (2.15), our model reduces to a single-eld model having the scalar potential V () in terms of the non-canonical scalar . The latter can be traded for a canonical scalar by a eld redenition, similarly to that of Eq. (1.7).

2 gGi j(, )ij V (, ),(2.8)

having the NLSM metric (i, j = 1, 2, the primes denote the

derivatives with respect to )

Gi j =

1 A2

2 f1 f 1 + 23 M3M 2 +A M2 f1 f 1 + 13 M3M

f1 f 1 + 13 M3M

3 2

(2.9)

13 M2 2

We would like to emphasize that the discovered existence of a at direction is automatic in the class of models under consideration, and it does not require supersymmetry. In a generic solution, two scalar elds and are going to evolve toward the at direction.

In order to make our model to be more specic and treatable for numerical calculations, in the following we eliminate the functional freedom above by choosing

f 2() = 1 + 2 and M2() = M2(1 + 2) (2.16)

with some (non-negative) coupling constants , , and M.

The choice (2.16) is also motivated by renormalizability. Though each of the quantum eld theories (2.1) and (2.13) is not renormalizable as a theory of quantum gravity, it still makes sense to demand the (limited) renormalizability of the quantized scalar sector in a classical (curved) gravitational background. Then the non-minimal couplings (2.16) naturally arise with the renormalization counterterms [26,27]. The Higgs potential (2.2) also ts the limited renormaliz-ability requirement.

The eld theory (2.13) has two real scalars (, ) minimally coupled to the Einstein gravity and having the scalar potential (2.14). The kinetic term of the scalar is canonically normalized, whereas the canonical term of the scalar has the -dependent factor. In the next sections we study two-eld ination in those models.

3 Classication of our model against the literature

Though our model (2.13) has the non-canonical kinetic term of , it falls into the class of the two-eld inationary models

in terms of the two scalars 1,2 = (, ), minimally coupled

to gravity in the physical (Einstein) frame, and having the scalar potential

V (, ) =

112M2 2 + A2VH(). (2.10)

The NLSM kinetic terms have no ghosts under the condition (2.6) because

det Gi j =

16M4 A3 . (2.11)

The NLSM in Eqs. (2.8) and (2.9) can be further simplied by considering A as the new independent scalar eld (instead of ) and doing the eld redenition

A = exp

3 , (2.12)

which leads to the canonical kinetic term of the scalar eld . We nd

(g)1/2L =

2 R +

2 g +

2 exp

3 g

4 M2() f 2() exp

3 2 exp 2

3 VH().

(2.13)

Hence, the full scalar potential V (, ) is given by

V (, ) =

4 M

2()

f 2() exp

3 2 + exp 2

3 VH().

(2.14)

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Eur. Phys. J. C (2016) 76 :26 Page 5 of 14 26

studied, e.g., in Ref. [28] in the slow-roll approximation and having the action

S = d

4xg

2 R+

and

I J =

1 3H2

V, V, V, V,

2 ()(

e2b()

2 ()(

) V (, ) .

. (3.12)

3.2 Correspondence of our model to the literature

In our case (2.13), we have to specify above that

b() =

(3.1)

As regards cosmological perturbations, their spectra and evolution in the theory (3.1), we employ the results of Refs. [2832] in the linear approximation with respect to the slow-roll parameters. To make our paper self-contained and complete, a derivation of the relevant equations is summarized in Appendix A. Those results are used in Sect. 5 for our numerical analysis.

3.1 Equations of motion

When assuming a spatially at FriedmannLematreRobertsonWalker (FLRW) Universe with the metric

ds2 = dt2 a(t)2dx2, (3.2)

the eld equations in the theory (3.1) take the form

+ 3H + V, = b,e2b

2 3, b, =

2 3, b, = 0, (3.13)

as well as

V, =

32 M2exp

3 exp

3 f 2 , (3.14)

V, =

32 M M,

f 2exp

3 2+

32 M2 f, 2 f 32 f exp

3 +(VH),

(3.15)

and

V, = M2exp

2, (3.3)

+ (3H + 2b, ) + e2bV, = 0, (3.4)

H2 =

1 3

3 2exp

3 f 2 , (3.16)

V, = 6Mexp

1 2

2 +

2e2b

2 + V , (3.5)

H =

3 M, exp

3 f 2 M f f, ,

(3.17)

2 + e2b

, (3.6)

where the dots stand for the time derivatives, the subscripts after the comma denote the derivatives with respect to the elds, and H = a/a is the Hubble function.

In two-eld ination the slow-roll parameters form 2 2

matrices, (I, J = (, ) = 1, 2),

I J

V, =

3 2

M2, + M M, f 2 exp 23 2

3 2

+ 4M M, f, + M2 f, 2 f 3 2 f exp 23

32 M2 f 2, 6 f 2 2exp

and I J

3 + (VH),, (3.18)

where we have used the scalar potential (2.14).

The equations of motion for cosmological perturbations are given by Eqs. (7.25) and (7.26), whose coefcients are listed in Eqs. (7.27), (7.28), (7.29), and (7.30) of Appendix A. Therefore (see Appendix A again), the perturbation (power) spectra are given by Eqs. (7.68), (7.69), and (7.70) at the horizon crossing, and by Eqs. (7.79), (7.80), and (7.81) on super-Hubble scales, with

3. (3.19)

, (3.7)

whose entries are given by

22H2 , = eb

2H2 , = e2b

22H2 , (3.8)

I J =

V,I J

3H2 , (3.9)

+ =

HH2 . (3.10)

In the case (3.1) one nds

I J =

2H2

2 eb

e2b

(3.11)

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4 Special case with VH = 0

A simple two-eld inationary model of the same type as dened in our Eq. (2.1), though with a mass term instead of the Higgs scalar potential and without non-minimal interactions to R or R2, was considered in Ref. [32]. It was found by numerical calculations in Ref. [32] that the Starobinsky ination is robust against that extension for a certain range of the ratio of two scalar masses. The model of Ref. [32] is in good agreement with the Planck data [13]. Multi-eld dynamics of Higgs ination in the presence of non-minimal couplings was analyzed in Ref. [33] where it was found that it is also in very good agreement with the Planck measurements.

In what follows, we restrict ourselves to the different case with VH = 0 (or in the limit 0), when the functions

f () and M() are given by Eq. (2.16), for simplicity. Then the scalar potential reads

V (, ) =

4 M2 1 +

In particular, the Hubble function squared of Eq. (3.5) reads

H2 =

2 +

1 6e

23 2 +1 4

1 + 2

(1 + 2) exp

3 2 . (4.6)

5 Numerical results

The two-eld scalar potential (4.1) is semi-positively denite. It reduces to the Starobinsky scalar potential (1.3) at a xed (or stabilized) , and to a power-law scalar potential V/M2 (as a sum of the 24 and the 26 terms) at a xed (or stabilized) .

The shape of the Starobinsky scalar potential at = = 0

is given in Fig. 1. In that case the scalar potential does not depend upon at all. The one-dimensional deformations of the scalar potential in the - and -directions are given by Fig. 2a, b, respectively.

The deformed scalar potential in the -direction (Fig. 2a) at = 0 essentially amounts to rescaling the M2 to the

M2(1+2), though the effect of stabilization of is already

visible in Fig. 2a. A change of the parameter at = 0

merely affects the amplitude of CMB uctuations that xes the effective scalaron mass, Meff = (3.0106)( 50Ne ). Hence,

the Starobinsky inationary pattern is very robust against changes of as long as 2 1 or, simply, when is much

less than 1. The situation does not signicantly change under small nite values of the parameter 0 < 1, as is illus

trated by Fig. 2b. When either , or , or both, grow well beyond 1, the eld is quickly stabilized, as is illustrated by Fig. 3a, b. However, our numerical calculations show that the ination becomes not viable by failing to get the observed value of the spectral (scalar) index ns. Therefore, in the following, we assume that both and are well below 1.

3 2 . (4.1)

The slow-roll matrices take the form

1 +

exp

I J =

2H2

2 e

, (4.2)

whereas the I J is the same as that in Eq. (3.12) with

M2V, = 1 +

exp

2 exp

3 (1 +

, (4.3)

M2V, = 6 exp

3 exp

3 1 +

6 1 +

exp

3 , (4.4)

M2V, =

1 + 2

exp

3 2

+122

1 + 2 +3

1 + 2 1 +2 322

1 + 2

122 exp

3 3 1 +

exp

+322

1 + 2 1 1 +2 exp

+922

1 + 2 322

1 + 2 1 1 +2 exp

3 .

(4.5)

Fig. 1 The scalar potential V/M2 of the Starobinsky model at = 0

and = 0. It serves as the starting point for deformations in the moduli

space (, )

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Eur. Phys. J. C (2016) 76 :26 Page 7 of 14 26

Fig. 2 a The scalar potential V/M2 at = 0 and = 1. b The scalar potential V/M2 at = 0.01 and = 0

Fig. 3 a The scalar potential V/M2 at = 1 and = 1000. b The scalar potential V/M2 at = 100 and = 10

Fig. 4 The scalar potential V/M2 at = 0.01 and = 0.001

As our primary example, we investigate the viable inationary model specied by the parameters = 0.01 and

= 0.001 in more detail below. The prole of its scalar

potential is given in Fig. 4.

The (time) running of the slow-roll parameters and in our special example is given by Fig. 5a, b, respectively. The spectral scalar index at the pivot scale is given by ns = 0.96

0.01. As to the tensor-to-scalar ratio r, we get r = 0.056

0.003. The spectral scalar index running s dns/d ln k is |s| < 0.05 in all our models.

To get those results, we used numerical solutions to the background equations of motion, whose graphs are given by Fig. 6a, b, for the elds and , respectively.

Our numerical calculations in this section support the qualitative conclusion that the Starobinsky ination is robust against the eld dependence in the non-minimal functions f () and M(), as long as the non-minimal coefcients and are much less than 1. In other words, the Starobinsky ination is stable against small deformations of the non-minimal couplings as long as those deformations are much less than of the order 1 (in Planck units). In the case of large deformations, ination persists but is not viable.

We also found that at the end of ination the scalaron eld oscillates near its minimum and thus contributes to (pre)heating, whereas the (matter) scalar eld does not, approaching a constant value. It can be already seen in Fig. 6a, b, but it is much better illustrated by our numerical ndings in Fig. 7a, b. Actually, the eld starts oscillating and thus contributing to the reheating only when the parameter is much larger than 1, however, it does not lead to a viable ination.

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26 Page 8 of 14 Eur. Phys. J. C (2016) 76 :26

Fig. 5 a The running slow-roll parameter at = 0.01 and = 0.001. b The running slow-roll parameter at = 0.01 and = 0.001

Fig. 6 a The running of at = 0.01 and = 0.001. b The running of at = 0.01 and = 0.001

The prole of the -solution does not signicantly change under varying parameters and . The behavior of the scalar during slow-roll and after is more sensitive to the values of the parameters and . In all cases we observe stabilization of after slow roll, as long as one of the parameters or is positive. It gives another manifestation of the robustness of the Starobinsky ination against small changes of the parameters and .

Finally, the numerical solutions to the perturbation equations for uctuations and on the background specied by Fig. 6, in our primary example with the parameters = 0.01 and = 0.001, are presented in Fig. 8.

6 Conclusion

We found that the Starobinsky ination is robust against mixing scalaron with another (matter) scalar via non-minimal interactions of the latter with both R and R2 terms in the original (Jordan) frame, as long as the non-minimal eld couplings are much smaller than one (in the Planck units). The non-minimal couplings were introduced by promoting the parameters of the original Starobinsky model to the (matter scalar) eld-dependent functions, under the additional

restriction of renormalizability of matter in the classical gravitational background.

We conrmed numerically that the inationary trajectory in our two-eld inationary models remains close to the single-eld attractor solution in the original Starobinsky model [1] under adding small non-minimal couplings to the R and R2 terms in Eq. (2.1). Our main statement is reected in the title of our paper by calling our two-eld inationary models the Starobinsky-like ones. Though our numerical solutions to the dynamical equations (Sect. 5) were obtained by using some initial conditions for ination, we found that the dependence of our solutions upon small changes in the initial conditions is weak and rather unimportant. It is related to the facts that (1) our numerical solutions also exhibit an attractor-type behavior (see e.g., Refs. [36,37] for more), and(2) our scalar potentials do not have ridges that are generically present in multi-eld ination caused by non-minimal couplings and whose presence leads to strong dependence upon the initial conditions at the onset of ination [38].

The two-eld Starobinsky-like ination becomes not viable when any of the non-minimal parameters is of the order one or larger. Our results are complementary to the ndings of Ref. [32] where the robustness of the Starobinsky ination was established in another two-eld Starobinsky-

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Eur. Phys. J. C (2016) 76 :26 Page 9 of 14 26

Fig. 7 a The typical behavior of the scalar eld near its minimum after ination. b The typical behavior of the scalar near its minimum after ination

Fig. 8 a The behavior of . b The behavior of

like limit with = = 0 and a non-vanishing mass term of

the matter scalar.

The main difference of our two-eld inationary models against the single-eld Starobinsky model is the presence of isocurvature perturbations. However, those perturbations turn out to be very small and (currently) undetectable. As was argued in Ref. [39], signicant isocurvature perturbations in generic multi-eld inationary models with non-minimal couplings may account for the observed low power in the CMB angular power spectrum of temperature anisotropies at low multipoles [40]. However, in our models the isocurvature perturbations are not amplied enough to be the reason for that observation.

Though we did not investigate primordial non-Gaussianities in our Starobinsky-like two-eld inationary models, we expect them to be negligible, like the original (single-eld) Starobinsky model.

The eld-dependent couplings are quite natural from the viewpoint of string theory where all coupling constants are given by expectation values of scalar elds. As regards the

physical meaning of our two scalars from the viewpoint of string theory, it is conceivable that scalaron is related to string theory dilaton, whereas another (matter) scalar is given by one of the moduli arising from superstring compactication. A detailed investigation of the possible connection of our models to string theory is beyond the scope of this paper.

Acknowledgments This work was supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, the special TMU Fund for International Research, and the Competitiveness Enhancement Program of the Tomsk Polytechnic University in Russia. The authors are grateful to D. Kaiser andS. Vagnozzi for discussions and correspondence, and to the referee for careful reading of our submission and critical remarks.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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26 Page 10 of 14 Eur. Phys. J. C (2016) 76 :26

Appendix A: cosmological perturbations, their power spectra and evolution at the horizon and super-Hubble scales

Linear perturbations

The standard form of scalar-perturbed spacetime metric in the longitudinal gauge (when the off-diagonal spatial components of the stress-energy tensor vanish) is given by

ds2 = (1 + 2 )dt2 a2(1 2 )dx2. (7.1)

One can decompose the scalar elds into their backgrounds and perturbations as follows:

(t, x) = (t) + (t, x), (7.2)

(t, x) = (t) + (t, x). (7.3)

The Fourier components of the perturbations are denoted by k(t) and k(t), respectively. When omitting the subscript k for simplicity, as is common in the literature, the perturbed KleinGordon equations of motion read

+ 3H + k2a2 + V, (b, + 2b2,) 2e2b

k2a2 + C Q + C Q = 0, (7.10)

where the background equations and the energy-momentum constraints above have been used in the notation [28],

C = 2e2bb2,

2 + 3

Q + 3H Q + 2b, Q + 2b, Q

e2b

22H2

42H2 e2bb,

2V,

H + V,, (7.11)

C = 3e2b

e4b

3 2H2

e2b

2H2 +

H +

e2b

H + V,,

(7.12)

C = 3e2b

e4b

4 2H2

e2b

2 2H2 +

H + e2bV,,

(7.13)

C = 3

e2b

32H2

2H2 + 2b,

2e2bb,V,

e2b

H +

H + e2bV,. (7.14)

Adiabatic and entropy perturbations

It is common in the literature to decompose cosmological linear scalar perturbations into two directions that are either parallel or orthogonal to the trajectory in the eld space [34]. The rst type of perturbations is called curvature (or adiabatic) perturbations, whereas the second type is called isocurvature (or entropy) perturbations, respectively. It can be done by introducing the linear combinations (they do not refer to new scalar elds)

cos +sin eb and s sin +cos eb,

(7.15)

where we have used the notation

, cos

+V, 2b,e2b = 4 2V, , (7.4)

+ (3H + 2b,) +

k2a2 + e2bV, + 2b,

+ e2b

(V, 2b,V,) + 2b,

2e2bV, . (7.5)

The Einstein equations lead to the energy and momentum constraints as follows:

+ H + H

= 4

k2 a2

+ e2b

+ b,e2b

2 + V, + V,

, sin

2 + e2b

+ H =

+ e2b

2. (7.16)

The corresponding MS variables Q +

H are

. (7.7)

In terms of the MukhanovSasaki (MS) variables [34],

Q +

H , (7.8)

the perturbed KleinGordon equations take the form

Q+3H Q2e2bb, Q+

k2a2 +C Q+C Q =0,

(7.9)

given by the linear combinations

Q cos Q+sin eb Q, s sin Q+cos eb Q.

(7.17)

The gauge-invariant quantity, known as the co-moving curvature perturbation [34], reads in terms of Q follows:

Q , (7.18)

H and Q +

123

Eur. Phys. J. C (2016) 76 :26 Page 11 of 14 26

while the renormalized entropy (isocurvature) perturbation [34] is given by

Perturbation spectra

We are now in a position to study the perturbation spectra of the two-eld ination and evolution of its perturbations. The power spectra of the adiabatic and entropy perturbations are given by the correlation functions [28]

k Qk =

s. (7.19)

In terms of the adiabatic and entropy vectors in eld space, dened by [28]

E I = (cos , eb sin ), E Is = ( sin , eb cos ),

I = {, } , (7.20) the corresponding rst order derivatives are

V = E I V,I , Vs = E Is V,I , (7.21)

and the second order derivatives are

V = E I E J V,I J , Vs = E I E Js V,I J , Vss = E Is E Js V,I J .(7.22)

Given the notation above, the background equations of motion in the adiabatic and entropy directions are

+ 3H + V = 0, (7.23)

sin , (7.24)

respectively. The equations of motion for the perturbations are given by

Q + 3H Q +

k2a2 + C Q +

22k3 PQ (k)(k k ), (7.31)

sksk =22k3 Ps(k)(k k ), (7.32)

k sk

22k3 CQs(k)(k k ). (7.33) The cosmologically important scales are given by (1) the horizon crossing (inside the Hubble scale) and (2) the scales over the Hubble scale, so that it is natural to evaluate the correlation functions at those scales, along the standard procedure [28].

Evolution of perturbations at the horizon crossing

In terms of the conformal time =

1a(t) dt and the new

variables

u = aQ , us = as, (7.34) Eqs. (7.25) and (7.26) can be rewritten

u +

2Vs

au s + k2

a + a2C u

2Vs

s + Css = 0,

(7.25)

a + a2Cs us = 0, (7.35)

u s

2Vs

s + 3H s +

k2a2 + Css s

au + k2

a + a2Css us

2Vs

Q + Cs Q = 0,

(7.26)

where we have again used the notation [28]

C = V

a + a2Cs u = 0, (7.36)

where the primes denote the derivative with respect to the conformal time .

In the slow-roll approximation these equations can be further simplied as

d2d2 + k2

2Vs

2 2 V

H + 3

42H2 b,

s2c V + (c2 + 1)s Vs

, (7.27)

Cs =

6H Vs

2 + 3

1 + 2E1

dd + M

1 2

u us

2V Vs

2 + 2Vs +

Vs H

=0,(7.37)

+2b

s3V c3Vs , (7.28)

Css = Vss

2 + b,(1 + s2)c V

+b,c2s Vs 2(b, + b2,), (7.29) Cs =

6H Vs

with the notation

E =

0 s

s 0

0 s3

s3 0

, (7.38)

6 + 3 4s

2s 3ss

H , (7.30) with s and c standing for the sin and cos , respectively.

2V Vs

2 +

3s2c 4s3

2s3 3c(1 + s2)

(7.39)

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26 Page 12 of 14 Eur. Phys. J. C (2016) 76 :26

and

2b, . (7.40)

In Eqs. (7.38) and (7.39) we kept only the linear terms with respect to b, because it is suppressed by the MPl and b, = 0 in our case (2.13). The terms proportional to are

written down separately, in order to emphasize the difference between the canonical and non-canonical kinetic terms.According to Ref. [28], it is convenient to introduce L and Q by

2L =

~R =

cos sin

sin cos

, (7.47)

so that

~R1 (E + M) ~R = Diag 1, 2

, (7.48)

where the star subscript refers to the horizon crossing.

Since R varies slowly around the Hubble crossing, one can replace R by R. When using the notation [28]

1 + 2 = 3 ( + ss 2 c) , (7.49)

(

2) sin 2 = 6

1 +M2 , (7.41)

and then rewrite Eq. (7.37) in the standard (in mathematical physics) form

u + 2Lu + Qu = 0. (7.42)

As the next step, again following Ref. [28], let us introduce the time-dependent matrix R which satises R = LR, and

the new vector v dened by u = Rv. Then the equation above

can be resolved for v as

v + R1 L2 L + Q R

v = 0, (7.43)

where

L2 L

, Q = k2

2 + 3

s s3

, (7.50)

(

2) cos 2 = 3

ss 2 + c(1 + 2s2)

(7.51)

at k = aH, and

w = ~R1Rv, (7.52)

and one can also rewrite Eq. (7.42) as

w A + k2

12 (2 + 3A) wA = 0 (A = 1, 2), (7.53)

with

A =

1 3

12 E (7.44)

in the linear order with respect to the slow-roll parameters. It follows

L2 L + Q k2

2 + 3

A. (7.54)

The solution to Eq. (7.53) with the proper asymptotic behavior reads [28]

wA =

1 +12 (E + M) , (7.45)

where the second term reads

12 (E + M) =

3 2

2 exp

i(A + 12) 2

H(1)A(k)eA(k)

(7.55)

2 + + s2 c s s 3

s s

3 ss c (1 + s

(7.46)

It is usually assumed in the literature that the slow-roll parameters vary slowly enough during the few e-folds when the inationary scale crosses the Hubble radius. In that case, one can replace the time-dependent matrix with its value at the Hubble crossing. In other words, the matrix on the r.h.s. of Eq. (7.46) is supposed to be evaluated at k = aH, so

that the remaining time dependence only exists in the overall coefcient 3/2. Then one can always diagonalize this matrix by using a time-independent rotation matrix,

2 )

in terms of the Hankel function H(1) of the rst kind and of the order A, where [28]

A =

94 + 3A, (7.56)

and eA, A = 1, 2, are the independent orthonormal (Gaus

sian) random variables,

eA(k) = 0, e A(k)eB(k )

AB(3)(k k ). (7.57)

Because of the independence of w1 and w2, the correlations of u and us around the Hubble crossing can be expressed in terms of Q and s as

123

Eur. Phys. J. C (2016) 76 :26 Page 13 of 14 26

a2 Q Q = cos2 w1w1 + sin2 w2w2 , (7.58)

a2 ss

= sin2

w1w1 + cos2

w2w2

CRS =

H2 2

(1 + k22)(2s3 2s)g

k aH

(7.70)

, (7.59)

sQ =12 sin 2 w1w1 w2w2 , (7.60)

where we have

wAwA =4 ()

H(1)A(k) 212k1(k)2 GA(k).

(7.61)

Evolution of perturbations on super-Hubble scales

A two-eld inationary model is reduced to a single-eld inationary model when the isocurvature perturbations are suppressed. Then the adiabatic spectrum takes the form

PSHR(k)

H4 4

. (7.71)

However, it does not apply to our model (Sect. 2) in a generic case where it does not reduce to the Higgs or the Starobinsky (single-eld) inationary model.

The existence of the isocurvature modes is a generic feature of two-eld inationary models, and it is going to affect adiabatic perturbations also during the super-Hubble scale evolution, so that Eq. (7.71) does not apply, in general. To get the power spectra and the correlation functions in that case, one should solve the coupled system of Eqs. (7.25) and (7.26). A numerical approach is the only way in most cases.

However, in some special cases, when the slow-roll approximation is at work, one may analytically solve the equations of motion on the super-Hubble scales too. The example considered in Ref. [28] was Eqs. (7.25) and (7.26) in the slow-roll approximation,

Q AH Q + B Hs and s DHs, (7.72)

where

A = + 2 cs2, (7.73)

B = 2s + 2s3 2

dd N 2s, (7.74)

D = ss + c(1 + s2). (7.75)

Equation (7.72) implies that adiabatic and isocurvature perturbations have a strong interaction unless the isocurvature perturbations rapidly decay. For constant values of (A, B, D), Eq. (7.72) can be solved as [28]

Q (N) eAN Q +

BD A

Therefore, after taking into account that

1 +

, (7.62)

one nds [28]

PQ =

2(1 2 ) cos2 G1(k) + sin2 G2(k) ,

(7.63)

Ps=

2(1 2 ) sin2 G1(k) + cos2 G2(k) ,

(7.64)

CQ s =

2(1 2 )sin 2

2 [G1(k) G2(k)] .

(7.65)

Given A 1, i.e. A 32 +A, one can further simplify

the result above, by expanding GA(x) as follows:

GA(x) =

2 x3

2 (x)

2 (1 + 2Ag(x))

= (1 + x2)(1 + 2Ag(x)), (7.66) where the new function has been introduced,

g(x) = Re

H(1)3

2 (x)

dH(1)(x)

d = 32

. (7.67)

It follows that the power spectra and the correlations of curvature and entropy perturbations are

PR =

H2 2

(1 + k22)

1 + 2 + (6 2 2s2c)g

k aH

(7.68)

eDN eAN

PS =

H2 2

(1 + k22)

1 2 + (2 2ss + 2(1 + s2)c)g

s, (7.76) s(N) eDN s, (7.77)

where the number N of the e-folds after the Hubble crossing has been introduced.

Given

k aH

(7.69)

eAN , (7.78)

123

26 Page 14 of 14 Eur. Phys. J. C (2016) 76 :26

one can easily nd the power spectra and the correlation functions as [28]

PSHR(N) PR + PS

2 e N 1 2

+2 CRS

e N 1 , (7.79)

PSHS(N) PSe2 N , (7.80)

CSHRS(N) CRSe N + PS

B e N e N 1

, (7.81)

where = DA, and the PR, PS and CRS are supposed

to be evaluated in the asymptotic limits of Eqs. (7.68), (7.69), and (7.70), respectively, i.e. at k 0.

Unfortunately, as already noticed in Ref. [28], the constant slow-roll approximation does not hold for many e-folds, and it breaks down long before the exit from ination. In another analytically treatable case, with the mass terms as the scalar potential and the canonical kinetic terms for scalars, the curvature and isocurvature perturbations were computed in Ref. [35].

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We consider an extension of the Starobinsky model, whose parameters are functions of an extra scalar field. Our motivation is to test the robustness (or sensitivity) of the Starobinsky inflation against mixing scalaron with another (matter) scalar field. We find that the extended Starobinsky model is (classically) equivalent to the two-field inflation, with the scalar potential having a flat direction. For the sake of fully explicit calculations, we perform a numerical scan of the parameter space. Our findings support the viability of the Starobinsky-like two-field inflation for a certain range of its parameters, which is characterized by the scalar index ......, the tensor-to-scalar ratio ......, and a small running of the scalar index at .......

Details

Title

Starobinsky-like two-field inflation

Author

Kaneda, Sho; Ketov, Sergei V

Pages

1-14

Publication year

2016

Publication date

Jan 2016

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-016-3888-0

ProQuest document ID

1758616730

SIF and Springer-Verlag Berlin Heidelberg 2016

Starobinsky-like two-field inflation

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