Inflation with a quartic potential in the framework of Einstein-Gauss-Bonnet gravity

Ekaterina O. Pozdeeva,1, ∗ Mayukh Raj Gangopadhyay,2, †

Mohammad Sami,3, 4, 5, 6, ‡ Alexey V. Toporensky,7, 8, § and Sergey Yu. Vernov1, ¶

1Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia2Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India

3International Center for Cosmology, Charusat University, Anand 388421, Gujarat, India4Maulana Azad National Urdu University, Gachibowli, Hyderabad 500032, India

5Institute for Advanced Physics and Mathematics, Zhejiang University of Technology,Hangzhou 310032, China

6Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan7Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetsky pr. 13, Moscow 119991, Russia

8Kazan Federal University, Kremlevskaya Street 18, Kazan 420008, Russia

We investigate inflationary dynamics in the framework of the Einstein-Gauss-Bonnet gravity. Inthe model under consideration, the inflaton field is nonminimally coupled to the Gauss-Bonnetcurvature invariant, so that the latter appears to be dynamically important. We consider a quarticpotential for the inflaton field, in particular, the one asymptotically connected to the Higgs inflation,and a wider class of coupling functions not considered in the earlier work. Keeping in mind theobservational bounds on the parameters — the amplitude of scalar perturbations As, spectral indexns and tensor-to-scalar ratio r, we demonstrate that the model a quartic potential and the proposedcoupling function is in agreement with observations.

PACS numbers: 98.80.-k, 98.80.Cq, 04.50.KdKeywords: Inflation, modified gravity, effective potential, slow-roll approximation

∗ pozdeeva@www-hep.sinp.msu.ru† mayukh@ctp-jamia.res.in‡ msami@jmi.ac.in§ atopor@rambler.ru¶ svernov@theory.sinp.msu.ru

arX

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-qc]

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2

I. INTRODUCTION

The current observations on Cosmic Microwave Background put tight constraints on models of inflation [1]. Thepopular models with quadratic and quartic potentials [2], that played an important role in the history of inflationarymodel building, already turn out to be incompatible with observations, as the tensor-to-scalar ratio of perturbationswhich these models predict is too large [3–5]. If one adheres to General Relativity, the field potential should be rathergently sloping in order to match with observations [3, 6, 7]. These restrictions however, can drastically be softened inmodified theories of gravity [4, 8–13].

A well-known example is provided by the Higgs field as an inflaton [11], which due to its large self-coupling givesrise to the large amplitude of scalar perturbations in the framework of the standard theory of gravity1. However, anonminimal coupling of the Higgs field with the curvature can significantly reduce the amplitude of scalar perturbationsbringing the model within the observational bounds. Unfortunately, the price for this is paid by the large numericalvalue of the dimensionless constant of the nonminimal coupling which sounds unnatural2. This motivates a search formore complicated scenarios of inflation, in particular, the one obtained by adding the term proportional to R2 to theEinstein-Hilbert Lagrangian. Another possibility is provided by the α-attractor formalism which also allows loweringof the tensor-to-scalar ratio of perturbations down to the ones consistent with observation [14, 15].

We hereby consider a scenario that uses a particular form of quadratic gravity, namely, the Gauss-Bonnet (GB)term added to the Einstein-Hilbert action. Being a total derivative, this term alone does not contribute to equationsof motion, it, however, becomes dynamically important if coupled with a function of the scalar field, ξ(φ). This mighthave important implications for inflation as well as for late time acceleration. On the other hand, the GB term arisesnaturally in the string theory framework as a quantum correction to the Einstein-Hilbert action [16–23].

A plethora of inflationary models with the GB term [24–42], including the ones with cosmological attractor con-structs [33, 42] have been discussed in the literature. The most actively studied models with GB coupling involves thefunction ξ inversely proportional to the scalar field potential [25–27, 32, 36, 37, 39, 40, 42]. Considering this specificform of the coupling function, it has been demonstrated that models based upon quadratic and quartic potentialscould be rescued: the GB term reduces the tensor-to-scalar ratio r to appropriate values consistent with observa-tion [26, 36]. Interestingly, the GB term does not change the value of the spectral index ns for the mentioned choice ofthe coupling function. To the best of our knowledge, there is no physical reason for such a choice of the function ξ(φ).The particular form of coupling function ξ(φ) is motivated by the consideration of simplicity. It would be plausibleto enlarge the class of coupling functions and check the observational viability of the respective models.

In the present paper, we investigate a wider class of theories involving GB coupling with coupling function ξ(φ)not related to the inflaton potential, and show that the above-mentioned tensions with observation can be removed.In this framework, it is possible to reduce the scalar amplitude such that the Higgs field coupled to GB term couldprovide reasonable perturbation parameters compatible with current observational limits. Note that we consider theminimal coupling between the scalar field and the scalar curvature.

A useful tool that allows us to obtain different inflationary models with the GB term is provided by the constructionof an effective potential. This function has been proposed to study stability of de Sitter solutions in models withnonminimal coupling [43, 44] and generalized to the models with the GB term [45]. In this paper, we use the formalismof the effective potential for investigations of inflationary parameters in the models under consideration.

II. THE SLOW-ROLL PARAMETERS AND THE LEADING ORDER EQUATIONS

In what follows, we shall consider the modified gravity model with the GB term, [26]:

S =

∫d4x√−g[UR− gµν

2∂µφ∂νφ− V (φ)− ξ(φ)

2G], (1)

where U is a positive constant, the functions V (φ), and ξ(φ) are differentiable ones, and G is the Gauss-Bonnet term:G = RµνρσR

µνρσ − 4RµνRµν +R2.

Application of the variation principle leads to the following system of equations in the spatially flat Friedmann

1 The tensor-to-scalar ratio of perturbations is also large in this case which also applies to the model-based upon quadratic potential.2 The numerical value of nonminimal coupling constant turns out to be about 50000 which is rather large compared to unity for a

dimensionless fundamental constant.

3

universe [29, 45]:

12UH2 = φ2 + 2V + 24ξH3, (2)

4UH = − φ2 + 4ξH2 + 4ξH(

2H −H2), (3)

φ = − 3Hφ− V ′ − 12ξ′H2(H +H2

), (4)

where H = a/a is the Hubble parameter, a(t) is the scale factor, dots and primes denote the derivatives with respectto the cosmic time t and the scalar field φ, respectively.

During inflation H(t) is always finite and positive, therefore, it is possible to use the dimensionless parameterN = ln(a/ae), where ae is a constant, as a new measure of time3.

Following Refs. [26, 29], we consider the slow-roll parameters:

ε1 = − H

H2= − d ln(H)

dN, εi+1 =

d ln |εi|dN

, i > 1,

δ1 =2

UHξ =

2

UH2ξ′

dφ

dN, δi+1 =

d ln |δi|dN

, i > 1,

where we use d/dt = H d/dN . The slow-roll approximation requires |εi| � 1 and |δi| � 1. We fix ae by the conditionε1 = 1.

The slow-roll conditions ε1 � 1, ε2 � 1, δ1 � 1, and δ2 � 1 allow to simplify Eqs. (2)–(4). Indeed,

δ2 =δ1Hδ1

=2ξ

Uδ1− ε1, (5)

so from |δ2| � 1 and |ε1| � 1 it follows |ξ| � |Hξ|. Using |δ1| � 1 and |δ2| � 1, we obtain from Eqs. (2) and (3):

12UH2 ' φ2 + 2V, (6)

4UH ' − φ2 − 4ξH3 = − φ(φ+ 4ξ′H3

). (7)

Using,

ε1 = − H

H2' φ2

3(φ2 + 2V )+

1

2δ1 � 1,

we obtain φ2 � 2V , and Eq. (6) takes the following form

6UH2 ' V. (8)

Taking the time derivative of this equation and using Eq. (7), we get

φ ' − V ′

3H− 4ξ′H3. (9)

Substituting this relation to Eq. (4), we get that |φ| ' |12ξ′H2H| � |12ξ′H4|. Thus, the slow-roll conditions resultto

φ2 � V, |φ| � |12ξ′H4|, 2|ξ|H � U, |ξ| � |ξ|H ,

so, the leading order equations in the slow-roll approximation have the following form:

H2 ' V

6U, (10)

H ' − φ2

4U− ξH3

U, (11)

φ ' − V ′ + 12ξ′H4

3H. (12)

In the following section, we briefly discuss the effective potential formalism to be used for analysing the inflationarydynamics.

3 Note that in many papers [26, 29, 41, 46] N = 0 corresponds to the beginning of inflation, whereas we fix N = 0 at the end of inflation.Also, there is an alternative definition of the e-folding number: N = − ln(a/ae), see [3, 32, 42].

4

III. THE EFFECTIVE POTENTIAL AND INFLATIONARY SCENARIOS

A. The slow-roll approximation

To analyze the stability of de Sitter solutions in model (1) the effective potential has been proposed in Ref. [45]:

Veff (φ) = − U2

V (φ)+

1

3ξ(φ). (13)

The effective potential is not defined in the case of V (φ) ≡ 0, but inflationary scenarios are always unstable in thiscase [47] (see also [35]). In this paper, we consider inflationary scenarios with positive potentials only: V (φ) > 0 duringinflation. The effective potential characterizes existence and stability of de Sitter solutions completely. It is howevernot enough to fully characterize quasi-de Sitter inflationary stage, and the potential V (φ) enters into expressions ofthe inflationary parameters (see below) as well. Nevertheless, keeping the effective potential in the correspondingformulae would be helpful, as we will see soon.

Using Eqs. (11) and (12), we get that the functions H(N) and φ(N) satisfy the following leading order equations:

dH

dN' − H

UV ′V ′eff , (14)

dφ

dN' − 2

V

UV ′eff . (15)

In terms of the effective potential the slow-roll parameters are as follows:

ε1 = − 1

2

d ln(V )

dN=V ′

UV ′eff , (16)

ε2 = − 2V

UV ′eff

[V ′′

V ′+V ′′effV ′eff

]= − 2V

UV ′eff

[ln(V ′V ′eff )

]′, (17)

δ1 = − 2V 2

3U3ξ′V ′eff , (18)

δ2 =− 2V

UV ′eff

[2V ′

V+V ′′effV ′eff

+ξ′′

ξ′

]

=− 2V

UV ′eff

[ln(V 2ξ′V ′eff )

]′.

(19)

So, |ε1| � 1 and |δ1| � 1 if V ′eff is small enough. It allows us to use the effective potential for construction of theinflationary scenarios in models with the GB term.

Using the known formulae [26, 32] for the tensor-to-scalar ratio r and the spectral index ns, we obtain:

r = 8|2ε1 − δ1| =4

U

[dφ

dN

]2= 16

V 2

U3

(V ′eff

)2, (20)

ns =1− 2ε1 −2ε1ε2 − δ1δ2

2ε1 − δ1= 1− 2ε1 −

d ln(r)

dN

=1 +d ln(V/r)

dN= 1 +

2

U

(2V V ′′eff + V ′V ′eff

).

(21)

Moreover, the spectral index ns can be presented via derivatives of the effective potential only:

ns = 1− d

dN

(ln

(dVeffdN

)). (22)

5

A standard way of the reconstruction of inflationary models [3, 32, 42] includes the assumption of explicit form ofthe inflationary parameter ns and r as functions of N . Formula (22) shows how the knowledge of ns(N) allows tocalculate Veff (N).

The expression for amplitude As in the leading order approximation is [29]:

As ≈H2

π2Ur≈ V

6π2U2r. (23)

In the slow-roll approximation, the e-folding number N can be presented as the following function of φ:

N(φ) =

φ∫φend

dN

dφdφ '

φend∫φ

U

2V V ′effdφ. (24)

By this definition, N < 0 during inflation. To get a suitable inflationary scenario we calculate inflationary parametersfor −65 < N < −50 and compare them with the observation data [1].

Integrating Eq. (15), one gets the function φ(N) is either in the analytic form, or in quadratures. We assume thatN = 0 at the end of inflation, and fix the value of the integration constant by the condition ε1(φ(0)) = 1. After this,we know εi(N) and δi(N) and the inflationary parameters.

B. Stability of de Sitter solutions

The effective potential Veff is a useful tool to seek de Sitter solutions and to determine their stability [45]. In thecase of the existence of a stable de Sitter solution, it is difficult to construct an inflationary scenario with a gracefulexit. One might consider models with unstable de Sitter solutions or without an exact de Sitter solution, but unstablequasi-de Sitter ones are more suitable.

In the case of a monomial potential V and a more complicated function ξ:

V = V0φn, ξ =

3U2

V0αφq +

(β +

3U2

V0

)φ−n, (25)

with arbitrary constants α, β, V0 > 0, n and q 6= −n, we have:

Veff =U2

V0φn

(αφq+n +

V0β

3U2

),

In the case of α = 0 and β 6= 0, we obtain ξ = C/V , where C = 3U2 + V0β, and Veff is proportional to ξ:

Veff =C − 3U2

3V=

β

3φn(26)

and these is no de Sitter solution, because the effective potential Veff has no extremum for n > 0. For n < 0 thereexist the only extremum for φ = 0, but the potential V is not finite at φ = 0. By the same reason, these is no deSitter solution with φdS 6= 0 in the case α 6= 0 and β = 0.

In the case of nonzero α and β, the de Sitter point is given by

φdS =

(nβV0

3U2αq

)1/(q+n)

. (27)

In the present paper, we focus on the case of q < 0 and n > 0. The choice is motivated from the fact that, in thiscase, the effective potential, which governs the slow-roll regime, is flatter than the potential V (φ). We demonstratelater that it can improve the situation with inflation in the simplest cases of massive and self-interacting potentials.In the case of q < 0 and n > 0, we should assume that βV0/α < 0 to get a de Sitter solution with a real φdS > 0.

At the de Sitter point, the second derivative of the effective potential is

V ′′eff (φdS) =nβ(q + n)

3φ2+ndS

. (28)

For positive values of n and φdS , the de Sitter solution is stable if β(n+ q) > 0 and unstable in the case β(n+ q) < 0.The case of the quartic potential V and a more complicated function ξ will be investigated in Section V.In the next sections, we demonstrate the usefulness of the effective potential for selected cases.

6

IV. INFLATIONARY PARAMETERS IN CASE OF MONOMIAL POTENTIAL WITH GB COUPLING

A. Application to the known model

The choice of the function ξ(φ) = C/V (φ), where C is a constant, is actively studied [25–27, 32, 36, 37, 39, 40, 42].In this case,

Veff =C − 3U2

3V, (29)

and the slow-roll parameters are as follows:

ε1 =

(3U2 − C

)V ′

2

3UV 2, ε2 =

4(C − 3U2

) (V V ′′ − V ′2

)3UV 2

,

δ1 =2C

3U2ε1 , δ2 = ε2 .

So, the inflationary parameters are

ns = 1 +2(3U2 − C

) (2V V ′′ − 3V ′

2)

3UV 2. (30)

r =16V ′

2 (3U2 − C

)29U3V 2

, (31)

Note that in the case C = 3U2, the slow-roll approximation does not work, because all slow-roll parameters areidentically equal to zero.

For V = V0φn, where V0 and n are constants, we integrate Eq. (24), taking into account ε1(φ(0)) = 1, and obtain

φ2(N) =n(4N − n)(C − 3U2)

3U. (32)

So,

ε1 = − n

4N − n, ε2 = − 4

4N − n, (33)

δ1 = − 2C n

3U2(4N − n), δ2 = − 4

4N − n, (34)

From Eqs. (30) and (31), we obtain:

ns = 1 +2(n+ 2)

4N − n, r =

∣∣∣∣16n(C − 3U2)

3U2(4N − n)

∣∣∣∣ , (35)

As =V0(4N − n)1+n/2

32π2(3U)n/2[n(C − 3U2)]1−n/2. (36)

In the case of a monomial potential V , adding of the GB term with ξ = C/V does not change ns, but changes Asand r. Let us note, that for n > 2, the combination C − 3U2 which is the numerator of the effective potential entersin the numerators of both As and r. This means that both the parameters can be made appropriately small if thecorresponding effective potential is small, independently of the magnitude of the actual potential V (φ), which allowsus to obtain the required values of spectral index and tensor-to-scalar ratio for the Higgs field, coupled appropriatelyto the GB term. Note also, that in this scenario we need not introduce a dimensionless parameter large in comparedto unity similar to the theory of Higgs field coupled to curvature [11]. However, we yet need to address the problemassociated with the spectral index ns.

7

Substituting −65 < N < −50, one can compare the inflationary parameters with the observation data. In casen = 4 and ε1 = ε2 = δ2, we obtain in the slow-roll approximation

r =

∣∣∣∣16(C − 3U2)

3U2(N − 1)

∣∣∣∣ , ns = 1 +3

N − 1, (37)

The observation [1]: ns = 0.9649 ± 0.0042 at 68% CL, implies that −96 < N < −75. At the same time the numberof e-foldings before the end of inflation at which observable perturbations were generated for the φ4 model withoutthe GB term has been estimated [48] as 64 and we do not think that the addition of the GB term can essentiallyincrease this number. By this reason, the model with the φ4 potential and ξ ∼ 1/φ4 is ruled out. In this paper, wefind such a function ξ(φ) that the model with V = V0φ

4 does not contradict the observation data. For n > 4, we alsoget contradictions with the observation data.

We complete the present subsection with a brief discussing on n = 2 case, when both ns, and As do not dependon C. For −65 < N < −55, one gets 0.9639 < ns < 0.9695 that is in a good agreement with observation. Sinceε2 > ε1 and δ2 > ε1, then we cannot use the slow-roll approximation up to the point N = 0, when ε1 = 1, but thisapproximation is valid for any N < −1/2 if |C| < 3U2. Note that choosing 1.5U2 < C < 4.5U2, one gets r < 0.0673that does not contradict the observation data. This means that it is possible to revive a massive scalar field inflation,as stated in [26, 40]. Note, that unlike the n > 2 case, the scalar field potential itself should ensure the correct valueof As, so the requirement that the mass of the scalar field should be of the order of 10−5MPl is necessary.

B. Models with monomial functions V and ξ

In the case of V = V0φn and ξ = ξ0φ

q, we get the following expressions for the slow-roll parameters:

ε1 = nUα qφq+n + n

φ2, (38)

ε2 = 2U2n− α qφq+n (n+ q − 2)

φ2, (39)

δ1 = − 2Uα qφq+n−2(α qφq+n + n

), (40)

δ2 = 2U−2qαφq+n (n+ q − 1)− n (n− 2 + q)

φ2(41)

and for the inflationary parameters:

ns = 1− 2U

φ2([3n+ 2q − 2]qαφn+q + n2 − 2n

),

r =16U

φ2(qαφn+q + n

)2.

The expression for amplitude is

As ≈V0 φ

n+2

12π2 (α qφq+n + n)2 . (42)

For n = 4 and q = −2, the slow-roll parameters (38)–(41) can be presented in the following form:

ε1 = −8U

(αφ2 − 2

)φ2

, ε2 =16U

φ2,

δ1 = − 8αU(αφ2 − 2

), δ2 = 8αU.

The condition for the end of inflation ε1(φend) = 1 is satisfied at the point φend =√

8/(1 + 4α).For the considering case, the tensor-to-scalar ratio and the spectral index of scalar perturbations can be presented

in the following form:

ns = 1 + 8αU − 48U

φ2, (43)

r =64U

(αφ2 − 2

)2φ2

. (44)

8

The expression for the amplitude is

A ≈ V0 φ6

384U3π2 (αφ2 − 2)2 . (45)

The e-folding number can be expressed as follows:

N =ln((8αU + 1)

(2− αφ2

)/2)

8αU, (46)

hence,

φ2 =2 (8αU − e8UαN + 1)

α (8αU + 1). (47)

The spectral index of scalar perturbations (43) is given by the expression:

ns =(8αU + 1)

(1− 16αU − e8UαN

)8αU − e8UαN + 1

, (48)

the tensor-to-scalar ratio (44) can be expressed as follows:

r =128αUe16UαN

(8αU + 1) (8αU − e8UαN + 1)

=16(16Uα+ ns − 1)2

3(8Uα− ns + 1).

(49)

For N = − 60, the spectral index ns ≈ 0.9506 at α = 10−4. For the same values of α and N the amplitude is:

As =V0(8Uα− e8UαN + 1

)3192π2U3α3(8Uα+ 1)e16UαN

≈ 62084.7 · V0 .

To get the required magnitude of the amplitude As, we chouse the corresponding value of V0. Substituting thesame, we can find ns and r as functions of α. The value of ns now does depend on the GB coupling, however, itappears to be impossible to get both ns and r in the observationally allowed range (see Fig. 1). This problem can becircumvented by considering a more complicated form of the coupling function. This consideration is the goal of thenext section.

V. THE φ4 POTENTIAL AND COMPLICATED FORMS OF THE GB INTERACTION

In this section, we shall investigate models with quartic potential for the GB coupling function of a more complicatedstructure.

A. The existence of de Sitter solutions

Let us consider a more general model,

V = λφ4, ξ = ξ2φ−2 + ξ4φ

−4 + ξ6φ−6, (50)

with arbitrary constants ξ2, ξ4, and ξ6. The dimensionless constant λ in the potential will be fixed to λ = 0.1 in orderto describe a large field approximation for the Higgs potential.

The condition V ′eff (φdS) = 0 gives the following values of φ2dS :

φ2dS =

−β ±

√β2 − 3ξ2ξ6ξ2

, at ξ2 6= 0,

− 3ξ62β

, at ξ2 = 0,

(51)

9

1.×10-4 2.×10-4 5.×10-4 0.001 0.002 0.0051.×10-9

1.5×10-9

2.×10-9

2.5×10-9

3.×10-9

α

As

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.000.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

ns

rFIG. 1. Inflationary parameters for the model with V = V0φ

4 and ξ = ξ2φ−2. Blue solid, red dashed and green dot curves

correspond to N = −55, N = −60, and N = −65 respectively. We fix U = 1/2. The contours correspond to the marginalizedjoint 68% and 95% CL. The cyan dash-dot lines are for the Planck’18 TT, TE, EE+ low E+ lensing data, whereas the blacksolid lines correspond to Planck’18 TT, TE, EE+ low E+ lensing + BK15+ BAO data. The pivot scale is fixed at usualk = 0.002MPc−1.

where β = ξ4 − 3U2/λ.

Without loss of generality, we consider φdS > 0 only4. The condition φ2dS > 0 gives restrictions on values of theparameters. If ξ2 = 0, then a de Sitter solution exists if and only if ξ6/β < 0. In the case ξ2 6= 0, we have the followingpossibilities:

1. If ξ6ξ2 > β2/3, then these is no de Sitter solution.

2. If 0 6 ξ6ξ2 6 β2/3 and ξ2β > 0, then these is no de Sitter solution.

3. If 0 < ξ6ξ2 < β2/3 and ξ2β < 0, then there exist two de Sitter solutions.

4. If ξ6ξ2 = β2/3 and ξ2β < 0, then there exists only one de Sitter solution.

5. If ξ6ξ2 < 0, then there exists only one de Sitter solution.

6. If ξ6 = 0 and ξ2β < 0, then there exists only one de Sitter solution.

The stability of a de Sitter point is defined by the sign of V ′′eff (φdS). In particular, if ξ2 = 0, then

V ′′eff (φdS) =64β4

81ξ36, (52)

so a de Sitter solution is stable at β < 0 and ξ6 > 0 and is unstable at β > 0 and ξ6 < 0. If ξ6 = 0, then de Sittersolution is stable at β > 0 and ξ2 < 0 and is unstable at β < 0 and ξ2 > 0. To get a suitable inflationary scenario wecan consider both the case with unstable de Sitter solution, and the case without de Sitter solution.

4 The case φdS = 0 is excluded because the function ξ is singular at φ = 0.

10

B. The inflationary parameters

For the model under consideration, we obtain the following expressions,

ε1 = − 8λ

3Uφ4(ξ2φ

4 + 2βφ2 + 3ξ6), ε2 = − 16λ

3Uφ4(βφ2 + 3ξ6

), (53)

δ1 = − 8λ

9U3φ6(λφ4ξ2 + (6U2 + 2λβ)φ2 + 3λξ6

) (φ4ξ2 + 2βφ2 + 3ξ6

), (54)

δ2 = − 8λ(−λξ22φ8 + 2(6U2β + 2λβ2 + 3λξ2ξ6)φ4 + 12ξ6(3U2 + 2λβ)φ2 + 27λξ26)

3Uφ4(λξ2φ4 + 6U2φ2 + 2λβφ2 + 3λξ6). (55)

The inflationary parameters are as follows:

ns = 1 +8λ(ξ2φ

4 + 6βφ2 + 15ξ6)

3Uφ4, r =

64λ2(ξ2φ

4 + 2βφ2 + 3ξ6)2

9U3φ6. (56)

In the generic case, when parameters ξ2, ξ6, and β are nonzero, analytical solutions cannot be obtained. Thusnumerical methods are being implemented to get the relationships between the inflationary parameters. In Fig. 2,one can see that for the e-foldings number N between −65 and −55, one can always get ns and r in the observationalrange for particular choice of ξ2. Here, we have taken ξ6 = −0.1 and U = 1/2. The value of β is taken to be β = −7.4.The choice of β is from the fact to keep As ∼ 2.1× 10−9. The parameter ξ2 is taken in the range 0 ≤ ξ2 ≤ 0.5.

FIG. 2. Parameter space of ns and r for the model with V = λφ4 and ξ = ξ2φ−2 + ξ4φ

−4 + ξ6φ−6 in the case β 6= 0. Blue

solid, red dashed and green dot curves correspond to N = −55, N = −60, and N = −65 respectively. The contours correspondto the marginalized joint 68% and 95% CL. The cyan dash-dot lines are for the Planck’18 TT, TE, EE+ low E+ lensing data,whereas the black solid lines correspond to Planck’18 TT, TE, EE+ low E+ lensing + BK15+ BAO data. The pivot scale isfixed at usual k = 0.002MPc−1.

It is also interesting to consider a few particular cases, when parameters ξ2, ξ6, or β are equal to zero. In these casessome analytic results can be obtained. In particular, if ξ6 = 0 and β = 0, then ε1 is a constant, hence, the slow-rollinflation is not possible in this case. We consider other particular cases in the next subsections of this section.

11

C. The case of β = 0

In the case of ξ6 6= 0 and β = 0, the inflationary parameters (56) are as follows:

ns = 1 +8λ(ξ2φ

4 + 15ξ6)

3Uφ4,

r =64λ2

(ξ2φ

4 + 3ξ6)2

9U3φ6.

(57)

The solution of Eq. (15) with an additional condition ε1(φ(0)) = 1 has the following form:

φ(N) = 4

√3ξ6(3Ue16λξ2N/(3U) − 8λξ2 − 3U)

ξ2(8λξ2 + 3U)(58)

FIG. 3. Parameter space of ns and r for the model with V = λφ4 and ξ = ξ2φ−2 + ξ4φ

−4 + ξ6φ−6 in the case β = 0. Blue solid,

red dashed, and green dot curves correspond to N = −55, N = −60, and N = −65 respectively. The contours correspond tothe marginalized joint 68% and 95% CL. The cyan dash-dot lines are for the Planck’18 TT, TE, EE+ low E+ lensing data,whereas the black solid lines correspond to Planck’18 TT, TE, EE+ low E+ lensing + BK15+ BAO data. The pivot scale isfixed at usual k = 0.002MPc−1.

Substituting (58) into expressions (57), we get:

ns = 1 +8λξ2(3Ue16λξ2N/(3U) + 32λξ2 + 12U)

3U(3Ue16λξ2N/(3U) − 8λξ2 − 3U), (59)

r =64√

3λ2ξ32ξ6e32λξ2N/(3U)√

(8λξ2 + 3U)

U(3Ue16λξ2N/(3U) − 8λξ2 − 3U)(8λξ2 + 3U)√ξ32ξ6(3Ue16λξ2N/(3U) − 8λξ2 − 3U)

(60)

The inflationary parameter ns does not depend on ξ6, but we cannot put ξ6 = 0, because φ(N) ≡ 0 in this case.In Fig. 3, we show the variation of r with ns for a particular choice of ξ6 = −0.1. For lower value of ξ6 the value of rgoes lower.

D. The case ξ6 = 0 and β 6= 0

In the case ξ6 = 0 the inflationary parameters (56) are

ns = 1 +8λ

3Uξ2 +

16λβ

Uφ2, r =

64λ2(ξ2φ2 + 2β)2

9U3φ2. (61)

12

We fix φend ≡ φ(0) = − (16βλ)/(8ξ2λ+ 3U) by the condition ε1 = 1, solve Eq. (24) and get:

φ2 = B

(eA1N

A1 + 1− 1

), (62)

ns = 1 +A1 +3BA1

φ2= 1 +A1 +

3A1(A1 + 1)

eA1N −A1 − 1

r =A2

1(φ2 +B)2

Uφ2=

A21Be2A1N

U(A1 + 1) (eA1N −A1 − 1)

As =V

6π2U2r=

λB

6Uπ2A21

(eA1N −A1 − 1

)3(A1 + 1)e2A1N

where A1 = 8λξ2/(3U) and B = 2β/ξ2.To get appropriate values for inflationary parameters we suppose N = −65 is a start point of inflation, λ = 0.1,

U = 1/2, A1 = −0.01517, and B = 2 · 10−10:

ns ≈ 0.9584, r ≈ 3.96 · 10−13, As ≈ 2.02 · 10−9. (63)

FIG. 4. Inflationary parameters for the model with V = λφ4 and ξ = ξ2φ−2 + ξ4φ

−4 in the case β 6= 0. In the left picture, blueplane corresponds to N = −55, red plane corresponds to N = −60, and green plane correspond to N = −65 [from the bottomupwards]. The z-axis corresponds to As. In the right picture, the contours correspond to the marginalized joint 68% and 95%CL. Blue solid, red dashed, and green dot curves correspond to N = −55, N = −60, and N = −65 respectively. The cyandash-dot lines are for the Planck’18 TT, TE, EE+ low E+ lensing data, whereas the black solid lines correspond to Planck’18TT, TE, EE+ low E+ lensing + BK15+ BAO data. The pivot scale is fixed at usual k = 0.002MPc−1.

Thus, if ξ2 = −0.0284 and β = −2.84 · 10−12, then we get appropriate inflationary parameters at N = −65.

E. The case ξ2 = 0

The behaviour of the inflationary parameters for this case is given in the Fig. 4. Thus, from Fig. 4, it is obviousthe observationally best suited case corresponds to N = −65 presented by the green line (plane).

13

-1.×10-22 -8.×10-23 -6.×10-23 -4.×10-23 -2.×10-23

1.×10-9

2.×10-9

3.×10-9

4.×10-9

5.×10-9

ξ6

As

-1.×10-22 -8.×10-23 -6.×10-23 -4.×10-23 -2.×10-23

5.×10-15

1.×10-14

1.5×10-14

2.×10-14

ξ6

r

FIG. 5. Inflationary parameters As (left) and r (right) for the model with V = V0φ4 and ξ = ξ4φ

−4 + ξ4φ−6. Blue solid, red

dashed and green dot curves correspond to N = −55, N = −60, and N = −65 respectively.

In the case of ξ6 > 0, there exists a stable de Sitter solution that can make such models unsuitable for the descriptionof the early Universe evolution. In the case of ξ6 < 0, one gets the following inflationary parameters:

r =64λ2

(2β φ2 + 3 ξ6

)29φ6U3

, (64)

ns = 1 +16β λ

Uφ2+

40λ ξ6Uφ4

, (65)

As =3Uφ10

128λπ2 (2β φ2 + 3 ξ6)2 . (66)

To get the following relation between the e-folding number N and the field φ:

N ' −φend∫φ

3Uφ3

4λ(2βφ2 + 3ξ6)dφ

=3Uφ2

16β λ−

9Uξ6 ln(2β φ2 + 3 ξ6

)32β2λ

+4β λ−

√−2λ (−8λβ2 + 9Uξ6)

8βλ(67)

+9Uξ632β2λ

ln

3 ξ6 +4(−4β λ+

√−2λ (−8λβ2 + 9Uξ6)

)β

3U

,

we use Eq. (24) and define the end point of inflation using the condition ε1(φ = φend) = 1:

φ2end =2

3U

(−4β λ+

√16λ2β2 − 18Uλξ6

).

Note that this result is correct for β 6= 0 only. The case β = 0 is considered separately.The inverse [in context of (67)] relation between field and e-folding number is rather complicated and requires

numerical considerations. However in particular case ξ4 = 3U2

λ (β = 0), all expressions can be simplified including therelations between field and e-folding number and vice versa:

N =φ4U

16λ ξ6+

1

2, φ4 =

8λ ξ6U

(2N − 1) . (68)

14

Therefore, in the case β = 0, we get the following inflationary parameters

ns = 1 +5

2N − 1, r =

2√

2λξ6√U3ξ6(2N − 1)3

, (69)

and

As =

√2 ((2N − 1)λ ξ6)

5/2

3π2U3/2λξ62 . (70)

In this case, the scalar spectral index is independent of ξ6. In the range, −65 ≤ N ≤ −55, the range of ns is0.961832 ≥ ns ≥ 0.954955. As usual in this case, we put U = 1/2 and λ = 0.1. The variation of As and r with respectto ξ6 is plotted in Fig. 5.

VI. CONCLUSIONS

In this paper, we have studied inflation in models with the GB coupling. The presence of the GB coupling cruciallymodifies the underlying dynamics of the system [20]. As demonstrated in the earlier work [45], the existence andstability of de Sitter solution in such models can be analysed in terms of the effective potential solely. As for theinflationary parameters, it appears that the effective potential is not enough to characterize them fully. Nevertheless,such a notation can be helpful to understand how these parameters vary in different models since the effective potentialenters in the most formulas derived to determine them. We have considered several examples of the power-law scalarfield potential V (φ) and the coupling function ξ(φ). In the case of V = V0φ

n and ξ = ξ0φ−n, the effective potential

can be made arbitrary small by choosing appropriate coupling coefficients. We show that under this condition, boththe amplitude of scalar perturbations and the tensor-to-scalar ratio can be turned to arbitrary small numbers if n > 2.Note, that this does not require smallness of V0. This means that the amplitude can be turned small required byobservations even for the Higgs field, since the Higgs potential can be approximated as a monomial one with n = 4and V0 = 0.1 for large φ. Note that in this case, we have not much freedom in choosing the coupling function — sincethe effective potential is a difference of two functions, one of which is proportional to the coupling function, and thesecond is inversely proportional to the potential, we should choose the coupling function to be almost equal to theinverse potential. An exact cancellation of these two terms in the effective potential does not give us a viable modelsince the scalar field dynamics is absent in this case. In the case of ”almost cancellation” the amplitude of scalarperturbations can be made small for the Higgs field. However, the spectral index ns, which does not depend upon ξ0in this case, turns out to narrowly missing the range of values admitted by observation for N < 66 e-folds.

If we abandon the assumption that the coupling function has the power index inverse to that of the potential, theeffective potential cannot be made small in general for large V and ξ. This means that we cannot use such modelsfor describing inflation with the potential too large to ensure the required small values of As similar to the case ofthe Higgs-driven inflation. However, if appropriate As is already ensured by the potential itself (for V = λφ4 thisrequires a very small λ), the coupling to GB term can improve the situation with the tensor-to-scalar ratio r. Inthis case, we have more freedom in the coupling function, though our choice is still restricted since we do not wantfor the slow-roll regime to end in a stable de Sitter solution. For a power-law potential proportional to φn and apositive coupling function proportional to φ−m we need n > m. An unstable de Sitter solution exists for such choice,and as V ′eff = 0 in any de Sitter point, the expression (20) indicates that the tensor-to-scalar ratio can be made

sufficiently small. We have considered a particular case of quartic potential (n = 4) with the m = 2 coupling function.The classic inflationary scenario for a minimally coupled scalar field with a quartic potential is now ruled out due toinappropriately large value of r, the problem can be addressed by invoking the GB coupling. The main remaininghurdle of the quartic potential with nonminimal coupling to the GB term is related to ns which can be made consistentwith observation only for coupling giving rise to inappropriately large r (see Fig. 1).

We show that the aforesaid difficulties can be circumvented by considering a more complicated form of the couplingfunction. We use

ξ = ξ2φ−2 + ξ4φ

−4 + ξ6φ−6 (71)

and demonstrate that the respective model fits the observational data [1] for all three considered observables: As,ns, and r. An appropriate choice of the coefficients allows us to obtain this fit for the Higgs field (see Figs. 2 & 3).The chosen form of the function ξ(φ) has a singularity at φ = 0 that may arise difficulties for the consideration theevolution of the Universe after inflation. This problem can be solved by the introduction of a small positive constant

15

M∗ and the consideration of the function

ξ∗ =ξ2

M2∗ + φ2

+ξ4

(M2∗ + φ2)

2 +ξ6

(M2∗ + φ2)

3 , (72)

where the value of the constant M∗ � φend. So, during inflation the function (71) is a good approximation for thefunction (72), whereas after inflation, when φ � M∗, this function is almost a constant and the behaviour of thiscosmological model is similar to the corresponding model without the GB term.

Another possible way out to improve upon φ4 model could be provided by invoking a usual nonminimal couplingwith the curvature R. The corresponding models without the GB term are studied in Refs. [9–11]; the frameworkis extended to the Einstein-Gauss-Bonnet gravity in Ref. [29]. It would be interesting to investigate the model withgeneric forms of coupling functions and we defer the same to our future study.

ACKNOWLEDGEMENTS

This work is partially supported by Indo-Russia Project: E.P., A.T., and S.V. are supported by the RFBR grant18-52-45016 and M.S. is supported by INT/RUS/ RFBR/P-315. A.T. is supported by Russian Government Programof Competitive Growth of Kazan Federal University. Work of M.R.G. is supported by the Department of Scienceand Technology, Government of India under the Grant Agreement number IF18-PH-228 (INSPIRE Faculty Award).M.R.G. wants to thank Nilanjana Kumar for useful discussions regarding numerical solutions.

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