10.1007@s10509-015-2608-9

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Astrophys Space Sci (2016) 361:19

DOI 10.1007/s10509-015-2608-9

O R I G I N A L A R T I C L E

The view of chaotic inflationary universe from f (R) gravity

M. Sharif 1·

Iqra Nawazish1

Received: 17 October 2015 / Accepted: 4 December 2015

© Springer Science+Business Media Dordrecht 2015

Abstract This paper is devoted to study the chaotic infla-

tionary scenario for isotropic and homogeneous flat universemodel in the framework of f(R) gravity. We investigate

different chaotic potential models such as quadratic, quar-

tic and fractional potentials. We explore quasi-de Sitter so-

lutions and formulate bounding values of inflaton field for

each model. We evaluate slow-roll parameters, number of

e-folds and other significant observational parameters like

spectral index, scalar power spectrum as well as tensor-

scalar ratio and discuss their consistency with Planck con-

straints. It is concluded that all considered models are com-

patible with Planck data whereas the value of tensor-scalar

ratio is larger than Planck’s result but for quadratic potential,

the measured value is very close to Planck constraint.

Keywords Inflation · Slow-roll approximation · f (R)

gravity

1 Introduction

One of the most revolutionary investigations of the last cen-

tury is to disclose origin of the universe which open a new

window of research. Observational data resolves this puz-

zle by putting forward a standard model named as big-bang.

This model also untangle story of the universe after its birth.

According to big-bang model, the universe was experiencing

decelerated expansion characterized by matter or radiation

B M. Sharif

[email protected]

I. Nawazish

[email protected]

1 Department of Mathematics, University of the Punjab,

Quaid-e-Azam Campus, Lahore 54590, Pakistan

dominated phase but this epoch has issues such as horizon,

monopole and flatness. These crucial problems compelled toassess an accelerating epoch in the early universe, referred

as inflation. The inflationary scenario is currently the most

leading paradigm to illustrate initial conditions of isotropic

and homogeneous universe.

Guth (1981) and Sato (1981) were pioneers who pro-

voked the idea of accelerated epoch after big-bang but it

faced some shortcomings which led to propose another ver-

sion of inflation, known as new inflation (Linde 1982). This

new inflation corresponds to chaotic inflation in which ini-

tially, the inflaton attains a large value but slowly rolls down

over the potential hill and gets closer to the origin of poten-tial. The chaotic inflationary model has many attractive fea-

tures as it describes an inflationary epoch when large quan-

tum fluctuations are present at the Planck time (Linde 1983).

Myrzakulov and Sebastiani (2012) studied cosmological in-

flation for inhomogeneous viscous fluids and discussed the

possibility to regain reheating phase. Sharif and Saleem

(2015a, 2015b) investigated warm inflation for intermediate

phase in general relativity (GR) and found cosmological pa-

rameters consistent with recent Planck and WMAP7 data.

They also explored dynamics of warm inflation via non-

Abelian gauge fields and obtained compatible observational

parameters for WMAP7 constraints.Recent observational evidences indicate that the universe

is again in an accelerating phase whose source is named

as “dark energy”. There are two remarkable approaches to

explain the puzzling nature of this energy either modify-

ing matter or gravity part of the Einstein-Hilbert action. The

most promising factor about this energy is its unknown na-

ture which gives rise to discuss the present inflating universe

in the context of modified theories of gravity via inflationary

models. The f (R) theory is the simplest modification of GR

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19 Page 2 of 8 M. Sharif, I. Nawazish

where the Ricci scalar R is replaced by an arbitrary function

f(R) (Sharif and Zubair 2012).

Bamba et al. (2014) investigated slow-roll and observa-

tional parameters of inflationary models through reconstruc-

tion methods in f (R) gravity. They analyzed different f (R)

models and concluded that power-law model gives the best

fit values compatible with BICEP2 and Planck observations.

Artymowski and Lalak (2014) obtained non-zero vacuum

energy of a scalar field by extending Starobinsky inflation-

ary model to R + αRn + β R2n and found compatible re-

sults for BICEP2 as well as Planck observations. Huang

(2014) analyzed polynomial inflationary f(R) model and

compatible values of tensor-scalar ratio as well as spectral

index for Planck constraints. Myrzakulov et al. (2015) stud-

ied fluid cosmology, scalar field theories and f(R) grav-

ity to reconstruct feasible inflationary models. Bamba and

Odintsov (2015) discussed inflationary scenario for modifi-

cation of gravity, R2

term in loop quantum cosmology aswell as quantum anomaly and concluded that tensor-scalar

ratio and spectral index of density perturbations are viable

for Planck data.

The chaotic inflation model attracted many researchers

to discuss number of fascinating phenomena of inflationary

cosmology such as superheavy particle production, preheat-

ing and primordial gravitational waves (Kofman et al. 1994;

Chung et al. 1998; Chung et al. 1999). Gao et al. (2014) ex-

plored fractional chaotic inflationary model in the context

of supergravity and discussed observational quantities for

various fractional exponents. Myrzakul et al. (2015) investi-

gated chaotic inflation for flat FRW metric and studied mas-

sive scalar field as well as massless self-interacting scalar

field in the background of higher derivative gravity theories.

They found that inflation is viable for massive scalar field

but it appears to be unrealistic for quartic potential. The

chaotic inflation has also been studied on brane to discuss

natural chaotic inflation along with supergravity (Maartens

et al. 2000; Kawasaki et al. 2000).

In this paper, we explore chaotic inflationary scenario for

power-law model of f (R) gravity and discuss its nature for

three different potential models. The structure of this pa-

per is organized as follows. Section 2 gives basic formalism

to construct Lagrangian and also briefly discuss inflation-

ary dynamics as well as recent observational data of Planck.

In Sect. 3, we formulate quasi-de Sitter solutions, slow-roll

parameters as well as observational quantities such as spec-

tral index, tensor-scalar ratio and scalar power spectrum for

quadratic, quartic and fractional potential models. Finally,

we conclude our results in the last section.

2 Basic formalism and inflationary dynamics

The action of f(R) gravity (Nojiri and Odintsov 2011) is

given by

A=

d 4x√ −g

f(R)

2κ2 +Lm

, (1)

where f is an arbitrary function of R and Lm is the mat-

ter Lagrangian which depends on a scalar field φ subject to

the effective scalar potential V (φ). Consequently, the above

action takes the form

A=

d 4x√ −g

f(R)

2κ2 − 1

2gµν ∂µφ∂ν φ − V(φ)

, (2)

where κ2 = 8πGN = 8π

M 2P l

, M 2P l = 1.2 × 1019 GeV is the

Planck mass. We consider flat FRW metric

ds 2 = −N 2(t)dt 2 + a2(t )dx2 + dy 2 + dz2

. (3)

The scale factor (a) and lapse function (N ) are functions of

cosmic time t . In order to determine Lagrangian of action

(2), we use Lagrangian multiplier approach (Capozziello

2002; Cognola et al. 2008) and introduce the Lagrangian

multiplier χ such as

A=

N a3

f

2κ2 − χ (R − R) − 1

2gµν∂µφ∂ν φ − V(φ)

dt,

(4)

where dynamical constraint (

¯R) and χ take the following

form

R = 6

N 2

aa

− a N

aN + a2

a2

, χ = f R

2κ2, (5)

f R = df dR

and dot represents time derivative. Integrating this

action by parts, we obtain the first order Lagrangian as fol-

lows

L(a, a , N, R, R,φ, φ)

=

1

2κ2Na3(f

−Rf R)

−

6a a2f R

N −

6a2a Rf RR

N

+ a3 φ2

2N − a3NV(φ). (6)

For a dynamical system, the Euler-Lagrange equation is

∂L

∂q i − d

dt

∂L

∂ q i

= 0,

where qi represent generalized coordinates. The variation

of Euler-Lagrange equation with respect to N(t) and a(t)

7/23/2019 10.1007@s10509-015-2608-9


The view of chaotic inflationary universe from f (R) gravity Page 3 of 8 19

yields

f − Rf R

2+ 3H 2f R + 3H Rf RR = κ2

φ2

2+ V(φ)

, (7)

f − Rf R

2+

2 H + 3H 2

f R + ( R + 2H R)f RR

+ R2f RRR = −κ2 ˙

φ2

2 − V(φ)

, (8)

where

H = a

a, R = 12H 2 + 6 H ,

R = 24H H + 6 H , N(t ) = 1.

(9)

The sum of kinetic energy, φ2

2 and potential energy, V(φ)

define effective energy density ρφ while their difference give

effective pressure pφ . Thus, the equation of state parameter

(EoS) becomes

ωφ =pφ

ρφ

=φ2

2 − V(φ)

φ2

2 + V(φ). (10)

The energy conservation law, ρφ + 3H (ρφ + pφ ) = 0, im-

plies that

φ + 3H φ + V (φ) = 0. (11)

This is known as Klein-Gordon (KG) equation, also referred

as scalar wave equation.

Inflation is described by a rapid exponential expansion of

the universe after the big-bang. It occurs when the scale fac-tor is accelerating or comoving Hubble length is decreasing

with time such as a (t ) > 0 or d dt

(aH)−1 < 0. In inflationary

era, strong energy condition is violated leading to

φ2 V (φ), φ 3H φ. (12)

Chaotic inflation is used to discuss the early inflating uni-

verse in which chaotic conditions originate some fluctuation

patches. This inflation occurs when inflaton field, φ M P l

as well as it is assumed to be negatively very large at the

beginning. Chaotic inflation ends if φ ∼ M P l and inflaton

moves towards the origin of potential and starts oscillating.

Due to this behavior of chaotic inflation, the models cor-

responding to this inflation are also referred as large field

models.

Inflation can also be described by quasi-de Sitter solution

when H H dS (constant), H dS represents the quasi-de Sit-

ter solution. In order to determine quasi-de Sitter solution of

chaotic inflation for action (2), we choose power-law f (R)

model (Allemandi et al. 2004) defined as

f(R) = f 0Rn, (13)

where f 0 and n are positive constants but n = 0, 1. For this

model, the field equations (7) and (8) reduce to

(1 − n)

2Rn + 3nRn−1H 2 + 3n(n − 1)H Rn−2R)

= κ2

f 0

φ2

2+ V(φ)

, (14)

(1 − n)

2Rn +

3H 2 + 2 H

nRn−1

+ n(n − 1)(2H R + R)Rn−2

+ n(n − 1)(n − 2)Rn−3R2 = −κ2

f 0

φ2

2− V(φ)

. (15)

The EoS parameter for (12) becomes ωφ −1 and hence

the quasi-de Sitter solution can be analyzed. The slow-roll

approximation is carried out when inflaton and matter or ra-

diation interactions are useless as well as kinetic energy is

much smaller than the potential energy (Kolb and Turner

1994). The slow-roll approximation technique is used to an-

alyze inflation via slow-roll parameters (, η) which take

the form

= − H

H 2, η = − H

H 2 − H

2H H ≡ 2 −

2H , (16)

where H is negative. These parameters can also be ex-

pressed (Lyth and Liddle 2009; Bamba et al. 2014) as

= 1

2κ2

V (φ)

V(φ)

2

, η = 1

κ2

V (φ)

V(φ)

= − φ

H φ .

During inflation, is very small but positive while inflating

universe vanishes when takes the value of unity. Similarly,

η should also be smaller than unity and this is only possible

when the following approximations are valid.

H

H 2

1,

H

2H H

1. (17)

Under the slow-roll approximation, Eq. (12) yields

3H φ −V (φ), 3H φ −V (φ)φ. (18)

In order to solve the issues of big-bang, there must exist

an accelerated epoch whose extent is described by number

of e-folds given as

N = ln

af

ai

= t f

t i

H(t)dt 3

φi

φf

H 2

V (φ)dφ, (19)

where af , ai , t f , t i , φf and φi are scale factors, time and

inflaton field, respectively at the end as well as at the be-

ginning of inflation. The horizon and flatness problems are

7/23/2019 10.1007@s10509-015-2608-9



resolved when the number of e-folds is about 60 (Liddle and

Lyth 2000). The amplitude of tensor and scalar perturbation

(2T

), (2R

), spectral index (ns ) and tensor-scalar ratio (r)

are defined (Linde 1990; Saski and Stewart 1996) as

2R =

H

φδφ

2

, δφ = H

2π,

2T = 8κ2

H

2π

2

, r = 2T

2R

, (20)

ns = 1 + d

dk

ln 2

R

= 1 − d

d N

ln 2

R

,

where δφ represents quantum fluctuations. These basic

definitions can also be rewritten as

2R = κ2H 2

8π 2, r = 16. (21)

According to the recent observations from Planck data, these

quantities yield viable results for ns

= 0.9666

±0.0062, r <

0.10 (95%CL) and 2R

10−9 (Ade et al. 2014).

3 Quasi-de Sitter solutions

In this section, we study quasi-de Sitter solutions, slow-roll

parameters as well as observational quantities such as spec-

tral index, tensor-scalar ratio and scalar power spectrum for

quadratic, quartic and fractional potentials.

3.1 Quadratic potential

Here, we discuss chaotic inflationary scenario for massivefree field defined by quadratic potential such as

V(φ) = m2φ2

2, (22)

where m > 0 and φ is assumed to be negatively large. Dur-

ing inflation, m φ whereas φ M P l which is taken to

ignore the quantum effects. In GR, the quasi-de Sitter solu-

tion yields

H 2dS =κ2V(φ)

3, H κ2V (φ)φ

6H dS

.

For quadratic potential, the above equation and KG equation

using slow-roll approximation give

H dS = −2

π

3

mφi

M P l

,

φ = φi +mM P l√

12π(t − t i ),

M P l√ 12π

< |φi | <M 2P l

2m

3

π.

(23)

Initially, the inflaton field is greater than the Planck mass for

H dS < M P l and kinetic energy remains small. The slow-roll

parameters and number of e-folds are

= η M 2P l

4π φ2i

, N 2π φ2i

M 2P l

. (24)

In order to determine the quasi-de Sitter solution for thiscase, we take Eq. (14) with H H dS and RdS 12H 2dS

which leads to

(1 − n)

2Rn

dS + 3nRn−1dS H 2dS =

κ2

f 0V (φ). (25)

Thus, the quasi-de Sitter solution takes the form

H dS =

16π

12nf 0(2 − n)

12n

mφi

M P l

1n

. (26)

The inflaton field and its bound is given as

φ = φi −m2

3

12nf 0(2 − n)

16π

12n

M P l

mφi

1n

(t − t i )φi , (27)

m2

9

n2

12nf 0(2 − n)

16π

M P l

m< |φi |

< M nP l

12nf 0(2 − n)

16π

M P l

m. (28)

To formulate the slow-roll parameters and e-folds, we

take first and second order time derivatives of Eq. (7) via

slow-roll approximation which yields

6H H f R − 72H 3H f RR = κ2V (φ)φ, (29)

6H H f R − 72H 3H f RR = κ2

2V (φ)φ − V (φ)H φ

.

(30)

For the considered power-law model, the above equations

give slow-roll parameters and e-folds as

= η m2

3n

12nf 0(2 − n)

16π

1n

M P l

mφi

2n

, (31)

N 3n

2m2 16π

12nf 0(2 − n) 1

n mφi

M P l

2n

. (32)

The spectral index, the tensor-scalar ratio and amplitude of

scalar power spectrum are given as follows

ns = 1 − 4m2

3n

12nf 0(2 − n)

16π

1n

M P l

mφi

2n

, (33)

r = 16m2

3n

12nf 0(2 − n)

16π

1n

M P l

mφi

2n

, (34)

7/23/2019 10.1007@s10509-015-2608-9



Table 1 r for different values of N

N Tensor-scalar ratio

55 0.145

60 0.133

65 0.123

70 0.114

72 0.111

2R = 3n

m2M 2P lπ

16π

12nf 0(2 − n)

2n

mφi

M P l

4n

. (35)

From Eqs. (32) and (33), the number of e-folds are 50 <

N < 73 and tensor-scalar ratio is given in Table 1.

It is found that tensor-scalar ratio is slightly larger than

the Planck’s value, r < 0.10 as N approaches to 73. The

boundary values of the field can also be expressed as

28m2

n

n2

12nf 0(2 − n)

16π

M P l

m< |φi |

<

41m2

n

n2

M nP l

12nf 0(2 − n)

16π

M P l

m. (36)

Equations (28) and (36) imply that

m <

n

41M P l . (37)

For a viable range of scalar power spectrum, we must have

m 10−6M P l . If boundary value of the field (φi ) is larger

than the Planck mass then number of e-folds are also very

large during inflation which imply that slow-roll parameters

should be small enough to make inflation realistic. In the

case of quadratic potential for power-law f(R) model, we

have = η = 12 N

which describes that these parameters are

very small as required and the results obtained in GR are

recovered for n = f 0 = 1.

3.2 Quartic potential

In this section, we study chaotic inflation via quartic poten-

tial which describes self-interaction of massless scalar field

as

V(φ) = λφ4

4, (38)

where λ is a positive coupling constant and φ is again neg-

ative as well as greater than the Planck mass. In GR, the

quasi-de Sitter solution and the bounding value of self-

interacting field are given as

H dS =

2λπ

3

φ2i

M P l

,

M P l√ 3π

< |φi | <

3

2λπ

14

M P l

3

π,

(39)

where M P l

λ14

1. The slow-roll parameters and e-folds take

the form

M 2P l

π φ2i

, η 3M 2P l

2π φ2i

, N πφ2i

M 2P l

. (40)

Since the field is negatively large, so number of e-folds

should be large enough to make slow-roll parameters very

small. We formulate the quasi-de Sitter solution and inflaton

field as well as its boundary value as follows

H dS =

8λπ

12nf 0(2 − n)

12n

φ2i

M P l

1n

, (41)

2λ

9

n2(2−n)

12nf 0(2 − n)

16π

12(2−n)

M 1

2−n

P l

< |φi | < M n+1

2P l

12nf 0(2 − n)

16π

14

. (42)

The viable inflation is obtained for valid slow-roll approxi-

mation which leads to slow-roll parameters and number of

e-folds as

2λφ2i

3n

12nf 0(2 − n)

8λπ

1n

M P l

φ2i

2n

, (43)

η λφ2i

n

12nf 0(2 − n)

8λπ

1n

M P l

φ2i

2n

, (44)

N 3n

2(2 − n)λφ2i

8λπ

12nf 0(2 − n)

1n

φ2i

M P l

2n

. (45)

The expressions for spectral index, tensor-scalar ratio and

amplitude of scalar power spectrum can be found from dif-

ferent relations of slow-roll parameters as follows

ns = 1 − 2λφ2i

n

12nf 0(2 − n)

8λπ

1n

M P l

φ2i

2n

, (46)

r = 32λφ2i

3n

12nf 0(2 − n)

8λπ

1n

M P l

φ2i

2n

, (47)

2R = 3n

2λπφ2i M 2P l

8λπ

12nf 0(2 − n)

2n

φ2i

M P l

4n

. (48)

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Table 2 N and r for different values of n

n Nu mb er of e-folds Tensor-scalar ratio

0.3 44 < N < 64 0.147

0.5 50 < N < 73 0.146

0.7 58 < N < 84 0.146

0.9 68 < N < 100 0.145

1.3 108 < N < 157 0.145

An alternate expression for self-interacting inflaton field is

68λ

n

n2(2−n)

12nf 0(2 − n)

8λπ

12(2−n)

M 1

2−n

P l < |φi |

<

43λ

n

n2(2−n)

12nf 0(2 − n)

8λπ

12(2−n)

M 1

2−n

P l . (49)

Comparing the above expression with Eq. (42), we obtain a

condition for coupling constant as

λn

4(2−n) <

n

43

n2(2−n)

8π

12nf 0(2 − n)

n4(2−n)

M n(1−n)2(2−n)

P l . (50)

The e-folds is found by using Eq. (45) in (46) as follows

N 3

(1 − ns )(2 − n), 0 < n < 2. (51)

For this interval of n, the number of e-folds N and tensor-

scalar ratio r are given in Table 2.

Table 2 indicates that the points from n = 0.3 to 0.9

provide a viable range of number of e-folds for ns

=0.0396, 0.0272 whereas for n ∈ (0, 2), the tensor-scalar ra-

tio is r ∼ 0.14 which is greater than the Planck constraint,

i.e., r < 0.10. In order to have a consistent range of ampli-

tude of scalar power spectrum, the coupling constant must

be λ−2 10−3. During inflation, the large number of e-

folds provides very small values of slow-roll parameters as

1(2−n) N

and η 32(2−n) N

. Thus for the quartic poten-

tial, the large number of e-folds, smaller values of slow-roll

parameters and viable range of amplitude of scalar power

spectrum yield realistic inflationary scenario whereas the

tensor-scalar ratio is inconsistent with Planck observational

value (Sharif and Saleem 2014). All the results obtained

for power-law f(R) model can be recovered by inserting

n = f 0 = 1.

3.3 Fractional potential

Here, we discuss chaotic inflationary scenario through an

example of massless large field defined by fractional poten-

tial such as

V(φ) = V 0φαβ , (52)

where α and β are positive integers while V 0 describes the

height relative to vacuum energy during inflation. In this

case, we take coprime fractional power and evaluate expres-

sions for quasi-de Sitter solution as well as boundary values

of massless scalar field as follows

H dS = 32V 0π

12nf 0(2 − n) φ

αβ

i

M 2P l

1

2n

, (53)

V 0α2

18β2

nβ12nf 0(2 − n)

32V 0π

β

M 2βP l

1α(1−n)−2nβ

< |φi | <

M

2(1−n)P l ×

12nf 0(2 − n)

32V 0π

βα

. (54)

The sufficient condition required to have realistic inflation-

ary process is given by slow-roll parameters and number of

e-folds as

V 0α2φ

α

β −2

6nβ2

12nf 0(2 − n)

32V 0π

M P l

φ2i

1

n, (55)

η V 0α2φαβ −2

4nβ2

12nf 0(2 − n)

32V 0π

M P l

φ2i

1n

, (56)

N 3nβ2φ2−α

β

V 0α2{(1 − n) + 2nβα}

32V 0π

12nf 0(2 − n)

φ

αβ

i

M 2P l

1n

.

(57)

With the help of these slow-roll parameters, we formulate

observational quantities such as spectral index, tensor-scalar

ratio and amplitude of scalar power spectrum for fractional

potential model given by

ns = 1 − V 0α2φαβ −2

2nβ2

12nf 0(2 − n)

32V 0π

M P l

φ2i

1n

, (58)

r = 16V 0α2φαβ −2

6nβ2

12nf 0(2 − n)

32V 0π

M P l

φ2i

1n

, (59)

2R = 6nβ2φ

2− αβ

V 0π α2M 2P l

32V 0π

12nf 0(2 − n)

M P l

φ2i

2n

. (60)

The bound of spectral index leads to an alternate expression

of considered inflaton field

31V 0α2

21nβ2

nβ12nf 0(2 − n)

32V 0π

β

M 2βP l

1α(1−n)+2nβ

< |φi |

<

21V 0α2

2nβ2

nβ12nf 0(2 − n)

32V 0π

β

M 2βP l

1α(1−n)+2nβ

,

which gives the following condition

7/23/2019 10.1007@s10509-015-2608-9



Table 3 N and r for different values of n and αβ = 10

3


0.3 43 < N < 62 0.146

0.5 47 < N < 69 0.144

0.7 52 < N < 76 0.146

0.9 59 < N < 86 0.145

1.3 79 < N < 114 0.146

1.5 94 < N < 137 0.145

Table 4 N and r for different values of n and αβ = 5

3


0.3 35 < N < 52 0.145

0.5 34 < N < 50 0.145

0.7 33 < N < 48 0.146

0.9 32 < N < 46 0.147

1.3 30 < N < 43 0.1471.5 29 < N < 42 0.146

V {(1−α)(1−n)−2nβ2}

0

<

2nβ2

21α2

nα12nf 0(2 − n)

32V 0π

{α(2−n)+2nβ}

× M 2{α+(1−n)(α(1−n)+2nβ)}P l . (61)

In order to analyze feasible range of inflationary scenario,

we express the number of e-folds and tensor-scalar ratio as

N 3

2(1 − ns ){(1 − n) + 2nβα}

,

r 8

N {(1 − n) + 2nβα}

,

(62)

where n > 0 and βα

is taken as inverse of coprime fractional

power of massless scalar field. Here, number of e-folds are

expressed in terms of spectral index while tensor-scalar ratio

depends on number of e-folds. We analyze these observa-

tional parameters for the same data points as in the previous

case.

Table 3 is evaluated for another value of αβ

to observe the

effect of these relative prime numbers.

In Table 3, we obtain a suitable range of number of e-

folds to sort out the horizon problem for n = 0.3 to 0.9 withαβ = 10

3 whereas tensor-scalar ratio r ∼ 0.14 is inconsistent

with Planck observational value. Table 4 represents that the

number of e-folds are getting smaller by increasing the val-

ues of fraction term αβ

as well as n while tensor-scalar ra-

tio is found to be larger than the Planck results. To discuss

the realistic nature of fractional chaotic inflation, we take

spectral index which gives alternate expressions of slow-roll

parameters in terms of number of e-folds as

1

2 N {(1 − n) + 2nβα}

, η 3

4 N {(1 − n) + 2nβα}

.

This indicates that these slow-roll parameters become very

small for large number of e-folds and for a huge growth of

massless scalar field. Thus, we have viable inflationary pro-

cess in the case of fractional potential model for power-law

f(R) model whereas tensor-scalar ratio does not ensure the

Planck results.

4 Concluding remarks

In this paper, we have studied inflationary scenario when

the inflaton field initiates from a large field value and then

rolls down towards origin where the field value is about to

vanish. Such inflationary process is known as chaotic infla-tion in which inflaton field is greater than M P l and ends

when inflaton field is nearly close to M P l . We have investi-

gated chaotic inflation in the framework of power-law model

of f(R) gravity. We have discussed inflation by quasi-de

Sitter expansion when Hubble parameter is smaller than

M P l and formulated quasi-de Sitter solutions for quadratic,

quartic and fractional potential models which are also re-

ferred as large field models. In quadratic potential, we have

studied massive non-interacting scalar field, in quartic po-

tential model we have analyzed massless self-interacting

scalar field and fractional potential model describes mass-

less scalar field with fractional exponents. For each potentialmodel, we have also found boundary values of inflaton field.

We have explored these models to investigate realistic

nature of chaotic inflation. For this purpose, we have for-

mulated slow-roll parameters, number of e-folds and other

observational quantities, i.e., spectral index, amplitude of

scalar power spectrum and tensor-scalar ratio. The results

are summarized as follows.

• For quadratic potential model, the number of e-folds are

found to be 50 < N < 73. Also, a significant relation-

ship between number of e-folds and slow-roll parameters

is = η = 1

2 N . The scalar power spectrum satisfies ob-servational value for m 10−6M P l and measured value

of tensor-scalar ratio r = 0.11 is very close to Planck con-

straint.

• In case of quartic potential, the feasible range of num-

ber of e-folds is obtained from n = 0.3 to 0.9 whereas

r ∼ 0.14 for all values of n. The slow-roll parameters

and number of e-folds are related as 1(2−n) N

and

η 32(2−n) N

. The coupling constant is constrained as

λ 10−3 which yields 2R

10−9.

7/23/2019 10.1007@s10509-015-2608-9



• We have calculated number of e-folds for αβ = 10

3 , 53

and

n ∈ (0.3, 1.5) in fractional potential model. For αβ = 10

3 ,

we have obtained compatible range of number of e-folds

for n = 0.3 to 0.9 while r ∼ 0.14. The value αβ = 5

3 yields

incompatible range of number of e-folds for all values of

n, r ∼ 0.14 and 2R

10−9 for Eq. (61).

It is concluded from the above results that we have small

values of slow-roll parameters for large number of e-folds

and inflaton field which make the inflationary process re-

alistic. Also, the number of e-folds and all other observa-

tional parameters are compatible with Planck constraints

whereas tensor-scalar ratio exceeds for Planck observation.

It is worth mentioning here that all our results can be recov-

ered for n = f 0 = 1.

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