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Inflation from R2 gravity: A new approach using nonlinear electrodynamics

Christian Corda a,b,⇑,1, Herman J. Mosquera Cuesta c,d,e,2

a International Institute for Theoretical Physics and Mathematics Einstein-Galilei, via Bruno Buozzi 47, 59100 Prato, Italyb Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682, USAc Instituto de Cosmologia, Relatividade e Astrofìsica (ICRA-BR), Centro Brasilero de Pesquisas Fisicas, Rua Dr. Xavier Sigaud 150, CEP 22290 - 180 Urca Rio de Janeiro - RJ Brazild ICRANet, International Coordinating Center, Piazza della Repubblica,10, 65122 Pescara, Italye Departamento de Fìsica, Universidade Vale do Acaraù, Av. da Universidade 850, Campus da Bet�ania, CEP 62.040-370, Sobral, Ceara, Brazil

a r t i c l e i n f o

Article history:Received 26 March 2009Received in revised form 22 November 2010Accepted 2 December 2010Available online 8 December 2010

Keywords:InflationNonlinear lagrangian

a b s t r a c t

We discuss another approach regarding the inflation from the R2 theory of gravity originally proposed byStarobinski. A non-singular early cosmology is proposed, where, adding a nonlinear electrodynamicsLagrangian to the high-order action, a bouncing is present and a power-law inflation is obtained. Inthe model the Ricci scalar R works like an inflaton field.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The accelerated expansion of the Universe that is currently pur-ported from observations of SNe Ia suggests that cosmologicaldynamics is dominated by a ‘‘new’’ substance of the universe con-stituents dubbed as Dark Energy, which is able to provide a largenegative pressure to account for the late-time accelerate expansion.This is the standard picture, in which such a new ingredient is con-sidered as a source of the right-hand-side of the field equations. It isposed that it should be some form of un-clustered non-zero vacuumenergy which, together with the clustered Dark Matter, drives theglobal dynamics. This is the so-called ‘‘concordance model’’ (KCDM)which gives, in agreement with the data analysis of the observationsof the Cosmic Microwave Background Radiation (CMBR), LymanLimit Systems (LLS) and type la supernovae (SNe Ia), a good frame-work for understanding the currently observed Universe. However,the KCDM presents several shortcomings as the well known ‘‘coin-cidence’’ and ‘‘cosmological constant’’ problems [1].

An alternative approach to explain the purported late-timeacceleration of the universe is to change the left hand side of thefield equations, and to inquire whether the observed cosmic

dynamics can be achieved by extending general relativity [2–4].In this different context, it is not required to search candidatesfor Dark Energy and Dark Matter, which until to date, have notbeen found, but rather it claims that only the ‘‘observed’’ ingredi-ents: curvature and baryon matter, have to be taken into account.Considering this point of view, one can posit that gravity is notscale-invariant [5]. In so doing, one allows for a room for alterna-tive theories to be opened [6–8]. In principle, interesting Dark En-ergy and Dark Matter models can be built by considering f(R)theories of gravity [5,9] (here R is the Ricci curvature scalar).

In this perspective, even the sensitive detectors of gravitationalwaves like bars and interferometers (i.e. those which are currentlyin operation and the ones which are in a phase of planning and pro-posal stages [10,11]), could, in principle, test the physical consis-tency of general relativity or of any other theory of gravitation.This is because in the context of Extended Theories of Gravityimportant differences with respect to general relativity show upafter studying the linearized theory [12–15].

In this paper, another approach regarding the inflation from theR2 theory of gravity, which is the simplest among f(R) theories andwas been originally proposed by Starobinski [16], is shown. A non-singular early cosmology is proposed, where, adding a nonlinearelectrodynamics Lagrangian to the high-order action, a bouncingis present and a power-law inflation is obtained. In the modelthe Ricci scalar R works like an inflaton field.

In the general picture of high order theories of gravity, recentlythe R2 theory has been analysed in various interesting frameworks,see [17,18] for example.

We recall that extensions of the traditional Maxwell electromag-netic Lagrangian, which take into account high order terms of the

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⇑ Corresponding author. Addresses: Associazione Scientifica Galileo Galilei, ViaPier Cironi 16-59100 PRATO, Italy. Institute for Basic Research, P. O. Box 1577, PalmHarbor, FL 34682, USA.

E-mail addresses: christian.corda@ego-gw.it (C. Corda), herman@icra.it (H.J.Mosquera Cuesta).

1 Partially supported by a Research Grant of The R.M. Santilli Foundations NumberRMS-TH-5735A2310.

2 Fellow of Fundação Cearense de Apoio ao Desenvolvimento Cientìfico eTecnològico (FUNCAP), Fortaleza, Ceara, Brazil.

Astroparticle Physics 34 (2011) 587–590

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electromagnetic scalar F, have been used in cosmological models[19], gravitational redshifts of neutron stars [20] and pulsars [21].Moreover, a particular nonlinear Lagrangian has been analysed inthe context of the Pioneer 10/11 spacecraft anomaly [22].

2. Action and lagrangian

Let us consider the high order action [16–18]

S ¼Z

d4xffiffiffiffiffiffiffi�gp

Rþ aR2 þ Lm

� �: ð1Þ

Such an Eq. (1) is a particular choice in respect to the wellknown canonical one of General Relativity (the Einstein–Hilbertaction [23]) which is

S ¼Z

d4xffiffiffiffiffiffiffi�gp

ðRþ LmÞ: ð2Þ

We are going to show that the action (1), applied to the Fried-mann–Lemaı̀tre–Robertson–Walker Cosmology, generates a non-singular inflationary phase of the Universe where the Ricci scalaracts like inflaton, and a bouncing is present, if Lm is the non linearelectrodynamics Lagrangian. Note that in this letter we work with8pG = 1, c = 1 and �h = 1.

Inflationary models of the early Universe were analysed in theearly and middles 1980’s (see [24] for a review), starting from anidea of Starobinski [16] and Guth [25]. These are cosmologicalmodels in which the Universe undergoes a brief phase of a very ra-pid expansion in early times. In this context the expansion could bepower-law or exponential in time. Inflationary models providesolutions to the horizon and flatness problems and contain a mech-anism which creates perturbations in all fields [24].

In Cosmology, the Universe is seen like a dynamic and thermo-dynamic system in which test masses (i.e. the ‘‘particles’’) are thegalaxies that are stellar systems with a number of the order of109 � 1011 stars [23]. Galaxies are located in clusters and superclusters, and observations show that, on cosmological scales, theirdistribution is uniform. This is also confirmed by the WMAP dataon the Cosmic Background Radiation [26,27]. These assumptioncan be summarized in the so called Cosmological Principle: theUniverse is homogeneous everywhere and isotropic around everypoint. Cosmological Principle simplifies the analysis of the largescale structure, because it implies that the proper distances be-tween any two galaxies is given by an universal scale factor whichis the same for any couple of galaxies [23].

In this framework, the cosmological line – element is the wellknown Friedmann–Lemaı̀tre–Robertson–Walker one, and for asake of simplicity we will consider the flat case, because the WMAPdata are in agreement with it [26,27]:

ds2 ¼ �dt2 þ a2ðdz2 þ dx2 þ dy2Þ: ð3Þ

Following [23] we also get

glm ¼

�1 0 0 00 þa2 0 00 0 þa2 00 0 0 þa2

; ð4Þ

ffiffiffiffiffiffiffi�gp ¼ a3; ð5Þ

and

R ¼ �61a

d _adtþ

_aa

� �2" #

: ð6Þ

One can use the Lagrange multipliers putting

S ¼ 2p2Z

dt a3ðRþ aR2Þ � b Rþ 6€aaþ 6

_aa

� �2" #

þ a3Lm

( ): ð7Þ

b can be obtained by varing the action in respect to R. It is

a3 @ðRþ aR2Þ@R

dR� bdR ¼ 0; ð8Þ

which gives

b ¼ a3 @ðRþ aR2Þ@R

¼ a3ð2aRþ 1Þ: ð9Þ

Thus, substituting in Eq. (7) one obtains

S¼2p2Z

dt �2a3aR2�6a2€að2aRþ1Þ�6að _aÞ2ð2aRþ1Þþa3Lm

n o:

ð10Þ

The term �6a2€að2aRþ 1Þ is critical as it contains a secondderivative of a. Let us integrate it. It is

�6Z

dta2€að2aRþ 1Þ ¼ �6a2 _að2aRþ 1Þ

þ 6Z

dt 2aa2 _a _Rþ 2að _aÞ2ð2aRþ 1Þh i

¼ 6Z

dt 2aa2 _a _Rþ 2að _aÞ2ð2aRþ 1Þh i

; ð11Þ

where we have taken into account that the term outside the integralis equal to zero as it is a pure divergence.

Substituting in Eq. (10), one gets

S ¼ 2p2Z

dt �a3aR2 þ 12aa2 _a _Rþ 6að _aÞ2ð2aRþ 1Þ þ a3Lm

n o:

ð12ÞThen, the Lagrangian is

L ¼ �a3aR2 þ 12aa2 _a _Rþ 6að _aÞ2ð2aRþ 1Þ þ a3Lm: ð13Þ

The energy function associated to the Lagrangian is [23]

EL ¼@L@ _a

_aþ @L@ _R

_R� L: ð14Þ

Combining Eq. (13) with Eq. (14), the conditionEL ¼ 0; ð15Þ

together with the definition of the Hubble constant, i.e. H ¼ 1a

dadt, and

with a little algebra gives

H2 ¼ Lm

3aR� H

_RR: ð16Þ

From the Euler–Lagrange equation for a and _a, i.e. [23]

@L@a¼ d

dt@L@ _a

� �; ð17Þ

one gets

€Rþ 3H _R ¼ 2Lm

3a: ð18Þ

An important question is where Eq. (15) comes from [29]. Ingeneral relativity, due to the reparametrization invariance of thetime coordinate, the total energy (including the contribution fromthe gravity sector) vanishes [29]. In the action (12), however, thereis not the reparametrization invariance because the total derivativeterms are dropped [29]. Then, one can think that the total energydoes not always vanish [29]. We clarify this point as it follows.Let us start by the original action (1) from which the action (12)arises. Let us consider the conformal transformation [30]

~gab ¼ e2Ugab; ð19Þ

where the conformal rescaling

e2U ¼ 2aRþ 1 ð20Þ

588 C. Corda, H.J. Mosquera Cuesta / Astroparticle Physics 34 (2011) 587–590

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has been chosen. By applying the conformal transformation (19) tothe action (1) the conformal equivalent Hilbert–Einstein action

A ¼ d4xffiffiffiffiffiffiffi�~g

p eR þ LðU;U;aÞ þ Lm

h ið21Þ

is obtained. LðU;U;aÞ is the conformal scalar field contribution de-rived from

eRab ¼ Rab þ 2 U;aU;b � gabU;dU;d � 1

2gabU

;d;d

� �; ð22Þ

and

eR ¼ e�2U þ R� 6�U� 6U;dU;d� �: ð23Þ

Clearly, the reparametrization invariance of the time coordinateis consistent with the new action (21) in the conformal Einsteinframe and the total energy (including the contribution from thegravity sector) vanishes in this case too. One could object thatthe energy in the conformal Einstein frame is different with respectto the energy in the original Jordan frame, but in Ref. [31] it hasbeen shown that the two conformal frames are energetically equiv-alent if, together with the conformal rescaling (19), times andlengths are rescaled as eU while the mass-energy is rescaled ase�U. This analysis permits to enable the condition of Eq. (15) inthe present discussion too.

3. Nonlinear electrodynamics lagrangian and Inflation

In order to show that our model admits a power law inflation-ary phase, we need to postulate some matter Lagrangian Lm whichcan perform the condition of inflation P <� q [24]. We will use thenon linear electrodynamics Lagrangian of [19], which is

Lm � �14

F þ c1F2 þ c2G2; ð24Þ

where F is the electromagnetic scalar, c1, c2 are two constants and,considering the electromagnetic field tensor Fab (see [23] the defini-tion of this object), G is defined like [19] G � 1

2 gablmFabFlm.The Lagrangian (24), differently from the one of the singular

Einstein–Maxwell Universe, performs a non-singular Universewith bouncing [19]. This is because the energy condition of singu-larity theorems [28] is not satisfied in the case of the non linearelectrodynamics Lagrangian (see [19] for details).

In fact, following [19], one uses the equation of state

p ¼ 13q� q�; ð25Þ

where

q� �163

c1B4 ð26Þ

(see Eqs. (15), (16) and (25) of [19]) and B is the magnetic field asso-ciated to F.

This equation of state is no longer given by the Maxwellian va-lue, thus, using Eq. (24), from Eqs. (16) and (18) one gets

B ¼ B0

2a2 ; ð27Þ

where B0 is a constant [19], and

_a2 ¼ B20

12aa2R1� 8c1B2

0

a4

!� 2Ha2

_RR; ð28Þ

which can be solved by suitably choosing the origin of time.

One gets

a2 ¼ B0ffiffiffiap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23

t2 þ 12c1� �r

: ð29Þ

This expression is not singular for c1 > 0. In this case we see thatat the instant t = 0 a minimum value of the scale factor is present:

a2min ¼

B0ffiffiffiap

ffiffiffiffiffiffiffiffi8c1

p: ð30Þ

This also implies that, for a value t ¼ffiffiffiffiffiffiffiffiffiffi12c1p

, the energy density�q reaches a maximum value qmax = 1/64c1. For smaller values of tthe energy density decreases, vanishing a t = 0, while the pressurebecomes negative [19].

In this way, the condition of inflation P � q⁄ < 0 [24] gives theinflationary solutions for Eqs. (16) and (18), if one assumes thatthe Ricci scalar R acts like inflaton:

RðtÞ ’ ð1þ Ht=bÞ2;ainf ðtÞ ’ ð1þHt=bÞwþ1=2

ð31Þ

with b ’w and

Hinf ’ffiffiffiffiffiffiffiL�m

q; ð32Þ

where L�m is the right hand side of Eq. (16) which is constant duringthe inflationary phase. The idea of considering the Ricci scalar as aneffective scalar field (scalaron) arises from Starobinski [16].

4. Conclusion remarks

Another approach regarding the inflation from the R2 theory ofgravity, which was originally proposed by Starobinsk, has beenanalysed. A non-singular early cosmology has been proposed,where, adding a nonlinear electrodynamics Lagrangian to thehigh-order action, a bouncing is present and a power-law inflationis obtained. In the model which has been discussed, the Ricci scalarR works like an inflaton field.

Acknowledgements

The authors thank Professor Mario Novello for useful discus-sions on the topics of this paper. We also thank an unknown ref-eree for precious advices and suggestions which permitted toimprove this paper.

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