This article discusses an alternative approach to inflation based on R2 gravity. A non-singular early cosmology is proposed where a bouncing occurs and power-law inflation is obtained. Adding a nonlinear electrodynamics Lagrangian to the high-order action generates this behavior, with the Ricci scalar R acting as the inflaton field. The model presents a Lagrangian and derives the associated energy function and field equations. It finds a solution where the Hubble parameter is related to the matter Lagrangian and time derivative of R.

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Astroparticle Physics 34 (2011) 587–590
Contents lists available at ScienceDirect
Astroparticle Physics
journal homepage: www.elsevier.com/locate/astropart
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, Italy
b
Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682, USA
c
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 Brazil
d
ICRANet, International Coordinating Center, Piazza della Repubblica,10, 65122 Pescara, Italy
e
Departamento de Fìsica, Universidade Vale do Acaraù, Av. da Universidade 850, Campus da Betania, CEP 62.040-370, Sobral, Ceara, Brazil
a r t i c l e i n f o a b s t r a c t
Article history: We discuss another approach regarding the inflation from the R2 theory of gravity originally proposed by
Received 26 March 2009 Starobinski. A non-singular early cosmology is proposed, where, adding a nonlinear electrodynamics
Received in revised form 22 November 2010 Lagrangian to the high-order action, a bouncing is present and a power-law inflation is obtained. In
Accepted 2 December 2010
the model the Ricci scalar R works like an inflaton field.
Available online 8 December 2010
Ó 2010 Elsevier B.V. All rights reserved.
Keywords:
Inflation
Nonlinear lagrangian
1. Introduction dynamics can be achieved by extending general relativity [2–4].
In this different context, it is not required to search candidates
The accelerated expansion of the Universe that is currently pur- for Dark Energy and Dark Matter, which until to date, have not
ported from observations of SNe Ia suggests that cosmological been found, but rather it claims that only the ‘‘observed’’ ingredi-
dynamics is dominated by a ‘‘new’’ substance of the universe con- ents: curvature and baryon matter, have to be taken into account.
stituents dubbed as Dark Energy, which is able to provide a large Considering this point of view, one can posit that gravity is not
negative pressure to account for the late-time accelerate expansion. scale-invariant [5]. In so doing, one allows for a room for alterna-
This is the standard picture, in which such a new ingredient is con- tive theories to be opened [6–8]. In principle, interesting Dark En-
sidered as a source of the right-hand-side of the field equations. It is ergy and Dark Matter models can be built by considering f(R)
posed that it should be some form of un-clustered non-zero vacuum theories of gravity [5,9] (here R is the Ricci curvature scalar).
energy which, together with the clustered Dark Matter, drives the In this perspective, even the sensitive detectors of gravitational
global dynamics. This is the so-called ‘‘concordance model’’ (KCDM) waves like bars and interferometers (i.e. those which are currently
which gives, in agreement with the data analysis of the observations in operation and the ones which are in a phase of planning and pro-
of the Cosmic Microwave Background Radiation (CMBR), Lyman posal stages [10,11]), could, in principle, test the physical consis-
Limit Systems (LLS) and type la supernovae (SNe Ia), a good frame- tency of general relativity or of any other theory of gravitation.
work for understanding the currently observed Universe. However, This is because in the context of Extended Theories of Gravity
the KCDM presents several shortcomings as the well known ‘‘coin- important differences with respect to general relativity show up
cidence’’ and ‘‘cosmological constant’’ problems [1]. after studying the linearized theory [12–15].
An alternative approach to explain the purported late-time In this paper, another approach regarding the inflation from the
acceleration of the universe is to change the left hand side of the R2 theory of gravity, which is the simplest among f(R) theories and
field equations, and to inquire whether the observed cosmic was been originally proposed by Starobinski [16], is shown. A non-
singular early cosmology is proposed, where, adding a nonlinear
⇑ Corresponding author. Addresses: Associazione Scientifica Galileo Galilei, Via electrodynamics Lagrangian to the high-order action, a bouncing
Pier Cironi 16-59100 PRATO, Italy. Institute for Basic Research, P. O. Box 1577, Palm is present and a power-law inflation is obtained. In the model
Harbor, FL 34682, USA. the Ricci scalar R works like an inflaton field.
E-mail addresses: christian.corda@ego-gw.it (C. Corda), herman@icra.it (H.J. In the general picture of high order theories of gravity, recently
Mosquera Cuesta). the R2 theory has been analysed in various interesting frameworks,
1
Partially supported by a Research Grant of The R.M. Santilli Foundations Number
RMS-TH-5735A2310.
see [17,18] for example.
2
Fellow of Fundação Cearense de Apoio ao Desenvolvimento Cientìfico e We recall that extensions of the traditional Maxwell electromag-
Tecnològico (FUNCAP), Fortaleza, Ceara, Brazil. netic Lagrangian, which take into account high order terms of the
0927-6505/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.astropartphys.2010.12.002

3. Author's personal copy
588 C. Corda, H.J. Mosquera Cuesta / Astroparticle Physics 34 (2011) 587–590
electromagnetic scalar F, have been used in cosmological models b can be obtained by varing the action in respect to R. It is
[19], gravitational redshifts of neutron stars [20] and pulsars [21].
@ðR þ aR2 Þ
Moreover, a particular nonlinear Lagrangian has been analysed in a3 dR À bdR ¼ 0; ð8Þ
the context of the Pioneer 10/11 spacecraft anomaly [22]. @R
which gives
2. Action and lagrangian
@ðR þ aR2 Þ
b ¼ a3 ¼ a3 ð2aR þ 1Þ: ð9Þ
Let us consider the high order action [16–18] @R
Z
ffi
4 pffiffiffiffiffiffi
Thus, substituting in Eq. (7) one obtains
S¼ d x Àg R þ aR2 þ Lm : ð1Þ Z n o
S ¼ 2p2 dt À2a3 aR2 À 6a2 að2aR þ 1Þ À 6aðaÞ2 ð2aR þ 1Þ þ a3 Lm :
€ _
Such an Eq. (1) is a particular choice in respect to the well
known canonical one of General Relativity (the Einstein–Hilbert ð10Þ
action [23]) which is
Z The term À6a2 að2aR þ 1Þ is critical as it contains a second
€
4 pffiffiffiffiffiffi
ffi
S¼ d x Àg ðR þ Lm Þ: ð2Þ derivative of a. Let us integrate it. It is
Z
2
We are going to show that the action (1), applied to the Fried- À6 dta að2aR þ 1Þ ¼ À6a2 að2aR þ 1Þ
€ _
`
mann–Lemaıtre–Robertson–Walker Cosmology, generates a non- Z h i
singular inflationary phase of the Universe where the Ricci scalar þ 6 dt 2aa2 aR þ 2aðaÞ2 ð2aR þ 1Þ
__ _
acts like inflaton, and a bouncing is present, if Lm is the non linear Z h i
electrodynamics Lagrangian. Note that in this letter we work with ¼ 6 dt 2aa2 aR þ 2aðaÞ2 ð2aR þ 1Þ ;
__ _ ð11Þ
8pG = 1, c = 1 and = 1.
h
Inflationary models of the early Universe were analysed in the where we have taken into account that the term outside the integral
early and middles 1980’s (see [24] for a review), starting from an is equal to zero as it is a pure divergence.
idea of Starobinski [16] and Guth [25]. These are cosmological Substituting in Eq. (10), one gets
models in which the Universe undergoes a brief phase of a very ra- Z n o
pid expansion in early times. In this context the expansion could be S ¼ 2p2 dt Àa3 aR2 þ 12aa2 aR þ 6aðaÞ2 ð2aR þ 1Þ þ a3 Lm :
__ _
power-law or exponential in time. Inflationary models provide
solutions to the horizon and flatness problems and contain a mech- ð12Þ
anism which creates perturbations in all fields [24]. Then, the Lagrangian is
In Cosmology, the Universe is seen like a dynamic and thermo-
dynamic system in which test masses (i.e. the ‘‘particles’’) are the L ¼ Àa3 aR2 þ 12aa2 aR þ 6aðaÞ2 ð2aR þ 1Þ þ a3 Lm :
__ _ ð13Þ
galaxies that are stellar systems with a number of the order of The energy function associated to the Lagrangian is [23]
109 À 1011 stars [23]. Galaxies are located in clusters and super
clusters, and observations show that, on cosmological scales, their @L @L _
EL ¼ _
aþ R À L: ð14Þ
distribution is uniform. This is also confirmed by the WMAP data _
@a _
@R
on the Cosmic Background Radiation [26,27]. These assumption Combining Eq. (13) with Eq. (14), the condition
can be summarized in the so called Cosmological Principle: the EL ¼ 0; ð15Þ
Universe is homogeneous everywhere and isotropic around every
1 da
point. Cosmological Principle simplifies the analysis of the large together with the definition of the Hubble constant, i.e. H ¼ a dt
, and
scale structure, because it implies that the proper distances be- with a little algebra gives
tween any two galaxies is given by an universal scale factor which _
Lm R
is the same for any couple of galaxies [23]. H2 ¼ ÀH : ð16Þ
3aR R
In this framework, the cosmological line – element is the well
`
known Friedmann–Lemaıtre–Robertson–Walker one, and for a _
From the Euler–Lagrange equation for a and a, i.e. [23]
sake of simplicity we will consider the flat case, because the WMAP
@L d @L
data are in agreement with it [26,27]: ¼ ; ð17Þ
_
@a dt @ a
2 2 2 2 2
ds ¼ Àdt þ a2 ðdz þ dx þ dy Þ: ð3Þ
one gets
Following [23] we also get
€ _ 2Lm :
R þ 3HR ¼ ð18Þ
À1 0 0 0 3a
2
0 þa 0 0
g lm ¼ ; ð4Þ An important question is where Eq. (15) comes from [29]. In
0 0 þa2 0 general relativity, due to the reparametrization invariance of the
0 0 0 þa2 time coordinate, the total energy (including the contribution from
the gravity sector) vanishes [29]. In the action (12), however, there
pffiffiffiffiffiffiffi
Àg ¼ a3 ; ð5Þ is not the reparametrization invariance because the total derivative
terms are dropped [29]. Then, one can think that the total energy
and
does not always vanish [29]. We clarify this point as it follows.
2 #
1 da_ _
a Let us start by the original action (1) from which the action (12)
R ¼ À6 þ : ð6Þ
a dt a arises. Let us consider the conformal transformation [30]
One can use the Lagrange multipliers putting g ab ¼ e2U g ab ;
~ ð19Þ
Z (
2 # )
€ _ where the conformal rescaling
a a
S ¼ 2p2 dt a3 ðR þ aR2 Þ À b R þ 6 þ 6 þ a3 Lm : ð7Þ
a a e2U ¼ 2aR þ 1 ð20Þ

4. Author's personal copy
C. Corda, H.J. Mosquera Cuesta / Astroparticle Physics 34 (2011) 587–590 589
has been chosen. By applying the conformal transformation (19) to One gets
the action (1) the conformal equivalent Hilbert–Einstein action rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffih i B0 2À 2 Á
4
~ e a2 ¼ pffiffiffi t þ 12c1 : ð29Þ
A ¼ d x Àg R þ LðU; U;a Þ þ Lm ð21Þ a 3
This expression is not singular for c1 0. In this case we see that
is obtained. LðU; U;a Þ is the conformal scalar field contribution de-
at the instant t = 0 a minimum value of the scale factor is present:
rived from
B0 pffiffiffiffiffiffiffiffi
e 1 a2 ¼ pffiffiffi 8c1 :
min ð30Þ
R ab ¼ Rab þ 2 U;a U;b À g ab U;d U;d À g ab U;d ;
;d ð22Þ a
2 pffiffiffiffiffiffiffiffiffiffi
This also implies that, for a value t ¼ 12c1 , the energy density
and Àq reaches a maximum value qmax = 1/64c1. For smaller values of t
À Á the energy density decreases, vanishing a t = 0, while the pressure
e
R ¼ eÀ2U þ R À 6ÃU À 6U;d U;d : ð23Þ becomes negative [19].
In this way, the condition of inflation P À q⁄ 0 [24] gives the
Clearly, the reparametrization invariance of the time coordinate inflationary solutions for Eqs. (16) and (18), if one assumes that
is consistent with the new action (21) in the conformal Einstein the Ricci scalar R acts like inflaton:
frame and the total energy (including the contribution from the
gravity sector) vanishes in this case too. One could object that RðtÞ ’ ð1 þ Ht=bÞ2 ;
the energy in the conformal Einstein frame is different with respect ð31Þ
ainf ðtÞ ’ ð1 þ Ht=bÞwþ1=2
to the energy in the original Jordan frame, but in Ref. [31] it has
been shown that the two conformal frames are energetically equiv- with b ’ w and
alent if, together with the conformal rescaling (19), times and qffiffiffiffiffiffiffi
lengths are rescaled as eU while the mass-energy is rescaled as Hinf ’ LÃ ;
m ð32Þ
eÀU. This analysis permits to enable the condition of Eq. (15) in
the present discussion too. where LÃ is the right hand side of Eq. (16) which is constant during
m
the inflationary phase. The idea of considering the Ricci scalar as an
3. Nonlinear electrodynamics lagrangian and Inflation effective scalar field (scalaron) arises from Starobinski [16].
In order to show that our model admits a power law inflation- 4. Conclusion remarks
ary phase, we need to postulate some matter Lagrangian Lm which
can perform the condition of inflation P À q [24]. We will use the Another approach regarding the inflation from the R2 theory of
non linear electrodynamics Lagrangian of [19], which is gravity, which was originally proposed by Starobinsk, has been
analysed. A non-singular early cosmology has been proposed,
1
Lm À F þ c1 F 2 þ c2 G2 ; ð24Þ where, adding a nonlinear electrodynamics Lagrangian to the
4
high-order action, a bouncing is present and a power-law inflation
where F is the electromagnetic scalar, c1, c2 are two constants and, is obtained. In the model which has been discussed, the Ricci scalar
considering the electromagnetic field tensor Fab (see [23] the defini- R works like an inflaton field.
tion of this object), G is defined like [19] G 1 gablm F ab F lm .
2
The Lagrangian (24), differently from the one of the singular Acknowledgements
Einstein–Maxwell Universe, performs a non-singular Universe
with bouncing [19]. This is because the energy condition of singu- The authors thank Professor Mario Novello for useful discus-
larity theorems [28] is not satisfied in the case of the non linear sions on the topics of this paper. We also thank an unknown ref-
electrodynamics Lagrangian (see [19] for details). eree for precious advices and suggestions which permitted to
In fact, following [19], one uses the equation of state improve this paper.
1
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