A Cosmological Scenario from the Starobinsky Model within the … · 2019. 7. 30. ·...

Research ArticleA Cosmological Scenario from the Starobinsky Model withinthe 119891(119877 119879) Formalism

P H R S Moraes12 P K Sahoo 3 G Ribeiro4 and R A C Correa 25

1Universita degli Studi di Napoli Federico II (UNINA) Dipartimento di Fisica Napoli 80126 Italy2Instituto Tecnologico de Aeronautica (ITA) Departamento de Fısica 12228-900 Sao Jose dos Campos SP Brazil3Department of Mathematics Birla Institute of Technology and Science Pilani Hyderabad Campus Hyderabad 500078 India4Universidade Estadual Paulista ldquoJulio de Mesquita Filhordquo (UNESP) Departamento de Fısica e Quımica12516-410 Guaratingueta SP Brazil

5Scuola Internazionale Superiore di Studi Avanzati (SISSA) Via Bonomea 265 34136 Trieste Italy

Correspondence should be addressed to R A C Correa fis04132gmailcom

Received 19 November 2018 Accepted 27 March 2019 Published 2 June 2019

Academic Editor Charalampos C Moustakidis

Copyright copy 2019 P H R S Moraes et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper we derive a novel cosmological model from the 119891(119877 119879) theory of gravitation for which 119877 is the Ricci scalar and 119879 isthe trace of the energy-momentum tensorWe consider the functional form119891(119877 119879) = 119891(119877)+119891(119879) with119891(119877) being the StarobinskyModel named 119877 + 1205721198772 and 119891(119879) = 2120574119879 with 120572 and 120574 being constants We show that a hybrid expansion law form for the scalefactor is a solution for the derived Friedmann-like equations In this way the model is able to predict both the decelerated and theaccelerated regimes of expansion of the universe with the transition redshift between these stages being in accordance with recentobservations We also apply the energy conditions to our material content solutions Such an application makes us able to obtainthe range of acceptability for the free parameters of the model named 120572 and 120574

1 Introduction

The119891(119877) theories of gravity [1 2] are an optimistic alternativeto the shortcomings General Relativity (GR) faces as theunderlying gravitational theory such as those discussed byPadmanabhan [3] Antoniadis et al [4] Demorest et al [5]Bull et al [6] They can account for the cosmic acceleration[7 8] with no need for a cosmological constant providinggood match between theory and cosmological observations[9ndash11]

Particularly in standard cosmology derived fromGR thedark energy and dark matter should compose sim 95 of theuniverse [12] but their nature is still dubious [13 14]

Another crucial trouble surrounding GR is the difficultyin quantizing it Attempts to do so have been proposed [15ndash17] and can in future provide us a robust and trustworthymodel of gravity-quantum mechanics unification

Meanwhile it is worthwhile to attempt to consider thepresence of quantum effects in gravitational theories Those

effects can rise from the consideration of terms proportionalto the trace of the energy-momentum tensor 119879 in the gravi-tational part of the 119891(119877) action yielding the 119891(119877 119879) gravitytheories [18] Those theories were also motivated by the factthat although 119891(119877) gravity is well behaved in cosmologicalscales the solar system regime seems to rule out most of the119891(119877) models proposed so far [19ndash22] Furthermore rotationcurves of spiral galaxies were constructed in 119891(119877) gravity butthe results did not favour the theory as it can be checkedin Chiba [23] Dolgov ampKawasaki [24] Olmo [25]The struc-ture and cosmological properties of the modified gravitystarting from119891(119877) theory to power-counting renormalizablecovariant gravity were presented by Nojiri amp Odintsov [26]The review in Nojiri et al [27] describes the cosmologicaldevelopments regarding inflation bounce and late-time evo-lution in 119891(119877) 119891(G) and 119891(T)modified theories of gravitywith G andT being the Gauss-Bonnet and torsion scalars

Despite its recent elaboration 119891(119877 119879) gravity has alreadybeen applied to a number of areas such as cosmology

HindawiAdvances in AstronomyVolume 2019 Article ID 8574798 8 pageshttpsdoiorg10115520198574798

2 Advances in Astronomy

([28ndash34] and astrophysics [35ndash42]) Particularly solar systemtests have been applied to 119891(119877 119879) gravity [43] and the darkmatter issue was analysed by Zaregonbadi et al [44] Thelate-time behaviour of cosmic fluids consisting of collisionalself-interacting dark matter and radiation was discussed byZubair et al [45] Moreover considering the metric and theaffine connection as independent field variables the Palatiniformulation of the 119891(119877 119879) gravity can be seen in [46 47]

By investigating the features of an 119891(119877 119879) or 119891(119877)modelone realizes the strong relation they have with the functionalform of the chosen functions for 119891(119877 119879) and 119891(119877) as well aswith their free parameters values In fact a reliable method toconstraint those ldquofreerdquo parameters to values that yield realisticmodels can be seen in Correa amp Moraes [48] and Correa etal [49] for these theories respectively

In the119891(119877) gravity a reliable and reputed functional formwas proposed by AA Starobinsky as [50 51]

119891 (119877) = 119877 + 1205721198772 (1)

which is known as Starobinsky Model (SM) with 120572 aconstant It predicts a quadratic correction of the Ricci scalarto be inserted in the gravitational part of the Einstein-Hilbertaction

SM has been deeply applied to the cosmological andastrophysical contexts in the literature Starobinsky showedthat a cosmological model obtained from (1) can satisfycosmological observational tests [51] On the other handthe model seems to predict an overproduction of scalaronsin the very early universe In an astrophysical context SMis also of great importance In Sharif amp Siddiqa [52] theauthors have explored the source of a gravitational radiationin SM by considering axially symmetric dissipative dustunder geodesic condition In Resco et al [53] it has beenshown that in SM it is possible to find neutron stars with 2119872⊙ which raises as an important alternative to some of the GRshortcomings mentioned above [4 5] In Astashenok et al[54] the macroscopical features of quark stars were obtained

Our proposal in this paper is to construct a cosmo-logical scenario from an 119891(119877 119879) functional form whose119877minusdependence is the same as in the SM ie with a quadraticextra contribution of 119877 as in (1) The 119879minusdependence willbe considered to be linear as 2120574119879 with 120574 being a constantTherefore we will take

119891 (119877 119879) = 119877 + 1205721198772 + 2120574119879 (2)

As far as the present authors know the SM has not beenconsidered for the 119877minusdependence within 119891(119877 119879) models forcosmological purposes so far only in the study of astrophys-ical compact objects [55ndash57] and wormholes [58 59] Webelieve this is due to the expected high nonlinearity of theresulting differential equation for the scale factor Anyhowthe consideration of linear material corrections together withquadratic geometrical terms can imply interesting outcomesin a cosmological perspective as it does in the astrophysicallevel

2 The 119891(119877119879)=119877+1205721198772+2120574119879 Gravity

Following the steps in Harko et al [18] we can write the119891(119877 119879) = 119877 + 1205721198772 + 2120574119879 gravity total action as

119878 = 116120587 int (119877 + 1205721198772 + 2120574119879)radicminus1198921198894119909+ int119871119898radicminus1198921198894119909

(3)

in which 119892 is the determinant of the metric 119892120583] 119871119898 is thematter Lagrangian and we are working with natural units

By taking 119871119898 = minus119901 with 119901 being the pressure of theuniverse the variational principle applied in (3) yields thefollowing field equations

12119866120583] + 120572(119877120583] minus 1

4119877119892120583] + 119892120583]◻ minus nabla120583nabla])119877= 4120587119879120583] + 120574 [119879120583] + (119901 + 1

2119879)119892120583]] (4)

In (4) 119866120583] is the Einstein tensor 119877120583] is the Ricci tensor119879120583] = diag(120588 minus119901 minus119901 minus119901) 120588 is the matter-energy densityof the universe and 119879 = 120588 minus 3119901 Still in (4) it can bestraightforwardly seen that the limit 120572 = 120574 = 0 recovers theGR field equations

Also the above choice for thematter Lagrangian is usuallyassumed in the literature as it can be checked in Harko et al[18] Moraes et al [35 38] among many others

3 The 119891(119877119879)=119877+1205721198772+2120574119879 Cosmology

Let us assume a flat Friedmann-Robertson-Walker metricin the field equations above Such a substitution yields thefollowing Friedmann-like equations

( 119886119886)2 + 6120572119866 (119886 119886 119886 ) = 8120587

3 120588 + 120574 (120588 minus 1199013 ) (5)

119886119886 + 1

2 ( 119886119886)2 + 6120572119878 (119886 119886 119886 )

= minus4120587119901 + 12120574 (120588 minus 119901)

(6)

where we are using the following definitions

119866 (119886 119886 119886 ) equiv 2 ( 119886119886)2 [ 119886

119886 + 119886 minus 3

2 ( 119886119886)2] minus ( 119886

119886)2 (7)

119878 (119886 119886 119886 ) equiv 32 [( 119886

119886)4 + 119886119886]

+ 2 [ 119886 1198862 minus 3 ( 119886

119886)2 119886119886] +

119886 (8)

In the equations above 119886 = 119886(119905) is the scale factor anddots represent time derivatives Once again the limit 120572 = 120574 =0 retrieves the standard formalism

Advances in Astronomy 3

In terms of the Hubble parameter 119867 = 119886119886 the values of120588 and 119901 from (5)-(6) are

120588 = 119865 (120574)2 [120574119865 (119867) minus (8120587 + 120574)119866 (119867)] (9)

119901 = 119865 (120574) [4120587119865 (119867) + 3120574119866 (119867)] (10)

with the following definitions

119865 (119867) equiv minus1198654 (119867)minus 6120572 [minus31198652 (119867) + 1198663 (119867) + 31198651 (119867)] (11)

119866 (119867) equiv 3 1198672 + 6 [1198664 (119867) + 1198653 (119867)] 120572 (12)

119865 (119867) equiv 1198654 (119867) + 181205721198652 (119867) minus 1198651 (119867) + + 22 (13)

119866 (119867) equiv minus1198672 + + 3120572 [1198661 (119867) + 1198662 (119867) + 21198675 + 61198673] (14)

For (9)-(10) we considered 119865(120574) = 1(321205872 + 16120587120574 + 1205742)Moreover 119865119894 and119866119894 with 119894 = 1 2 3 4 are functions of119867 andits time derivatives expressed by the following

1198651 (119867) = 1198672 (1 + 4) (15)

1198652 (119867) = 1198674 minus 4119867 (16)

1198653 (119867) = 2 (1198675 + 31198673) (17)

1198654 (119867) = 31198672 + 2 (18)

1198661 (119867) = 1198674 + (3 + 7) (19)

1198662 (119867) = minus12119867 + 1198672 (minus3 minus 12 + 2) minus 2 (20)

1198663 (119867) = 3 (1 + 2) + 2 (21)

1198664 (119867) = minus21198674 minus 2 + 21198672 (22)

We can consider as a solution for (9)-(10) the scale factorin the hybrid expansion law form [60]

119886 (119905) = 119890119898119905119905119899 (23)

with 119898 and 119899 being constants It can be seen that such a formfor the scale factor consists of a product of power law andexponential law functions Equation (23) mimics the powerlaw and de-Sitter cosmologies as particular cases and can pre-dict the transition from a decelerated to an accelerated regimeof the universe expansion It has been applied to Brans-Dickemodels inOzgur et al [60] yielding observational constraintsto 119898 and 119899

From (23) the Hubble and deceleration parameters are

119867 = 119898 + 119899119905 (24)

119902 = minus 1198861198861198672 = minus1 + 119899

(119898119905 + 119899)2 (25)

deSitter (m = 027 n = 0)minus10

minus08

minus06

minus04

minus02

00

02

04

q

0 1 2 3 4 5minus1z

m = 027 n = 075m = 028 n = 075m = 0 n = 075

Figure 1 Deceleration parameter evolution in redshift for differentvalues of 119898 and 119899

such that the deceleration parameter is defined in such a waythat its negative values describe an accelerated expansion ofthe universe

We know that the universe not always has acceleratedits expansion [7 8] The accelerated regime of expansion isconsidered to be a late-time phenomenon

In thisway one can choose the constants119898 and 119899 in such away that the power law dominates over exponential law in theearly universe and the exponential law dominates over powerlaw at late times Therefore the decelerated and acceleratedregimes of the universe expansion can be respectively welldescribed as well as the transition between these regimes

From (25) it is clear that there is a transition fromdeceleration to acceleration phases of the universe expansionat 119905 = (1119898)(minus119899plusmnradic119899) with 0 lt 119899 lt 1 Since the negativity ofthe second term leads to a negative time which indicates annonphysical situation we conclude that the cosmic transitionmay have occurred at 119905 = (radic119899 minus 119899)119898

From 119886(119905)1198860 = 1(1 + 119911) with 1198860 = 1 being the presentvalue of the scale factor and 119911 being the redshift we obtainthe following time-redshift relation

119905 = 119899119898119882[119898

119899 ( 11 + 119911)1119899] (26)

where 119882 denotes the Lambert function (also known asldquoproduct logarithmrdquo)

Using (26) we can plot the deceleration parameter withrespect to the redshift which can be appreciated in Figure 1in which the values chosen for119898 and 119899 are in agreement withthe observational constraints found in Ozgur et al [60]

Plotting 119902 as a redshift function has the advantage ofchecking the reliability of the model through the redshiftvalue in which the decelerated-accelerated expansion of theuniverse transition occurs We will call the transition redshiftas 119911119905119903 and in our model it can be seen that it depends directlyon the parameter 119898 From Figure 1 the transition occursat 119911119905119903 = 05836 06777 corresponding to 119898 = 025 027


respectively The values of the transition redshift 119911119905119903 for ourmodel are in accordance with the observational data as onecan check in Capozziello et al [61 62] Farooq et al [63]

Now let us write the solutions for the material content ofour model named 120588 and 119901 Using (24) in (9)-(10) we have

120588 = 119865 (120574)2 [minus120574 (P1 +P2) +P3

+ 3 (8120587 + 120574) (P4 + P5 +P6)] (27)

119901 = 119865 (120574) [minus (G1 +G2 +G3 +G4) +G5] (28)

where we are using the definitions

P1 equiv 32120572119899F3 (119905)3 times 4F1 (119905) [1 minus F1 (119905)2 ]

+ 119899 [1 minus F3 (119905)minus22 ] minus 2 (29)

P2 equiv 18120572F2 (119905)4 [F2 (119905)minus2 minus 1] (30)

P3 equiv 3F2 (119905)2 minus 2119899F3 (119905)2 (31)

P4 equiv 6120572119899F3 (119905)5 2F1 (119905)2 [2 minus 3F1 (119905)]minus 119899F3 (119905)minus1 (32)

P5 equiv 12120572F2 (119905)5 [1 minus F2 (119905)minus1] (33)

G1 equiv 120572119899F3 (119905)4F1 (119905)2 9120574F3 (119905) [6F1 (119905) minus 4]minus 2Fminus11 (119905) 119866 (120574) + 119866 (120574) (34)

G2 equiv 119899F3 (119905)2 [1205724119866 (120574) minus 3120574 minus 8120587] (35)

G3 equiv 1205721198992F3 (119905)2 [119866 (120574) + 18120574] + 120572119866 (120574) (36)

G4 equiv 120572119899F3 (119905)4 119866 (120574) (37)

G5 equiv F2 (119905)2 times 9120572F2 (119905)2 [minus2120574F2 (119905) minus 120574 minus 8120587]minus 120572

4119866 (120574) + 3120574 + 12120587 (38)

with 119866(120574) = minus108120574 minus 288120587 and F119895 with 119895 = 1 2 3 beingfunctions of 119905 only defined as

F1 (119905) = 119898119905 + 119899 (39)

F2 (119905) = 119898 + 119899119905 (40)

F3 (119905) = 1119905 (41)

The evolution of the energy density pressure and corre-sponding equation of state (EoS) parameter 120596 = 119901120588 with119898 = 027 119899 = 075 are shown in Figures 2ndash4 in which thetime units are Gyr

2 4 6 8 100t

002

004

006

008

010

012

014

016

= -01 = -01 = -005 = -01

Figure 2 Evolution of 120588 with time

2 4 6 8 100t

= -01 = -01 = -005 = -01

minus005

minus004

minus003

minus002

minus001

000

001

002

p


4 Energy Conditions

Energy conditions (ECs) in the context of a wide class ofcovariant theories including GR are relations one demandsfor the energy-momentum tensor of matter to satisfy in orderto try to capture the idea that ldquoenergy should be positiverdquo Byimposing so one can obtain constraints to the free parametersof the concerned model

The standard point-wise ECs are [64ndash66]

Null energy condition (NEC) 120588 + 119901 ge 0Weak energy condition (WEC) 120588 ge 0 120588 + 119901 ge 0Strong energy condition (SEC) 120588 + 3119901 ge 0Dominant energy condition (DEC) 120588 ge |119901|

Generally the above ECs are formulated from the Raychaud-huri equation which describes the behaviour of space-liketime-like or light-like curves in gravity

In the present model the energy-momentum tensor hasthe form of a perfect fluid So we will use the above relations


= -01 = -01 = -005 = -01

1 2 3 4 5 6 70t

minus06

minus04

minus02

00


10

05

000

5

10

minus4

minus2

0

t

Figure 5 Evolution of 120588

for analysing the ECs in 119891(119877 119879) theory The behaviour of theECs with 119898 = 027 119899 = 075 and 120572 = minus002 is given inFigures 5ndash8

5 Discussion and Conclusions

In this article we have proposed an 119891(119877 119879) cosmologicalmodel For the functional form of the function 119891(119877 119879) weinvestigated a quadratic correction to119877 as in the well-knownSM together with a linear term on 119879 The substitution ofsuch an 119891(119877 119879) function in the gravitational action generatesextra terms in the field equations (4) and consequently inthe Friedmann-like equations (5) and (6) Those extra termsbring some new information regarding the dynamics of theuniverse as we discuss below

Our solution for the scale factor 119886(119905) is a hybrid expansionlaw well described in Ozgur et al [60] From (23) we haveobtained the cosmological parameters namely Hubble factorand deceleration parameter Specifically for the decelerationparameter we could separate it in two phases one describingthe decelerated and the other describing the accelerated

15

10

05

000

5

10

minus4

minus2

0

+p

t

Figure 6 Validation of NEC

0

3

2

1

0

5

10

minus4

minus2

0

+3p

t

Figure 7 Violation of SEC

regime of the universe expansion This can be well-checkedin Figure 1 in which the transition redshift between these twostages agrees with observational data Moreover one can alsonote that remarkably the present (119911 = 0) values for 119902 namedminus0178 for 119898 = 027 and minus0157 for 119898 = 025 are also inagreement with observational data [12]

We have also obtained solutions for the material contentof the universe named 120588 and 119901 (see (27)-(41)) In Figures2ndash4 we plot the evolution of 120588 119901 and 120596 the EoS parameterThe values chosen for 120572 and 120574 respect the energy conditionsoutcomes presented in Section 4 In Figure 3 we see that thepressure of the universe starts in positive values and thenassumes negative values In standard model of cosmology anegative pressure fluid is exactly the mechanism responsiblefor accelerating the universe expansion In the present modelsuch a behaviour for the pressure was naturally obtained

It is also worth stressing that the EoS parameter shows atransition from a decelerated to an accelerated regime of theexpansion of the universe This can well be seen in Figure 4by recalling that from standard cosmology the latter regimemay happen if 120596 lt minus13 [67] Moreover it can be seenfrom such a figure that as time passes by 120596 997888rarr minus1 in


minus05

10

05

00

0

5

10

minus4

minus2

0

minusp

t

Figure 8 Validation of DEC

accordance with recent observational data on the cosmicmicrowave background temperature fluctuations [12]

Furthermore in Figures 5ndash8 we plotted the ECs from thematerial solutions of our cosmological model Those figureswere plotted in terms of 120574 for fixed 120572 = minus002 They showthe validation of WEC NEC and DEC with a wide range ofacceptable values for 120574

On the other hand Figure 7 shows violation of SECHowever as it has been deeply discussed by Barcelo amp Visser[68] an early and late-time accelerating universe must violateSEC

As a further work we can look for the 119891(119877 119879) = 119877 +1205721198772 + 2120574119879 gravity universe at very early times particularlyinvestigating the production of scalarons in this modelMoraesamp Santos [30] have shown that the trace of the energy-momentum tensor contribution in the theory is higher for theearly universe when compared to the late-time contributionAccording to Starobinsky [51] somemechanism should workin the early universe to prohibit the scalaron overproductionwithin SM The high contribution of the terms proportionalto 119879 in the early universe may well be this mechanism

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

P H R S Moraes thanks Sao Paulo Research Foundation(FAPESP) grant 201508476-0 for financial support P KSahoo acknowledges DST New Delhi India for providingfacilities through DST-FIST Lab Department of Mathemat-ics where a part of this work was done R A C Correa ispartially supported by FAPESP (Foundation for Support to

Research of the State of Sao Paulo) under grants numbers201603276-5 and 201726646-5

References

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[2] A De Felice and S J Tsujikawa ldquof(R) theoriesrdquo Living Reviewsin Relativity vol 13 no 3 pp 1ndash161 2010

[3] T Padmanabhan ldquoCosmological constantmdashthe weight of thevacuumrdquo Physics Reports vol 380 no 5-6 pp 235ndash320 2003

[4] J Antoniadis P C C Freire N Wex et al ldquoA massive pulsar ina compact relativistic binaryrdquo Science vol 340 p 448 2013

[5] P BDemorest T Pennucci SM RansomM S E Roberts andJW T Hessels ldquoA two-solar-mass neutron star measured usingShapiro delayrdquo Nature vol 467 no 7319 pp 1081ndash1083 2010

[6] P Bull Y Akrami J Adamek et al ldquoBeyondΛCDM problemssolutions and the road aheadrdquo Physics of the Dark Universe vol12 p 56 2016

[7] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 pp 1009ndash1038 1998

[8] S Perlmutter G Aldering and G Goldhaber ldquoMeasurementsofΩ andΛ from42 high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 pp 565ndash586 1999

[9] S Tsujikawa ldquoObservational signatures of f (R) dark energymodels that satisfy cosmological and local gravity constraintsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 77 no 2 Article ID 023507 2008

[10] S Capozziello V F Cardone and A Troisi ldquoReconciling darkenergy models with f(R) theoriesrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 71 no 4 2005

[11] S Nojiri and S D Odintsov ldquoModified f(R) gravity consistentwith realistic cosmology from matter dominated epoch todark energy universerdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 Article ID 086005 2006

[12] G Hinshaw D Larson and E Komatsu ldquoNine-yearWilkinsonMicrowave Anisotropy Probe (WMAP) observations cosmo-logical parameter resultsrdquoTheAstrophysical Journal SupplementSeries vol 208 no 2 p 19 2013

[13] J Evslin ldquoModel-independent dark energy equation of statefrom unanchored baryon acoustic oscillationsrdquo Physics of theDark Universe vol 13 pp 126ndash131 2016

[14] Y Zhang ldquoSelf-interacting darkmatter without direct detectionconstraintsrdquoPhysics of theDarkUniverse vol 15 pp 82ndash89 2017

[15] E S Fradkin and A A Tseytlin ldquoQuantum string theory effec-tive actionrdquo Nuclear Physics B Theoretical Phenomenologicaland Experimental High Energy Physics Quantum Field Theoryand Statistical Systems vol 261 no 1 pp 1ndash27 1985

[16] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[17] D Friedan EMartinec and S Shenker ldquoConformal invariancesupersymmetry and string theoryrdquo Nuclear Physics B Theoret-ical Phenomenological and Experimental High Energy PhysicsQuantum FieldTheory and Statistical Systems vol 271 no 1 pp93ndash165 1986

[18] T Harko F S N Lobo S Nojiri and S D Odintsov ldquoF(RT) gravityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 84 no 2 Article ID 024020 2011


[19] A L Erickcek T L Smith and M Kamionkowski ldquoSolarsystem tests do rule out 1R gravityrdquoPhysical ReviewD ParticlesFields Gravitation and Cosmology vol 74 no 12 Article ID121501 2006

[20] T Chiba T L Smith and A L Erickcek ldquoSolar Systemconstraints to generalf(R)gravityrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 75 no 12 Article ID124014 2007

[21] S Capozziello A Stabile and A Troisi ldquoNewtonian limit of119891(119877) gravityrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 76 no 10 Article ID 104019 2007

[22] G J Olmo ldquoLimit to general relativity in 119891(119877) theories ofgravityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 75 no 2 Article ID 023511 2007

[23] T Chiba ldquo1R gravity and scalar-tensor gravityrdquo Physics LettersB vol 575 no 1-2 pp 1ndash3 2003

[24] A D Dolgov and M Kawasaki ldquoCan modified gravity explainaccelerated cosmic expansionrdquo Physics Letters B vol 573 pp1ndash4 2003

[25] G J Olmo ldquoPost-Newtonian constraints onf(R) cosmologiesin metric and Palatini formalismrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 72 no 8 Article ID083505 2005

[26] S Nojiri and S D Odintsov ldquoUnified cosmic history in mod-ified gravity from F(R)F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 pp 59ndash144 2011

[27] S Nojiri S D Odintsov and V K Oikonomou ldquoModifiedgravity theories on a nutshell inflation bounce and late-timeevolutionrdquo Physics Reports vol 692 pp 1ndash104 2017

[28] P H R S Moraes ldquoCosmological solutions from inducedmatter model applied to 5Df(R T) gravity and the shrinkingof the extra coordinaterdquo European Physical Journal C vol 75 p168 2015

[29] P H Moraes G Ribeiro and R A Correa ldquoA transitionfrom a decelerated to an accelerated phase of the universeexpansion from the simplest non-trivial polynomial function119879 of in the 119891(119877 119879) formalismrdquo Astrophysics and Space ScienceAn International Journal of Astronomy Astrophysics and SpaceScience vol 361 no 7 p 227 2016

[30] P H R S Moraes and J R L Santos ldquoA complete cosmologicalscenario from f(R T120601) gravity theoryrdquo European PhysicalJournal C vol 76 p 60 2016

[31] P H R S Moraes and P K Sahoo ldquoThe simplest non-minimalmattergeometry coupling in thef(R T)cosmologyrdquoTheEuropean Physical Journal C vol 77 p 480 2017

[32] R Myrzakulov ldquoFRW cosmology in F (RT) gravityrdquo TheEuropean Physical Journal C vol 72 article 2203 2012

[33] D R K Reddy R Santikumar and R L Naidu ldquoBianchi type-III cosmological model in f(RT) theory of gravityrdquoAstrophysicsand Space Science vol 342 no 1 pp 249ndash252 2012

[34] V Singh and C P Singh ldquoFriedmann cosmology with mattercreation in modified f(RT) gravityrdquo International Journal ofTheoretical Physics vol 55 no 2 pp 1257ndash1273 2016

[35] P H Moraes J D Arbanil and M Malheiro ldquoStellar equi-librium configurations of compact stars in 119891(119877 119879) theory ofgravityrdquo Journal of Cosmology and Astroparticle Physics no 6p 005 2016

[36] M E Alves P H Moraes J C de Araujo and M MalheiroldquoGravitational waves in f(RT120593) and f(RT) theories of gravityrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 94 no 2 Article ID 024032 2016

[37] P H Moraes and P K Sahoo ldquoModeling wormholes in 119891(119877 119879)gravityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 96 no 4 Article ID 044038 2017

[38] P HMoraes R A Correa and R V Lobato ldquoAnalytical generalsolutions for static wormholes in 119891(119877 119879) gravityrdquo Journal ofCosmology and Astroparticle Physics vol 07 029 pages 2017

[39] A Das F Rahaman B K Guha and S Ray ldquoCompact starsinf(R T)gravityrdquo European Physical Journal C vol 76 p 6542016

[40] M Sharif and Z Yousaf ldquoDynamical analysis of self-gravitatingstars in f(RT) gravityrdquo Astrophysics and Space Science vol 354no 2 pp 471ndash479 2014

[41] Z Yousaf M Ilyas andM Z Bhatti ldquoInfluence of modificationof gravity on spherical wormhole modelsrdquo Modern PhysicsLetters A vol 32 no 30 Article ID 1750163 2017

[42] E H Baffou M J Houndjo and I G Salako ldquoViscousgeneralized Chaplygin gas interacting with f(R T) gravityrdquoInternational Journal of Geometric Methods in Modern Physicsvol 14 no 4 Article ID 1750051 p 14 2017

[43] H Shabani and M Farhoudi ldquoCosmological and solar systemconsequences of f(RT) gravity modelsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 90 Article ID044031 2014

[44] R Zaregonbadi M Farhoudi and N Riazi ldquoDark matter fromf(RT) gravityrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 94 no 8 p 084052 2016

[45] M Zubair ldquoImpact of collisional matter on the late-timedynamics of f(R T) gravityrdquo Symmetry vol 10 p 463 2018

[46] J Wu G Li T Harko and S Liang ldquoPalatini formulation off(R T) gravity theory and its cosmological implicationsrdquo TheEuropean Physical Journal C vol 78 no 5 p 430 2018

[47] E Barrientos F S Lobo S Mendoza G J Olmo and DRubiera-Garcia ldquoMetric-affine 119891(119877 119879) theories of gravity andtheir applicationsrdquo Physical Review D Particles Fields Gravita-tion and Cosmology vol 97 no 10 Article ID 104041 2018

[48] R A C Correa andPH R SMoraes ldquoConfigurational entropyinf(R T)brane modelsrdquo European Physical Journal C vol 76 p100 2016

[49] R A Correa P HMoraes A de Souza Dutra and R da RochaldquoInformation content in 119865(119877) brane models with nonconstantcurvaturerdquo Physical Review D Particles Fields Gravitation andCosmology vol 92 no 12 Article ID 126005 2015

[50] AA Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[51] A A Starobinsky ldquoDisappearing cosmological constant in f(R)gravityrdquo JETP Letters vol 86 no 3 pp 157ndash163 2007

[52] M Sharif and A Siddiqa ldquoAxial dissipative dust as a sourceof gravitational radiation in f(R) gravityrdquo Physics of the DarkUniverse vol 15 pp 105ndash113 2017

[53] M Aparicio Resco A de la Cruz-Dombriz F J Llanes Estradaand V Zapatero Castrillo ldquoOn neutron stars in f(R) theoriesSmall radii large masses and large energy emitted in a mergerrdquoPhysics of the Dark Universe vol 13 pp 147ndash161 2016

[54] A V Astashenok S Capozziello and S D Odintsov ldquoNonper-turbative models of quark stars in f(R) gravityrdquo Physics LettersB vol 742 pp 160ndash166 2015

[55] M Zubair and I Noureen ldquoEvolution of axially symmetricanisotropic sources in f(RT) gravityrdquoEuropean Physical JournalC vol 75 p 265 2015


[56] I Noureen and M Zubair ldquoDynamical instability and expan-sion-free condition inf(R T)gravityrdquo European Physical JournalC vol 75 p 62 2015

[57] I Noureen M Zubair A A Bhatti and G Abbas ldquoShear-freecondition and dynamical instability inf(R T) gravityrdquo EuropeanPhysical Journal C vol 75 p 323 2015

[58] M Zubair S Waheed and Y Ahmad ldquoStatic spherically sym-metric wormholes in f(R T) gravityrdquo The European PhysicalJournal C vol 76 no 8 p 444 2016

[59] Z Yousaf M Ilyas andM Zaeem-ul-Haq Bhatti ldquoStatic spher-ical wormhole models in f (R T) gravityrdquoTheEuropean PhysicalJournal Plus vol 132 p 268 2017

[60] O Akarsu S Kumar R Myrzakulov M Sami and L Xu ldquoCos-mology with hybrid expansion law scalar field reconstructionof cosmic history and observational constraintsrdquo Journal ofCosmology and Astroparticle Physics vol 01 p 022 2014

[61] S Capozziello O Luongo and N E Saridakis ldquoCosmographicbounds on the cosmological deceleration-acceleration transi-tion redshift inf(R)gravityrdquo Physical Review D vol 90 ArticleID 044016 2014

[62] S Capozziello O Luongo and E N Saridakis ldquoTransitionredshift in 119891(119879) cosmology and observational constraintsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 91 no 12 Article ID 124037 2015

[63] O Farooq F R Madiyar S Crandall and B Ratra ldquoHubbleparameter measurement constraints on the redshift of thedeceleration-acceleration transition dynamical dark energyand space curvaturerdquoThe Astrophysical Journal vol 835 no 12017

[64] S W Hawking and G F R Ellis The Large Scale Structure ofSpace-Time Cambridge University Press Cambridge UK 1973

[65] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[66] M Visser Lorentzian wormholes AIP Series in Computationaland Applied Mathematical Physics American Institute ofPhysics New York NY USA 1995

[67] B Ryden Introduction to Cosmology Addison Wesley SanFrancisco Mass USA 2003

[68] C Barcelo and M Visser ldquoTwilight for the energy conditionsrdquoInternational Journal of Modern Physics D Gravitation Astro-physics Cosmology vol 11 no 10 pp 1553ndash1560 2002

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([28ndash34] and astrophysics [35ndash42]) Particularly solar systemtests have been applied to 119891(119877 119879) gravity [43] and the darkmatter issue was analysed by Zaregonbadi et al [44] Thelate-time behaviour of cosmic fluids consisting of collisionalself-interacting dark matter and radiation was discussed byZubair et al [45] Moreover considering the metric and theaffine connection as independent field variables the Palatiniformulation of the 119891(119877 119879) gravity can be seen in [46 47]

By investigating the features of an 119891(119877 119879) or 119891(119877)modelone realizes the strong relation they have with the functionalform of the chosen functions for 119891(119877 119879) and 119891(119877) as well aswith their free parameters values In fact a reliable method toconstraint those ldquofreerdquo parameters to values that yield realisticmodels can be seen in Correa amp Moraes [48] and Correa etal [49] for these theories respectively

In the119891(119877) gravity a reliable and reputed functional formwas proposed by AA Starobinsky as [50 51]

119891 (119877) = 119877 + 1205721198772 (1)

which is known as Starobinsky Model (SM) with 120572 aconstant It predicts a quadratic correction of the Ricci scalarto be inserted in the gravitational part of the Einstein-Hilbertaction

SM has been deeply applied to the cosmological andastrophysical contexts in the literature Starobinsky showedthat a cosmological model obtained from (1) can satisfycosmological observational tests [51] On the other handthe model seems to predict an overproduction of scalaronsin the very early universe In an astrophysical context SMis also of great importance In Sharif amp Siddiqa [52] theauthors have explored the source of a gravitational radiationin SM by considering axially symmetric dissipative dustunder geodesic condition In Resco et al [53] it has beenshown that in SM it is possible to find neutron stars with 2119872⊙ which raises as an important alternative to some of the GRshortcomings mentioned above [4 5] In Astashenok et al[54] the macroscopical features of quark stars were obtained

Our proposal in this paper is to construct a cosmo-logical scenario from an 119891(119877 119879) functional form whose119877minusdependence is the same as in the SM ie with a quadraticextra contribution of 119877 as in (1) The 119879minusdependence willbe considered to be linear as 2120574119879 with 120574 being a constantTherefore we will take

119891 (119877 119879) = 119877 + 1205721198772 + 2120574119879 (2)

As far as the present authors know the SM has not beenconsidered for the 119877minusdependence within 119891(119877 119879) models forcosmological purposes so far only in the study of astrophys-ical compact objects [55ndash57] and wormholes [58 59] Webelieve this is due to the expected high nonlinearity of theresulting differential equation for the scale factor Anyhowthe consideration of linear material corrections together withquadratic geometrical terms can imply interesting outcomesin a cosmological perspective as it does in the astrophysicallevel

2 The 119891(119877119879)=119877+1205721198772+2120574119879 Gravity

Following the steps in Harko et al [18] we can write the119891(119877 119879) = 119877 + 1205721198772 + 2120574119879 gravity total action as

119878 = 116120587 int (119877 + 1205721198772 + 2120574119879)radicminus1198921198894119909+ int119871119898radicminus1198921198894119909

(3)

in which 119892 is the determinant of the metric 119892120583] 119871119898 is thematter Lagrangian and we are working with natural units

By taking 119871119898 = minus119901 with 119901 being the pressure of theuniverse the variational principle applied in (3) yields thefollowing field equations

12119866120583] + 120572(119877120583] minus 1

4119877119892120583] + 119892120583]◻ minus nabla120583nabla])119877= 4120587119879120583] + 120574 [119879120583] + (119901 + 1

2119879)119892120583]] (4)

In (4) 119866120583] is the Einstein tensor 119877120583] is the Ricci tensor119879120583] = diag(120588 minus119901 minus119901 minus119901) 120588 is the matter-energy densityof the universe and 119879 = 120588 minus 3119901 Still in (4) it can bestraightforwardly seen that the limit 120572 = 120574 = 0 recovers theGR field equations

Also the above choice for thematter Lagrangian is usuallyassumed in the literature as it can be checked in Harko et al[18] Moraes et al [35 38] among many others

3 The 119891(119877119879)=119877+1205721198772+2120574119879 Cosmology

Let us assume a flat Friedmann-Robertson-Walker metricin the field equations above Such a substitution yields thefollowing Friedmann-like equations

( 119886119886)2 + 6120572119866 (119886 119886 119886 ) = 8120587

3 120588 + 120574 (120588 minus 1199013 ) (5)

119886119886 + 1

2 ( 119886119886)2 + 6120572119878 (119886 119886 119886 )

= minus4120587119901 + 12120574 (120588 minus 119901)

(6)

where we are using the following definitions

119866 (119886 119886 119886 ) equiv 2 ( 119886119886)2 [ 119886

119886 + 119886 minus 3

2 ( 119886119886)2] minus ( 119886

119886)2 (7)

119878 (119886 119886 119886 ) equiv 32 [( 119886

119886)4 + 119886119886]

+ 2 [ 119886 1198862 minus 3 ( 119886

119886)2 119886119886] +

119886 (8)

In the equations above 119886 = 119886(119905) is the scale factor anddots represent time derivatives Once again the limit 120572 = 120574 =0 retrieves the standard formalism



120588 = 119865 (120574)2 [120574119865 (119867) minus (8120587 + 120574)119866 (119867)] (9)

119901 = 119865 (120574) [4120587119865 (119867) + 3120574119866 (119867)] (10)



119866 (119867) equiv 3 1198672 + 6 [1198664 (119867) + 1198653 (119867)] 120572 (12)

119865 (119867) equiv 1198654 (119867) + 181205721198652 (119867) minus 1198651 (119867) + + 22 (13)

119866 (119867) equiv minus1198672 + + 3120572 [1198661 (119867) + 1198662 (119867) + 21198675 + 61198673] (14)


1198651 (119867) = 1198672 (1 + 4) (15)

1198652 (119867) = 1198674 minus 4119867 (16)

1198653 (119867) = 2 (1198675 + 31198673) (17)

1198654 (119867) = 31198672 + 2 (18)

1198661 (119867) = 1198674 + (3 + 7) (19)


1198663 (119867) = 3 (1 + 2) + 2 (21)

1198664 (119867) = minus21198674 minus 2 + 21198672 (22)


119886 (119905) = 119890119898119905119905119899 (23)



119867 = 119898 + 119899119905 (24)

119902 = minus 1198861198861198672 = minus1 + 119899

(119898119905 + 119899)2 (25)


minus08

minus06

minus04

minus02

00

02

04

q

0 1 2 3 4 5minus1z

m = 027 n = 075m = 028 n = 075m = 0 n = 075







119905 = 119899119898119882[119898

119899 ( 11 + 119911)1119899] (26)







120588 = 119865 (120574)2 [minus120574 (P1 +P2) +P3

+ 3 (8120587 + 120574) (P4 + P5 +P6)] (27)

119901 = 119865 (120574) [minus (G1 +G2 +G3 +G4) +G5] (28)





P3 equiv 3F2 (119905)2 minus 2119899F3 (119905)2 (31)





G3 equiv 1205721198992F3 (119905)2 [119866 (120574) + 18120574] + 120572119866 (120574) (36)

G4 equiv 120572119899F3 (119905)4 119866 (120574) (37)


4119866 (120574) + 3120574 + 12120587 (38)


F1 (119905) = 119898119905 + 119899 (39)

F2 (119905) = 119898 + 119899119905 (40)

F3 (119905) = 1119905 (41)


2 4 6 8 100t

002

004

006

008

010

012

014

016

= -01 = -01 = -005 = -01


2 4 6 8 100t

= -01 = -01 = -005 = -01

minus005

minus004

minus003

minus002

minus001

000

001

002

p


4 Energy Conditions







= -01 = -01 = -005 = -01

1 2 3 4 5 6 70t

minus06

minus04

minus02

00


10

05

000

5

10

minus4

minus2

0

t






15

10

05

000

5

10

minus4

minus2

0

+p

t


0

3

2

1

0

5

10

minus4

minus2

0

+3p

t






minus05

10

05

00

0

5

10

minus4

minus2

0

minusp

t






Data Availability




Acknowledgments



References










































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







120588 = 119865 (120574)2 [120574119865 (119867) minus (8120587 + 120574)119866 (119867)] (9)

119901 = 119865 (120574) [4120587119865 (119867) + 3120574119866 (119867)] (10)



119866 (119867) equiv 3 1198672 + 6 [1198664 (119867) + 1198653 (119867)] 120572 (12)

119865 (119867) equiv 1198654 (119867) + 181205721198652 (119867) minus 1198651 (119867) + + 22 (13)

119866 (119867) equiv minus1198672 + + 3120572 [1198661 (119867) + 1198662 (119867) + 21198675 + 61198673] (14)


1198651 (119867) = 1198672 (1 + 4) (15)

1198652 (119867) = 1198674 minus 4119867 (16)

1198653 (119867) = 2 (1198675 + 31198673) (17)

1198654 (119867) = 31198672 + 2 (18)

1198661 (119867) = 1198674 + (3 + 7) (19)


1198663 (119867) = 3 (1 + 2) + 2 (21)

1198664 (119867) = minus21198674 minus 2 + 21198672 (22)


119886 (119905) = 119890119898119905119905119899 (23)



119867 = 119898 + 119899119905 (24)

119902 = minus 1198861198861198672 = minus1 + 119899

(119898119905 + 119899)2 (25)


minus08

minus06

minus04

minus02

00

02

04

q

0 1 2 3 4 5minus1z

m = 027 n = 075m = 028 n = 075m = 0 n = 075







119905 = 119899119898119882[119898

119899 ( 11 + 119911)1119899] (26)







120588 = 119865 (120574)2 [minus120574 (P1 +P2) +P3

+ 3 (8120587 + 120574) (P4 + P5 +P6)] (27)

119901 = 119865 (120574) [minus (G1 +G2 +G3 +G4) +G5] (28)





P3 equiv 3F2 (119905)2 minus 2119899F3 (119905)2 (31)





G3 equiv 1205721198992F3 (119905)2 [119866 (120574) + 18120574] + 120572119866 (120574) (36)

G4 equiv 120572119899F3 (119905)4 119866 (120574) (37)


4119866 (120574) + 3120574 + 12120587 (38)


F1 (119905) = 119898119905 + 119899 (39)

F2 (119905) = 119898 + 119899119905 (40)

F3 (119905) = 1119905 (41)


2 4 6 8 100t

002

004

006

008

010

012

014

016

= -01 = -01 = -005 = -01


2 4 6 8 100t

= -01 = -01 = -005 = -01

minus005

minus004

minus003

minus002

minus001

000

001

002

p


4 Energy Conditions







= -01 = -01 = -005 = -01

1 2 3 4 5 6 70t

minus06

minus04

minus02

00


10

05

000

5

10

minus4

minus2

0

t






15

10

05

000

5

10

minus4

minus2

0

+p

t


0

3

2

1

0

5

10

minus4

minus2

0

+3p

t






minus05

10

05

00

0

5

10

minus4

minus2

0

minusp

t






Data Availability




Acknowledgments



References










































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








120588 = 119865 (120574)2 [minus120574 (P1 +P2) +P3

+ 3 (8120587 + 120574) (P4 + P5 +P6)] (27)

119901 = 119865 (120574) [minus (G1 +G2 +G3 +G4) +G5] (28)





P3 equiv 3F2 (119905)2 minus 2119899F3 (119905)2 (31)





G3 equiv 1205721198992F3 (119905)2 [119866 (120574) + 18120574] + 120572119866 (120574) (36)

G4 equiv 120572119899F3 (119905)4 119866 (120574) (37)


4119866 (120574) + 3120574 + 12120587 (38)


F1 (119905) = 119898119905 + 119899 (39)

F2 (119905) = 119898 + 119899119905 (40)

F3 (119905) = 1119905 (41)


2 4 6 8 100t

002

004

006

008

010

012

014

016

= -01 = -01 = -005 = -01


2 4 6 8 100t

= -01 = -01 = -005 = -01

minus005

minus004

minus003

minus002

minus001

000

001

002

p


4 Energy Conditions







= -01 = -01 = -005 = -01

1 2 3 4 5 6 70t

minus06

minus04

minus02

00


10

05

000

5

10

minus4

minus2

0

t






15

10

05

000

5

10

minus4

minus2

0

+p

t


0

3

2

1

0

5

10

minus4

minus2

0

+3p

t






minus05

10

05

00

0

5

10

minus4

minus2

0

minusp

t






Data Availability




Acknowledgments



References










































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






= -01 = -01 = -005 = -01

1 2 3 4 5 6 70t

minus06

minus04

minus02

00


10

05

000

5

10

minus4

minus2

0

t






15

10

05

000

5

10

minus4

minus2

0

+p

t


0

3

2

1

0

5

10

minus4

minus2

0

+3p

t






minus05

10

05

00

0

5

10

minus4

minus2

0

minusp

t






Data Availability




Acknowledgments



References










































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






minus05

10

05

00

0

5

10

minus4

minus2

0

minusp

t






Data Availability




Acknowledgments



References










































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery




























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





A Cosmological Scenario from the Starobinsky Model within the … · 2019. 7. 30. ·...

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Transcript of A Cosmological Scenario from the Starobinsky Model within the … · 2019. 7. 30. ·...