We prove that the field equations of the Starobinsky model for inflation in a
Friedmann-Lema\^{\i}tre-Robertson-Walker constitute an integrable system as the
field equations pass the singularity test. The analytical solution in terms of
a Painlev\'{e} Series for the Starobinsky model is presented for the case of
zero and nonzero spatial curvature. In both cases the leading-order term
describes the radiation era provided by the corresponding higher-order theory.Comment: 5 pages, references added, to appear in EPJ

Paliathanasis, Andronikos

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Abstract We prove that the field equations of the Starobinsky model for inflation in a Friedmann–Lemaître–Robertson–Walker metric constitute an integrable system. The analytical solution in terms of a Painlevé series for the Starobinsky model is presented for the case of zero and nonzero spatial curvature. In both cases the leading-order term describes the radiation era provided by the corresponding higher-order theory

Andronikos Paliathanasis

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Analytic solution of the Starobinsky model for inflation

We prove that the field equations of the Starobinsky model for inflation in a Friedmann–Lemaître–Robertson–Walker metric constitute an integrable system. The analytical solution in terms of a Painlevé series for the Starobinsky model is presented for the case of zero and nonzero spatial curvature. In both cases the leading-order term describes the radiation era provided by the corresponding higher-order theory

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Eur. Phys. J. C (2017) 77:438DOI 10.1140/epjc/s10052-017-5009-0Regular Article - Theoretical PhysicsAnalytic solution of the Starobinsky model for inflationAndronikos Paliathanasis1,2,a1 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile2 Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South AfricaReceived: 21 April 2017 / Accepted: 20 June 2017 / Published online: 1 July 2017© The Author(s) 2017. This article is an open access publicationAbstract We prove that the field equations of the Starobin-sky model for inflation in a Friedmann–Lemaître–Robertson–Walker metric constitute an integrable system. The analyti-cal solution in terms of a Painlevé series for the Starobinskymodel is presented for the case of zero and nonzero spa-tial curvature. In both cases the leading-order term describesthe radiation era provided by the corresponding higher-ordertheory.1 IntroductionIn the so-called modified/extended theories of gravity [1]new dynamical terms, of geometric origin, are introducedwhich force the evolution of the gravitational field equationsin order to explain various phenomena which were studiedby recent observations [2,3]. However, in the modified grav-itational theories the new terms increase the complexity ofthe field equations and even in the simplest models, such asthat of an isotropic and homogeneous universe, the existenceof an analytical solution is not obvious. Although numericalmethods can be applied to approximate the evolution of thefield equations, that is not sufficient for the complete studyof a theory; the analysis of the critical points it is not suf-ficient to provide us with information for the evolution of asystem far from the critical points. Consequently the exis-tence of analytical solutions for the field equations has leadto the application of various techniques from the analysis ofdynamical systems for the study of the integrability.1One of the simplest modifications of the Einstein–Hilbertaction which consider quantum corrections is the Starobinskymodel of inflation [9] with action integralS =∫d4x√−g(R + q R2) +∫d4x√−gLm, (1)1 For the application of the invariant transformations in modifiedtheories see for instance [4–8] and the references therein.a e-mail: anpaliat@phys.uoa.grwhere R2 describes the quantum-gravitational effects in theearly universe and Lm is the Lagrangian of the matter source.The latter action integral corresponds to the family of the so-called quadratic theories instance [10–12].The gravitational field equations are of fourth order and inthe case of a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe with line element2ds2 = −dt2 + a2(t)(dx2 + dy2 + dz2) (2)are calculated to be3(a˙a)2− 54q(a˙a)4+ 18q(2(a˙a)2a¨a−(a¨a)2+2(a˙a)2a(3))= ρm (3)and2a¨a+(a˙a)2+ 18q((a˙a)4+ a¨a2)+ 24q(a˙a2a(3)−3(a˙a)2a¨a)+ 12q a(4)a= −pm, (4)where ρm , pm are the energy density and the pressure of thematter source. In the case of vacuum the field equations (3)and (4) admit an unstable (special) nonsingular solution [9].Moreover, it is important to mention that, when the equationof state parameter of the matter source, ρm, pm , is that ofan ideal gas, i.e., pm = (γ − 1) ρm , then Eq. (4) can beintegrated to (3), while in general the conservation law ρ˙m +3H (ρm + pm) = 0 holds.The action integral (1) corresponds to the f (R) theo-ries of gravity [13], where f (R) = R + q R2, while amore general consideration of the Starobinsky model is the32 We have assumed that the lapse function in the FLRW line elementis constant, i.e., N (t) = 1 and a (t) denotes the scale factor.3 For reviews in f (R)-gravity see for instance [14,15], while someobservational constraints can be found in [16–19].123438 Page 2 of 4 Eur. Phys. J. C (2017) 77 :438f (R) = R + q Rn theory4 [26] or [27]. Exact solutionsof power-law f (R) theories can be found in [28,29]. Theimportance of the Starobinsky model (1) is that it providesan inflationary scenario which is favored by the observations[30]. Furthermore, it has been shown that various modelsof inflation are identical to the Starobinsky model when theinflationary phase takes place [31] whereas the Mixmasteruniverse provides nonchaotic trajectories [32].The introduction of a Lagrange multiplier in f (R) theo-ries [33] can be used to reduce the order of the theory froma fourth-order to a second-order theory by increasing at thesame time the number of degrees of freedom [34]. In partic-ular a new field is introduced which is equivalent to that ofa Brans–Dicke scalar field with zero Brans–Dicke parame-ter [14,35] the so-called O’Hanlon theory [36]. Thereforethe field equations in a FLRW background form a two-dimensional canonical Hamiltonian system which describesa particle moving in a flat space while the potential whichforces the evolution of the system is related with the formof the f (R) function. Because the scalar field description isthat of a Brans–Dicke field the theory is defined in the Jordanframe. Hence under a conformal transformation a minimallycoupled scalar field is defined and the theory is defined nowin the Einstein frame.5 Therefore the Starobinsky model canbe seen as a mechanical model providing a minimally scalarfield [39] to drive the inflationary phase of the universe; fora review see [40].By using the property that the field equations describe acanonical Hamiltonian system various functions f (R) havebeen determined in which the field equations admit conser-vation laws which are linear or quadratic in the momentum[8], while recently in [41] it was found that the cosmologicalmodel f (R) = R + q Rn passes the singularity test and isintegrable for some values of the power n. However, the casen = 2, which is that of the Starobinsky model, has been ruledout and the main reason is that for n = 2 the field equationsadmit singular special solutions following from the Rn term.This is in contrast to the Starobinsky model, in which the R2term provides a nonsingular solution as mentioned above.A specific f (R) theory provides a de Sitter universe ifthere exists R = R0 such that the Barrow–Ottewill [42] con-dition holds,R0 f ′ (R0) − 2 f (R0) = 0. (5)It is straightforward to see that, for arbitrary R0, that is, R0 →R, the latter condition can be seen as a first-order differential4 There are a plethora of physical theories which have been inspired bythe Starobinsky model of inflation such as in SUGRA or in other grav-itational theories, for instance see [20–25] and the references therein.5 For a discussion between these two frames see [37,38] and the ref-erences therein.equation with solution the quadratic function f (R) = f0 R2,where f0 is a constant of integration.2 Integrability of the field equationsIn the case of the vacuum the field equations in f (R) = R2theory are2a2a˙a(3) − 3(a˙)4 − aa¨(aa¨ − 2a˙2) = 0 (6)and2a3a(4) + 4a2a˙a(3) + 3 (a˙)4 + 3aa¨(aa¨ − 4(a˙)2) = 0. (7)The two equations are not independent and derivation of (6)gives the fourth-order equation (7). There are various waysin which Eq. (6) can be written as a first-order ordinary dif-ferential equation.6 If we select the new dependent variablew = 1ududv and independent variable v, where u = a˙, v = a,then Eq. (6) becomes the following Riccati equation:2dwdv+ 3w2 + 2wv− 3v2= 0 (8)with solution w (v) = v3−w0v(v3+w0) , where w0 is a constant ofintegration. Therefore it follows that H(t)H0 = (v20a−32 +a 32 ) 23 ,where H (t) = a˙a, and for initial conditions such that v0 = 0provides the closed-form solution a (t)  t−1. That is notthe unique case. In order to see that consider now the newvariables {x, y} = {H (t) , ddt (H (t))}; Eq. (6) takes the formof the linear equation2dydx− y + 6x2 = 0 (9)with solution y (x) = −2x2 +u1√x , that is,∫ dHu1√H−2a2 =(t − t0) where, in the limit u1 = 0, it gives H (t) = 12(t−t0) ,that is, a (t) = a0√(t − t0). This is an ideal gas solutionwhich mimics a radiation solution, while it is a singular (spe-cial) solution. This singular solution is used below in orderto prove the integrability of the Starobinsky model. The exis-tence of the radiation solution is not a surprise in the sensethat f (R)-gravity can always provide a radiation epoch in theevolution of the universe [43]. However, the radiation solu-tion has been investigated before in a higher-order theorywhich includes the Starobinsky term as also other terms fol-low from the Gauss–Bonnet invariant in [44–46]. Moreoverthe radiation solution in quadratic theories has been foundthat can describes a past isotropic singularity for the BianchiI universe [47].The method that we apply is that of the singularity anal-ysis and specifically we follow the ARS algorithm [48–50].6 The field equations (6), (7) admit as point symmetries the ∂t , t∂t anda∂a vector fields which form the {2A1 ⊗ A1} Lie algebra.123Eur. Phys. J. C (2017) 77 :438 Page 3 of 4 438Singularity analysis is a powerful method which has beenapplied in cosmological studies for the reconstruction of theanalytical solution of various models [41,51–54]. We omitthe properties of the singularity analysis and we refer thereader to the extended review [55].We continue by firstly applying the method for thequadratic theory f (R) = R2 and consider now Eq. (7).We find that the leading-order behavior is the power-lawsolution a (t) = a0τ 1/2, where τ = t − t0 and t0 denotes theposition of the singularity. The application of the ARS algo-rithm shows the resonances to be s1 = −1 , s2 = 0 , s3 = 32and s4 = 52 , which means that the analytic solution isexpressed by the right Painlevé series [56]a (t) = a0τ 12 + a1τ + a2τ 32 + a3τ 2 +∞∑i=4a4τ1+i2 , (10)where the constants of integration are a0, a3, a5 and theposition of the singularity is t0. However, with the use of(6) we find that a5 = 0, while the calculation of the firstcoefficient constants gives the solutiona (t) = a0τ 12 + a3τ 2 + 1932(a3)2a0τ72 + 17264(a3)3(a0)2 τ5+∞∑j=10a jτ1+ j2 . (11)For the field equations of the Starobinsky model we applythe same algorithm and we find the same resonances as thoseof the quadratic model, which means that the analytic solutionis given by Eq. (10) or specifically, by calculation of the firstnine coefficient constants, the solution isa (t) = a0τ 12 + a3τ 2 − a072q τ52 + a5τ 3 + 1932(a3)2a0τ72− 5252a3qτ 4 +(41259200q2+ a3a516a0)τ92+(17264(a3)3(a0)2 −a5132q)τ 5 +∞∑j=10a¯ jτ1+ j2 , (12)where the constants of integration are again the coefficientsa0, a3, a5 and the position of the singularity is t0, while theconstraint equation (3) gives a5 = 0 or, if we assume theexistence of a dust fluid, that is, pm = 0 and ρm = ρm0a−3,it follows that ρm0 = 3152 qa5 (a0)2. In the latter scenario itis important to mention that the term a0t1/2 describes theleading-order behavior.From the values of the resonances it is easy to see thatthe radiation solution is an unstable solution,7 while the field7 For a discussion of the relation between the values of the resonancesand the stability of the leading-order behavior see [57].equations of the Starobinsky model for inflation in a spa-tially flat FLRW spacetime pass the singularity test and areintegrable.3 DiscussionSingularity analysis is a powerful method to study the inte-grability of dynamical systems. However, it has a basic disad-vantage in that it is coordinate dependent. That is the reasonthat the Starobinsky model did not pass the singularity analy-sis in the consideration of [41]. The reason is that in the spaceof variables {a, R}, in which usually f (R)-gravity is referredto, the leading-order behavior, a (t) = a0t 12 , provides a sin-gular behavior for only one of the dynamical variables, whilefor the Ricci scalar it is a constant. However, we overpassedthat problem by working directly with the fourth-order dif-ferential equation and without using the Lagrange multiplier.We now consider the case of nonzero spatially curvedspacetime. Hence for the action integral (1) the field equa-tions are derived,ρm0a3= 3(a˙a)2− 54q(a˙a)4+ 18q(2(a˙a)2a¨a−(a¨a)2+ 2(a˙a)2a(3))+ k2a2− 6qka2(a¨a)+ qk2a4, (13)0 = 2 a¨a+(a˙a)2+ 18q((a˙a)4+ a¨a2)+ 24q(a˙a2a(3) − 3(a˙a)2a¨a)+ 12q a(4)a+ k6a2+ 2qka2((a˙a)2− 2 a¨a)− qk26a4,(14)where for the matter source we assumed that of a dust fluid.We apply the ARS algorithm and we find that the solutionis expressed again by the right Painlevé series (11) wherenow the coefficient constants depend also upon the curvaturek. For instance the first terms of the solution area (t) = a0τ 12 − k12a0 τ32 + a3τ 2 −(a072q+ k2288 (a0)3)τ52+ a5τ 3 +∞∑r=6a¯ jτ1+r2 (15)where from (13) it follows that ρm0 = 3152 qa5 (a0)2 +30a3qk.123438 Page 4 of 4 Eur. Phys. J. C (2017) 77 :438We conclude that the Starobinsky model for inflation in aFLRW spacetime with or without spatial curvature is an inte-grable system. Last but not least, from the singularity analy-sis we found that the radiation era is described by a unstablepoint which is in agreement with the dynamical analysis fora higher-order theory [44,45].Acknowledgements The author acknowledges financial support ofFONDECYT Grant No. 3160121 and thanks the Durban Universityof Technology for the hospitality provided while part of this work wasperformed.Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.References1. T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rep. 513, 1(2012)2. S. Capozziello, Int. J Mod. Phys. D 11, 483 (2002)3. K. Koyama, Rep. Prog. Phys. 79, 046902 (2016)4. S. Capozziello, E. Piedipalumbo, C. Rubano, P. Scudellaro, Phys.Rev. D. 80, 104030 (2009)5. Y. Zhang, Y.-G. Gong, Z.-H. Zhu, Phys. Lett. B 688, 13 (2010)6. B. Vakili, Phys. Lett. B 664, 16 (2008)7. A. Paliathanasis, M. Tsamparlis, S. Basilakos, Phys. Rev. D 84,123514 (2011)8. A. Paliathanasis, Class. Quantum Gravity 33, 075012 (2016)9. A.A. Starobinsky, Phys. Lett. B 91, 99 (1980)10. H. Nariai, K. Tomita, Prog. Theor. Phys. 46, 776 (1971)11. G.V. Bicknell, J. Phys. A.: Math. Nucl. Gen. 7, 1061 (1974)12. J.D. Barrow, Nucl. Phys. B 296, 679 (1988)13. H.A. Buchdahl, Mon. Not. R. Astron. Soc. 150, 1 (1970)14. T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82, 451 (2010)15. S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011)16. J. Santos, J.S. Alcaniz, F.C. Carvalho, N. Pires, Phys. Lett. B 669,14 (2008)17. F.C. Carvalho, E.M. Santos, J.S. Alcaniz, J. Santos, JCAP 0806,008 (2008)18. Y.-S. Song, H. Peiris, W. Hu, Phys. Rev. D 76, 063517 (2007)19. R. Nunes, S. Pan, E.N. Saridakis, E.M. Abreu, JCAP 1701, 008(2017)20. F. Farakos, A. Kehagias, A. Riotto, Nucl. Phys. B. 876, 187 (2013)21. R. Gannouji, H. Nandan, N. Dadhich, JCAP 11, 051 (2011)22. G.K. Chakravarty, S. Mohanty, Phys. Lett. B 746, 242 (2015)23. A. Addazi, M.Y. Khlopov, Phys. Lett. B 766, 17 (2017)24. K. Kamada, J. Yokoyama, Phys. Rev. D 90, 103520 (2014)25. I. Garg, S. Mohanty, Phys. Lett. B 751, 7 (2015)26. S.M. Carroll, V. Duvvuri, M. Trodden, M.S. Turner, Phys. Rev. D70, 043528 (2004)27. Q.-G. Huang, JCAP 02, 035 (2014)28. T. Clifton, J.D. Barrow, Class. Quant. Grav. 23, 2951 (2006)29. T. Clifton, J.D. Barrow, Phys. Rev. D 72, 103005 (2005)30. P.A.R. Ade et al. (Planck 2015 Collaboration), A&A 594, A20(2016)31. A. Kehagias, A.M. Dizgah, A. Riotto, Phys. Rev. 89, 043527 (2014)32. R. Moriconi, G. Montani, S. Capozziello, Phys. Rev. D 90, 101503(2014)33. S. Capozziello, J. Matsumoto, S. Nojiri, S.D. Odintsov, Phys. Lett.B 693, 198 (2010)34. B. Whitt, Phys. Lett. B 145, 175 (1984)35. A. De Felice, S. Tsujikawa, Living Rev. Relativ. 13, 3 (2010)36. J. O’Hanlon, Phys. Rev. Lett. 29, 137 (1972)37. N. Banerjee, B. Majumder, Phys. Lett. B 754, 129 (2016)38. M. Postma, M. Volponi, Phys. Rev. D 90, 103516 (2014)39. J.D. Barrow, S. Cotsakis, Phys. Lett. B 214, 515 (1988)40. K. Bamba, S.D. Odintsov, Symmetry 7, 220 (2015)41. A. Paliathanasis, P.G.L. Leach, Phys. Lett. A 380, 2815 (2016)42. J.D. Barrow, A.C. Ottewill, J. Phys. A: Math. Gen. 16, 2757 (1983)43. L. Amendola, D. Polarski, S. Tsujikawa, Phys. Rev. Lett. 98,131302 (2007)44. S. Cotsakis, G. Kolionis, A. Tsokaros, Phys. Lett. B 721, 1 (2013)45. S. Cotsakis, S. Kadry, G. Kolionis, A. Tsokaros, Phys. Lett. B 755,387 (2016)46. J.D. Barrow, J. Middleton, Phys. Rev. D 75, 123515 (2007)47. J.D. Barrow, S. Hervik, Phys. Rev. D 74, 124017 (2006)48. M.J. Ablowitz, A. Ramani, H. Segur, Lettere al Nuovo Cimento23, 333 (1978)49. M.J. Ablowitz, A. Ramani, H. Segur, J. Math. Phys. 21, 715 (1980)50. M.J. Ablowitz, A. Ramani, H. Segur, J. Math. Phys. 21, 1006 (1980)51. S. Cotsakis, J. Demaret, Y. De Rop, L. Querella, Phys. Rev. D 48,4595 (1993)52. J. Miritzis, P.G.L. Leach, S. Cotsakis, Gravit. Cosmol. 6, 282 (2000)53. A. Paliathanasis, J.D. Barrow, P.G.L. Leach, Phys. Rev. D 94,023525 (2016)54. J. Latta, G. Leon, A. Paliathanasis, JCAP 1611, 051 (2016)55. A. Ramani, B. Grammaticos, T. Bountis, Phys. Rep. 180, 159(1989)56. M.R. Feix, C. Géronimi, L. Cairó, P.G.L. Leach, R.L. Lemmer,S.É. Bouquet, J. Phys. A: Math. Gen. 30, 7437 (1997)57. A. Paliathanasis, P.G.L. Leach, Phys. Lett. A 381, 1277 (2017)123

Analytic Solution of the Starobinsky Model for Inflation

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