Phase space of modified Gauss-Bonnet gravity. Sante Carloni ∗ Centro Multidisciplinar de Astrofísica - CENTRA, Instituto Superior Tecnico - IST, Universidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001, Portugal José P. Mimoso † Faculdade de Ciências, Departamento de Física & Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, Ed. C8, Campo Grande, 1749-016 Lisboa, Portugal (Dated: January 3, 2017) WeinvestigatetheevolutionofnonvacuumFriedmann-Lemaître-Robertson-Walker(FLRW)with any spatial curvature in the context of Gauss-Bonnet gravity. The analysis employs a new method whichenablesustoexplorethephasespaceofanyspecifictheoryofthisclass. Weconsiderseveral examples, discussing the transition from a decelerating into an acceleration universe within these theories. We also deduce from the dynamical equations some general conditions on the form of the action which guarantee the presence of specific behaviours like the the emergence of accelerated expansion. As in f(R) gravity, our analysis shows that there is a set of initial conditions for which thesemodelshaveafinitetimesingularitywhichcanbeanattractor. Thepresenceofthisinstability alsointheGauss-Bonnetgravityistobeascribedtothefourth-orderderivativeinthefieldequations, i.e., is the direct consequence of the higher order of the equations. 7 1 0 I. INTRODUCTION 2 n The search for a purely geometrical description of dark energy has led the research community to the analysis of a a number of possible extensions of General Relativity (GR). From the now “classical” f(R) [1] and scalar tensor [2] J theories to more complex extensions which involve the presence of torsion or of more complicated invariants in the 1 gravitational actions [3]. These theories have been thoroughly studied and have revealed interesting features as well as a number of problems connected to different kinds of instabilities. Among those theories which include terms of ] c fourth order in the derivative of the metric, the so-called modified Gauss-Bonnet gravity has been shown to have q interesting properties. - r The idea of these theories comes from the concept of brane words, which are in turn derived from string theories. g In five dimensions one considers a Gauss-Bonnet (GB) term which is normally non minimally coupled with a scalar [ (modulus or dilaton) field. One can shown that the induced theory on four dimensions is able to generate the de 1 Sitter solution and has other relevant cosmological properties [4, 5]. v In[6]anewclassoftheoriesofgravity,dubbedGauss-Bonnetgravity,wasproposedinwhichtheGauss-Bonnetterm 1 appears in four dimensions. As it is well known, the Gauss-Bonnet invariant is a total divergence in four dimensions 3 and therefore a linear GB term would be irrelevant in this case. The new class of theories overcomes this issue 2 introducing in the Hilbert-Einstein action a generic non linear function of the Gauss-Bonnet term. The cosmology 0 of Gauss-Bonnet gravity has been thoroughly studied [7–10]. Some of the members of this class of theories were 0 . shown to be able to pass the Solar System tests and to possess cosmological solutions producing cosmic acceleration 1 [11–13]. The same authors also found that linear cosmological pertubations of a spatially flat background contains 0 some instabilities [14]. Although this is clearly a problem, one must remember that the dynamics of higher order 7 1 theories depends on the value of the spatial curvature in a more complex way that standard (GR). This means that : evenifk isalmostzero,asrecentlyfoundbythePlanckcollaboration[15],thedynamicalfeaturesofthesemodelscan v be completely different, especially at perturbative level (see eqs. (13) below for a glimpse of this). This fact justifies i X a further analysis of these theories which takes into account spatial curvature. r Gauss-Bonnet gravity, like many other extensions of GR, has a structure that makes it difficult to understand the a physics of thesemodels. In thisrespect, the use ofsemi quantitative techniques, likethe dynamical systemsapproach (DSA) [16], can help in clarifying details of the dynamics of cosmological models based on these theories and other modifications of GR [17] which are not obvious. The use of DSA for this purpose is now fairly standard, however the analysisisoftenlimitedtobasicaspectsofthemethod, likeforexamplethechoiceofvariablesandtheclosuredothe dynamical system. Recently, a new strategy to alleviate these problems was proposed in the context of f(R)-gravity ∗ E-mail: [email protected] † E-mail: [email protected] 2 [18]. With respect to the previously proposed approaches, the new method has the advantage to be applicable to any form of the function f and leads to a clearer (albeit not complete) understanding of expanding cosmologies. In this work, we will propose a similar technique to treat Gauss Bonnet cosmologies in which the action is the sum oftheHilbert-Einsteintermandageneric(nonlinear)functionoftheGaussBonnetinvariant. Asinthecaseoff(R) gravity the method will allow to treat any forms of the function f and will enable us to characterise the nature of the attractors (if any) for the cosmologies Gauss-Bonnet gravity. We will apply this technique to a number of versions of this theory that have been deemed interesting and/or compatible with the present data. The results of this study will be the starting point for a more complete analysis of the dynamics of linear perturbations, which will be pursued elsewhere. Unless otherwise specified, natural units ((cid:126) = c = k = G = 1) will be used throughout this paper, Latin indices B run from 0 to 3. The symbol ∇ represents the usual covariant derivative and ∂ corresponds to partial differentiation. We use the (+,−,−,−) signature and the Riemann tensor is defined by Ra =Wa −Wa +We Wa −Wf Wa , (1) bcd bd,c bc,d bd ce bc df where the Wa are the Christoffel symbols (i.e. symmetric in the lower indices), defined by bd 1 Wa = gae(g +g −g ) . (2) bd 2 be,d ed,b bd,e The Ricci tensor is obtained by contracting the first and the third indices R =gcdR . (3) ab acbd Finally the Hilbert–Einstein action in the presence of matter is given by (cid:90) √ (cid:20) 1 (cid:21) S = dx4 −g R+L , (4) 2κ m where κ=8π and has the dimension of the inverse of a length square. II. BASIC EQUATIONS The Action for modified Gauss-Bonnet gravity reads (cid:90) √ (cid:20)R (cid:21) S = d4x −g +f(G) +S (gµν,ψ), (5) 2κ M where S (gµν,ψ) is the matter action and ψ collectively denotes the matter fields. The Gauss-Bonnet invariant is M defined as G ≡R2−4R Rµν +R Rµναβ. (6) µν µναβ Varying the action with respect to the metric yields the field equations: G +8(cid:2)R +R g −R g −R g +R g + R(g g −g g )(cid:3)∇ρ∇σf(cid:48)+(Gf(cid:48)−f)g =κT , (7) µν µρνσ ρν σµ ρσ νµ µν σρ µσ νρ 2 µν σρ µσ νρ µν µν where the prime represnts the derivative with respect to G and G =R −(1/2)Rg is the Einstein-tensor. µν µν µν The matter energy-momentum tensor is defined as usual as √ 2 δ( −gL ) T =−√ m . (8) µν −g δ(gµν) Our treatment will consider Friedmann Lemaître Robertson Walker (FLRW) metric: (cid:20) dr2 (cid:21) ds2 =−dt2+a2(t) +r2(dθ2+sin2θdφ2) , (9) 1−kr2 where a is the scale factor and k the spatial curvature. We also assume that the cosmic fluid is a prefect fluid with equation of state p=wµ with 0≤w ≤1, where µ and p, respectively are the energy density and pressure measured by the co-moving observer, and where we assume w to be constant. In this metric, the Gauss-Bonnet invariant is (cid:18) (cid:19) k G =24 H2+ (H˙ +H2) (10) a2 3 where H = a˙ is the Hubble factor. This expression is particularly important because it connects the sign of G to the a sign of the deceleration factor. For k ≥ 0, any cosmology which transits between acceleration and deceleration will have to change the sign of G. If k <0, this is not necessarily true. In this respect Gauss-Bonnet cosmology depends in an even more crucial way on the spatial curvature than GR. III. THE DYNAMICAL SYSTEMS APPROACH. First, we introduce a constant χ such that the products Rχ and Gχ2 are dimensionless and we redefine all the 0 0 0 constants appearing in the action (5) to obtain (cid:90) √ S = (cid:8)χ R+f(cid:0)Gχ2(cid:1)+ L (g ,Φ)(cid:9) −g d4x (11) 0 0 m µν where now all the constants (with the exception of χ ) in the action are assumed dimensionless. 0 Now defining the parameters ... H˙ H¨ H˙2 H H˙3 H˙H¨ q= , j= − , s= +3 −4 . (12) H2 H3 H4 H4 H6 H5 the cosmological equations can be written as 3χ (cid:18)H2+ k (cid:19)−µ+f −24H2(q+1)(cid:18)H2+ k (cid:19)f(cid:48)+(cid:26)3226H4(cid:0)j+q2−2(cid:1)k2 0 a2 a2 a4 (cid:27) k +3227H6(5j+2q(3+1)−9) +2634H8[j+q(3q+4)] f(cid:48)(cid:48) =0 (13) a2 (cid:18) (cid:19) 1 f k χ H2(q+1)+ µ(3w+1)+ −24H2(q+1) +H2 f(cid:48) 0 3 3 a2 (cid:26) (cid:18) (cid:19)(cid:18) (cid:19) k k + 2532H4 +H2 +9H2 s+2532H8[j(92q+29)+q2(87q+131)−28q] a2 a2 k2 k (cid:27) +2532H4[j(4q−3)+(q−3)q2+6(q−1)] +2632H6[j(24q−11)+q(2q(5q−3)−31)+25] f(cid:48)(cid:48) a4 a2 +(cid:26)2833H6(cid:0)j+q2−2(cid:1)2 k3 +2833H8(cid:0)j+q2−2(cid:1)[11j+q(15q+8)−18]k2 a6 a4 (cid:27) k +2833H10[j+q(3q+4)][19j+q(21q+4)−36] +2835H12[j+q(3q+4)]2 f(cid:48)(cid:48)(cid:48) =0 (14) a2 Here the non trivial role of the spatial curvature is particularly evident: the k = 0 equations are very different from the full ones. In order to analyse the phase space let us define the expansion normalized variables G k µ G= , K= , Ω= , 3H4 a2H2 3H2χ0 (15) J=j Q=q, A=χ H2. 0 We also define the logarithmic (dimensionless) “time variable” N =lna. Note that in choosing this time variable we are assuming that we represent the phase space for H >0, i.e., we are considering only expanding cosmologies. It is important to stress that, since G has the same sign of G, a sufficient condition for the transition between deceleration and deceleration can only be obtained if G=0 is not an invariant submanifold of the phase space. The phase space associated to the equations (13) is then described by the autonomous system dG G(cid:18) G (cid:19) 8[K+X−GY−Ω+1] = 8− − , dN 2 K+1 (K+9)Z dΩ (cid:20) G (cid:21) =Ω +(3w+1) , dN 4(K+1) (16) dK KG =− dN 4K+4 dA A(cid:18) G (cid:19) = −8 , dN 4 K+1 4 together with the two constraints 0=(9+K)(cid:2)J+K(cid:0)J+Q2−2(cid:1)Z+Q(3Q+4)(cid:3)+K+X−GY−Ω+1 (17) G=8(1+K)(1+Q). (18) In all the equations above we have defined f X= (19) 3χ H2 0 f(cid:48)H2 Y = (20) χ 0 326H6f(cid:48)(cid:48) Z= (21) χ 0 Note that choosing the variables above, the system is singular for K+1 = 0. Such singularity is originated by our choice of coordinates: selecting for example the variable Q instead of G it can be eliminated to obtain the system dQ (Q2−2)K+Q(3Q+4) K−Ω+1+X−8(K+1)(Q+1)Y =−Q2− − , dN (K+1) (K+1)(K+9)Z dΩ =−Ω[2Q+3(w+1)], dN (22) dK =−2K(1+Q) dN dA =AQ, dN where now X,Y,Z are function of Q,K. The two systems are equivalent when one is away from K = −1. Since in the examples we consider there is no special point for K=−1, and moreover there is no appreciable difference in the structure of the fixed points, we will perform the analysis using Eqs. (16). However one must stress that K = −1 is nevertheless of interest because, since both G and its derivatives become zero, this case represents a state in which the theory effectively becomes of order two. Therefore the system (22) can be useful to explore this property of Gauss-Bonnet gravity. The solutions associated to the fixed points can be derived using the modified Raychaudhury equation (the second of the Eqs. (13)) in the variables (12). In a fixed point we have 1 d3H =s =− 1 (cid:110)2(K +9)T (cid:2)J +K (cid:0)J +Q2−2(cid:1)+Q (3Q +4)(cid:3)2+2Q H dN3 ∗ (K +1)(K +9)Z ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (23) +(1+3w)Ω +2X −2G Y +2+29Z J +2Z Q [46J +K (2J (K +12)−3K −31)] ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ +K Z [50−22J −3J K +6K ]+Z [K (K +20)+87]Q3+Z [131−3K (K +4)]Q2−28Z Q (cid:9) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ where 2832H10f(cid:48)(cid:48)(cid:48) T= (24) χ 0 and an asterisk represents the value of a quantity at a fixed point. The general solution for the above equation is (cid:40)H +H N +H N2 s∗ =0 0 1 2 H = H0e−pN +ep2N (cid:16)H1cospN2√3 +H2sinpN2√3(cid:17) s∗ (cid:54)=0 (25) √ where p = −3s∗ and the H are integration constants. Using the definition of N the above equations translate in i equations for the scale factor a˙ (cid:40)H0+H1lna+H2(lna)2 s∗ =0 = (cid:104) (cid:16) √ (cid:17) (cid:16) √ (cid:17)(cid:105) (26) a H0a−p+ap/2 H1cos p23lna +H2sin p23lna s∗ (cid:54)=0 In the fist case the equation can be solved exactly to give (cid:40)(cid:112)4H H −H2 (cid:20)1 (cid:113) (cid:21) H (cid:41) a(t)=a exp 2 0 1 tan (t−t ) 4H H −H2 − 1 . (27) 0 2H 2 0 2 0 1 2H 2 2 5 which was already found in the case of f(R)-gravity [18]. A major problem one finds when these solutions are found is associated with the determination of the integration constants. Such constants are important as they concur to determine the nature of the solution. For example, the solution given by (27) will have a finite time singularity if 4H H −H2 > 0. One way in which the value of these 2 0 1 constants can be inferred is to consider the value of the initial condition of the orbits that approach this point. IV. EXAMPLES Wewillnowapplythemachinerypresentedabovetoanumberofphysicallyrelevanttheories. Wewillthencompare our results with those found in literature. A. Gn-gravity As a first example, let us consider the model f =αχ2nGn (28) 0 we have X=3n−1αA2n−1Gn, (29) Y =3n−1nαA2n−1Gn−1 (30) Z=26 3n−1(n−1)nαA2n−1Gn−2 (31) T=28 3n−1(n−2)(n−1)nαA2n−1Gn−3. (32) where the constraint (18) holds. The dynamical equations read dG (cid:20) (cid:18) 1 1 (cid:19) 22−n31−nA1−2n(K−Ω+1)G1−n(cid:21) =G 4−G + + , dN K+1 4n(K+9) n(1−n)α(K+9) dΩ (cid:20) G (cid:21) =Ω +(3w+1) , dN 4(K+1) (33) dK KG =− dN 4(K+1) dA A(cid:18) G (cid:19) = −8 , dN 4 K+1 The system admits four invariant submanifolds (K = 0,Ω = 0,A = 0,G = 0). These results imply that no global attractor exists in general. The fixed subspaces for this system are indicated in Table I. We have a line of fixed points L for K = K ,Ω = 0 0,A = 0,G = 0 for 2+3n < 0. Other fixed points exist only at specific intervals of n. For example, the point A is present for 2−3n>0, whereas B is present only for (n−1)α>0. The points on the line L are associated with the solution given by the equation a˙ H (cid:20) (cid:18)1√ (cid:19) (cid:18)1√ (cid:19)(cid:21) = 1 +a1/2 H sin 3loga +H cos 3loga . (34) a a 2 2 3 2 For point A we have (cid:34) (cid:32) √ (cid:33) (cid:32) √ (cid:33)(cid:35) a˙ √1−36n (1−36n) 3 (1−36n) 3 a =H1a 32933n2+1 +a1/2 H2sin √321033n2+1loga +H3cos √321033n2+1loga . (35) whereas B corresponds to solution (27). A numerical integration of the equation (34) and (35) is given in Figures 1 and 3. The stability of all the fixed points can be deduced using the standard Hartmann-Grobmann theorem [19] and it is indicated in Table I. The points on L are all unstable and A and B can be attractors or saddles depending on the values of n and α. 6 Because of the presence of the invariant submanifold there is no orbit that can represent a transition between deceleratedandacceleratedcosmologies. Inthisrespectthistheoryisnotusefultomodeldarkenergyandcouldonly be used as patch model for eternal inflation. Even in this case, however, these models can have the same issues of f(R) gravity, i.e. the onset of a finite time singularity. Our conclusions are consistent with the results in [11, 13]. a 15 10 5 t 1 2 3 4 Figure 1: Numerical solution of equation (34). The constants H have all been chosen to be one and the initial condition is i a(0)=0.01. a 40 30 20 10 t 5 10 15 20 Figure 2: Numerical solution of equation (35). The constants H have all been chosen to be one and the initial condition is i a(0)=0.01. B. The De Felice-Tsujikawa models. In Ref. [12] some particular forms of f(G) were proposed which, based on exact and perturbative arguments, were considered to be cosmologically viable. In our formulation such models can be written as f(G)=αχ2Garctan(cid:0)χ2G(cid:1)−βχ , 0 0 0 f(G)=αχ2Garctan(cid:0)χ2G(cid:1)−βχ −γln(cid:0)1+χ2G(cid:1), (36) 0 0 0 0 f(G)=αln(cid:2)cosh(cid:0)χ2G(cid:1)(cid:3)−βχ , 0 0 whereβ,differentlyformαandγ,isadimensionalconstant. Alltheseconstantareconsideredpositive. Thepresence of β requires the introduction of a new variable B=β/H2 whose dynamic equation is dB 3B(G−8K−8) =− . (37) dN 4(K+1) 7 a 2.5 2.0 1.5 1.0 0.5 t 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 3: Numerical solution of equation (27). The constants H have all been chosen in such a way that 4H H −H2 > 0 i 0 2 1 and the initial condition is a(0) = 0.01. The solution presents a finite time singularity. If the above condition is violated the solution approaches a static universe as in the case of (34) an (35). Table I: Fixed subspaces of f(G) = αGn and their associated solutions. Here a = H +H +H , A is the solution of the 0 1 2 3 ∗ algebraic equation A stays for attractor, F for attractive focus, S for saddle. A Coordinates {G,K,Ω,A} Scale Factor Existence Stability L {0,K ,0,0} Solution of (34) 2−3n>0 Unstable 0 A (cid:110) 2532 ,0,0,0(cid:111) Solution of (35) 2−3n>0 A n<0 36n−1 S 0<n<2/3 A 72 ≤n< 1 B (cid:110)8,0,0,(cid:2)3n−18n(n−1)α(cid:3)1−12n(cid:111) (27) (n−1)α>0 F 1036<9 n< 722 A 1369 S otherwise By including this additional equation in the system (16), we are ready to explore these models. 1. Moldel 1: f(G)=αχ2Garctan(cid:0)χ2G(cid:1)−βχ 0 0 0 For the first form of f above we have X=αAGarctan(cid:0)3A2G(cid:1)−B, (38) Y =αA(cid:20)arctan(cid:0)3A2G(cid:1)+ 3A2G (cid:21) (39) 9A4G2+1 α327A3 Z= (40) (9A4G2+1)2 α21133A7G T=− . (41) (9A4G2+1)3 8 and the dynamical equations read dG = 1G(cid:18)8− G (cid:19)+ 9A4G2+1 (cid:8)1+K−Ω−B−3A3G2[α+3A(B+Ω)]+9A4G2(K+1)(cid:9), dN 2 K+1 48αA3(K+9) dΩ (cid:20) G (cid:21) =Ω +(3w+1) , dN 4(K+1) dK KG =− , (42) dN 4(K+1) dA A(cid:18) G (cid:19) = −8 , dN 4 K+1 dB 3B(G−8K−8) =− . dN 4(K+1) The system admits three invariant submanifolds (K = 0,Ω = 0,A = 0). The last one, depending on the values of the parameters, can be singular. These results imply that no global attractor exists in general, but they allow the possibility of a transition between decelerated and accelerated cosmologies. The system (42) presents only a two dimensional fixed point subspace with coordinates (cid:26) 192αA3 (cid:27) S ={G ,K ,Ω ,A ,B }= 8,0,0,A ,1− ∗ (43) ∗ ∗ ∗ ∗ ∗ ∗ 576A4+1 ∗ which corresponds to the solution (27). Note that, by definition B > 0 and this implies that this subspace can exist only for α> 576A4∗+1. 192A3 ∗ The stability of the fixed points on the surface S can be obtained using the Hartmann-Grobmann theorem [19]. A plot of the real part of the eigenvalues is given in Figure 4. It is evident that the point on S can be saddles or attractors depending on the values of α. An easier way to visualize the phase space dynamics is to define the variable 3A(cid:0)576A4−192αA3+1(cid:1) Y=B− (44) A2(576A4+1) and analyse the corresponding dynamical system: dG = 1G(cid:18)8− G (cid:19)+ 9A4G2+1 (cid:8)1+K−Ω−3A3G2(α+3AΩ)+9A4G2(K+1)(cid:9) dN 2 K+1 48αA3(K+9) (cid:0)9A4G2+1(cid:1)2 (cid:18)AY+3 12 (cid:19) + − , A(K+9) 48αA3 576A4+1 dΩ (cid:20) G (cid:21) =Ω +(3w+1) , dN 4(K+1) (45) dK KG =− , dN 4(K+1) dA A(cid:18) G (cid:19) = −8 , dN 4 K+1 dY = 3(−G+8K+8) (cid:104)(cid:0)576A4+1(cid:1)2(AY+2)−192αA3(cid:0)576A4+5(cid:1)(cid:105). dN 4A(576A4+1)2(K+1) which presents only one line of fixed points ({8,0,0,A ,0}) rather than a surface and shows clearly that there is no ∗ heteroclinic orbit that connects the unstable points to the stable points on the line. This implies that there cannot be an orbit in which an unstable phase characterised by (27) is followed by a stable phase characterized by the same solution. This implies that the theory has an accelerated expansion attractor, but in order to conclude that a transition deceleration/accelerationispossibleoneshouldlookatthenumericalevolutionoftheorbits. Thisresultisconsistent with what was found in [13], in which one of such transitions has been observed by numerical integration. Thereisnogeneralwaytodeducethesizeofthesetofinitialconditionsthatleadstocosmicacceleration,however, by inspection, we can deduce that the smaller the value of the parameter α the larger the initial conditions that lead to the cosmic acceleration transition. 9 In this model the presence of the cosmological constant does not seem to change too much the dynamics. When B = 0 the fixed subspace reduces to a one dimensional space of fixed points associated to the solution (27). Their character can be once again either that of a saddle or of an attractor, depending on the value of α. Again the bigger the parameter α the lower the number of stable point in the one dimensional space. The only concrete difference between this model and the more general one is the fact that the equations to solve are easier in the presence of the cosmological constant. 30 20 10 * 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -10 -20 -30 Figure 4: Plot of the real part of the eigenvalues of the fixed points of the line L for f(G)=αχ2Garctan(cid:0)χ2G(cid:1)−βχ2. Here α 0 0 0 has been chosen to be 3 and w=0. 2. Model 2: f(G)=αχ2Garctan(cid:0)χ2G(cid:1)−γln(cid:0)1+χ2G(cid:1)−βχ 0 0 0 0 For this form of f above we have γlog(cid:0)9A4G2+1(cid:1) X=− +αAGtan−1(cid:0)3A2G(cid:1)−B, (46) 3A Y =αAtan−1(cid:0)3A2G(cid:1)+ 3A3G(α−2γ) (47) 9A4G2+1 384A3(cid:0)α+γ(cid:0)9A4G2−1(cid:1)(cid:1) Z= (48) (9A4G2+1)2 27648A7G(cid:0)2α+9A4γG2−3γ(cid:1) T=− . (49) (9A4G2+1)3 and the dynamical equations read dG = 1G(cid:18)8− G (cid:19)+ 9A4G2+1 (cid:8)1+K−Ω−B−3A3G2[α+3A(B+Ω)]+9A4G2(K+1)(cid:9) dN 2 K+1 48αA3(K+9) γ(cid:0)9A4G2+1(cid:1)2 + ln(cid:0)9A4G2+1(cid:1), 144A4(K+9)(α−γ+9A4γG2) dΩ (cid:20) G (cid:21) =Ω +(3w+1) , dN 4(K+1) (50) dK KG =− , dN 4(K+1) dA A(cid:18) G (cid:19) = −8 , dN 4 K+1 dB 3B(G−8K−8) =− . dN 4(K+1) 10 The system above differs for the previous one only by just one term. It has essentially the same general features in terms of invariant submanifold and fixed points. The two dimensional fixed point subspace has coordinates (cid:40) 192αA3 (cid:34) 384A3 ln(cid:0)576A4+1(cid:1)(cid:35)(cid:41) L={G ,K ,Ω ,A ,B }= 8,0,0,A ,1− ∗ +γ ∗ − ∗ (51) ∗ ∗ ∗ ∗ ∗ ∗ 576A4+1 576A4+1 3A ∗ ∗ ∗ the solution associate to these points is (27). Using the Hartmann-Grobmann theorem it is easy to prove that, also in this case, the fixed points on the line can be either attractors or saddles, with the difference that now the stability depends on γ other than α. Therefore the same picture of the previous model seems to emerge in this case. The logarithmic correction seems to be irrelevant on Friedmannian dynamics, but will surely be relevant at perturbative level. 3. Model 3: f(G)=αln(cid:2)cosh(cid:0)χ2G(cid:1)(cid:3)−βχ 0 0 For this form of f above we have X= α ln(cid:2)cosh(cid:0)3A2G(cid:1)(cid:3)−B, (52) 3A Y =αAtanh(cid:0)3A2G(cid:1), (53) Z=3α26A3sech2(cid:0)3A2G(cid:1), (54) T=−2932αA5tanh(cid:0)3A2G(cid:1)sech2(cid:0)3A2G(cid:1) . (55) and the dynamical equations read dG 1 (cid:18) G (cid:19) cosh2(cid:0)3A2G(cid:1)(cid:8)3A(B−K+Ω−1)−αln(cid:2)cosh(cid:0)3A2G(cid:1)(cid:3)(cid:9) = G 8− + dN 2 K+1 72αA4(K+9) Gsinh(cid:0)3A2G(cid:1)cosh(cid:0)3A2G(cid:1) + 24A2(K+9) dΩ (cid:20) G (cid:21) =Ω +(3w+1) , dN 4(K+1) (56) dK KG =− , dN 4(K+1) dA A(cid:18) G (cid:19) = −8 , dN 4 K+1 dB 3B(cid:18) G (cid:19) =− −8 . dN 2 K+1 The system admits again three invariant submanifolds: K = 0,Ω = 0,A = 0, and the latter can be singular. These results imply that, in general, no global attractor exists also for this model. The system (56) presents only a two dimensional fixed point subspace with coordinates (cid:40) αlog(cid:0)cosh(cid:0)24A2(cid:1)(cid:1)(cid:41) L={G ,K ,Ω ,A ,B }= 8,0,0,A ,1−8αA tanh(cid:0)24A2(cid:1)+ ∗ (57) ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ 3A ∗ as in the case of Model 2, the solution associated to this fixed point is (27). The stability of the fixed points is of the same type to the two previous models, depending essentially on the value of the variable α. In Figure 5 we represent a plot of the eigenvalues of these points. Inspiteofthedifferentanalyticalform,theobserveddynamicsessentiallyisthesameasintheprevioustwomodels. This should not be surprising as all these models have been explicitly designed to have the same characteristics. Our results confirm this fact. As for model 2, differences between the behaviour of these cosmologies will likely become evident at perturbative level.