A new approach to the analysis of the phase space of f(R)-gravity
PPrepared for submission to JCAP
A new approach to the analysis ofthe phase space of f ( R ) -gravity. S. Carloni a a Centro Multidisciplinar de Astrofisica - CENTRA, Instituto Superior Tecnico - IST, Uni-versidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001, Portugal.E-mail: [email protected]
Abstract.
We propose a new dynamical system formalism for the analysis of f ( R ) cosmolo-gies. The new approach eliminates the need for cumbersome inversions to close the dynamicalsystem and allows the analysis of the phase space of f ( R ) -gravity models which cannot beinvestigated using the standard technique. Differently form previously proposed similar tech-niques, the new method is constructed in such a way to associate to the fixed points scalefactors, which contain four integration constants (i.e. solutions of fourth order differentialequations). In this way a new light is shed on the physical meaning of the fixed points.We apply this technique to some f ( R ) Lagrangians relevant for inflationary and dark energymodels. a r X i v : . [ g r- q c ] O c t ontents f ( R ) gravity (in brief ). 44 The new Dynamical Systems Approach. 5 R n -gravity 105.2 The case f ( R ) = R + αR n n = 1 . 25 Since the first formulation of General Relativity (GR), many extensions of the original Ein-stein equations have been investigated. The reasons of the interest in such theories are quitedisparate: from the first attempts to unify geometrically the electromagnetic and gravitationalinteraction started by Weyl [1], to the understanding of the corrections to the gravitational ac-tion typical of quantum field theory in curved spacetime and fundamental unification schemes[2], to the attempts to give a complete geometric explanation of the dark phenomenology.Among these extensions, the class of theories called f ( R ) -gravity [3] is the simplestrealisation of higher order gravity (order four) and has an important role as a natural modelfor inflation [4]. More recently, it was shown to have also an interesting (and very debated)role as geometrical dark energy model [8]. In dealing with these theories, the necessity to solvefourth order differential equations is the source of the difficulties in the true understanding ofthe features of their cosmology. Such problems were the origin of the development of a seriesof methods to indirectly analyse the physical properties of f ( R ) cosmologies.Dynamical System Approach (DSA) has proven to be one of the most effective of suchmethods. DSA has a number of different realisations [5, 6]. In the following we shall considerthe one proposed in the 1970’s by Collins [7] and successively developed by Wainwright, Ellisand Uggla. This version of DSA is defined in terms of dimensionless, expansion normalisedvariables and allows a very clear physical interpretation of the results. DSA has had a key rolein analysing Bianchi models [9] and minimally coupled scalar tensor [10] in the context of GRand has allowed the exploration of the cosmology of a number of modifications of Einstein’stheory [11, 12]. In [11], the model f ( R ) = χR n was analysed with this method for the fisttime. Later on the method was extended in [13, 14] to the case of a generic f ( R ) . DSA– 1 –llowed for the first time to analyse in detail the phase space for f ( R ) cosmologies makingsome generals statements on these cosmologies and revealing a number of interesting attractorsolutions.In spite of this success the method above has some unsatisfactory aspects. First of allthe possibility to analyse a given f ( R ) theory depends on the exact resolution of an algebraicequation of generic order, or, in the most complicated cases, a transcendental equation. Suchoperation not only limits the set of possible Lagrangians which can be analysed with DSA,but also introduces a number of singularities, so that in many cases the dynamical system isnot of class C .Another important problem is that the choice of the dynamical system variables wasmade in such a way to obtain in the fixed points solutions with only two integration constants.This implies that these solutions correspond to the general solutions of the fourth ordercosmological model where two integration constants have been set to zero. This is problematicbecause without knowledge of the full solution in a fixed point it is impossible to characterisethe correct behaviour of these cosmologies when these fixed points are nodes.In addition, since the dynamical system variables are not always independent from eachother, the dynamical system might present fixed points which can correspond to inconsistentconditions in the cosmological equations i.e. to have f ( R ) = 0 and R = 0 for a function f for which f (0) (cid:54) = 0 . The presence of these points must be a spurious effect due to the way inwhich he DSA is constructed.In this paper we propose a new DSA to deal with the cosmology of f ( R ) gravity. Thismethod is built in such a way to avoid the necessity of solving exactly algebraic/transcendentalequations and to give, in the fixed points, solutions of the cosmological equations whichcontain four integration constants. We will show that the new formulation contains theresults of the old method but reveals unsuspected additional features of the evolution of f ( R ) cosmologies.The paper is organised as follow. In section 2 we will give the basic equations. Insection 3 we will introduce the original DSA for f ( R ) -gravity. Section 4 is dedicated to theconstruction of the new DSA, and in Section 5 the new method is applied to some interestingmodel of f ( R ) gravity. Section 6 is dedicated to the conclusions.Unless otherwise specified, natural units ( (cid:126) = c = k B = 8 πG = 1 ) will be used through-out this paper, Latin indices run from 0 to 3. The symbol ∇ represents the usual covariantderivative and ∂ corresponds to partial differentiation. We use the ( − , + , + , +) signature andthe Riemann tensor is defined by R abcd = W abd,c − W abc,d + W ebd W ace − W f bc W adf , (1.1)where the W abd are the Christoffel symbols (i.e. symmetric in the lower indices), defined by W abd = 12 g ae ( g be,d + g ed,b − g bd,e ) . (1.2)The Ricci tensor is obtained by contracting the first and the third indices R ab = g cd R acbd . (1.3)Finally the Hilbert–Einstein action in the presence of matter is given by A = (cid:90) dx √− g (cid:20) R + L m (cid:21) . (1.4)– 2 – Basic Equations
In this paper we are going to deal only with metric f ( R ) theories, which are characterized bythe Action S = (cid:90) d x √− g (cid:2) f ( R, ¯ α, ¯ β... ) + L m (cid:3) , (2.1)where L m represents the matter contribution. In general the function f is considered ananalytic function of the Ricci scalar R and contains a set of additional dimensional parametersindicated by barred Greek letters.Varying the action with respect to the metric gives the generalisation of the Einsteinequations: f (cid:48) G ab = T mab + 12 g ab (cid:0) f − Rf (cid:48) (cid:1) + ( g ca g db − g ab g cd ) ∇ c ∇ d f (cid:48) , (2.2)where G ab is the Einstein tensor, f = f ( R, ¯ α, ¯ β... ) , f (cid:48) = df ( R, ¯ α, ¯ β... ) dR , and T Mab = 2 √− g δ ( √− g L m ) δg ab represents the stress energy tensor of standard matter. These equations reduce to the standardEinstein field equations when f ( R, ¯ α, ¯ β... ) = ¯ αR with ¯ α = 1 / .Our treatment will consider only homogeneous and isotropic spacetimes i.e. FriedmannLemaître Robertson Walker (FLRW) metrics: ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) , (2.3)where a is the scale factor and k the spatial curvature. We also assume that the cosmic fluidis a prefect fluid with equation of state p = wµ with ≤ w ≤ . It is common to write thefield equations (2.2) in the metric (2.3) as two equation resembling the Raychaudhuri and theFriedmann equations in GR H + ka = 13 f (cid:48) (cid:26) (cid:2) f (cid:48) R − f (cid:3) − H ˙ f (cid:48) + µ m (cid:27) , H + H + ka = − f (cid:48) (cid:26) (cid:2) f (cid:48) R − f (cid:3) + ¨ f (cid:48) − H ˙ f (cid:48) + p m (cid:27) , (2.4)where R = 6 (cid:18) ˙ H + 2 H + ka (cid:19) , (2.5) H ≡ ˙ a/a , the prime represents the derivative with respect to R and the “dot" is the derivativewith respect to t . The two equations (2.4) are not independent: the second can be obtainedderiving the first with respect to the cosmic time t once the Bianchi identities for T mµν isconsidered. In FLRW these identities take the form ˙ µ m + 3 H ( µ m + p m ) = 0 , (2.6)which is the same as the GR energy conservation equation for the cosmic fluid.– 3 – The Original Dynamical Systems Approach for f ( R ) gravity (in brief ). Using the equations above we can formulate the cosmic evolution in terms of dynamical sys-tems [13, 14] (beware of the differences in signature!). Introducing the general dimensionlessvariables : x = ˙ f (cid:48) Hf (cid:48) , y = R H , z = f H f (cid:48) , Ω = µ m H f (cid:48) , K = ka H , (3.1)where µ m represents the energy density of a perfect fluid that is present in the model. Ascustomary, we also define the logarithmic (dimensionless) “time variable” N = ln a . Notethat in choosing this time variable we are assuming that we represent the phase space for H > i.e. we are considering only expanding cosmologies . In this variables the cosmologicalequations (2.4) are equivalent to the autonomous system: dxdN = x (Ω − z − − x + 2( y − z ) + (1 − w )Ω ,dydN = y [(Γ − x + 2Ω − z + 2] , (3.2) dzdN = z (2 − x − z + 2Ω) z + x y Υ ,d Ω dN = Ω (2Ω − x − z − − w ) , K + x + z − y − Ω . The quantity Υ is defined as Υ ≡ f (cid:48) Rf (cid:48)(cid:48) . (3.3)Since Υ is a function of R only, the problem of obtaining Υ = Υ( x, y, z, Ω) is reduced to theproblem of writing R = R ( x, y, z, Ω) . This can be achieved from the definitions (3.1): yz = Rf (cid:48) f . (3.4)Solving the above equation for R allows one to write R in terms of y and z and close thesystem (3.2). It is clear that the properties of (3.4) determine the possibility of closing (andtherefore analysing) the system (3.2) as well as some of the properties of this system i.e. thedifferential structure. One example is f ( R ) = R p exp( qR ) for which Υ = yzy − pz . (3.5)In this case it is evident that the system is not C (1) as the curve y − pz = 0 is singular.This fact can have serious repercussions on the properties of the flow [14]. The contracting case can also be considered using the time variable M = − N . Since by definition almostall the variables (3.1) are invariant under a change of sign of H the phase space for H < will have a strictresemblance with the one with H > . The H < part of the phase space can be important to analyse, forexample bouncing scenarios. In the following we will not consider this part of the phase space focusing onlyon expanding cosmologies, leaving this analysis for a future work. – 4 – The new Dynamical Systems Approach.
We will now start to construct the new approach. Before we define the dynamical systemvariables we will need, however, two preliminary steps. The first one will concern the formof the action and the second one will be the introduction of new (cosmic) parameters. These(re-)definitions will be the cornerstones of the new method.
In dealing with dynamical system it is crucial to gain an understanding (and control) overthe dimensional structure of the theory we are considering. The reason is that the number ofdynamical system variables needed for DSA will also depend on the number of dimensionalconstants present in the theory. To construct the new method, therefore, we will rewritethe action (2.1) in a special form. In particular we will introduce a constant R such thatthe product RR is dimensionless. In addition, we will also introduce some dimensionlessparameter in the form of Greek letters which will represent the ratio between the couplingconstant of the additional invariants in the theory and (a power of) R . In this way the (2.1)can be written as S = (cid:90) d x √− g [ f ( RR , α... ) + L m ] , (4.1)where f has the same properties of the one in (2.1) and R will be assumed non-negative.The main reason behind the formulation above is that in this way any f ( R ) action containsonly one dimensional constant. Therefore, instead of defining a dynamical variable for eachdimensional constant ( ¯ α, ¯ β, ... ) , we only need one dynamical variable related to R to analysethe phase space of actions of any complexity. In addition, this setting prevents the appearanceof fixed points not consistent with the cosmological equations which are typical of the originalDSA. We will use this formulation (4.1) of the action as a starting point in the constructionof our new DSA. Looking for a different way to constrain the properties of the cosmic fluids, Visser proposeda set of cosmic (or cosmographic) parameters [15–18] q = − ¨ aa H − , j = ¨ aa H − , s = .... aa H − , (4.2)with which one is able to characterize completely a cosmological model. These quantities aredirectly related with the Taylor development of the scale factor and this property determinesalso our capability to measure them. With few exceptions [16], in the case of GR only thelowest order cosmic parameter have been so far fully exploited, but in the case of f ( R ) -gravitythe situation is different: higher order parameters become crucial and can be used to char-acterize the evolution of many important cosmological phenomena e.g. structure formation[19]. However, after a quick look to the f ( R ) cosmological equations written in terms of thesequantities (e.g. [19, 20]), one soon realizes that this type of cosmographic parameters arenot always the ideal objects to work with. The same happens when one tries to use them toformulate a dynamical system approach: although in principle the (4.2) are the ideal objectsto construct the DSA they do not constitute always an advantageous set of variables.– 5 –or this reason, it is necessary to look for new sets of parameters which share thestructure of the original cosmographic parameters (4.2) but, at the same time, are moresuitable to deal with our specific problem. One possibility is to use the Hubble parameter todefine the variables: ¯ q = ˙ HH , ¯ = ¨ HH ˙ H , ¯ s = ... HH ˙ H . (4.3)The variables above have been used in [21] to propose new ways to perform the reconstructionof exact cosmological solutions. Unfortunately also the (4.3) do not prove useful to implementa more powerful dynamical system approach. Another possibility, which will be adopted inthe following, is to define q = ˙ HH , j = ¨ HH − ˙ H H , s = ... HH + 3 ˙ H H − H ¨ HH . (4.4)This choice, apparently cumbersome, appears much simpler in terms of the logarithmic time N : q = H ,N H , j = H ,NN H , s = H ,NNN H . (4.5)The (4.5) will be the second cornerstone of the mode we intend to propose.In terms of q , j , s the Ricci scalar and its derivatives read R = 6 (cid:20) ( q + 2) H + ka (cid:21) , (4.6) ˙ R = 6 H (cid:26) [ j + q ( q + 4)] H − ka (cid:27) , (4.7) ¨ R = 6 H (cid:26)(cid:2) s + 4 j ( q + 1) + ( q + 8) q (cid:3) H + 2(2 − q ) ka (cid:27) , (4.8)and the cosmological equations (2.4) can be written as H (1 + q ) + µ f (cid:48) + 12 ka H f (cid:48)(cid:48) f (cid:48) − f f (cid:48) − H f (cid:48)(cid:48) f (cid:48) [ j + q ( q + 4)] = 0 ,H + ka − µ f (cid:48) (1 + 3 w ) − f f (cid:48) ++ H f (cid:48)(cid:48) f (cid:48) (cid:26) H [ s + j (4 q + 5) + q ( q + 9) + 4 q ] + 6(1 − q ) ka (cid:27) ++ 36 H f (3) f (cid:48) (cid:20) − kH ( j + q + 4 q ) a + 92 H ( j + q + 4 q ) − k a (cid:21) , (4.9)respectively. Note that now the cosmological equations are equations for j and s instead of H and ˙ H . In fact, in (4.9), this last quantity has been substituted according to (4.4).In the system (4.9) the Ricci scalar only remains present in the function f . This meansthat the equations above should be supplied with an additional constraint given by the (4.6).This relation will allow to simplify considerably the final dynamical system. The first step to obtain and autonomous system of first order differential equations corre-sponding to the (4.9) is the definition of the dynamical variables. In the case of single fluid– 6 –osmology , we choose the set of variables: R = R H , K = ka H , Ω = µ H f (cid:48) , J = j , Q = 32 q , A = R H . (4.10)Note that the variable associated to matter does not coincide exactly with the matter densityparameter. This is a manifestation of the non-minimal coupling of matter and gravitationtypical of f ( R ) -gravity. Also, because of our definition of R , the variable A will be alwaysnon-negative. In addition to the above variables, we will introduce, like in the original DSA,the logarithmic time variable N .With this choice the cosmological equations (4.9) are completely equivalent to the au-tonomous system: d R dN = 49 Q ( Q − R + 9) − R + 4 J + 4 ,d Ω dN = Ω18 { [9 R − Q + 9 Q + 9 J + 9)] Y − Q + 9( w + 1)] } ,d J dN = Z Y { R − Q ( Q + 9) + 9( J + 1)] } ++ 13 Y [6 X + 4 Q − R + 3Ω(1 + 3 w ) + 6]++ 154 { − Q ) R + 2(3 + 2 Q )[ Q ( Q + 15) + 9(5 J − } ,d Q dN = 6 J − Q ,d K dN = − K (cid:18) Q (cid:19) ,d A dN = 43 AQ , (4.11)together with the two constraints − K + R − X − (cid:20)(cid:18) Q (cid:19) Q − R J + 1 (cid:21) Y , (4.12) R = K + 23 Q + 2 . (4.13)The first corresponds to the Hamiltonian (Friedmann) constraint, which guarantees the con-servation of matter energy, and the second is simply the definition of the Ricci scalar. Thefunctions X = X ( A , R ) , Y = Y ( A , R ) , Z = Z ( A , R ) are defined respectively as X ( A , R ) = f ( R , A , α, ... )6 H f (cid:48) ( R , A , α, ... ) , Y ( A , R ) = 24 H f (cid:48)(cid:48) ( R , A , α, ... ) f (cid:48) ( R , A , α, ... ) , Z ( A , R ) = 96 H f (cid:48)(cid:48)(cid:48) ( R , A , α, ... ) f (cid:48) ( R , A , α, ... ) . (4.14) The approach can be trivially generalised to the multi-fluid case by adding a suitable number of Ω variableseach corresponding to the energy density of the given fluids. Such generalization does not add anything tothe understanding of the method and it will not be pursued here. – 7 –hese quantities represent the part of the system which depends on the form of the Lagrangian(4.1). Note that, differently from the approaches presented in [13, 14], in this version of theDynamical Systems Approach no resolution of algebraic equation is required to close thesystem. This means that with the technique proposed here we can in principle analyze all f ( R ) models.The two constraints allow us to eliminate two variables ( J and Q ) and reduce the totalsystem to: d R dN = 2 R ( K − R + 2) − Y ( X + K − R − Ω + 1) ,d Ω dN = Ω(2 − w + X + 3 K − R − Ω) ,d K dN = 2 K ( K − R + 1) ,d A dN = − A (2 + K − R ) . (4.15)It is clear that this system posses a minimum of three invariant submanifolds: (i) Ω = 0 , (ii) K = 0 and (iii) A = 0 . Depending on the form of Y one can also have a fourth invariantsubmanifold in R = 0 . These invariant submanifolds can be imagined as “parts" of the phasespace which have the property that any orbit that starts in them is trapped. They can havea very specific physical meaning. For example, the existence of the invariant submanifold K = 0 tells us that if we start in an orbit with zero spatial curvature we cannot evolvetowards a positive or negative curvature parts of the phase space and that the existence of Ω = 0 implies that a vacuum cosmology remains vacuum (i.e. standard matter cannot becreated or destroyed). The submanifold R = 0 represents, instead, the case in which the Ricciscalar is identically zero. Since the equation for R contains terms with Y at denominator,this submanifold can be singular. Finally, the submanifold A = 0 corresponds to the case inwhich the constant R is zero and it is of more difficult physical interpretation. It can bethought to correspond to the case in which the gravitational part of the action is identicallyzero i.e. there is no gravitational interaction . Since A appears also in Y , the submanifold A = 0 can also be singular.The singular character of the invariant submanifolds R = 0 and A = 0 , however, doesnot imply the absence of fixed points. The existence of fixed points that belong to a singularmanifold is one of the exotic properties of dynamical systems which are not of order C(1).The coordinates of (and therefore the solution associated with) these points can be explainedin terms of the limiting form of f for R, R → . In fact, close to the fixed points in the R = 0 and A = 0 manifolds the phase space of a given theory f and the one of its limit for R, R → can be considered isomorphic and will admit the same fixed points. For thesefixed points the stability analysis will not necessarily be the standard one. For example, someof the eigenvalues of these points might diverge. We will impose the condition that actualfixed points will exist if the equations at least of class C (2) i.e. that the dynamical systemequations, the cosmological equation and the eigenvalues will be finite for a given fixed point. At this point one might think that reformulating the dynamical system approach distinguishing thedimensional constant in front of the Hilbert-Einstein term and the one(s) of the higher order invariant(s)might ease the interpretation of this kind of invariant submanifolds. Indeed this can be done, but it does notadd anything to the phase space analysis. For this reason we will rather keep using the approach presentedabove, which is more compact. – 8 –ith the above setting, we are ready to analyse the phase space for a given form of f ( R ) (and therefore a form of the functions X , Y , Z ). The analysis can be done usingthe classical dynamical system theory i.e. obtaining the fixed points imposing that the N -derivatives of the dynamical variables are zero and using the Hartman-Grobman theorem todetermine their stability . Naturally, since the phase space is not compact one will need toperform an asymptotic analysis in order to obtain a full description of the dynamics. Suchtask, however, will be left for a future work.The solutions associated to the fixed points can be derived writing the modified Ray-chaudhuri equation in a fixed point, s = 1 H d HdN = 427 Q ∗ [(2 Q ∗ + 33) Q ∗ − R ∗ + 27] + − J ∗ (cid:26) Z ∗ Y ∗ [2( Q ∗ + 9) Q ∗ + 9(2 − R ∗ )] + 8 Q ∗ + 15 (cid:27) + − Z ∗ Y ∗ [2( Q ∗ + 9) Q ∗ + 9(2 − R ∗ )] + 8 Y ∗ [2 Q ∗ − R ∗ + 1)]+ − X ∗ Y ∗ − J ∗ Z ∗ Y ∗ + 4Ω ∗ Y ∗ (1 + 3 w ) + 2(2 − R ∗ ) , (4.16)where the asterisk indicates the value of a variable in a fixed point . In spite of being athird order differential equation the (4.16) is always solvable exactly for H . In addition, since(4.16) is a fourth order equation in a , the solution associated to the fixed points solutioncontains four integration constants (and up to three different “regimes”). As we will see, inthe case in which a theory can be treated with both the new and the original method, thereis correspondence between the fixed points. This suggests that the original method implicitlysets to zero some integration constants. We will discover that this can hide information onthe actual meaning of the fixed point, particularly when it is a sink or a source. The otherkey cosmological quantities can be deduced by the cosmological equations once the (4.16) issolved. Specifically one has, from (2.6), µ = a − w ) . (4.17)In literature one often defines the barotropic factor of the high order corrections consideredas an effective fluid.Unfortunately, it will not always be possible to integrate the (4.16) exactly up to anexpression for the scale factor. In this case we will rely on considerations on the structure ofthe equation for a and numerical integrations to obtain the behaviour of the scale factor andthe other key quantities. For this reason we will limit ourselves to give the solutions for thescale factor in terms of plots. We should stress here that some general conclusions on the system (4.15) could be drawn without speci-fying the form of f . However this can be dangerous. Consider for example a case in which the function Y isproportional to R − . In this case the system would admit a fixed point R = 0 , Ω = 0 , K = 0 , A = 0 whichwould be not necessarily present in other cases. For this reason we will refrain form such speculations andwill work only on a given forma on f . It is clear that, as some free parameters enter in the dynamical equations there will be values of theseparameters for which the eigenvalues of the fixed points are zero. Such phenomena are called bifurcations (seee.g. [29]) and we will not explore them here. It is important to stress here that the (4.16) represents the form that the Raychaudhuri equation takesarbitrary close to a fixed point. The meaning of the solutions associated to the fixed points can only beunderstood correctly in this way. The representation of these solutions that will be given in the followingchapter has the only purpose of clarify the nature of the solution of such approximated equations. – 9 –
Examples
In this section we are going to apply the method described above to some specific models. Wewill start with two simple ones ( f ( R ) = R n , f ( R ) = R + αR n ) that are treatable also withthe original DSA highlighting common features and differences between these two methods.After that, we will explore other two models (the Starobinsky and the Hu-Sawicki models)that are more physically interesting, but cannot be analysed with the original DSA. R n -gravity As a first check for our new formalism let us consider the first model that has been analysedwith the original DSA [11]. This model, which is also called sometimes “ R n -gravity”, ischaracterized by an action in which the Ricci scalar appears as a generic power rather thanlinearly and it constitutes the simplest fourth order modification of GR.Putting the action in the form of Section 4.1 we have A = (cid:90) d x √− g [ R n R n + L m ] . (5.1)Using the original DSA it was realised that for specific values of the parameter n there could beorbits which naturally present a transition between decelerated and accelerated expansion [11].This model was subsequently subjected to more detailed studies which involved cosmologicalperturbations [19, 22–24] as well as astrophysical and cosmological tests (e.g. [25–27]). Theresult of these investigations and other physical considerations points to the fact that R n -gravity is inconsistent with multi-scale observations and should be considered only a toymodel.It is known that in terms of the original DSA, the case f ∝ R n is degenerate: twodynamical system variables ( y and z ) coincide. The special character of R n -gravity is presentalso in the new approach: since the term R n appear as a factor of R n the equation for A isdecoupled and can be excluded. Substituting in the remaining equations the expression for X , Y and Z X = R n , Y = 4( n − R , Z = 8( n − n − R , (5.2)we obtain d R dN = R (cid:26) K − R ) − n − (cid:20) K + (cid:18) n − (cid:19) R − Ω + 1 (cid:21)(cid:27) ,d Ω dN = Ω (cid:20) K + (cid:18) n − (cid:19) R − Ω + 2 − w (cid:21) ,d K dN = 2 K ( K + R + 1) . (5.3)The system presents in general three invariant submanifolds: K = 0 , Ω = 0 and R = 0 . Table1 contains the standard fixed points of (5.3), together with their associated solution. Thestability of the fixed points is shown in the case w = 0 in Table 2.– 10 –he solutions associated to the fixed points deserve further discussion. Since now weare solving the full (4.16) these solutions will be specified by a linear differential equation in a which is more complex than the one of the original DSA. For example for the points A , B and C we have ˙ aa = H a + a / (cid:20) H sin (cid:18) √ a (cid:19) + H cos (cid:18) √ a (cid:19)(cid:21) . (5.4)This equation can be solved exactly (but almost always implicitly) only in the case in whichtwo of the constants H i are zero. For H and H zero one has, for example, a = a ( t − t ) , (5.5)in the other cases the solution can only be expressed in terms of inverses of hypergeometricfunctions. Looking at the nature of equation (5.4) it is clear that for small a the power lawbehaviour is the dominant component of the solution whereas for large a the hypergeometricbehaviour is dominant. A numerical integration of the equation (5.4) is given in Figure 1. It Figure 1 . Numerical solution of equation (5.4). The constants H i have all been chosen to be oneand the initial condition is a (0) = 0 . . is clear that the solution has sigmoid behaviour: after a power law growth of the type (5.5),the expansion rate starts, at first, to increase and then to decrease, to approach eventuallya constant. This results is confirmed by the numerical check of the first derivative of thesolution.The solutions associated to the points D and E are given by the equation ˙ aa = H a + a (cid:104) H sin (cid:16) √ a (cid:17) + H cos (cid:16) √ a (cid:17)(cid:105) . (5.6)As before, this equation can be solved exactly only in the case in which two of the constants H i are zero. For H and H zero one has, for example, a = a ( t − t ) , (5.7)– 11 – μ Figure 2 . Numerical solution for the energy density of (5.4). The constants H i have all been chosento be one and the initial condition is a (0) = 0 . . Figure 3 . Numerical solution of equation (5.6). The constants H i have all been chosen to be oneand the initial condition is a (0) = 0 . . which is dominant only for small a . The numerical integration (Figure 3) shows that also inthis case the solution approaches a constant a late time in a manner similar to the previousone. Both these solutions present two inflection points i.e. changes in the sign of the expansionrate of the universe.For F and G instead one has, respectively, ˙ aa = ( H + H + H ) a n − n − n +1 , (5.8)and ˙ aa = ( H + H + H ) a − w +1)2 n , (5.9)– 12 – .5 1.0 1.5 2.0 2.5 3.0 t0.101101001000 μ Figure 4 . Numerical solution for the energy density of (5.6). The constants H i have all been chosento be one and the initial condition is a (0) = 0 . . which can be integrated exactly to give a pure power law behaviour. Table 1 . Fixed points of f ( R ) = χR n and their associated solutions. Here a = H + H + H . Point Coordinates { R , K , Ω } Scale Factor
A { , − , } (5.4) B { , − , − − w } (5.4) C { n (1 − n ) , n − n − , } (5.4) D { , , − w } (5.6) E { , , } (5.6) F (cid:110) (5 − n ) n n − n +2 , , (cid:111) a = a ( t − t ) (1 − n )(1 − n ) n − G (cid:110) − n +3 w n , , n w +8 n − nw − n +3 w +32 n (cid:111) a = a ( t − t ) n w +1) It is instructive to compare the results above with the ones of [11]. It is evident thatthe fixed points we have obtained, correspond to the ones found in [11]. In particular thesolutions associated to the fixed points are coincident when H and H are set to zero. Thisresult shows how the original DSA would give at best an incomplete result. For example in(5.4), since the a − term is only relevant at small a , if A , B , C are attractors the cosmology– 13 – able 2 . Stability of the fixed points of f ( R ) = χR n in the case w = 0 . Here A stays for attractor,R for repeller, S for saddle. Point n < (cid:0) − √ (cid:1) (cid:0) − √ (cid:1) < n < < n < / / < n < A S S S S B S S S S C S A S S D R R R R E S S S S F A S S A G S S S SPoint < n < / / < n < / / < n < (cid:0) √ (cid:1) n > (cid:0) √ (cid:1) A S S S S B S S S S C S S A S D S R R R E S S S S F R S S A G S S S Swill tend to become static after a phase of accelerated expansion. In terms of the stability,instead, the two methods show a complete consistency: the nature of the fixed points isthe same as the one in [11]. Specifically, comparing the ranges of stability the possibility ofa transition between almost Friedmann (point F ) and power law inflation/dark energy era(point G ) is present also with the new method. In fact, since both the fixed points are onthe K = 0 invariant submanifold, we can check this result explicitly plotting this part of thephase space (see Figure 5). It is evident that there is, in complete agreement with [11], a set– 14 –f initial condition in which the transition appears. The stability analysis results in Table 2guarantees that this is the case also for orbits in the full phase space. Figure 5 . Plot of a section of the invariant submanifold K = 0 for the model f ( R ) = χR n . Here theabscissa represents the variable R and the ordinate the variable Ω . f ( R ) = R + αR n Let us now consider another important model for fourth order gravity: the one in which ageneric power of the Ricci scalar is added to the Hilbert-Einstein term. Historically this isthe most studied form of fourth order gravity because of its appearance in several quantumgravity calculations [28]. In cosmology the model with n = 2 has gained particular attentionas a model of geometric inflation [4].The Lagrangian for this class of theories can be written in the form showed in Section4.1 as f ( R R, α ) = R R + αR n R n . (5.10)Note that for n < this theory is not defined in R = 0 and in R = 0 , but in fact we will seethat, due to the presence of derivatives of f with respect to R in the cosmological equations(and therefore in the dynamical system) we will have divergences also in other intervals of n .Substituting the f above in the expression for X , Y and Z we obtain X = R n + ( n − AR n ( AR + αn n − A n R n ) , Y = 4( n − R (cid:20) − ARAR + αn n − A n R n (cid:21) , Z = 8( n − n − R (cid:20) − ARAR + αn n − A n R n (cid:21) . (5.11)– 15 –ubstituting in the general system (4.15) we obtain d R dN = R (cid:2) n (2 n − K − (cid:0) n + 3 n + 1 (cid:1) R + n Ω + 4 n − n (cid:3) n ( n − − ( K − Ω + 1)6 n − α ( n − n A n − R n − ,d Ω dN = Ω (cid:20) K + (cid:18) n − (cid:19) R − Ω + 2 − w − ( n − AR n ( AR + αn n − A n R n ) (cid:21) ,d K dN = 2 K ( K − R + 1) ,d A dN = − A (2 + K − R ) . (5.12)This system admits four invariant submainfolds ( K = 0 , Ω = 0 , A = 0 , R = 0 ). The last twoinvariant submanifolds can be singular, but they can still contain some fixed points. As said,the presence of fixed points is based on the requirement of convergence of the cosmologicalequations and the eigenvalues of the fixed points. This implies that some fixed points willonly exist for certain values of n .The list of the fixed points, their associated solutions and the interval of existence ofthe fixed points are given in Table 3. Note that the points A − G are the same of the onesfound in the case of R n -gravity. The presence of this type of fixed points can be understoodthinking that nearby A = 0 the function f can be approximated with its limit for small R .For the values of n for which these fixed points exist the approximate function is R n R n andtherefore the fixed points of the case f ∝ R n R n appear also in the phase space of this theory.On top of the points A − G of the previous case we have some additional ones, whichwe will name H i . These points will exist if α ( n − > (remember that the variable A isdefined to be non negative) and their number depends on the value of n .In the H i the scale factor is described by the equation ˙ aa = H + 3 H log a + 9 H log a, (5.13)which admits the exact solution a ( t ) = a exp (cid:40) (cid:112) H H − H H tan (cid:20)
12 ( t − t ) (cid:113) H H − H (cid:21) − H H (cid:41) . (5.14)For H H − H > this solution is monotonically growing with two inflection points at t = t ∗ , = t ± arcsin (cid:18) − √ H H − H H (cid:19)(cid:112) H H − H + 2 kπ, k ∈ N . (5.15)and presents a discontinuity in t = ¯ t = t − π (cid:112) H H − H + kπ, k ∈ N . (5.16)For t → ¯ t + this solution approach to zero, whereas when t → ¯ t − the solution presents avertical asymptote. – 16 – able 3 . Fixed points of f ( R ) = R + αR n with their interval of existence and their associatedsolutions. Point Coordinates { R , K , Ω , A } Scale Factor Existence
A { , − , , } (5.4) n < / B { , − , − − w, } (5.4) n < / C { n (1 − n ) , n − n − , , } (5.4) n < D { , , − w, } (5.6) n < / E { , , , } (5.6) n < / F (cid:110) (5 − n ) n n − n +2 , , , (cid:111) a = a ( t − t ) (1 − n )(1 − n ) n − n < G (cid:8) − n +3 w n , a = a ( t − t ) n w +1) n < , n w +8 n − nw − n +3 w +32 n , (cid:111) H i (cid:110) , , , − n (cid:112) α ( n − (cid:111) (5.13) α ( n − > In the case H H − H < the (5.13) is instead not periodic and approaches a constant.The expanding or contracting character depends on the sign of the quantity P = − H + (cid:112) H − H H H , (5.17) P > implies a growing scale factor and P < a decaying one. Plots of the solution (5.13)can be found in Figure 6.In the case H H − H > the solution (5.14) presents features which are physicallyvery interesting. For times close to ¯ t + the solution grows exponentially, then around t ∗ itchanges into a decelerated expansion with non-constant deceleration factor. After a coastingphase around t ∗ , the decelerated expansion is followed by a new accelerated expansion phase.In a finite time (at t = ¯ t − ), however, the solution becomes singular in the sense that a andits derivatives as well as the Ricci scalar diverges at this specific time. This is a well knownproperty of f ( R ) -gravity [31, 32], but in this context one is able to appreciate both thisdrawback of the theory and its potential as a model that unify inflation and dark energy.Note that for H , = 0 the solution (5.14) reduce to the standard de Sitter solution.This fact on one hand connects the points H i to the de Sitter fixed point of the original DSA.On the other hand gives a hint of the true meaning of de Sitter solutions in the framework of f ( R ) -gravity. These models in fact present also other types of singularities, like the “weak singularities” in [33] or the – 17 – .5 1.0 1.5 2.0 2.5 3.0 3.5 t0.51.01.52.02.5a (a) Plot of a period of (5.13) in the case H H − H > . Here H = 1 , H = 3 , H = 2 and a = 1 . (b) Plot of (5.13) in the case H H − H < . Theconstants H i have all been chosen so that P > and a = 1 (c) Plot of (5.13) in the case H H − H < . Theconstants H i have all been chosen so that P < and a = 1 Figure 6 . Plots of (5.13) illustrating the different behaviour that this solution can represent. ones found in [34]. – 18 – - ( t ) - - - ln ( μ ) Figure 7 . Numerical solution for the energy density of (5.13) in the case H H − H > . Here H = 1 , H = 3 , H = 2 and a = 1 . The stability of the fixed points can be calculated as in the previous case using theHartman Grobman theorem, and it is illustrated in Table 4. It is tempting to attempt ageneral derivation for the stability of points H i , however numerical inspection shows thatsuch analysis might be unreliable. The calculations show that when points H i exist there areonly two options for their stability: they can be either a saddle or an attractor (or their focuscounterpart). In Table 4 we report some samples of the stability of points H i for differentvalues of the parameters α and n . Examples of the phase space for these theories in the formof the invariant submanifold R , A is given in Figures 8. Since this model can be treatedalso with the original DSA it is useful to make a comparison between the results we obtainedabove and the ones in [14]. Differently from the case of R n gravity the phase space obtainedby the two methods is not the same. In particular, the original DSA returns a phase spacewith many more fixed points. The origin of this difference is probably to be attributed to thechoice of variables of the original DSA. For the common points one can compare the results onthe stability and it is easy to verify complete consistency. For example, the de Sitter solutionthat in [14] is associated with E ∗ has a stability that coincide with the point H of the presentanalysis.In the case n = 2 the new DSA returns a phase space with no finite fixed points. It isknown that the case n = 2 in the theory (5.10) presents significant physical differences withrespect to the other model of this class [28], and it is to be expected that the phase space willreflect these differences. The fact that the phase space does not present a point of type H does not necessarily imply that the model does not have de Sitter solutions (we know in factthat they are present [30]). A careful analysis of the equations shows that the fixed point inthis case is asymptotic ( A → ∞ ) and it is therefore excluded by the present analysis.It is interesting to note that one of the conclusions in [31] is that to avoid the singularityone either has to recur to special initial conditions or to add additional curvature invariants.This result seems consistent with our findings. In order to avoid the singularity one either hasto control the initial conditions (by setting to zero some of the constant H i ), the values of theparameters like in Figure 5.2, or hope that adding additional curvature invariants the fixedpoints H will become irrelevant (in the same way of what happens with the case n = 2 above).– 19 – able 4 . Stability of the fixed points of f ( R ) = R + αR n in the case w = 0 . Here n is the smallestreal solution of the equation n − n + 417 n −
81 = 0 , A stays for attractor, R for repeller, Sfor saddle, F S for saddle focus. The fixed points appear in the table only if they exist in at least oneof the intervals of the parameter n indicated. Point n < (cid:0) − √ (cid:1) (cid:0) − √ (cid:1) < n < < n < n n < n < / A S S S S B F S F S F S F S C S S S S D S S S S E S S S S F A S S S G F S F S F S SPoint / < n < n > C S NA F S NA G F S NA Table 5 . Stability of the fixed points H i of f ( R ) = R + αR n in the case w = 0 . Here A stays forattractor, S for saddle, F A for saddle focus and NA represents the absence of fixed points. In thecases considered there is only one fixed point H . n = − n = − n = 3 / n = 5 / α = − S S NA NA α = − S S NA NA α = 1 A A NA NA α = 2 NA NA S S– 20 – a) Plot of the R > section of the phase space inthe case n = 1 / α = − .(b) The case n = 3 α = 1 / . Figure 8 . Samples of the invariant submanifold
Ω = 0 , K = 0 for the theory f ( R ) = R + αR n .The values of the parameters have been chosen to give the best graphical representation of the phasespace. The issue is that the feature of the inflation dark energy connection seems to be inextricablytied to the approach to the singularity. Thus, in order to generate cosmic acceleration withoutincurring in the singularity requires that other dynamical mechanisms/forms of the function f have to be found. The Starobinsky model is one of the most important class of models of Dark Energy based onfourth order gravitation [35]. The basic idea is to construct, via a function of the Ricci scalar,an effective cosmological constant which is relevant in curved spacetimes, but approaches zero– 21 –n the case of flat spacetime. The function f of this model is given by f ( R , R ) = R + λ ¯ R (cid:34)(cid:18) R ¯ R (cid:19) − n − (cid:35) , (5.18)with n and λ positive and ¯ R of the order of the inverse of the present value of the cosmologicalconstant. The parameter λ and the value of n are related by an algebraic equation. In [35]considerations on the Solar System constraints and cosmological linear perturbation theorywere used to find that one should have n ≥ and λ (cid:38) . .Following Section 4.1 we can write the Lagrangian (5.18) for this class of models as f ( R ) = R R + α (cid:104)(cid:0) β R R (cid:1) − n − (cid:105) , (5.19)where the parameters α, β are related to the ones in (5.18) by the relations λ = α | β | R and ¯ R = ( | β | R ) − .The functions X , Y , Z are X = α + 12 αβ AR [3 AR (1 + 2 n ) − αn ]6 A (cid:104) (1 + 6 A β R ) n +1 − α A βn R (cid:105) + 6 AR − α A , Y = 3 2 αβn A (cid:0) A β (2 n + 1) R − (cid:1) (1 + 6 β A R ) (cid:104) (1 + 6 A β R ) n +1 − α A βn R (cid:105) , Z = − αβ n ( n + 1) A R (cid:0) A β (2 n + 1) R − (cid:1) (1 + 6 β A R ) (cid:104) (1 + 6 A β R ) n +1 − α A βn R (cid:105) , (5.20)and the dynamical system (4.15) becomes d R dN = − n A (2 n A R + 6 A R − (cid:8) β A R [ n ( K − R + Ω + 3) + R ]6 β R A [2 n (3 R + Ω − K (4 n + 3) + 4 n ( R − −
5) + R ] (cid:9) − [6 A ( K − Ω + 1) − α ] (cid:0) A β R + 1 (cid:1) n +2 αβn A [36 A β (2 n + 1) R − ,d Ω dN = α Ω { β AR [3 AR (2 n + 1) − αn ] + 1 } A (cid:104) (1 + 6 A β R ) n +1 − α A βn R (cid:105) − Ω { A [6(2 + 3 w ) − K + 12 R + 6Ω] + α } A ,d K dN = 2 K ( K − R + 1) ,d A dN = − A (2 + K − R ) . (5.21)The system presents three invariant submanifolds: K = 0 , Ω = 0 , A = 0 , although the lastsubmanifold is singular. Note that, differently form the previous case, none of the fixed points A − G is present here. This can be explained looking at the limit of the action for R → . Inthis case in fact the action reduces to R R rather than R n R n and therefore the fixed pointsof R n -gravity are not present. – 22 –ven if this system is valid for any value of the parameters, it is not possible to find itsfixed points analytically : this task would entail the resolution of an algebraic equation oforder n for which no general analytical solution is known. We therefore refer to the boundsin [35] and we set from now on n = 2 and β, α = 1 . In this case the system admits threefixed points with real coordinates (see Table 6). One of them is on the singular submanifoldand presents singular eigenvalues so that it does not appear in Table 6. The other two ( H and H ) are both associated to a solution of the type (5.13).The stability analysis reveals that the character of the points H i is in general different.In our particular case, only one of these point is an attractor whereas the other is unstable.Since the phase space contains invariant submanifolds, we can conclude that for these values ofthe parameters there is only a specific set of initial conditions which lead to the attractor H .Therefore, also in the case of the Starobinsky model there is the possibility that the cosmologywill evolve towards a singularity, but this occurrence depends strictly on the choice of theinitial conditions. In orbits which do not approach H , the fate of the cosmological modelsdepends on the presence of asymptotic attractors. Table 6 . Fixed points for the Starobinsky model and their stability in the case n = 2 and α = β = 1 .Here A is the only positive real solution of the equation (6 A − (cid:0) A + 1 (cid:1) + 144 A (cid:0) A + 4 (cid:1) +1 = 0 different form / , S represents a saddle and F A is an attractive focus. Point Coordinates { R , K , Ω , A } Scale Factor Stability H (cid:8) , , , (cid:9) (5.13) S H { , , , A } (5.13) F A A plot of the section of the invariant submanifold K = 0 , Ω = 0 which contains H and H is given in Figure 9. As a last example we consider a model for geometric DE proposed by Hu and Sawicki [36].The model was designed to be able to reproduce cosmic acceleration without the explicit intro-duction of a cosmological constant and, at the same time, to be compatible with cosmologicaland Solar System tests.The action for the Hu-Sawicki model can be written as (4.1) setting f ( R , R ) = R R − αR n R n βR n R n , (5.22)where the parameter n is chosen to be positive, α > and β > . With this choice of f the Of course the calculation could be done numerically, but we do not perform such task here. – 23 – igure 9 . The section of the invariant submanifold K = 0 , Ω = 0 of the phase space of the Starobinskymodel which contains H and H in the case n = 2 , R > and β, α = 1 . functions X , Y , Z are given by X = 6 β AR − α A β + α A R (cid:2) β n A n R n (1 + n ) − αn n − A n − R n − + 1 (cid:3) AR + 6 n A n R n (12 A β R − αn ) + β n +1 A n +1 R n +1 , Y = − n [12 AR − n A n R n ( α − A β R + αn )]6 AR [6 AR + 6 n A n R n (12 A β R − αn ) + β n +1 A n +1 R n +1 ] − n R ( β n A n R n + 1) , Z = 2 n AR (cid:2) β n A n R n ( n + 1) − α n − A n − R n − ( n + 2 + 3 n ) + 1 (cid:3) R [6 AR + 6 n A n R n (12 A β R − αn ) + β n +1 A n +1 R n +1 ]+ 16 n R ( β n A n R n + 1) − n ( n + 1) R ( β n A n R n + 1) . (5.23)– 24 –ubstituting in (4.15), we obtain the dynamical system d R dN = R αn ( n + 1) (cid:8) αn (3 K − Ω + 5) + R [ α − A β ( K − Ω + 1)] + 2 αn ( K − R + 2)+ αn (5 K − R − Ω + 9) − n A β R ( K − Ω + 1) (cid:9) + 2 n R { [24 A βn R − α K ( n − K − Ω + 1) + α ( n − R } αβ n ( n − n + 1) A n R n − α ( n − + β n A n R n +2 [ α − A β (1 + Ω + K )] αn ( n + 1) + 6 − n A − n R − n αn ( n −
1) ( K − Ω + 1) ,d Ω dN = α R Ω [( β ( n + 1) − nα )6 n n A n R n + 1]6 β AR ( β n A n R n + 1) − αβn n A n R n − Ω (cid:18) Ω − K + 2 R + 3 w − α A β (cid:19) ,d K dN = 2 K ( K − R + 1) ,d A dN = − A (2 + K − R ) , (5.24)This system can present four invariant submanifolds ( K = 0 , Ω = 0 , A = 0 and R = 0 ),but the last one exists only for < n < . The A = 0 submanifold can be singular. Thishappens for n > . Differently from the Starobinsky model, the phase space in this casecontains different fixed points depending on the value of the parameter n . In particular, theconditions β > and A > limits strongly the number of fixed point belonging to the class H . The list of fixed points for n (cid:54) = 1 is given in Table 7. The case n = 1 requires a specialtreatment and will be covered in a separate subsection.As usual, the conditions of existence for the fixed points in Table 7 have been determinedasking that the dynamical system should be at least of class C (2) in the fixed point. Inone case (the point C ) there are values of n for which the Jacobian is divergent, but theeigenvalues converge. We refer as interval of existence of the point C the one of convergenceof the eigenvalues. Within the interval of existence of the fixed points the stability can bedetermined with the standard methods. The results are given in Table 8. As in the case of f ( R ) = R + αR n the stability of H i needs to be calculated case by case, but it can only bea saddle or an attractor of their focus counterparts. An example of the stability of points H i for different combinations of the values of α , β and n in the case of dust ( w = 0 ) is given inTable 8In this case the points C and H are the only possible finite attractors for the cosmology.The first point, however, only exists for < n < . Therefore for n > , we are in the samesituation of the Starobinsky model: only a careful choice of the initial conditions could avoidthe singularity of solution (5.13). n = 1 . It is interesting to note that the dynamical system (5.24) is not defined for n = 1 , whichmeans that the dynamical system in this case needs to be obtained re-deriving X , Y , Z and– 25 – able 7 . Fixed points of the Hu-Sawicki model for n (cid:54) = 1 and their associated solutions. Here A represents the positive solutions of the equation given in the last line of the Table. Point Coordinates { R , K , Ω , A } Scale Factor Existence
A { , − , , } (5.4) < n < B { , − , − − w, } (5.4) < n < C { n (1 − n ) , n − n − , , } (5.4) < n < D { , , − w, } (5.6) < n < E { , , , } (5.6) < n < / F (cid:110) (5 − n ) n n − n +2 , , , (cid:111) a = a ( t − t ) (1 − n )(1 − n ) n − < n < H i { , , , A } (5.13) − n A − n { A +2 n +1 β n A n (6 A β − α )+ β n A n [36 A β + α ( n − n A n [36 A β + α ( n − } αn [ β n ( n +1) A n − n +1] = 0 the equations (4.15). This procedure yields X = R − R β AR + 1) , Y = 8 R (6 β AR + 1) − R (3 β AR + 1) , Z = − R (6 β AR + 1) + 16 R (6 β AR + 1) + 16 R (3 β AR + 1) , (5.25)– 26 – able 8 . Stability of the fixed points of the Hu-Sawicki model in the case n (cid:54) = 1 and α (cid:54) = 1 . Here Astays for attractor, S for saddle and F A for attractive focus. The value of the A coordinate of H isapproximated. Point < n < / / < n < n > A S S NA B S S NA C F A F A NA D S S NA E S NA NA F S S NA G S S NA ( n, α, β ) Coordinates of H i Stability (2 , , / { , , , . } S { , , , . } F A (3 , , / { , , , . } F A { , , , . } S (4 , , / { , , , . } S { , , , . } F A and, therefore, d R dN = 112 αβ A (cid:8) α K (1 + 30 β AR ) − ( K + 1)(6 β AR + 1) + α [18 β AR (2 β AR − R + 3) + 1] (cid:9) + Ω(6 β AR + 1) [1 − α + 12 β AR (1 + 3 β AR )]12 αβ A ,d Ω dN = Ω α − (6 β AR + 1) { (3 K − R − w + 2)[ α − − A β R (3 A β R + 1)]+6 α A β R (cid:9) − Ω ,d K dN = 2 K ( K − R + 1) ,d A dN = − A (2 + K − R ) . (5.26)Differently from the system (5.24), (5.26) does not present the invariant submanifold R = 0 , and the submanifold A = 0 is singular. The properties of the phase space depends onthe values of α .In particular, for α (cid:54) = 1 the phase space contains three fixed points of the type H and anadditional one H Ω = { R = 2 , K = 0 , Ω = − (4 + 3 w ) , A = − β } . However, all of these pointshave A < for α and β positive and have to be discarded. Therefore, this version of the Hu-Sawicki model presents a finite phase space structure analogous to the one of f ( R ) = R + αR .The difference is that the Hu-Sawicki model dose not necessarily have asymptotic fixed pointsof the type H . – 27 –or α = 1 , the situation is very different. In this case the function f reduces to f ( R, R ) = βR R βR R , (5.27)i.e. the Hilbert Einstein term is eliminated. The phase space for this case contains thepoints A (which for n = 1 coincides with C ) to E . Three fixed points of the type H exist,but two of them have A < for β positive and have to be discarded. The third one hascoordinates { R = 2 , K = 0 , Ω = 0 , A = 0 } . In addition, two new fixed points with coordinates I = { R = 4 , K = 3 , Ω = 0 , A = 0 } and L = { R = (5 − w ) , K = 0 , Ω = − (1 + w ) , A = 0 } are present.The modified Raychaudhuri equation (4.16) returns the same solutions for the fixedpoints A - E . The solution associated to I is (5.4) whereas the one associated to L is a = a ( t − t ) w ) . (5.28)The stability of the fixed points in this case can be calculated in the standard way andwith the exception of H all appear to be unstable (see Table 9). Point H has instead one zeroeigenvalue and the analysis of its stability requires the use of the Center Manifold Theorem(CMT) [37]. To apply this theorem the first step is to write the system (5.26) in the form d A dN = C A + F ( A , Y ) (5.29) d Y dN = P Y + G ( A , Y ) (5.30)where Y = { K , ¯ R , Ω } , ¯ R = 3(2 − R ) − β A , C corresponds to the linear pat of the equationfor A , the vector P to the linear part of the equation of Y and F and the vector G representthe non–linear part of the equation for A and Y . The CMT tells us that the behaviour of thefixed points is determined by the solution h of the equation d A dN = C A + F ( A , h ( A )) (5.31)where the vector function h ( A ) is given by h (cid:48) ( A ) [ C A + F ( A , h ( A )] − [ P h ( A ) + G ( A , h ( A ))] = 0 (5.32)Approximating the function h ( A ) with its Taylor series one finds that h ( A ) = a A + ... (5.33)where a is a vector that depends on the parameter β . Since the first non-zero term in theprevious expression is quadratic, (5.31) implies that point H has the stability of a saddle-node.In Figure 10 we plot an example of the invariant submanifold K = 0 , Ω = 0 corresponding tothis case.
In this paper we have presented a new approach to analyse the finite phase space of thecosmology of f ( R ) -gravity. The new method can be applied to any form of the function f – 28 – able 9 . Stability of the fixed points of f the Hu-Sawicki model in the case n = 1 and α = 1 . HereA stays for attractor, S for saddle, F A for attractive focus and SN for non hyperbolic saddle-node. Point Coordinates { R , K , Ω , A } Scale factor Stability
A { , − , , } (5.4) S B { , − , − − w, } (5.4) S D { , , − w, } (5.6) S E { , , , } (5.6) S H { , , , } (5.6) SN I { , , , } (5.13) S L (cid:8) (5 − w ) , , − (1 + w ) , (cid:9) a = a ( t − t ) w ) S Figure 10 . The section of the invariant submanifold K = 0 , Ω = 0 in the case n = 1 , α = 1 , β = for the Hu-Sawicki model. Note that on the A > part of this invariant submanifold H appears anattractor, but has a saddle character in the unphysical A < of the phase space (not representedhere). which is analytical in R without the need of cumbersome inversions. The phase space obtainedis more regular than the one obtained with the original dynamical systems approach, althoughit is not possible to eliminate singularities in a complete way as they are due to the essential– 29 –orm of the cosmological equations. The new method also naturally excludes the fixed pointswhich represent states incompatible with the definition of the dynamical system variables.Therefore the new DSA returns a phase space which matches in a closer way the actualevolution of the cosmological equations.Using the idea of higher order cosmological parameters (like “jolt” and “snap”), thenew DSA is able to associate to the fixed points full solutions of the cosmological equations(in the sense of solutions with four integration constants). Among the fixed points foundin our analysis, the points labeled H i are surely the most interesting. They correspondto the dominance of the curvature terms and their number depends on the value of keyparameters appearing in the function f . The scale factor in H is a transcendental functionwith the remarkable property to be able to combine an initial exponential expansion a phaseof decelerated expansion and a final accelerated expansion phase. Its existence on one handdescribes clearly the role of f ( R ) -gravity as model for double inflation or a unified model forinflation, standard cosmology and dark energy. On the other, however, it confirms clearlythat f ( R ) cosmologies with these properties can run into finite time singularities. This isindicated by the fact that in many of the cases we have analysed, the solution H is the onlyattractive fixed point of the phase space (although not a global attractor). It is easy toprove with our method, and in accordance to the results in literature (see e.g. [31, 33]) thatadding a special set of additional invariants one might avoid this scenario, but in general suchavoidance requires fine-tuning.It is important to be careful in considering the nature of the points H . Their associatedsolution (5.13) contains integration constant which can take any value and it is in generalrelated to the constants appearing in f . This means that the fact that a theory posses anattractor H does not mean that the full behaviour (5.13) is necessarily realised: one couldhave, for example, that for specific values of the coupling constants and of the H i vanishes.If H and H are zero then H represents a standard de Sitter solution. This depends on thevalue of the coupling constant of the model as well as the initial condition for the model.However it is evident that the nature of (5.13) has repercussions on the understanding of theactual meaning and role of the de Sitter solutions in the context of f ( R ) - gravity.The DSA proposed has first been verified on the simple case of f ( R ) = R n and we founda complete agreement in terms of fixed points and stability with the original DSA in [11].However, the solutions associated to the fixed points present significative differences. In fact,even if the solutions obtained with the new method reduce to the ones of the old DSA settingto zero two of the four integration constants, it appears clear that their (time-)asymptoticbehaviour is different. This means that, for example, when the fixed points are attractors, theparts of the solution excluded in the original DSA can be dominant. In this respect, therefore,the original DSA returns incomplete information on the cosmology of f ( R ) -gravity.As another test of the new DSA, we have considered another form of f that was analysedwith the original dynamical system method: f ( R ) = R + αR n . In this case, it turns out thatthe only difference between the two treatment is in the number of fixed points: the new methodexcludes fixed points that give conditions inconsistent with the definition of the variables. Theremaining fixed points, however, present a stability that is completely equivalent. The theory f ( R ) = R + αR n is also the simplest theory that shows points of the type H in the finitephase space. An interesting exception in this respect is the case n = 2 and it is natural toexpect that this difference is related somehow to the special properties of this model.As last steps we have applied the new DSA to two models which were not analysablewith the original method, but at the same time constitute important theoretical models for– 30 –nflation and/or dark energy: the Starobinsky and the Hu-Sawicki models.The Starobinsky model, unfortunately, cannot be treated in general due to the complex-ity of the algebraic equations needed to determine the fixed points. We have therefore limitedour analysis to the case in which the value of the parameters are chosen to be compatiblewith Solar System and cosmological perturbations constraints given in [35]. In this specificcase, we obtain a phase space in which only two fixed points of the type H appear. Sinceonly one of the points is an attractor we can conclude that the scenario of solution (5.14) ispossible in this model, but it is not achievable for general initial conditions.The case of the Hu-Sawicki model is more involved. Only for specific intervals of theparameter n the phase space admits fixed points different form the type H . The situationis complicated by the fact that the points belong to singular submanifold. The case n = 1 has to be treated separately and reserves a number of surprises. For example, in the case n = 1 , α (cid:54) = 1 the phase space has no finite fixed points, much in the same way of the case f ( R ) = R + αR . The two models, however, differ in the asymptotic structure of the phasespace. In the case n = 1 , α = 1 we found, instead, that only the (unique) point H is alwayseffectively an attractor.The new DSA seems therefore to be very efficient in uncovering the features of interesting f ( R ) cosmologies. However, as all methodologies, the new DSA presents also a series ofdrawbacks. For example, one would like to be able to consider in an easier way the GR-likestates for the cosmology to be able to find (if they exist) "mimicking behaviours" of thesetheories, not dissimilar to the isotropization mechanism already found in scalar tensor gravity[38, 39]. In addition, the impossibility to define a set of compact variables and the fact thatour results point clearly to the presence of asymptotic fixed points makes the present analysisincomplete. The resolution of these issues in the context of the new DSA will be the focus offuture studies. Acknowledgements
This work was supported by the Fundação para a Ciência e Tecnologia through projectIF/00250/2013. The author would like to thank Dr S. Vignolo for useful discussions.
References [1] Weil, H.
Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin ,GA II, 29-42, [31], 465. See also O’ Raifeartaigh “The Dawning of Gauge Theory” (1997),24-37.[2] J. F. Donoghue, talk given at Advanced School on Effective Theories 25 Jun - 1 Jul 1995.Almunecar, Spain [gr-qc/9512024].[3] See, for example, the following reviews: T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. (2010) 451 [arXiv:0805.1726 [gr-qc]]; S. Capozziello and M. Francaviglia, Gen. Rel. Grav. (2008) 357 [arXiv:0706.1146 [astro-ph]]; S. Nojiri and S. D. Odintsov, Phys. Rept. (2011)59 [arXiv:1011.0544 [gr-qc]].[4] A. A. Starobinsky, Phys. Lett. B (1980) 99.[5] V. A. Belinsky, I. M. Khalatnikov, Zh. Eksp. Teor, Fiz. , 1700 (1969), Sov. Phys. JETP ,911 (1969); Zh. Eksp. Teor, Fiz. , 2163 (1969), Sov. Phys. JETP , 1174 (1970); Zh. Eksp. Teor, Fiz. , 314 (1970), Sov. Phys. JETP , 169 (1970); V. A. Belinsky,I. M. Khalatnikov, and E. M. Lifshitz, Usp. Fiz. Nauk , 463 (1970),
Advances in Physics – 31 – , 525 (1970); V. A. Belinsky, E. M. Lifshitz, and I. M. Khalatnikov, Zh. Eksp. Teor. Fiz. , 1969 (1971), Sov. Phys. JETP , 1061 (1971).[6] C. W. Misner, Phys. Rev. Lett. , 1071 (1969).[7] Collins, C.B., Stewart, J.M. (1971) Monthly Notices Roy. Astron. Soc. 153: pp. 419[8] see e.g. S. Capozziello, S. Carloni and A. Troisi, Recent Res. Dev. Astron. Astrophys. (2003)625 [astro-ph/0303041]; S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev.D (2004) 043528 [astro-ph/0306438]; S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth.Mod. Phys. (2007) 115 [hep-th/0601213].[9] See e.g. J.Wainwright, G. F. R. Ellis, “Dynamical systems in cosmology” CambridgeUniversity Press (2005) and references therein.[10] E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D [11] S. Carloni, P. K. S. Dunsby, S. Capozziello, A. Troisi, Class. Quant. Grav. , 4839-4868(2005). [gr-qc/0410046].[12] J. A. Leach, S. Carloni and P. K. S. Dunsby, Class. Quant. Grav. (2006) 4915[gr-qc/0603012]; S. Carloni and P. K. S. Dunsby, J. Phys. A (2007) 6919 [gr-qc/0611122];M. Abdelwahab, S. Carloni and P K. S. Dunsby, Class. Quant. Grav. (2008) 135002[arXiv:0706.1375 [gr-qc]]; S. Carloni, A. Troisi and P. K. S. Dunsby, Gen. Rel. Grav. (2009) 1757 [arXiv:0706.0452 [gr-qc]]; X. Roy, T. Buchert, S. Carloni and N. Obadia, Class.Quant. Grav. (2011) 165004 [arXiv:1103.1146 [gr-qc]]; A. Bonanno and S. Carloni, New J.Phys. (2012) 025008 [arXiv:1112.4613 [gr-qc]]; S. Carloni, S. Vignolo and L. Fabbri, Class.Quant. Grav. (2013) 205010 [arXiv:1303.5828 [gr-qc]]. S. Carloni, S. Vignolo and R. Cianci,Class. Quant. Grav. (2014) 185007 [arXiv:1401.0473 [gr-qc]]. S. Carloni, E. Elizalde andP. J. Silva, Class. Quant. Grav. (2010) 045004 [arXiv:0909.2219 [hep-th]]; G. Leon andE. N. Saridakis, JCAP (2015) 04, 031 [arXiv:1501.00488 [gr-qc]]; M. A. Skugoreva,E. N. Saridakis and A. V. Toporensky, Phys. Rev. D (2015) 4, 044023 [arXiv:1412.1502[gr-qc]]; G. Kofinas, G. Leon and E. N. Saridakis, Class. Quant. Grav. (2014) 175011[arXiv:1404.7100 [gr-qc]]; C. R. Fadragas, G. Leon and E. N. Saridakis, Class. Quant. Grav. (2014) 075018 [arXiv:1308.1658 [gr-qc]]; G. Leon and E. N. Saridakis, JCAP (2013) 025[arXiv:1211.3088 [astro-ph.CO]]; G. Leon and E. N. Saridakis, Class. Quant. Grav. (2011)065008 [arXiv:1007.3956 [gr-qc]]; Fiziev, P. and Georgieva, D. Phys. Rev. D et al. , Phys. Rev. D75 , 083504 (2007).[gr-qc/0612180].[14] S. Carloni, A. Troisi, P. K. S. Dunsby, Gen. Rel. Grav. , 1757-1776 (2009). [arXiv:0706.0452[gr-qc]].[15] M. Visser, Class. Quant. Grav. (2004) 2603 [arXiv:gr-qc/0309109].[16] V. Sahni, T. D. Saini, A. A. Starobinsky and U. Alam, JETP Lett. (2003) 201 [Pisma Zh.Eksp. Teor. Fiz. (2003) 249] [astro-ph/0201498].[17] U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. (2003) 1057 [astro-ph/0303009].[18] M. Dunajski and G. Gibbons, Class. Quant. Grav. (2008) 235012 [arXiv:0807.0207 [gr-qc]].[19] K. N. Ananda, S. Carloni and P. K. S. Dunsby, Class. Quant. Grav. 26: 235018 (2007)[arXiv:0809.3673 [astro-ph]].[20] S. Capozziello, V. F. Cardone, V. Salzano, Phys. Rev. D78 , 063504 (2008). [arXiv:0802.1583[astro-ph]].[21] S. Carloni, R. Goswami and P. K. S. Dunsby, Class. Quant. Grav. (2012) 135012[arXiv:1005.1840 [gr-qc]]. – 32 –
22] K. N. Ananda, S. Carloni, P. K. S. Dunsby, Phys. Rev.
D77 , 024033 (2008). [arXiv:0708.2258[gr-qc]].[23] S. Carloni, P. K. S. Dunsby, A. Troisi, Phys. Rev.
D77 , 024024 (2008). [arXiv:0707.0106[gr-qc]].[24] S. Carloni, Open Astron. J. (2010) 76 [arXiv:1002.3868 [gr-qc]].[25] S. Capozziello, V. F. Cardone, S. Carloni et al. , Int. J. Mod. Phys. D12 , 1969-1982 (2003).[astro-ph/0307018].[26] S. Capozziello, V. F. Cardone, A. -Troisi, Mon. Not. Roy. Astron. Soc. , 1423-1440 (2007).[astro-ph/0603522].[27] T. Clifton, J. D. Barrow, Phys. Rev.
D72 , 103005 (2005). [gr-qc/0509059].[28] I. L. Buchbinder, S. Odintsov, L. Shapiro “Effective action in quantum gravity” CRC Press(1992)[29] M. W. Hirsch, S. Smale, R. L. Devaney,
Differential equations, dynamical systems, and anintroduction to chaos , Academic press, 2012.[30] J. D. Barrow and A. C. Ottewill, J. Phys. A (1983) 2757.[31] S. Capozziello, M. De Laurentis, S. Nojiri and S. D. Odintsov, Phys. Rev. D (2009) 124007[arXiv:0903.2753 [hep-th]].[32] S. Nojiri and S. D. Odintsov, Phys. Rev. D (2008) 046006 [arXiv:0804.3519 [hep-th]];K. Bamba, S. Nojiri and S. D. Odintsov, JCAP (2008) 045 [arXiv:0807.2575 [hep-th]].[33] S. A. Appleby, R. A. Battye and A. A. Starobinsky, JCAP (2010) 005 [arXiv:0909.1737[astro-ph.CO]].[34] A. D. Dolgov and M. Kawasaki, Phys. Lett. B (2003) 1 [astro-ph/0307285].[35] A. A. Starobinsky, JETP Lett. (2007) 157 [arXiv:0706.2041 [astro-ph]].[36] W. Hu and I. Sawicki, Phys. Rev. D (2007) 064004 [arXiv:0705.1158 [astro-ph]].[37] J.Car, “Aplications of Center manifold Theory”, Springer-Verlag, NewYork,1981.[38] J. Garcia-Bellido and M. Quiros, Phys. Lett. B 243 (1990) 45; T. Damour and K. Nordtvedt,Phys. Rev. D 48 (1993) 3436.[39] S. Carloni, S. Capozziello, J. A. Leach and P. K. S. Dunsby, Class. Quant. Grav. (2008)035008 [gr-qc/0701009].(2008)035008 [gr-qc/0701009].