TTensor f ( R ) theory of gravity Tomasz Stachowiak å Department of Applied Mathematics and Physics,Graduate School of Informatics, Kyoto University,606-8501 Kyoto, Japan
August 7, 2017
Abstract
I propose an alternative f ( R ) theory of gravity constructed by applying the function f directly tothe Ricci tensor instead of the Ricci scalar. The main goal of this study is to derive the resultingmodified Einstein equations for the metric case with Levi-Civita connection, as well as for thegeneral nonmetric connection with torsion. The modification is then applied to the Robertson-Walkermetric so that the cosmological evolution corresponding to the standard model can be studied.An appealing feature is that even in the vacuum case, scenarios without initial singularity andexponential expansion can be recovered. Finally, formulae for possible observational tests are given. The foundation of the present work is to consider a modified Lagrangian (density), which dependsfunctionally on the full Ricci tensor R ab , not just on its trace R as is the case in the so-called f ( R )theories of gravity. The principles of relativity require that this modification be obtained covariantly,and not component-wise, so writing f ( R ab ) could be misleading. Since f will be a tensor-valuedfunction, for the sake of distinction from the usual f ( R ) theory, the extension will be referred to astensor f ( R ).The motivation in both cases is the same – the inclusion of higher-order-of-curvature effects whichcan classically be ignored, but which lead to important modifications in other regimes. Most notably,the Starobinsky inflation model [1] induced by quadratic terms is a particularly important result inthis spirit. Although initially introduced on quantum gravity grounds, with corrections built fromvarious contractions of the Ricci tensor, it is now often considered in the language of quadratic f ( R )theories [2].Despite the initial similarity, the tensor f ( R ) gravity presented here differs considerably from theusual one, and the goal of this article is to focus first on the development of this new theory, with acomparative study left for future work. Accordingly, the notation and mathematical setting will begiven as well as the modified Einstein equations. Not to stop at the abstract level I will also considerpossible applications to cosmology, with a view to nonsingular evolution, and provide basic formulaeto be used in observational cosmology. å [email protected] a r X i v : . [ g r- q c ] A ug otable differences and similarities with the ordinary f ( R ) theory will be pointed out throughoutthe derivations in Sections 2, 4, and 5, but for a more complete, general review of the standardapproach, the reader might want to consult review articles [3], [4], or [5] and references therein. In the usual f ( R ) theories one postulates the Lagrangian L = f (tr[ R ]) = f (cid:161) R ac g ca (cid:162) , (1)with the summation convention used, and the covariant metric tensor denoted by g ac . On purelyabstract grounds, the order in which f and trace appear is not fixed, so instead of the above I willconsider the Lagrangian to be L g = tr[ f ( R )] = [ f ( R )] ac g ca , (2)where the square brackets are used to indicate elements of a matrix, and the bare symbol R has torefer to the tensor not the scalar, as explained below.A similar idea has been studied before by Borowiec [6, 7], but it differed from the present workin two ways. First, it used a torsionless metric and second, the Lagrangian depended on polynomialinvariants of the Ricci tensor tr[ R k ]. Such scalars formed with powers k higher than the space-timedimension can be reduced to the lower ones by using the characteristic polynomial. However, thiscannot, in general, be done explicitly for transcendental functions – i.e., when one needs to use aninfinite series of powers of R . What is more, the coefficients of the characteristic polynomial themselvesdepend on the components of R , leading to an unwieldy expression of an original function of R interms of a function of the invariants tr[ R k ]. The present work aims at overcoming this problem, andalso at including connections with the most general torsion and nonmetricity.To proceed with the general treatment, the first thing to settle is what tensors and operators to use,and in particular how to interpret f ( R ). Power series immediately come to mind, so what is neededis a representation of R such that it can be composed with itself by matrix multiplication consistentwith relativistic index contraction. In other words, R should be an endomorphism, for instance on thetangent bundle over the space-time.To treat R as such an endomorphism, mixed indices have to be used so that the result composition R ab R b c is again a mixed-indices tensor of the same valence. R ab would do as well, but with the formerchoice the eigenvalue problem can be written as R ab v b = λ v a , (3)i.e., for eigenvectors rather than eigenforms, which seems more natural. The two are still equivalentthrough the musical isomorphism, and such an R is a self-adjoint operator with regard to the metric 〈 u , R ( v ) 〉 = u a g ab R b c v c = u a R ca v c = R ba u a g bc v c = 〈 R ( u ), v 〉 , (4)provided that R ab is symmetric, which is the case for the Levi-Civita connection. When one allows forthe torsion to be nonzero the above requires a generalization given in Section 4.In the bracket-component notation, f should act on R considered as a linear operator with matrixelements [ R ] ab , and should also give as the result an operator, whose elements are denoted by[ f ( R )] ab . For example, for the composition with itself it is convenient to write [ R · R ] ab = [ R ] ab , sothe superscript 2 refers to the operator power, not a component. Accordingly, R will signify the (1,1)2alence tensor, and for the Ricci scalar the contraction R aa or R will be used. After “bracketing,” theindex notation is recovered, which allows for raising and lowering; for brevity, the brackets will beomitted in the simplest cases such as [ R ] ab = R ab .For any analytic f : (cid:82) → (cid:82) the following definition of the matrix function f ∗ : (cid:82) n → (cid:82) n can beused : [ f ∗ ( R )] ab : = ∞ (cid:88) n = f n [ R n ] ab = f (cid:49) ab + f R ab + f R as R sb + f R as R st R tb + ··· , (5)where f ( ξ ) = ∞ (cid:88) n = f n ξ n , (6)and the sums are written explicitly, as they are not tensor contractions ( R n is an operator power asexplained above). The above requires that the spectral radius of ρ ( R ) = lim n →∞ (cid:107) R n (cid:107) n be less than theradius of convergence of the series f ( ξ ).For example, when f = exp, the above two Lagrangians are L = exp( R ) = + R + R + R + ···= + R aa +
12! ( R aa ) +
13! ( R aa ) + ··· , L g = tr (cid:183) (cid:49) + R + R + R + ··· (cid:184) = d + R aa + R ab R ba + R ab R b c R ca + ··· , (7)where d is the dimension of the space-time. Thus the first essential deviation appears at the quadraticlevel and is proportional to f ( R ab R ba − ( R aa ) ) if the same f is used in both approaches. The differenceis also evident when the Lagrangians are written in terms of the eigenvalues of R : L = f (cid:195)(cid:88) i λ i (cid:33) vs L g = (cid:88) i f ( λ i ). (8)A degeneracy in λ i might then lead to the same theories, e.g., when the traceless Ricci tensorvanishes: ˆ R ab : = R ab − d R (cid:49) ab =
0. The Ricci tensor is then proportional to the identity matrix and[ f ( R )] aa = d f ( R / d ), which, up to a simple rescaling of f , is the same as the Lagrangian L . However,one has to be careful when making such substitutions directly in the action, because R is determinedonly after having solved the Einstein equations. If the assumption ˆ R = f is determined, there is no freedom of choice for its constant term f , whichnaturally corresponds to the cosmological constant. In other words, in such a nonperturbativeinterpretation, its value is tied to the whole expansion and cannot be adjusted independently. Theexpansion around R = f is almost linear, then higher-order terms can beignored in the weak field limit leading to the Einstein-Hilbert action and a small perturbation ofgeneral relativity. 3lthough intuitive, the above definition is not very convenient when a function is real analyticbut has complex singularities like tanh( ξ ). A definition better suited for the situation at hand is anelegant generalization of Cauchy’s formula f ( R ) : = π i (cid:90) C ( ξ (cid:49) − R ) − f ( ξ ) d ξ , (9)for a contour C which encloses the spectrum of R but not the singularities of f ( ξ ). The two definitionsagree for fairly general assumptions, and for a function that is real on the real axis, the matrix f ( R )will also be real [8].The dimension of f ( R ) affects how the function is given, because R has the units of curvature, andso should the Lagrangian. At first, it seems two constants are necessary to give f ( R ) = C ˜ f ( R / C ) interms of a function ˜ f which only contains dimensionless parameters, but this can be rewritten as f ( R ) = C ˜ f (cid:181) C C RC (cid:182) → C ˜ f ( R / C ), (10)with a redefined dimensionless ˜ f . The remaining constant C can then be further rescaled using thecosmological or the Hubble constant depending on context – this is done in Section 5.Having defined tr[ f ( R )], the total action, including the matter Lagrangian L M , is taken to be S = (cid:90) (cid:181) π G L g + L M (cid:182) (cid:112)− g d x , (11)where G is the gravitational constant, and the modified Einstein equations can then be obtained inone of the two standard ways. One is to assume the Levi-Civita connection and take the metric as thedynamical variable; the other is to consider both the metric and the connection as dynamical. Theformer is called the metric and the latter the Palatini formulation (or, more generally, metric-affine).In both cases the variation of the f ( R ) term is needed, and the second definition of a tensor functionallows us to easily calculate it as δ tr[ f ( R )] = tr π i (cid:90) C ( ξ (cid:49) − R ) − δ R ( ξ (cid:49) − R ) − f ( ξ ) d ξ = tr π i (cid:90) C ( ξ (cid:49) − R ) − f ( ξ ) δ R d ξ = tr π i (cid:90) C ( ξ (cid:49) − R ) − f (cid:48) ( ξ ) d ξ δ R = tr (cid:163) f (cid:48) ( R ) δ R (cid:164) , (12)where the cyclic property of tracetr[ X X ... X k ] = tr[ X k X X ... X k − ], (13)was used in the first line, and integration by parts in the second. Reexpressing δ R with δ g and δ Γ toarrive at the modified Einstein equation is the subject of the next two sections. In what follows, f and f ∗ can safely be treated as the same object, so the star will be dropped. .1 Definitions and notation To shortly review the conventions used, the covariant derivative and the connection coefficients ina basis { e a } are related through ∇ e a e b = Γ cba e c , (14)so that for a coordinate basis e a = ∂ a one has ∇ a X b = ∂ a X b + Γ b ca X c . (15)As Γ will not in general be symmetric in the lower indices, care needs to be taken regarding theirorder. The antisymmetric part of the connection defines the torsion as T ( X , Y ) : = ∇ X Y − ∇ Y X − [ X , Y ] = e a T abc X b Y c , (16)and in a coordinate basis, where [ ∂ a , ∂ b ] =
0, it follows that T abc = Γ acb − Γ abc . (17)The Riemann tensor is given by R ( X , Y ) Z : = ∇ [ X ∇ Y ] Z − ∇ [ X , Y ] Z = e d R dabc Z a X b Y c , (18)or, in term of components in a coordinate basis, R dabc = ∂ b Γ dac − ∂ c Γ dab + Γ d sb Γ sac − Γ d sc Γ sab , (19)and the Ricci tensor is the contraction R ab : = R cacb . (20)Note, then that although R ab is constructed solely with the connection (curvature), for the operator R ab = g ac R cb the metric is necessary. Finally, the signature will be taken to be ( − , + , + , + ), and thespeed of light equal to unity, so that coordinates have the dimension of length, and the metric itself isdimensionless. The natural connection solely determined by the metric through ∇ a g bc = T abc = δ g , is δ Γ cba = g cd ( ∇ b δ g ad + ∇ a δ g db − ∇ d δ g ba ), (21)and in turn for the covariant Ricci tensor one has δ R ab = ∇ c ( δ Γ cab ) − ∇ b ( δ Γ cac ), (22) The brackets involving vectors denote commutation not antisymmetrization – i.e., there is no prefactor of . δ R ab = (cid:179) ∇ d ∇ b δ g ad + ∇ d ∇ a δ g db − ∇ d ∇ d δ g ba (cid:180) +− (cid:179) g cd ∇ b ∇ a δ g dc + ∇ b ∇ d δ g ad − ∇ b ∇ d g cd δ g ca (cid:180) = (cid:179) ∇ d ∇ b δ g ad + ∇ d ∇ a δ g bd − (cid:3) δ g ab − g cd ∇ b ∇ a δ g cd (cid:180) . (23)Next, by observing that 0 = δ ( (cid:49) ac ) = g bc δ g ab + g ab δ g bc , (24)the variation of the operator R becomes δ R ab = g ac ( δ R cb − R sb δ g cs ), (25)leading to δ (cid:161) tr[ f ( R )] (cid:112)− g (cid:162) = (cid:179) [ f (cid:48) ( R )] ab δ R ba + tr[ f ( R )] g bd δ g bd (cid:180) (cid:112)− g = (cid:179) [ f (cid:48) ( R )] ac δ R ca − [ R f (cid:48) ( R )] cd δ g dc + tr[ f ( R )] g bd δ g bd (cid:180) (cid:112)− g . (26)The variation δ R ab of (23) can be substituted into the above, and due to (cid:112)− g ∇ a X a = ∂ a (cid:161) (cid:112)− gX a (cid:162) each term containing the covariant derivative can be integrated by parts provided that the variationsvanish at the boundary or that the boundary is empty. The result is δ (cid:161) tr[ f ( R )] (cid:112)− g (cid:162) = (cid:179) ∇ c ∇ d [ f (cid:48) ( R )] cb − (cid:3) [ f (cid:48) ( R )] bd − ∇ a ∇ c [ f (cid:48) ( R )] ac g bd − [ R f (cid:48) ( R )] bd + [ f ( R )] aa g bd (cid:180) (cid:112)− g δ g bd . (27)Finally, defining the stress-energy tensor T by δ ( (cid:112)− g L M ) δ g bd = : 12 T bd (cid:112)− g , (28)the condition δ S = (cid:3) [ f (cid:48) ( R )] bd − ∇ c ∇ b [ f (cid:48) ( R )] cd + ∇ a ∇ c [ f (cid:48) ( R )] ac g bd + [ R f (cid:48) ( R )] bd − tr[ f ( R )] g bd = π G T bd . (29)As can be seen, the last two terms on the left-hand side reduce to the standard Einstein tensor for f = Id, whereas the other terms are zero since f (cid:48) = In the more general case, the connection is independent of the metric, and there are two assump-tions that can be relaxed here: vanishing torsion and metric compatibility. In general the connectioncan be decomposed into the sum Γ abc = (cid:101) Γ abc + K abc − C abc , K abc : = −
12 ( T abc + T bca − T cab ), (30)6here (cid:101) Γ is the Levi-Civita connection for g , K is called the contorsion tensor, and C describes thenonmetricity C abc : =
12 ( ∇ c g ab + ∇ b g ca − ∇ a g bc ). (31)Accordingly, the variation of the Ricci tensor is now δ R ab = ∇ c ( δ Γ cab ) − ∇ b ( δ Γ cac ) − T d bc δ Γ cad , (32)and neither the connection coefficients nor the Ricci tensor are symmetric in the lower indices. Theeigenvalues of R might not be real any more, in which case they appear in conjugate pairs. This meansthat the trace of f ( R ) will still be real, for real analytic f .There is, however, a possible natural generalization, because of the following identity R ab = R ba + ∇ a T c cb + T c cd T dab , (33)which leads to the introduction of a new tensor, which is the symmetric part of R , S ab : = R ab −
12 ( ∇ a T c cb + T c cd T dab ). (34)These tensors have the same trace so there is no need for S ab in the standard f ( R ) theories – the tracecancels the imaginary parts of the conjugate pairs of the eigenvalues. Here, the situation is different,because the function f is applied to the eigenvalues of R before the trace is taken, so although thefinal result is real, it also depends on the imaginary parts. The other reasons and equations for the f ( S ) variant are given following the f ( R ) derivation below.In contrast to the preceding section, only first derivatives are present in the action, and theintegration by parts requires an additional term, because the torsion affects the expression forcovariant divergence: (cid:112)− g ∇ a (cid:161) X a (cid:162) = ∂ a (cid:161) (cid:112)− g X a (cid:162) + (cid:112)− g (cid:179) T bba − C bba (cid:180) X a . (35)The total variation of the Lagrangian then becomes δ (cid:161) tr[ f ( R )] (cid:112)− g (cid:162) = (cid:179) P ba δ R ab − [ R f (cid:48) ( R )] db δ g bd + [ f ( R )] aa g bd δ g bd (cid:180) (cid:112)− g = (cid:179) tr[ f ( R )] g bd − [ R f (cid:48) ( R )] db (cid:180) δ g bd (cid:112)− g + (cid:179) ∇ b P ba (cid:49) d c − ∇ c P da − T d bc P ba + ( C ssb − T ssb ) P ba (cid:49) d c − ( C ssc − T ssc ) P da (cid:180) δ Γ cad (cid:112)− g , (36)where the derivative tensor is denoted by P ab : = [ f (cid:48) ( R )] ab for brevity.In addition to the stress-energy tensor T , a new quantity is necessary to reflect the fact thatmatter fields can, in general, depend on the connection – if only through the covariant derivative. Thehyper-momentum tensor is defined thus: (cid:112)− g Q abc : = δ (cid:161) (cid:112)− g L M (cid:162) δ Γ abc , (37) The underline denotes the sum over cyclic permutations. π G T bd = [ R f (cid:48) ( R )] ( db ) − tr[ f ( R )] g bd ,8 π GQ cad = ∇ b (cid:179) (cid:49) [ b c P d ] a (cid:180) − ( T − C ) ssb (cid:49) [ b c P d ] a − T d cb P ba , (38)where the symmetrization is necessary, because the variation δ g bd is symmetric, even though R bd isnot.The second set of equations can be simplified if an auxiliary connection is defined to beˆ Γ abc : = Γ abc − (cid:49) ab ( T − C ) ssc , (39)and using the associated covariant derivative ˆ ∇ , the second set of Einstein equations reads8 π GQ cad = ˆ ∇ b (cid:179) (cid:49) [ b c P d ] a (cid:180) − T d cb P ba . (40)Additionally, contraction over the pair of indices { cd } leads to3 ˆ ∇ b P ba + T ssb P ba + π GQ sas =
0, (41)which allows us to rewrite the main equations as[
R f (cid:48) ( R )] ( db ) − tr[ f ( R )] g bd = π G T bd ,ˆ ∇ c P da − (cid:179) T d cb − T ssb (cid:49) d c (cid:180) P ba = π G (cid:179) Q cad − Q sas (cid:49) d c (cid:180) . (42)As in the ordinary f ( R ) formulation, the torsion equations become algebraic for the Einstein-Hilbert case f ( R ) = R because P ab = [ f (cid:48) ( R )] ab = g ab , so that derivatives of Γ only appear in R . Further,if the matter fields are such that Q abc ≡
0, contractions of the torsion equations give3 C ssa = − C ass = T ssa ,2 C ( ad ) c = T dac + T ss ( a g c ) d (43)This means that if T ssa =
0, then 2 C ( ad ) c = T dac , and it follows immediately from (30) that K abc = C abc .But that, by definition, means the connection must be the Levi-Civita one.In other words, for f ( R ) = R , zero hyper-momentum and totally antisymmetric torsion, the theorybecomes standard general relativity. Note that for this to happen it is not necessary to assume zerotorsion from the beginning, just that all its traces vanish.Since the Ricci tensor is, in general, no longer symmetric, the tensor P cannot be used directly todefine a new metric for which equation (42) would define a metric connection. In the standard f ( R )theories, the tensor that enters is R itself, and it can be decomposed into (anti)symmetric parts at thelevel of the Einstein equations, as the function f is applied only to its trace, and all f ( R aa ) terms arejust scalars.Here, the situation is different in that even in the first set of equations the symmetrization isapplied to R f (cid:48) ( R ), not to R , and the second set of equations contains f ( R ), not f (cid:48) ( R ). Because even forthe second power one has g bc X c ( d X a ) b (cid:54)= X ( ab ) g bc X ( cd ) , symmetrizing the equations would not lead toa single distinguished tensor to be used as the new metric. Moreover, even though the components of R are real, it seems natural to consider a self-adjoint matrix, for which the action is directly related tothe eigenvalues as in (8). 8hese problems could be overcome by constructing the action with the symmetric tensor S ,introduced before, whose variation is simply δ S ab = ( δ R ab + δ R ba ). The derivation is essentially thesame as in (36), and the difference is that the tensor contracted with δ g is already symmetric, so theEinstein equations are 8 π G T bd = [ S f (cid:48) ( S )] db − tr[ f ( S )] g bd ,8 π GQ cad = ˆ ∇ b (cid:179) (cid:49) [ b c P d ] a (cid:180) − T d cb P ba , (44)where now, by a slight abuse of notation, P ab = [ f (cid:48) ( S )] ab , and the auxiliary covariant derivative is theone given by equation (39).As before, the trace can be used to rewrite the second equation as (42), and following the samereasoning as for the standard f ( R ) derivation, the torsionless connection with no hyper-momentumyields ˆ ∇ c P da =
0. (45)This would indicate that ˆ Γ is the Levi-Civita connection for the metric P da , but the situation iscomplicated by the fact that the tensor P = f (cid:48) ( R ) is not conformally related to the original metric g , sothe signature might not be the same, and the determinant of g is not directly proportional to that of P ; also, raising of indices in P does not amount to matrix inversion. It should also be kept in mindthat with the standard extension of covariant derivative to tensor densities, which uses (cid:112) | g | to cancelthe weight, the above equation can be rewritten as1 (cid:112) | g | ˆ ∇ c (cid:179)(cid:112) | g | P da (cid:180) =
0, (46)but this is not equivalent to 1 (cid:112) | det P | ˆ ∇ c (cid:179)(cid:112) | det P | P da (cid:180) =
0, (47)unless (cid:112) det P is used to extend ˆ ∇ to densities. Without specifying which extension is used, thecondition ∇ c ( (cid:112) | g | g ab ) = P , the morefundamental equation (45) is better as an indication of a metric connection in the present case.With some effort, the Christoffel formula can be used to express ˆ Γ as a function of derivatives of P ,but the derivatives of the connection coefficients are still involved in the nonlinear term f (cid:48) ( S ). Thequestion is then whether they can be eliminated with the help of the remaining equations.In the standard approach, the first set of the Einstein equations (44) can, in principle, be used tosolve for the Ricci scalar and accordingly simplify the second set by using the Ricci tensor associatedwith the new metric and its Levi-Civita connection [3]. Here, one would have to solve nonlinearequations for the whole tensor S in order to eliminate the connection in the same manner. At present,it appears that this path of investigation is not applicable, because the equations involve full tensors R or S , not just their traces. The standard cosmological model is the basic example that needs to be considered in order to gaininsight into the applicability of the proposed modification. The model assumes spatial homogeneity9nd isotropy, requiring the Robertson-Walker geometry, which in spherical coordinates { t , r , θ , ϕ } hasthe metric d s = − d t + a ( t ) (cid:181) d r − kr + r d Ω (cid:182) = g ab d x a d x b , (48)where d Ω = d θ + sin θ d ϕ is the standard metric on the unit sphere. The final assumption in thisfirst attempt at modified cosmology will be that the RW metric provides the only dynamical variable –the scale factor a ( t ) – the connection is that of Levi-Civita and the metric formalism can be used.Accordingly, the matter source will be taken to be a homogeneous perfect fluid with density ρ andpressure p , so that the stress energy tensor is T ab = p g ab + ( p + ρ ) u a u b , (49)where the four-velocity in these coordinates is just u = ∂ t .There are then effectively only two modified Einstein equations, one of third order and one offourth corresponding to the T and T components of (29) respectively. However, the latter followsfrom the derivative of the former, which is the generalization of the Friedmann equation H (cid:161) f (cid:48)(cid:48) ( λ ) + f (cid:48)(cid:48) ( λ ) (cid:162) d λ d t = π G ρ + H (cid:161) ( λ − λ ) f (cid:48)(cid:48) ( λ ) + f (cid:48) ( λ ) − f (cid:48) ( λ ) (cid:162) + λ (cid:161) f (cid:48) ( λ ) + f (cid:48) ( λ ) (cid:162) − f ( λ ) − f ( λ ) (50)where λ are the eigenvalues of R λ = aa , λ = H + ka + .. aa , (51) H is the Hubble “constant” H = . a / a , and the overdot denotes the time derivative.The present value of the constant, H : = H (0), is customarily used to obtain dimensionless quanti-ties and, as discussed in Section 2, there is still an unspecified constant in the function f . Although H has the suitable dimension, it will not do as C , because the function f should be a fundamentalquantity valid for all gravitational actions, not just the FRW cosmology, and thus cannot be definedwith such specific constants. Instead, C will become a physical parameter of the new theory, and theHubble constant H will serve to provide the dimensionless counterpart c : = C H − .Of course, the roles could be reversed, with C used instead of H , but for initial clarity it is betterto keep with the convention of rescaling densities, time, etc., with H . The dimensionless eigenvaluesare then α : = λ H − , β : = λ H − , (52)which gives e.g. f ( λ ) = f (cid:161) α H − (cid:162) and leads to further simplification H − f ( λ ) = c ˜ f ( α / c ) = : F ( α ), (53)and similarly for β . The main equation can then be rewritten as h (3 F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β )) d α d τ = Ω + h (cid:161) ( β − α ) F (cid:48)(cid:48) ( β ) + F (cid:48) ( β ) − F (cid:48) ( α ) (cid:162) + α ( F (cid:48) ( α ) + F (cid:48) ( β )) − F ( α ) − F ( β ), (54)10here h , the density parameter and dimensionless time are defined by h : = HH , Ω : = π G ρ H and τ : = H t . (55)The function F can then be specified with any suitable number of dimensionless parametersincluding c . It could be considered to be given a priori by some elementary function like A sin( B ξ ), ordefined by infinitely many expansion coefficients as the series (6). Yet to consider such coefficients asindependent parameters would be to multiply entities beyond necessity, so I will adopt the formerapproach here.A quantitative reason can also be given for this, in anticipation of the observational analysis.Finding the coefficients from the data would undoubtedly lead to better and better fits as the numberof coefficients increases, but such a fit would come with a huge cost as measured by the Akaike orBayesian information criteria, which are now standard tools of observational cosmology [9, 10].As for the nature of parameters in the present case, some more information can be gleaned from thezeroth- and first-order expansions of F , as they reproduce the standard model with the cosmologicalconstant. The general form is F ( ξ ) = F + F ξ , but the overall rescaling of the Lagrangian is notimportant, and taking F = H = π G ρ − ka − H F , (56)upon identifying the cosmological constant Λ = − H F . In terms of the original function f , this meansthat f = − Λ /2, and it suggests that the cosmological constant itself could be used as a fundamentaldimensional quantity by f ( ξ ) = Λ f (cid:181) ξ Λ (cid:182) , (57)with ˜ f carrying no other free parameters. Using the respective density parameter Ω Λ : = Λ /3, thismeans that F ( α ) = H f ( λ ) = Ω Λ f (cid:181) α Ω Λ (cid:182) , (58)where the expansion of ˜ f is then necessarily restricted to˜ f ( ξ ) = − + ξ + O ( ξ ). (59)Turning now to the dynamics of this model, a minimal set of variables yielding a closed system canbe built from the derivatives of a ( t ), or rather their rescaled versions h and α , which are identicallyrelated by α =
3( . h + h ). Also, the other of the eigenvalues can be eliminated through β = h + Ω k a + α Ω k : = kH , (60)although for shorter notation it will be better to keep the symbol β and understand it as a function of a , h and α , which will be the replacements for . a , .. a and ... a . As before, ξ is just an auxiliary independent variable used to define functions and their rescalings. ∇ a T ab = ρ is expressible interms of a if one assumes an equation of state p = ( γ − ρ =⇒ ρ ( t ) = ρ a ( t ) − γ =⇒ Ω = (cid:88) j Ω j a − γ j . (61)Finally, introducing W : = h (cid:161) ( β − α ) F (cid:48)(cid:48) ( β ) + F (cid:48) ( β ) − F (cid:48) ( α ) (cid:162) + α ( F (cid:48) ( α ) + F (cid:48) ( β )) − F ( α ) − F ( β ), (62)for the sake of brevity, a dynamical system with three degrees of freedom described by the variables { α , h , a } is obtained: . α = Ω + W ( α , β , h , a ) (cid:161) F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β ) (cid:162) h = : v ( α , h , a ),. h = α − h = : v ( α , h , a ),. a = ah = : v ( α , h , a ), (63)where the dot now refers to the new time τ . Note that the denominator of v would only be identicallyzero for the purely linear F , which is the standard general relativity. The form of v and v is dictatedby the definition of h and the essential dynamics lies with v . This is also where we find the differencein complexity between the new theory and f ( R ), for which v and v are the same, but the firstequation would read . α = β − α ) h + Ω − F (3 β + α ) + α F (cid:48) (3 β + α )12 hF (cid:48)(cid:48) (3 β + α ) . (64)The difference between the two equations in the simplest quadratic case F ( ξ ) = − Ω Λ + ξ + F ξ isjust ( Ω Λ + Ω − h )/(2 F h ), which is nonzero exactly when the evolution deviates from the Friedmannequation. As was mentioned in Section 2, if ˆ R vanishes, then a simple rescaling of F also leads to thesame equations but in this particular geometry the condition is very restrictive. For flat universes (asin the examples below) the only solutions with this property are the de Sitter ones, h = const, whichdo not exhaust all possible solutions, even when Ω =
0. On the other hand, the difference disappearscompletely if we take different functions: F ( ξ ) = F + ξ + F ξ for f ( R ) and (cid:101) F ( ξ ) = F + ξ + F ξ for f ( R ); the theories are equivalent for the Robertson-Walker geometry at the quadratic level, evenwhen ˆ R (cid:54)=
0. However, no such simple relation could be found for cubic terms.A general feature of the main system (63) is that if the geometry is flat, i.e., k = Ω , but instead consider the higher-order terms of F as some sort of field imitatingmatter. For example, if F ( ξ ) = − Ω f + ξ + F ξ , the main equation (54) becomes h = Ω f + F (cid:179) . h − h .. h − h . h + h (cid:180) , (65)so that Ω f acts as dark energy and the F term acts as effective material content.Another general, and problematic, feature of the . α equation is the singularity at h =
0, i.e., whenexpansion changes to contraction and vice versa. This is not a singularity of equation (54) and canlead to a valid solution provided that the numerator of v vanishes as well. Thus, care has to betaken when using the dynamical system form, because the singularities might simply signify that theleft-hand side of the original equation is zero, and vice versa: a zero of v might in fact be a singularityof the original equation (63). 12 .1 Examples of cosmological models A very basic example illustrating these features is to take a flat, empty universe and assume theexponential function F ( ξ ) = Ω f e ξ Ω f − = ξ + ξ Ω f + O ( ξ ), (66)which includes the linear action, but no cosmological constant in the usual sense. The specific form of v is then v (cid:181) Ω f α , (cid:113) Ω f h , a (cid:182) = − α + e α (3 α − h − + h ( h + (1 − α ) h + α − (cid:113) Ω f (3e α + e h ) h , (67)where the additional factors in the arguments are only introduced to shorten the formula. It is stillessentially transcendental, so one has to resort to qualitative analysis first to locate the points andregions of interest. This can be done with the help of Figure 1, which shows the planar vector field( v , v ) together with the locations of singular lines and zeros of the right-hand side v (left panel), andthe phase portrait constructed from typical trajectories (right panel); the particular value of Ω f = - - - h α A A B α = h - - - - h α A A B Figure 1:
The vector field (63) and its phase space diagram for a flat empty universe with exponential Lagrangianfunction (66) and Ω f = v respectively. The blue parabola corresponds to v =
0. Because of huge variation, the vector lengths are notdrawn to scale to better show the discontinuity of direction at the singular line.
The left and right saddle points A and A correspond to time-reversed de Sitter and standard deSitter solutions, respectively, and their positions ( h ,3 h ) are given by h = ± (cid:112) Ω f w /3, where w isthe positive solution of e − w + w =
1. 13he singular critical point B could be considered as a static solution because it lies on the singularline h =
0, but also on the W = B is not well defined, as it depends on the path.Importantly, there are no periodic orbits on either side of B , as the line h = B into two elliptic sectors of opening π . The “closed” trajectories have B as theirlimit point, so they are asymptotically static both in the past and in the future.More physically realistic evolutions here seem to consist of trajectories that are attracted by A and subsequently scattered along the unstable direction towards infinity. These are expandinguniverses with ever increasing acceleration, and also with initial singularity, which can be read fromthe phase portrait: going back back in time, the trajectory has increasingly negative α , and discardingthe exponentially small terms for large h and α the right-hand side is approximately. α = h , . h = − h , (68)making α and h diverge in finite (negative) time.There are also two mixed cases – i.e., trajectories going from a big bang becoming asymptoticallystatic as they tend to B and vice versa: asymptotically static in the past, but then getting scatteredby A into accelerated expansion. These exemplary behaviours of the scale factor and the Hubbleconstant are plotted in Figure 2. Note that the time integration constant τ such that a ( τ ) = h ( τ ) =
1, so it is adjusted for each trajectory for better visibility in this andsubsequent graphs. t - a t h Figure 2:
Behaviour of a and h for typical flat, empty universes corresponding to (66). Blue and orange curvesare trajectories which have infinite h in the past, but the former escapes to infinite h while the latter is trappedby B . Green and red curves both start at B in the infinite past, but the former escapes while the latter isrecaptured. It is probably more instructive to consider a more intricate model, which is furnished by taking arational function ˜ f = − + ξ − ξ =⇒ F ( ξ ) = − Ω Λ + ξ + ξ Ω Λ + O ( ξ ), (69)which includes the constant term, so it can be identified with the cosmological constant as in (58).Note that if the series were to be used, different expansions in different regions would be required.14he reduction of the resulting powers of R with the characteristic polynomial would have to be carriedout separately, which would lead to cumbersome expressions – if it were possible to obtain closed onesat all.Direct substitution of this F into (63) produces a v which is several lines long, so it is perhaps bestto skip its specific form and, similarly to before, view the vector field and the various singular lines ofthe phase space; they are shown in the left panel of Figure 3. The picture is now considerably morecomplex, with many more singular points of type B , for which both the numerator and denominatorin . α vanish. These points signify a possible crossings through the otherwise impassable barriersindicated by the red lines.There are still only two critical points A , A located at (cid:179) ∓ (cid:113) Ω Λ , Ω Λ (cid:180) , which are asymptoticequilibria, and as before, they correspond to time-reversed de Sitter and standard de Sitter solutions,respectively. However, as the phase diagram of Figure 3 shows, there are now two heteroclinictrajectories connecting them, one through B at (cid:161) Ω Λ (cid:162) and the other through B at (cid:161) Ω Λ (cid:162) . - - - - h α A A B B B B B B α = h - - - - h α A A B B B B B B Figure 3:
The vector field (63) and its phase portrait for a flat empty universe with rational F given by (69) andwith Ω Λ =
3. The field is singular on the red curves. The green curves represent a vanishing numerator of v ,and the blue parabola corresponds to . h =
0. Because of huge variation, the vector lengths are not drawn to scaleto better show the discontinuity of direction at the singular lines.
There is a complication here, not present in the previous example, though. The horizontal greenlines at ± Ω Λ are singularities of F , and so also of the Friedmann equation, but they cancel out in v ,resulting in the straight-line trajectories. These are not singularities of curvature either, because α and h remain finite, so if one considers the action principle as purely formal to obtain the dynamicalequations, these solutions could have some physical meaning.A similar situation is found for the pair B and B located at (cid:179) ± (cid:112) Ω Λ , − Ω Λ (cid:180) , except that thewhole line can be thought of as just one trajectory for which h goes from ∞ to −∞ in finite time. On15oth lines, the second equation . h = v can be integrated to give h = (cid:113) ± Ω Λ tanh (cid:181)(cid:113) ± Ω Λ ( τ − τ ) (cid:182) =⇒ a = cosh (cid:181)(cid:113) ± Ω Λ ( τ − τ ) (cid:182) (70)where the integration constant τ can be complex, giving in effect three types of functions: tangent forthe trajectory on the lower line, hyperbolic tangent for the A A segment, and hyperbolic cotangentfor the trajectories on the upper line that escape to ±∞ . The dependence of the scale factor and h on time for these cases is shown in Figure 4. Additionally, the trajectories coming from infinityqualitatively reflect the behaviour of the generic trajectories in the respective region in Figure 3; inparticular, the past singularity is reached in finite time. - - τ a - - τ - - - h Figure 4:
Behaviour of a and h for the singular lines α = ± Ω Λ of Figure 3. The blue trajectory goes through B B B , the orange one connects A to A , and the green one has A and A as limit points. Outside the singular lines, there are the two special heteroclinic orbits: from A through B to A and from B to A . The first is possible, because the equation can be regularized by considering α as afunction of h so that α (cid:48) ( h ) = v / v , which leads to a local expansion at B α = Ω Λ − h + O (cid:161) h (cid:162) . (71)This trajectory is similar to the one through B but avoids the problem of singular action. The secondcase, upon closer inspection, also admits continuation through B , as is revealed by switching again to h : = h − (cid:112) Ω Λ as the independent variable. The series for α can then be found α = − Ω Λ − (cid:112) h + O (cid:161) h (cid:162) . (72)Both of these solutions are shown in Figure 5, the first is probably the best candidate for a “bounce”universe, and the second has a big-bang singularity.Looking more closely at the behaviour at infinity also reveals an asymptotic relation of the form α ∼ − h , which, together with the two previous expansions, suggests looking for the equation of theextended separatrix involving α + h . Indeed, it turns out that there is a parabola through B , A , B , A B given by U : = α + h − Ω Λ =
0, (73)16 - - τ a - - - τ - - h Figure 5:
The nontrivial heteroclinic trajectories of Figure 3: the blue line corresponds to the one connecting A and A through B , and the orange line to the one coming from infinity to A through B . which is an invariant set, i.e, d U d τ (cid:175)(cid:175)(cid:175)(cid:175) U = =
0, (74)as can be checked by direct substitution.Eliminating α from U = h = Ω Λ − h , which again givestrigonometric solutions for h and a akin to (70) – in particular, for the big-bang type a = sinh (cid:181)(cid:113) Ω Λ ( τ − τ ) (cid:182) ∼ ( τ − τ ) , (75)which is the behaviour of the standard Friedmann cosmology with the so-called stiff matter character-ized by p = ρ . The same equation of state holds also for a minimally coupled massless scalar field φ for which the energy density is just the kinetic term ρ = ˙ φ , or approximately when the potentialterm can be neglected: ˙ φ (cid:192) V ( φ ). This suggests a correspondence analogous to that of standard R theories, which are conformally equivalent to scalar field cosmologies [11].The introduction of matter through a nonzero Ω term means that the system (63) can no longer besimply visualized on a plane, but particular solutions can still easily be obtained numerically. Themost important ingredient would be dust matter ( γ = γ = ), but sincethe latter constitutes a tiny fraction of Ω in the standard Λ CDM model, Ω = Ω m a − was assumed inthe numerical integration. Thus, this particular model will depart from reality close to the Big Bangby ignoring the radiationr-dominated GUT era and the inflationary phase, when the value of Λ ismuch larger than the Λ CDM one used below.A surprising property to notice is that the parabola (73) is still an invariant set, and accordinglyequation (75) gives a bing-bang solution also with dust. This is due to the singular nature of thedenominator in the Ω / F (cid:48)(cid:48) term in v . Although this means that the stiff matter component dominatesin the earliest epochs, the “effective equation of state” p / ρ changes with time as the de Sitter state isreached. By analogy with the standard Friedmann cosmology, one can eliminate p from the secondEinstein equation to obtain the time-dependent adiabatic index as γ τ = (cid:181) − a .. a . a (cid:182) = (cid:181) − α h (cid:182) . (76)17his function can be used to compare the behaviour of the density for the present model and thecorresponding Friedmann equation including the stiff matter term, i.e., h = Ω Λ + Ω m a − + Ω s a − , Ω Λ + Ω m + Ω s =
1. (77)The comparison is shown in Figure 6. τ / / / γ Figure 6:
The time-dependent index of the equation of state p = ( γ − ρ . The orange line corresponds to the Λ CDM model (77) with Ω Λ = Ω m = Ω m = Ω s = F given by (69), Ω Λ = Ω m = A : the former through B and the latterfrom the right of the first quadrant (see Figure 3). In the present case, there is no constraint on the sum of all the Ω terms, and Ω Λ and Ω m neednot be the same as in the Λ CDM model, because the Einstein equations are different. The parametervalues are both subject to estimation from observations, but for the present qualitative comparison onecan use the asymptotic behaviour of (75): a ∼ exp( (cid:112) Ω Λ /2 τ ), which should correspond to the relevantasymptotics of Λ CDM, i.e., a ∼ exp( (cid:112) τ ), so that Ω Λ = f ( R ) equations.At any rate, the comparison shows that the universe whose trajectory lies in the first quadrant(Figure 3) and tends to the de Sitter attractor A has γ = a − ) must be cancelled close to the initial singularity, so that only the a − term matters instead. Ithappens due to the trajectory approaching the horizontal singular line of α = Ω Λ /2 so asymptoticallythe solution (70) holds and a ∼ ( τ − τ ). The transition from cosmic strings directly to exponentialexpansion makes this class of trajectories unlikely as physical models.The heteroclinic trajectory (green in Figure 6) is unchanged by dust with γ ≈ Λ CDMwith stiff matter (blue), but the agreement is much better than in the previous scenario. The shaperesembles more that of the standard Λ CDM (orange) in that there is no cusp, although different typesof matter dominate initially. This in itself is not an obstacle, as it is unlikely that classical GR anddustlike matter determine the initial singularity anyway, and in the bouncing scenarios a (cid:48)(cid:48) (0) =
0, sothat γ could even tend to infinity. 18n interesting analogy here is that the heteroclinic trajectory is unchanged by the addition ofdust, so that it can be thought of as defined purely by the geometry and the function f ( R ) – quiteas the cosmological constant can be thought of as a geometric term rather than an actual materialcomponent. In both cases, such content-independent gravity only makes sense as a model for the latehomogeneous universe, not at smaller scales like black holes. Note also that this particular example(69) was deliberately chosen with a singularity so that it cannot be treated perturbatively. By itself, itmay not be a replacement for Λ CDM, but its most prominent feature, the invariant manifold U = F ( ξ ) is that the phase space is cut into several regions by the red lines and the trajectories cannot becontinued through them even with local analysis because the vector field’s directions are opposite oneach side. Nevertheless, A is a steady state attractor for almost the whole first quadrant, and thereare two heteroclinic scenarios without singularities.This behaviour is more pronounced when one considers more peculiar setups – for example, withthe periodic Lagrangian F ( ξ ) = Ω Λ (cid:181) − + tan (cid:181) ξ Ω Λ (cid:182)(cid:182) . (78)Because F enters the equations with the rescaled eigenvalues α and β as its arguments, it is moreconvenient to eliminate h and use the eigenvalues as the dependent variables. In order to do that, arescaled time d σ : = d τ / h can be used, giving for the flat case d α d σ = Ω + W F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β ) ,d β d σ = Ω + W F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β ) − ( α − β ) (cid:161) α − β (cid:162) . (79)This setup gives rise to a period cell structure of the phase space, as seen in Figure 7, and there areinfinitely many critical points and heteroclinic orbits to choose from.At present this cannot be considered to be more than a toy model, but it hints at the possibilityof constructing a phase space with compartments for different epochs of evolution separated by thesingular lines and transitions taking place through the critical points. The behaviours of h and a would need to be recovered from that of α and β in order to give physical interpretation, and at firstglance, it is hard to judge whether the complexity comes from the choice of dependent variables, or isan intrinsic feature of the tensor f ( R ) theory.The determination of the actual (real, if one can call it that) F ( ξ ), or f , is a question in itself, andat present it is hard to imagine what other fundamental theory could provide it. At the very leastit should be constrained by observations, but some new approach will be required not to merely fitsubsequent polynomial approximations of a series if one wants to recover the complete function. In order to assess the applicability of the proposed construction one must turn to observationalcosmology. The detailed numerical analysis is outside the scope of this article and will be deferred tofuture work. Nevertheless, some preparatory analysis is straightforward and can be given here.19 - - - - - α β Figure 7:
The vector field and the singular lines for the system of equations (79) with trigonometric F ( ξ ) of (78)and Ω Λ = The standard cosmological test relies on the supernovae Ia data and the relationship between theredshift and luminosity. In the Friedmann case, there is a direct relation between H and the redshift,so the integration of time and distance is straightforward. Here, the equations involve up to the thirdderivative of the scale factor, so another route needs to be taken: for small redshifts, a series formulabinding various expansion coefficients can be given, while in the general case, the dynamical systemhas to be integrated.Recall first that the redshift is linked to the scale factor by z + = a − , for a (0) = r is d L = r (1 + z ). Provided, then,that r can be expressed by z , this will allow us to calculate the apparent luminosity and relate toobservations [12].The required expression follows from the condition of the null geodesic: d s =
0, which for themetric (48) gives directly r = (cid:112) k sin (cid:181) (cid:112) k (cid:90) d ta (cid:182) , (80)20here a limit is understood for k =
0. Assuming that a or z are monotonic functions of t that can beused for parametrization of the light path, the above can be rewritten as d L = + zH (cid:112) Ω k sin (cid:112) Ω k z (cid:90) d zh . (81)In the standard model, H is simply given as a function of z by the Friedmann equation, and theintegral can even be explicitly calculated by means of elliptic functions [13]. As mentioned above,this cannot be done here, but following [13], the main equation can be used to give constraints of thehigher characteristicss – the deceleration parameter q and the jerk j : q : = − .. aa . a = − α h , j : = ... aa . a = . α h − q . (82)A change of the independent variable from t (or τ ) to z immediately givesd h d z = + q + z h , d q d z = j − q − q + z , (83)which then allows us to expand h in the integral (81) in powers of z , so that the whole expression canbe expanded as d L = zH (cid:195) + − q z − + j − q − q + Ω k z + O ( z ) (cid:33) . (84)For small z , this provides a means to finding H , q and j from the luminosity data, but one alsohas to take into account that these parameters are not independent. In the standard model, q can beeliminated because h (cid:48) ( z ) is an explicit function of z and the density parameters Ω . Similarly here, thejerk is constrained by the main equation, which for this purpose becomes j + q = Ω + W ( α , β , h , a )3(3 F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β )) h , (85)with α = − h q , β = h (2 − q ) + Ω k (1 + z ) . (86)So, given the function F , the constraint on j is j + q = Ω + W ( − q ,2 − q + Ω k ,1,1)3(3 F (cid:48)(cid:48) ( − q ) + F (cid:48)(cid:48) (2 − q + Ω k )) . (87)Finally, to obtain the luminosity distance for larger redshifts, where a series expansion is notpracticable, an augmented dynamical system is a straightforward solution. Assuming again that z canbe used as the independent variable, as is the case in exponential expansion, a dynamical equation for d L is necessary instead of the integral (81).The null geodesic condition gives d r d z = (cid:112) − kr H , (88)and denoting the dimensionless distance by l = H d L leads tod l d z = l + z + (cid:112) (1 + z ) − Ω k l h , (89)21hile the basic system now reads d α d z = − Ω + W (cid:161) α , β , h ,(1 + z ) − (cid:162) (1 + z )(3 F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β )) h ,d h d z = h − α + z ) h . (90)Because z has become the independent variable, this system is non-autonomous and only two-dimensional (regardless of k and Ω ). Even in the Friedmann case, for more complex H ( z ), the integral(81) has to be obtained numerically. The only complication here is that three ordinary differentialequations need to be integrated; their initial conditions follow from the definitions l (0) = α (0) = − q , h (0) =
1. (91)
The main modification of the gravitational action proposed here is to include terms nonlinearin curvature, but going further than polynomials, so that rational functions with a finite radius ofconvergence or even transcendental functions can be used. Additionally, instead of considering just afunction of the Ricci scalar f (tr[ R ]), the whole tensor can be treated as an argument, and the tracetaken at the very end to produce a scalar Lagrangian density tr[ f ( R )]. In the case of transcendentalfunctions, this considerably changes the results, when compared to the ordinary f ( R ) theories.With a view to fully general treatment, such as including spin, the presented derivation is valid foraffine connections with nonvanishing torsion and without the assumption of metricity. An importantconsequence is that for nonsymmetric Ricci tensors one can no longer introduce an obvious metricconformal to the original g ab . This stems from the nonlinear functions of the Ricci tensor entering theequations, instead of just functions of the Ricci scalar multiplying R ab or g ab .Despite the difficulties, workable equations can be derived and applied to the Robertson-Walkergeometry so that the analogue of the standard cosmological model may be studied. As is generally thecase, the modified Einstein equations are of higher order, and instead of one Friedmann equation, onehas a three-dimensional dynamical system.An obvious complication is that the dynamical variables enter the equations both inside andoutside the transcendental functions, which leaves little hope for explicit solutions. Nevertheless,these models are within reach and if the function f is determined from other fundamental principles,the dynamics and observational consequences can still be effectively analysed, as shown here.The analysis of phase portraits for both rational and transcendental f reveals critical points whichare attractors and which correspond to de Sitter solutions. More importantly, there also exist non-singular “big bounce” evolutions, which are heteroclinic trajectories, and explicit solutions for themcan be given. For the dynamical systems to be two-dimensional it was assumed that the curvature waszero and no ordinary matter was present. On the one hand this allows for a complete visualizationof the phase diagram, but on the other, it limits the physical applicability. Still, the late or presentUniverse with accelerated expansion can be modelled as the de Sitter attractor, while for the bigbounce solutions the scale factor does not approach zero, so that matter density never dominates andneglecting it is justifiable.If dustlike matter is included, the separatrix of the above simplified rational model survives andthe same explicit solutions hold. One still has both big bounce and big bang solutions, not unlike those22f Λ CDM with stiff matter. In general, matter changes the early evolution around the separatrix butnot on it. Thus, the next possible step in constructing a viable model seems to be identifying f ( R ) suchthat it also has an invariant submanifold, but which depends on Ω m , not just on the geometry and Λ .In any case, the elegant feature here is that the cosmological constant can appear naturallybecause of how the theory is constructed – it is identified with the constant term of f ( R ). Yet, evenwhen this term was zero ( f = exp − k could lead to more interesting results still. For example, seeing how one of the scenarios imitatesstiff matter, it will be interesting to ask if such cosmologies can be equivalent to standard generalrelativity with a scalar field, similarly to the ordinary R case. It is also the quadratic f ( R ) case forthe Robertson-Walker geometry, when there is an equivalence with the ordinary f ( R ), although itdoes not seem to extend to higher orders. Another convergence is found when the traceless Ricci tensorvanishes, so that R ab is proportional to g ab and the Einstein equations for both theories coincide.However, as the examples show, even for an empty universe this might correspond only to fixed points,not to general solutions of the full theory.With a view to future work, some observational formulae are also given, so that the basic cosmolog-ical tests can be applied. A comparison to the standard model is in order to help guide the subsequenttheoretical developments. Specifically some constraints on the function f should be obtained. Thecrudest way would be to fit the first coefficients of its expansion, but of course there is no hope inrecovering the whole series this way.Rather, one might want to approach the problem by trying to fit a differential equation satisfied by f . Already for linear differential equations with rational coefficients this would reduce the number ofparameters to finite, while at the same time allowing for a the vast family of (confluent) hypergeometricfunctions and their generalizations.Future investigations could also address the question of reduction of the order of the dynamicalsystem (63). For the Einstein-Hilbert action, the third derivative of the scale factor does not enter,and only the Friedmann equation, which is a relation between H and a , is left. Here, the equationinvolving the third derivative of the scale factor, or . α , would be reduced if 3 F (cid:48)(cid:48) ( α ) + F (cid:48)(cid:48) ( β ) =
0. Forindependent α and β this happens only if F is linear, so that GR is recovered.If, on the other hand, there is a relation β = ψ ( α ), then a nontrivial solution to the functionalequation 3 J ( α ) + J ( ψ ( α )) = F is determined by F (cid:48)(cid:48) ( ξ ) = J ( ξ ).Ideally however, the function f should be mainly constrained by experiment not just the simplicityof the resulting equations. If this theory passes the basic cosmological tests, analysing it in a widercontext of gravitational physics will help address this issue. Questions of instabilities will have to beanswered, although as suggested by [3], the Palatini approach, applicable here, provides a settingto avoid at least the Ostrogradski instability. In general, issues such as ghost fields, semiclassicalstability and post-Newtonian (Solar System) tests will be required, and hopefully undertaken, toascertain the overall viability of the presented extension. References [1] Alexei A. Starobinsky, “A new type of isotropic cosmological models without singularity”, Phys.Lett. B , 1, 99-102 (1980). 232] Gianluca Allemandi, Andrzej Borowiec and Mauro Francaviglia, “Accelerated cosmological modelsin Ricci squared gravity”, Phys. Rev. D , 103503 (2004).[3] Gonzalo J. Olmo, “Introduction to Palatini theories of gravity and nonsingular cosmologies”,chapter 7 in Open Questions in Cosmology , ed. G.J. Olmo, InTech Publishing (2012).[4] Shin’ichi Nojiri and Sergei D. Odintsov, “Unified cosmic history in modified gravity: From F(R)theory to Lorentz non-invariant models”, Physics Reports, 505, 2–4, 59–144 (2011).[5] Thomas P. Satiriou and Valerio Faraoni, “ f ( R ) theories of gravity”, Reviews of Modern Physics,82(1), 451 (2010).[6] Andrzej Borowiec, Nonlinear Lagrangians of the Ricci Type, arXiv preprint gr-qc/9906043 (1999).[7] Andrzej Borowiec, “Metric-polynomial structures and gravitational Lagrangians”, in Institute ofPhysics Conference Series , 241–244, (2002).[8] Nicholas J. Higham, Functions of Matrices , SIAM (2008).[9] Andrew R. Liddle, “How many cosmological parameters?”, Mon. Not. R. Astron. Soc. , L49–L53(2004).[10] Aleksandra Kurek and Marek Szydłowski, “The Λ CDM Model in the Lead–A Bayesian Cosmo-logical Model Comparison”, The Astrophysical Journal 675(1), 1 (2008).[11] Brian Whitt, “Fourth-order gravity as general relativity plus matter”, Phys. Lett. B , 3–4,176–178 (1984).[12] Steven Weinberg,
Cosmology , Oxford University Press (2008).[13] Mariusz P. Dabrowski and Tomasz Stachowiak, “Phantom Friedmann cosmologies and higher-order characteristics of ezpansion”, Ann. Phys. , 771–812 (2006).[14] Mariusz P. Dabrowski, “Oscillating Friedman Cosmology”, Ann. Phys.248