Avoiding the cosmological constant issue in a class of phenomenologically viable F(R,{\cal G}) theories
aa r X i v : . [ g r- q c ] M a r Avoiding the cosmological constant issue in a class of phenomenologically viable F ( R, G ) theories Israel Quiros
1, a Dpto. Ingenier´ıa Civil, Divisi´on de Ingenier´ıa,Universidad de Guanajuato, Gto., CP 36000, M´exico.
In this paper we investigate a class of phenomenologically viable F ( R, G ) theories that are ableto avoid the cosmological constant issue. While the absence of ghosts and other kinds of instabilityissues is of prime importance, other reasonable requirements such as vanishing effective (low curva-ture) cosmological constant, including the flat space as a stable vacuum solution, are also imposedon the viable models. These are free of the cosmological constant problem thanks to the followingoutstanding feature: the de Sitter space is an attractor of the asymptotic cosmological dynamics,with the resulting constant Hubble rate being unrelated both to the energy density of vacuum andto the low-curvature effective cosmological constant. I. INTRODUCTION
The cosmological constant problem (CCP) [1–8] is one of the current unsolved puzzles in fundamental physics. Inthe most widespread version of the issue, the challenge is to explain the origin of the large discrepancy between thetheoretically predicted value of the energy density of vacuum ρ theorvac ∼ GeV and the observed value ρ obsvac ∼ − GeV .In this paper we shall not search for a solution to the CCP, that has shown to be a very complex issue withstrong roots in the particle’s physics sector of the field theory. Instead, we shall look for theoretically consistentmodifications of general relativity (GR) that are able to avoid the issue. The absence of the problem can be analternative explanation to the unsolved puzzle. In order to ensure this goal, the following necessary and sufficientconditions should be satisfied. • Necessary condition : de Sitter space with constant Hubble rate H = H , where H is its present value, shouldbe an attractor of the asymptotic dynamics of the related Friedmann-Robertson-Walker (FRW) cosmologicalmodel. • Sufficient condition : H should be unrelated both to the quantum vacuum energy density and to the low-curvature effective cosmological constant (if different from the energy density of vacuum).The necessary condition ensures that, no matter which modification of GR one is dealing with, at present it shouldbe indistinguishable from the ΛCDM cosmological model, and that this de Sitter stage is a natural outcome of thecosmological evolution, quite independent of the chosen initial conditions. The sufficient condition assures that thepresent (observed) value of the Hubble rate H ≈ − h yr − , where h is a dimensionless parameter in the range0 . < ∼ h < ∼ .
82 or, in Planck units: H ∼ − GeV, has nothing to do neither with the vacuum energy density ρ vac ∼ ρ Pl ∼ GeV → H vac ∼ M Pl ∼ GeV, nor with the effective (low-curvature) cosmological constant Λ eff (assuming that these are not coincident), which may be assumed to be vanishing if flat space is to be a solution ofthe equations of motion (EOM).The phenomenologically viable models should satisfy additional reasonable consistency requirements:1. The theoretical framework should be free of ghosts and other harmful instabilities.2. For sufficiently small curvature the theory should be indistinguishable from GR with an effective – presumablyvery small or even vanishing – cosmological constant.3. Although in curved space the energy density of the quantum vacuum must be non-vanishing, in flat space itshould be zero due to some (yet undiscovered) symmetry. Hence, flat space should be a stable solution of thevacuum EOM. a Electronic address: iquiros@fisica.ugto.mx
The first consistency requirement above is an unavoidable theoretical criterion that any viable model of actual physicalprocesses should respect. Ghosts that arise in modified gravity theories describe physical excitations that are drawnas external lines in Feynman diagrams [9]. The existence of physical ghost leads, eventually, to either the existence ofnegative norm states or to negative energy eigenstates. Hence, one is faced either with problems for the formulationof a consistent quantum theory or with catastrophic instabilities when the ghost couples to conventional matter fields.The second requirement impacts directly the phenomenological viability of the theoretical framework. It reflects ourbelief, deeply rooted in the existing amount of experimental evidence, that any modification of gravity in the Solarsystem, at leading order, must be very close to GR. In this regard, the third requirement is a consequence of ourunderstanding that weak gravity may be viewed as a small deformation of Minkowski space or, in other words, that,for an isolated source of gravity, the space is asymptotically flat as in GR. One example of a theory that certainlydoes not satisfy one the conditions stated above: the sufficient condition for avoidance of the cosmological constantissue, is precisely general relativity.Physically motivated modifications of GR can be based on the inclusion of higher-order curvature operators. Indeed,such a generalization might be considered desirable as it will cause the graviton propagator to fall off more quickly in theUV, thereby improving the renormalisability properties [9]. Modifying gravity in this way, however, also has a numberof drawbacks. In particular, it can introduce instabilities into the theory, such as ghost-like degrees of freedom. Onephysically motivated example of the inclusion of higher-order curvature invariants (and their admissible combinations)is string theory. The string effective action contains an infinite, well organized and ghost-free series of higher curvaturecorrections to the leading Einstein gravity [10–16]. One way to incorporate the quadratic contributions to the effectiveaction while keeping the theory ghost-free is to consider the Gauss-Bonnet invariant [17–21]: G = R − R µν R µν + R µνσλ R µνσλ . However, since this term in the action amounts to a total derivative that does not affect the equationsof motion, the only way in which it may affect the local dynamics of fields in 4 dimensions, is to dynamically coupleit through, for instance, a scalar field [17], or to consider general functions F ( G ) in the action [18].In this paper we shall investigate a class of F ( R, G ) models where the curvature invariants enter in the followingcombination: αR + β G ⇒ F ( R, G ) = F ( R + c G ), where c = β/α is a constant. We shall study, in particular, a Born-Infeld (BI) inspired class of models where the mentioned combination is within a square root [22–26]. This particularclass of Lagrangian obeys the necessary and sufficient conditions for avoidance of the CCP as well as the additionalreasonable consistency requirements stated above so that, the cosmological constant issue is effectively avoided.The paper has been organized in the following way. In the next section II we discuss the fundamentals of F ( R, G ) = F ( R + c G ) theories, including the equations of motion and the small curvature limit. Then, in section III, we checkthe F ( R, G ) theories in general to the absence of ghosts. The absence of ghosts due to the anisotropy of spaceis linked with the specific form of theories F ( R, G ) = F ( R + c G ). In section IV we concentrate in this particularclass, focusing to the BI inspired model that satisfies the necessary and sufficient conditions for the avoidance of thecosmological constant issue, as well as the additional requirements for phenomenological viability. The asymptoticcosmological dynamics of the class of BI inspired models is investigated in section V for the particular case when thevacuum has vanishing energy density, while the asymptotic dynamics of the general case when the energy density ofvacuum is non-vanishing, is discussed in section VI. In both cases the late time de Sitter attractor is identified andfully characterized. The way in which the cosmological constant problem is avoided in the chosen class of models, isdiscussed in section VII. Other interesting aspects of the model are discussed in section VIII while brief conclusionsare given in section IX. II. F ( R, G ) MODIFICATIONS OF GRAVITY
Here we shall focus in F ( R, G ) theories of the kind F ( Lovelock ) gravity [27, 28], i. e., F ( R, G ) = F ( αR + β G ) , (1)where α and β are parameters with mass dimensions M − and M − , respectively. Hence, the following relationshipstake place: F G = βα F R , F G R = F R G , F RR = αβ F G R , F GG = βα F R G , (2)where F R ≡ ∂F/∂R , F G R ≡ ∂ F/∂ G ∂R , etc.We consider an action of the form: S = 12 κ Z d x √− g (cid:2) F ( R, G ) + 2 ǫκ µ (cid:3) , (3)where µ and κ are free parameters with the dimension of mass and inverse mass, respectively, while ǫ = ±
1. TheEOM that are derived from the above action – plus a matter piece action – read [29, 30]: G µν + Σ curv µν = κ h T ( m ) µν + ǫµ g µν i , (4)where we have defined the effective gravitational coupling κ ≡ κ F R , (5)while Σ curv µν = − (1 + βα R ) F R ( ∇ µ ∇ ν − g µν ∇ ) F R + 12 (cid:18) R + βα G − FF R (cid:19) g µν + 4 βαF R (cid:0) R λµ ∇ λ ∇ ν + R λν ∇ λ ∇ µ − R µν ∇ (cid:1) F R + 4 βα ( R µλνσ − g µν R λσ ) ∇ λ ∇ σ F R F R , (6)comprises the contribution coming from the fourth-order curvature terms, ∇ ≡ g µν ∇ µ ∇ ν and T ( m ) µν is the stress-energy tensor of the matter degrees of freedom. While deriving the EOM (6) we have taken into account the relation-ships (2). The trace of (4):3 ∇ F R F R − βα G λσ ∇ λ ∇ σ F R F R + R + 2 βα G − FF R = κ F R h T ( m ) + 4 ǫµ i , (7)where T ( m ) = g µν T ( m ) µν is the trace of the stress-energy tensor of matter, amounts to an additional dynamical equationon the variable F R . A. Small curvature limit ( R ≈ G ≈ ) Let us to expand the function F ( R, G ) at small curvature up to fourth-order curvature terms [25]: F ( R, G ) = F + F R R + 12 F RR R + F G G , (8)where F = F ( R, G ) = F (0 , F R = ∂F/∂R | (0 , , etc. The following effective (low curvature) action is retrieved: S eff = M Z d x √− g (cid:18) R − eff + 16 m R + βα G (cid:19) , (9)where M = F R κ , Λ eff = − F F R − ǫµ M , m = F R F RR . (10)The Gauss-Bonnet term in (9) amounts to a total divergence so that it does not modify the EOM and may be safelyomitted.The exchange of the extra scalar degree of freedom with mass m between two test particles with masses m and m , modifies the Newtonian gravitational potential through an additional Yukawa interaction: V ( r ) = − G N m m r h α exp (cid:16) − m c ¯ h r (cid:17)i , where c is the speed of light, ¯ h is the reduced Planck constant and m = ¯ hcλ = 1 . λ × − GeV , (11)with the length scale λ in µ m. According to [31] the gravitational-strength Yukawa interactions are limited to ranges λ < . µ m with 95% confidence, so that m > × − M Pl . Hence the following bound is to be satisfied: m = F R F RR > . × − M . (12)It should be stressed that the effective action (9), which coincides with the one for the Starobinsky model [32–36]with a non-vanishing cosmological constant, is correct only for small curvatures down to scales of the order: R m ∼ R ≪ α ⇒ R ≪ m α , G ∼ αβ R ≪ β . (13)For much smaller curvature the effective action just coincides with the Einstein-Hilbert action: S EHeff = M Z d x √− g ( R − eff ) , (14)since, as R →
0, the related curvature quantities R and G vanish faster than R . III. GHOST FREEDOM
In order to investigate the propagating degrees of freedom, we shall study the linearization of the action (3) aroundmaximally symmetric spaces of constant curvature R [26]. In this case G = R /
6. We shall expand the action upto terms quadratic in the curvature, so that terms like ( R − R ) , ( R − R ) ( G − G ) and higher, will be omitted. Wehave that: F ( R, G ) = ˜ F + ˜ F R ( R − R ) + 12 ˜ F RR ( R − R ) + ˜ F G ( G − G ) + O (3) , where ˜ F ≡ F ( R , G ), ˜ F R ≡ F R ( R , G ), etc. If reorganize the above equation we can write it in more compact form(in the given approximation): F ( R, G ) = ξ + ζ R + υ R + ω G , (15)where we have introduced the following identifications: ξ ≡ ˜ F − R ˜ F R + 12 R ˜ F RR − R ˜ F G ,ζ ≡ ˜ F R − R ˜ F RR , υ ≡ ˜ F RR , ω ≡ ˜ F G . If substitute the above expansion back into the action (3) we get: S = M Z d x √− g (cid:18) R −
2Λ + 16 m R (cid:19) , (16)where M = ζ κ , Λ = − ξ + 2 ǫκ µ ζ , m = ζ υ = ˜ F R − R ˜ F RR F RR , (17)and the term under the integral ∝ G has been omitted since it amounts to a total derivative. It is a well-known factthat the linearization (16) is associated with three propagating degrees of freedom [37, 38]: the two polarizations ofthe (massless) graviton and a massive scalar mode with mass squared m . In order to avoid a tachyon instability itis then required that: m ≥ ⇒ ˜ F R > R ˜ F RR , ˜ F RR > . (18)As it was for the expansion around flat space, the present linearization is correct for small departures from de Sitterspace with constant curvature R , down to the scale R ∼ m . For much smaller curvature scales R ≪ m , the action(16) reduces to the Einstein-Hilbert action. A. Ghosts due to anisotropy of space
Even if the F ( R, G ) modified theory of gravity is free of ghosts when linearized around maximally symmetric spaces,when other less symmetric backgrounds such as anisotropic spaces, are considered, it is not for granted that the theorywill be free of ghost in this latter case. In Ref. [39] the study of linear perturbation theory for general F ( R, G ) wascarried out over an empty anisotropic background of the Kasner-type: ds = − dt + a ( t ) dx + b ( t )( dy + dz ) , (19)where a ( t ) and b ( t ) are the scale factors, in order to show that, within general F ( R, G ) theories, an anisotropicbackground has ghost degrees of freedom, which are absent on Friedmann-Robertson-Walker (FRW) backgrounds.Their study revealed that on this background the number of independent propagating degrees of freedom is four.It reduces to three on FRW backgrounds, since one mode becomes highly massive and decouples from the physicalspectrum. The ghost mode is inevitable unless the following condition is fulfilled [39]: (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( χ, ξ ) ∂ ( R, G ) (cid:12)(cid:12)(cid:12)(cid:12) = F RR F GG − ( F R G ) = 0 , (20)where χ , ξ are auxiliary fields introduced in the study of [39]. If the above condition is fulfilled, then perturbationsof χ and ξ are not independent. In general backgrounds this is true if [39] L = χ ( φ ) R + ξ ( φ ) G − V ( φ ). But this isnot the only possibility left to avoid the ghosts due to anisotropy.Actually, for theories of the kind we consider in this paper: F ( R, G ) = F ( R + c G ), where c = β/α is a free constant,the condition (20) is fulfilled since, for this class of theories: F G = cF R and F GG = cF R G = cF G R = c F RR , as seenfrom (2). Hence, for the latter more general class of fourth-order theories, ghosts due to anisotropy of space areabsent. B. Scalar perturbations and a modification of the dispersion relation
By the same reason as above, i. e., that the relationships (2) take place, neither a strong instability nor superluminalpropagation occurs due to a modification of the dispersion relation found in [40]. In this reference the authors performa general study of cosmological perturbations in vacuum for general F ( R, G ) theories. They found a modification ofthe dispersion relation for scalar perturbations, in comparison with previous similar studies [41, 42], that leads tounwanted – either unstable or tachyonic – behavior. Actually, in [40] the following non-standard wave equation wasobtained for the gauge invariant (Fourier) field Φ:1 a Q ∂ t (cid:16) a Q ˙Φ (cid:17) + B k a Φ + B k a Φ = 0 , (21)where Q = Q ( t ), B = B ( t ) and B = B ( t ) are time-dependent parameters and k is the wavenumber of theperturbation. In this study the degrees of freedom Φ and ˙Φ are enough to describe the behavior of the metricperturbations. The above wave equation equation contains a term proportional to k , which does not vanish ingeneric F ( R, G ) theories. This term is responsible for non-standard behavior of the scalar metric perturbations.For instance, if B were negative, then the Friedmann-Robertson-Walker (FRW) space were unstable on small scales(short wavelength limit) [40]. If B were positive instead, up to the leading term the group velocity v g ( k ) ≈ √ B k/a .It exceeds the speed of light for modes above the critical wavenumber k c = a/ √ B . Hence, the propagation of shortwavelength modes eventually (inevitably) becomes superluminal. This is true, except for the above mentioned specialcases where (equation (6.18) of Ref. [40]): F RR F GG − F R G F G R = F RR F GG − ( F R G ) = 0 . (22)The small wavelength modes inevitably either suffer from strong instability or undergo superluminal propagation.In the kind of theories we are investigating here (1), thanks to the relationship (2), the condition for absence ofinstability/tachyonic behavior (22), is identically fulfilled. This means that the above discussed kind of instability isnot present in the models of our interest. C. Absence of other instabilities
Among the most dangerous instabilities, when higher-curvature corrections of gravity are considered, is the so calledDolgov-Kawasaki (matter) instability [43–46]. This instability, which is specially important in the f ( R ) theories sincethe curvature scalar R is a dynamical degree of freedom [45, 46], is of special importance in the present setup as well.The stability criterion in this case requires that dκ dR = − κ F RR F R < . (23)Hence, the Dolgov-Kawasaki instability is avoided only for non-negative F RR ≥
0. If the R -derivative of κ in (23)were positive, the effective gravitational coupling increased with the curvature, so that, at larger curvature gravitybecomes stronger which then implies that R itself generates a larger curvature through the trace equation (7). Inother words, a positive feedback mechanism acts to destabilize the theory [26, 46].In addition to the above stability criteria, a constraint coming from requiring positivity of the effective gravitationalcoupling: κ = κ F R > , (24)is also to be satisfied.As shown in [40], for models of the class F ( R, G ) = F ( R + c G ), like the ones we are interested in here, the waveequation (21) for the scalar perturbations is given by:1 a Q ∂ t (cid:16) a Q ˙Φ (cid:17) + c s k a Φ = 0 , (25)where the squared sound speed is defined in the following way: c s = 1 + 8 β ˙ H/α βH /α . This term corresponds to fourth order spatial derivative in real space and is not a spurious result due to a bad choice of gauge since Φis gauge invariant.
We should require non-negative squared sound speed c s ≥ F ( R, G ) model, for the squared speed of propagation of the tensor modes one gets [40]: c T = F R + 4 β ¨ F R /αF R + 4 βH ˙ F R /α . The absence of ghosts requires that F R + 4 βH ˙ F R /α >
0, while, in order to avoid the Laplacian instability: c T ≥ F ( R, G ) theories are phenomenologically viable options for the description of our universe. In particular,the choice in (1) makes these theories very attractive possibilities for viable fourth-order theories of gravity since allof the mentioned instabilities may be avoided in a given subspace of the parameter’s space. IV. POWER-LAW F ( R + c G ) MODELS OF MODIFIED GRAVITY
Here we study a three-parametric class of models of the kind F ( R + c G ) modified gravity and check them tostability and phenomenological viability. For the present choice of the F ( R, G ) modification of gravity, several sourcesof instability such as ghosts due to anisotropy of space and non-standard behavior of the scalar metric perturbations– potentially leading to superluminal propagation of short wavelength modes – are eliminated. However, avoidanceof other kinds of instability such as the Dolgov-Kawasaki and Laplacian instabilities, as well as the requirement ofpositivity of the effective gravitational coupling, lead to additional constraints on the parameter space.In the present case we choose the following power-law function F ( R, G ): F ( R, G ) = − λ (1 − αR − β G ) ν , (26)where λ , α and β are free constants with mass dimensions M , M − and M − , respectively, while ν is a dimensionlessconstant. Notice that, although there are four free parameters in (26), the parameter λ may be combined with κ in(3), so that the resulting F ( R, G ) is actually a three-parametric function. We have that: F R = αλ ν (1 − αR − β G ) ν − , F G = βλ ν (1 − αR − β G ) ν − ,F R G = F G R = − αβλ ν ( ν −
1) (1 − αR − β G ) ν − ,F RR = − α λ ν ( ν −
1) (1 − αR − β G ) ν − , F GG = − β λ ν ( ν −
1) (1 − αR − β G ) ν − , (27)so that the relationships (2) are satisfied. In what follows we shall assume that the following constraint on thecurvature quantities is satisfied: 1 − αR − β G ≥ . (28)Under the above assumption, for the three-parametric class of function (26), absence of the Dolgov-Kawasaki instabilityand positivity of the effective gravitational coupling – requirements (23) and (24), respectively – amount to thefollowing conditions: dκ dR = κ ( ν −
1) (1 − αR − β G ) − ν λ ν < ,κ = κ (1 − αR − β G ) − ν αλ ν > . (29)Hence, phenomenologically viable theories of this type require that α > < ν < F ( R, G ) models of the kind [25, 48]: F ( R, G ) = − λ p − αR − β G , (30)fall into the above phenomenologically viable class of models of modified gravity, when we set ν = 1 /
2. In whatfollows we shall focus in the investigation, specifically, of this two-parametric class of models.
V. ASYMPTOTIC DYNAMICS OF BI-INSPIRED F ( R, G ) COSMOLOGY
Here we shall investigate the cosmological dynamics of the BI-inspired F ( R, G ) model (30) with action (3), in aFRW background space with flat spatial sections, whose line-element reads: ds = − dt + a ( t ) δ ik dx i dx j , i, j = 1 , , . (31)In this case we have that: R = 6 ˙ H + 12 H , G = 24 H (cid:16) ˙ H + H (cid:17) , (32)where H = ˙ a/a is the Hubble parameter.For the F ( αR + β G ) class of function the FRW equations of motion (4) read:3 H + 3 H ˙ F R F R (cid:18) βα H (cid:19) − (cid:18) R + βα G − FF R (cid:19) = κ F R (cid:0) ρ m − ǫµ (cid:1) , (33)¨ F R F R = − κ ( ω m + 1) ρ m F R (cid:16) βα H (cid:17) + H ˙ F R F R − (cid:16) βα H ˙ F R F R (cid:17) βα H ˙ H, (34)where ρ m and p m = ω m ρ m are the energy density and pressure of the matter fluid, while ω m is its equation of state(EOS) parameter, respectively. In the present case the trace equation (7) is not an independent equation so that wedo not write it. The above EOM-s can be written in the following alternative way:˙ HH = − m − Ω µ −
21 + 4 βα H − ˙ F R HF R + 13 αH (1 + 4 βα H ) , (35)¨ F R H F R = − ω m + 1)Ω m βα H + ˙ F R HF R − h βα H (cid:16) ˙ F R HF R (cid:17)i βα H ˙ HH , (36)where we have introduced the dimensionless energy densities of matter and of µ :Ω m ≡ κ ρ m F R H , Ω µ ≡ ǫκ µ F R H . (37)Notice that, for ǫ = +1 the constant µ contributes a negative energy density. This is not problematic since theeffective cosmological constant at low curvature isΛ eff = λ − ǫκ µ αλ , (38)as seen from (10). Hence, as long as λ ≥ ǫκ µ , Λ eff is a non-negative quantity even if ǫ = +1.For the choice (30), the assumption (28) is not an independent requirement but a constraint on the physicalviability of the resulting cosmological model. Hence, for the specific model of interest here, this constraint amountsto a phenomenological bond which, in FRW space, can be written in the following way:˙ HH ≤ αH (cid:16) βα H (cid:17) − (cid:16) βα H (cid:17) βα H . (39) A. Simplified dynamical system: matter vacuum with vanishing vacuum energy
In what follows, for simplicity, we shall investigate the particular case when the density of matter vanishes Ω m = 0(vacuum case) and µ = 0. This means that at small curvature: S = M Z d x √− g (cid:18) R − α + α R (cid:19) , (40)where M = αλ / κ . Although this is not the most general situation in which we may have even a vanishingeffective cosmological constant, anyway the basic features of the model are preserved. In order to perform the asymptotic dynamics analysis of this model, let us introduce the following dimensionless(bounded) variables of some state space: x = 11 + 4 βα H ⇒ βα H = 1 − xx , ≤ x ≤ ,y ± ≡ ˙ F R HF R ± ˙ F R ⇒ " ˙ F R HF R ± = y ± ∓ y ± , − ≤ y − ≤ , ≤ y + ≤ , (41)where the whole phase space is covered by the bounded variables x ∈ [0 ,
1] and y = y − ∪ y + ∈ [ − , x ′ = − x (1 − x ) " ˙ HH ± ,y ′± = (1 ∓ y ± ) " ¨ F R H F R ± − y ± (1 ∓ y ± ) " ˙ HH ± − y ± , (42)where " ˙ HH ± = − − x − y ± ∓ y ± + 4 βx α (1 − x ) , " ¨ F R H F R ± = y ± ∓ y ± − x (1 ∓ y ± ) + (1 − x ) y ± ]1 ∓ y ± " ˙ HH ± , (43)and the prime denotes derivative with respect to the time variable N = ln a . Notice that there are two differentdynamical systems in (42); one for the choice of the ’+’ sign and another one for the choice ’ − ’. However, thesedescribe a unique phase space spanned by the variables x and y = y − ∪ y + .The model (30) is phenomenologically viable only if the function F = F ( R, G ) is a real quantity, i. e., if thecondition (39) is fulfilled. For the present case, in terms of the variables x , y the latter reads: " ˙ HH ± ≤ − − x + 2 βx α (1 − x ) , (44)or y ± ≥ y ±∗ , where Recall that for non-vanishing µ , the cosmological constant at small curvature: Λ eff = ( λ − ǫκ µ ) /αλ , can be made as small as onedesires by properly arranging the parameters λ and µ if ǫ = +1. For instance, by letting λ = 2 κ µ + δλ , where δλ is a very smallquantity. For a compact introduction to the dynamical systems analysis close to this presentation see [49]. FIG. 1: Phase portraits of the dynamical system (42) for different (positive) values of the free parameters α and β . From leftto the right ( α, β ): (1 , ,
1) and (1 , . β/α equals 10, 1 and 10 − , respectively. The’gray’ region is unphysical since the condition (39) is not satisfied. The critical points are represented by the small (red) solidcircles. The thick dash-dot (blue) curve represents the condition ˙ H/H = −
1. Hence, the critical points that are located belowthis curve represent accelerated expansion. y −∗ = − x (1 − x ) + β α x (1 − x )(1 + x ) − β α x , y + ∗ = − x (1 − x ) + β α x (1 − x ) + β α x . (45)Hence, the physically meaningful phase space corresponds to the following region of the plane: Ψ = Ψ − ∪ Ψ +2D ,Ψ − = (cid:8) ( x, y ) : 0 ≤ x ≤ , − ≤ y ≤ , y ≥ y −∗ (cid:9) , Ψ +2D = (cid:8) ( x, y ) : 0 ≤ x ≤ , ≤ y ≤ , y ≥ y + ∗ (cid:9) . (46)From the first equation in (43) it also follows that the expansion is accelerated ( q = − − ˙ H/H <
0) if y < y −† for − ≤ y − ≤
0, or if y < y + † for 0 ≤ y + ≤
1, where: y −† = − x (1 − x ) + β α x (1 − x )(1 + 2 x ) − β α x , y + † = − x (1 − x ) + β α x (1 − x )(1 − x ) + β α x . (47)The phase portraits corresponding to the dynamical system (42) are shown in FIG. 1, for different values of thefree parameters α and β . The different orbits appearing in these phase portraits are generated by given sets ofinitial conditions ( x i (0) , y i (0)). Each orbit may be associated with a whole cosmic history, starting (possibly) in apast attractor (origin of the given evolutionary pattern) and ending up in a future attractor (destiny of the cosmicevolution). The ’gray’ region is unphysical since the condition (39) is not fulfilled. Hence, this region is excluded fromthe phase space Ψ. The thick dash-dot curves represent the condition q = − − ˙ H/H = 0. In consequence, points inΨ that are located below these curves represent accelerated expansion. In what follows we consider only non-negative β -s. The parameter α is also positive due to the requirements of absence of the Dolgov-Kawasaki instability and ofpositivity of the gravitational coupling as we have discussed before. The case with negative Gauss-Bonnet coupling( β <
0) has been studied in detail in [48].
B. Critical points of the dynamical system
Below we list those isolated critical points P i : ( x i , y i ) of the dynamical system (42) in Ψ , that are located withinthe phenomenologically viable region, together with their main properties. These points are marked by small (red) In what follows, without loss of generality, we call a given point of the phase space as a “critical point” only if it is located within thephenomenologically viable region of the phase space, i. e., if it is in Ψ , no matter whether it is, mathematically speaking, a critical Origin . The point P O : (0 ,
0) is the global past attractor in the phase space Ψ since the eigenvalues of thelinearization matrix for this point: λ = 2 and λ = 4, are both positive. At this point: x = 0 ⇒ H ≫ α/β, y = 0 ⇒ ˙ F R HF R = − ˙ FHF → . (48)Besides, the function F is undefined at this equilibrium point. Since at P O , the deceleration parameter q = − − ˙ H/H = 0, then: ˙ HH → − ⇒ H = t − ⇒ a ∝ t. (49)This means that the evolution of the Universe starts in a big-bang singularity where a ( t ) → H ≈ − H →−∞ .2. Transient stages . • Point P : (0 ,
1) is a saddle critical point since the eigenvalues of the corresponding linearization matrixare of different sign. Hence, this point is associated with a transient state of the cosmic evolution. It ischaracterized by a very high curvature with H ≫ α/β and ˙ F R /HF R undefined: y → ⇒ ˙ F R HF R → ∞ ( H > , (50)which means, in turn, that F R →
0, i. e., that F → ∞ . This latter limit implies that, at least,˙ HH < − ⇒ q > H > , (51)i. e., this point represents a transient stage of decelerated expansion. As a matter of fact, since at P , x = 0 and y = 1, then (see first equation in (43)):˙ HH → −∞ ⇒ q → ∞ . (52)This point should be associated with a curvature singularity. • The point P : (1 , x = 11 + 4 βα H = 1 , y = 1 ⇒ ˙ F R HF R → ±∞ , (53)represents a transient cosmic stage with low curvature H ≪ α/β (the numerical investigation reveals thatthis is a saddle equilibrium point as well). It is seen from the first equation in (43) that at P , due to thecompetition between the negative (first) and the positive (last) terms, the quantity ˙ H/H is undefined. Asa matter of fact this equilibrium state represents a turning point in what regards to the peace of the cosmicexpansion: It is seen from FIG. 1 that, as the given orbit evolves in the vicinity of P in the route fromthe past into the future attractors, the cosmic history turns from decelerating into accelerating expansion.Recall that the curve (47) corresponding to q = 0, joints the points P O and P . point of the dynamical system. Destiny . The point P dS : p β/ α , ! , is a future attractor since the eigenvalues of the corresponding linearization matrix: λ = − λ = −
4, areboth negative. It is the global future attractor in Ψ . This critical point represents a de Sitter solution since q = − ⇒ ˙ H = 0 ⇒ H = H . It is a de Sitter attractor with constant Hubble rate squared: H = α β r β α − ! . (54) VI. THREE DIMENSIONAL PHASE SPACE DYNAMICS AND THE DE SITTER SOLUTIONS
In the above section we have investigated in detail the asymptotic dynamics of the BI inspired F ( R, G ) theory (30)in the simplified case when µ = 0, i. e., vanishing vacuum energy density (we are investigating the vacuum caseexclusively, i. e., Ω m = 0). In this simplified case the asymptotic dynamics is described in a 2-dimensional (2D) phasespace which means, in turn, that the mathematical handling is simpler. However, the assumption that µ = 0, meansthat at small curvature, where the effective theory is given by the action: S = M Z d x √− g (cid:18) R − α + α R (cid:19) , we are not able to set the cosmological constant 1 /α to any negligible small value – as required by the observations –without forcing a unnaturally large coupling to the higher-curvature contribution. This is why, in the present section,we shall consider a non-vanishing vacuum energy µ = 0. This amounts to increasing the dimension of the phasespace from 2D to 3D. The corresponding mathematical handling is by far more complex. Our strategy to simplifythe mathematics y to focus, exclusively, in the de Sitter solutions which are the ones that can be associated with thelate-time accelerated expansion of the universe. A. Dynamical system and the de Sitter critical points
In the present case where µ = 0, in addition to the phase space variables x , y in (41), it is convenient to introducethe new bounded variable: u = Ω µ Ω µ + ǫ ⇒ Ω µ = ǫu − u , (55)where 0 ≤ u ≤ ǫ = ±
1. The resulting 3D dynamical system reads: x ′ = − x (1 − x ) " ˙ HH ± ,y ′± = (1 ∓ y ± ) " ¨ F R H F R ± − y ± − y ± (1 ∓ y ± ) " ˙ HH ± ,u ′ = − u (1 − u ) ( y ± ∓ y ± + 2 " ˙ HH ± ) , (56)where, as before, the prime denotes derivative with respect to the time variable N = ln a . Besides:3 " ˙ HH ± = − − (cid:20) ǫ − u − u (cid:21) x − y ± ∓ y ± + 4 βx α (1 − x ) , " ¨ F R H F R ± = y ± ∓ y ± − x (1 ∓ y ± ) + (1 − x ) y ± ]1 ∓ y ± " ˙ HH ± . (57)In this section we shall focus in the de Sitter solutions exclusively. In order to check whether the correspondingcritical points are within the phenomenologically viable region, it is required that F ≥
0, where F is defined in (30).Hence, ˙ HH (cid:18) βα H (cid:19) + 2 + 4 βα H ≤ αH , but, since at the de Sitter points ˙ H = 0, then, for these critical points to be in the physically meaningful region it isrequired that: 1 p β/ α ≤ x ≤ . (58)At the de Sitter point, for y ≥
0, the equations (56) and (57) become: x ′ = 0 , y ′ = y (1 − y ) , u ′ = u (1 − u ) yy − , (59)while for negative y < x ′ = 0 , y ′ = y, u ′ = u ( u − yy + 1 . (60)
1. de Sitter critical manifold
For y = 0 one obtains a critical manifold: P dS = ( ( x, , u ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p β/ α ≤ x ≤ , u ∗ = u ∗ ( x ) ) , (61)where we have defined u ∗ ( x ) = (1 − x )(1 + 2 x ) − β α x (1 − x )[1 + (2 − ǫ ) x ] − β α x . Depending on location (smaller or larger u -values), points in P dS can be either saddle critical points or local attractorsinstead. For points in P dS we have that,12 βH + 9 αH − F R − ǫακ µ F R = 0 . (62)On the other hand, since y = 0 ⇒ ˙ F R = 0, one gets that F R = ˜ F R =const., where the value of the constant ˜ F R depends on the initial conditions. From (62) we obtain the following second-order algebraic equation: This is a consequence of P dS being a manifold instead of an isolated critical point. H + 3 α β H − ˜ F R − ǫακ µ β ˜ F R = 0 , whose real root is: H ( F ) = 3 α β vuut β α − ǫακ µ ˜ F R ! − . (63)Given that for a set of points in P dS : the saddle points, the de Sitter solution is a transient stage, one may thinkthat these can be associated with primordial inflation. However, as we shall see below, this is not the case since, ifassociate the late-time attractor P dS (see below) with the late-time inflationary stage, there is not possible to get therequired amount of e-foldings of inflation. Hence, points in the above de Sitter manifold are to be associated withintermediate to late-time cosmological evolution.
2. Isolated de Sitter attractor
The other possibility is for y = 1 /
2, where we are led with the isolated attractor: P +dS : p β/ α , , ! . (64)At this point we have that (54): H = α β r β α − ! . This isolated attractor is to be associated with the late-time stage of accelerated expansion of the universe. It can bechecked that the ratio between H ( F ) in (63) and H above, at large α ≫ H ( F ) H ≈ − ǫακ µ ˜ F R ! . Hence, the saddle points in the manifold P dS can not be associated with primordial inflation since there is not possibleto get the required amount of inflation. B. The 3D phase portrait
The 3D phase space where to look for phenomenologically viable behavior of the dynamical system (56) is definedin the following way: Ψ = (cid:8) ( x, y, u ) : 0 ≤ x ≤ , − ≤ y ≤ , ≤ u ≤ , u ≤ u ±∗ (cid:9) , (65)where u ±∗ = h x (1 − x ) − β α x i (1 ∓ y ± ) + (1 − x ) y ± h (1 − ǫ ) x (1 − x ) − β α x i (1 ∓ y ± ) + (1 − x ) y ± , FIG. 2: Phase portrait of the dynamical system (56) for ǫ = +1 and the following values of the free parameters: α = 1, β = 1.Two sets of orbits that are generated by chosen initial data are shown in different colors. The boundary of the physicallymeaningful phase space F = 0, where F is defined in (30) – surface with contours – is included. In the left and center figuresthe orientation is given by [ θ, φ, ψ ] = [ − o , o , o ] and [ θ, φ, ψ ] = [130 o , o , − o ], respectively, where θ , φ and ψ are theEuler angles. Meanwhile, in the right figure the x, y -projection is shown: [ θ, φ, ψ ] = [ − o , o , o ]. The isolated point representsthe de Sitter attractor P dS , while the dotted curved segment represents the de Sitter critical manifold P dS . and, as before, y = y − ∪ y + .The 3D phase portrait of the dynamical system (56) is shown in FIG. 2 for ǫ = 1 and the following values ofthe free parameters: α = 1, β = 1. Two sets of orbits that are generated by chosen initial data sets, are shown indifferent colors. A surface with contours representing the boundary of the physically meaningful phase space: F = 0,where F is defined in (30), has been included in the figure. In the left and center figures the orientation is givenby [ θ, φ, ψ ] = [ − o , o , o ] and [ θ, φ, ψ ] = [130 o , o , − o ], respectively, where θ , φ and ψ are the Euler angles.Meanwhile, in the right figures the x, y -projection is shown: [ θ, φ, ψ ] = [ − o , o , o ]. The isolated point in each figurerepresents the de Sitter attractor P dS , while the dotted curved segment represents the de Sitter critical manifold P dS .Notice that the phase space orbits end up either at the isolated de Sitter attractor P dS , or at local attractor deSitter points in the manifold P dS (uppermost points in the manifold). The green orbits end up, precisely, at theuppermost points of P dS , while navy orbits end up at the isolated de Sitter attractor. In any case the state towardsthe universe is attracted which corresponds to de Sitter expansion a ( t ) ∝ exp H t , where H is given either by (63)or by (54). VII. AVOIDANCE OF THE COSMOLOGICAL CONSTANT ISSUE IN THE PRESENT SETUP
Perhaps the most interesting solution in the phenomenologically viable subspace of the phase space is the de Sitterattractor P dS (here we should add the de Sitter points in the critical manifold P dS which exist only in the casewhere µ is non-vanishing). The de Sitter attractor solutions are interesting in the present model because these entailthat, at late time, the FRW universe described by the theory (3), (30), is almost indistinguishable from the ΛCDMcosmological model. One should not be surprised by this result since, at low curvature, a non-vanishing cosmologicalconstant Λ eff = ( λ − ǫκ µ ) /αλ , arises in this model. The surprising result is that, at the de Sitter attractor H = Λ eff /
3, which challenges our intuition.In order to fix ideas, momentarily, we shall choose ǫ = +1. Let us to set λ = 2 κ µ in (3). Under this choice ourmodel coincides with the one previously investigated in [25], where an exact cancellation mechanism of the cosmologicalconstant has been applied. Actually, under the above choice the effective cosmological constant at low curvature,Λ eff = 0, exactly vanishes. This includes, as a particular case, the flat space ( R = G = 0), for which: The asymptotic dynamics of this model was studied in [48] for negative Gauss-Bonnet coupling. F ( R, G ) = − λ p − αR − β G , F R = − αλ F ⇒ F (0 ,
0) = − λ , F R (0 ,
0) = αλ . In this particular case, assuming vacuum background, we have that G µν = 0 ( R µν = 0), while the fourth-ordercurvature contributions Σ curv µν in (6), amount to:Σ curv µν (0 ,
0) = − F (0 , F R (0 , g µν = 1 α g µν , so that the EOM (4) for vacuum: Σ curv µν (0 ,
0) = κ µ F R (0 , g µν = 2 κ µ αλ g µν , (66)become an identity after our choice 2 κ µ = λ . Hence, as long as both F R (0 ,
0) = αλ / > F RR (0 ,
0) = α λ / > F ( R ) model investigated in [7], where the flat space was a unstable solution of the equations of motion.From equation (4) it is seen that, for our above choice ǫ = +1, the vacuum energy density is a negative quantity: ρ vac = − µ . But this is not problematic since, as mentioned above, the effective energy density of vacuum at smallcurvature vanishes. The fact we want to underline here is that the energy density of vacuum ρ vac , the effectivecosmological constant Λ eff in the low-curvature regime and the present value of the Hubble rate H , are unrelatedquantities in our setup. Actually, for the de Sitter attractor P dS (64), that arises in the phase space corresponding toour cosmological model, the constant expansion rate reads: H = α β r β α − ! . (67)It has nothing to do neither with ρ vac nor with Λ eff . Actually, given that Λ eff = 0 thanks to our choice ( λ = 2 κ µ ),and that the de Sitter attractor P dS arises no matter whether µ = 0, as in Sec. V, or µ = 0, as in Sec. VI,the constant Hubble rate (67) is independent of the vacuum energy density ρ vac = − µ , as well as of the effective(low-curvature) cosmological constant Λ eff = 0. This non-trivial fact is at the core of the avoidance of the cosmologicalconstant issue in the present setup.It is apparent from above that the theory (3) with F ( R, G ) given by (30) and λ = 2 κ µ ( ǫ = +1), satisfiesthe necessary and sufficient conditions discussed in the introduction (Sec. I), as well as the additional reasonablerequirements, that are to be satisfied in order to have a phenomenologically satisfactory theory of gravity where thecosmological constant problem does not arise. There are, however, certain observational constraints that should besatisfied as well. Take, for instance, the bond imposed on the mass of the scalar perturbation [31] around flat space(12): m = F R (0 , F RR (0 ,
0) = 23 α > . × − M ⇒ α < M − . (68)Let us consider two limiting situations. A. Vanishing Gauss-Bonnet contribution
Assume, first, that the dimensionless quantity β/α ≪ β → F ( R ) theory. In this case from (67) it follows that: H ≈ α > − M , which means that the observational constraint H ∼ − M on the present value of the Hubble rate, can not besatisfied unless one gives up the requirement (68). As a matter of fact this requirement can be smoothed out or even7 FIG. 3: Plots of the squared speed of propagation of scalar (left) and tensor (right) perturbations vs the coordinates of thephase plane – c s = c s ( x, y ) and c T = c T ( x, y ), respectively – for the orbits in the center panel of FIG. 1 ( µ = 0, α = β = 1).The dashed parts of the curves mean that the squared speed of propagation is negative, signaling the occurrence of a Laplacianinstability. avoided due to the chameleon effect [50, 51]: The effective mass of the scalar degree of freedom may be a function ofthe local background curvature or, equivalently, of the energy density of the local environment, so that it can be largeat Solar System and terrestrial curvatures and densities and small at cosmological scales. In other words: it may beshort ranged in the Solar System and become long ranged at cosmological densities, thus affecting the cosmologicaldynamics. It has been shown that for metric F ( R ) theories the chameleon effect plays an important role [46, 52–54].In this F ( R ) limit of the present formalism there are not Laplacian or gradient instabilities. B. Dominant Gauss-Bonnet contribution
The other limiting situation β/α ≫ α →
0, is when the Gauss-Bonnet term dominates the latetimes cosmological dynamics. From (67) it follows that: H ≈ √ β , so that the experimental bond (68) is avoided in this case. The price to pay for evading the constraint (68) is thestrong Gauss-Bonnet coupling required: β ∼ M − . This sets the scale of smallness of the Gauss-Bonnet term |G| much below 1 /β ∼ − M , i. e., the scale of small curvature p |G| is far below the present value of thecurvature of the Universe ∼ − M . This is why, in this limit, the present cosmological dynamics is dictated bythe Gauss-Bonnet interaction. At much smaller curvature scales, |G| ≪ /β , the Gauss-Bonnet term amounts to atotal derivative and may be safely removed, so that the coupling β does not play any role.In both limiting situations the resulting physical picture is one in which the energy density of vacuum is of the orderof the Plack mass to the 4th power, with vanishing effective (low-curvature) cosmological constant because flat spaceis a stable solution of the vacuum EOM, and a FRW de Sitter expansion with the required (present day) Hubble rate H , is the global future attractor.8 VIII. DISCUSSION
The absence of ghosts and instabilities such as: ghosts due to anisotropy of space or to linear perturbations aroundspherically symmetric static background [47], tachyonic, Dolgov-Kawasaki and graviton instabilities, in the presentmodel is a consequence of the choice of F ( R, G ) in (30). This makes of the theory (3), with F ( R, G ) given by (30), avery attractive possibility for a viable fourth-order theory of gravity.Unfortunately, but for the formal limit β →
0, the model is not free of other kinds of problems such as the Laplacianor gradient instability. Their absence would either impose additional requirements on the physical phase space, as wellas on the space of parameters of the theory, or require of additional modifications to the original set up. Accordingto [29, 40], the squared speed of propagation of the scalar and tensor modes (gravitational waves) in general F ( R, G )theories, are given by: c s = 1 + 8 βα ˙ H βα H = 1 + 2(1 − x ) ˙ HH , (69)and c T = F R + 4 βα ¨ F R F R + 4 βα H ˙ F R = 1 − − x ) ˙ HH , (70)respectively. Notice that c s + c T = 2. For the absence of Laplacian instabilities it is required that both c s ≥ c T ≥
0, i. e., that − − x ) ≤ ˙ HH ≤ − x ) . (71)Whether c s > c T >
1, the speed of propagation is superluminal. However, this is not a problem as it doesnot directly violate causality on the cosmological FRW background [29]. On the contrary, it is a real problem wheneither c s < c T < F ( R, G )theories. This includes our present set up. Let us to consider the two limiting situations studied in section VII: i) the F ( R ) limit where β → α →
0. In the former case we have that c s = c T = 1 , while in the latter limiting situation: c s = 1 + 2 ˙ HH , c T = 1 − HH , confirming that in the β → α → c s = c s ( x, y ) (left) and of c T = c T ( x, y ) (right) are shown for several orbits of the dynamical system (42),corresponding to the simplified case when µ = 0 (see sub-section V A), and to the choice of free parameters: α = β = 1(middle panel of FIG. 1). It is seen that, although c T is always a positive quantity, the squared speed of the scalarperturbations c s is negative along parts of the orbits, signaling that Laplacian instabilities are inevitable. This hasbeen associated with matter instabilities that arise at small scales and large redshifts in F ( R, G ) theories [29, 55].One possibility to deal with these unwanted instabilities can be based on the method developed in [19], where theauthors proposed a procedure to eliminate ghosts and to obtain viable F ( R, G ) models. The method is based on theintroduction of an auxiliary scalar field into the F ( R, G ) action. Then, in order to make the scalar mode not a ghost,a canonical kinetic term may be introduced in the action. After this, it is possible to obtain second-order equationsof motion and to impose suitable initial conditions determining a regular (unique) ghost-free evolution. Anotherpossibility can be to introduce non-minimal coupling to a scalar field φ of the following form (compare with (26)):9 F ( φ, R, G ) = − λ (cid:0) − αφ − R − K ( φ ) − β G (cid:1) ν , where K ( φ ) is a kinetic term for the scalar field. The coupled metric and scalar perturbations might conspire tocounteract the effects of the destabilizing Gauss-Bonnet contributions. We do not expect that the present setupmay account for a realistic description of our Universe, instead, it may be viewed as a toy model showing that thecosmological constant issue, if not solved, at least may be evaded.Although our model complies the requirements mentioned in the introduction, nevertheless, there should be othermodels of modified gravity that may fulfill these requirements as well. The proposal investigated in [7] seems to bean example of that. In that reference the author studies F ( R ) models given by: F ( R ) = R + λR "(cid:18) R R (cid:19) − n − , (72)where n , λ > R of the order of the presently observed effective cosmological constant, are the free parameters.Then, F (0) = 0 (the cosmological constant “disappears” in flat space) and R µν = 0 is always a solution of the EOMin the absence of matter. For R ≫ R , F ( R ) = R − ∗ , where Λ ∗ = λR /
2. The model has de Sitter solutions with R = x R , where x is the maximal root of a given algebraic equation (equation (6) of [7]). This model is free of theDolgov-Kawasaki instability, unfortunately, given that F RR <
0, flat space is unstable. For n ≥ F ( R ) → F ( R, G ). I. e.,from the start a non-vanishing vacuum energy density ∝ µ should be considered. Besides, it should be demonstratedthat the de Sitter solution with R = x R is a future attractor in the phase space of the model. To our knowledgethese items have not been investigated yet. IX. CONCLUSION
In this paper we have explored a class of Born-Infeld inspired F ( R, G ) models of modified gravity of type F ( R, G ) ∝√L Lovelock (see equation (30)): F ( R, G ) = − λ p − αR − β G . This two-parametric class of theory is free of most of the unwanted ghosts and instabilities, but for the Laplacianinstability which is inevitable in F ( R, G ) models in general. This calls for further modifications of (30) through, forinstance, the procedure proposed in [19].Models of the kind we have investigated here are very interesting alternatives to GR, not only because the inclusionof higher-order curvature operators is dictated by renormalization [37, 38] and by the formulation of GR as an effectivetheory [14, 15], but because the CCP may be avoided in these models. It should be mentioned that the full asymptoticdynamics of the present model has been investigated in [48]. In that reference, however, only the case with negativeGauss-Bonnet coupling ( β <
0) was considered. Although this latter case was not investigated here, the results of ourpresent study also apply to this case.It is necessary to mention, also, that the formal limit when in model (30) β →
0, i. e., if neglect the Gauss-Bonnetterm, will not satisfy the observational constraints coming from cosmology, in particular that the present value ofthe Hubble rate H ∼ − M Pl , unless the chameleon effect plays an important part in the origin of the effectivemass of the propagating scalar degree of freedom. The advantage of this limiting case is that Laplacian instabilitydoes not arise. Alternatively, when the Gauss-Bonnet interaction plays a significant role in the late time dynamics,the observational constraints of cosmological origin are met but the model should be improved in order to avoid thearising of gradient instability.Although the class of theory (30) meets the necessary and sufficient conditions to avoid the cosmological constantissue, there remain issues with the occurrence of Laplacian instabilities. Hence, in the last instance one may thinkof the present setup as a class of toy models that serve to show that, an alternative to explain the huge discrepancybetween the theoretically predicted and the observed values of the cosmological constant, does exist. It consists justin evading the problem.0 Acknowledgments
The author thanks FORDECYT-PRONACES-CONACYT for support of the present research under grant CF-MG-2558591. [1] S. Weinberg, Rev. Mod. Phys. (1989) 1-23[2] P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. (2003) 559-606 [e-Print: astro-ph/0207347][3] T. Padmanabhan, Phys. Rept. (2003) 235-320 [e-Print: hep-th/0212290][4] I. Zlatev, L.M. Wang, P.J. Steinhardt, Phys. Rev. Lett. (1999) 896-899 [e-Print: astro-ph/9807002][5] S.M. Carroll, Living Rev. Rel. (2001) 1 [e-Print: astro-ph/0004075][6] V. Sahni, A.A. Starobinsky, Int. J. Mod. Phys. D (2000) 373-444 [e-Print: astro-ph/9904398][7] A.A. Starobinsky, JETP Lett. (2007) 157-163 [e-Print: 0706.2041][8] J. Dreitlein, Phys. Rev. Lett. (1974) 1243-1244[9] T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rept. (2012) 1-189 [e-Print: 1106.2476][10] G. ’t Hooft, M.J.G. Veltman, Ann. Inst. H. Poincare Phys. Theor. A (1974) 69-94[11] M.H. Goroff, A. Sagnotti, Nucl. Phys. B (1986) 709-736[12] D.J. Gross, E. Witten, Nucl. Phys. B (1986) 1-10[13] S. Deser, A.N. Redlich, Phys. Lett. B (1986) 350; Phys. Lett. B (1987) 461 (erratum)[14] J.F. Donoghue, Phys. Rev. Lett. (1994) 2996-2999 [e-Print: gr-qc/9310024][15] J.F. Donoghue, Phys. Rev. D (1994) 3874-3888 [e-Print: gr-qc/9405057][16] L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. Lust, A. Riotto, Fortsch. Phys. (2016) 176-189 [e-Print: 1505.07657][17] S. Nojiri, S.D. Odintsov, Phys. Rev. D (2005) 123509 [e-Print: hep-th/0504052][18] S. Nojiri, S.D. Odintsov, Phys. Lett. B (2005) 1-6 [e-Print: hep-th/0508049][19] S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Phys. Rev. D (2019) 044050 [e-Print: 1811.07790][20] G. Calcagni, B. de Carlos, A. De Felice, Nucl. Phys. B (2006) 404-438 [e-Print: hep-th/0604201][21] P.G.S. Fernandes, P. Carrilho, T. Clifton, D.J. Mulryne, Phys. Rev. D (2020) 024025 [e-Print: 2004.08362][22] M. Born, L. Infeld, Proc. Roy. Soc. Lond. A (1934) 425-451[23] R. Ferraro, Franco Fiorini, Phys. Rev. D (2007) 084031 [e-Print: gr-qc/0610067][24] J. Beltran Jimenez, L. Heisenberg, G.J. Olmo, D. Rubiera-Garcia, Phys. Rept. (2018) 1-129 [e-Print: 1704.03351][25] D. Comelli, Phys. Rev. D (2005) 064018 [e-Print: gr-qc/0505088][26] I. Quiros, L.A. Urena-Lopez, Phys. Rev. D (2010) 044002 [e-Print: 1004.1719][27] D. Lovelock, J. Math. Phys. 12 (1971) 498-501; J. Math. Phys. (1972) 874-876[28] P. Bueno, P.A. Cano, A.O. Lasso, P.F. Ram´ırez, JHEP (2016) 028 [e-Print: 1602.07310][29] A. De Felice, J.M. Gerard, T. Suyama, Phys. Rev. D (2010) 063526 [e-Print: 1005.1958][30] S.D. Odintsov, V.K. Oikonomou, S. Banerjee, Nucl. Phys. B (2019) 935-956 [e-Print: 1807.00335][31] J.G. Lee, E.G. Adelberger, T.S. Cook, S.M. Fleischer, B.R. Heckel, Phys. Rev. Lett. (2020) 101101 [e-Print: 2002.11761][32] A.A. Starobinsky, Phys. Lett. B (1980) 99-102; Adv. Ser. Astrophys. Cosmol. (1987) 130-133[33] A.A. Starobinsky, Phys. Lett. B (1982) 175-178[34] A. Vilenkin, Phys. Rev. D (1985) 2511-2521[35] A.A. Starobinsky, JETP Lett. (2007) 157-163 [e-Print: 0706.2041][36] A. Kehagias, A. Moradinezhad Dizgah, A. Riotto, Phys. Rev. D (2014) 4, 043527 [e-Print: 1312.1155][37] K.S. Stelle, Phys. Rev. D (1977) 953-969; Gen. Rel. Grav. (1978) 353-371[38] A. Hindawi, B.A. Ovrut, D. Waldram, Phys. Rev. D (1996) 5583-5596 [e-Print: hep-th/9509142]; Phys. Rev. D (1996) 5597-5608 [e-Print: hep-th/9509147][39] A. De Felice, T. Tanaka, Prog. Theor. Phys. (2010) 503-515 [e-Print: 1006.4399][40] A. De Felice, T. Suyama, JCAP (2009) 034 [e-Print: 0904.2092][41] C. Cartier, J.C. Hwang, E.J. Copeland, Phys. Rev. D (2001) 103504 [e-Print: astro-ph/0106197][42] J.C. Hwang, H. Noh, Phys. Rev. D (2005) 063536 [e-Print: gr-qc/0412126][43] A.D. Dolgov, M. Kawasaki, Phys. Lett. B (2003) 1-4 [e-Print: astro-ph/0307285][44] S. Nojiri, S.D. Odintsov, Phys. Rev. D (2003) 123512 [e-Print: hep-th/0307288]; Gen. Rel. Grav. (2004) 1765-1780[e-Print: hep-th/0308176][45] V. Faraoni, Phys. Rev. D (2007) 067302 [e-Print: gr-qc/0703044]; Phys. Rev. D (2006) 104017 [e-Print:astro-ph/0610734][46] T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. (2010) 451-497 [e-Print: 0805.1726][47] A. De Felice, T. Suyama, T. Tanaka, Phys. Rev. D (2011) 104035 [e-Print: 1102.1521][48] R. Garcia-Salcedo, T. Gonzalez, C. Moreno, Y. Napoles, Y. Leyva, I. Quiros, JCAP (2010) 027 [e-Print: 0912.5048][49] R. Garc´ıa-Salcedo, T. Gonzalez, F.A. Horta-Rangel, I. Quiros, D. Sanchez-Guzm´an, Eur. J. Phys. (2015) 025008[e-Print: 1501.04851][50] J. Khoury, A. Weltman, Phys. Rev. Lett. (2004) 171104 [e-Print: astro-ph/0309300]; Phys. Rev. D (2004) 044026[e-Print: astro-ph/0309411] [51] I. Quiros, R. Garc´ıa-Salcedo, T. Gonzalez, F. Antonio Horta-Rangel, Phys. Rev. D (2015) 044055 [e-Print: 1506.05420][52] J.A.R. Cembranos, Phys. Rev. D (2006) 064029 [e-Print: gr-qc/0507039][53] I. Navarro, K. Van Acoleyen, JCAP (2007) 022 [e-Print: gr-qc/0611127][54] T. Faulkner, M. Tegmark, E.F. Bunn, Y. Mao, Phys. Rev. D (2007) 063505 [e-Print: astro-ph/0612569][55] A. De Felice, D.F. Mota, S. Tsujikawa, Phys. Rev. D81