Cosmological dynamics and dark energy from non-local infrared modifications of gravity
CCosmological dynamics and dark energyfrom non-local infrared modifications of gravity
Stefano Foffa, Michele Maggiore and Ermis Mitsou
D´epartement de Physique Th´eorique and Center for Astroparticle Physics,Universit´e de Gen`eve, 24 quai Ansermet, CH–1211 Gen`eve 4, Switzerland
Abstract
We perform a detailed study of the cosmological dynamics of a recently proposedinfrared modification of the Einstein equations, based on the introduction of a non-local term constructed with m g µν (cid:50) − R , where m is a mass parameter. The theorygenerates automatically a dynamical dark energy component, that can reproduce theobserved value of the dark energy density without introducing a cosmological constant.Fixing m so to reproduce the observed value Ω DE (cid:39) .
68, and writing w ( a ) = w +(1 − a ) w a , the model provides a neat prediction for the equation of state parameters ofdark energy, w (cid:39) − .
042 and w a (cid:39) − . (cid:50) − , one can extend the construction so to define a more generalfamily of non-local models. However, in a first approximation this turns out to beequivalent to adding an explicit cosmological constant term on top of the dynamicaldark energy component. This leads to an extended model with two parameters, Ω Λ and m . Even in this extension the EOS parameter w is always on the phantom side,in the range − . < ∼ w ≤ −
1, and there is a prediction for the relation between w and w a . a r X i v : . [ h e p - t h ] N ov Introduction
The study of modifications of General Relativity (GR) at cosmological scales has gainedmuch impetus in recent years, as one of the most promising directions for understandingthe origin of the observed acceleration of the Universe. The interest for such infrared(IR) modifications was initially spurred by the DGP model [1], which indeed has a self-accelerating solution [2, 3]. The viability of this specific proposal was eventually ruled outby the existence of a ghost instability [4–8], but the search for consistent IR modificationsof GR and the study of their cosmological consequences has been developed in variousdifferent directions. In particular, recent years have seen significant developments towardthe construction of a consistent theory of massive gravity [9–22] (see [23] for a review), andthe study of its cosmological consequences [24–31]. Another aspect of this intense activityis that various independent lines of reasoning seems to point toward the relevance of someform of non-locality for the dark energy problem. Non-local operators that modify GR inthe far IR appear in the degravitation proposal [32, 33] (see also [34–36]). A non-localcosmological model based on a non-local action has been proposed in [37], and has beenfurther studied in a number of recent papers, see e.g. [38–48]. Another interesting non-local model has been studied in [49–51]. Non-local gravity models have also been studiedas UV modifications of GR, see e.g. [52–56].In [57] it has been proposed a non-local modification of Einstein equation of the form G µν − m (cid:0) (cid:50) − G µν (cid:1) T = 8 πG T µν . (1.1)We use the notation (cid:50) to denote the d’Alembertian operator g µν ∇ µ ∇ ν with respect tothe metric g µν , and (cid:50) − is its inverse computed using the retarded Green’s function, asrequired by causality. The superscript T denotes the extraction of the transverse part ofthe tensor, which exploits the fact that, in a generic curved space-time, any symmetrictensor S µν can be decomposed as S µν = S T µν + 12 ( ∇ µ S ν + ∇ ν S µ ) , (1.2)where ∇ µ S T µν = 0 [58, 59]. The extraction of the transverse part of a tensor is a non-localoperation. For instance in flat space, where ∇ µ → ∂ µ , it is easy to show that the inversionof eq. (1.2) is S T µν = S µν − (cid:50) ( ∂ µ ∂ ρ S ρν + ∂ ν ∂ ρ S ρµ ) + 1 (cid:50) ∂ µ ∂ ν ∂ ρ ∂ σ S ρσ . (1.3)Because of its non-local nature, the transverse part of a tensor does not appear in theclassical equations of motion of a local theory. In eq. (1.1), however, we already have anexplicit (cid:50) − operator, so we have already payed the price of non-locality, and the use ofthe transverse part of a tensor becomes natural. Again, because of causality, we use theretarded Green’s function to define the non-local operators than enter in the extraction ofthe transverse part.Equation (1.1) can be seen as a refinement of the original degravitation idea proposedin [32, 33], which was based on an equation of the form (cid:18) − m (cid:50) (cid:19) G µν = 8 πG T µν (1.4)1with m a constant or, more generally, a function m ( (cid:50) ); this generalization could also beapplied to eq. (1.1), using an operator m ( (cid:50) ret ) inside the transverse-part operation). Ashortcoming of eq. (1.4) is that, since the covariant derivative does not commute with (cid:50) − ,the left-hand side of eq. (1.4) is not transverse, and hence ∇ µ T µν (cid:54) = 0. In contrast, theleft-hand side of eq. (1.1) is transverse by construction, so the energy-momentum tensoris automatically conserved. Observe furthermore that the use of the retarded Green’sfunction in eq. (1.1) ensures causality. However , the presence of a retarded propagatoralready at the level of the equations of motion (rather than, as usual, just in their solution),has important consequences for the conceptual meaning of such equations. As we discussin detail in [60], it implies that such non-local equations should not be understood as theequation of motion of a non-local QFT, but rather as effective classical equations derivedfrom some classical or quantum averaging of a more fundamental local theory.Equation (1.1) can be further generalized to G µν − m (cid:20) b (cid:0) (cid:50) − G µν (cid:1) T + b d − d (cid:0) g µν (cid:50) − R (cid:1) T (cid:21) = 8 πG T µν , (1.5)where b , b are arbitrary coefficients, and for the moment we work for generality in d spatial dimensions. The factor ( d − / (2 d ), is a convenient normalization of the b coef-ficient. In particular, in [61] has been studied the model with b = 0 , b = 1, and it hasbeen found that it has particularly interesting cosmological properties.The purpose of the present paper is to elaborate in more detail on the cosmologicalresults presented in [61]. We will also discuss in some detail the consequences of thefact that different definitions of the (cid:50) − operator are possible. Indeed, the most generalsolution of an equation such as (cid:50) f = j is f ( x ) = ( (cid:50) − j )( x ) ≡ f hom ( x ) + (cid:90) d d +1 x (cid:48) (cid:112) − g ( x (cid:48) ) G ( x ; x (cid:48) ) j ( x (cid:48) ) , (1.6)where f hom ( x ) is any solution of (cid:50) f hom = 0 and G ( x ; x (cid:48) ) is any Green’s function of the (cid:50) operator. To define our non-local model we must specify what definition of (cid:50) − weuse, i.e. we must specify the Green’s function and the corresponding solution of thehomogeneous equation. We will always use the retarded Green’s function. Still, in aFriedmann-Robertson-Walker (FRW) spacetime, there remains a freedom due to the factthat there is no obvious initial time where the convolution with the Green’s function starts.If we consider a model that, in the early Universe, starts from a radiation dominated(RD) phase, we can for instance start to convolution deep in RD (e.g., even at t = 0,as done in [37]). However, if we consider a model whose evolution begins in an earlierinflationary phase, the convolution will rather start at the beginning of the inflationaryphase. Once extrapolated into the RD phase, this different definition of (cid:50) − will generatea non-vanishing homogeneous solutions, that depends on the earlier history. As we willsee, in FRW in a first approximation this freedom turns out to be equivalent to thefreedom of introducing an explicit cosmological constant term. We will also discuss howthe introduction of auxiliary fields allows us to put these non-local models in a localform. In this “localized” form the parameters labeling different definitions of the (cid:50) − In contrast, the original degravitation proposal [32] was presented as an acausal modification of gravityat cosmological distances. (cid:50) − operator is defined so that theassociated homogeneous solution in the RD phase is set to zero. In sect. 4 we discuss howthe definition of the (cid:50) − operator can be extended and we show that, when one writesthe model in terms of auxiliary fields, this extension is reflected into the initial conditionsof the auxiliary fields. Apart from allowing us to identify a more general class of models,the discussion in this section is important also for understanding the issue of the stabilityof the solution within a given non-local model. As we will see, the local formulationputs together the space of solutions of all these different models. As a result, apparentinstabilities of a solution in the local formulation do not correspond necessarily to actualinstabilities in the original non-local model, since they correspond to moving from thesolution of a given non-local model toward the solutions of a different non-local model.In a related paper [60] we discuss in greater generality the conceptual issues raised bythese non-local equations, in particular in connection with apparent ghost-like degrees offreedom that seem to emerge from these models, and we show that such apparent ghostsare spurious and do not represent propagating degrees of freedom of the theory. Modelsof the form (1.5) with b (cid:54) = 0 seem less viable because of cosmological instabilities, and weexamine them in App. A. Our notation and convention are as in [57]. In particular, weuse the signature η µν = ( − , + , + , +). We now set b = 0 , b = 1 in eq. (1.5), i.e. we study the model given by G µν − m d − d (cid:0) g µν (cid:50) − R (cid:1) T = 8 πG T µν . (2.1)First of all, we need to give a precise definition of the (cid:50) − operator, i.e. we must assign theGreen’s function and the corresponding homogeneous solution in eq. (1.6). We directlyspecialize to a spatially flat FRW metric in d spatial dimensions, ds = − dt + a ( t ) d x .In this section we follow [37] and we define( (cid:50) − R )( t ) = − (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t ∗ dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) , (2.2)where t ∗ is some initial value of time, that we take here in RD. As we discuss in [60], a non-local equation such as (2.1), which involves the retarded inverse d’Alembertian, should beunderstood as an effective equation, obtained from some classical or quantum averagingof an underlying fundamental theory. Then, t ∗ can be interpreted as a value of time wheresuch an effective description becomes appropriate, and eq. (2.2) is only valid for t > t ∗ .Observe that, since in RD the Ricci scalar R vanishes, this definition is independent of3he exact value of t ∗ , as long as it is deep in RD. With this definition, also (cid:50) − R vanishesduring RD, and only becomes active in the subsequent matter dominated (MD) phase.In FRW, on a scalar f ( t ), we have (cid:50) f = − a − d ∂ ( a d ∂ f ), so one immediately verifiesthat eq. (2.2) indeed provides a possible inversion of the (cid:50) − operator. This inversioncorresponds to a retarded Green’s function, as we see from the fact that the integrationis only over times t (cid:48)(cid:48) and t (cid:48) smaller than t . Equivalently, we can rewrite eq. (2.2) as( (cid:50) − R )( t ) = − (cid:90) ∞ t ∗ dt (cid:48) θ ( t − t (cid:48) ) 1 a d ( t (cid:48) ) (cid:90) ∞ t ∗ dt (cid:48)(cid:48) θ ( t (cid:48) − t (cid:48)(cid:48) ) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) , (2.3)which can be rearranged in the form( (cid:50) − R )( t ) = (cid:90) ∞ t ∗ dt (cid:48) G ret ( t ; t (cid:48) ) R ( t (cid:48) ) , (2.4)where G ret ( t ; t (cid:48) ) = − θ ( t − t (cid:48) ) a d ( t (cid:48) ) (cid:90) tt (cid:48) dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) . (2.5)In sect. 4 we will study a more general class of models, in which we add a general solutionof the homogeneous equations to the definition (2.2) and we will find, quite remarkably,that the above freedom basically amounts to the possibility of introducing in the theory acosmological constant term.A similar issue of definition of non-local operators arises when we compute the trans-verse part in eq. (2.1). To extract the transverse part we proceed as in [57, 61]. Weintroduce a scalar field U from U ≡ − (cid:50) − R ≡ (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t ∗ dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) . (2.6)We then define S µν = − U g µν , and we split S µν as in eq. (1.2). To determine S µ we apply ∇ µ to both sides of this equation, obtaining (cid:50) S ν + ∇ µ ∇ ν S µ = − ∂ ν U . (2.7)We must therefore invert the operator ( δ µν (cid:50) + ∇ µ ∇ ν ). In FRW this inversion simplifiesconsiderably. Indeed, the three-vector S i vanishes because there is no preferred spatialdirection, while from the ν = 0 component of eq. (2.7) we get a differential equation for S , ¨ S + dH ˙ S − dH S = ˙ U . (2.8)In this case we must therefore invert the operator D = ∂ + dH∂ − dH . (2.9)Denoting by D ret ( t ; t (cid:48) ) the retarded Green’s function of this operator, the definition anal-ogous to (2.4) is S ( t ) = (cid:90) ∞ t ∗ dt (cid:48) D ret ( t ; t (cid:48) ) ˙ U ( t (cid:48) ) , (2.10)4.e. we set again to zero the solutions of the associated homogeneous equation D f = 0.We will refer to the non-local model that makes use of these definitions of (cid:50) − and D − as the “minimal model”.We can now write down the cosmological equations governing this model. Since theenergy-momentum tensor in eq. (2.1) is conserved by construction, the cosmological evo-lution is determined by the (0 ,
0) component of eq. (2.1), i.e. the Friedmann equation. H − m d ( U − ˙ S ) = 16 πGd ( d − ρ . (2.11)More explicitly, inserting the definitions (2.4) and (2.10), our non-local model is definedby the integro-differential equation H + m d (cid:20)(cid:90) ∞ t ∗ dt (cid:48) G ret ( t ; t (cid:48) ) R ( t (cid:48) ) − ∂ t (cid:90) ∞ t ∗ dt (cid:48) D ret ( t ; t (cid:48) ) ∂ t (cid:48) (cid:90) ∞ t ∗ dt (cid:48)(cid:48) G ret ( t (cid:48) ; t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) (cid:21) = 16 πGd ( d − ρ . (2.12) To evolve the equation numerically it can be convenient to transform the integro-differentialequation (2.1) into a set of local equations. This can be obtained using the auxiliary fields U ( t ) and S ( t ) defined above. Equation (2.6) can be written as (cid:50) U = − R so, togetherwith eqs. (2.8) and (2.11), we have three differential equations for the three functions { H ( t ) , U ( t ) , S ( t ) } . The retarded prescriptions in eqs. (2.6) and (2.10) are automaticallytaken into account by assigning initial conditions on U ( t ) and S ( t ) at an initial time t ∗ and integrating the equations forward in time.To integrate, we must then assign U, ˙ U , S and ˙ S at t = t ∗ . In turn, these initialconditions are uniquely specified by the definitions of the (cid:50) − and D − operators givenin eqs. (2.6) and (2.10), and in particular by the choice of the associated homogeneoussolutions, which here we have set to zero. Thus, from eq. (2.6) we have U ( t ∗ ) = 0.Furthermore, eq. (2.6) gives ˙ U ( t ) = 1 a d ( t ) (cid:90) tt ∗ dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) , (3.1)and therefore also ˙ U ( t ∗ ) = 0. Similarly, the retarded nature of D ret ( t ; t (cid:48) ) in eq. (2.10)implies that S ( t ∗ ) = 0. Furthermore, writing D ret ( t ; t (cid:48) ) = θ ( t − t (cid:48) ) g ( t ; t (cid:48) ), we have˙ S ( t ) = (cid:90) ∞ t ∗ dt (cid:48) (cid:2) δ ( t − t (cid:48) ) g ( t ; t (cid:48) ) + θ ( t − t (cid:48) ) ∂ t g ( t ; t (cid:48) ) (cid:3) ˙ U ( t (cid:48) )= g ( t ; t ) ˙ U ( t ) + (cid:90) tt ∗ dt (cid:48) ∂ t g ( t ; t (cid:48) ) ˙ U ( t (cid:48) ) . (3.2)5n t = t ∗ this vanishes, because ˙ U ( t ∗ ) = 0. In summary, the original integro-differentialequation (2.12) is equivalent to the coupled system of differential equations H − m d ( U − ˙ S ) = 16 πGd ( d − ρ (3.3)¨ U + dH ˙ U = 2 d ˙ H + d ( d + 1) H , (3.4)¨ S + dH ˙ S − dH S = ˙ U , (3.5)(where we used the fact that, in FRW with generic d , R = 2 d ˙ H + d ( d + 1) H ), togetherwith the initial conditions U ( t ∗ ) = ˙ U ( t ∗ ) = S ( t ∗ ) = ˙ S ( t ∗ ) = 0 . (3.6)It is important to stress that the initial conditions on the auxiliary fields U and S arefixed, once we give the definition of the (cid:50) − and D − operators in the original non-localmodel. Taking these initial conditions as free parameters is incorrect. In other words, thespace of solutions of the local system (3.3)–(3.5), with generic initial conditions on U and S , is much larger than the space of solutions of the original non-local equation. Differentchoice of initial conditions on U and S correspond to different choices of the homogeneoussolutions associated to eqs. (3.4) and (3.5), i.e. of the equations (cid:50) U = 0 and D S = 0,which corresponds to different choices of the homogeneous functions used to define the (cid:50) − and D − in the original non-local model. Any given definition of (cid:50) − and D − fixes acorresponding solution of the homogeneous solutions associated to eqs. (3.4) and (3.5). Ifone forgets this simple but important point, one can easily fall into the mistake of believingthat the solutions of (cid:50) U = 0 and D S = 0 represent scalar propagating degrees of freedomof the original non-local model. The fact that these degrees of freedom are spurious, andare an artifact of the “localization” procedure, has been recognized recently by variousauthors in similar non-local models [46,51,62,63]. The issue is even more important in flatMinkowski space, where these spurious degrees of freedom include a ghost. This wouldlead to the erroneous conclusion that the quantum vacuum of these theories is unstable.In fact, there is no propagating degree of freedom associated to the ghost. In the flat-space case the solutions of the associated homogeneous equation (cid:50) U = 0 are of coursejust plane wave. However, the coefficients a k and a ∗ k of these plane-wave solutions are notfree parameters that, at the quantum level, can be promoted to annihilation and creationoperators of a quantum field. Simply, they are fixed once the definition of the (cid:50) − operatoris given (e.g. to a k = a ∗ k = 0), and do not parametrize degrees of freedom of the originalnon-local theory (see also the more extended discussion in [60]). Having clarified this important conceptual point, we can now use the local form of theequations to study the cosmological evolution. We take ρ equal to the sum of the matterdensity ρ M and the radiation density ρ R , and we henceforth restrict to d = 3 spatialdimensions. We do not add by hand a cosmological constant term ρ Λ , since our aimis to investigate whether a viable dynamical dark energy (DE) component emerges au-tomatically from the term proportional to the mass m . It is also convenient to define6 = U − ˙ S , since this is the quantity that appears in eq. (2.11), and use { H, U, Y } asindependent variables. We also define ρ DE ( t ) = ρ γY ( x ) , (3.7)where ρ = 3 H / (8 πG ), and γ ≡ m H . (3.8)Then eq. (2.11) becomes H ( t ) = 8 πG ρ M ( t ) + ρ R ( t ) + ρ DE ( t )] . (3.9)Thus, the term proportional to m plays the role of a dynamical dark energy. In orderto deal with dimensionless quantities only we define as usual h ( t ) = H ( t ) /H , Ω i ( t ) = ρ i ( t ) /ρ c ( t ) (where ρ c ( t ) = 3 H ( t ) / (8 πG ) and i labels radiation, matter and dark energy),and we use the notations Ω M ≡ Ω M ( t ), Ω R ≡ Ω R ( t ), Ω DE ≡ Ω DE ( t ). We find usefulto parametrize the temporal evolution using the variable x ≡ ln a ( t ) instead of t , and wedenote df /dx = f (cid:48) . Then, we get [61] h ( x ) = Ω M e − x + Ω R e − x + γY ( x ) , (3.10)where the evolution of Y ( x ) is obtained from the coupled system of equations Y (cid:48)(cid:48) + (3 − ζ ) Y (cid:48) − ζ ) Y = 3 U (cid:48) − ζ ) U , (3.11) U (cid:48)(cid:48) + (3 + ζ ) U (cid:48) = 6(2 + ζ ) , (3.12)and ζ is given by ζ ( x ) ≡ h (cid:48) h = − M e − x + 4Ω R e − x − γY (cid:48) M e − x + Ω R e − x + γY ) . (3.13)The initial conditions (3.6), together with the definition Y = U − ˙ S , imply that Y ( t ∗ ) = 0.Furthermore, using eq. (3.5), we see that eq. (3.6) also implies that ¨ S ( t ∗ ) = 0, andtherefore also ˙ Y ( t ∗ ) = 0. Thus, the initial conditions corresponding to the original integro-differential equation (2.12) are U ( t ∗ ) = U (cid:48) ( t ∗ ) = Y ( t ∗ ) = Y (cid:48) ( t ∗ ) = 0 . (3.14)Observe that ζ ( x ) is related to the total equation of state (EOS) parameter w ( t ), definedby p ( t ) = w ( t ) ρ ( t ), where p = (cid:80) i p i , ρ = (cid:80) i ρ i (and, again, i labels radiation, matter anddark energy). Combining energy-momentum conservation ˙ ρ + 3(1 + w ) Hρ = 0 with theFriedmann equation H = (8 πG/ ρ we get in fact ˙ H/H = − (3 / w ( t )] or, using x as time evolution variable and observing that ˙ H/H = H (cid:48) /H , ζ ( x ) = −
32 [1 + w ( x )] . (3.15)We finally define the dark energy equation-of-state (EOS) parameter w DE ( x ) from˙ ρ DE + 3(1 + w DE ) Hρ DE = 0 . (3.16)Observing that ˙ ρ = Hρ (cid:48) we get w DE ( x ) = − − Y (cid:48) ( x )3 Y ( x ) . (3.17)The EOS parameter of this dark energy component is therefore close to − | Y (cid:48) / Y | (cid:28) .3 Perturbative solutions and stability The above equations are highly non-linear, because the function Y ( x ) and its derivativeappears also in ζ ( x ). As discussed in [61] it is useful to begin by studying a perturbativeregime, where the contribution of Y ( x ) to ζ ( x ) is negligible. In particular we expect thatthis will be true in the early Universe (i.e. at x large and negative) so that we recoverstandard cosmology at early times. We therefore assume that, as x → −∞ , ζ ( x ) (cid:39) − M e − x + 4Ω R e − x M e − x + Ω R e − x ) , (3.18)and we check a posteriori the self-consistency of the procedure. In this case, in each givenera ζ ( x ) can be further approximated by a constant ζ , with ζ = − ζ = − / U isgiven by [61] U ( x ) = 6(2 + ζ )3 + ζ x + u + u e − (3+ ζ ) x , (3.19)where the coefficients u , u parametrize the general solution of the homogeneous equation U (cid:48)(cid:48) + (3 + ζ ) U = 0. For later use, we study here the perturbative solution with genericinitial conditions, and we will later impose the initial conditions (3.14) appropriate to ourproblem. Plugging eq. (3.19) into eq. (3.11) and solving for Y ( x ) we get [61] Y ( x ) = − ζ ) ζ (3 + ζ )(1 + ζ ) + 6(2 + ζ )3 + ζ x + u − ζ ) u ζ + 3 ζ − e − (3+ ζ ) x + a e α + x + a e α − x , (3.20)where α ± = 12 (cid:20) − ζ ± (cid:113)
21 + 6 ζ + ζ (cid:21) . (3.21)Observe that in RD ζ = − U and Y vanish. Thisis a consequence of the fact that in RD the Ricci scalar vanishes, so (cid:50) U = 0 and the onlycontributions to U and to ( U g µν ) T come from the solutions of the homogeneous equations.The inhomogeneous solution is self-consistent with our perturbative approach. Indeed, ina pure RD phase it just vanishes, and in a generic epoch, as x → −∞ , Y ( x ) ∝ x so itscontribution to ζ ( x ) is anyhow negligible compared to the term Ω M e − x and Ω R e − x ineq. (3.13).Specializing now the case in which the evolution is started at a value x = x ∗ deep inRD, we see that at the initial time the inhomogeneous solution vanishes and therefore U ( x ∗ ) = u + u e − x ∗ , (3.22) Y ( x ∗ ) = u + a e α + x ∗ + a e α − x ∗ . (3.23)Imposing the initial conditions (3.14) in RD therefore amounts to setting u = u = a = a = 0, i.e. we set to zero the solution of the homogeneous equations in RD.In sect. 4 we will study what happens if we rather start the evolution in an earlier phase,such as an earlier inflationary epoch. Observe that, in a generic epoch, the homogeneoussolutions for U are always stable (as long as ζ ≥ −
3, i.e. w ≤
1, which is always the case).8he homogeneous solution for Y is stable as long as both α + ≤ α − ≤
0. This givesthe condition ζ ≤ −
1, i.e. w ≡ − − ζ ≥ − , (3.24)which is satisfied in RD and MD. In particular, in RD α ± = (1 / − ± √
13) and in MD α ± = ( − ± √ /
4. However, the condition w < − / a , a , the perturbative solution is unstable. However, asdiscussed above (and as we will discuss again in detail in sect. 4) the initial conditionsare in one-to-one correspondence with the definition of the non-local operators in theoriginal non-local model. Thus, if we start the evolution in an earlier inflationary era,and we define the non-local operators (cid:50) − and D − so that their associated homogeneoussolutions vanish, we must set a = a = 0 in eq. (3.20) in the perturbative solution validduring the inflationary era. With this definition of the non-local model, the exponentiallygrowing homogeneous solutions are simply not solutions of the original non-local integro-differential equation, and are an artifact due to the fact that the space of solutions of thelocal form of the equations is larger than the space of solutions of the original non-localmodel. In turn, setting to zero the homogeneous solutions during the inflationary era willgenerate non-zero homogeneous solutions during the subsequent RD era, whose effect willbe studied in sect. 4. We now integrate eqs. (3.10)–(3.13) numerically. Since the initial conditions on
U, Y arefixed by the definition of (cid:50) − and D − in the original non-local model, the only freeparameter is γ , plus of course the values of Ω M and Ω R that enter through eq. (3.10).However, just as in ΛCDM the parameters Ω Λ , Ω M and Ω R are related by the conditionΩ M + Ω R + Ω Λ = 1, similarly here γ, Ω M and Ω R are related by the condition that, at x = 0, Ω M + Ω R + γY (0) = 1. In other words, since by definition h ( x ) = H ( x ) /H ,the only consistent solutions are those that satisfy h (0) = 1. We set Ω M and Ω R to thePlanck best-fit values Ω M = 0 . R = 4 . × − h − , h = 0 . Λ = 0). The appropriate value of γ must then be determined by trials and errors, sinceΩ DE = γY (0) and the evolution of Y ( x ) depends on γ itself through the dependence of ζ ( x ) on γ . We find that, having set Ω M = 0 . h = 0 . R = 4 . × − h − ,the required value is γ = 0 . | Ω M + Ω R + Ω DE − | < − ). This corresponds to m/H = 3 γ / (cid:39) . H . Alternatively we could start from the equations in the form (3.3)–(3.5), fix an initial value x ∗ , say deepin RD, assign γ as well as the values of ρ M ( x ∗ ) and ρ R ( x ∗ ), and let the system evolve forward in time(again, the initial conditions on U and S are uniquely fixed by the definition of the non-local operators inthe original non-local model). The present value of time t (or, equivalently, the value x ) is then identifiedby the condition that H ( x ) reaches the observed value H . Each value of { γ, ρ M ( x ∗ ) , ρ R ( x ∗ ) } producesa given matter and dark energy content at x = x . The values of γ , ρ M ( x ∗ ) and ρ R ( x ∗ ) could then bechosen, by trial and errors, so to obtain the desired values of Ω M and Ω R today. However, passing to thedimensionless quantity h ( x ) and fixing directly Ω M and Ω R in eq. (3.10) to the desired values is a muchmore effective way of proceeding, since then we must vary just a single parameter γ . The reason is thatin the three-dimensional space spanned by γ , ρ M ( x ∗ ) and ρ R ( x ∗ ) there are degeneracies, due to the factthat two models with different values of { γ, ρ M ( x ∗ ) , ρ R ( x ∗ ) } can reach the same values of H at differentvalues of x . Imposing that the present time is at x = 0 removes this degeneracy. (cid:45) (cid:45) U (cid:72) x (cid:76) (cid:45) (cid:45) (cid:45) Γ Y (cid:72) x (cid:76) (cid:45) (cid:45) (cid:45) Ρ i (cid:72) x (cid:76) (cid:144) Ρ t o t (cid:72) x (cid:76) (cid:87) D E (cid:72) z (cid:76) (cid:144) (cid:87) M (cid:72) z (cid:76) Figure 1: Upper panels: the functions U ( x ) and γY ( x ) from the numerical integration ofthe exact equations (blue solid lines), and the corresponding perturbative solutions (dashedred); we use γ = 0 . i = ρ i ( x ) /ρ c ( x ) for i = R (green, dot-dashed) i = M (red, dashed) and i = DE (blue solid line). Lower rightpanel: the ratio Ω DE ( z ) / Ω M ( z ), shown as a function of the redshift z = e − x −
1, in ournon-local model (blue solid line) and in ΛCDM (red dashed line).The result of the numerical integration of eqs. (3.11)–(3.13) is shown in the upperpanels of Fig. 1 (blue solid lines). The red dashed lines give the corresponding perturbativesolutions, that could be obtained analytically by matching the solution (3.19), (3.20) acrossthe RD-MD transition or, more simply, directly by numerical integration of eqs. (3.11) and(3.12), setting γ = 0 in eq. (3.13).The behavior of γY ( x ) is particularly interesting, since γY ( x ) is equal to the darkenergy density ρ DE ( x ) (normalized to ρ , see eq. (3.7)). In the RD phase it remains zero,while in the MD phase it begins to grow according to the perturbative solution, and finallyit becomes large and begins to dominate near the present epoch. It then decreases andgoes to zero in the future, roughly as a − / = e − x/ . Even if it goes to zero, at large x this dark energy density still remains the dominant component, since it only decreasesapproximately as a − / , while the matter density decreases as a − . In the lower left panelof Fig. 1 we show the energy fractions Ω i ( t ) = ρ i ( t ) /ρ c ( t ) for i = radiation, matter anddark energy. In the the lower right panel we show the ratio Ω DE ( z ) / Ω M ( z ) as a functionof the the redshift z = e − x − z = 1 . m (or, equivalently, γ ) from the condition Ω DE = 1 − Ω M − Ω R , we have no morefree parameters and the time evolution of Ω DE ( x ) is uniquely fixed, so we get a predictionfor evolution the of ρ DE ( x ) with x . This information can be compactly summarized usingfitting functions, as we now discuss. In principle the function ρ DE ( x ) /ρ = γY ( x ) computed above by numerical integrationof the differential equations, and displayed in Fig. 1, contains all the information on theevolution of the DE density. However, in practice it is convenient to “coarse grain” theinformation contained in Fig. 1, expressing it in terms of a fitting function that containsjust a few parameters that can be directly compared to observations. Since the functionΩ DE ( x ) is negligible in the early Universe (as we see from the lower left panel in Fig. 1),it is actually sufficient to find a parametrization that fits it well in the recent cosmologicalepoch, where it start to become important. In general, the most appropriate fittingfunction and the corresponding best-fit parameters will depend on the range of valuesof x = ln a that we consider. It is useful to distinguish different case, also to have an ideaof the stability of the fit. We first consider the region − < x <
0, corresponding to redshifts 0 < z < ∼ .
72. Weuse the standard fitting function [65, 66] w DE ( a ) = w + (1 − a ) w a , (3.25)where a = e x . We define ∆ w as the difference between the value of the numerical expres-sion and this fitting function, and we minimize with respect to w and w a the quantity χ = (cid:90) − dx (∆ w ) ( x ) . (3.26)We find that the best-fit values are w = − . w a = − . w DE ( x ) determined numerically (blue solid line) and the fittingfunction (3.25) with these best-fit values (red, dashed). For later purposes, we also showin this figure these functions in the region 0 < x <
1. We see that this fitting function isno longer accurate for x >
0, which however corresponds to the future and it is thereforenot relevant for the comparison with observations. In contrast, in the region − ≤ x < | ∆ w/w | ≤ × − , so in this region this fitting function should bequite accurate for most purposes.To assess the robustness of these best-fit values under changes of the cosmologicalparameters we have repeated the numerical integration changing Ω M , readjusting themass m so that Ω M + Ω R + γY (0) = 1, and repeating the fitting procedure. We changeΩ M in the interval [0 . , . M = 0 . w , w a we get w = − . w a = − . M = 0 .
033 we get w = − . w a = − . w and w a are unaffected. We11 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 2: Left: the numerical values of w DE ( x ) (blue solid line) compared to the function w DE ( a ) = w + (1 − a ) w a with w = − . w a = − . − < x <
1. Right: the value of ∆ w/w , in the region − < x < w DE ( a ) = w + (1 − a q ) w a , (3.27)again restricting for the moment to the region − < x <
0. Taking also q as a free fittingparameter gives the best-fit values w = − . w a = − . q = 1 . χ is practically irrelevant, so the introduction of q as a new fitting parameter in this case is not justified.In conclusion, in the region − < x < w = − . , w a = − . , (3.28)where we quoted the number of digits which is stable under changes in Ω M in the intervalΩ M ∈ [0 . , . w , w a ) plane are, at 95% c.l. w = − . +0 . − . and w a < .
32 [64]. Actually, sinceour prediction for w a is such that | w a | (cid:28)
1, it is meaningful to compare directly with theresult of ref. [64] for a constant w DE , which is much more stringent. The result obtainedcombining Planck+WP+SNLS is w DE = − . +0 . − . while Planck+WP+Union2.1 gives w DE = − . ± .
17. The prediction given in eq. (3.28) is therefore consistent with thePlanck result, and on the phantom side. The fact that the EOS parameter is on the phantom side is generically a consequenceof the fact that in our model the DE density starts from zero in RD and then grows duringMD. Thus, in this regime ρ DE > ρ DE >
0, and then eq. (3.16) implies (1 + w DE ) < The region − < x <
0. At x = − z (cid:39) .
72) we have Ω DE ( x ) / Ω M ( x ) (cid:39) .
09, which is small but not completely negligible, anddepending on the type of cosmological observations that one might wish to use for testingthe model, it can be useful to have a fitting function that works accurately down to lowervalues of x , e.g. down to the value x = − z (cid:39) Of course, a full comparison with the Planck data also requires the computation of the cosmologicalperturbations in our model. Work on this is in progress. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 3: Left panel: the EOS parameter w DE ( x ) in the region − < x < w = − . w a = − . w = − . w a = − . w/w from the fit (3.29).where Ω DE ( x ) / Ω M ( x ) (cid:39) . × − . At even more negative values of x the effect of darkenergy becomes even smaller, and in most situations a more accurate parametrization willprobably not be needed.The fitting function (3.25) works quite well, as we have seen, for − < x <
0, but goesastray for x < −
1, as we can see from the left panel in Fig. 3. In contrast, a good fittingfunction over the whole range − < x < w DE ( a ) = w − ¯ w a ln a . (3.29)Minimizing χ = (cid:82) − dx (∆ w ) with respect to w and ¯ w a we get w = − . w a = − . | ∆ w/w | ≤ . × − over the whole interval − < x <
0, as we can see fromthe right panel in Fig. 3. However, comparing with the right panel in Fig. 2 we see thatin the region − < x < Finally, we fit w DE ( x ) in the region − < x <
1. Of course this is to some extentacademic, since only the region x ≤
0, i.e. our past, is relevant for comparison withobservations, but this exercise is still instructive to get a general understanding of howthe fitting function can depend on the range considered. In this case, we see from theleft panel of Fig. 2 that the standard fit (3.25) is no longer accurate, and at x > w , w a , q ) so to minimize χ = (cid:82) − dx (∆ w ) . This gives w = − . w a = − . q = 1 . w/w ≤ × − over the range − < x < ρ DE ( x ) are obtained as usual from energy-momentum conservation ∂ t ρ DE + 3[1 + w DE ( x )] Hρ DE = 0, which integrates to ρ DE ( x ) = ρ DE (0) exp (cid:26) − (cid:90) x dx (cid:48) [1 + w DE ( x (cid:48) )] (cid:27) . (3.30)13n the region − < x < ρ DE ( x ) = ρ DE (0) e − w ) x − w a [ x − ( e x − . (3.31)Note that, for | x | (cid:28) ρ DE ( x ) (cid:39) ρ DE (0) e − w ) x +(3 / w a x and the term O ( x ) can beneglected, giving back the usual behavior ρ DE ( x ) (cid:39) ρ DE (0) e − w ) x = ρ DE (0) a − w ) .In the region − < x < − ρ DE ( x ) = ρ DE (0) e − w ) x +(3 /
2) ¯ w a x = ρ DE (0) e − w )( x − w x ) , (3.32)where w = ¯ w a / [2(1 + w )] (cid:39) . The cosmological model discussed above makes use of a specific definition of the (cid:50) − and D − operators, given in eqs. (2.6) and (2.10). More generally, we could study the evolution(starting again from RD) of a non-local model in which the (cid:50) − and D − operators, appliedto a function F ( t ), are defined by( (cid:50) − F )( t ) ≡ f ( t ) + (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t ∗ dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) F ( t (cid:48)(cid:48) ) , (4.1)( D − F )( t ) ≡ g ( t ) + (cid:90) ∞ t ∗ dt (cid:48) D ret ( t ; t (cid:48) ) F ( t (cid:48) ) , (4.2)where f ( t ) is a given solution of (cid:50) f = 0 and g ( t ) is a given solution of D g = 0. To motivatethe introduction of these homogeneous solutions, consider a cosmological model that startsfrom an earlier phase (for instance an inflationary phase) followed by RD and then MD. Inthis case it could be more natural to define (cid:50) − setting to zero the homogeneous solutionat the beginning of the inflationary era, that we denote as t = t i ( (cid:50) − R )( t ) = − (cid:90) tt i dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t i dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) . (4.3)If we denote by t ∗ the value of cosmic time when the inflationary epoch ends and RDstarts (so t ∗ > t i ), and we compute the value of ( (cid:50) − R )( t ) during RD using the definition(4.3), we have( (cid:50) − R )( t ) = − (cid:90) t ∗ t i dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t i dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) − (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t i dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) , (4.4)where we have split (cid:82) tt i dt (cid:48) = (cid:82) t ∗ t i dt (cid:48) + (cid:82) tt ∗ dt (cid:48) . The first integral is just a number, c ≡ − (cid:90) t ∗ t i dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t i dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) . (4.5)14n the second integral, in contrast, t (cid:48) > t ∗ , and we can use the fact that R ( t (cid:48)(cid:48) ) = 0 in RD,i.e. for t (cid:48)(cid:48) > t ∗ , so (cid:82) t (cid:48) t i dt (cid:48)(cid:48) can be replaced by (cid:82) t ∗ t i dt (cid:48)(cid:48) . Then we find that during RD, ratherthan having ( (cid:50) − R )( t ) = 0 as with the definition (2.2), we now have( (cid:50) − R )( t ) = c + c f ( t ) , (4.6)where c = − (cid:82) t ∗ t i dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) and f ( t ) = (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) . (4.7)Observe that both the constant c and the function f are solutions of the homogeneousequation (cid:50) f = 0, as it is clear writing (cid:50) = − a − d ∂ ( a d ∂ ). Thus, in RD this definition of (cid:50) − includes a given homogeneous solution.The same point is also easily understood in terms of the perturbative solutions for U and Y given in eqs. (3.19) and (3.20). If for instance we define the non-local operatorsso that u = u = a = a = 0 in the inflationary perturbative solution, during theinflationary phase we have U ( x ) = 6(2 + ζ infl0 )3 + ζ infl0 x , (4.8) Y ( x ) = − ζ infl0 ) ζ infl0 (3 + ζ infl0 )(1 + ζ infl0 ) + 6(2 + ζ infl0 )3 + ζ infl0 x (4.9)where ζ infl0 is the constant value of ζ ( x ) during inflation. At the inflation-RD transition,this solution will smoothly match to a perturbative RD solution, obtained setting ζ = − U ( x ) = u R0 + u R1 e − x , (4.10) Y ( x ) = u R0 + a R1 e α R+ x + a R2 e α R − x . (4.11)where α R ± = (1 / − ± √ u R0 , u R1 , a R1 , a R2 can be deter-mined analytically imposing the continuity of the functions and of their derivatives at thetransition, and will be non-zero.The above discussion shows that, in general, even if we are interested only in thecosmological evolution starting from RD, we should in general use the definitions of (cid:50) − and D − given in eqs. (4.1) and (4.2), allowing in general for a given homogeneous solutionboth in (cid:50) − and in D − , i.e. in FRW we can in general define U ( t ) ≡ − (cid:50) − R ≡ U hom ( t ) + (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t ∗ dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) , (4.12) S ( t ) ≡ D − ˙ U ≡ S , hom ( t ) + (cid:90) ∞ t ∗ dt (cid:48) D ret ( t ; t (cid:48) ) ˙ U ( t (cid:48) ) , (4.13)where U hom ( t ) and S , hom ( t ) are given solution of the homogeneous equation. These aredetermined once we assign the values of U, U (cid:48) , Y and Y (cid:48) at some initial time x in . Inturn, the set of values { U ( x in ) , U (cid:48) ( x in ) , Y ( x in ) , Y (cid:48) ( x in ) } can be rewritten as the set of15 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
10 0 100.00.51.01.5 x Γ Y (cid:72) x (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
10 0 100.00.51.01.5 x Γ Y (cid:72) x (cid:76) Figure 4: Left panel: γY ( x ), choosing the initial conditions on the perturbative solution(3.19,3.20) with u = u = a = a = 0 = 0 (blue solid line) compared to the solutionobtained setting, at x in = − Y ( x in ) = 10 and Y (cid:48) ( x in ) = α + Y ( x in ) (red dashed line).Right panel: the same with Y ( x in ) = 10 .values taken by the perturbative solutions, { U pert ( x in ) , U (cid:48) pert ( x in ) , Y pert ( x in ) , Y (cid:48) pert ( x in ) } , fora suitable choice of the parameters u , u , a , a that appears in eqs. (3.19) and (3.20). Theadvantage of using the set { u , u , a , a } to parametrize the space of initial conditions isthat in the early phase of the evolution, when we are still deep in the RD phase, the modesproportionals to u , u , a , a evolve independently, according to eqs. (3.19) and (3.20). Inparticular, in RD u , a and a are associated to exponentially decaying modes, so (inthe space of solutions of the local model) along these directions of the parameter spacethe solution with initial conditions u = a = a = 0 is an attractor. Thus, along thesedirections even relatively large initial deviations of Y ( x ) from the unperturbed solution canbe reabsorbed by the evolution. This is illustrated in Fig. 4, where we start at x in = − Y ( x in ) (10 on the left panel and 10 on the right panel),and with the initial value Y (cid:48) ( x in ) chosen equal to α + Y ( x in ), so that at the initial time wehave excited the mode e α + x in eq. (3.20). However, since α + < γY ( x ) is always completely negligible with respect to the radiation densiy in theearly Universe). The mode e α − x decays even faster, since also α − <
0, and | α − | > | α + | .The situation is different for u , which corresponds to a marginally stable mode. Inconclusion, if we discard the exponentially decaying modes associated to { u , a , a } , theonly interesting generalization of our non-local model is obtained defining, in RD, − (cid:50) − R ≡ u + (cid:90) tt ∗ dt (cid:48) a d ( t (cid:48) ) (cid:90) t (cid:48) t ∗ dt (cid:48)(cid:48) a d ( t (cid:48)(cid:48) ) R ( t (cid:48)(cid:48) ) , (4.14)while in the definition (4.2) we can still keep S , hom ( t ) = 0. We now explore the physicalmeaning and the cosmological consequences of this modification.16 .2 Modification of (cid:50) − and the cosmological constant Quite interestingly, the introduction of u is equivalent to introducing an explicit cosmo-logical constant term in the non-local model. Indeed, let us write − (cid:50) − R ≡ u − (cid:50) − R , (4.15)where (cid:50) − R is the new definition of the retarded (cid:50) − operator given in eq. (4.14) while (cid:50) − R is our “old” definition (2.2). The model which uses the new definition is governedby the equation G µν − m d − d (cid:0) g µν (cid:50) − R (cid:1) T = 8 πG T µν , (4.16)which, using eq. (4.15), becomes G µν − m d − d (cid:0) g µν (cid:50) − R (cid:1) T = 8 πG T µν − Λ g µν , (4.17)with Λ = [( d − / d ] m u . We have therefore re-introduced a cosmological constant!Writing ρ Λ = Λ / (8 πG ) and Ω Λ = ρ Λ /ρ , in d = 3 we getΩ Λ = m u H = γu . (4.18)The result is quite interesting because it shows that, once we discard the modes that areexponentially decaying during RD, the whole freedom in the definition of the non-localoperators (cid:50) − and D − boils down to the possibility of introducing an explicit cosmologicalconstant in the equations, with a values determined by γ and by the initial conditions onthe auxiliary field U . In the next section we will study the cosmological evolution of thismore general class of models. u (cid:54) = 0 The evolution equations, with the definition (4.14) of the (cid:50) − operator, are still given byeqs. (3.10)–(3.13), except that now the initial conditions on U and Y in RD are U ( x in ) = Y ( x in ) = u and U (cid:48) ( x in ) = Y (cid:48) ( x in ) = 0, as we see from eqs. (3.19) and (3.20) with ζ = − u of order one is illustrated in the left panel of Fig. 5, wherewe compare the solution of the previous section, found choosing the initial conditions onthe perturbative solution (3.19,3.20) with u = 0, to the solution found setting u = 4.In both cases we adjust γ so to maintain fixed Ω M = 0 . γ = 0 . u = 0 and γ = 0 . u = 4. We see that, when u > u is to increase w from the value − .
042 that it has for u = 0, toward avalue closer to −
1, but still on the phantom side (for u = 4, we get w = − . u is formally equivalent to Let us stress again that, even if from the point of view of the local formulation u enters throughthe initial conditions, at the level of the original non-local formulation each value of u defines a different theory. Each theory will be characterized by its own value of γ , required in order to get Ω M = 0 . (cid:45) (cid:45) Γ Y (cid:72) x (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) w Figure 5: Left panel: γY ( x ), choosing the initial conditions on the perturbative solution(3.19,3.20) with u = 0 (blue solid line) and with u = 4 (red dashed line). In both caseswe adjusts γ so to maintain fixed Ω M = 0 . u = 0 (blue solid line) and u = 4 (red dashed line).introducing a cosmological constant, with Ω Λ = γu , on top of which evolves a dynamicaldark energy, with the sum of these two components still constrained to take the value0 .
68 today. Therefore the value of w is shifted toward the value − u , e.g. u = 400. In this case we must choose γ = 0 . M = 0 . w (cid:39) − . u → + ∞ , the model approaches more and more ΛCDM.In fact, in order to keep the contribution Ω Λ = γu at a value smaller or equal than 0 . u → ∞ we must tune γ →
0; correspondingly, the dynamical contribution γY ( x ) to the DE goes to zero, and in the limit u → ∞ we remain with a ΛCDM modelwith γu kept fixed at the value 0.68. At u > ∼ w and w a are well fitted by w (cid:39) − − Au , w a (cid:39) − Bu (4.19)with A (cid:39) . B (cid:39) .
1, and therefore w a (cid:39) BA (1 + w ) (cid:39) . w ) . (4.20)So, in the more general class of model parametrized by u , the prediction (3.28) changes.Nevertheless, we see that even in this more general class of models the EOS parametertoday is always on the phantom side, and while u spans the whole range u ∈ [0 , ∞ ), theprediction for w remains in the rather narrow range [ − . , − w and w a in which u is eliminated, and which therefore remains as apure prediction of the model.It is also interesting to explore the region u <
0. In this case we are effectivelyadding a negative value of Ω Λ . The resulting evolution is given in right panel of Fig. 6 for u = −
10. In this case we find w (cid:39) − . w a (cid:39) − .
32. Observe that, for u <
0, thevalue of w is shifted even more toward the phantom side. Varying u we find that for u lower than a critical value u c (cid:39) −
12 it is no longer possible to obtain γY (0) = 0 . (cid:45) (cid:45) Γ Y (cid:72) x (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Γ Y (cid:72) x (cid:76) Figure 6: Left: γY ( x ), choosing the initial conditions on the perturbative solution with u = 0 (blue solid line) and with u = 400 (red dashed line). Right: the same for u = 0(blue solid line) and u = −
10 (red dashed line). (cid:248)(cid:248) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) w w a Figure 7: The values of the pair ( w , w a ) obtained for different u (dots). The starcorresponds to the model with u = 0.This is due to the fact that the function Y ( x ) can only begin to rise at the beginning ofthe MD era, and if at this epoch it starts from a value below a critical one, it cannot risefast enough to attain the required value in x = 0. This again has the effect of limiting thepossible range of predictions for w and w a in our model.In Fig. 7 we show the values of the pair ( w , w a ) obtained for different u (as definedby fitting to the function w ( a ) = w + (1 − a ) w a in the region − < x < w = − . , w a = − . u = 0. Toward itsright we have displayed the points computed for u = 2 , , , , , , , , ,
300 and400, while at its left we have shown the points obtained for u = − , − , . . . , − , − . , − w a = a (1 + w ) + b (1 + w ) + c (1 + w ) , (4.21)with a (cid:39) . b (cid:39) − .
386 and c (cid:39) − . u c (cid:39) −
12, a fit of the form w ( a ) = w + (1 − a ) w a inthe range − < x < Y ( x )starts from negative values, it must cross the horizontal axis somewhere in order to reacha positive value at x = 0, as in the right-panel of Fig. 6. At this point Y (cid:48) /Y diverges, and19herefore also w ( x ). If this happens within the interval [ − , w = w (0), w a = − w (cid:48) (0) (which, for values of u ≥ − < x < u = u c (cid:39) −
12 for which we can obtain an evolution such that γY (0) = 0 . w (cid:39) − . , w a (cid:39) − . , (4.22)which we take as the most extreme prediction of our model. This is however just a smallcorner of our parameter space. As we see from Fig. 7, in the rest of the parameter space w and w a vary over a much more narrow range. u from an earlier inflationary phase In the above analysis, the value of u during RD has been taken as a free parameter.However, we have seen that such a non-zero value is naturally generated by the evolutionfrom a pre-existing inflationary phase, and it is interesting to try to estimate it in terms ofthe parameters of such a phase. Consider then a model that starts in an earlier inflationaryphase, followed by RD and MD. As we saw in eq. (3.24), during an inflationary phasethe coefficient α + in eq. (3.20) is positive, and the corresponding homogeneous solution isunstable. However, if the (cid:50) − operator is defined setting to zero the homogeneous solutionsduring the inflationary phase, the solutions e α ± x are spurious, and are simply not solutionsof the original non-local equation. Thus, in the space of solutions of the original non-localequation, the perturbative inhomogeneous solution during the inflationary phase is stable,even if in the space of solutions of the differential equations of the local formulation it isunstable. The unstable direction is a spurious solutions, which has been introduced bythe localization procedure.As shown in eqs. (4.3)–(4.6), this definition of the (cid:50) − operator in the inflationaryphase will generate a non-vanishing homogeneous solution in RD, and we can find thecorresponding values of the constants u R0 , u R1 , a R1 and a R2 . This exercise should be takenwith some care, because we are assuming that the non-local massive gravity model that weare considering is valid up to the energies, such as 10 GeV, where inflation takes place.Actually, as discussed in [60, 61], the non-local equation (2.1) should be understood as aneffective classical equation obtained from some form of classical or quantum smoothing ina more fundamental theory, so the model might be modified in the UV well before suchscales are reached. Furthermore, we must anyhow make assumptions on the values of U and Y at the beginning of inflation. With these caveats, lets us define the non-localoperators so that the solution (3.20) in the inflationary phase has u = a = a = 0, andchoose u so that U ( x ) = 0 at the beginning of inflation, x = x i . Setting for simplicity ζ infl0 = 0 in eq. (4.8) (which corresponds to a phase of de Sitter inflation), eqs. (3.19) and(3.20) then give U ( x ) = Y ( x ) = 4( x − x i ) (4.23)during inflation. Denoting by x = x f the value of x where inflation ends and RD begins,we have U ( x f ) = Y ( x f ) = 4∆ N , where ∆ N = x f − x i is the number of inflationary e-folds.Depending on the energy scale at which inflation takes place, the minimum number ofe-folds required for a successful de Sitter inflationary model ranges from ∆ N (cid:39)
67 for an20nflationary scale at 10 GeV, to ∆ N (cid:39)
37 for inflation at the TeV. This gives a minimumvalue of U ( x f ) (cid:39) N = 67 we have U ( x f ) = 268.Performing the matching to the analytic RD solution we find that during RD U ( x ) = u R0 − e − ( x − x f ) , (4.24) Y ( x ) = u R0 + c e α + ( x − x f ) + c e α − ( x − x f ) . (4.25)where u R0 = 4(∆ N + 1) (cid:39) N , (4.26)and c , c = O (1). Since in RD α ± <
0, all exponentials decay, and this solution is quicklyattracted toward the solution with u R1 = a R1 = a R2 = 0. Thus, the subsequent evolution isidentical to that obtained setting initial condition in the RD phase such that u R0 is givenby eq. (4.26) while u R1 = a R1 = a R2 = 0. Since u R0 > ∼
100 we can use the fit (4.19) and (underthe hypothesis specified above) we get a prediction for w and w a in terms of the numberof inflationary e-folds, w (cid:39) − −
18 ∆
N , w a (cid:39) −
140 ∆
N , (4.27)as well as the relation (4.20) (or, more accurately, the relation (4.21)) between w and w a . Observe that in this case w will be very close to −
1. For ∆ N = 67, eq. (4.27) gives w (cid:39) − . N = 37 we get w (cid:39) − . In this paper we have analyzed the cosmological consequences of a non-local generalizationof GR that, in the far IR, involves the addition to the Einstein equations of a term propor-tional to the transverse parts of g µν (cid:50) − R , eq. (2.1). This models can be seen broadly as aclassical theory of massive gravity, in the sense that GR is deformed by the introductionof a mass parameter (although, as discussed in [61], the graviton in this theory remainsmassless!). A rather appealing feature of the g µν (cid:50) − R model is that it is highly predictive,since, contrary to typical scalar-tensor theories, f ( R ) theories, Rf ( (cid:50) − R ) theories, etc.,we do not have arbitrary functions of the scalar field or of the curvature that enter themodel, and which are normally chosen so to have the desired cosmological behavior. Inour model in a first approximation we only have one free parameter, the mass scale m ,or equivalently the dimensionless parameter γ = m / (9 H ), which replaces the parameterΩ Λ in ΛCDM. We have also seen that the model can be extended adding the most generalsolution of the homogeneous equations (cid:50) U = 0 and D S = 0 in the definition of the (cid:50) − and D − operators. In the end, this amounts to adding to the corresponding local systema more general set of initial condition, parameterized by four variables u , u , a , a . How-ever, u , a , a parametrize irrelevant directions in the space of solutions. It is thereforenatural to consider a “next-to-minimal” model, in which only u is retained. We havefound that the introduction of u is equivalent to adding a cosmological constant on topof the dynamical dark energy components. In this sense, the existence of this marginallystable direction of parameter space is not surprising. It is clear that, given a cosmolog-ical model that produces a dynamical dark energy, we can always put on top of it thecontribution of a cosmological constant. 21t the level of background evolutions, these models provide quite interesting predic-tions:1. In the case u = 0, as already shown in [61], once we fix γ so to reproduce theobserved value of the DE density today, Ω DE (cid:39) .
68, we have no more freedom, andwe get a sharp prediction for the dark energy equation of state. Writing w ( a ) = w + (1 − a ) w a , we get w = − .
042 and w a = − . w so close to −
1. Second, the result is on thephantom side, as suggested, at the 2 σ level, by the Planck results [64].2. In the models parametrized by m and u we have one more free parameter andtherefore, unavoidably, less predictivity. Nevertheless, even opening this new direc-tion in parameter space, the predictions of the model remain quite sharp. First ofall, the parameter w always remain on the phantom side. As u sweeps the range u ∈ [0 , ∞ ), the prediction for w remains in the rather narrow range [ − . , − − u → + ∞ . Similarly, w a moves monotonicallyfrom the value that it has for u = 0, w a = − . w a = 0 for u → ∞ . As u → ∞ we then approach the ΛCDM point ( w = − , w a = 0).We have also changed u toward negative values and found that, below a criticalvalue u (cid:39) −
12, it is no longer possible to obtain Ω DE (cid:39) .
68 today. In the range ofallowed negative values of u , w goes even more toward the phantom side, reachinga minimum value w (cid:39) − .
33, with w a (cid:39) − . w on the phantom side,in the relatively narrow range − . < ∼ w ≤ −
1. Furthermore, if w is measuredwith sufficient accuracy and is within this range, we can deduce from it the valueof u and therefore get a pure prediction for w a . Equivalently, our model predicts arelation between the observed values of w and w a , which is displayed in Fig. 7 andfitted in eq. (4.21).The phantom value of w that we find is quite suggestive, in view of the Planck results.The target of the Euclid mission is to reach a precision of 0.01 on w and of 0.1 on w a [67].At this level of precision, we will have a very stringent test of the prediction given ineq. (3.28), or more generally of the relation between w a and w given in Fig. 7. Acknowledgments.
We thank Luca Amendola, Ed Copeland, Yves Dirian, ValeriaPettorino and Christof Wetterich for useful discussions. Our work is supported by theFonds National Suisse.
A Cosmological dynamics of the (cid:50) − G µν model. A.1 Cosmological evolution equations
In this appendix we consider the model obtained setting b = 1 , b = 0 in eq. (1.5), G µν − m (cid:0) (cid:50) − G µν (cid:1) T = 8 πG T µν . (A.1)22e now define S µν ≡ (cid:50) − G µν , (A.2)and we split it into its transverse and longitudinal parts, as in eq. (1.2). We can thenrewrite eq. (A.1) as a pair of differential equations G µν − m S T µν = 8 πG T µν , (A.3) (cid:50) S µν = G µν . (A.4)In the case of FRW, the procedure for extracting the transverse part from S µν and reducingeqs. (A.3) and (A.4) to a system of ordinary differential equations has been describedin [57]. In FRW, for symmetry reasons the only non-vanishing components of the tensor S µν are S ( t ) and S ii ( t ) (where the sum over i is understood). Similarly, the only non-vanishing component of the vector S µ that enters in eq. (1.2) is S . Using the combinations U = S + S ii , V = S − d S ii , (A.5)as well as W = − ( d + 1) S , one finds a system of four coupled equations for the fourfunctions H, U, V, W . Specializing henceforth to d = 3, the result is [57] H + m (cid:16) U + 3 V − ˙ W (cid:17) = 8 πG ρ M + ρ R ) , (A.6)¨ W + 3 H ˙ W − H W = ˙ U + 3 ˙ V + 12 HV , (A.7)¨ U + 3 H ˙ U = 6 ˙ H + 12 H , (A.8)¨ V + 3 H ˙ V − H V = − H , (A.9)where, as in sect. 2, we have taken ρ equal to the sum of the matter density ρ M and theradiation density ρ R . We then define ρ DE ( t ) ≡ m πG ( ˙ W − U − V ) , (A.10)so eq. (A.6) takes again the form H ( t ) = 8 πG ρ M ( t ) + ρ R ( t ) + ρ DE ( t )] . (A.11)We pass again to dimensionless variables as in sect. 2, we use x = ln a ( t ) instead of t , andwe also trade W for a field Y defined by Y = ˙ W − U − V = hW (cid:48) − U − V . Then theFriedmann equation reads h ( x ) = Ω M e − x + Ω R e − x + γY ( x ) , (A.12)where now γ = m / (12 H ), and the evolution of Y ( x ) is obtained from the coupled systemof equations Y (cid:48)(cid:48) + (3 − ζ ) Y (cid:48) − ζ ) Y = − U (cid:48) + 3(1 + ζ ) U + 3 V (cid:48) + 3(3 − ζ ) V , (A.13) U (cid:48)(cid:48) + (3 + ζ ) U (cid:48) = 6(2 + ζ ) , (A.14) V (cid:48)(cid:48) + (3 + ζ ) V (cid:48) − V = − ζ , (A.15)23here ζ ( x ) ≡ h (cid:48) h = −
12 3Ω M e − x + 4Ω R e − x − γY (cid:48) ( x )Ω M e − x + Ω R e − x + γY ( x ) . (A.16)Just as with the (cid:50) − R model of sect. 2, we see from these equations that, at the levelof background evolution, compared to ΛCDM the cosmological constant term is replacedby a dark energy term with ρ DE ( x ) = ρ γY ( x ) or, in terms of the dark energy fractionΩ DE ( x ), Ω DE ( x ) ≡ ρ DE ( x ) ρ c ( x ) = γY ( x ) h ( x ) , (A.17)and the dynamics of Y ( x ) is governed by the coupled system of equations (A.13)-(A.15).The effective EOS parameter of this dark energy component is again defined by ˙ ρ DE +3(1 + w DE ) Hρ DE = 0, which gives again eq. (3.17). A.2 Perturbative solutions and instabilities
Neglecting the contribution of Y to ζ and setting ζ ( x ) (cid:39) ζ the equations for U and V become U (cid:48)(cid:48) + (3 + ζ ) U (cid:48) = 6(2 + ζ ) , (A.18) V (cid:48)(cid:48) + (3 + ζ ) V (cid:48) − V = − ζ , (A.19)whose solution is (see also [68]) U ( x ) = 6(2 + ζ )(3 + ζ ) x + u + u e − (3+ ζ ) x , (A.20) V ( x ) = ζ v e β + x + v e β − x , (A.21)where β ± = − ζ ± (cid:115)(cid:18) ζ (cid:19) + 8 . (A.22)In particular, during RD, β ± = (1 / − ± √ β ± = (1 / − ± √ Y is obtained plugging eqs. (A.20) and (A.21) into eq. (A.13) (with ζ ( x )replaced by ζ ) and is of the form Y ( x ) = c + c x + 3 u (1 + ζ ) + c u e − (3+ ζ ) x + c v e β + x + c v e β − x + y e α + x + y e α − x . (A.23)The coefficients c , . . . , c are functions of ζ easily obtained by direct substitution (andwhose relatively cumbersome expression we will not need below). The terms proportionalto y , y are the general solution of the homogeneous equation Y (cid:48)(cid:48) + (3 − ζ ) Y (cid:48) − ζ ) Y = 0, and α ± = (1 / − ζ ± (cid:112)
21 + 6 ζ + ζ ]. The solutions of the inhomogeneousequations obtained setting u = u = v = v = y = y = 0 are self-consistent with theperturbative approach, since at early times (i.e. as x → −∞ ) Y ( x ) ∝ x , so its contributionto ζ ( x ) in eq. (A.16) is indeed negligible compared to Ω M e − x and Ω R e − x . Therefore theyprovide a solution of the equations that gives back standard cosmology at early times.24n sect. 2 we found that, for the (cid:50) − R model, the homogeneous solutions are stable(or, in the case of u , marginally stable) both in RD and MD. There is a potential insta-bility if there is an earlier inflationary phase, which can however be avoided assigning theappropriate boundary conditions that exclude them during inflation. In contrast, in thismodel the homogeneous solution for V ( x ) associated to the mode e β + x is unstable bothin RD and in MD, since in both regimes β + >
0. This instability makes it impossible toobtain a convincing evolution during RD and MD. If we start the evolution from an earlierinflationary era, even setting to zero the homogeneous solution during this epoch, once weenter in RD and we match the perturbative solution during inflation with the perturbativesolution during RD, the homogeneous solutions of the RD era will be generated, and willquickly lead to an instability of the system. We could start the evolution from the RDera, assigning there initial conditions that amount to setting to zero the homogeneous RDsolution, but in any case the instability will show up in MD. We have indeed checked thisbehavior with the numerical integration of the exact equations (A.12)–(A.15). The insta-bility is triggered by the exponentially growing mode of V ( x ) but, since V ( x ) couples toall other functions, it leads to an instability also in the functions U and Y and then in theHubble parameter h ( x ), which leads to an early phase of accelerated expansion that screwsup the standard RD and MD epochs. The conclusion is that the model with b = 1 , b = 0is not cosmologically viable, since already at the level of background evolution it cannotreproduce standard cosmology at early times. This conclusion extends to all models ofthe form (1.5) as long as b (cid:54) = 0, i.e. as long as the operator (cid:50) − G µν is present, since itsinclusion automatically brings in the function V ( x ) which is responsible for the instability.The fact that tensor non-localities generically brings instabilities has also been recentlyfound, in a different non-local model, in [47]. References [1] G. Dvali, G. Gabadadze, and M. Porrati, “4-D gravity on a brane in 5-D Minkowskispace,”
Phys.Lett.
B485 (2000) 208–214, hep-th/0005016 .[2] C. Deffayet, “Cosmology on a brane in Minkowski bulk,”
Phys.Lett.
B502 (2001)199–208, hep-th/0010186 .[3] C. Deffayet, G. Dvali, and G. Gabadadze, “Accelerated universe from gravityleaking to extra dimensions,”
Phys.Rev.
D65 (2002) 044023, astro-ph/0105068 . Furthermore, the subsequent exponential decrease of the solution during RD and MD still allows usto obtain a sensible cosmological evolution even if there is an exponentially growing term during inflation.Observe also that, despite this exponential growth, the DE density ρ DE ( x ) remains utterly negligiblecompared to ρ c ( x ) in the inflationary as well as in the subsequent RD phases. Alternatively, one could set the graviton mass to extremely small values compared to H , so to suppressthe instability and therefore the early beginning of the acceleration era, as suggested in [68]. This howeveris not very appealing since the required value of m depends strongly on the point x in where we set theinitial conditions. As we move x in toward −∞ , the required graviton mass becomes smaller and smaller,in order to suppress the exponential growth for a longer time. Furthermore, even in this way it is notpossible to obtain a viable DE model. As found in [68], with a value m (cid:39) − H one can suppress thegrowth of the instability during RD, for an evolution starting at a redshift z ∼ , and one can obtain aDE of order of the observed value today, but its EOS parameter today, w , turns out to be between − . − .
5, which is not consistent with the present cosmological observations.
JHEP (2003) 029, hep-th/0303116 .[5] A. Nicolis and R. Rattazzi, “Classical and quantum consistency of the DGP model,”
JHEP (2004) 059, hep-th/0404159 .[6] D. Gorbunov, K. Koyama, and S. Sibiryakov, “More on ghosts in DGP model,”
Phys.Rev.
D73 (2006) 044016, hep-th/0512097 .[7] C. Charmousis, R. Gregory, N. Kaloper, and A. Padilla, “DGP Specteroscopy,”
JHEP (2006) 066, hep-th/0604086 .[8] K. Izumi, K. Koyama, and T. Tanaka, “Unexorcized ghost in DGP brane world,”
JHEP (2007) 053, hep-th/0610282 .[9] C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action,”
Phys.Rev.
D82 (2010) 044020, .[10] C. de Rham, G. Gabadadze, and A. J. Tolley, “Resummation of Massive Gravity,”
Phys.Rev.Lett. (2011) 231101, .[11] C. de Rham and G. Gabadadze, “Selftuned Massive Spin-2,”
Phys.Lett.
B693 (2010) 334–338, .[12] C. de Rham, G. Gabadadze, and A. J. Tolley, “Ghost free Massive Gravity in theSt¨uckelberg language,”
Phys.Lett.
B711 (2012) 190–195, .[13] C. de Rham, G. Gabadadze, and A. J. Tolley, “Helicity Decomposition of Ghost-freeMassive Gravity,”
JHEP (2011) 093, .[14] S. Hassan and R. A. Rosen, “Resolving the Ghost Problem in non-Linear MassiveGravity,”
Phys.Rev.Lett. (2012) 041101, .[15] S. Hassan and R. A. Rosen, “On Non-Linear Actions for Massive Gravity,”
JHEP (2011) 009, .[16] S. Hassan, R. A. Rosen, and A. Schmidt-May, “Ghost-free Massive Gravity with aGeneral Reference Metric,”
JHEP (2012) 026, .[17] S. Hassan and R. A. Rosen, “Confirmation of the Secondary Constraint andAbsence of Ghost in Massive Gravity and Bimetric Gravity,”
JHEP (2012)123, .[18] S. Hassan, A. Schmidt-May, and M. von Strauss, “Proof of Consistency of NonlinearMassive Gravity in the St¨uckelberg Formulation,”
Phys.Lett.
B715 (2012) 335–339, .[19] D. Comelli, M. Crisostomi, F. Nesti, and L. Pilo, “Degrees of Freedom in MassiveGravity,”
Phys.Rev.
D86 (2012) 101502, .2620] M. Jaccard, M. Maggiore, and E. Mitsou, “Bardeen variables and hidden gaugesymmetries in linearized massive gravity,”
Phys.Rev.
D87 (2013) 044017, .[21] D. Comelli, F. Nesti, and L. Pilo, “Massive gravity: a General Analysis,” .[22] P. Guarato and R. Durrer, “Perturbations for massive gravity theories,” .[23] K. Hinterbichler, “Theoretical Aspects of Massive Gravity,”
Rev.Mod.Phys. (2012) 671–710, .[24] C. de Rham, G. Gabadadze, L. Heisenberg, and D. Pirtskhalava, “CosmicAcceleration and the Helicity-0 Graviton,” Phys.Rev.
D83 (2011) 103516, .[25] K. Koyama, G. Niz, and G. Tasinato, “Analytic solutions in non-linear massivegravity,”
Phys.Rev.Lett. (2011) 131101, .[26] K. Koyama, G. Niz, and G. Tasinato, “Strong interactions and exact solutions innon-linear massive gravity,”
Phys.Rev.
D84 (2011) 064033, .[27] T. Nieuwenhuizen, “Exact Schwarzschild-de Sitter black holes in a family of massivegravity models,”
Phys.Rev.
D84 (2011) 024038, .[28] A. H. Chamseddine and M. S. Volkov, “Cosmological solutions with massivegravitons,”
Phys.Lett.
B704 (2011) 652–654, .[29] G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, and A. J.Tolley, “Massive Cosmologies,”
Phys.Rev.
D84 (2011) 124046, .[30] A. De Felice, A. E. Gumrukcuoglu, C. Lin, and S. Mukohyama, “On the cosmologyof massive gravity,” .[31] G. Tasinato, K. Koyama, and G. Niz, “Exact Solutions in Massive Gravity,” .[32] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, and G. Gabadadze, “Nonlocalmodification of gravity and the cosmological constant problem,” hep-th/0209227 .[33] G. Dvali, S. Hofmann, and J. Khoury, “Degravitation of the cosmological constantand graviton width,”
Phys.Rev.
D76 (2007) 084006, hep-th/0703027 .[34] G. Dvali and G. Gabadadze, “Gravity on a brane in infinite volume extra space,”
Phys.Rev.
D63 (2001) 065007, hep-th/0008054 .[35] G. Dvali, G. Gabadadze, and M. Shifman, “Diluting cosmological constant ininfinite volume extra dimensions,”
Phys.Rev.
D67 (2003) 044020, hep-th/0202174 .[36] G. Dvali, “Predictive Power of Strong Coupling in Theories with Large DistanceModified Gravity,”
New J.Phys. (2006) 326, hep-th/0610013 .2737] S. Deser and R. Woodard, “Nonlocal Cosmology,” Phys.Rev.Lett. (2007) 111301, .[38] T. Koivisto, “Dynamics of Nonlocal Cosmology,” Phys.Rev.
D77 (2008) 123513, .[39] T. Koivisto, “Newtonian limit of nonlocal cosmology,”
Phys.Rev.
D78 (2008)123505, .[40] S. Capozziello, E. Elizalde, S. Nojiri, and S. D. Odintsov, “Accelerating cosmologiesfrom non-local higher-derivative gravity,”
Phys.Lett.
B671 (2009) 193–198, .[41] E. Elizalde, E. Pozdeeva, and S. Y. Vernov, “De Sitter Universe in Non-localGravity,”
Phys.Rev.
D85 (2012) 044002, .[42] Y. Zhang and M. Sasaki, “Screening of cosmological constant in non-localcosmology,”
Int.J.Mod.Phys.
D21 (2012) 1250006, .[43] E. Elizalde, E. Pozdeeva, and S. Y. Vernov, “Reconstruction Procedure in NonlocalModels,”
Class.Quant.Grav. (2013) 035002, .[44] S. Park and S. Dodelson, “Structure formation in a nonlocally modified gravitymodel,” Phys.Rev.
D87 (2013) 024003, .[45] K. Bamba, S. Nojiri, S. D. Odintsov, and M. Sasaki, “Screening of cosmologicalconstant for De Sitter Universe in non-local gravity, phantom-divide crossing andfinite-time future singularities,”
Gen.Rel.Grav. (2012) 1321–1356, .[46] S. Deser and R. Woodard, “Observational Viability and Stability of NonlocalCosmology,” .[47] P. G. Ferreira and A. L. Maroto, “A few cosmological implications of tensornonlocalities,” .[48] S. Dodelson and S. Park, “Nonlocal Gravity and Structure in the Universe,” .[49] A. Barvinsky, “Nonlocal action for long distance modifications of gravity theory,” Phys.Lett.
B572 (2003) 109–116, hep-th/0304229 .[50] A. Barvinsky, “Dark energy and dark matter from nonlocal ghost-free gravitytheory,”
Phys.Lett.
B710 (2012) 12–16, .[51] A. O. Barvinsky, “Serendipitous discoveries in nonlocal gravity theory,”
Phys.Rev.
D85 (2012) 104018, .[52] H. Hamber and R. M. Williams, “Nonlocal effective gravitational field equationsand the running of Newton’s G,”
Phys.Rev.
D72 (2005) 044026, hep-th/0507017 .[53] J. Khoury, “Fading gravity and self-inflation,”
Phys.Rev.
D76 (2007) 123513, hep-th/0612052 . 2854] T. Biswas, T. Koivisto, and A. Mazumdar, “Towards a resolution of thecosmological singularity in non-local higher derivative theories of gravity,”
JCAP (2010) 008, .[55] L. Modesto, “Super-renormalizable Quantum Gravity,”
Phys.Rev.
D86 (2012)044005, .[56] F. Briscese, A. Marciano, L. Modesto, and E. N. Saridakis, “Inflation in(Super-)renormalizable Gravity,”
Phys.Rev.
D87 (2013) 083507, .[57] M. Jaccard, M. Maggiore, and E. Mitsou, “A non-local theory of massive gravity,”
Phys.Rev.
D88 (2013) 044033, .[58] S. Deser, “Covariant Decomposition and the Gravitational Cauchy Problem,”
Ann.Inst.Henri Poincare (1967) 149.[59] J. J. York, “Covariant decompositions of symmetric tensors in the theory ofgravitation,” Ann.Inst.Henri Poincare (1974) 319.[60] S. Foffa, M. Maggiore, and E. Mitsou, “Apparent ghosts and spurious degrees offreedom in non-local theories,” .[61] M. Maggiore, “Phantom dark energy from non-local massive gravity,” .[62] N. Koshelev, “Comments on scalar-tensor representation of nonlocally correctedgravity,” Grav.Cosmol. (2009) 220–223, .[63] T. S. Koivisto, “Cosmology of modified (but second order) gravity,” AIP Conf.Proc. (2010) 79–96, .[64]
Planck Collaboration
Collaboration, P. Ade et. al. , “Planck 2013 results. XVI.Cosmological parameters,” .[65] M. Chevallier and D. Polarski, “Accelerating universes with scaling dark matter,”
Int.J.Mod.Phys.
D10 (2001) 213–224, gr-qc/0009008 .[66] E. V. Linder, “Exploring the expansion history of the universe,”
Phys.Rev.Lett. (2003) 091301, astro-ph/0208512 .[67] Euclid Theory Working Group
Collaboration, L. Amendola et. al. , “Cosmologyand fundamental physics with the Euclid satellite,”
Living Rev.Rel. (2013) 6, .[68] L. Modesto and S. Tsujikawa, “Non-local massive gravity,”1307.6968