Emergence of dynamical dark energy from polynomial $f(R)$ theory in Palatini formalism

We consider FRW cosmology in $f(R)= R+ \gamma R^2+\delta R^3$ modified framework. The Palatini approach reduces its dynamics to the simple generalization of Friedmann equation. Thus we study the dynamics in two-dimensional phase space with some details. After reformulation of the model in the Einstein frame, it reduces to the FRW cosmological model with a homogeneous scalar field and vanishing kinetic energy term. This potential determines the running cosmological constant term as a function of the Ricci scalar. As a result we obtain the emergent dark energy parametrization from the covariant theory. We study also singularities of the model and demonstrate that in the Einstein frame some undesirable singularities disappear.


Introduction
A variety of explanations have been proposed for the Universe accelerating expansion at the current epoch. Among them, the idea of positive cosmological constant Λ, as the simplest candidates, seems to be viable. However, it is only an economical description (with the help of one free parameter) of observational facts rather than an effective explanation. The simplest alternative candidate for the constant cosmological parameter being a key element in the standard cosmological model (called ΛCDM model) is a time-dependent (or running) cosmological term. It is crucial for avoiding fine-tuning and coincidence problems [1,2].
It would be nice to derive the dynamics of the running cosmological term as an emergent phenomenon from a more fundamental theory, for example from the string theory or from the first principles of quantum mechanics [3]. In this context, it is important to formulate a dynamical cosmological term without violating the covariance of the action. For example, models with a slowly rolling homogeneous cosmological scalar field, provide a popular alternative to the standard time-independent cosmological constant. We can study the simultaneous evolution of the background expansion and an evolution of the scalar field with the self-interacting potential [4].
In this paper we are going to push forward an idea of the emergent running cosmological term from a covariant theory [5]. Parametrization of the cosmological term is derived directly from a formulation of the model in the Einstein frame by means of the Palatini variational approach. In analogy with Starobinsky's purely metric formulation [6], we obtain the parametrization of the cosmological term directly from the potential of the scalar field which appears after formulation of the specific FRW model in the Einstein frame. As a next step we investigate the dynamics of the model with such a form of the dark energy.
In this letter, we demonstrate how f (R) model is modified in the Palatini formulation. Our construction provides a simple model of an evolving dark energy (running cosmological term) to explain a dynamical relaxation of the vacuum energy (gravitational repulsive pressure) to a very small value today (cosmological constant problem [7]). This model, when studied in the Einstein frame, leads also to a small deviation from the w = −1 prediction of the non-running dark energy.

Cosmological equations for the polynomial f (R) theory in the Palatini formalism
The Palatini gravity action for f (R) gravity is given by whereR is the generalized Ricci scalar [8,9]. From the action (1) we get where T µν is energy momentum tensor and∇ α is the covariant derivative calculated with respect to Γ. If we take the trace of Eq. (2), we get a structural equation, which is given by where T = g µν T µν . We assume the FRW metric in the following form where a(t) is the scale factor, k is a constant of spatial curvature (k = 0, ±1) and t is the cosmological time. Thereafter, we assume the flat model (k = 0).
We assume the energy-momentum tensor for perfect fluid where p = wρ with w = const. The conservation condition T µ ν;µ = 0 [10] givesρ where H is the Hubble function and ρ m is the density of baryonic and dark matter which is assumed to be in the form of dust (w = 0). In our paper the function f (R) is assumed in the polynomial form as where γ i are some dimensionful parameters. Therefore, we introduce more convenient dimensionless functions and parameters where H 0 is the present value of Hubble function, Ω m,0 = ρm,0 For the sake of generality (following the standard cosmological model) the presence of the cosmological constant is also assumed.

For the function (8) the structural equation (4) is in the following form
The Friedmann equation for the function (8) has the following form 3. Singularities in the polynomial f (R) theory in the Palatini formalism The Friedmann equation (11) can be rewritten in an equivalent form |b| d dt is a new parametrization of time (this parametrization is not a diffeomorphism) and The potential V (a) can be used to construction of a phase space portrait. In this case the phase space is twodimensional (a, a ) : The dynamical system has the following form We assume that the potential function, except some isolated (singular) points, belongs to the class C 2 (R + ). The example phase portraits for the dynamical system (15)- (16) are presented in figures 1, 2 and 3. The evolution of a universe can be treated as a motion of a fictitious particle of unit mass in the potential V (a). Here a(t) plays the role of a positional variable. Equation of motion (16) assumes the form analogous to the Newtonian equation of motion. In this case the lines In our model, there are two types of singularities: the freeze and sudden singularity. They are a consequence of the Palatini formalism. We get the freeze singularity when b + d/2 = 0. The sudden singularity appears when b = 0 or b + d/2 is equal the infinity.
For the case when the positive part of f (R) dominates after the domination of the negative part of f (R), it is possible that two freeze singularities appear. This situation is presented in figure 4 for f (R) =R + 10 −2R2 − 10 −6R3 . In this case they appear two freeze singularities and one sudden singularity. The evolution of b(a)+ d(a) 2 , which corresponds with figure 4, is presented in figure 5. Note that, for values of scale factor, for which function b(a) + d(a) 2 has roots, the freeze singularities appear. V (a) potential, which corresponds with figure 4, is presented in figure 6 and 7.   (1) and (2) are the saddle type and critical points (3) and (4) are the center type. The red dashed line presents the sudden singularity. The black dashed lines present the freeze singularities. The grey color marks the non-physical domain (f (R) < 0). The reds trajectories represent the path of evolution for the flat universe. These trajectories seperate the domain with the negative curvature (k = −1) from the domain with the positive curvature (k = +1). The scale factor is expressed in the logarithmic scale.

Singularities in the Palatini
For the special case of polynomial f (R) =R+γR 2 +δR 3 , one gets the following structural equation where Ω γ = 3γH 2 0 and Ω δ = 9δH 4 0 .  (2), (3) and (4) which represent the static Einstein universes. Critical point (2) is the saddle type and critical points (3) and (4) are the center type. The black dashed lines present the freeze singularities. The scale factor is expressed in the logarithmic scale. The homoclinic orbits represent the bouncing models, which evolution starts and ends at the Einstein Universe (critical point 2). In the domain bounded by the homoclinic orbits the oscilating models present without the initial singularity.
The Friedmann equation takes the form where The condition for appearance of the freeze singuarity is b + d 2 = 0 and in this case it has the form  where For the sudden singularity the condition b = 0 provides the equation which has the following solutions

The Palatini approach in the Einstein frame
If f (R) = 0 then the action (1) can be rewritten in dynamically equivalent form of the first order Palatini gravitational action [11,12,13] S(g µν , Γ λ ρσ , χ) = Let Φ = f (χ) be a scalar field, where χ =R. Then the action (25) takes the form where the potential U (Φ) is given as with Φ = df (χ) dχ andR ≡ χ = dU (Φ) dΦ . After the Palatini variation of the action (26) we get the following equations of motion As a consequence of (28b) the connectionΓ is a metric connection for a new (conformally related) metricḡ µν = Φg µν ; thusR µν =R µν ,R =ḡ µνR µν = Φ −1R andḡ µνR = g µνR . The g-trace of (28a) gives a new structural equation Equations (28a) and (28c) can be rewritten in the following formR In this case, the structural equation is given by the following formula The action for the metricḡ µν and the scalar field Φ can be recast to the Einstein frame form with non-minimal coupling between Φ andḡ µν [13,14]).
The metricḡ µν takes the standard FRW form where dt = Φ(t) 1 2 dt and a new scale factorā(t) = Φ(t) 1 2 a(t). In the case of the barotropic matter, the cosmological equations are and w =p m /ρ m = p m /ρ m . In this case, the conservation equations has the following forṁ Let us consider our Palatini f (R) = n i=1 γ iR i model in the Einstein frame, where γ 1 = 1. The potentialŪ is given by the following formulā The scalar field Φ can be parametrized byR in the following way The relation betweenŪ andR for f (R) =R + γR 2 + δR 3 case is presented in Fig. 8.
In this frame, two scenarios of cosmic evolution may appear. In the first one the evolution of the universe starts from the generalized sudden singularity. The second case is when it starts from the freeze singularity. The diagrams of the corresponding Newtonian potentials V (ā) are presented in Figs. 9 and 10. We can use the potential V (ā) to construct phase space portraits analogous to the ones in section 3 (see Figs. 11 and 12).
The evolution of the scalar field potentialŪ (t), which plays a role of dynamical cosmological constant, is presented in Fig. 13 for the case with the generalized sudden singularity. Note that for the late time the potentialŪ (t) is constant. The evolution ofŪ (t), for the case when the freeze singularity appears, is presented in Fig. 14. For the late time the potentialŪ (ā) can be approximated as From the structural equation (32) for f (R) =R +γR 2 + δR 3 case, we get the parameterization of the dust matter density with respect toR It is interesting that in the Einstein frame the interaction between dark matter and dark energy naturally appears as a physical phenomenon. This interaction modifies the original scaling law for dust matter by a function (t).

Conclusions
The main goal of the paper was to point out some advantages of formulation the Palatini FRW cosmology in the Einstein frame. The most crucial one is that in the Einstein frame the parametrization of dark energy is uniquely determined. In general it is obtained in the covariant form as a function of the Ricci scalar.
It is well know that scalar-tensor theories of gravity can be formulated both in the Jordan as well as in the Einstein frame. These frames are conformally related [15]. We also know that the formulations of a scalar-tensor theory in two different conformal frames although mathematically equivalent are physically inequivalent.
Faraoni and Gunzing gives a simple argument which favours the Einstein frame over the Jordan frame because in the latter one should potentially detect the time-dependent amplification induced by gravitational waves [16]. An analogous problem has been dedected in f (R) gravity that the Jordan frames could be physically nonequivalent, although they are connected by a conformal transformation [17,18]. In principle, there are two types of admissible arguments for favouring one frame over another one: the one coming from observations (for example astronomical observations) or theoretical nature (e.g. showing that some obstacles or pathologies will vanishing in privileged frame).
From our investigation of the model in Einstein frame we obtained that some pathologies like degenerated multiple freeze singularities [19] disappear in a generic case. The big bang singularity is replaced by the singularity of finite scale factor. Because the potentialŪ (R(t)) is constant for the late time, in the case when matter is negligible, the inflation appears like in the case f (R) =R + γR 2 [19].
There are also some other advantages when transforming to Einstein frame, namely that in this frame one naturally obtains the formula on dynamical dark energy which is going at late time toward cosmological constant. It is important that corresponding parametrization of dark energy is not postulated ad hock but it emerges from the first principles -which is the formulation of the problem in the Einstein frame. It is important that the parametrization of dark energy (energy density as well as a pressure) in terms of the Ricci scalar is given in a covariant form from the structure equation.
After transition to the Einstein frame the model evolution is governed by the Friedmann equation with two interacting fluids: dark energy and dark matter. This . Critical point (1) represents the static Einstein universe and is a saddle. The black dashed line presents the freeze singularity. The grey color presents the non-physical domain (ā <ās). The red trajectories represent the path of evolution for the flat universe. These trajectories separate the domain with the negative curvature (k = −1) from the domain with the positive curvature (k = +1). The scale factor is expressed in the logarithmic scale.
interaction modifies the standard scaling of the redshift relation for dark matter.