Cosmic acceleration from modified gravity with Palatini formalism
aa r X i v : . [ g r- q c ] F e b Prepared for submission to JCAP
Cosmic acceleration from modifiedgravity with Palatini formalism
Andrzej Borowiec, a Micha l Kamionka, b Aleksandra Kurek c andMarek Szyd lowski c,d a Institute of Theoretical Physics, University of Wroc lawpl. Maksa Borna 9, 50-204 Wroc law, Poland. b Astronomical Institute, University of Wroc lawul. Kopernika 11, 51-622 Wroc law, Poland. c Astronomical Observatory, Jagiellonian Universityul. Orla 171, 30-244 Krak´ow, Poland. d Mark Kac Complex Systems Research Centre, Jagiellonian Universityul. Reymonta 4, 30-059 Krak´ow, Poland.E-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
We study new FRW type cosmological models of modified gravity treated on thebackground of Palatini approach. These models are generalization of Einstein gravity bythe presence of a scalar field non-minimally coupled to the curvature. The models employStarobinsky’s term in the Lagrangian and dust matter. Therefore, as a by-product, an ex-hausted cosmological analysis of general relativity amended by quadratic term is presented.We investigate dynamics of our models, confront them with the currently available astro-physical data as well as against ΛCDM model. We have used the dynamical system methodsin order to investigate dynamics of the models. It reveals the presence of a final suddensingularity. Fitting free parameters we have demonstrated by statistical analysis that thisclass of models is in a very good agreement with the data (including CMB measurements) aswell as with the standard ΛCDM model predictions. One has to use statefinder diagnosticin order to discriminate among them. Therefore Bayesian methods of model selection havebeen employed in order to indicate preferred model. Only in the light of CMB data theconcordance model remains invincible.
Keywords: modified gravity, cosmological simulations, dark energy theory, cosmic singular-ity
ArXiv ePrint: ontents – 1 –
Introduction
As it is well-know a cosmological constant was the first and the simplest modification ofgeneral relativity performed by Einstein himself in order to stop cosmic expansion. Todaythe so-called Standard or Concordance (denoted also as ΛCDM or LCDM: Lambda Cold DarkMatter) Cosmological Model, which is based on this modification, turns out to be the bestfitted model with respect to a huge amount of high precision currently available astrophysicaldata. In particular it properly describes a present day cosmic acceleration [1] by means ofdark components: Dark Energy and Dark Matter. However ΛCDM model suffers for essentialtheoretical problems (e.g. well-known coincidence and fine tuning problems), specially relatedto a primordial stage of cosmic evolution [2]. Some of these problems can be cured by moresophisticated modifications. Among them the so-called f ( R )-gravity models constitute a hugefamily [3, 4] including also models based on Palatini formulation [5]. Quite recently there isa renewed interest in the Paltini modified gravity [6] which treats a metric and torsion-freeconnection as independent variables, see e.g. [5, 7, 8] for more details. Resulting equationsof motion remain second order as in the Einstein gravity case.Modification of Einstein’s General Relativity becomes a viable way to address the ac-celerated expansion of the Universe as well as dark matter and dark energy problems inmodern cosmology (see e.g. [2] and references therein). This includes modified theories witha non-trivial gravitational coupling [9–13]. Viable non-minimal models unifying early-timeinflation with late-time acceleration have been, particularly, discussed in [12]. Apart of cos-mological viability another justification for modified gravity should be taken seriously intoaccount: curvature corrections appear naturally as a low energy limits of quantum gravity,quantization on curved background and/or effects from extra dimensional physics.Both purely metric as well as Palatini f ( R )- gravity can be further extended by adding(scalar) field non-minimally coupled to the curvature [9–11]. In cosmological settings Palatiniformalism gives rise, similarly to Einstein gravity, to the first order autonomous differentialequation for a scale factor which can be recast into the form of Friedmann equation. Thisenables us to analyze the corresponding cosmological models as a one-dimensional particle-like Hamiltonian system with entire dynamics encoded in an explicitly obtained effectivepotential function. Such approach simplifies meaningly computer simulations and numericalanalysis. Before doing these one has to constrain model parameters by using astrophysicaldata [14–17] and CosmoNest code [18]. Simple Palatini based cosmological models has beenpreviously tested against various sets of cosmological data in [19]. Very recently it hasbeen shown that gravitational redshift of galaxies in clusters is consistent both with generalrelativity and some models of f ( R ) − modified gravity [20].The paper is organized as follows. In Section 2 we recall some basic facts about Palatinigravity non-minimally coupled to dilaton-like field. Its cosmological application is consideredin Section 3: particularly, general expression for generalized Friedmann equation is derivedtherein. Two classes of new cosmological models stemming from two different solutions of theso-called master equation are issued in Sections 4 and 5. In Section 6 we describe estimationof models parameters by astrophysical data followed by Bayesian method of model selec-tion provided in Section 7. Dynamical analysis of our models by means of one-dimensionalparticle-like Hamiltonian system with a Newtonian type effective potential function is sub-ject of our investigations in Section 8. In order to visualize the dynamics we watch plots ofpotential functions and draw two-dimensional phase space trajectories for the correspondingmodels. Section 9 treats about additional statefinder diagnostics. We end up with summary– 2 –f obtained results and general conclusions presented in Section 10.In this paper we took the first step in testing the kinematical sector of the cosmologicalmodels of modified gravity, i.e. we apply cosmographic analysis to the background dynamic.The next step will be the studying of matter perturbations in the class of models underconsideration. In these future investigations the results of the present paper will be used asa starting point. This allows to exploit the advantages of Bayesian methods [21]. The main object of our considerations in the present article is a cosmological application ofsome non-minimally coupled scalar-tensor Lagrangians of the type L = √ g ( f ( R ) + F ( R ) L d ) + L mat (2.1)treated within the Palatini approach as in [9]. Hereafter we set L d = − g µν ∂ µ φ∂ ν φ theLagrangian for a free-scalar (massless) dilaton-like field φ and L mat represents a matterLagrangian. Because of Palatini formalism R is a scalar R = R ( g, Γ) = g µν R µν (Γ) composedof the metric g and the Ricci tensor R µν (Γ) of the symmetric ( ≡ torsionless) connectionΓ (for more details concerning the Palatini formalism see e.g. [7, 8]). Therefore ( g, Γ) aredynamical variables. Particularly, the metric g is used for raising and lowering indices.We began with recalling some general formulae already developed in [9], both f ( R ) and F ( R ) are assumed to be analytical functions of R . Equations of motion for gravitationalfields (Γ , g ) can be recast [9] into the form of the generalized Einstein equations R µν ( h ) ≡ R µν ( bg ) = g µα P αν (2.2)(see also [8]), where R µν ( h ) is now the Ricci tensor of the new conformally related metric h = bg with the conformal factor b specified below. A (1 ,
1) tensor P µν is defined by P µν = c b δ µν − F ( R ) b T d µν + 1 b T mat µν (2.3)and contains matter and dilaton stress-energy tensors: T mat µν = δL mat δg µν ; T d µν = δL d δg µν . Here onerespectively has ( c = ( f ( R ) + F ( R ) L d ) = ( L − L mat ) / √ gb = f ′ ( R ) + F ′ ( R ) L d (2.4)where a prime denotes the derivative with respect to R .Field equations for the scalar field φ is ∂ ν ( √ gF ( R ) g µν ∂ µ φ ) = 0 (2.5)Dynamics of the system (2.1) is controlled by the so-called master equation bR = 2 c − F ( R ) L d + τ (2.6)obtained by contraction of (2.2), where we set τ = g µν T mat µν for a trace of the matter stress-energy tensor. In more explicit form it reads as2 f ( R ) − f ′ ( R ) R + τ = (cid:0) F ′ ( R ) R − F ( R ) (cid:1) L d (2.7)These reproduce the same field equations as treated in [9]. Throughout the paper we shall work with units c = 8 πG = 1. The metric signature is ( − + ++) . One can easily add both mass and potential interaction for φ , cf. [9]. – 3 – Cosmology from the generalized Einstein equations
Assuming the Cosmological Principle to hold we take the physical metric g to be a Friedmann-Robertson-Walker (FRW) metric gg = − dt + a ( t ) h − κr dr + r (cid:16) dθ + sin ( θ ) dϕ (cid:17)i (3.1)where a ≡ a ( t ) is a scale factor and κ is the space curvature ( κ = 0 , , − T matµν of the universe is described by a non-interacting mixture ofperfect fluids. We denote by w i the corresponding barotropic coefficients. Each species isrepresented by the stress-energy tensor T ( i ) µν = ( ρ i + p i ) u µ u ν + p i g µν satisfying a metric (withthe Christoffel connection of g ) conservation equation ∇ ( g ) µ T ( i ) µν = 0 (see [22]). This givesrise to the standard relations between pressure and density (equation of state) p i = w i ρ i and ρ i = η i a − w i ) .Let us recall that for the standard cosmological model based on the standard Einstein-Hilbert Lagrangian (considered both in the purely metric as well as Palatini formalisms) L EH = √ gR + L mat (3.2)the corresponding Friedmann equation, ensuing from Einstein’s field equations, takes theform H + κa ≡ ˙ a a + κa = 13 X i η i a − w i ) (3.3)when coupled to (non-interacting) multi-component non-interacting barotropic perfect fluids,where η i a − w i ) represents a perfect fluid component with an equation of state (EoS) pa-rameter w i . Here H = ˙ aa denotes a Hubble parameter. This is due to the fact that geometrycontributes to the r.h.s. of the Friedmann equation through (cf. 2.7) R = − τ = X i (1 − w i ) η i a − w i ) ≡ X i (1 − w i ) ρ i (3.4)For example, the preferred ΛCDM model can be defined by three fluid components: thecosmological constant w Λ = −
1, dust w dust = 0 and radiation w rad = assuming thespacial flatness condition κ = 0. (As a matter of fact the spacial curvature term κa − canbe also mimicked by barotropic fluid w curv = − .) The radiation component which hasno contribution to the trace τ can be practically neglected due to extremely small valueΩ rad ∼ − . Alternatively, instead of introducing cosmological constant via perfect fluidDark Energy component, one can, following Einstein, modify the Einstein-Hilbert Lagrangian(3.2): L EH → L Λ EH = √ g ( R − L mat .On the other hand we have that the field equation for the scalar field φ ≡ φ ( t ) is ddt ( √ gF ( R ) ˙ φ ) = 0, so that √ gF ( R ) ˙ φ = const and consequently gF ( R ) L d = A = const.This simply implies that F ( R ) L d = A a − (3.5)with an arbitrary positive integration constant A (see (2.5)). It means that this term behavesas a stiff matter component ( w stiff = 1). – 4 –ssuming perfect fluid matter as a source, the generalized Einstein equations rewritesunder the form ˙ aa + ˙ b b ! + κa = F ( R ) L d b + c b + X i (1 + 3 w i ) η i b a − w i ) (3.6)It becomes a generalized Friedmann equation for the ordinary Hubble parameter H ≡ ˙ aa ifwe take into account that from (2.7) and (3.5) the scalars R and L d are implicit functions ofthe scale factor a . Further calculations give rise to the decomposition H = K ( a ) G ( a ) (3.7)where G ( a ) = f + 2 F L d X i (1 + 3 w i ) η i a − w i ) − κ ( f ′ + F ′ L d ) a − (3.8)and the function K ( a ) is defined by K ( a ) = 2( f ′ + F ′ L d ) (cid:20) f ′ − F ′ L d + 3[˜ τ + 2( F ′ R − F ) L d ][ f ′′ + ( F ′′ − F − ( F ′ ) ) L d ] f ′′ R − f ′ + [ F ′′ R + 2 F ′ − F − ( F ′ ) R ] L d (cid:21) − (3.9)where ˜ τ = P i ( w i + 1)(1 − w i ) η i a − w i ) for the multi-component fluid considered before:˙ τ = 3 H ˜ τ . As explained above the parameters, R, L d , and their functions, e.g. f ( R ) , F ′ ( R ) L d ,etc. become through equations (2.7), (3.5) implicitly dependent of the scale factor a . In thisway, the generalized Friedmann provides an autonomous system of first order ordinary differ-ential equation for the scale factor a (se the next Section for more details). The decomposition(3.7) is furnished in such a way that for standard cosmology: b = 1 , c = R, L d = 0 one has K ( a ) = 1 and one recovers the standard Friedmann equation (3.3), and particularly ΛCDMmodel as well.Our objective here is to investigate a possible cosmological applications and confrontagainst astrophysical data of the following subclass of gravitational Lagrangians (2.1) L = √ g (cid:16) R + αR + βR δ + γR σ L d (cid:17) + L mat (3.10)where α, β, γ, δ, σ are free parameters of the theory. It should to be observed that the grav-itational part f ( R ) contains the so-called Starobinsky quadratic term R [23] with some R δ contribution. In the limit α , β , γ → α, β, γ are dimensionfull with the corresponding dimensions satisfying[ R ] = [ αR ] = .. etc.. In fact, a numerical value of the constant γ is unessential since it canbe incorporated into the scalar field: φ → | γ | φ . Therefore, we further assume that it takesa discreet values γ = 0 , ±
1. Such re-scaling would not be possible if one admits nontrivialself-interaction with nonvanishing potential energy U ( φ ) for the scalar field.Following a common strategy particularly applicable within the Palatini formalism (see[7, 8, 10]) one firstly finds out an exact solution of the master equation (2.7). It allows toconstruct explicit cosmological model based on the generalized Friedmann equation. For thispurpose and in order to reduce a number of independent parameters, we shall assume throughthe rest of this paper that universe is spatially flat ( κ = 0) and filled exclusively with themost natural dust matter component only: ˜ τ = ρ = − τ = ηa − = 2 f − f ′ R + ( F − F ′ R ) L d .– 5 –odification comes from the geometric part of the theory. In this case formulae (3.7)-(3.9)simplify to more readable form H = 2( f ′ + F ′ L d ) [3 f − f ′ R + (3 F − F ′ R ) L d ]3 h f ′ − F ′ L d + f − f ′ R +( F ′ R − F ) L d ][ f ′′ +( F ′′ − F − ( F ′ ) ) L d ] f ′′ R − f ′ +[ F ′′ R +2 F ′ − F − ( F ′ ) R ] L d i (3.11)This reconstructs the ΛCDM model under the choice f = R − , F = 0, which is the limit α = γ = 0 , δ = − , β = − α = β = γ = 0. The first model we wish to be considered is based on the solution of (2.7), (3.5) resemblingthe Einstein–DeSitter universe (cf. 3.4) R = ρ = ηa − (4.1)provided that the integration constant A (see (3.5)) and parameter σ take the values A = γ β ( δ − δ η , σ = − δ (4.2)where δ = 0 ,
1. Positivity of A can be ensured by an appropriate choice of γ = ± R = 4Λ + ηa − instead (4.1). The conformalfactor b reads now b = 1 + 2 αηa − + 3 δ − δ βη δ a − δ (4.3)As a consequence, we have obtained the generalized Friedmann equations (3.7)-(3.9) H = K ( a ) G ( a )under the form G ( a ) = 23 ηa − + αη a − − − δ δ βη δ +1 a − δ +1) (4.4)and K ( a ) = 2 + 4 αηa − − − δδ βη δ a − δ h − αηa − − (1 − δ )(2 − δ ) δ βη δ a − δ i . When α = 0 and β = 0 the Friedmann equation takes the simplest form H = ηa − , i.e.the same as in Einstein general relativity with a pure dust matter. As we already mentionedit reconstructs the (flat) Einstein–DeSitter model.Before proceeding further let us observe that the scaling properties of (3.6) are analogousto that in standard cosmology (3.3). It entitles us to introduce dimensionless cosmological(density like) parameters: the standard Ω ,m = η H as well the new one: Ω ,β = βη δ ,Ω ,α = αη corresponding to new parameters α , β in the Lagrangian (3.10). With this set ofvariables the generalized Friedmann equation rewrites under the form (cid:18) HH (cid:19) = K ( z ) G ( z ) = 2 + 4Ω ,α (1 + z ) − − δδ Ω ,β (1 + z ) δ h − ,α (1 + z ) − (1 − δ )(2 − δ ) δ Ω ,β (1 + z ) δ i × – 6 – (cid:20) ,m (1 + z ) + Ω ,α Ω ,m (1 + z ) − − δδ Ω ,β Ω ,m (1 + z ) δ +1) (cid:21) (4.5)where redshift 1 + z = a − . The number of free parameters ( α, β, δ, η ) = (Ω ,α , Ω ,β , δ, Ω ,m )to be fitted by experimental data is 4. There are constrained by the following conditions: δ = 0 , − ,
1, Ω ,m ∈ h , i .Above equation can be rewritten in a more convenient for computer simulation form (cid:18) HH (cid:19) = Ω ,m K ( z ) ˜ G ( z ) = Ω ,m ,α (1 + z ) − − δδ Ω ,β (1 + z ) δ h − ,α (1 + z ) − (1 − δ )(2 − δ ) δ Ω ,β (1 + z ) δ i ×× (cid:20) z ) + Ω ,α (1 + z ) − − δδ Ω ,β (1 + z ) δ +1) (cid:21) (4.6)Now normalization constraint can be simply set byΩ ,m K (0) ˜ G (0) = 1 . (4.7)There is a number of interesting subcases, e.g, δ = , which we shall not study here. How-ever the case of Einstein gravity supplemented by the quadratic Starobinsky term correspondsto β = 0: L ES = √ g ( R + αR ) + L mat . In this simplest case one gets (cid:18) HH (cid:19) = Ω ,m ,α (1 + z ) [2 − ,α (1 + z ) ] (cid:2) z ) + Ω ,α (1 + z ) (cid:3) (4.8)Cosmology for (4.8) has been also recently investigated in [24] (see also [25]). We will see bycomparison against experimental data that this model does not differ qualitatively from theΛCDM, i.e. as expected it contains dynamical cosmological constant which becomes activein recent times. Adding dilaton field φ makes it even more quantitatively similar providedwe shall use the same standard solution (4.1). Yet another models can be determined by the solution R = ξa − δ , A = γ δ (cid:20) η (1 − δ ) β (cid:21) , σ = 2 δ (5.1)where ξ ≡ h η (1 − δ ) β i δ . One needs β (1 − δ ) , γδ >
0. Now the parameter β cannot vanish.The conformal factor b reads b = 1 + 4 δ δ + 2 αξa − δ + β (1 + δ ) ξ δ a − δ δ (5.2)As a consequence, we obtain components constituting the generalized Friedmann equations(3.7)-(3.9) (cf. (3.11)) under the form G ( a ) = 1 + δ δ ξa − δ + 13 αξ a − δ + 2 − δ − δ ) ηa − – 7 – ( a ) = δδ + 4 αξa − δ + 2 β (1 + δ ) ξ δ a − δ δ h δδ + 2 δ − δ αξa − δ + β (2 − δ ) ξ δ a − δ δ i (5.3)Introducing, as before, dimensionless density parameters: Ω ,m = η H , Ω ,β = ξ H = H h η (1 − δ ) β i δ , Ω ,α = αH Ω ,β one gets in terms of the redshift 1 + z = a − (cid:18) HH (cid:19) = K ( z ) G ( z ) = δδ + 12Ω ,α (1 + z ) δ + 2 δ − δ Ω ,m Ω − ,β (1 + z ) δ δ h δδ + 6 δ − δ Ω ,α (1 + z ) δ + − δ − δ Ω ,m Ω − ,β (1 + z ) δ δ i ×× (cid:20) δδ Ω ,β (1 + z ) δ + 3Ω ,α Ω ,β (1 + z ) δ + 2 − δ − δ Ω ,m (1 + z ) (cid:21) (5.4)The number of free parameters ( α, β, δ, η ) = (Ω ,α , Ω ,β , δ, Ω ,m ) to be fitted by experimentaldata is 4. There are constrained by the following conditions: (1 − δ ) β, η, Ω ,β >
0, Ω ,m ∈h , i , δ = 0 , − , ,c = Ω ,m Ω − ,β allows us to rewrite the last equation in a more convenientform (cid:18) HH (cid:19) = Ω ,β K ( z ) ˜ G ( z ) = Ω ,β δδ + 12Ω ,α (1 + z ) δ + 2 δ − δ Ω ,c (1 + z ) δ δ h δδ + 6 δ − δ Ω ,α (1 + z ) δ + − δ − δ Ω ,c (1 + z ) δ δ i ×× (cid:20) δδ (1 + z ) δ + 3Ω ,α (1 + z ) δ + 2 − δ − δ Ω ,c (1 + z ) (cid:21) (5.5)Then normalization constraint can be simply set byΩ ,β K (0) ˜ G (0) = 1 . (5.6)Again in order to better control the role of Starobinsky term one can switch it off by setting α = 0. As it has been mentioned before the parameter β cannot vanish now.One can summarize this part by concluding that we have obtained for numerical analysisfour new cosmological models which will be further on denoted correspondingly as I, I α =0 and II, II α =0 . Models I and II have three free parameters to be fitted by experimentaldata. Models I α =0 and II α =0 have two such parameters. These models are formulated notonly in terms of Lagrangian functions but also by giving explicit form for the correspondingFriedmann equations. Finally, the simplest case I β =0 describes cosmological model based on R Palatini modified gravity without dilaton field which has similarly to ΛCDM only one freeparameter. Such quadratic gravity models have been extensively studded in the literature(4.8) by using different methods than the one we have employed here. The reason is that theexplicit form of the corresponding Friedmann equation for this case (see (4.8)) has not beenused except [24].
In the Bayesian approach to estimation model parameters (i.e. best fit values and credibleintervals) one uses a posterior probability density function (pdf), which is defined in thefollowing way: P ( ¯Θ | D, M ) = P ( D | ¯Θ , M ) P ( ¯Θ | M ) P ( D | M ) . (6.1)– 8 –Θ is the vector of model M parameters and P ( D | ¯Θ , M ) ≡ L is the likelihood function formodel M . The so called prior pdf for model parameters P ( ¯Θ | M ) should be estimated beforethe data D comes into analysis and should involve information which we have gathered inearlier studies, e.g. with different data sets or on theoretical grounds. Finally P ( D | M ) ≡ E is the so-called evidence, which could be ignored in model constraint analysis. The best fitvalues can be estimated using the maximum of the joined posterior pdf (6.1) (i.e. its mode).One can also consider marginalized posterior pdf: P ( θ i | D, M ) = Z P ( ¯Θ | D, M )d ¯ φ, (6.2)where ¯Θ = { θ i , ¯ φ } , and use its mode or mean and credible interval (usually defined as intervalwhich involves 68% or 95% of the probability).In order to estimate the parameters of our models we use supernovae (SNIa) data [14],the observational H ( z ) data [15], the measurement of the baryon acoustic oscillations (BAO)from the SDSS luminous red galaxies [16] and information coming from CMB [17].We use a sample of N = 557 data [14], which consist of data from Union [26], SupernovaCosmology Project [14], SDSS SN Survey [27] and CfA3 [28]. The likelihood function isdefined in the following way: L SN ∝ exp " − X i ( µ theor i − µ obs i ) σ i , (6.3)where: σ i is the total measurement error, µ obs i = m i − M is the measured value ( m i –apparentmagnitude, M –absolute magnitude of SNIa), µ theori = 5 log D Li + M = 5 log d Li + 25, M = − H + 25 and D Li = H d Li , where d Li is the luminosity distance given by d Li = (1 + z i ) c R z i dz ′ H ( z ′ ) (with the assumption k = 0). In this paper the likelihood as afunction independent of H has been used (which is obtained after analytical marginalizationof formula (6.3) over H ).We use constraints coming from 13 measurements of Hubble function at different red-shifts z . Those data points are obtained by two observational methods. The first method isbased on the measurements of spectroscopic ages of red galaxies [29], while the second oneis based on the measurements of BAO scale in radial direction [30]. For the H ( z ) data thelikelihood function is given by: L H z ∝ exp " − X i ( H ( z i ) − H i ) σ i , where H ( z i ) is the Hubble function, H i denotes observational data.We also use information coming from the so called BAO A parameter, which is relatedto the Baryon Acoustic Oscillations scale measured in the redshift space power spectrum ofluminous red galaxies (LRG) from the Sloan Digital Sky Survey (SDSS) [16]. For BAO Aparameter data the likelihood function is characterized by: L BAO ∝ exp (cid:20) − ( A theor − A obs ) σ A (cid:21) , (6.4)where A theor = p Ω m , (cid:16) H ( z A ) H (cid:17) − h z A R z A H H ( z ) dz i and A obs = 0 . ± .
017 for z A = 0 . ,c δ Ω ,β Ω ,m χ TOT / ,α = 0, Ω ,c ∈ < − , > , δ ∈ (0 , > .
572 0 .
997 0 .
254 0 .
652 294 . ,α = − ,c ∈ < − , > , δ ∈ (0 , > .
748 0 .
553 0 .
012 0 .
045 278 . ,α = − ,c ∈ < − , > , δ ∈ (0 , > .
226 0 .
538 0 .
005 0 .
024 278 . ,α = − ,c ∈ < − , > , δ ∈ (0 , > .
469 0 .
524 0 .
009 0 .
013 278 . ,α = − ,c ∈ < − , > , δ ∈ (0 , > .
693 0 .
508 0 .
020 0 .
019 278 . Table 1 . Comparison of estimated parameters Ω ,c , δ , Ω ,β and Ω ,m for model II with fixed valueof Ω ,α . It shows that essential parameters as δ or Ω ,m behaves stable under a wide range of Ω ,α provided Ω ,α = 0. Finally, we use constraints coming from CMB temperature power spectrum, ie. CMB R shift parameter [31], which is related to the angular diameter distance ( D A ( z ∗ )) to the lastscattering surface: R = √ Ω m H c (1 + z ∗ ) D A ( z ∗ ) . (6.5)The likelihood function has the following form: L CMB ∝ exp (cid:20) −
12 ( R − R obs ) σ A (cid:21) , (6.6)where R obs = 1 .
725 and σ − A = 6825 .
27 for z ∗ = 1091 . L T OT is characterized by: L T OT = L SN L H z L BAO L CMB . (6.7)We assume flat prior probabilities for model parameters (second column of Table 3) . Theprior probabilities because of constraints coming from previous estimations of the parameterswere calculated using N = 192 SNIa data sample [32]. Additionally we have assumed that H = 74 . kms − M pc − ] [33] and Ω m ∈ [0 , I α =0 , I β =0 , I, II α =0 , II as well as for ΛCDM estimated by CosmoNest package. Parameters Ω ,β , Ω ,m for models Iand parameter Ω ,m for models II are calculated using equations (4.7), (5.6) and estimatedvalues of the remaining parameters. We consider two cases: estimations with data sets com-ing from late universe (i.e. SNIa, H(z) and BAO) and estimations including also informationfrom early universe (i.e. SNIa, H(z), BAO and CMB). Top part of the table relates to thefirst case, while the bottom relates to the second one. The best fit values correspond to themean of the marginalized posterior pdf (68% credible intervals are also shown, estimation ofsample variance is shown in cases where the credible interval has misleading values). Finallymode of joined posterior pdf is given (values in brackets) and the corresponding value of χ (table 2) (see next Section for explanations).In order to get better insight into the role of quadratic term we have made severalestimations for fixed values of Ω α . The results are gathered in table 1. They differ significantlywhen α = 0 and α = 0. In the last case the concrete values of Ω α have secondary meaning.Moreover, the switching off R term leads to the cosmology which is not longer similar toΛCDM (cf. Fig 11, 9). – 10 –
30 −20 −10 0 −10 0 10 0 0.5 1 0 0.5 1 Ω m Ω β −30 −20 −10 0−10010 δ −30 −20 −10 000.51 Ω α Ω m −30 −20 −10 000.51 −10 0 1000.51 Ω β −10 0 1000.51 δ Figure 1 . Constraints of the parameters of model I – estimations without CMB data. In 2D plotssolid lines are the 68% and 95% confidence intervals from the marginalized probabilities. The colorsreflect the mean likelihood of the sample. In 1D plots solid lines show marginalized probabilities ofthe sample, dotted lines are mean likelihood. For numerical results see Table 3, No 3. At the fig. 1, 3, 2, 4 we show constraints of the parameters of our models. In the 2D plotssolid lines are the 68% and 95% confidence intervals from the marginalized probabilities. Thecolors reflect the mean likelihood of the sample. In the 1D plots solid lines show marginalizedprobabilities of the sample, dotted lines are mean likelihood. It is seen that it’s not possibleto constrain properly parameter Ω ,α using available data (cf. table 1).It is interesting to compare (Fig 5) a posterior probability density functions of Ω ,m for new models with the one for ΛCDM model (see table 3 for the corresponding best fitvalues and credible intervals). As one can conclude only for model I (all three parametersfitted with and without CMB data) it remains at the same level when compared with ΛCDMmodel. It should be remarked that in some cases, best fit value for Ω ,m is small and closeto the well-known amount of barionic matter Ω b ∼ ,
05 (cf. table 3 row no 1, 5, 10, 11).– 11 –
30 −20 −10 0 0 5 0 0.1 0.2 0 0.5 Ω m Ω β −30 −20 −10 002468 δ −30 −20 −10 000.10.2 Ω α Ω m −30 −20 −10 000.20.4 0 4 800.10.2 Ω β δ Figure 2 . Constraints of the parameters of model I – estimations including CMB data. In 2D plotssolid lines are the 68% and 95% confidence intervals from the marginalized probabilities. The colorsreflect the mean likelihood of the sample. In 1D plots solid lines show marginalized probabilities ofthe sample, dotted lines are mean likelihood. For numerical results see Table 3, No 9. Finally, at the fig 6 Hubble’s diagrams for all our models are shown and comparedwith ΛCDM. Most of them are practically indistinguishable from the concordance model onthe level of Hubble diagrams. Moreover, the χ − test gives mostly comparable results andin few cases priority belongs to new models provided that CMB data are neglected. It isclearly seen that only for the special case II α =0 (fitting made with and without CMB data),the corresponding plots considerable differ from all remaining ones. These motivates us forfurther analysis and employing Bayesian model selection methods. Bayesian theory enables to compare investigated models, i.e. enables to show which one is thebest (most probable) in the light of analysed data. Let us consider the posterior probabilityfor model indexed by i ( M i ): P ( M i | D ) = P ( D | M i ) P ( M i ) P ( D ) . (7.1)– 12 –
60 −40 −20 0.5 1 0 0.02 0 0.02 Ω m δ −60 −40 −200.40.60.81 Ω β −60 −40 −2000.010.020.03 Ω α Ω m −60 −40 −2000.010.020.03 0.5 100.010.020.03 δ Ω β Figure 3 . Constraints of the parameters of model II – estimations without CMB data. In 2D plotssolid lines are the 68% and 95% confidence intervals from the marginalized probabilities. The colorsreflect the mean likelihood of the sample. In 1D plots solid lines show marginalized probabilities ofthe sample, dotted lines are mean likelihood. For numerical results see Table 3, No 5. P ( M i ) is the prior probability for the model under investigation, P ( D ) is normalizationconstant, P ( D | M i ) is the marginalized likelihood (also called evidence) and is given by: P ( D | M i ) = Z P ( D | ¯Θ , M i ) P ( ¯Θ | M i )d ¯Θ . (7.2)It is convenient to choose the base model M b and compare all models from investigatedset with respect to this base model by considering the ratio of posterior probabilities (posteriorodds): P ( M b | D ) P ( M i | D ) = P ( M b ) P ( M i ) P ( D | M b ) P ( D | M i ) = P ( M b ) P ( M i ) B bi . When all considered models are equally prior probable the posterior odds is reduced tothe evidence ratio, so called Bayes Factor ( B bi ). Its value can be simply interpreted asthe strength of evidence in favour of base model : 0 < ln B bi < < ln B bi < . . < ln B bi < B bi > B bi = − ln B ib the negative values of ln B bi should be interpreted in favour of model underinvestigation. – 13 –
60 −50 −40 0.4 0.6 0.8 0 0.01 0.02 0.02 0.04 Ω m δ −60 −50 −400.40.60.8 Ω β −60 −50 −4000.010.02 Ω α Ω m −60 −50 −400.010.020.030.04 0.4 0.6 0.800.010.02 δ Ω β Figure 4 . Constraints of the parameters of model II – estimations including CMB data. In 2D plotssolid lines are the 68% and 95% confidence intervals from the marginalized probabilities. The colorsreflect the mean likelihood of the sample. In 1D plots solid lines show marginalized probabilities ofthe sample, dotted lines are mean likelihood. For numerical results see Table 3, No 11. In Table 2 one can find values of logarithm of Bayesian Factor for new models, whichwere calculated with respect to the base ΛCDM model using CosmoNest code. The valueswere averaged from five runs.As one can generally conclude models based on solution II are not supported by dataused in analysis: we found weak and strong evidence in favor of ΛCDM model (for both withand without CMB data cases). When considering Model I one can find out that there isweak evidence in favor of it with respect to ΛCDM model in the light of observations fromlate Universe. When information from early Universe is included, i.e. CMB data point istaken into account, the conclusion change: moderate evidence is found in favor of ΛCDMmodel. Amazingly, the special case of quadratic gravity ( I β =0 ) is not supported by the data,in both cases the evidence in favor of the base model is strong. The result for model I α =0 isinconclusive when the data from late Universe are considered. Admitting information fromCMB gives us strong evidence in favor of ΛCDM model.– 14 – Ω m,0 Ω m,0 Ω m,0 Ω m,0 Ω m,0 Ω m,0 Ω m,0 Ω m,0 Ω m,0 Ω m,0 Figure 5 . Posterior probability density functions of Ω m, parameter for all cases (red lines). Firstrow correspond to models I, second to models II. First column and forth: β = 0 (i.e. quadraticgravity); second and fifth: α = 0 . In third and sixth column 3 parameters were fitted. Black curvescorrespond to ΛCDM model. Left panel: parameters were fitted using SnIa, H z and BAO data. Rightpanel includes CMB data. In some cases Ω m, is the same order as Ω barionic ∼ .
04. For numericalvalues see Table 3.
34 36 38 40 42 44 46 0.01 0.1 1Union2 datamodel I, special case Ω α =0model I, special case Ω β =0model Imodel II, special case Ω α =0model II Λ CDM model 34 36 38 40 42 44 46 0.01 0.1 1Union2 datamodel I, special case Ω α =0model I, special case Ω β =0model Imodel II, special case Ω α =0model II Λ CDM model
Figure 6 . Comparison of Hubble’s diagrams for model I (green) and II (magenta). Grey linedenotes special case of quadratic gravity I β =0 . Blue ( I α =0 ) and light blue ( II α =0 ) lines denote mostdivergent with respect to ΛCDM (black) models without Starobinsky’s term. Left panel correspondsto estimation without CMB data, right panel relates to estimation including CMB data. Having fixed free parameters of our models on can think about their dynamical propertiesencoded in a time evolution of the scale factor. In fact, the dynamics of our models is deter-mined by very complicated Friedmann type equations which constitute first order ordinary(non-linear) autonomous differential equations on the scale factor. Moreover r.h.s. of thegeneralized Friedmann equation (4.5,5.4) is rather rational than polynomial function (cf.(3.3)) of a . While obtaining exact solutions of these equations is very difficult (if possibleat all) it is enough to apply qualitative and numerical methods of analysis of differentialequations. The main goal of this method is instead of studying individual trajectories of thesystem under consideration to describe a geometrical structure of a phase space. The main– 15 –odel Estimation without CMB data Estimation with CMB dataln B Λ CDM,Model χ T OT / B Λ CDM,Model χ T OT / I α =0 − . ± . .
287 453 . ± . . I β =0 . ± . .
142 35 . ± . . I − . ± . .
400 4 . ± . . II α =0 . ± . .
233 169 . ± . . II . ± . .
237 206 . ± . . .
583 0 276 . Table 2 . Values of the logarithm of Bayesian Factor together with the corresponding χ / advantage is the possibility of investigating entire evolution, represented by trajectories inthe phase space, for all admissible initial conditions. The phase space (the phase plane ( a, ˙ a )in our case) is organized through critical points and trajectories. As a result we obtain aglobal phase portrait of the system illustrating the stability of special solutions as well as itsgeneric properties among all evolutional paths in the phase space.It would be useful to represent the dynamics of the cosmological models in terms ofdynamical system theory. For this aim let us re-parametrize the original time variable in(4.5,5.4) to the new re-scaled variable, say τ such that dτ = | H | dt and define an effectivepotential function V ( a ) = − a K ( a ) G ( a ). Then we obtain that dynamics is reduced to thedynamics of a fictitious particle of unit mass moving on a half line in the effective potential V ( a ) with energy level E κ = − κ/
2, where κ = 0 , ± (cid:18) dadτ (cid:19) + V ( a ) = E κ (8.1)The system is defined in the part of the configuration space in which E κ − V is non-negative.Therefore the dynamics of cosmological model is governed by dynamical system dadτ = x (8.2) dxdτ = − dVda . (8.3)of Newtonian type. Critical points of this system correspond to extremes on a diagram of theeffective potential. Because the stability of the critical point depends on the eigenvalues ofa linearization matrix (solutions of the characteristic equation) calculated at this point oneobtains two possibilities (a trace of the linearization matrix is always vanishing). If one hasa maximum then the critical point is a saddle. In contrast a minimum on the diagram of thepotential function means that we have a center type of critical point. Critical points providestationary solutions: stable for minimum and unstable otherwise.In fact dynamics of the system can be read off from the graph of potential function itself(in cosmological setting the effective potential is non-positive: V ≤ E = constant (mightbe negative). The difference E − V ≥ κ will determine different cosmic evolution. Such visualization is complementary tothe one which offers phase portrait. - - - - - - @ a D V LCDM H a L V H a L - - - - @ a D V LCDM H a L V H a L Figure 7 . The diagram of the effective potential in particle–like representation of cosmic dynamicsfor model I versus ΛCDM model (left picture relates to estimation without CMB data, the rightrelates to estimation employing CMB data; table 3, No 3, 6, 9, 12). Note that till the present epochtwo potential plots almost coincide. Particulary, one can observe decelerating BB era. Maximum ofthe potential function corresponds to Einstein’s unstable static solution (saddle point). Discrepanciesbecome important in the future time: e.g. discontinuities of the potential functions (vertical, redlines) denote that V → −∞ , i.e. ˙ a → ∞ for a → a final . It turns out to be finite–time (sudden)singularity. In any case the shadowed region below the graph is forbidden for the motion. - - - - @ a D V LCDM H a L V H a L - - - - @ a D V LCDM H a L V H a L Figure 8 . The diagram of the effective potential in particle like representation of cosmic dynamic forthe model of quadratic gravity I β =0 versus ΛCDM model (left picture relates to estimations withoutCMB data, the right relates to estimations employing CMB data; Table 3, no 2, 6, 8, 12). Maximum ofthe potential function corresponds to unstable static solution (saddle point). Again, until the presentepoch there is no striking differences between plots. One can observe finite–size sudden singularity inthe near future (vertical, red lines). In any case the shadowed region below the potential is forbiddenfor the motion. – 17 – - - - - - - @ a D V LCDM H a L V H a L - - - - @ a D V LCDM H a L V H a L Figure 9 . The diagram of the effective potential in particle like representation of cosmic dynamicfor the model I α =0 versus ΛCDM model (left picture relates to estimations without CMB data, theright relates to estimations employing CMB data; Table 3, no 1, 6, 7, 12). Both pictures representdecelerating, till the present epoch, universe with BB initial singularity. Now discrepancies withΛCDM model are more evident. In any case the shadowed region below the potential is forbidden forthe motion. - - - - - - @ a D V LCDM H a L V H a L - - - - - - @ a D V LCDM H a L V H a L Figure 10 . The diagram of the effective potential in particle–like representation of cosmic dynamicsfor model II versus ΛCDM model (left picture relates to estimation without CMB data, the rightrelates to estimation with CMB data; Table 3, no 5, 6, 11, 12). The evolution of the model isrepresented through the energy level. Therefore both models are bouncing type. The universe iscontracting, reaches the minimal size and then is expanding with acceleration. Notice that some partof the potential plot coincide with ΛCDM model. - - - @ a D V H a L V LCDM H a L V H a L - - - - - - @ a D V LCDM H a L V H a L Figure 11 . The diagram of the effective potential in particle like representation of cosmic dynamicsfor the model II α =0 versus ΛCDM model (left picture relates to estimation without CMB data, theright relates to estimation with CMB data; Table 3, no 4, 6, 10, 12). One can observe big discrepancieswith respect to the concordance model. Moreover, for the first time, adding CMB data changes thetype of expansion from accelerating to not accelerating. – 18 –n the diagrams we have illustrated the effective potential functions (Fig. 7, 9, 8, 10,11) as well as generic samples of phase portraits for our models (Fig. 12, 13). We foundthat although analytical formulae for effective potential functions for various model are verydifferent their numerical values and therefore a shape of the corresponding graph might bealmost the same. As it was explained above similar shapes give rise to similar dynamicalbehavior. This is what happens in our case. - - - - @ a D a a Figure 12 . The diagram of the effective potential and the corresponding phase portrait of the model I for estimations using CMB data. However for estimations without CMB data the phase portraitlooks similar. We marked as a bolded trajectory of the flat model determined by the energy constraint E = 0. The phase space is divided by this trajectory on two domains at which lie closed and openmodels. We have situated the only possible critical point, the saddle ( a static ,
0) which represents theEinstein static universe. The vertical red line a = a static passing through the saddle critical pointdivides each trajectory into two parts: decelerating ( V ( a ) is a growing function of its argument) andaccelerating ( V ( a ) is a decreasing function of the scale factor) eras. This part of the phase portraitis topologically equivalent to phase portrait of ΛCDM model. Particulary, the Bing Bang era isdecelerating. .– 19 – .2 0.4 0.6 0.8 1.0 1.2 - - - - @ a D a a Figure 13 . The diagram of effective potential and the corresponding phase portrait on the plain ( a, ˙ a )for model II – estimations including CMB data. (In the case without CMB data the phase portraitlooks similar.) The phase portrait shows all evolutional paths for all admissible initial conditions. Wemark as bolded the trajectory of the flat model determined by the energy constraint E = 0. Thephase space is divided by this trajectory on two domains at which lie closed and open models. Notethat all solutions are bouncing type. In contrast to ΛCDM model on can notice accelerating scenariofrom the very beginning. .Let us discuss the plots of the effective potentials for new models in more details. Themodel I has reached very good numerical agreement with ΛCDM for a wide range of the scalefactor a : from the very beginning till the present time (fig. 7). After Big Bang the universeexpansion slows down (deceleration epoch) till some critical value (equal to static solution)and since then speeds up. This cosmic acceleration lasts forever for the case of ΛCDM.In contrast, the model I predicts finite size future singularity since the effective potentialfunction (as well as velocity plot) has a pole at a finalI = 2 . I β =0 ), the shape of the potential function is the same as formodel I with poorer numerical coincidence with ΛCDM (fig. 8): a finalI β =0 = 1 . II whichoffers a model of permanently accelerating universe of bouncing type. After a short period ofstrong acceleration (possible inflation) there is an era of low acceleration followed by ΛCDMepoch. A characteristic for models I and ΛCDM saddle point is not present for the case ofmodel II . Finite size future singularity of the potential function appears at a finalII = 2 . I , I β =0 and II , arenegligible. Similarity to ΛCDM is lost for both models with α = 0 (see fig. 9, 11) whichmakes this case difficult to accept. Models of modified gravity enlarge zoo of possible cosmicevolutions because of the presence of finite time final size singularity of the potential functionat some finite value of the scale factor. In this paper we have studied and confronted against astrophysical data as well as against thestandard ΛCDM model five cosmological models obtained from two solutions presented inSections 4, 5. For a deeper examination and comparison of properties of our models we haveused various cosmological parameters as: deceleration parameter q ( a ), effective equation ofstate w eff ( a ), JERK j ( a ), SNAP s ( a ) (see e.g. [39, 40]). For a later convenience we shallexpress them in terms of the effective potential V ( a ) q ( t ) = − a d adt (cid:20) a dadt (cid:21) − , ⇔ q ( a ) = − aV ′ ( a )2 V ( a ) . (9.1) w eff ( a ) = 13 [2 q ( a ) − . (9.2) j ( t ) = + 1 a d adt (cid:20) a dadt (cid:21) − , ⇔ j ( a ) = a V ′′ ( a )2 V ( a ) . (9.3) s ( t ) = + 1 a d adt (cid:20) a dadt (cid:21) − , ⇔ s ( a ) = a V ′′′ ( a )2 V ( a ) + a V ′′ ( a ) V ′ ( a )4 V ( a ) . (9.4)where V ′ ( a ) = dVda . Therefore, explicit analytic form of the potential function is also helpfulfor determination of these diagnostics. Below we shall summarize the results of our analysis,for details see fig. 14, 15, 16.Deceleration parameter q ( a ) for ΛCDM, in contrast to other models, goes asymptoticallyto −
1. However, the present day values of q ( a ) for new models are not much different,particulary q I β =0 ( a = 1) < q I ( a = 1) < q Λ CDM ( a = 1) < q II ( a = 1). Only for two modelswith α = 0 the present day values o q ( a = 1) are visibly greater (see fig. 14 and table 3).One notices that the values of q ( a ) are almost the same at the point a ≈ .
75 for all of themodels except the ones with α = 0. At the beginning q ( a = 0) ≈ . II for which one has q II (0) = − .
02 instead. It means a matter dominated epoch at thebeginning of cosmic evolution (cf. below for plots of w eff ( a )).Very similar situation is for effective equation of state parameter since it is algebraicallyrelated to q ( a ) (cf. (9.2)). For ΛCDM model lim a →∞ w eff ( a ) = − w eff,I β =0 ( a = 1) 75 for plots representing four models: I , I β =0 , II and ΛCDM. Thin,vertical line denotes present time. The model parameters were fitted using SN, H z , BAO and CMBdata (Table 3, No 7–12). @ a D L CDMmodel IImodel II, W Α = W Β = W Α = - - @ a D L CDMmodel IImodel II, W Α = W Β = W Α = Figure 15 . Plots of parameters JERK j ( a ) (left panel) and SNAP s ( a ) (right panel) for investigatedmodels. From the figure one can see that different models predict different present-day values ofJERK and SNAP. Unfortunately estimations of these parameters are beyond our present observationalpossibilities. However some recent analysis support the current values of jerk bigger than 2 [40]. Thin,vertical line denotes present epoch. The model parameters were fitted using SNIa, H z , BAO and CMBdata (Table 3, No 7–12). (see bottom fig. 14). At a = 0 there is w eff,I β =0 ≈ w eff,I ≈ w eff,I α =0 ≈ w eff,II α =0 ≈ w eff, Λ CDM = 0. This explicitly indicates early matter dominated epoch noticed alreadybefore for these models.As it can be seen on the fig. 15, that JERK parameter j ( a ) for ΛCDM model isconstant j Λ CDM = 1 and for a < . I α =0 , I and I β =0 approximate this value.Similarly, j II α =0 ≈ . .5 1.0 1.5 2.0 z - @ z D L CDMmodel IImodel II, W Α = W Β = W Α = Figure 16 . Plots of parameter Om ( z ) allowing to answer the question on nature of dark energy[41]. For ΛCDM (black line) we have Om ( z ) = constant which means that cosmological constantis the best description of dark energy. For models I β =0 (gray) and I α =0 (blue) the function Om isincreasing what means that the best model for dark energy is provided by a phantom. Remainingmodels have decreasing Om functions and therefore they are of quintessence type. However, one cannotice that both I and I β =0 models have a long period of being of cosmological constant type. Themodel parameters were fitted using SN, H z , BAO and CMB data (Table 3, No 7–12). In order to better understand properties of our models we shall use Om diagnostic from[41]: Om ( z ) ≡ H ( z ) /H − z ) − , z > . (9.5)which is designed to answer the question of nature of dark energy. For ΛCDM model we have Om ( z ) = constant (see black line on fig. 16) which means that cosmological constant Λ isthe best description of dark energy. For the model I β =0 the function Om is increasing whatmeans that the best model for dark energy is provided by a phantom. Remaining modelshave decreasing Om function and therefore they are of quintessence type. However, one cannotice that two models I and I β =0 have a long period of being of cosmological constant type. 10 Summary and conclusions In this paper we have found, and then analyzed by fitting to experimental data, two familiesof cosmological models based on two different solutions of Palatini modified gravity equippedwith non-minimal curvature coupling to (free) scalar dilaton-like field. We assume Cosmolog-ical Principle to hold and standard spatially flat FRW metric with dust matter as a source.Our analysis reveals how rich is class of cosmological models offered by modified gravity. Be-side of two main models denoted respectively as I and II we have investigated three specialcases I β =0 , I α =0 and II α =0 corresponding to reduced lagrangian functions. Particularly, asa by-product, we have employed quadratic a la’ Starobinsky gravity, I β =0 , for descriptionof the cosmic acceleration. All models labeled by I correspond to the solution (4.1) whichis the same solution as in Einstein gravity provided with FRW metric and dust matter. Inthe case of standard gravity this solution leads, after solving Friedmann equation, to decel-erating expansion. Modification of gravitational Lagrangian (e.g. by adding cosmologicalconstant) modifies Friedmann equation and thence give rise to accelerating expansion. Incontrast, models labeled by II are based on the solution (5.1) which has no correspondence– 23 –o standard gravity. Therefore the limit β I , in spite of much more complicated Friedmann equation, qualitativelyand quantitatively mimics the ΛCDM one (Ω = 0 . 77) from the very beginning of cosmicevolution (Big Bang) until the recent time. As it can be seen, on various plots, effectivepotential, deceleration and effective equation of state parameters (fig. 7, 14) are almost thesame until the present time ( a = 1). Differences appear for diagnostics employing third andforth order derivatives of the scale factor (fig 15). These properties would be helpful for futurediscrimination between models. Both models exhibit existence of the initial singularity, but incontrast to ΛCDM, our model predicts the (final) finite-size sudden (finite-time) singularityat the point a = 2 . a = 0 . , ( z = 0 . q = − . 814 and w ,eff = − . 876 are within the expectedestimations. The Hubble diagram for this case is practically indistinguishable from ΛCDMdiagram, cf. Fig. 6. Moreover, the estimation with CMB data added does not provide anyessential (qualitative nor quantitative) changes to this model.At the fig. 5 are shown posterior probability density functions for Ω ,m parameter forall of our models. It’s clearly seen that density function for model I almost entirely overlapssuch function for ΛCDM model (see also table 3 for the corresponding best fit values ofΩ ,m ). One can also see that inclusion to our estimations CMB data makes possible properestimation of Ω ,m for the following cases: I β =0 , II α =0 and II. In particular, for models IIamount of Ω ,m is one order less than for ΛCDM and it has the order of barionic matter.The second our model gives rise to permanently accelerating Universe without initialsingularities: Big Bang scenario is replaced by Big Bounce. The potential plot has no max-imum and there is no saddle point at the phase portrait (fig 12). The lack of the initialsingularity is the main advantage of this model in comparison with ΛCDM model. The coin-cidence with ΛCDM effective potential holds for a ∈ (0 . , . − (fig 14), which corresponds to domination of the spatialcurvature at the initial phase of Universe’s evolution. For the present time the decelerationparameter q = − . 529 and effective equation of state w ,eff = − . a = 2 . α is negative provided that β = 0. In contrast for I β =0 onehas α > a a Figure 17 . The phase portrait for quadratic gravity model I β =0 on the plain ( a, ˙ a ) – estimationsincluding CMB data (see fig. (8)). One can observe Big Bang singularity, saddle point correspondingto static solution as well as behaviour of the system near sudden singularity (red vertical line). Boldedtrajectory of the flat model is determined by energy level E = 0. .The Bayesian framework of model selection has been used for comparison theoreticalmodels with the concordance ΛCDM one. We investigated Bayes Factor to show whichmodel is the best (most probable) in the light of the astronomical data. We have found thatwhile some special cases of theoretical models becomes a weak evidence in favor of it overΛCDM if observations from current epoch are used (SN, H(z), BAO data). Inclusion of theinformation coming from early epochs (CMB data ) changes this situation because we obtainstrong evidence in favor of Standard Cosmological Model.Our investigations here have aimed to distinguish the favorable model by cosmographyof the FLRW background metric in the sample of theoretical models. Because of plenitude ofdynamical scenarios, an introductory selection of sample of theoretical models was necessaryand it accounts in final Bayesian inference. In the Bayesian framework adding of new obser-vations is natural for improving models parameters. It means that the effects of cosmologicalperturbations in this class of models have not been considered here and this important taskis postponed for future investigations. This will allow to enlarge the discriminatory tools forfurther analysis. For example, a sound speed of the fluctuations for the quadratic gravitymodel I β =0 as calculated in [24] is c s = Ω ,α Ω ,α − a (10.1)which in our case Ω ,α ∼ c s > 1. Thus such a model shouldbe, in principle, rejected. The dynamics of cosmological perturbations in extended gravitymodels, including Palatini formulation, have been studied in number of papers (see e.g.[43] and references therein) both for a matter density as well as for the background metric.However the contributions coming from non-minimal dilaton-curvature couplings one dealshere have not been developed yet (cf. [44] in the context of other scalar-tensor theories) andwill be a subject of our future investigations.– 25 – cknowledgments A.B. gratefully acknowledges interesting discussions with G. Allemandi, S. Capozziello, M.Francaviglia and S. Odintsov during a preliminary stage of this project. A.B. is supportedby the Polish NCN grant 2011/B/S12/03354. References [1] S. Perlmutter et al. [ Supernova Cosmology Project Collaboration ], “Measurements of Omegaand Lambda from 42 high redshift supernovae,” Astrophys. J. , 565-586 (1999).[astro-ph/9812133];A. G. Riess et al. [ Supernova Search Team Collaboration ], “Observational evidence fromsupernovae for an accelerating universe and a cosmological constant,” Astron. J. , 1009-1038(1998). [astro-ph/9805201].[2] S. M. Carroll, “The Cosmological constant,” Living Rev. Rel. , 1 (2001). [astro-ph/0004075];N. Straumann, “On the cosmological constant problems and the astronomical evidence for ahomogeneous energy density with negative pressure,” In *Duplantier, B. (ed.) et al.: Vacuumenergy, renormalization* 7-51. [astro-ph/0203330];T. Padmanabhan, “Cosmological constant: The Weight of the vacuum,” Phys. Rept. ,235-320 (2003). [hep-th/0212290];V. Sahni, “Dark matter and dark energy,” Lect. Notes Phys. , 141-180 (2004).[astro-ph/0403324];E. J. Copeland, M. Sami, S. Tsujikawa, “Dynamics of dark energy,” Int. J. Mod. Phys. D15 ,1753-1936 (2006). [hep-th/0603057];M. Li, X. -D. Li, S. Wang, Y. Wang, “Dark Energy,” Commun. Theor. Phys. , 525-604 (2011).[arXiv:1103.5870 [astro-ph.CO]]. T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, “ModifiedGravity and Cosmology,” [arXiv:1106.2476 [astro-ph.CO]];[3] S. Nojiri, S. D. Odintsov, “Modified gravity with negative and positive powers of the curvature:Unification of the inflation and of the cosmic acceleration,” Phys. Rev. D68 , 123512 (2003).[hep-th/0307288];S. Nojiri, S. D. Odintsov, “Modified gravity with ln R terms and cosmic acceleration,” Gen. Rel.Grav. , 1765-1780 (2004). [hep-th/0308176];S. Nojiri, S. D. Odintsov, “The Minimal curvature of the universe in modified gravity andconformal anomaly resolution of the instabilities,” Mod. Phys. Lett. A19 , 627-638 (2004).[hep-th/0310045].[4] S. Nojiri, S. D. Odintsov, “Introduction to modified gravity and gravitational alternative fordark energy,” [hep-th/0601213];S. Capozziello, M. Francaviglia, “Extended Theories of Gravity and their Cosmological andAstrophysical Applications,” Gen. Rel. Grav. , 357-420 (2008). [arXiv:0706.1146 [astro-ph]];A. De Felice, S. Tsujikawa, “f(R) theories,” Living Rev. Rel. , 3 (2010). [arXiv:1002.4928[gr-qc]];T. P. Sotiriou, V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys. , 451-497 (2010).[arXiv:0805.1726 [gr-qc]];S. Nojiri, S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory toLorentz non-invariant models,” Phys. Rept. , 59-144 (2011). [arXiv:1011.0544 [gr-qc]];S. Tsujikawa, “Modified gravity models of dark energy,” Lect. Notes Phys. , 99-145 (2010).[arXiv:1101.0191 [gr-qc]];S. Capozziello, M. De Laurentis, “Extended Theories of Gravity,” [arXiv:1108.6266 [gr-qc]].[5] G. J. Olmo, “Palatini Approach to Modified Gravity: f(R) Theories and Beyond,” Int. J. Mod.Phys. D , 413 (2011). [arXiv:1101.3864 [gr-qc]]. – 26 – 6] G. J. Olmo and H. Sanchis-Alepuz, “Hamiltonian Formulation of Palatini f(R) theories a laBrans-Dicke,” Phys. Rev. D , 104036 (2011). [arXiv:1101.3403 [gr-qc]];G. J. Olmo, H. Sanchis-Alepuz, S. Tripathi, “Enriched Phenomenology in Extended PalatiniTheories,” [arXiv:1002.3920 [gr-qc]];C. Barragan, G. J. Olmo and H. Sanchis-Alepuz, “Avoiding the Big Bang Singularity withPalatini f(R) Theories,” [arXiv:1002.3919 [gr-qc]];M. Campista, B. Santos, J. Santos and J. S. Alcaniz, “Cosmological Consequences of ExponentialGravity in Palatini Formalism,” Phys. Lett. B , 320 (2011). [arXiv:1012.3943 [astro-ph.CO]];F. Bauer, “Filtering out the cosmological constant in the Palatini formalism of modified gravity,”Gen. Rel. Grav. , 1733-1757 (2011). [arXiv:1007.2546 [gr-qc]];G. J. Olmo, H. Sanchis-Alepuz and S. Tripathi, “Enriched Phenomenology in Extended PalatiniTheories,” [arXiv:1002.3920 [gr-qc]];S. Li, H. R. Yu and T. J. Zhang, “Statefinder diagnosis for the Palatini f ( R ) gravity theories,”[arXiv:1002.3867 [astro-ph.CO]];T. Harko, T. S. Koivisto and F. S. N. Lobo, “Palatini formulation of modified gravity with anonminimal curvature-matter coupling,” Mod. Phys. Lett. A , 1467 (2011). [arXiv:1007.4415[gr-qc]];S. Capozziello, F. Darabi and D. Vernieri, “Equivalence between Palatini and metric formalismsof f(R)-gravity by divergence free current,” Mod. Phys. Lett. A , 65 (2011). [arXiv:1006.0454[gr-qc]];T. Koivisto, “Viable Palatini-f(R) cosmologies with generalized dark matter,” Phys. Rev. D ,043527 (2007). [arXiv:0706.0974 [astro-ph]];[7] M. Ferraris, M. Francaviglia, I. Volovich, “Universal gravitational equations,” Nuovo Cim. B108 , 1313-1317 (1993);M. Ferraris, M. Francaviglia, I. Volovich, “The Universality of vacuum Einstein equations withcosmological constant,” Class. Quant. Grav. , 1505-1517 (1994). [gr-qc/9303007];A. Borowiec, M. Ferraris, M. Francaviglia, I. Volovich, “Universality of Einstein equations for theRicci squared Lagrangians,” Class. Quant. Grav. , 43-55 (1998). [arXiv:gr-qc/9611067 [gr-qc]];A. Borowiec, M. Francaviglia, I. Volovich, “Topology change and signature change in non-linearfirst-order gravity,” Int. J. Geom. Meth. Mod. Phys. , 647-667 (2007).[8] G. Allemandi, A. Borowiec, M. Francaviglia, “Accelerated cosmological models in first ordernonlinear gravity,” Phys. Rev. D70 , 043524 (2004). [arXiv:hep-th/0403264 [hep-th]];G. Allemandi, A. Borowiec, M. Francaviglia, “Accelerated cosmological models in Ricci squaredgravity,” Phys. Rev. D70 , 103503 (2004). [arXiv:hep-th/0407090 [hep-th]].[9] S. Nojiri, S. D. Odintsov, “Gravity assisted dark energy dominance and cosmic acceleration,”Phys. Lett. B599 , 137-142 (2004). [astro-ph/0403622];G. Allemandi, M. Francaviglia, M. L. Ruggiero, A. Tartaglia, “Post-Newtonian parameters fromalternative theories of gravity,” Gen. Rel. Grav. , 1891-1904 (2005). [gr-qc/0506123].[10] A. Borowiec, “From Dark Energy to Dark Matter via Non-Minimal Coupling,”[arXiv:0812.4383 [gr-qc]].[11] O. Bertolami, J. Paramos, “On the non-trivial gravitational coupling to matter,” Class. Quant.Grav. , 245017 (2008). [arXiv:0805.1241 [gr-qc]];O. Bertolami, J. Paramos, T. Harko, F. S. N. Lobo, “Non-minimal curvature-matter couplings inmodified gravity,” [arXiv:0811.2876 [gr-qc]];O. Bertolami, P. Frazao, J. Paramos, “Reheating via a generalized non-minimal coupling ofcurvature to matter,” Phys. Rev. D83 , 044010 (2011). [arXiv:1010.2698 [gr-qc]];J. Paramos, O. Bertolami, “Mimicking the cosmological constant: constant curvature sphericalsolutions in a non-minimally coupled model,” [arXiv:1107.0225 [gr-qc]].[12] S. Nojiri, S. D. Odintsov, P. V. Tretyakov, “From inflation to dark energy in the non-minimalmodified gravity,” Prog. Theor. Phys. Suppl. , 81-89 (2008). [arXiv:0710.5232 [hep-th]]. – 27 – 13] D. Puetzfeld and Y. N. Obukhov, “On the motion of test bodies in theories with non-minimalcoupling,” Phys. Rev. D , 121501 (2008). [arXiv:0811.0913 [astro-ph]].[14] R. Amanullah et al. , “Spectra and Light Curves of Six Type Ia Supernovae at 0.511 ¡ z ¡ 1.12and the Union2 Compilation,” Astrophys. J. , 712 (2010). [arXiv:1004.1711 [astro-ph.CO]].[15] J. Simon, L. Verde, R. Jimenez, “Constraints on the redshift dependence of the dark energypotential,” Phys. Rev. D71 , 123001 (2005). [astro-ph/0412269].[16] D. J. Eisenstein et al. [ SDSS Collaboration ], “Detection of the baryon acoustic peak in thelarge-scale correlation function of SDSS luminous red galaxies,” Astrophys. J. , 560-574(2005). [astro-ph/0501171];W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol, J. A. Peacock, A. C. Pope, A. S. Szalay,“Measuring the Baryon Acoustic Oscillation scale using the SDSS and 2dFGRS,” Mon. Not. Roy.Astron. Soc. , 1053-1066 (2007). [arXiv:0705.3323 [astro-ph]];B. A. Reid et al. [ SDSS Collaboration ], “Baryon Acoustic Oscillations in the Sloan Digital SkySurvey Data Release 7 Galaxy Sample,” Mon. Not. Roy. Astron. Soc. , 2148-2168 (2010).[arXiv:0907.1660 [astro-ph.CO]].[17] E. Komatsu et al. [ WMAP Collaboration ], “Seven-Year Wilkinson Microwave AnisotropyProbe (WMAP) Observations: Cosmological Interpretation,” Astrophys. J. Suppl. , 18(2011). [arXiv:1001.4538 [astro-ph.CO]].[18] http://cosmonest.org/[19] S. Capozziello, V. F. Cardone, M. Francaviglia, “f(R) Theories of gravity in Palatini approachmatched with observations,” Gen. Rel. Grav. , 711-734 (2006). [astro-ph/0410135];T. P. Sotiriou, “Constraining f(R) gravity in the Palatini formalism,” Class. Quant. Grav. ,1253-1267 (2006). [gr-qc/0512017];M. Amarzguioui, O. Elgaroy, D. F. Mota, T. Multamaki, “Cosmological constraints on f(r)gravity theories within the palatini approach,” Astron. Astrophys. , 707-714 (2006).[astro-ph/0510519];A. Borowiec, W. Godlowski, M. Szydlowski, “Accelerated cosmological models in modified gravitytested by distant supernovae snia data,” Phys. Rev. D74 , 043502 (2006). [astro-ph/0602526];A. Borowiec, W. Godlowski, M. Szydlowski, “Dark matter and dark energy as a effects ofModified Gravity,” [astro-ph/0607639];S. Fay, R. Tavakol and S. Tsujikawa, “f(R) gravity theories in Palatini formalism: Cosmologicaldynamics and observational constraints,” Phys. Rev. D , 063509 (2007).[arXiv:astro-ph/0701479].[20] R. Wojtak, S.H. Hansen, J. Hjorth, “Gravitational redshift of galaxies in clusters as predictedby general relativity,“ Nature :567-569 (2011) [arXiv:1109.6571 [astro-ph.CO]].[21] R. Trotta, “Bayes in the sky: Bayesian inference and model selection in cosmology,” Contemp.Phys. , 71 (2008) [arXiv:0803.4089 [astro-ph]].[22] T. Koivisto, “Covariant conservation of energy momentum in modified gravities,” Class. Quant.Grav. , 4289-4296 (2006). [gr-qc/0505128].[23] A. A. Starobinsky, “Isotropization of arbitrary cosmological expansion given an effectivecosmological constant,” JETP Lett. , 66-69 (1983);A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys.Lett. B91 , 99-102 (1980).[24] T. S. Koivisto, “Bouncing Palatini cosmologies and their perturbations,” Phys. Rev. D ,044022 (2010). [arXiv:1004.4298 [gr-qc]].[25] G. F. R. Ellis, J. Murugan, C. G. Tsagas, “The Emergent universe: An Explicit construction,”Class. Quant. Grav. , 233-250 (2004). [gr-qc/0307112];D. Muller and S. D. P. Vitenti, “About Starobinsky inflation,” Phys. Rev. D , 083516 (2006). – 28 – arXiv:gr-qc/0606018];C. Corda, “An oscillating Universe from the linearized R theory of gravity,” Gen. Rel. Grav. ,2201-2212 (2008). [arXiv:0802.2523 [astro-ph]];G. J. Olmo, “Palatini Actions and Quantum Gravity Phenomenology,” [arXiv:1101.2841 [gr-qc]];C. Corda, H. J. Mosquera Cuesta, R. L. Gomez, “High-energy scalarons in R gravity as a modelfor Dark Matter in galaxies,” [arXiv:1105.0147 [gr-qc]];F. Bauer, “The cosmological constant filter without big bang singularity,” [arXiv:1108.0875[gr-qc]].[26] M. Kowalski et al. [ Supernova Cosmology Project Collaboration ], “Improved CosmologicalConstraints from New, Old and Combined Supernova Datasets,” Astrophys. J. , 749-778(2008). [arXiv:0804.4142 [astro-ph]].[27] J. A. Holtzman, J. Marriner, R. Kessler, M. Sako, B. Dilday, J. A. Frieman, D. P. Schneider,B. Bassett et al. , “The Sloan Digital Sky Survey-II Photometry and Supernova IA Light Curvesfrom the 2005 Data,” Astron. J. , 2306-2320 (2008). [arXiv:0908.4277 [astro-ph.CO]].[28] M. Hicken, W. M. Wood-Vasey, S. Blondin, P. Challis, S. Jha, P. L. Kelly, A. Rest,R. P. Kirshner, “Improved Dark Energy Constraints from 100 New CfA Supernova Type Ia LightCurves,” Astrophys. J. , 1097-1140 (2009). [arXiv:0901.4804 [astro-ph.CO]].[29] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, S. A. Stanford, “Cosmic Chronometers:Constraining the Equation of State of Dark Energy. I: H(z) Measurements,” JCAP , 008(2010). [arXiv:0907.3149 [astro-ph.CO]].[30] E. Gaztanaga, A. Cabre, L. Hui, “Clustering of Luminous Red Galaxies IV: Baryon AcousticPeak in the Line-of-Sight Direction and a Direct Measurement of H(z),” Mon. Not. Roy. Astron.Soc. , 1663-1680 (2009). [arXiv:0807.3551 [astro-ph]].[31] J. R. Bond, G. Efstathiou, M. Tegmark, “Forecasting cosmic parameter errors from microwavebackground anisotropy experiments,” Mon. Not. Roy. Astron. Soc. , L33-L41 (1997).[astro-ph/9702100].[32] T. M. Davis, E. Mortsell, J. Sollerman, A. C. Becker, S. Blondin, P. Challis, A. Clocchiatti,A. V. Filippenko et al. , “Scrutinizing Exotic Cosmological Models Using ESSENCE SupernovaData Combined with Other Cosmological Probes,” Astrophys. J. , 716-725 (2007).[astro-ph/0701510].[33] A. G. Riess et al. , “A Redetermination of the Hubble Constant with the Hubble SpaceTelescope from a Differential Distance Ladder.” Astrophys. J. , 539 (2009). [arXiv:0905.0695[astro-ph.CO]].[34] P. Mukherjee, D. Parkinson, A. R. Liddle, “A nested sampling algorithm for cosmologicalmodel selection,” Astrophys. J. , L51-L54 (2006). [astro-ph/0508461].[35] P. Mukherjee, D. Parkinson, P. S. Corasaniti, A. R. Liddle, M. Kunz, “Model selection as ascience driver for dark energy surveys,” Mon. Not. Roy. Astron. Soc. , 1725-1734 (2006).[astro-ph/0512484].[36] D. Parkinson, P. Mukherjee, A.R. Liddle, “A Bayesian model selection analysis of WMAP3,”Phys. Rev. D73 , 123523 (2006). [astro-ph/0605003].[37] D. L. Wiltshire, “Gravitational energy as dark energy: Cosmic structure and apparentacceleration,” [arXiv:1102.2045 [astro-ph.CO]].[38] Y. Shtanov and V. Sahni, “Unusual cosmological singularities in brane world models,” Class.Quant. Grav. , L101 (2002) [arXiv:gr-qc/0204040];J. D. Barrow, “Sudden future singularities,” Class. Quant. Grav. , L79-L82 (2004).[gr-qc/0403084];S. Nojiri, S. D. Odintsov, S. Tsujikawa, “Properties of singularities in (phantom) dark energyuniverse,” Phys. Rev. D71 , 063004 (2005). [hep-th/0501025]; – 29 – . Tretyakov, A. Toporensky, Y. Shtanov, V. Sahni, “Quantum effects, soft singularities and thefate of the universe in a braneworld cosmology,” Class. Quant. Grav. , 3259-3274 (2006).[gr-qc/0510104].[39] V. Sahni, T. D. Saini, A. A. Starobinsky and U. Alam, “Statefinder: A New geometricaldiagnostic of dark energy,” JETP Lett. , 201 (2003) [Pisma Zh. Eksp. Teor. Fiz. , 249(2003)] [arXiv:astro-ph/0201498];U. Alam, V. Sahni, T. D. Saini, A. A. Starobinsky, “Exploring the expanding universe and darkenergy using the Statefinder diagnostic,” Mon. Not. Roy. Astron. Soc. , 1057 (2003).[astro-ph/0303009];M. Visser, “Jerk and the cosmological equation of state,” Class. Quant. Grav. , 2603-2616(2004). [gr-qc/0309109];Y. .L. Bolotin, O. A. Lemets, D. A. Yerokhin, “Expanding Universe: slowdown or speedup?,”[arXiv:1108.0203 [astro-ph.CO]].[40] D. Rapetti, S. W. Allen, M. A. Amin, R. D. Blandford, “A kinematical approach to darkenergy studies,” Mon. Not. Roy. Astron. Soc. , 1510-1520 (2007). [astro-ph/0605683].[41] V. Sahni, A. Shafieloo, A. A. Starobinsky, “Two new diagnostics of dark energy,” Phys. Rev. D78 , 103502 (2008). [arXiv:0807.3548 [astro-ph]].[42] J. Khoury and A. Weltman, “Chameleon Fields: Awaiting Surprises for Tests of Gravity inSpace,” Phys. Rev. Lett. , 171104 (2004). [arXiv:astro-ph/0309300];J. Khoury and A. Weltman, “Chameleon Cosmology,” Phys. Rev. D , 044026 (2004).[arXiv:astro-ph/0309411];T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao, “Constraining f(R) gravity as a scalar tensortheory,” Phys. Rev. D , 063505 (2007). [arXiv:astro-ph/0612569].[43] K. Uddin, J. ELidsey and R. Tavakol, “Cosmological perturbations in Palatini modifiedgravity,” Class. Quant. Grav. , 3951 (2007) [arXiv:0705.0232 [gr-qc]];Y. Gong, “The growth factor parameterization and modified gravity,” Phys. Rev. D , 123010(2008) [arXiv:0808.1316 [astro-ph]];H. Wei, “Growth Index of DGP Model and Current Growth Rate Data,” Phys. Lett. B , 1(2008) [arXiv:0802.4122 [astro-ph]];H. Motohashi, A. A. Starobinsky and J. ’i. Yokoyama, “Analytic solution for matter densityperturbations in a class of viable cosmological f(R) models,” Int. J. Mod. Phys. D , 1731 (2009)[arXiv:0905.0730 [astro-ph.CO]];S. Tsujikawa, K. Uddin and R. Tavakol, “Density perturbations in f(R) gravity theories in metricand Palatini formalisms,” Phys. Rev. D , 043007 (2008) [arXiv:0712.0082 [astro-ph]];S. Tsujikawa, “Matter density perturbations and effective gravitational constant in modifiedgravity models of dark energy,” Phys. Rev. D , 023514 (2007) [arXiv:0705.1032 [astro-ph]];S. Baghram and S. Rahvar, “Structure formation in f ( R ) gravity: A distinguishing probe betweenthe dark energy and modified gravity,” JCAP , 008 (2010) [arXiv:1004.3360 [astro-ph.CO]].[44] T. Koivisto and H. Kurki-Suonio, “Cosmological perturbations in the palatini formulation ofmodified gravity,” Class. Quant. Grav. , 2355 (2006) [astro-ph/0509422];N. Tamanini and C. R. Contaldi, “Inflationary Perturbations in Palatini Generalised Gravity,”Phys. Rev. D , 044018 (2011) [arXiv:1010.0689 [gr-qc]];F. Bauer and D. A. Demir, “Inflation with Non-Minimal Coupling: Metric versus PalatiniFormulations,” Phys. Lett. B , 222 (2008) [arXiv:0803.2664 [hep-ph]]. – 30 – odels I: equation (4.6) - the parameters estimated without CMB dataΩ ,α Ω ,β δ Ω ,m q w eff, α = 0, Ω ,β ∈ < − , > , δ ∈ (0 , > - 4 . ± . . . +0 . − . (0 . . ± . . − . − . ,α ∈ < , > , β = 0 4 . ± . . − − . +0 . − . (0 . − . − . ,α ∈ < − , > , Ω ,β ∈ < − , > , δ ∈ (0 , > − . +3 . − . ( − . . +4 . − . (2 . . +0 . − . (0 . . ± . . − . − . ,α Ω ,c δ Ω ,β Ω ,m q w eff, α = 0, Ω ,c ∈ < − , > , δ ∈ (0 , 1) - 2 . +0 . − . (0 . . ± . . . ± . . . ± . . − . − . ,α ∈ < − , − > , Ω ,c ∈ < − , > , δ ∈ (0 , − . +5 . − . ( − . . +0 . − . (0 . . +0 . − . (0 . . ± . . . +0 . − . (0 . − . − . H /H = 1 − Ω ,m + Ω ,m (1 + z ) - the parameter estimated without CMB dataΩ ,m q w eff, ,m ∈ < , > . +0 . − . (0 . − . − . ,α Ω ,β δ Ω ,m q w eff, α = 0, Ω ,β ∈ < − , > , δ ∈ (0 , > − . ± . . . ± . . . +0 . − . (0 . . − . ,α ∈ < , > , β = 0 3 . ± . . − − . +0 . − . (0 . − . − . ,α ∈ < − , > , Ω ,β ∈ < − , > , δ ∈ (0 , > − . +7 . − . ( − . . +1 . − . (1 . . +0 . − . (0 . . ± . . − . − . ,α Ω ,c δ Ω ,β Ω ,m q w eff, α = 0, Ω ,c ∈ < − , > , δ ∈ (0 , 1) - 0 . ± . . . +0 . − . (0 . . +0 . − . (1 . . +0 . − . (0 . − . − . ,α ∈ < − , − > , Ω ,c ∈ < − , > , δ ∈ (0 , − . +3 . − . ( − . . +0 . − . (2 . . +0 . − . (0 . . +0 . − . (0 . . +0 . − . (0 . − . − . H /H = 1 − Ω ,m + Ω ,m (1 + z ) - the parameter estimated with CMB data addedΩ ,m q w eff, 12 Ω ,m ∈ < , > . +0 . − . (0 . − . − . Table 3 . The values of estimated parameters (mean of the marginalized posterior probabilities and 68% credible intervals or sample square rootsof variance, together with mode of the joined posterior probabilities, shown in brackets) for all discussed models. Model I α =0 corresponds to rowsNo 1, 7; model I β =0 : No 2, 8; model I : No 3, 9; model II α =0 : No 4, 10; model II : No 5, 11; ΛCDM: No 6, 12. Computations were made using U nion H z + BAO data. We compare estimations without CMB data (top part of the table) with the one employing CMB data (bottom part).data. We compare estimations without CMB data (top part of the table) with the one employing CMB data (bottom part).