The normalized Hubble parameter is given by (see eq.2): scalar field with a negative kinetic energy. Now using eq.(2) the normalized Hubble parameter becomes Comparing the QP-models with the observational data (we sample Ω$_{m}$ ∈ [0 . 1 , 1] and w ∈ $^{[}$−2 $^{,}$− 0 . 4]) we find that the best fit values are Ω$_{m}$ = 0 . 28 ± 0 . 02 and w = − 1 . 02 ± 0 . 06 with χ 2 $_{tot}$(Ω$_{m}$ , w ) /dof ≃ 309 . 2 / 307 in very good agreement with the 5 years WMAP data Komatsu et al. (2009). Also Davis et al. (2007) utilizing the Essence-SNIa+BAO+CMB and a Bayesian statis- tics found Ω$_{m}$ = 0 . 27 ± 0 . 04, while Kowalski et al. (2008) using the Union08-SNIa+BAO+CMB obtained Ω$_{m}$ ≃ 0 . 285 +0 . 03 − 0 $_{03}$. Obviously, our results coincide within 1 σ errors. It is worth noting that the concordance Λ- cosmology can be described by QP models with w strictly equal to -1. In this case we find: Ω$_{m}$ = 0 . $^{28}$± 0 . 02 with χ 2 $_{tot}$(Ω$_{m}$ ) /dof ≃ 309 . 3 / 308. where w$_{0}$ and w$_{1}$ are constants. We sample the un- known parameters as follows: w$_{0}$ ∈ $^{[}$−2 $^{,}$− 0 . 4] and w$_{1}$ ∈ $^{[}$−2 . 6 , 2 . 6]. We find that for the prior of Ω$_{m}$ = 0 . 28 the overall likelihood function peaks at w$_{0}$ = − 1 . 1 +0 22 − 0 . 16 and w$_{1}$ = 0 . 60 +0 . 62 − 1 54 while the corresponding χ 2 $_{tot}$( w$_{0}$, w$_{1}$ ) /dof is 307.6/307. In this modified cosmological model, we use a sim- ple parametrization for the Ricci scalar which is based on a Taylor expansion around the present time: R = r$_{0}$ + r$_{1}$ (1 − a ) [for more details see Linder 2004]. It is interesting to point that at the early epochs the cosmic evolution tends asymptotically to be matter dominated. In this framework, we have 3 E $^{2}$( a ) = Ω$_{m}$ a − $^{$^{3}$+(1}$− Ω$_{m}$) a − 3(1+ w 0 + w 1 $^{)}$e3 w 1 ( $^{a}$− 1) , (13) E $^{2}$( a ) = Ω$_{m}$ a − 3 $^{+ (1}$− Ω$_{m}$) a − 3(1+ w ) . (9) E $^{2}$( a ) = { a 4( r 0 + r 1 − $^{1)}$e4 r 1 $^{(1}$− a ) a ≥ a$_{t}$ Ω$_{m}$ a − 3 a < a$_{t}$ (14) w ( a ) = $^{1}$− $^{4}$R 3 [ $^{1}$− Ω$_{m}$e − ∫ 1 $^{$_{a}$(1}$− $^{4}$R)( dy/y ) ] − 1 . (15) 4.4. The low Ricci dark energy - LRDE model 4.2. The Braneworld Gravity - BRG model In the context of a braneworld cosmology (hereafter BRG) the accelerated expansion of the universe can be explained by a modification of gravity in which gravity itself becomes weak at very large distances (close to the Hubble scale) due to the fact that our four dimensional brane survives into an extra dimensional manifold (Def- fayet, Dvali, & Cabadadze 2002). The interesting point in this scenario is that the corresponding functional form of the normalized Hubble flow, eq. (4), is affected only by one free parameter (Ω$_{m}$). Notice, that the quantity δH 2 is given by where a$_{t}$ = 1 − (1 − 4 r$_{0}$ ) / 4 r$_{1}$ . The matter density pa- rameter at the present epoch, is related with the above constants via Ω$_{m}$ = ( 4 r 0 − 4 r 1 − 1 4 r 1 ) 4 r 0 +4 r 1 − $^{1}$e $^{1}$−4 r 0 . The equation of state parameter that corresponds to the cur- rent geometrical dark energy model is given by Note, that we sample the unknown parameters as follows: r$_{0}$ ∈ [0 . 5 , 1 . 5] and r$_{1}$ ∈ $^{[}$−2 . 4 $^{,}$− 0 . 1] and here are the results: r$_{0}$ = 0 . 82 ± 0 . 04 and r$_{1}$ = − 0 . 74 +0 . 10 − 0 08 (Ω$_{m}$ ≃ 0 . 28) with χ 2 $_{tot}$( r$_{0}$, r$_{1}$ ) /dof ≃ 309 . 8 / 307. where Ω$_{bw}$ $^{= (1}$− Ω$_{m}$) $^{2}$/ 4. The geometrical dark energy equation of state parameter (see eq.5) reduces to δH 2 = 2Ω$_{bw}$ + 2 √ Ω$_{bw}$ √ Ω$_{m}$ a − 3 + Ω$_{bw}$ , (10) w ( a ) = − 1 1 + Ω$_{m}$( a ) , (11) 4.5. The high Ricci dark energy - HRDE model A different than the previously described geometrical method was defined by Linder & Cahn (2007), in which the Ricci scalar at high redshifts evolves as where Ω$_{m}$( a ) ≡ Ω$_{m}$ a − $^{3}$/E$^{2}$( a ). The joint likelihood anal- ysis provides a best fit value of Ω$_{m}$ = 0 . $^{24}$±0 . 02, but the fit is rather poor χ 2 $_{tot}$(Ω$_{m}$ ) /dof ≃ 369 / 308. In this model we use the Chevalier-Polarski-Linder (Chevallier & Polarski 2001; Linder 2003, hereafter CPL) parametrization. The dark energy equation of state parameter is defined as a first order Taylor expansion around the present epoch: where β = Ω − 1 m − 1. Using the same sampling as in the QP-models, the joint likelihood peaks at Ω$_{m}$ = 0 . 28 ± 0 . 03 and w$_{1}$ = − 1 . 02 ± 0 . 1 with χ 2 $_{tot}$(Ω$_{m}$ , w$_{1}$ ) /dof ≃ 309 . 2 / 307. To this end, the effective equation of state parameter is related to E ( a ) according to eq.(3). E $^{2}$( a ) = Ω$_{m}$ a − $^{3}$( 1 + βa − 3 w 1 ) − lnΩ$_{m}$ / ln(1+ β ) , (17) R ≃ 1 4 [ $^{1}$− 3 w$_{1}$ δH 2 H 2 ] , (16) w ( a ) = w$_{0}$ + w$_{1}$ $^{(1}$− a ) . (12) 4.3. The parametric Dark Energy model - CPL model where δH 2 = E $^{2}$( a $^{)}$−Ω$_{m}$ a − $^{3}$. In this geometrical pattern the normalized Hubble flow becomes: