2 from the WMAP 5-years data (Komatsu et al. 2009) is R = 1 . 71 ± 0 . 019 at z$_{ls}$ = 1090 [or a$_{ls}$ = (1 + z$_{ls}$ ) − 1 ≃ 9 . $^{17}$× 10 − $^{4}$]. In this case, the χ 2 cmb function is given point of view, it can be shown that instead of using the exact Hubble flow through a modification of the Fried- mann equation we can utilize a Hubble flow that looks like the nominal one (see eq.1). The key point here is to consider that the accelerated expansion of the uni- verse can be attributed to a “geometrical” dark energy component. Due to the fact that the matter density in the universe (baryonic+dark) can not accelerate the cos- mic expansion, we perform the following parametrization (Linder & Jenkins 2003; Linder 2004): Finally, we use the Union08 sample of 307 supernovae of Kowalski et al. (2008). The corresponding χ 2 SNIa func- tion becomes: where a$_{i}$ = (1 + z$_{i}$ ) − 1 is the observed scale factor of the Universe, z$_{i}$ is the observed redshift, µ is the distance modulus µ = m − M = 5log d$_{L}$ + 25 and d$_{L}$ ( a, p ) is the luminosity distance d$_{L}$ ( a, p ) = c a ∫ 1 a d y y $^{2}$H ( y ) where c is the speed of light. We can combine the above probes by using a joint likelihood analysis: L $_{tot}$( p ) = L BAO × L cmb × L SNIa or χ 2 $_{tot}$( p ) = χ 2 BAO + χ 2 $_{cmb}$+ χ 2 SNIa in order to put even further constraints on the parameter space used. Note, that we define the likelihood estimator 1 as: L j ∝ $^{exp[}$− χ 2 $_{j}$/ 2]. From now on, for the modified cosmological models we will use the above formulation. Obviously, with the aid of the latter approach we include any modification to the Friedmann equation of general relativity in the last term of eq.(4). Now from eqs.(3, 4) we can derive, after some algebra, the “geometrical” dark energy equation of state χ 2 $_{cmb}$( p ) = [ R ( p $^{)}$− 1 . 71] 2 0 . 019 2 (7) χ 2 $_{SNIa}$( p ) = 307 ∑ i =1 [ µ $^{th}$( a$_{i}$, p $^{)}$− µ $^{obs}$( a$_{i}$ ) σ$_{i}$ ] 2 . (8) E $^{2}$( a ) = H $^{2}$( a ) H 2 0 = Ω$_{m}$ $_{a}$− 3 + δH 2 . (4) w ( a ) = − $^{1}$− 1 3 d ln δH 2 d ln a . (5) χ 2 $_{BAO}$( p ) = [ A ( p $^{)}$− 0 . 469] 2 0 . 017 2 (6) 4. CONSTRAINTS ON THE FLAT DARK ENERGY MODELS 3. LIKELIHOOD ANALYSIS 4.1. Constant equation of state - QP model In this section, we consider a large family of flat dark energy models and with the aid of the above cosmologi- cally relevant observational data, we attempt to put tight constraints on their free parameters. In the following subsections, we briefly present these cosmological mod- els which trace differently the nature of the dark energy. We will use various cosmological observations in order to constrain the dark energy models described in section 4. First of all, we use the Baryonic Acoustic Oscilla- tions (BAOs). BAOs are produced by pressure (acoustic) waves in the photon-baryon plasma in the early universe, generated by dark matter overdensities. Evidence of this excess was recently found in the clustering properties of the luminous SDSS red-galaxies (Eisenstein et al. 2005; Padmanabhan et al. 2007) and it can provide a “stan- dard ruler” with which we can constraint the dark energy models. In particular, we use the following estimator: A ( p ) = √$_{Ω}$$_{m}$ [ z 2 $_{s}$E ( a$_{s}$ )] 1 / 3 [ ∫ 1 a$_{s}$ da a $^{2}$E ( a ) ] 2 / $^{3}$, measured from the SDSS data to be A = 0 . $^{469}$± 0 . 017, where z$_{s}$ = 0 . 35 [or a$_{s}$ = (1 + z$_{s}$ ) − 1 ≃ 0 . 75]. Therefore, the corresponding χ 2 BAO function is simply written In this case the equation of state is constant (see for a review, Peebles & Ratra 2003; hereafter QP-models). Such dark energy models do not have much physical mo- tivation. In particular, a constant equation of state pa- rameter requires a fine tuning of the potential and ki- netic energies of the real scalar field. Despite the lat- ter problem, these dark energy models have been used in the literature due to their simplicity. Notice that dark energy models with a canonical kinetic term have − 1 ≤ w < − 1 / 3. On the other hand, models of phantom dark energy ( w < − 1) require exotic nature, such as a where p is a vector containing the cosmological parame- ters that we want to fit. On the other hand, a very accurate and deep geo- metrical probe of dark energy is the angular scale of the sound horizon at the last scattering surface as en- coded in the location l TT 1 of the first peak of the Cos- mic Microwave Background (CMB) temperature per- turbation spectrum. This probe is described by the CMB shift parameter (Bond, Efstathiou & Tegmark, 1997; Nesseris & Perivolaropoulos 2007) and is defined as R = √$_{Ω}$$_{m}$ ∫ 1 a$_{ls}$ da a $^{2}$E ( a $_{)}$. The shift parameter measured 1 Likelihoods are normalized to their maximum values. Note, that the step of sampling is 0.01 and the errors of the fitted parameters represent 2 σ uncertainties. Note that the overall number of data points used is N tot = 309 and the degrees of freedom: dof = N tot − n fit , with n fit the number of fitted parameters, which vary for the different models.