Probing a Steep EoS for Dark Energy with latest observations

We present a parametrization for the Dark Energy Equation of State "EoS" which has a rich structure, performing a transition at pivotal redshift $z_T$ between the present day value $w_0$ to an early time $w_i=w_a+w_0\equiv w(z\gg0)$ with a steepness given in terms of $q$ parameter. The proposed parametrization is $w=w_0+w_a(z/z_T)^q/(1+(z/z_T))^q$, with $w_0$, $w_i$, $q$ and $z_T$ constant parameters. It reduces to the widely used EoS $w=w_0+w_a(1-a)$ for $z_T=q=1$. This transition is motivated by scalar field dynamics such as for example quintessence models. We study if a late time transition is favored by BAO measurements combined with local determination of $H_0$ and information from the CMB. According to our results, an EoS with a present value of $w_0 = -0.92$ and a high redshift value $w_i =-0.99$, featuring a transition at $z_T = 0.28$ with an exponent $q = 9.97$ was favored by data coming from local dynamics of the Universe (BAO combined with $H_0$ determination). We find that a dynamical DE model allows to simultaneously fit $H_0$ from local determinations and Planck CMB measurements, alleviating the tension obtained in a $\Lambda$CDM model. Additionally to this analysis we solved numerically the evolution of matter over-densities in the presence of dark energy both at background level and when its perturbations were considered. We show that the presence of a steep transition in the DE EoS gets imprinted into the evolution of matter overdensities and that the addition of an effective sound speed term does not erase such feature.