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Curvature dark energy reconstruction through different cosmographic distance definitions

In the context of $f(\mathcal{R})$ gravity, dark energy is a geometrical fluid with negative equation of state. Since the function $f(\mathcal{R})$ is not known \emph{a priori}, the need of a model independent reconstruction of its shape represents a relevant technique to determine which $f(\mathcal{R})$ model is really favored with respect to others. To this aim, we relate cosmography to a generic $f(\mathcal R)$ and its derivatives in order to provide a model independent investigation at redshift $z \sim 0$. Our analysis is based on the use of three different cosmological distance definitions, in order to alleviate the duality problem, i.e. the problem of which cosmological distance to use with specific cosmic data sets. We therefore consider the luminosity, $d_L$, flux, $d_F$, and angular, $d_A$, distances and we find numerical constraints by the Union 2.1 supernovae compilation and measurement of baryonic acoustic oscillations, at $z_{BAO}=0.35$. We notice that all distances reduce to the same expression, i.e. $d_{L;F;A}\sim\frac{1}{\mathcal H_0}z$, at first order. Thus, to fix the cosmographic series of observables, we impose the initial value of $H_0$ by fitting $\mathcal H_0$ through supernovae only, in the redshift regime $z<0.4$. We find that the pressure of curvature dark energy fluid is slightly lower than the one related to the cosmological constant. This indicates that a possible evolving curvature dark energy realistically fills the current universe. Moreover, the combined use of $d_L,d_F$ and $d_A$ shows that the sign of the acceleration parameter agrees with theoretical bounds, while its variation, namely the jerk parameter, is compatible with $j_0>1$. Finally, we infer the functional form of $f(\mathcal{R})$ by means of a truncated polynomial approximation, in terms of fourth order scale factor $a(t)$.