The dynamics of background evolution and structure formation in phase space: a semi-cosmographic reconstruction

The Baryon Acoustic Oscillation (BAO) feature, imprinted in the transverse and radial clustering of dark matter tracers, enables the simultaneous measurement of the angular diameter distance $D_A(z)$ and the Hubble parameter $H(z)$ at a given redshift. Further, measuring the redshift space anisotropy (RSD) allows us to measure the combination $f_8(z)\equiv f\sigma_8(z)$. Motivated by this, we simultaneously study the dynamics of background evolution and structure formation in an abstract phase space of dynamical quantities: $x = H_0 D_A/c$, $p = dx/dz$, and $f_8$. We adopt a semi-cosmographic approach, whereby we do not pre-assume any specific dark energy model to integrate the dynamical system. The Luminosity distance is expanded as a Pad\'e rational approximation in the variable $(1+z)^{1/2}$. The dynamical system is solved by using a semi-cosmographic equation of state, which incorporates the dark matter density parameter along with the parameters of the Pad\'e expansion. The semi-cosmographic $D_A(z), H(z)$ and $f\sigma_8(z)$, thus obtained, are fitted with BAO and RSD data from the SDSS IV. The reconstructed phase trajectories in the $3D$ $(x,p,f_8)$ space are used to reconstruct some diagnostics of background cosmology and structure formation. At low redshifts, a discernible departure from the $\Lambda$CDM model is observed. The geometry of the phase trajectories in the projected spaces allows us to identify three key redshifts where future observations may be directed for a better understanding of cosmic tensions and anomalies.