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

TL;DR

This work reframes cosmic evolution as a dynamical flow in a 3D phase space with $x=H_0 D_A/c$, $p=dx/dz$, and $f_8=f\sigma_8$, using a semi-cosmographic Padé approach to model $D_L(z)$ and derive a corresponding $w^P(z)$. By solving coupled background and growth equations without committing to a specific dark energy model, the authors fit BAO+RSD data from SDSS-IV to reconstruct $D_A(z)$, $H(z)$, and $f_8(z)$ and map out phase trajectories in $(x,p,f_8)$. They identify three important redshifts, $z_1$, $z_2$, and $z_f$, where phase-space projections yield direct observational handles on $E(z)$, the angular diameter distance maximum, and the peak of $f_8(z)$, respectively. The analysis finds low-$z$ deviations from $\Lambda$CDM and places emphasis on the diagnostic $\mathcal{S}(z)$ for growth versus expansion, concluding that current data do not resolve the Hubble tension, but future surveys like DESI DR2 and 21-cm experiments could tighten constraints at the identified redshifts and illuminate cosmic tensions.

Abstract

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 $f_8(z)\equiv fσ_8(z)$. 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 Padé 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 Padé expansion. The semi-cosmographic $D_A(z), H(z)$ and $fσ_8(z)$, 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 $Λ$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.

Figures (6)

• Figure 1: The cosmological evolution in 3-D $(x,p,f_8)$ space for $\Lambda$CDM model with Planck18 parameters is shown along with the projections on the $(x, p)$, $(x, f_8)$ and $(p, f_8)$ spaces. The 3D ellipsoids shown above represent the SDSS IV covariance data measured at their respective redshifts. We also show the straight line of consistency in Eq. \ref{['eq:consistency']} in the $(x,p)$ plane.
• Figure 3: The schematic flowchart for the semi-cosmographic method for estimating cosmological parameters.
• Figure 4: Reconstruction of BAO+RSD observables using the semi-cosmographic method on mock data sets. In the insets, we show the residual after subtracting the mean $\Lambda$CDM from the reconstructed functions. The upper panel (a) corresponds to the reconstruction using mock data, where the data is drawn from a multivariate distribution with a mean corresponding to the $\Lambda$CDM model, and the covariance matrix is taken from SDSS IV. In the second panel (b), the covariance matrix is reduced to 50% of the original, to see the reduction of reconstruction error. In the lower panel (c), we have introduced data at two more intermediate redshifts to study the effect of data sparsity and additional data.
• Figure 5: The posterior distributions of the semi-cosmographic parameters ($H_0, \Omega_{m0}, \alpha, \beta, \gamma, \sigma_8$) after marginalizing over $r_d$. The corresponding 2D correlation plot shows the $68.27\%$, $95.45\%$, and $99.73\%$ confidence contours.
• Figure 6: The left part of the panel, figures (a), (b), (c) and (d) shows the SDSS data fitted with the semi-cosmographic Padé and $\Lambda$CDM model using MCMC analysis. The right part of the panel, figures (e), (f), (g), and (h) show the reconstruction of some diagnostics of background cosmology for semi-cosmographic Padé and the $\Lambda$CDM model with their $1\sigma$, $2\sigma$, and $3\sigma$ errors.