Connecting early and late epochs by f(z)CDM cosmography

TL;DR

The paper addresses tensions in the standard $\Lambda$CDM framework by proposing a model-independent cosmographic approach, $f(z)$CDM, that replaces the cosmological constant with a Padé-based function $f(z)$ to connect late-time expansion with early-Universe constraints from the CMBR and BAO. It develops the Padé cosmography using a $P_{(2,2)}$ form for $E(z)=H(z)/H_0$ and introduces $f(z)$CDM with $\Omega_f f(z)$ driving late-time dynamics, avoiding a priori assumptions on $\Omega_m$. Through joint analyses with Planck, BAO, Pantheon, $H_0$ priors, and cosmic clocks, the study finds a noticeable degeneracy between $\Omega_m$ and cosmographic parameters, and shows that a third-order cosmographic expansion can yield a better $\chi^2$ than $\Lambda$CDM on low-redshift data, highlighting the need for higher-order corrections. The work underscores the CMB temperature power spectrum’s sensitivity to cosmographic choices and outlines a path toward a self-consistent, cross-epoch cosmology, with future surveys poised to tighten constraints and test the framework against tensions in the standard model.

Abstract

The cosmographic approach is gaining considerable interest as a model-independent technique able to describe the late expansion of the universe. Indeed, given only the observational assumption of the cosmological principle, it allows to study the today observed accelerated evolution of the Hubble flow without assuming specific cosmological models. In general, cosmography is used to reconstruct the Hubble parameter as a function of the redshift, assuming an arbitrary fiducial value for the current matter density, $Ω_m$, and analysing low redshift cosmological data. Here we propose a different strategy, linking together the parametric cosmographic behavior of the late universe expansion with the small scale universe. In this way, we do not need to assume any "a priori" values for the cosmological parameters, since these are constrained at early epochs using both the Cosmic Microwave Background Radiation (CMBR) and Baryonic Acoustic Oscillation (BAO) data. In order to test this strategy, we describe the late expansion of the universe using the Padé polynomials. This approach is discussed in the light of the recent $H(z)$ values indicators, combined with Supernovae Pantheon sample, galaxy clustering and early universe data, as CMBR and BAO. We found an interesting dependence of the current matter density value with cosmographic parameters, proving the inaccuracy of setting the value of $Ω_m$ in cosmographic analyses, and a non-negligible effect of the cosmographic parameters on the CMBR temperature anisotropy power spectrum. Finally, we found that the cosmographic series, truncated at third order, shows a better $χ^2$ best fit value then the vanilla $Λ$CDM model. This can be interpreted as the requirement that higher order corrections have to be considered to correctly describe low redshift data and remove the degeneration of the models.

Figures (8)

• Figure 1: Luminosity distance in terms of redshift in the case of low-redshift approximation of Eq.(\ref{['eq:DL_lowz']}) and the Taylor expansion of Eq.(\ref{['eq:luminosity distance']}) truncated at first order, with $q_0=-1/2$, in comparison with two toy models and data from Supernova Cosmology Project "Union2.1" SNeIa compilation Amanullah:2010vvSuzuki:2011hu
• Figure 2: $H(z)$ evolution expansion of a toy model using Eq.(\ref{['eq:H_DE']}) up to $z=1$ and the minimal $\Lambda$CDM model afterwards. For this model (red line) we assume $q_0=-0.6$, $j_0=1.97$ and $s_0=l_0=0$. For comparison, we also show the minimal $\Lambda$CDM model (blue curve) and cosmographic Padé $H(z)$ series (yellow curve), ever using the same choice of cosmographic parameters values.
• Figure 3: $H(z)$ evolution expansion of $f(z)$CDM model, where $j_0$ is fixed to $1.97$ and $s_0=l_0=0$, while different values of deceleration parameter are assumed, as $q_0=-0.5$ (magenta curve), $q_0=-1$ (green curve), $q_0=-1.5$ (red curve). For comparison, the minimal $\Lambda$CDM model is drawn with a black solid line and the Cosmic Clock data of Tab.\ref{['tab:Hz']} are reported. In this plot, we assume $\Omega_m = 0.3$, $\Omega_{\Lambda} = 1-\Omega_m$ and $H_0 = 70$ Km/s/Mpc.
• Figure 4: C.L of f(z)CDM model with one parameter extension using base-2015 dataset. The deceleration parameter is free to vary, while $j_0$ is fixed in 2 (red line) and 6 (blue line), and $s_0=l_0=0$.
• Figure 5: Temperature Anisotropy power spectrum for the best fit f(z)CDM model, with one parameter extension, constrained with base-2015 joint data. Superimpose data are from Planck 2015 release.