A proposal to improve the accuracy of cosmological observables and address the Hubble tension problem

Context. Cosmological observational programs often compare their data not only with Lambda cold dark matter (Lambda CDM), but also with extensions applying dynamical models of dark energy (DE), whose time-dependent equation of state (EoS) parameters w differ from that of a cosmological constant. We found a degeneracy in the customary computational procedure for the expansion history of cosmological models once dynamical models of DE models were applied. This degeneracy, given the Planck-based Hubble constant H-0, provides an infinite number of cosmological models reproducing the Planck-measured cosmic microwave background (CMB) spectrum, including the one with a cosmological constant. Moreover, this degeneracy biases the comparison of Lambda CDM with dynamical DE extensions. Aims. We present a complementary computational approach, that breaks this degeneracy in the computation of the expansion history of models with a dynamical DE component: the "fixed early densities (EDs)" approach evolves cosmological models from the early Universe to the present, in contrast to the customary "fixed H-0" approach, which evolves cosmological models in reverse order. Although there are no equations to determine these EDs from first principles, we find they are accurately approximated by the Lambda CDM model. Methods. We implemented a refined procedure, applying both approaches, in an amended version of the code CLASS, where we focused on representative dynamical DE models using the Chevallier-Polarski-Linder (CPL) parametrization, studying cases with monotonically increasing and decreasing w over cosmic time. Results. Our results reveal that a dynamical DE model with a decreasing w of the form w(a) = - 0.9 + 0.1(1 - a) could provide a resolution to the Hubble tension problem. Moreover, we find that combining the fixed EDs approach and the customary fixed H-0 approach, while requesting to yield consistent results and being in agreement with observations across cosmic time, can serve as a kind of consistency check for cosmological models with a dynamical model of DE. Finally, we argue that implementing our proposed consistency check for cosmological models within current Markov chain Monte Carlo (MCMC) methods will increase the accuracy of inferred cosmological parameters significantly, in particular for extensions to Lambda CDM. Conclusions. Using our complementary computational scheme, we find characteristic signatures in the late expansion histories of cosmological models, allowing a phenomenological discrimination of DE candidates and a possible resolution to the Hubble tension, by ongoing and future observational programs.