Analytical polyΛCDM dynamics

In this paper, we study the $\varphi$CDM and the $poly\Lambda$CDM models and compare them to the standard concordance model, the vanilla $\Lambda$CDM.

Analytical solutions for the $\varphi$CDM model, vital for precision, show sub-% agreement with $\Lambda$CDM due to selected energy density components, yet larger differences from numerical solutions of the same equations underscore their reliability over numerical integration of stiff systems.

The $poly\Lambda$CDM model is a phenomenological modified gravity model. We analyse their respective epoch evolutions, perform a detailed analytical and numerical dynamical analysis for each, and conduct a comparative study between the frameworks. Our work revitalizes these models by incorporating systems with an expanded set of variables, enabling us to derive analytical expressions for the energy density ratios of all species in both models. Moreover, the $poly\Lambda$CDM model we present is both more comprehensive and streamlined compared to existing studies. It captures all known cosmic epochs, including radiation, matter, and dark energy phases, and more exotic ones. The $poly\Lambda$CDM model has the cosmological constant term dominance in the far future epoch.

For the $poly\Lambda$CDM model, we find the global transition from a pure reflector of the dark energy dark matter exchange dominant component, to the saddle reflector points of matter, radiation, curvature, modified gravity components, and then we transit to the SVT modified gravity component attractor-saddle epoch, which also transits to the final cosmological constant, attractor epoch, in the far future. Note that there is saddle transition between the modified gravity dominant component and the SVT modified gravity component. Note that there is saddle transition between the SVT modified gravity and the cosmological constant component in the far future.

Our findings show that all three models effectively describe the widely accepted cosmic evolution scenario and align with current observations. While all $\Lambda$CDM, $\varphi$CDM, and $poly\Lambda$CDM reproduce the qualitative and quantitative behaviour of the universe’s species across epochs, the $poly\Lambda$CDM model exhibits a richer phenomenology, offering deeper insights into cosmic evolution.