Analytical $polyΛ$CDM dynamics

Abstract

We develop a novel analytical dynamical analysis to derive precise energy density ratio evolutions for the $φ$CDM and $polyΛ$CDM models, comparing them to the standard $Λ$CDM model and validating against numerical solutions. Analytical solutions for the quintessence, i.e. $φ$CDM, show sub percent agreement with $Λ$CDM with greater reliability than numerical integration of stiff systems. The $polyΛ$CDM model, a phenomenological modified gravity framework, captures radiation, matter, dark energy, and exotic epochs, offering a streamlined yet comprehensive alternative to existing studies. Its dynamics reveal a global transition from a dark energy-dark matter exchange reflector, through saddle points of matter, radiation, curvature, and modified gravity, to an SVT modified gravity attractor-saddle, and finally to a cosmological constant attractor in the far future, with saddle transitions between modified gravity components. The $polyΛ$CDM model integrates modified gravity models, using dynamical analysis to distinguish observationally viable critical points and differentiate gravity epochs. All three models align with observed cosmic evolution, but $polyΛ$CDM richer phenomenology provides deeper insights into modified gravity dynamics. Code available at GitHub.

Figures (10)

• Figure 1: We illustrate the analytical with a constant line and the numerical solutions with dotted line of the model of $\phi$CDM cosmology, in the case which $\texttt{var-zero-is}=3 \times 10^{-17}$. In the third column in the label we describe the intersection point between two function variables, and the corresponding redshift, for example $m(N) \cap r(N) = (-7.3, 0.5)$, $z_{\rm mr} = 1484.5$ means that matter and radiation meet at $N=-7.3$, in an energy density of matter-radiation equality $\Omega_{\rm mr} = 0.5$, at a corresponding redshift of matter-radiation equality $z_{\rm mr} = 1484.5$ [See section ]
• Figure 2: As Fig. , but we zoom in so that we observe clearly the evolution of the scalar kinetetic term. We illustrate the analytical with a constant line and the numerical solutions with dotted line of the model of $\phi$CDM cosmology, in the case which $\texttt{var-zero-is}=3 \times 10^{-17}$. [See section ]
• Figure 3: We present the comparison between the analytical and the numerical evolution of the different species of energy density ratios as a function of the lapse function, $\Omega_s(N)$. [See section ]
• Figure 4: Zoomed in analytical comparison of energy density evolution of $\phi$CDM vs $\Lambda$CDM. [See section ]
• Figure 5: Energy density ratio of different species, $\Omega_s$ versus lapse time, $N$, of $poly\Lambda$CDM. [See section ]