$Λ(t)$CDM Model: Cosmological Implications and Dynamical System Analysis

TL;DR

This study tests a time-varying cosmological constant model, Λ(t)CDM, in which Λ(t)=α' a^{-2}+β H^{2}+λ_*, against DESI DR2 BAO, multiple SNe Ia compilations, and CMB shift parameters using MCMC. It finds a dataset-dependent sign change in the dark sector interaction Q(z), with vacuum decay into dark matter at low redshift and the reverse at high redshift, and identifies three dynamical-system critical points that describe transitions from matter domination to late-time acceleration. The dynamical-system analysis shows a viable cosmological evolution with a saddle point for matter-era behavior, a de Sitter attractor for certain parameter ranges, and a transitional regime, while Bayesian evidence strongly supports Λ(t)CDM over ΛCDM. Observational fits yield Ω_m0≈0.30–0.31, Ω_Λ0≈0.70, and ω_eff≈-0.68 to -0.70, indicating accelerated expansion driven by a dynamical dark energy component, though the model does not fully resolve the Hubble tension. The work highlights the utility of combining dynamical-systems methods with current data to explore nonstandard dark energy scenarios and outlines future steps including full CMB spectra and N-body analyses.

Abstract

We investigated a time-varying cosmological constant model using recent BAO measurements from DESI DR2, combined with Type Ia supernova samples (Pantheon$^{+}$, DES-Dovekie, and Union3) and CMB shift parameters, to constrain the $Λ(t)$CDM model parameters via Markov Chain Monte Carlo analysis. We find that the interaction term $Q(z)$ shows a sign change for all dataset combinations by crossing $Q(z)=0$, depending on the choice of the dataset: at low redshift $Q(z)<0$, indicating vacuum energy decaying into dark matter, while at high redshift $Q(z)>0$, corresponding to dark matter decaying into vacuum energy. The dynamical system analysis found three critical points, namely $P_1,P_2$, and $P_3$ respectively. The resulting critical points, determined by the underlying cosmological parameters, correspond to distinct epochs in cosmic evolution. Depending on the parameter combinations, these points characterize various cosmological phases, ranging from an accelerated stiff matter-dominated era to late-time accelerated expansion. The stability of each critical point is analyzed using linear stability theory, with the relevant physical constraints on the cosmological parameters duly incorporated throughout the analysis. For each dataset combinations, the $Λ(t)$CDM model predicts that $ω_0 > -1$, showing a preference for dynamical dark energy over the cosmological constant scenario with $ω_0 = -1$. Consequently, the model exhibits a transition phase in the range $N \equiv \log a(t) \approx -0.51$ to $-0.48$ and predicts $q_0$ in the range $-0.54$ to $-0.52$, with the precise transition point depending on the choice of dataset. Finally, the Bayesian evidence shows strong support for the $Λ(t)$CDM model over $Λ$CDM

Figures (5)

• Figure 1: This figure shows the confidence contours at the $1\sigma$ and $2\sigma$ levels for the parameters of the $\Lambda(t)$CDM model obtained using DESI DR2 BAO data combined with CMB shift parameters and different SNe Ia samples (Pantheon$^{+}$, DES-Dovekie, and Union3).
• Figure 2: This figure shows the evolution of the interaction term $Q(z)$ as a function of redshift for the $\Lambda(t)$CDM model using DESI DR2 BAO data combined with CMB shift parameters and different SNe Ia samples samples (Pantheon$^{+}$, DES-Dovekie, and Union3). The horizontal dashed line denotes $Q(z)=0$, shows the redshift at which the interaction changes sign for each dataset combination.
• Figure 3: Phase space trajectories in $\left(x,y\right)$ plane corresponding to the critical points for constrained parameter $\alpha'$ and $\beta$
• Figure 4: This figure shows the numerical solutions of Eqs. \ref{['eq22']}, \ref{['eq010']}, and \ref{['eq0010']} for the evolution of the density parameters $\Omega_m$ and $\Omega_\Lambda$ as functions of $N \equiv \log a(t)$. The first, second, and third columns correspond to the dataset combinations DESI DR2 + CMB + Pantheon$^{+}$, DESI DR2 + CMB + DES-Dovekie, and DESI DR2 + CMB + Union3, respectively. The vertical line at $N=0$ denotes the present epoch, while $N<0$ and $N>0$ represent the past and future cosmic evolution, respectively.
• Figure 5: This figure shows the numerical solutions of Eqs. \ref{['eq22']}, \ref{['eq010']}, and \ref{['eq0010']} for the effective EoS parameter $\omega_{\mathrm{eff}}$ and the deceleration parameter $q$ as functions of $N \equiv \log a(t)$. The first, second, and third columns correspond to the dataset combinations DESI DR2 + CMB + Pantheon$^{+}$, DESI DR2 + CMB + DES-Dovekie, and DESI DR2 + CMB + Union3, respectively. The vertical line at $N=0$ represents the present epoch, with $N<0$ and $N>0$ indicating the past and future epochs, respectively.