Λ_sCDM cosmology from a type-II minimally modified gravity

We have successfully integrated Λ_s Cold Dark Matter (CDM), a promising scenario for alleviating major cosmological tensions, into a concrete theoretical framework by endowing it with a specific Lagrangian from the VCDM model, a type-II minimally modified gravity. This promotes the scenario to a fully predictive model (dubbed Λ_s VCDM) that specifies the cosmological evolution self-consistently, including through the late-time anti-de Sitter (AdS)-to-de Sitter (dS) transition epoch. In this theory, an auxiliary scalar field generates an effective cosmological constant in the Friedmann equation not only when endowed with a constant potential, but also when endowed with a linear potential. This property allows an abrupt mirror AdS-to-dS transition to be realized via a piecewise-linear potential, implemented as a sudden change in slope at a junction. To remove the associated sudden (type-II) singularity and ensure stable evolution, we smooth the junction using a blended sigmoid interpolant, obtaining rapid but continuous transitions. We identify two qualitatively distinct smooth mirror AdS-to-dS realizations of Λ_s: (i) an agitated transition, in which the potential interpolates between equal-magnitude AdS and dS plateaus and Λ_s generically develops a central bump; and (ii) a quiescent transition, in which the potential remains continuous but changes slope across the transition layer, so that Λ_s(a) can remain monotone (possibly with shallow entrance/exit shoulders) and a central bump is not automatic. Depending on the transition type and sharpness, a finite-width transition can induce a transient accelerated-expansion interval (ä > 0) around the transition redshift (z ~ 1.5–2), in addition to the present-day accelerated expansion (for z ≲ 0.6 as in ΛCDM), and, if the background enters a region where V,_ϕϕ >2/3, a nested super-acceleration (Ḣ > 0) episode (and hence a bump in H). These distinct transient expansion histories can imprint characteristic signatures on both background and perturbation evolution; while the linear perturbation system is, in form, identical to that of ΛCDM, the scalar sector is modified through a Ḣ-dependent relation, with deviations localized primarily to the transition epoch. Our construction therefore enables a self-consistent observational assessment of smooth Λ_sCDM realizations and motivates dedicated multiprobe analyses to test transition dynamics and reassess cosmological tensions. Further work is warranted to assess whether Λ_sCDM can emerge as a credible extension of the concordance model, or at least as a useful guide for exploring its potential revisions.