Transition dynamic in the LSCDM model: Implications for bound cosmic structures

Abstract

We explore the predictions of $\Lambda_{\rm s}$CDM, a novel framework suggesting a rapid anti-de Sitter (AdS) to de Sitter (dS) vacua transition in the late Universe, on bound cosmic structures. In its simplest version, $\Lambda_{\rm s}$ abruptly switches sign from negative to positive, attaining its present-day value at a redshift of ${z_\dagger\sim 2}$ i.e., $\Lambda_{\rm s} \equiv \Lambda{\rm sgn}(z_{\dagger}-z)$. We will show that in the case of an abrupt sign-switching cosmological constant, there occurs a type II (sudden) singularity at the transition redshift, $z_{\dagger}$, where the total pressure of the universe diverges to infinity and the total energy density remains constant and finite. To avoid type II singularity, one can ``smooth-out'' the sudden sign-switch and describe it by using sigmoid functions (e.g., $\tanh$, logistic). However, since this correction would introduce an additional parameter ($\sigma$) to the model, we decided to examine the scenario in which the sign change of the cosmological constant is abrupt. This will also allow us to study the behavior of structure formation in the most extreme case without adding an extra parameter to our analysis. We will start our analysis by studying the spherical collapse model for a universe that contains dust (consisting of cold dark matter and baryons) and cosmological constant ($\Lambda$). For this universe, we will derive the equations describing the dynamics of the overdensity as a function of the background universe. Due to the shell crossing---and consequently the breakdown of the homogeneity and isotropy after the turnaround---, one cannot use the Friedmann equations (i.e., spherical collapse model) to describe the dynamics of the overdensity. Thus, we must refer to the semi-Newtonian approach and use the virialization condition to describe its dynamics. In the next step, we will extend our analysis of the spherical collapse model to include $\Lambda_{\rm s}$CDM, by incorporating the sign-switching cosmological constant ($\Lambda_{\rm s}$) into our calculations. To understand this process more clearly, we will separate our discussion into three parts. In the first part, we will study the evolution of the overdensity, if it enters turnaround under the effect of the positive cosmological constant (i.e., $\Lambda_{\rm s} \equiv +\Lambda$). In the second part, we will discuss the dynamics of the overdensity, if it enters turnaround under the effect of the negative cosmological constant (i.e., $\Lambda_{\rm s} \equiv -\Lambda$). In the third and final part, we will discuss the halos that completely virializes before the AdS-dS transition, and study the effect of the type II singularity on the bounded cosmic structures. At a first glance, it's clear that depending on the time of the transition, the overdensity will be effected differently. In summary, we can identify three primary influences which effects the structure formation in the $\Lambda_{\rm s}$CDM model: (i) the negative cosmological constant (AdS) phase for $z > z_\dagger$, (ii) the abrupt transition marked by a type II (sudden) singularity, leading to a sudden increase in the universe's expansion rate at $z=z_\dagger$, and (iii) an increased expansion rate in the late universe under a positive cosmological constant for $z < z_\dagger$, compared to $\Lambda$CDM. We find that the virialization process of cosmic structures, and consequently their matter overdensity, varies depending on whether the AdS-dS transition precedes or follows the turnaround. Specifically, structures virialize with either increased or reduced matter overdensity compared to the Planck/$\Lambda$CDM model, contingent on the timing of the transition. Despite its profound nature, the singularity exerts only relatively weak effects on such systems, thereby reinforcing the model's viability in this context.