Metastable Cosmological Constant and Gravitational Bubbles: Ultra-Late-Time Transitions in Modified Gravity

Abstract

The observed cosmological constant may originate as the minimum value $U_{min}$ of a scalar field potential, where the scalar field is frozen due to a large mass. If this vacuum is metastable, it may decay to a true vacuum either at present or in the future. Assuming its decay rate $Γ$ is comparable to the Hubble expansion rate $H_0$, we estimate the scale of true vacuum bubbles and analyze their evolution. We find that their initial formation scale is sub-millimeter and their tension causes rapid collapse if $m \gtrsim 1.7 \cdot 10^{-3}\, eV$. For smaller masses, the bubbles expand at the speed of light. We extend our analysis to scalar-tensor theories with non-minimal coupling, finding that the nucleation scale of gravitational constant bubbles remains consistent with the sub-millimeter regime of General Relativity. The critical mass scale remains around $10^{-3}\,eV$. A theoretical estimate at redshift $z_{obs} \sim 0.01$ suggests an observable bubble radius of $\sim 50$ Mpc, implying a gravitational transition triggered $\sim 300$ Myr ago, with a present-day size approaching $100$ Mpc. Additionally, we explore mass ranges ($m < 10^{-3}\,eV$) and non-minimal coupling $ξ$ ranges ($10^{-8}\,eV^{2-n} - 10^{-1}\,eV^{2-n}$) that lead to a variation $ΔG/G_N$ within the $1\%-7\%$ range. We assume non-minimal coupling of the form $F(ϕ)=1/κ- ξϕ^n$, with $κ=8πG_N$ and $2 \leq n \leq 9$. Finally, we review various local physics or/and transition based proposed solutions to the Hubble tension, including ultra-late-time transitional models ($z \sim 0.01$), screened fifth-force mechanisms, and the $Λ_{\rm s}$CDM model, which features a transition at $z \sim 2$. We discuss observational hints supporting these scenarios and the theoretical challenges they face.