Defocusing dark energy: Raychaudhuri diagnostics beyond $w<-1/3$ and the phantom divide

Abstract

In general relativity, cosmic acceleration is timelike defocusing of the comoving congruence and requires a negative total active gravitational mass density, $\mathcal{M}_{\rm tot}=\rho_{\rm tot}+3p_{\rm tot}<0$. The criterion $w\equiv p/\rho<-1/3$ diagnoses sector repulsion only for $\rho>0$: the inequality reverses for $\rho<0$, and ratio variables are ill-defined at $\rho=0$ even when the stress-energy tensor is finite. For sign-changing effective dark energy (DE), as in $\Lambda_{\rm s}$CDM-type histories, we instead use the signed density $\rho_{\rm de}$ and two branch-independent combinations. The regular null energy condition (NEC) boundary $\mathcal{I}_{\rm de}=\rho_{\rm de}+p_{\rm de}=0$ replaces the phantom divide $w_{\rm de}=-1$, while $\mathcal{M}_{\rm de}=\rho_{\rm de}+3p_{\rm de}<0$ governs sector-level Raychaudhuri repulsion. For a separately conserved DE sector with a smooth negative-to-positive density crossing of finite odd order $n$ at $z_\dagger$, we prove that $\mathcal{I}_{\rm de}$ and $\mathcal{M}_{\rm de}$ are negative in a punctured neighborhood and non-positive at the crossing, while $w_{\rm de}$ develops a kinematic pole with universal residue $n(1+z_\dagger)/3$. If $\mathcal{M}_{\rm de}>0$ at some sufficiently high redshift, continuity requires at least one repulsion boundary $z_{\rm rep}>z_\dagger$: the sector is already repulsive while $\rho_{\rm de}<0$. We derive the exact range of $\rho_{\rm de}'(z_\dagger)$ for acceleration at the crossing with the total NEC satisfied. Under the stated single-impulse and stationary-point assumptions, the deceleration parameter has one or three sign-changing zeros. A smooth $\Lambda_{\rm s}$CDM profile, an exponential infrared $f(T)$ model, and the minimal phantom brane illustrate the results. These results motivate organizing late-time inference around $(\rho,p,\mathcal{I},\mathcal{M})$ rather than around $w$ alone.