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, ${\mathcal{M}}_{\mathrm{tot}}=\ddot{I}{}_{\mathrm{tot}}+3p_{\mathrm{tot}}<0$. The criterion $w\equiv p/\ddot{I}<-1/3$ diagnoses sector repulsion only for $\ddot{I}>0$: the inequality reverses for $\ddot{I}<0$, and ratio variables are ill-defined at $\ddot{I}=0$ even when the stress-energy tensor is finite. For sign-changing effective dark energy (DE), as in $\hat{I}{›}_{\mathrm{s}}$CDM-type histories, we instead use the signed density $\ddot{I}{}_{\mathrm{de}}$ and two branch-independent combinations. The regular null energy condition (NEC) boundary ${\mathcal{I}}_{\mathrm{de}}=\ddot{I}{}_{\mathrm{de}}+p_{\mathrm{de}}=0$ replaces the phantom divide $w_{\mathrm{de}}=-1$, while ${\mathcal{M}}_{\mathrm{de}}=\ddot{I}{}_{\mathrm{de}}+3p_{\mathrm{de}}<0$ 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 $z_{†}$, we prove that ${\mathcal{I}}_{\mathrm{de}}$ and ${\mathcal{M}}_{\mathrm{de}}$ are negative in a punctured neighborhood and non-positive at the crossing, while $w_{\mathrm{de}}$ develops a kinematic pole with universal residue $n(1+z_{†})/3$. If ${\mathcal{M}}_{\mathrm{de}}>0$ at some sufficiently high redshift, continuity requires at least one repulsion boundary $z_{\mathrm{rep}}>z_{†}$: the sector is already repulsive while $\ddot{I}{}_{\mathrm{de}}<0$. We derive the exact range of $\ddot{I}{}_{\mathrm{de}}^{\prime }(z_{†})$ 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 $\hat{I}{›}_{\mathrm{s}}$CDM profile, an exponential infrared $f(T)$ model, and the minimal phantom brane illustrate the results. These results motivate organizing late-time inference around $(\ddot{I},p,\mathcal{I},\mathcal{M})$ rather than around $w$ alone.

28 pages, 3 figures, 1 table