Quintessential dark energy crossing the phantom divide

Motivated by recent results from the DESI collaboration, we explore two classes of quintessence models that can give rise to crossing of the dark energy equation of state through the ``phantom divide'' $w=-1$. These are models with Lagrangians that involve higher powers of the kinetic energy $\dot\phi^2$, or where the dark matter (DM) mass is a function of $\phi$. Both have similar features with respect to the reconstructed redshift-dependent $w(z)$: moderate tuning of parameters is required to achieve the desired shape, and it is difficult or impossible for $w(z)$ to continue evolving smoothly as $z$ becomes large. Nevertheless, they give a strong improvement over $\Lambda$CDM in fitting the data. We point out that models of coupled dark matter and dark energy that cross the phantom divide are under pressure from constraints on long-range DM forces. They rule out the simplest renormalizable coupling of scalar DM to quintessence, but leave the fermionic case marginally allowed, while exponentially coupled models are safe from current constraints.