Effective Field Theory for Freezing Gravity with Minimally Coupled Matter
Freezing gravity (FG), a phantom-crossing dark energy model, was recently proposed in the absence of matter. In this work, we study FG minimally coupled to a perfect fluid through a detailed cosmological perturbation analysis and derive the conditions for the absence of ghost and gradient instabilities. The ghost stability conditions are obtained analytically without taking the high-$k$ limit, yielding the ghost-free range of scales and a finite cutoff when it exists. In the branch where a finite cutoff is already present in vacuum, we find that matter coupling lowers it relative to its vacuum value. Moreover, in the high-$k$ regime, the gradient stability condition guarantees a positive effective gravitational coupling for cold dark matter. We further show that two characteristic properties of FG persist in the presence of matter. First, the background and perturbation sectors remain separated, in the sense that they are controlled by independent sets of parameters, allowing FG to realize arbitrary background evolutions, including phantom crossing, while maintaining stable perturbation dynamics. Second, the scalar degree of freedom of FG becomes non-dynamical in the large-scale limit at linear order, while propagating with a finite speed of sound on small scales. Whether this behavior persists to arbitrarily high orders in perturbation theory, thereby avoiding the potential strong-coupling issue, requires further investigation through a nonlinear perturbation analysis. To make contact with large-scale structure and gravitational lensing observables, we also derive the effective gravitational coupling and gravitational slip within the quasi-static regime and express our results in terms of the EFT parameters.