Determine the finite-frequency perturbation behavior of vacuum slow-roll and phantom-like branches

Determine whether the vacuum slow-roll and phantom-like branches with nonzero slow-roll parameter in Gravity from Entropy possess viable gravitational-wave propagation through an independent perturbation analysis, beyond the strict-principal tensor condition established for their backgrounds.

Background

The paper shows that evolving flat vacuum slow-roll and phantom-like branches with epsilon not equal to zero fail the strict-principal real-characteristic condition for tensor perturbations. However, failure of this strict-principal condition does not by itself resolve every aspect of their perturbative behavior, particularly outside the strict-principal approximation. The authors note that the earlier work introducing those branches left their perturbation analysis unresolved.