Fragility of stealth solutions in mimetic gravity

Abstract

We study a broad class of constrained mimetic-type extensions of general relativity with action $S=\int {\mathrm{d}}^{4}x\sqrt{-g}(R/2+\lambda C[g,\Psi ]+{\mathcal{L}}_{\mathrm{m}})$, where $R$ is the Ricci scalar, $\lambda$ is a Lagrange multiplier, $C[g,\Psi ]$ is a scalar functional of the metric and generic field content $\Psi$ (possibly involving $\Psi$ and its covariant derivatives) and ${\mathcal{L}}_{\mathrm{m}}$ is the matter Lagrangian. The branch $\bar{\lambda }\to 0$, with the bar denoting a background value, provides a simple screening-like limit in which the constrained sector decouples, as in cosmological realizations where $\bar{\lambda }$ is typically nonzero on large scales while locally one expects $\bar{\lambda }\simeq 0$. On the exactly stealth branch $\bar{\lambda }=0$, the constrained sector drops out of the background dynamics, so, on domains where a background profile $\bar{\Psi }$ satisfying $\bar{C}=0$ exists, the theory admits the corresponding general relativity geometries as stealth solutions. As an explicit realization of this mechanism, we consider the scalar field case, where $C=g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi \pm 1=0$ becomes a Hamilton-Jacobi equation selecting geodesic congruences; in this setting, we study spherically symmetric solutions and construct a stealth Kerr profile using Carter separability. We then show, at the general level, that the $\bar{\lambda }=0$ branch is perturbatively degenerate with general relativity: the constrained sector contributes to the dynamics only through terms weighted by $\bar{\lambda }$, which vanish on the stealth branch, while still imposing an infinite hierarchy of constraints on the fluctuations. Consequently, the $\bar{\lambda }\to 0$ limit is generically non-uniform, making the would-be screening perturbatively pathological.

19 pages