Theories with limited extrinsic curvature and a nonsingular anisotropic universe

We propose a class of theories that can limit scalars constructed from the extrinsic curvature. Applied to cosmology, this framework allows us to control not only the Hubble parameter but also anisotropies without the problem of Ostrogradsky ghost, which is in sharp contrast to the case of limiting spacetime curvature scalars. Our theory can be viewed as a generalization of mimetic and cuscuton theories (thus clarifying their relation), which are known to possess a structure that limits only the Hubble parameter on homogeneous and isotropic backgrounds. As an application of our framework, we construct a model where both anisotropies and the Hubble parameter are kept finite at any stage in the evolution of the universe in the diagonal Bianchi type I setup. The universe starts from a constant-anisotropy phase and recovers Einstein gravity at low energies. We also show that the cosmological solution is stable against a wide class of perturbation wavenumbers, though instabilities may remain for arbitrary initial conditions.