The Theoretical Landscape of Mimetic Gravity: A Comprehensive Review
Mimetic gravity has emerged as a compelling extension of General Relativity (GR), originally motivated by the attempt to isolate the conformal degree of freedom of the gravitational field. By reparametrizing the physical metric in terms of an auxiliary metric and a scalar field, the theory naturally gives rise to a longitudinal degree of freedom that mimics the behavior of cold dark matter. This review provides a comprehensive survey of the theoretical landscape of mimetic gravity and its multifaceted applications to cosmology and high-energy physics. We begin by examining the original formulation and addressing the fundamental question of its equivalence to GR, highlighting how a singular disformal transformation introduces new physical degrees of freedom. We then explore minimal generalizations that lead to unified cosmological models, including mimetic matter scenarios and extensions into $f(R, \phi)$ gravity, which allow for the reconstruction of any desired expansion history. Significant attention is given to the ``limiting curvature'' hypothesis through $f(\Box \phi)$ modifications, providing a classical mechanism for resolving cosmological and black hole singularities. We critically assess the challenges facing the theory, specifically the gradient and ghost instabilities identified in cosmological perturbations, and discuss modern resolutions such as ghost-free mimetic massive gravity and covariant formulations of Ho\v{r}ava gravity. Finally, we discuss the role of the mimetic field in the early universe, specifically in the context of asymptotically free gravity and the resolution of the self-reproduction problem in inflation.