Uniqueness and nonexistence of spacelike translating solitons in GRW spacetimes

We study the uniqueness and nonexistence of $ n $-dimensional spacelike translating solitons of the mean curvature flow immersed in a generalized Robertson-Walker (briefly GRW) spacetime $ -I\times_fM^n $ obeying appropriated curvature constraints on the Riemannian fiber $ M^n $ which also involve the warping function $ f $, via a sharp Bochner type inequality jointly with several maximum principles related to a drift Laplacian naturally attached to such a spacelike translating soliton. In this setting, we also derive new Calabi-Bernstein type results concerning the mean curvature equation attached to entire spacelike translating graphs constructed over the Riemannian fiber of a GRW spacetime. Furthermore, applications to some standard models of GRW spacetimes, like the Einstein-de Sitter and steady state type spacetimes, are also presented.