Anisotropic Domain Walls and Cosmologies

Abstract

We investigate anisotropic domain wall and cosmological solutions in $D$-dimensional gravitational theories of arbitrary spacetime signature coupled to scalar fields with a scalar potential. We identify a class of scalar potentials for which the second-order equations of motion admit a first-order formulation. The associated bosonic flows allow non-trivial Ricci-flat worldvolumes, while their Killing-spinor realization further requires the worldvolume to admit a non-vanishing parallel spinor. Five-dimensional ${\cal N}=2$ gauged supergravity provides a concrete realization of this construction, with the worldvolume geometry determined by its signature and giving rise to pp-wave, hyper-Kähler, and hypersymplectic geometries. We further construct a general class of anisotropic solutions depending on a single variable, with the anisotropy encoded in a constant traceless matrix. A particular branch arises when a subset of scalar fields contribute through their kinetic terms but do not enter the scalar potential; these fields follow geodesics on the scalar manifold. As an explicit application, we recover the Kasner-dilaton-AdS solutions and show that, in the appropriate limit, they satisfy the first-order flow equations.