  The $k$-essence theory is a prototypical class of scalar-field models that
already gives rich phenomenology and has been a target of extensive studies in
cosmology. General forms of shift-symmetric $k$-essence are known to suffer
from formation of caustics in a planar-symmetric configuration, with the only
exceptions of canonical and DBI-/cuscuton-type kinetic terms. With this in
mind, we seek for multi-field caustic-free completions of a general class of
shift-symmetric $k$-essence models in this paper. The field space in UV
theories is naturally curved, and we introduce the scale of the curvature as
the parameter that controls the mass of the heavy field(s) that would be
integrated out in the process of EFT reduction. By numerical methods, we
demonstrate that the introduction of a heavy field indeed resolves the caustic
problem by invoking its motion near the would-be caustic formation. We further
study the cosmological application of the model. By expanding the equations
with respect to the curvature scale of the field space, we prove that the EFT
reduction is successfully done by taking the limit of infinite curvature, both
for the background and perturbation, with gravity included. The next
leading-order computation is consistently conducted and shows that the EFT
reduction breaks down in the limit of vanishing sound speed of the
perturbation.
