  Within the braneworld paradigm the tachyonic scalar field has been used to
generate models that attempt to solve some of the open problems that physics
faces nowadays, both in cosmology and high energy physics as well. When these
field configurations are produced by the interplay of higher dimensional warped
gravity with some matter content, braneworld models must prove to be {\it
stable} under the whole set of small fluctuations of the gravitational and
matter fields background, among other consistency tests. Here we present a
complete proof of the stability under scalar perturbations of tachyonic thick
braneworlds with an embedded maximally symmetric 4D space-time, revealing its
physical consistency. This family of models contains a recently reported
tachyonic de Sitter thick braneworld which possesses a series of appealing
properties. These features encompass complete regularity, asymptotic flatness
(instead of being asymptotically dS or AdS) even when it contains a negative
bulk cosmological constant, a relevant 3-brane with dS metric which naturally
arises from the full set of field equations of the 5D background (it is not
imposed), qualitatively describing the inflationary epochs of our Universe, and
a graviton spectrum with a single zero mode bound state that accounts for the
4D graviton localised on the brane and is separated from the continuum of
Kaluza-Klein massive graviton excitations by a mass gap. Gauge vector fields
with a single massless bound state in its mass spectrum are also localised on
this braneworld model a fact that allows us to recover the Coulomb's law of our
4D world. All these properties of the above referred tachyonic braneworld
together with the positive stability analysis provided in this work, constitute
a firm step towards the construction of realistic cosmological models within
the braneworld paradigm.
