  Heterotic $M$-theory consists of a five-dimensional manifold of the form
$S^1/\mathbf{Z}_2 \times M_{4}$. It has been shown that one of the two orbifold
planes, the "observable" sector, can have a low energy particle spectrum which
is precisely the $N=1$ supersymmetric standard model with three right-handed
neutrino chiral supermultiplets. The other orbifold plane constitutes a
"hidden" sector which, since its communication with the observable sector is
suppressed, will be ignored in this paper. However, the finite fifth-dimension
allows for the existence of three-brane solitons which, in order to render the
vacuum anomaly free, must appear. That is, heterotic $M$-theory provides a
natural framework for brane-world cosmological scenarios coupled to realistic
particle physics. The complete worldvolume action of such three-branes is
unknown. Here, treating these solitons as probe branes, we construct their
scalar worldvolume Lagrangian as a derivative expansion of the heterotic DBI
action. In analogy with similar calculations in the $M_{5}$ and $AdS_{5}$
context, this leads to the construction of "heterotic Galileons". However,
realistic vacua of heterotic $M$-theory are necessarily $N=1$ supersymmetric in
four dimensions. Hence, we proceed to supersymmetrize the three-brane
worldvolume action, first in flat superspace and then extend the results to
$N=1$ supergravity. Such a worldvolume action may lead to interesting
cosmology, such as "bouncing" universe models, by allowing for the violation of
the Null Energy Condition (NEC).
