  We study the phenomenon of cosmological particle production of Dirac fermions
in a Friedman-Robertson-Walker spacetime, focusing on a (1+1)-dimensional case
in which the evolution of the scale factor is set by the equations of
Jackiw-Teitelboim gravity. As a first step towards a quantum simulation of this
phenomenon, we consider two possible lattice regularizations, which allow us to
explore the interplay of particle production and topological phenomena in
spacetimes with a boundary. In particular, for a Wilson-type discretization of
the Dirac field, the asymptotic Minkowski vacua connected by the intermediate
expansion corresponds to symmetry-protected topological groundstates, and have
a boundary manifestation in the form of zero-modes exponentially localized to
the spatial boundaries. We show that particle production can also populate
these zero modes, which contrasts with the situation with a na\"ive-fermion
discretization, in which conformal zero-mass fields exhibit no particle
production. We present a scheme for the quantum simulation of this
gravitational analogue by means of ultra-cold atoms in Raman optical lattices,
which requires real-time control of the Raman-beam detuning according to the
scale factor of the simulated spacetime, as well as band-mapping measurements.
