  In this paper we adapt the mathematical machinery presented in \cite{P1} to
get, by means of the Laplace-Beltrami operator, the discrete spectrum of the
Hamiltonian of the Schr\"{o}dinger operator perturbed by an actractive 3D delta
interaction in a Friedmann flat universe. In particular, as a consequence of
the treatment in \cite{P1} suitable for a Minkowski spacetime, the discrete
spectrum of the ground state and the first exited state in the above mentioned
cosmic framework can be regained. Thus, the coupling constant $\lambda$ must be
choosen as a function of the cosmic comooving time $t$ as ${\lambda}/a^{2}(t)$,
with ${\lambda}$ be the one of the static Hamiltonian studied in \cite{P1}. In
this way we can introduce a time dependent delta interaction which is relevant
in a primordial universe, where $a(t)\rightarrow 0$ and becomes negligible at
late times, with $a(t)>>1$. We investigate, with the so obtained model, quantum
effects provided by point interactions in a strong gravitational regime near
the big bang. In particular, as a physically interesting application, we
present a method to depict, in a semi-classical approximation, a test particle
in a (non commutative) quantum spacetime where, thanks to Planckian effects,
the initial classical singularity disappears and as a consequnce a ground state
with negative energy emerges. Conversely, in a scenario where the scale factor
$a(t)$ follows the classical trajectory, this ground state is instable and thus
physically cannot be carried out.
