problem of writing a complete form for the operator (\[path\]) in a RW spacetime background in the Dirichlet picture as an open issue. Nevertheless, for RW backgrounds at large times, ignoring curvature effects proves to be a satisfactory approximation, and in such a case one may consider the vertex operator (\[path\]) as a sufficient description for the physical vertex operator of a $D$-particle. As we shall show below, the results of such analyses appear reasonable and interesting enough to encourage further studies along this direction.

For times long after the event, the trajectory $y^i(t)$ will be that of free motion in the curved space time under consideration. In the flat space time case, this trajectory was a straight line [@dparticle; @kmw; @szabo], and in the more general case here it will be simply the associated [*geodesic*]{}. Let us now determine its form, which will be essential in what follows.

The space time assumes the form: $$\label{rwmetric}
ds^2 = -dt^2 + a(t)^2 (dX^i)^2$$ where $a(t)$ is the RW scale factor. We shall work with expanding RW space times with scale factors $$a(t) =a_0 t^p, \qquad p \in R^+$$ The geodesic equations in this case read: $$\begin{aligned}
{\ddot t} + pt^{2p-1}({\dot y}^i)^2 