&=& 0 \nonumber \\
{\ddot y} + 2\frac{p}{t}({\dot y}^i) {\dot t}&=& 0\end{aligned}$$ where the dot denotes differentiation with respect to the proper time $\tau$ of the $D$-particle.

With initial conditions $y^i(t_0)=0$, and $dy^i/dt (t_0) \equiv v^i$, one easily finds that, for long times $t \gg t_0$ after the event, the solution acquires the form: $$\label{pathexpre} 
y^i(t) =\frac{v^i}{1-2p}\left(t^{1-2p}t_0^{2p} - t_0 \right) + 
{\cal O}(t^{1-4p}), \qquad t  \gg t_0$$

To leading order in $t$, therefore, the appropriate vertex operator (\[path\]), describing the recoil of the $D$-particle, is: $$\label{path2} 
V=\int _{\partial \Sigma} a_0^2 \frac{v^i}{1-2p}\Theta (t-t_0)\left(tt_0^{2p}-t_0t^{2p}\right)
\partial_n X^i$$ where $\Theta (t-t_0)$ is the Heaviside step function, expressing an instantaneous action ([*impulse*]{}) on the $D$-particle at $t=t_0$ [@kmw; @recoil]. As we shall see later on, such deformed $\sigma$-models may be viewed as providing rather generic mathematical prototypes for models involving phase transitions at early stages of the Universe, leading effectively to time-varying speed of light. In the context of the present work, therefore, we shall be rather vague as far as the precise physical significance of the operator (\[path2\]) is concerned, and merely exploit the consequences of such deformations for the expansion of the RW spacetime after time $t_0$, from both a mathematical and physical 