_0^z \frac{d z'}{(1 + z')\,\sqrt{\Omega_\Lambda + \Omega_m (1 + z')^3}}\right)~.$$ It is clear that, in order to for $n_{\rm short}(z)$ to be reduced significantly, e.g., by two orders of magnitude, at the epoch $z \simeq 0.9$ of the GRB 090510 [@grb090510], while keeping $n^\star ={\cal O}(1)$, one must consider the magnitude of $v$ as well as the string scale $\ell_s$. For instance, for string energy scales of the order of TeV, i.e., string time scales $\ell_s/c = 10^{-27}~s$, one must consider a brane velocity $v \le \sqrt{10} \times 10^{-11} \, c$, which is not implausible for a slowly moving D-brane at a late era of the Universe [^5]. This is compatible with the constraint on $v$ obtained from inflation in [@emninfl], namely $v^2 \le 1.48 \times 10^{-5}\,g_s^{-1}$, where $g_s < 1$ for the weak string coupling we assume here. On the other hand, if we assume a 9-volume $V_9 = (K \ell_s)^9$: $K \sim 10^3$ and $\ell_s \sim 10^{-17}$/GeV, then the brane velocity $v \le 10^{-4} \, c$. In our model, due to the friction induced on the D-brane by the bulk D-particles, one would expect that the late-epoch brane velocity should be much smaller than during the inflationary era immediately following a D-brane collision.

Towards D3-Foam Phenomenology
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The above discussion was in the context of type-IIA string theories, but may easily be extended to compactified type-IIB theories of D-particle foam [@li], which may be of interest for low-energy Standard Model phenomenology. In this case, as suggested in [@li], one may construct ‘effective’ D-particle defects by compactifying D3-branes on three-cycles. In the foam model considered in [@li], the rôle of the brane Universe was played by D7-branes compactified appropriately on four-cycles. The foam was provided by compactified D3-brane ‘D-particles’, in such a way that there is on average a D-particle in each three-dimensional volume, $V_{A3}$, in the large Minkowski space dimensions of the D7-brane. We recall that, although in the