  Biroli et al.'s extension of the standard mode-coupling theory to
inhomogeneous equilibrium states [Phys. Rev. Lett. 97, 195701 (2006)] allowed
them to identify a characteristic length scale that diverges upon approaching
the mode-coupling transition. We present a numerical investigation of this
length scale. To this end we derive and numerically solve equations of motion
for coefficients in the small q expansion of the dynamic susceptibility
$\chi_{\mathbf{q}}(\mathbf{k};t)$ that describes the change of the system's
dynamics due to an external inhomogeneous potential. We study the dependence of
the characteristic length scale on time, wave-vector, and on the distance from
the mode-coupling transition. We verify scaling predictions of Biroli et al. In
addition, we find that the numerical value of the diverging length scale
qualitatively agrees with lengths obtained from four-point correlation
functions. We show that the diverging length scale has very weak k dependence,
which contrasts with very strong $k$ dependence of the $q\to 0$ limit of the
susceptibility, $\chi_{\mathbf{q}=0}(\mathbf{k};t)$. Finally, we compare the
diverging length obtained from the small q expansion to that resulting from an
isotropic approximation applied to the equation of motion for the dynamic
susceptibility $\chi_{\mathbf{q}}(\mathbf{k};t)$.
