$p > 1$ one faces the problem of having cosmological horizons (cf. (\[horizon\])), which recently has attracted considerable attention in view of the impossibility of defining a consistent scattering $S$-matrix for asymptotic states [@challenge; @emnsmatrix]. In this case the operator ${\cal D}^{(q-2)}$ is [*not subleading*]{} and one has an [*extended (higher-order) logarithmic algebra*]{} defined by (\[elalg\]). It is interesting to remark that now the logarithmic world-sheet terms in the coefficient of the ${\cal D}^{(q-2)}$ operator imply that the limit $z \to w$ is plagued by ultraviolet world-sheet divergences, and hence the world-sheet conformal invariance is spoiled. This necessitates Liouville dressing, in order to restore the conformal symmetry [@ddk]. As we shall show later, in such cases with horizons the recoil of the D-particle may induce a non-trivial backreaction on the spacetime geometry, which results in an effective spacetime in which the horizons [*disappear*]{}. This happens, as we shall discuss later, in the context of Liouville strings with the identification of the Liouville mode with time.

We now turn to a study of the correlators of the various ${\cal D}^{(q)}$ operators, which will complete the study of the associated logarithmic algebras, in analogy with the flat target-space case of [@kmw]. From 