case $p > 1$, i.e. $q > 2$ there is a non-trivial cosmological [*horizon*]{}, which as we shall see requires special treatment from a conformal symmetry viewpoint.

We commence with the no-horizon case, $1 < q \le 2$. We first notice that the linear in $t$ term in (\[path2\]) leads to the conventional logarithmic algebra, discussed in [@kmw], corresponding to a pair of impulse (‘recoil’) operators $C,D$. The main point of our discussion below is a study of the $t^{2p}$ terms in (\[path2\]), and their connection to other logarithmic algebras. Indeed, we observe that a logarithmic algebra [@lcft; @kogan; @kmw] can be obtained for these terms of the operators, if we define ${\cal D} \equiv {\cal D}^{(q)}$ and ${\cal C} \equiv q\epsilon {\cal D}^{(q-1)}$. In this case we have the following OPE with $T$: $$\begin{aligned}
\label{tdope}
&~& (z-w)^2~T\cdot {\cal D}=(1-\frac{\epsilon^2}{2}){\cal D} + {\cal C}, \nonumber \\
&~& (z-w)^2~T\cdot {\cal C}=(1-\frac{\epsilon^2}{2}){\cal C} + {\cal O}([t-t_0]^{q-2}),\end{aligned}$$ where throughout this work we ignore terms with negative powers in $t-t_0$ (e.g. of order $q-2$ and higher), for large $t\gg t_0$. Notice that in the case $q < 1$ (i.e. $p < 1/2$) the ${\cal C}$ operator defined above is absent.

In the second case 