&~& + \epsilon (2p+q)2p~t_0 - (4p^2 + 4pq - 2p)\}
{\cal D}^{(q-2)} + {\cal O}([t-t_0]^{q-3})
\label{onshell})\end{aligned}$$ From the above we observe that the on-shell operators become marginal as they should, given that an on-shell theory ought to be conformal. Moreover, and more important, the world-sheet divergences [*disappear*]{} upon imposing the condition $$\label{xi0}  
\epsilon^{2}{\rm ln}(L/a)^2 = \xi_0 = 
{\rm constant~independent~of}~\epsilon,~a,~L$$ where $L$ ($a$) is the world-sheet (ultraviolet) infrared cut-off on the world sheet. As we shall discuss later on, this condition will be of importance for the closure of the logarithmic algebra, which characterizes the fixed point [@kmw]. Hence, the conformal invariance is preserved by the on-shell states, any dependence from it being associated with [*off-shell*]{} states.

We next notice that, in the context of the RW metric (\[rwmetric\]), there are two cases of expanding universes, one corresponding to $0< p \le 1$, and the other to $p > 1$. Whenever $p \le 1$ (which notably incorporates the cases of both radiation and matter dominated Universes) there is [*no horizon*]{}, given that the latter is given by: $$\label{horizon}
 \delta (t) = a(t)\int_{t_0}^\infty \frac{dt'}{a(t')}$$ In this case the relevant value for $q$ is $q=2p \le 2$. On the other hand, for the 