  In [1] a new cosmological model is proposed with no big bang singularity in
the past, though past geodesically incomplete. This model starts with an
inflationary era, follows with a stiff matter dominated period and evolves to
accelerated expansion in an asymptotically de Sitter regime in a realistic
fashion. The big bang singularity is replaced by a directional singularity.
This singularity cannot be reached by comoving observers, since it would take
them an infinite proper time lapse to go back to it. On the contrary, observers
with nonzero linear momentum have the singularity at finite proper time in
their past, though arbitrarily large. Hence, the time lapse from the initial
singularity can be as long as desired, even infinity, depending on the linear
momentum of the observer. This conclusion applies to similar inflationary
models. Due to the interest of these models, we address here the properties of
such singularities.
