  The Selberg trace formula is specified for cosmological billiards in $4=3+1$
spacetime dimensions. The spectral formula is rewritten as an exact sum over
the initial conditions for the Einstein field equations for which periodic
orbits are implied. For this, a suitable density of measure invariant under the
billiard maps has been defined, within the statistics implied by the BKL
paradigm. The trace formula has also been specified for the stochastic limit of
the dynamics, where the sum over initial conditions has been demonstrated to be
equivalent to a sum over suitable symmetry operations on the generators of the
groups that define the billiard dynamics, and acquires different features for
the different statistical maps.
  Evidence for scars at the quantum regime is provided. The validity of the
Selberg trace formula at the classical level and in the quantum regime enforces
the validity of the semiclassical descriptions of these systems, thus offering
further elements for the comparison of quantum-gravity effects and the present
observed structure of the universe. This procedure also constitutes a new
approach in hyperbolic geometry for the application of the Selberg trace
formula for a chaotic system whose orbits are associated to precise statistical
distributions, for both billiard tables corresponding to the desymmetrized
fundamental domain and to that a a congruence subgroup of it.
