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The Selberg trace formula is specified for cosmological billiards in $4=3+1$spacetime dimensions. The spectral formula is rewritten as an exact sum overthe initial conditions for the Einstein field equations for which periodicorbits are implied. For this, a suitable density of measure invariant under thebilliard maps has been defined, within the statistics implied by the BKLparadigm. The trace formula has also been specified for the stochastic limit ofthe dynamics, where the sum over initial conditions has been demonstrated to beequivalent to a sum over suitable symmetry operations on the generators of thegroups that define the billiard dynamics, and acquires different features forthe different statistical maps.  Evidence for scars at the quantum regime is provided. The validity of theSelberg trace formula at the classical level and in the quantum regime enforcesthe validity of the semiclassical descriptions of these systems, thus offeringfurther elements for the comparison of quantum-gravity effects and the presentobserved structure of the universe. This procedure also constitutes a newapproach in hyperbolic geometry for the application of the Selberg traceformula for a chaotic system whose orbits are associated to precise statisticaldistributions, for both billiard tables corresponding to the desymmetrizedfundamental domain and to that a a congruence subgroup of it.

Periodic orbits in cosmological billiards: the Selberg trace formula for  asymptotic Bianchi IX universes, evidence for scars in the wavefunction of  the quantum universe and large-scale structure anisotropies of the present  universe