  Assuming that Quantum Mechanics is universal and that it can be applied over
all scales, then the Universe is allowed to be in a quantum superposition of
states, where each of them can correspond to a different space-time geometry.
How can one then describe the emergence of the classical, well-defined geometry
that we observe? Considering that the decoherence-driven quantum-to-classical
transition relies on external physical entities, this process cannot account
for the emergence of the classical behaviour of the Universe. Here, we show how
models of spontaneous collapse of the wavefunction can offer a viable mechanism
for explaining such an emergence. We apply it to a simple General Relativity
dynamical model for gravity and a perfect fluid. We show that, by starting from
a general quantum superposition of different geometries, the collapse dynamics
leads to a single geometry, thus providing a possible mechanism for the
quantum-to-classical transition of the Universe. Similarly, when applying our
dynamics to the physically-equivalent Parametrised Unimodular gravity model, we
obtain a collapse on the basis of the cosmological constant, where eventually
one precise value is selected, thus providing also a viable explanation for the
cosmological constant problem. Our formalism can be easily applied to other
quantum cosmological models where we can choose a well-defined clock variable.
