Measure problem in cosmology

The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. We review this construction and show how one can visualize the measure in terms of a “magnetic flux” of solutions through phase space. Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat Friedmann-Lemaitre-Robertson-Walker universes. We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. The resulting measure is independent of time and of the choice of coordinates on the space of fields. We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. One such quantity is the probability of inflation in simple scalar field models. We find that, according to our implementation of the canonical measure, the probability for N e-folds of inflation in single-field, slow-roll models is suppressed by of order exp⁡(-3N) and we discuss the implications of this result.