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Monte Carlo simulation of cosmologies with dust

The quantum theory of the Friedmann cosmological model with dust and cosmological constant ($Λ$) is not exactly solvable analytically. We apply Path Integral Monte Carlo (PIMC) techniques to study its quantum dynamics using the physical Hamiltonian corresponding to the dust field as a clock. We study (i) quantum fluctuations around classical paths and (ii) formulate the analogues of the no-boundary and tunnelling proposals and simulate the ground state wave functions. For $Λ< 0$ a unique ground state wave function exists. For $Λ> 0$ the physical Hamiltonian is not bounded below, but the path integral for the propagator is convergent over a range of Euclidean time $T$. We investigate the properties of the convergent propagator. The path integral can be made convergent for all values of $T$ by restricting the integral over paths with action greater than equal to zero. We explore the consequences of such a choice.