In this work, we present a complete analysis of the quantisation of the classical Brans-Dicke Theory using the method of affine quantisation in the Hamiltonian description of the theory. The affine quantisation method is based on the symmetry of the phase-space of the system, in this case the (positive) half-plane, which is identified with the affine group. We consider a Friedmann-Lemaître-Robertson-Walker spacetime, and since the scale factor is always positive, the affine method seems to be more suited than the canonical quantisation for our Quantum Cosmology. We find the wave function of the Brans-Dicke universe, and its energy spectrum. A smooth bounce is expected at the semi-classical level in the quantum phase-space portrait. We also address the problem of equivalence between the Jordan and Einstein frames.

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Strong lensing time delay measurements provide a valuable and almost model-independent tool for cosmological investigations. In this work we recognize that they also carry information on the strength of the gravitational coupling at the redshift of the lens, and thus could be in principle used to test the equivalence principle on extragalactic scales. For the case of an extended lens with a static mass distribution we explicitly derive an analytical relation between Ġ/G and the relative variation of the time delay. For illustrative purpose, we apply our formula to the light curves of multiple images of the quasar DES J0408–5354 and simulated ones, which results in weak constraints on the variation of Ġ/G of order 10−1–10−2 yr−1 in the best scenario. Finally, we briefly discuss how those constraints can be improved in the next future.

We investigate cosmological models described by a scalar field with an exponential potential, and apply the stochastic formalism, which allows us to study how quantum field fluctuations give rise to stochastic noise. This modifies the classical dynamics of the scalar field at large scales, above a coarse-graining scale. In particular we explore how quantum field fluctuations perturb the equation of state on large scales which can lead to a quantum instability of the classical collapse solution below the Planck scale in the case of a pressureless collapse.