Connection between cosmological time and the constants of nature

We examine in greater detail the proposal that time is the conjugate of the constants of nature. Fundamentally distinct times are associated with different constants, a situation often found in ``relational time'' settings. We show how in regions dominated by a single constant the Hamiltonian constraint can be reframed as a Schr\"odinger equation in the corresponding time, solved in the connection representation by outgoing-only monochromatic plane waves moving in a ``space'' that generalizes the Chern-Simons functional (valid for the equation of state w=-1) for other w. We pay special attention to the issues of unitarity and the measure employed for the inner product. Normalizable superpositions can be built, including solitons, ``light-rays'' and coherent/squeezed states saturating a Heisenberg uncertainty relation between constants and their times. A healthy classical limit is obtained for factorizable coherent states, both in monofluid and multifluid situations. For the latter, we show how to deal with transition regions, where one is passing on the baton from one time to another, and investigate the fate of the subdominant clock. For this purpose minisuperspace is best seen as a dispersive medium, with packets moving with a group speed distinct from the phase speed. We show that the motion of the packets' peaks reproduces the classical limit even during the transition periods, and for subdominant clocks once the transition is over. Deviations from the coherent/semiclassical limit are expected in these cases, however. The fact that we have recently transitioned from a decelerating to an accelerating Universe renders this proposal potentially testable, as explored elsewhere.