Inflaton and time in the matter-gravity system

The emergence of time in the matter-gravity system is addressed within the context of the inflationary paradigm. A quantum minisuperspace-homogeneous minimally coupled inflaton system is studied with suitable initial conditions leading to inflation and the system is approximately solved in the limit for a large scale factor. Subsequently normal matter (either nonhomogeneous inflaton modes or lighter matter) is introduced as a perturbation and it is seen that its presence requires the coarse averaging of a gravitational wave function (which oscillates at trans-Planckian frequencies) having suitable initial conditions. Such a wave function, which is common for all types of normal matter, is associated with a “time density” in the sense that its modulus is related to the amount of time spent in a given interval (or the rate of flow of time). One is then finally led to an effective evolution equation (Schrödinger Schwinger-Tomonaga) for “normal” matter. An analogy with the emergence of a temperature in statistical mechanics is also pointed out.