Inflaton and time in the matter-gravity system

Abstract

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 large scale factor. Subsequently normal matter (either non homogeneous 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.

The Born–Oppenheimer method, quantum gravity and matter

“…Let us first note that (23) is equal to (19). Furthermore, on comparing (24) and (26) and eliminating Ĥ k 0 , one obtains Eq.…”

“…Let us examine it in more detail [19]. We start from the quantum (WDW) Einstein equation for a flat FRW universe in the coordinate (a, φ) representation:…”

“…It is difficult to find a solution to Eq. (38) however following our previous work [19] we solve it by factorising…”

“…The above approach followed by BKK in [16] is what the authors also call a "BO scheme" because, on collecting the different powers of m, the total wave-function is splitted into a minisuperspace (homogeneous) part satisfying Eq. (19) and an inhomogeneous part of higher order in m.…”

“…Concerning this approach in Section 3 we shall limit ourselves to a brief description of it and observe that even if it has nothing in common, both technically and physically, with our first, traditional BO method, it does bear some resemblance with an alternative (non BO) approach which we have previously examined [19]. It consisted of the study of a quantum matter-gravity system containing a minimally coupled massive homogeneous scalar field which is known to lead to inflation.…”