New scale factor measure

Abstract

The computation of probabilities in an eternally inflating universe requires a regulator or ''measure.'' The scale factor time measure truncates the Universe when a congruence of timelike geodesics has expanded by a fixed volume factor. This definition breaks down if the generating congruence is contracting-a serious limitation that excludes from consideration gravitationally bound regions such as our own. Here we propose a closely related regulator which is well defined in the entire spacetime. The new scale factor cutoff restricts to events with a scale factor below a given value. Since the scale factor vanishes at caustics and crunches, this cutoff always includes an infinite number of disconnected future regions. We show that this does not lead to divergences. The resulting measure combines desirable features of the old scale factor cutoff and of the light-cone time cutoff, while eliminating some of the disadvantages of each.

New local duals in eternal inflation

“…We show that, with particular initial conditions, it is equivalent to the New Scale Factor Cutoff [18]. This generalizes to arbitrary eternally inflating spacetimes the duality between the (long) fat geodesic and (old) scale factor time cutoff discovered in Ref.…”

“…If geodesics are terminated at the first conjugate point 9 , this procedure assigns a unique scale factor parameter to every event in the future of Σ 0 [18] A Short Fat Geodesic is a fat geodesic restricted to values of the scale factor parameter larger than that at Σ 0 , which we may choose to be zero. Thus, it consists of the portions of the fat geodesic along which neighboring geodesics are farther away than they are on Σ 0 .…”

“…3, we define the Short Fat Geodesic measure, and we show that it is the local dual to the recently proposed New Scale Factor Cutoff [18]. This generalizes to arbitrary eternally inflating spacetimes the known duality between the fat geodesic and the scale factor time cutoff [17], which originally applied only to everywhere-expanding multiverse regions.…”

“…Examples include the proper time cutoff [8][9][10][11][12], where T is the proper time along geodesics in a congruence; the scale factor time cutoff [9,[13][14][15][16][17][18], where T measures the local expansion of geodesics; and the light-cone time cutoff [19][20][21], where T is determined by the size of the future light-cone of an event.…”

“…The duality derived below is more general and applies to arbitrary eternally inflating universes. If the expanding phase is sufficiently long, the terminal point where η = 0 can be less than one Planck time from the caustic [18]. This is expected to be generic if the initial conditions are dominated by a long-lived metastable de Sitter vacuum.…”