38,688 research outputs found
Order-invariant measures on fixed causal sets
A causal set is a countably infinite poset in which every element is above
finitely many others; causal sets are exactly the posets that have a linear
extension with the order-type of the natural numbers -- we call such a linear
extension a {\em natural extension}. We study probability measures on the set
of natural extensions of a causal set, especially those measures having the
property of {\em order-invariance}: if we condition on the set of the bottom
elements of the natural extension, each possible ordering among these
elements is equally likely. We give sufficient conditions for the existence and
uniqueness of an order-invariant measure on the set of natural extensions of a
causal set.Comment: 25 pages; to appear in Combinatorics, Probability and Computin
The CMB and the measure of the multiverse
In the context of eternal inflation, cosmological predictions depend on the
choice of measure to regulate the diverging spacetime volume. The spectrum of
inflationary perturbations is no exception, as we demonstrate by comparing the
predictions of the fat geodesic and causal patch measures. To highlight the
effect of the measure---as opposed to any effects related to a possible
landscape of vacua---we take the cosmological model, including the model of
inflation, to be fixed. We also condition on the average CMB temperature
accompanying the measurement. Both measures predict a 1-point expectation value
for the gauge-invariant Newtonian potential, which takes the form of a
(scale-dependent) monopole, in addition to a related contribution to the
3-point correlation function, with the detailed form of these quantities
differing between the measures. However, for both measures both effects are
well within cosmic variance. Our results make clear the theoretical relevance
of the measure, and at the same time validate the standard inflationary
predictions in the context of eternal inflation.Comment: 28 pages; v2: reference added, some clarification
Causal Set Dynamics: A Toy Model
We construct a quantum measure on the power set of non-cyclic oriented graphs
of N points, drawing inspiration from 1-dimensional directed percolation.
Quantum interference patterns lead to properties which do not appear to have
any analogue in classical percolation. Most notably, instead of the single
phase transition of classical percolation, the quantum model displays two
distinct crossover points. Between these two points, spacetime questions such
as "does the network percolate" have no definite or probabilistic answer.Comment: 28 pages incl. 5 figure
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