Measure problem (cosmology)
The measure problem in cosmology concerns how to compute the ratios of universes of different types within a multiverse.
[1] Measures can be evaluated by whether they predict observed physical constants, as well as whether they avoid counterintuitive implications, such as the youngness paradox or Boltzmann brains.
[2] While dozens of measures have been proposed,[3]: 2 few physicists consider the problem to be solved.
[4] Infinite multiverse theories are becoming increasingly popular, but because they involve infinitely many instances of different types of universes, it is unclear how to compute the fractions of each type of universe.
[4] Alan Guth put it this way:[4] Sean M. Carroll offered another informal example:[1] Different procedures for computing the limit of this fraction yield wildly different answers.
[1] One way to illustrate how different regularization methods produce different answers is to calculate the limit of the fraction of sets of positive integers that are even.
Everyone agrees that the first sequence, ordered by increasing size of the integers, seems more natural.
Similarly, many physicists agree that the "proper-time cutoff measure" (below) seems the simplest and most natural method of regularization.
This measure has the advantage of being stationary in the sense that probabilities remain the same over time in the limit of large
This lopsidedness continues, until the most numerous observers resembling us are "Boltzmann babies" formed by improbable fluctuations in the hot, very early, Universe.
Therefore, physicists reject the simple proper-time cutoff as a failed hypothesis.
[3]: 1 This approach can be generalized to a family of measures in which a small region grows as
, which avoids the youngness paradox by not giving greater weight to regions that retain high energy density for long periods.
yields an "oldness paradox" in which most life is predicted to exist in cold, empty space as Boltzmann brains rather than as the evolved creatures with orderly experiences that we seem to be.
[7] This measure has also been shown to produce good agreement with observational values of the cosmological constant.
[8] The stationary measure proceeds from the observation that different processes achieve stationarity of
[3]: 2 For instance, different regions of the universe can be compared based on time since star formation began.
[3]: 3 Andrei Linde and coauthors have suggested that the stationary measure avoids both the youngness paradox and Boltzmann brains.
[2] However, the stationary measure predicts extreme (either very large or very small) values of the primordial density contrast
The causal diamond is the finite four-volume formed by intersecting the future light cone of an observer crossing the reheating hypersurface with the past light cone of the point where the observer has exited a given vacuum.
For such an observer, Bayes' theorem may appear to break down over this timescale due to anthropic selection effects; this hypothetical breakdown is sometimes called the "Guth–Vanchurin paradox".
One proposed resolution to the paradox is to posit a physical "end of time" that has a fifty percent chance of occurring in the next few billion years.
Another, overlapping, proposal is to posit that an observer no longer physically exists when it passes outside a given causal patch, similar to models where a particle is destroyed or ceases to exist when it falls through a black hole's event horizon.
[11][12] Guth and Vanchurin have pushed back on such "end of time" proposals, stating that while "(later) stages of my life will contribute (less) to multiversal averages" than earlier stages, this paradox need not be interpreted as a physical "end of time".
The literature proposes at least five possible resolutions:[13][14] Guth and Vanchurin hypothesize that standard probability theories might be incorrect, which would have counterintuitive consequences.
