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On the other hand, if one took a many-worlds version of this quantum cosmology
theory, one would have to weight the ?worlds? (classical universes) by something
like the number of observers within them. One would expect this number to be
proportional to the volume of space at the time and other conditions when observers
can exist (other factors being equal) [10]. Therefore, in the many-worlds version one
would multiply the quantum measure given above for the ?worlds? (the ?bare?
probability distribution for universe configurations [11]) by something like V to get
the measure for observations (the ?observational? probability distribution [11]).
The result, V exp [(4.5/m2)/(lnm3V +1.5 ln lnm3V )], is then suciently rapidly
rising with large m3V that the part with large m3V , even if cut o at m3V of
order exp (4.5/m2), dominates over the exponentially large peak near the minimum
value of m3V . There is thus enough space for the no-boundary proposal to be
6 consistent with our observations of a large and expanding universe [11], but this
argument implicitly assumed a many-worlds version of the no-boundary proposal. A
similar assumption had been made earlier in the broader context of eternal stochastic
inflation [10]. In a single-history version, it seems plausible that the Hartle-Hawking
?no-boundary? quantum state may collapse with nearly unit probability to a classical
universe configuration that only lasts of the order of the Compton wavelength of the
inflaton scalar field, presumably far too short to be consistent with our observations.
This suggestive evidence against a single-history quantum cosmology theory is
of course not yet conclusive, since we do not yet know what the quantum state of
the universe is. Indeed, the ?tunneling? wavefunction proposals of Vilenkin, Linde,
and others [12] predict that the ?bare? quantum measure for small universes is
exponentially suppressed, rather than enhanced as discussed above for the Hartle-
Hawking ?no-boundary? proposal. The ?tunneling? proposals would thus apparently
be consistent with our observations whether one used a many-worlds version or a
single-history version. But the possibility is open that increased theoretical understanding
of quantum cosmology may lead us to favor a quantum theory, such as
the ?no-boundary? one may turn out to be when it is better understood, that is
consistent with our observations only in its many-worlds version rather than in its
single-history version.
Another tentative piece of observational evidence in favor of many-worlds quantum
theory is a comparison with the calculation [13] of likely values of the cosmological
constant. If the assumptions of that paper are correct, and if the ?subuniverses?
used there are the ?worlds? used here (?terms in the state vector? [13]) rather than
different spacetime regions within one ?world? (?local bangs? [13]), then our observational
evidence of the cosmological constant is consistent with many-worlds
quantum theory but not with single-history quantum theory. However, we need a
better understanding of physics to know whether the assumptions are correct (such
as the assumption that ?the cosmological constant takes a variety of values in different
?subuniverses? ? [13]).
Therefore, it may turn out, when we better understand fundamental physics and
quantum cosmology, that the observational evidence of the expansion of the universe
and of the cosmological constant may lead us to favor many-world quantum theories
over single-history quantum theories.

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