Physics Letters B 669 (2008) 197–200

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Physics Letters B

www.elsevier.com/locate/physletb

Is our universe decaying at an astronomical rate?

Don N. Page

Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2G7

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 August 2008Accepted 18 August 2008Available online 27 August 2008Editor: M. Cvetic

PACS:98.80.Qc04.60.-m95.30.Sf98.80.-k

Unless our universe is decaying at an astronomical rate (i.e., on the present cosmological timescaleof Gigayears, rather than on the quantum recurrence timescale of googolplexes), it would apparentlyproduce an infinite number of observers per comoving volume by thermal or vacuum fluctuations(Boltzmann brains). If the number of ordinary observers per comoving volume is finite, this scenarioseems to imply zero likelihood for us to be ordinary observers and minuscule likelihoods for ouractual observations. Hence, our observations suggest that this scenario is incorrect and that perhaps ouruniverse is decaying at an astronomical rate.

© 2008 Elsevier B.V. All rights reserved.

“The most incomprehensible thing about the world is that itis comprehensible”, according to Einstein. This mystery has both aphilosophical level1 and a scientific level. The scientific level of themystery is the question of how observers within the universe haveordered observations and comprehensions of the universe.

If observers within the universe were sufficiently dominated bythose that are thermal or vacuum fluctuations (Boltzmann brains[1–3] or BBs), rather than by ordinary observers (OOs) that arisefrom nonequilibrium processes like Darwinian evolution by natu-ral selection, then the probability would be very near unity that arandom observer would be such a BB. However, the observationsand thoughts of such observers would be very unlikely to have thedegree of order we experience. Therefore, if our observations canbe interpreted to be those of random observers, our ordered obser-vations would be statistical evidence against any theory in whichBBs greatly dominate. A theory in which the universe lasts too longafter a finite period of OOs is in danger of being such a theory thatis statistically inconsistent with observation.

This argument, a cosmic variant of the doomsday argument [4],uses a version of the Copernican Principle or Vilenkin’s Principle ofMediocrity (PM) [5], such as Bostrom’s [6] Strong Self-Sampling As-sumption (SSSA): “One should reason as if one’s present observer-moment were a random sample from the set of all observer-moments in its reference class”. This is similar to how I mighttoday state my Conditional Aesthemic Principle (CAP) [7]: “Unlessone has compelling contrary evidence, one should reason as ifone’s conscious perception were a random sample from the set of

E-mail address: don@phys.ualberta.ca.1 A theistic explanation is that the universe was created by an omniscient God

who comprehends fully and who created humans in His image to comprehend par-tially.

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.08.039

all conscious perceptions”. I would argue that the reference classof all observer-moments (which I would call conscious perceptions,each being all that one is consciously aware of at once) should bethe universal class of all observer-moments, but for the presentLetter it is sufficient to take any reasonable restriction of the refer-ence class.

For example, consider the reference class of scientists observingthe galaxy–galaxy correlation function (GGCF).2 This would almostcertainly be only a very tiny fraction of all observer-moments forBBs and presumably a much larger (but still small) fraction of allobserver-moments for OOs. However, if the BBs sufficiently dom-inate, most scientists observing the GGCF would be BBs ratherthan OOs. Then within this reference class, we could compare thelikelihoods that various theories give the observed GGCF. For BBtheories that predict that almost all observer-moments within thisreference class are BBs, we would expect that almost all of the“observed” GGCFs would be highly disordered, so the likelihood ofour observed GGCF would be very much smaller than in OO the-ories in which most observer-moments within the same referenceclass are OOs. Therefore, unless we take the prior probability forBB theories to be extremely near unity, our observed GGCF wouldbe strong statistical evidence that the posterior probability for BBtheories should be taken to be very small.

In simple terms, our observations of order (e.g., a comprehen-sible world) statistically rule out BB theories in which observersformed by thermal or vacuum fluctuations greatly dominate overordinary observers.

If we have a theory for a finite-sized universe that has ordinaryobservers for only a finite period of time (e.g., during the lifetimeof stars and nearby planets where the ordinary observers evolve),

2 I am grateful to James Hartle for this example.

198 D.N. Page / Physics Letters B 669 (2008) 197–200

each of which makes only a finite number of observations (per-haps mostly ordered), then the universe would have only a finitenumber of ordinary observers with their largely ordered observa-tions. On the other hand, if such a theory predicts that the universelasts for an infinite amount of time, then one would expect fromvacuum fluctuations an infinite number of observers (almost allextremely short-lived, with very little ordered memory) and obser-vations (almost entirely with minuscule order or comprehension).Such a BB theory would give a much smaller likelihood of ouractual observations than an OO theory (in which BBs do not dom-inate over OOs) and so would be discredited.

Therefore, a good theory for a finite-sized universe apparentlyshould also predict that it have a finite lifetime. (For example, thiswas a property of the k = +1 Friedmann–Robertson–Walker dustmodel universes.)

For an infinite universe (infinite spatial volume in one con-nected region, or an infinite number of connected regions in asuitable multiverse picture of the universe [8]), which is predictedto be produced by eternal inflation [9–11], the argument is not soclear. In this case one could get an infinite number of both ordi-nary observations and disordered observations, and then there maybe different ways of taking the ratio to give the likelihoods of thepossible observations within each reference class [12–20].

For example, one may [13] define the probability P j for an ob-servation to be the product P j = p j f j of the probability p j for aparticular pocket universe and of the probability f j of the obser-vation within that pocket. Here I shall assume that f j is regulatedby taking the ratio of different kinds of observations within a fixedcomoving volume of the pocket universe. This would give the rightanswer for any finite bubble universe, no matter how large, but ofcourse it is an untested ansatz to apply this procedure to an openor infinite bubble universe.

With this assumption, the prospect of BB production then leadsone to conclude that any model universe should not last forever ifit has only a finite time period where ordinary observers dominate[21], though it must be noted that other regularization schemesdo not lead to this conclusion [13,18,19]. However, here we shallassume observations within a given pocket universe are regulatedby taking a finite comoving volume.

The next question is what limits on the lifetime can be deducedfrom this argument. In [21] it was implicitly assumed that the uni-verse lasted for some definite time t and then ended. Then therequirement was that the number of vacuum fluctuation observa-tions per comoving volume during that time not greatly exceed thenumber of ordered observations during the finite time that ordi-nary observers exist. For any power-law expansion with exponentof order unity, and using a conservative action of 1050 for a 1 kgbrain to last 0.1 seconds, I predicted [21] that the universe wouldnot last past t ∼ e1050

years, and for an universe that continues togrow exponentially with a doubling time of the order of 10 Gyr, Ipredicted that the universe would not last past about 1060 years.If instead one used what I now believe is a more realistic actionof 1042 for a 1 kg brain to be separated by 30 cm from the corre-sponding antimatter [22–24], then the corresponding times wouldbe t ∼ e1042

years and 1052 years respectively.Here I wish to emphasize that the expected lifetime should be

much shorter if the universe is expanding exponentially and justhas a certain decay rate for tunneling into oblivion. Then the de-cay rate should be sufficient to prevent the expectation value ofthe surviving 4-volume, per comoving 3-volume, from divergingand leading to an infinite expectation value of vacuum fluctuationobservations per comoving 3-volume. Since this minimum decayrate is set by the asymptotic cosmological expansion rate of theuniverse, it may be said to be astronomical (huge on the scale ofthe quantum recurrence time [25]).

Suppose [23] that the decay of the universe proceeds at therate, per 4-volume, of A for the nucleation of a small bubble thatthen expands at practically the speed of light, destroying every-thing within the causal future of the bubble nucleation event. Thenif one takes some event p within the background spacetime, theprobability that the spacetime would have survived to that eventis P (p) = e−AV 4(p) , where V 4(p) is the spacetime 4-volume tothe past of the event p in the background spacetime. Then therequirement that there not be an infinite expectation value of vac-uum fluctuation observations within a finite comoving 3-volumeis the requirement that the 4-volume within the comoving region,weighted by the survival probability P (p) for each point, be fi-nite rather than infinite. For an asymptotically de Sitter backgroundspacetime with cosmological constant Λ, the expectation value ofthe 4-volume of the surviving spacetime is finite if and only if [23]

A > Amin = 9

4πH4

Λ = Λ2

4π� (20 Gyr)−4 = e−562.5, (1)

using H0 = 72 ± 8 km s−1 Mpc−1 from the Hubble Space Telescopekey project [26] and ΩΛ = 0.72 ± 0.04 from the third-year WMAPresults of [27] to get HΛ = H0

√ΩΛ .

For a universe that is spatially flat and has its energy den-sity dominated by the cosmological constant and by nonrelativisticmatter, as ours now seems to be, its k = 0 FRW metric may bewritten as

ds2 = T 2[−dτ 2 + (sinh4/3 τ

)(dr2 + r2 dΩ2)], (2)

where T = 2/(3HΛ). This gives a survival probability

P (τ ) = exp

[− 16

27

A

Amin

τ∫0

dx sinh2 x

( τ∫x

dy

sinh2/3 y

)3]. (3)

With the present earth population of nearly 7 billion, this wouldgive a minimal expected death rate of about 7 persons per cen-tury. (Of course, it could not be 7 persons in one century, but all7 billion with a probability of about one in a billion per century.)It also gives an upper limit on the present half-life of our universeof [23] 19 Gyr.

Although there are no observations that directly rule out thesuggestion here that our universe is decaying at an astronomicalrate (more likely than not to decay within 20 billion years), thissuggestion is surprising and would itself result in various myster-ies. For example, it would leave it unexplained why the decay rateA is greater than Amin ≈ (20 Gyr)−4 and yet not so great to makeit highly improbable that our universe has lasted as long as it has.There might be a factor of 1000 window for the allowed A, butif, for example, the decay rate is given by Eq. (5.20) of [28], thistranslates into only a 0.6% variation in the allowed value of thegravitino mass [23], which seems unnaturally restrictive. Further-more, even if the annihilation rate A can be within this range forour part of the multiverse, it would still leave it unexplained whyit is not less than Amin in some other part of the multiverse thatalso allows observers to be produced by vacuum fluctuations. If itwere less in any such part of the multiverse, then it would seemthat part would have an infinite number of vacuum fluctuation ob-servations that is in danger of swamping the ordered observationsin our part.

Because of these potential problems with the prediction sug-gested here (that the universe seems likely to decay within 20 bil-lion years), one might ask how this prediction could be circum-vented.

One obvious idea is that the current acceleration of the universeis not due to a cosmological constant that would last forever if theuniverse itself did not decay away. Perhaps the current accelerationis instead caused by the energy density of a scalar field that is

D.N. Page / Physics Letters B 669 (2008) 197–200 199

slowly rolling down a gentle slope of its potential [21,29]. However,this seems to raise its own issue of fine tuning, since although theobservership selection effect can perhaps explain the small valueof the potential, it does not seem to give any obvious explanationof why the slope should also be small, unless the scalar field isactually sitting at the bottom of a potential minimum (effectivelya cosmological constant).

Another possibility is that the normalization employed in thisLetter to get a finite number of ordinary observers, namely torestrict to a finite comoving volume, might not be the correct pro-cedure if our universe really has infinite spatial volume. This isindeed the conclusion of several authors [13,18,19].

For example, Bousso and Freivogel [18] have recently arguedthat the paradox does not arise in the local description [30] (unlessthe decay time is much, much longer) and is evidence against theglobal description of the multiverse. On the other hand, both Linde[19] and Vilenkin [13] have more recently given examples of howone may avoid the dreaded “invasion of Boltzmann brains” withintheir global description.

Linde’s solution [19] is analogous to the following situation [23].Consider imaginary humans who have a ‘youthful’ phase of 100years of life with frequent and mostly ordered observations, fol-lowed by a ‘senile’ phase of trillions of years of infrequent andmostly disordered observations. Assume that the trillions of yearsare sufficient to give many more total ‘senile’ disordered obser-vations than ‘youthful’ ordered observations for each human. Onemight think that most observations would then be disordered,so that someone’s having an ordered observation (which wouldthus be very unlikely under this scenario) could count as evidenceagainst the theory giving this scenario. However, if the populationgrowth rate of such humans is sufficiently high that at each timethe number of youthful humans and their ordered observationsoutnumbers the senile humans and their disordered observations,Linde’s solution is that at each time the probability is higher thatan observation would be ordered than that it would be disordered.

With the same aim, Vilenkin argues [13] that since BBs areequilibrium quantum fluctuation processes, they should be lumpedwith bubbles in being calculated by the factor p j in P j = p j f jrather than by the factor f j , which he suggests should be re-stricted to observations formed by nonequilibrium processes. Thenhe notes that when BBs are counted in p j , they are dominated bybio-friendly bubbles that are also counted in p j , since each suchopen bubble gives an infinite universe and an infinite number ofOOs. He concludes, “As a result, freak observers [BBs] get a vanish-ing relative weight.”

I agree that these solutions are possible ways to regulate the in-finity of observations that occur in a universe that expands forever,but they do not yet seem very natural [20]. The problem arisesfrom the fact that if the youthful humans or OOs are always atlate times to outnumber the senile humans or BBs, the populationof these fictitious humans, or the volume of the universe in theoriginal example, must continue growing forever, producing an in-finite number of both youthful and senile fictitious humans or ofboth OOs and BBs in cosmology. Then it is ambiguous how onetakes the ratio, which is the fundamental problem with trying tosolve the measure problem in theories with eternal inflation.

Of course, the fact that we have ordered observations and arealmost certainly not Boltzmann brains is strong evidence againstwhat I have here proposed as a natural way of using volumeweighting in the global viewpoint (unless the universe has a half-life less than 20 billion years [23]). So in comparison with theobservations, my proposal of regulating by comoving volume isdefinitely worse than the other prescriptions [13,18,19], unless theuniverse really is decaying at an astronomical rate.

So if the prediction suggested in this paper (that our universeseems likely to decay within 20 billion years) is wrong, it may be

part of our general lack of understanding of the measure in themultiverse. On the other hand, despite the fine-tuning problemsmentioned above, it is not obvious to me that it really is wrong,so one might want to take it seriously unless and until some otherreally natural way is found to avoid our ordered observations beingswamped by disordered observations from vacuum fluctuations.

One might ask what the observable effects would be of the de-cay of the universe, if ordinary observers like us could otherwisesurvive for times long in comparison with 20 billion years.

First of all, the destruction of the universe would likely occurby a very thin bubble wall traveling extremely close to the speedof light, so no one would be able to see it coming to dread the im-minent destruction. Furthermore, the destruction of all we know(our nearly flat spacetime, as well as all of its contents of parti-cles and fields) would happen so fast that there is not likely to benearly enough time for any signals of pain to reach our brains. Andno grieving survivors will be left behind. So in this way it wouldbe the most humanely possible execution.

Furthermore, in an Everett many-worlds version of quantumtheory (e.g., [7]), the universe will always persist in some frac-tion of the Everett worlds (better, in some measure), but it is justthat the fraction or measure will decrease asymptotically towardzero. This means that there is always some positive measure forobservers to survive until any arbitrarily late fixed time, so onecould never absolutely rule out a decaying universe by observa-tions at any finite time, though sufficiently late ordinary observerswould have good statistical grounds for doubting the astronomicaldecay rate suggested here.

In any case, the decrease in the measure of the universe that Iam suggesting here takes such a long time that it should not causeanyone to worry (except to try to find a solution to the huge sci-entific mystery of the measure for the string landscape or othermultiverse theory). However, it is interesting that the discovery ofthe cosmic acceleration [31,32] may not teach us that the universewill certainly last much longer than the possible finite lifetimes ofk = +1 matter-dominated FRW models previously considered, butit may instead have the implication that our universe is actuallydecaying astronomically faster than what was previously consid-ered.

Acknowledgements

I have benefited greatly from numerous email discussions, espe-cially with Andrei Linde and Alex Vilenkin on eternal inflation andwith Jim Hartle and Mark Srednicki on probability, but also withAndreas Albrecht, Anthony Aguirre, Nick Bostrom, Raphael Bousso,Bernard Carr, Sean Carroll, David Coule, William Lane Craig, GeorgeEllis, Gary Gibbons, Steve Giddings, J. Richard Gott, Pavel Krtouš,John Leslie, Don Marolf, Joe Polchinski, Martin Rees, Michael Salem,Glenn Starkman, Lenny Susskind, Neil Turok, and Bill Unruh. Finan-cial support was provided by the Natural Sciences and EngineeringResearch Council of Canada.

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