Andrei Linde

1. Counting and measuring universes Andrei Linde

2. Contents: Why inflationary multiverse? Various measures Counting worlds and minds

3. Why multiverse ? It was proposed more than 25 years ago. Why so much interest NOW ?

4. Historically, the question was opposite: Why UNIverse? Uniformityof our world is explained by inflation: Exponential stretching of the new-born universe makes it almost exactly uniform. However, inflationary fluctuations eternally produce new parts of the universe with different properties. Inflationary universe becomes a multiverse

5. Eternal inflation and string theory landscape An enormously large number of possible types of compactification which exist e.g. in the theories of superstrings should be considered not as a difficulty but as a virtue of these theories, since it increases the probability of the existence of mini-universes in which life of our type may appear. A.L. 1986 Various versions of this idea appeared in the papers of Alex Vilenkin, A.L. and Andrei Sakharov since 1982.

6. Why did it take 20 years until people started taking this idea seriously? 1. Inflationary theoryreceived strong observational support 2. Acceleration of the universeand existence of dark energy (cosmological constant) was firmly established 3. String theorycould not explain these observational data. This was a creative crisis which was resolved in 2003 with finding the mechanism of vacuum stabilization in string theory (KKLT construction and other related mechanisms).

7. 4. Immediately after that, we learned that this mechanism allows vacuum to be in10500different states. This established the picture of inflationary multiverse consisting of infinitely many “universes” of 10500 types (string theory landscape). Lerche, Lust and Schellekens 1987, Bousso, Polchinski 2000, After KKLT stabilization: Douglas 2003, Susskind 2003 5. These developments provided the framework for solving the cosmological constant problem using anthropic principle A.L. 1984, CC as a function of fluxes A. Sakharov 1984, CC as a function of compactification S. Weinberg 1987, anthropic bound on CC Recent treatment in a series of papers by Weinberg, Vilenkin, Garriga, Bousso

8. Measure problem There are two basic ways to benefit from anthropic considerations: 1. Minimal: On the basis of already established facts find whether the new facts do not look too surprising, so one can concentrate on other problems until some manifest inconsistency is found. 2. Ambitious: Prove that we live in the best, most probable of all worlds. This program may or may not work. It requires establishing a probability measure in a universe having indefinitely large volume. We must learn how to compare infinities.

9. Examples of measures Volume-weighted: a) proper time cutoff (A.L., Mezhlumian, Bellido 1993, 1994, Vilenkin 1994) b) scale factor cutoff (A.L., Mezhlumian, Bellido 1993, 1994, Vilenkin 1994) c) pocket based (Vilenkin, Garriga and collaborators) d) stationary measure (A.L. 2006, A.L., Vanchurin and Winitzki 2008) Youngness paradox, ruled out Difficulties with Boltzmann brains Q-problem, BBs? Q-problem 2. Comoving (Starobinsky 1985, A.L. 1986) 3. Causal diamond or causal patch (Bousso 2006) Other measures by Garriga, Vilenkin, Susskind, Hartle… Difficulties with Boltzmann brains Difficulties with Boltzmann brains

10. Measures for non-eternalinflation The goal: To test and interpret various suggestions in the situations where one does not need to compare infinities. A toy model: a closed universe or a compact open or flat universe with a potential V short inflation long inflation collapse collapse

11. t Proper time cutoff. The number of stars is proportional to the volume, same as in eternal inflation t a a

12. t Scale factor cutoff at a given moment compares stars to nothing. If integrated over all times, it compares all stars to all stars; the result is proportional to the amount of inflation. But in the eternal inflation case, the result does notdepend on the amount of inflation, so there is no smooth matching between non-eternal and eternal inflation t a a

13. t Comoving and causal diamond measures count only a part of all stars. If inflation is non-eternal and short, the causal measures give results which depend on the amount of inflation. t Stationary measure compares stars to stars. The results are proportional to the amount of slow-roll inflation, both for eternal and for non-eternal inflation a

14. How many different universes are in the multiverse ? Garriga and Vilenkin 2001, A.L. and Vanchurin 2009 There are perhaps ~10500 vacua in string theory landscape. Douglas 2003 If these vacua appear as a result of bubble formation which is not followed by slow roll inflation, then each of these vacua is equally unimportant, because nobody can live there. Thus one could think that the number of possibilities is much smaller than 10500. However, in fact, the number of different universes which may emerge as a result of slow roll inflation is much greater than 10500.

15. The universe at the end of inflation consists of e3Nindependent domains of size H-1, in each of which the scalar field jumps either upwards or downwards. In this sense, each of the stringy vacua experiencing N e-folds of a slow rollinflation can produce different geometries. For N > 60 we get This is much greater than the number of vacua in the landscape.

16. The same result can be obtained from the expression for the entropy of inflationary perturbations obtained by Mukhanov et al in 1992. This result provides an independent meaning for the number N of e-folds of the slow-roll inflation and the number of different universes in terms of entropy of inflationary perturbations:

17. For the simplest models of eternal chaotic inflation, the total number of different locally-Friedmann universes is proportional to where N is the number of e-folds aftereternal inflation. For the simplest models of chaotic inflation with a polynomial potential, N ~S, where S is dS Gibbons-Hawking entropy of the universe at the end of chaotic inflation. In particular, for the simplest theory of a massive scalar field with a quadratic potential, This is much greater than the number of vacua in string theory. A.L., V. Vanchurin, to appear

18. However, an observer inside a vacuum with the cosmological constant can access only a small part of these universes, For this bound diverges until it reaches the eternal inflation bound discussed above. For observers living at a cosmological time t there is a related bound, even for vanishing : For the present value of the cosmological constant one has This is much smaller than the number following from eternal inflation, but it is still much greater than the number of vacua in string theory.

19. The number of different universes which could be registered by a local observer is much smaller Remember that the total Hamiltonian of the whole universe including an observer identically vanishes. Therefore, according to DeWitt 1967, the whole universe does not evolve in time. An unobserved universe is time-independent, i.e. dead. Time evolution is possible only for the “rest of the universe,” i.e. for the universe minus an observer. The rest of the universe has energy eigenstates equal to – En, where En are the energy eigenstates of an observer. An observer and the rest of the universe form a mutually consistent pair, and should be studied as such.

20. From the perspective of any local observer, the total number of different observable universes is bounded by Here m is the mass of an observer and R is its size. For a human observer with m ~ 100 kg and R ~ 1 meter one has Moreover, a typical human observer during his lifetime can receive about 1016 or 1017 bits of information. This means that a typical observer cannot receive information about more than different universes. It is still much greater than 10500.

21. The estimates given in this talk provide a dramatic increase of the probability that two identical copies of Alex Vilenkin will ever meet. The typical distance separating two identical (from the point of view of any human observer) copies of Vilenkin must be smaller than That is why we should take very seriously the results of the recent cosmological observations suggesting that this rare event did already happen.

22. Happy birthdays, Alexes !!!!