Ｍ ultiverse and the Naturalness Problem Hikaru KAWAI 2012/ 12/ 4 at Osaka University

Contents 1. Low energy effective theory of quantum gravity / string theory 2.Wave function of the multiverse 3. Naturalness and Big Fix I will discuss the low energy effective action of quantum gravity and the possibility of solving the naturalness problem by using it.

1. Low energy effective theory of quantum gravity / string theory

Low energy effective theory of quantum gravity / string theory is obtained by integrating out the short distance physics. why not Because of the symmetry, it should be Usually, action is additive. Is that all? (Sugawara ‘82 )

An observer finds only a local field theory. The coupling constants are determined by the state or the history of the universe. Actually, in quantum gravity or matrix model, there are some mechanisms that the low energy effective action becomes : local operators By replacing each factor to its expectation value as is usually done in the mean field approximation, we have “multi local action”

Consider Euclidean path integral which involves the summation over topologies, Then there should be a wormhole-like configuration in which a thin tube connects two points on the universe. Here, the two points may belong to either the same universe or the different universe. If we see such configuration from the side of the large universe(s), it looks like two small punctures. But the effect of a small puncture is equivalent to an insert ion of a local operator. Multi local action from wormhole (1) Coleman ‘88

Summing over the number of wormholes, we have bifurcated wormholes ⇒ cubic terms, quartic terms, … Therefore, a wormhole contribute to the path integral can be expressed as Thus wormholes contribute to the path integral is Multi local action from wormhole (2)

The effective action becomes a factorized form By introducing the Laplace transform Coupling constants are not merely constant but to be integrated. we can express the path integral as Multi local action from wormhole (3)

Various possibilities for the emergence of space-time (1) A μ as the space-time coordinates mutually commuting A μ ⇒ space-time Multi local action from IIB matrix model (1) (2) A μ as non-commutative space-time non-commutative A μ ⇒ NC space-time (3) A μ as derivatives Asano, Tsuchiya and HK

embedding in ten matrices If the covariant derivative on a manifold acts on the regular representation field, its action can be decomposed into D scalars. : the Clebsh-Gordan coefficients r : regular representation : regular representation field on D-dim manifold M Hanada, HK, Kimura Multi local action from IIB matrix model (2)

We can calculate the low energy effective action by using the background field method, and we obtain Multi local action from IIB matrix model (3) in the Lorentzian space time We consider Lorentzian theory from now on.

the effective action of quantum gravity/string is given by More precisely, the path integral is given by the coupling constants are determined by the state We have seen Coupling constants are not merely constant but to be integrated.

the question What physics does the multi local action describe? Are some special values of the coupling constants favored? (Naturalness problem) In the following, as an ad hoc assumption, we postulate the probabilistic interpretation for the multiverse wave function.

2. Wave function of the multiverse

the multiverse Multiverse that includes indefinite number of large universes appears naturally in quantum gravity / string theory. matrix model ・ Each block represents a large universe. ・ quantum gravity Okada, HK

Wave Function of the multiverse The multiverse sate is a superposition of N-verses.

Probabilistic interpretation We assume a kind of probabilistic interpretation: the probability of finding N universes with size and the coupling constants represents

Probability distribution of is chosen in such a way that is maximized, irrespectively of

Wave function of a universe (1) We assume that the universe emerges with a small size ε with probability amplitude.

is a superposition of the universe with various age, + + +… gives the probability of finding a universe of age. Wave function of a universe (2)

infrared cutoff We introduce an infrared cutoff for the size of universes. ceases to exist bounces back or ∞ finite Wave function of a universe (3)

We have seen the coupling constants are chosen in such a way that the lifetime of the universe becomes maximum. What values of the coupling constants make the lifetime maximum? the question

3. Naturalness and Big Fix

Cosmological constant Λ ～ curvature ～ energy density (extremely small) What value of Λ maximizes the life time of the universe? For simplicity, we assume the S 3 topology. The cosmological constant in the far future is predicted to be very small.

The other couplings (Big Fix) (1) The exponent is divergent, and regulated by the IR cutoff : BIG FIX are determined in such a way that the entropy of the universe at the late stages is maximized. assuming all matters decay to radiation

The other couplings (Big Fix) (2) However, some of the couplings can be determined without knowing the detail of the cosmological evolution. Quantities like are almost determined by the low energy physics. ⇒ If the low energy physics and the cosmological evolution is understood, we can calculate theoretically, and all of the renormalized couplings are in principle determined by maximizing it. case 1. Symmetry example 1. It becomes important only after the QCD phase transition. 2. The effective action at QCD scale is invariant under ⇒ is minimum or maximum at at least locally. Nielsen, Ninomiya

The other couplings (Big Fix) (3) case 2. End point example Higgs coupling 1. Some couplings are bounded. 2. The entropy of the universe can be monotonic in them. ⇒ The coupling is fixed to the end point. A scenario for. Fix v h to the observed value and vary assuming the leptogenesis ⇒ sphaleron process ⇒ baryon number ⇒ radiation from baryon decay ⇒ Higgs mass is at stability bound, the stability bound. without SUSY

The other couplings (Big Fix) (4) non-trivial example QCD coupling or proton mass We assume that dark matters decay faster than protons,. If the curvature term balances with matter (protons) before the proton decay, the universe bounce back when the protons decay. The earlier the protons decay, the less C rad remains. C rad is maximized if the curvature term balances with the energy density when the protons decay.

The other couplings (Big Fix) (5) (cont’d) The curvature term balances with the energy density when the protons decay. ⇒ ⇒ in our universe Cf. ⇒ Reasonable?

4. Summary

Summary In the quantum gravity or string theory, the low energy effective action is not a simple local one but the multi-local one. The multiverse naturally appears, and it becomes a superposition of states with various values of the coupling constants. The coupling constants are fixed is such a way that the lifetime of the universe is maximized. For example the cosmological constant in the far future is predicted to be very small:. Some scenario other than the probabilistic interpretation. Comparison of different dimensional space-time? Generalization to the landscape? Future problems The Higgs mass is predicted at its lower bound.

Appendix A. Partition function

We consider the partition function of the multiverse:

the question How to evaluate the partition function for a single universe:

The path integral for a universe Question: Is there a natural choice for them? If the initial and final states are given, the path integral is well defined.

Initial state For the initial state, we assume that the universe emerges with a small size ε.

Final state: case 1 For the final state, we have two possibilities. finite The universe shrinks back. We assume the final state is The partition function

Final state: case 2 (1) ∞ The universe keeps expanding. It is not clear how to define the path integral for the universe: If we take, the path integral includes the history of the universe for the corresponding time. Therefore as a trial we take

Final state: case 2 (2) Then the partition function becomes If, the result does not depend on except for the phase which should be there in the classical limit: ad hoc assumption

Under this ad hoc assumption, we have the partition function for a universe ∞ finite Then the integration for the multiverse partition has an infinite peak at, which again indicates that the cosmological constant at the late stages of the universe vanishes.

Big Fix Then the multiverse partition function is given by BIG FIX are determined in such a way that the entropy at the late stages of the universe is maximized. For simplicity we assume the topology of the space and that all matters decay to radiation at the late stages.