Eternal Inflation and Eternal Inflation and Sinks in the LandscapeSinks in the Landscape

Eternal Inflation and Eternal Inflation and Sinks in the LandscapeSinks in the Landscape

Andrei Linde

Andrei Linde

New New InflationInflationNew New InflationInflation

V

ChaoticChaotic InflationInflation ChaoticChaotic InflationInflation

Eternal InflationEternal Inflation

Hybrid Hybrid InflationInflation Hybrid Hybrid InflationInflation

Inflation in string theoryInflation in string theory Inflation in string theoryInflation in string theory

KKLMMT brane-anti-brane inflation

Racetrack modular inflation

D3/D7 brane inflation

DBI inflation (non-minimal kinetic terms)

Initial conditions for Initial conditions for inflation:inflation:

Initial conditions for Initial conditions for inflation:inflation:

In the simplest chaotic inflation model, eternal inflation begins at the Planck density, if the potential energy is greater than the kinetic and gradient energy in a smallest possible domain of a Planckian size.

In the models where inflation is possible only at a small energy density (new inflation, hybrid inflation) the probability of inflation is not suppressed if the universe is flat or open but compact, e.g. like a torus.

A.L. 1986A.L. 1986

Zeldovich and Starobinsky 1984; Zeldovich and Starobinsky 1984; A.L. 2004A.L. 2004

Recent probability measure proposed by Gibbons and Turok disfavors the probability of inflation. It requires that at the Planck time the potential energy density must be 12 orders of magnitude smaller than the kinetic energy. I am unaware of any physical mechanism that would enforce this requirement.

A general problem with any measure on a set of classical trajectories is that it cannot describe the transition from the initial quantum regime at super-Planckian densities, so it can describe anything but… initial conditions.

In the string landscape scenario, with many dS vacua stabilized by KKLT mechanism, one either directly rolls down to the state where life of our type is impossible (AdS, 10D Minkowski space), or enters the state of eternal inflation. This essentially eliminates the problem of initial conditions, replacing it by the problem of the probability measure for eternal inflation.

Let 10Let 101000 1000 flowers flowers blossomblossom Let 10Let 101000 1000 flowers flowers blossomblossom

< < 00< < 00

= = 00= = 00

> 0> 0> 0> 0

Predictions of Predictions of Inflation:Inflation:

Predictions of Predictions of Inflation:Inflation:

1) The universe should be homogeneous, isotropic and flat, = 1 + O(10-4) [

Observations: the universe is

homogeneous, isotropic and flat, = 1 +

O(10-2)

• Inflationary perturbations should be gaussian and adiabatic, with flat spectrum, ns = 1+ O(10

-1)

Observations: perturbations are gaussian and adiabatic, with flat spectrum, ns = 1 + O(10

-2)

Any Any alternatives?alternatives? Any Any alternatives?alternatives?

1) Pre-big bang and 1) Pre-big bang and ekpyrotic/cyclic scenarioekpyrotic/cyclic scenario Lots of problems. It is difficult to Lots of problems. It is difficult to

say what these models actually predict say what these models actually predict since they involve a transition through since they involve a transition through the singularity. the singularity.

2) String gas cosmology2) String gas cosmology

Does not solve flatness problem, but Does not solve flatness problem, but was claimed to produce perturbations with was claimed to produce perturbations with n = 1n = 1. .

Nayeri, Vafa, Brandenberger, Patil, 2005-2006Nayeri, Vafa, Brandenberger, Patil, 2005-2006

A more detailed investigation A more detailed investigation demonstrated that the perturbations demonstrated that the perturbations produced in this model have spectrum with produced in this model have spectrum with n = 5n = 5, which is ruled out by observations., which is ruled out by observations. Kaloper, Kofman, A.L. and Mukhanov, Kaloper, Kofman, A.L. and Mukhanov, hep-th/0608200

Brandenberger et al, Brandenberger et al, hep-th/0608186

Inflationary MultiverseInflationary Multiverse Inflationary MultiverseInflationary MultiverseFor a long time, people believed in the cosmological principle, which asserted that the universe is everywhere the same.This principle is no longer required. Inflationary universe may consist of many parts with different properties depending on the local values of the scalar fields, compactifications, etc.

Example: SUSY landscapeExample: SUSY landscape Example: SUSY landscapeExample: SUSY landscape

V

SU(5) SU(3)xSU(2)xU(1)

SU(4)xU(1)

Weinberg 1982: Supersymmetry forbids tunneling from SU(5) to SU(3)xSU(2)XU(1). This implied that we cannot break SU(5) symmetry. A.L. 1983: Inflation solves this problem. Inflationary fluctuations bring us to each of the three minima. Inflation make each of the parts of the universe exponentially big. We can live only in the SU(3)xSU(2)xU(1) minimum.

Supersymmetric SU(5)

Inflation and Cosmological Inflation and Cosmological ConstantConstant

Inflation and Cosmological Inflation and Cosmological ConstantConstant

1) Anthropic solutions of the CC problem using inflation and fluxes of antisymmetric tensor fields (A.L. 1984), multiplicity of KK vacua (Sakharov 1984), and slowly evolving scalar field (Banks 1984, A.L. 1986). All of us considered it obvious that we cannot live in the universe with

2) Derivation of the anthropic constraint

Weinberg 1987; Martel, Shapiro, Weinberg 1997; Vilenkin, Garriga 1999, 2001, 2005

Three crucial steps in finding the anthropic solution of the CC problem:

Inflation and Cosmological Inflation and Cosmological ConstantConstant

Inflation and Cosmological Inflation and Cosmological ConstantConstant

3) String theory landscape

Multiplicity of (unstable) vacua:Lerche, Lust and Schellekens 1987: 101500 vacuum states

Bousso, Polchinski 2000; Feng, March-Russell, Sethi, Wilczek 2000

Vacuum stabilization:KKLT 2003, Susskind 2003, Douglas 2003,…

perhaps 101000 metastable dS vacuum states - still counting…

Latest anthropic Latest anthropic

constraints on constraints on Latest anthropic Latest anthropic

constraints on constraints on Aguirre, Rees, Tegmark, and Wilczek, astro-ph/0511774

observed observed valuevalue

Dark EnergyDark Energy (Cosmological (Cosmological Constant) is about 73% of the Constant) is about 73% of the cosmic piecosmic pie

Dark EnergyDark Energy (Cosmological (Cosmological Constant) is about 73% of the Constant) is about 73% of the cosmic piecosmic pie

What’s about What’s about Dark MatterDark Matter, , another 23% of the pie? Why another 23% of the pie? Why there is 5 times more dark there is 5 times more dark matter than ordinary matter?matter than ordinary matter?

What’s about What’s about Dark MatterDark Matter, , another 23% of the pie? Why another 23% of the pie? Why there is 5 times more dark there is 5 times more dark matter than ordinary matter?matter than ordinary matter?

Example:Example: Dark matter in the Dark matter in the axion fieldaxion fieldExample:Example: Dark matter in the Dark matter in the axion fieldaxion fieldStandard lore: If the axion mass is smaller than 10-5 eV, the amount of dark matter in the axion field contradicts observations, for a typical initial value of the axion field.Anthropic argument: Due to inflationary fluctuations, the amount of the axion dark matter is a CONTINUOUS RANDOM PARAMETER. We can live only in those parts of the universe where the initial value of the axion field was sufficiently small (A.L. 1988). Recently this possibility was analyzed by Aguirre, Rees, Tegmark, and Wilczek.

Latest anthropic constraints on Latest anthropic constraints on

Dark MatterDark Matter Latest anthropic constraints on Latest anthropic constraints on

Dark MatterDark MatterAguirre, Rees, Tegmark, and Wilczek, astro-ph/0511774

The situation with Dark Matter is even better than The situation with Dark Matter is even better than with the CC !with the CC !

observed observed valuevalue

Two types of Two types of questions:questions: Two types of Two types of questions:questions:What is a What is a typical historytypical history of a of a given point in an eternally given point in an eternally inflating universe? inflating universe? (That is what (That is what we studied so far.)we studied so far.)

What is a What is a typical populationtypical population ? ? (Volume-weighted distributions.)(Volume-weighted distributions.)

Let us apply the same logic Let us apply the same logic to inflationto inflationLet us apply the same logic Let us apply the same logic to inflationto inflation

Example:Example: Example:Example:

1) What is a typical age when any 1) What is a typical age when any particular scientist makes his best particular scientist makes his best discoveries?discoveries?

2) What is the average age of 2) What is the average age of scientists making greatest scientists making greatest discoveries?discoveries?

Both questions make sense, but the answers are Both questions make sense, but the answers are significantly different because of the significantly different because of the exponentially growing population of scientists.exponentially growing population of scientists.

Chaotic inflation requires > Mp. This condition can be easily satisfied in the scalar field theory plus gravity, but it is difficult to construct inflationary models with > Mp in supergravity and string theory. It took almost 20 years to construct a natural version of chaotic inflation in SUGRA. In string theory we started doing it only 3 years ago, with limited success.Let us imagine that just one out of 101000 of string theory vacua allows existence of the field ~ 100, in units of Mp. The volume produced during chaotic

inflation in this universe is proportional to e ~ 1010000 .

Therefore a part of the universe in such an improbable and “unnatural” state will have more volume (and contain more observers like us) than all other string theory vacua combined.

HOWEVER…HOWEVER…HOWEVER…HOWEVER…

What will change if we study What will change if we study physical volume?physical volume?What will change if we study What will change if we study physical volume?physical volume?

Consider a more complicated Consider a more complicated

potential :potential :

Consider a more complicated Consider a more complicated

potential :potential :

Naively, one could expect that each coefficient in

this sum is O(1). However, |V0| < 10-120, otherwise

we would not be around. A.L. 1984, S. Weinberg 1987

For a quadratic potential, one should have m ~ 10-5 to account for the smallness of the amplitude of

density perturbations H~ 10-5

Is there any reason for all other parameters to be small? Specifically, we must have

A simple argument:A simple argument:A simple argument:A simple argument:Suppose that the upper bound on the inflaton field is given by the condition that the potential energy is smaller than Planckian, V < 1. In addition, the effective gravitational constant should not blow up. In this case

In these models the total growth of volume of the universe during inflation (ignoring eternal inflation, which will not affect the final conclusion) is

For a purely quadratic model, the volume is proportional to

But for the theory the volume is much smaller:

But in this case at the end of inflation, when

one has

The greatest growth by a factor of occurs for

Thus, if we can have a choice of inflationary parameters (e.g. in string landscape scenario), then the simplest chaotic inflation scenario is the best. (See also the talk by Kachru.)

This may explain why chaotic inflation is so simple: A power-law fine-tuning of the parameters gives us an exponential growth of volume, which is maximal for a purely quadratic potential

Probabilities in the Probabilities in the LandscapeLandscapeProbabilities in the Probabilities in the LandscapeLandscapeWe must find all possible vacua We must find all possible vacua (statistics), and all possible (statistics), and all possible continuous parameters (out-of-continuous parameters (out-of-equilibrium cosmological dynamics).equilibrium cosmological dynamics).

Douglas Douglas 20032003We must also find a way to compare We must also find a way to compare the probability to live in each of the probability to live in each of these states.these states.

A.Linde, D. Linde, Mezhlumian, Bellido A.Linde, D. Linde, Mezhlumian, Bellido 1994; Vilenkin 1995; Garriga, 1994; Vilenkin 1995; Garriga, Schwarz-Perlov, Vilenkin, Winitzki, Schwarz-Perlov, Vilenkin, Winitzki, 2005 2005

Example:Example: Two dS vacuaTwo dS vacuaExample:Example: Two dS vacuaTwo dS vacua

is dS is dS entropyentropy

This isThis is the square of the Hartle-Hawking wave the square of the Hartle-Hawking wave functionfunction, which tells that the fraction of the , which tells that the fraction of the comoving volume of the universe with the comoving volume of the universe with the cosmological constant Vcosmological constant Vi i = = is proportional tois proportional to

PP00

PP11

PPii is the probability is the probability

to find a given point to find a given point in the vacuum in the vacuum dSdSii

In this context the HH wave function describes In this context the HH wave function describes the the ground state of the universeground state of the universe, it has no relation to , it has no relation to creation of the universe “from nothing”, and it does creation of the universe “from nothing”, and it does not require any modifications recently discussed in not require any modifications recently discussed in the literature.the literature.

This does This does notnot mean that the probability to mean that the probability to live in the state with a cosmological live in the state with a cosmological constant constant is given byis given by . .Life appears soon after Life appears soon after the new bubble the new bubble formation, which is formation, which is proportional to the proportional to the probability current. And probability current. And the currents in both the currents in both directions are equal to directions are equal to each other, each other, independently of the independently of the vacuum energy:vacuum energy:

PP00

PP11

PPii is the probability is the probability

to find a given point to find a given point in the vacuum in the vacuum dSdSii

That is why one may expect a flat prior That is why one may expect a flat prior probability distribution to live in the probability distribution to live in the bubbles with different cosmological bubbles with different cosmological constants.constants.But in string theory all vacua are But in string theory all vacua are metastable. What are the implications of metastable. What are the implications of this metastability?this metastability?

Metastable vacua in string theoryMetastable vacua in string theory(string theory landscape)(string theory landscape)

Metastable vacua in string theoryMetastable vacua in string theory(string theory landscape)(string theory landscape)

Based on the KKLT scenarioBased on the KKLT scenarioBased on the KKLT scenarioBased on the KKLT scenario

AdS minimumAdS minimum Metastable dS minimumMetastable dS minimum

Kachru, Kallosh, A.L., Trivedi 2003Kachru, Kallosh, A.L., Trivedi 2003

1) Start with a theory with the runaway potential

2) Bend this potential down due to (nonperturbative) quantum effects

3) Uplift the minimum to the state with positive vacuum energy by adding a positive energy of an anti-D3 brane in warped Calabi-Yau space

100 150 200 250 300 350 400s

-2

-1.5

-1

-0.5

0.5V

100 150 200 250 300 350 400s

0.2

0.4

0.6

0.8

1

1.2

V

KKLT potential always has a Minkowski KKLT potential always has a Minkowski minimum minimum (Dine-Seiberg vacuum)(Dine-Seiberg vacuum)

Now we will study decay from dS to the Now we will study decay from dS to the collapsing vacua with negative vacuum collapsing vacua with negative vacuum energy.energy.

Probability of Probability of tunneling from dS to tunneling from dS to Minkowski space Minkowski space typically is somewhat typically is somewhat greater than greater than

ee-S-S ~ e ~ e--1/1/ ~ e ~ e--

1010120120

After the tunneling, the After the tunneling, the field does not jump back field does not jump back - no recycling.- no recycling.

Tunneling from an uplifted supersymmetric AdS vacuum

Tunneling to the Tunneling to the sinksink Tunneling to the Tunneling to the sinksinkFor the tunneling from an uplifted AdS For the tunneling from an uplifted AdS vacuum, the final result depends not on vacuum, the final result depends not on VVdSdS but on but on |V|VAdSAdS|| prior to the uplifting. In prior to the uplifting. In the class of the KKLT models that we the class of the KKLT models that we explored, explored, |V|VAdSAdS|| is related to SUSY breaking is related to SUSY breaking (to the square of the gravitino mass) after (to the square of the gravitino mass) after the AdS uplifting, and is of the orderthe AdS uplifting, and is of the order

In the simplest SUSY models, the rate In the simplest SUSY models, the rate of a decay to a sink is eof a decay to a sink is e1010120120

times times greater than the probability to jump greater than the probability to jump from our vacuum to a higher dS from our vacuum to a higher dS vacuum.vacuum.

Ceresole, Dall’Agata, Giryavets, Kallosh, A.L., hep-th/0605266

Two dS vacua and AdS Two dS vacua and AdS sinksinkTwo dS vacua and AdS Two dS vacua and AdS sinksink Parts of dS space Parts of dS space

tunneling to space tunneling to space with negative V with negative V rapidly collapse and rapidly collapse and drop out of drop out of equilibriumequilibrium (one-way (one-way road to hell). road to hell). Therefore instead of Therefore instead of the detailed balance the detailed balance equations, one has equations, one has flow equations:flow equations:

Ceresole, Dall’Agata, Giryavets, Kallosh, A.L., hep-th/0605266

PP00

PP11

Narrow and Wide Narrow and Wide Sinks Sinks Narrow and Wide Narrow and Wide Sinks Sinks

If the decay to the sink is slower than the decay of If the decay to the sink is slower than the decay of the upper dS vacuum to the lower dS vacuum, then the the upper dS vacuum to the lower dS vacuum, then the probability distribution is given by the Hartle-probability distribution is given by the Hartle-Hawking expression, despite the vacuum instability and Hawking expression, despite the vacuum instability and the general probability flow down. the general probability flow down. In this case the In this case the probability currents between different dS states probability currents between different dS states remain equal, which suggests flat prior probability remain equal, which suggests flat prior probability distribution for the cosmological constant.distribution for the cosmological constant.

On the other hand, if the decay to the sink is very On the other hand, if the decay to the sink is very fast, one will have an inverted probability fast, one will have an inverted probability distribution, and the CC will take its smallest value.distribution, and the CC will take its smallest value.

In the string landscape scenario In the string landscape scenario we we do not study the ground state of the do not study the ground state of the universeuniverse, as we did before. Instead , as we did before. Instead of that, of that, we study the universe with we study the universe with many holes in the groundmany holes in the ground..

Conclusion:Conclusion: Conclusion:Conclusion:

Instead of studying Instead of studying static static probabilitiesprobabilities, like in a pond with , like in a pond with still water, we study still water, we study probability probability currentscurrents, like in a river dividing , like in a river dividing into many streams. into many streams. In other words, in addition to In other words, in addition to exploring exploring vacuum statisticsvacuum statistics, we also , we also explore explore vacuum dynamicsvacuum dynamics, including , including irreversible vacuum decayirreversible vacuum decay and the and the growth of the volumegrowth of the volume of different of different parts of the universe.parts of the universe.

The main conclusion:

It is unusual, it is It is unusual, it is complicated, it has many complicated, it has many

unsolved problems, one may unsolved problems, one may like it or hate it, but we like it or hate it, but we must learn how to live in a must learn how to live in a world where many different world where many different possibilities are availablepossibilities are available

It is unusual, it is It is unusual, it is complicated, it has many complicated, it has many

unsolved problems, one may unsolved problems, one may like it or hate it, but we like it or hate it, but we must learn how to live in a must learn how to live in a world where many different world where many different possibilities are availablepossibilities are available

IT IS IT IS SCIENCESCIENCEIT IS IT IS SCIENCESCIENCE