The Cosmological Constant Problem and the Multiverse of

The Cosmological Constant Problemand the Multiverse of String Theory

Raphael Bousso

Berkeley Center for Theoretical PhysicsUniversity of California, Berkeley

PiTP, 3rd lecture, IAS, 29 July 2011

The (Old) Cosmological Constant Problem

Why the cosmological constant problem is hard

Recent observations

Of ducks and unicorns

The Landscape of String Theory

Landscape Statistics

Cosmology: Eternal inflation and the Multiverse

The Measure Problem

Einstein’s cosmological constant

The cosmological constant problem began its life as anambiguity in the general theory of relativity:

Rµν −12

Rgµν + Λgµν = 8πGTµν

Λ introduces a length scale into GR,

LΛ =

√3|Λ|

,

which is (roughly) the largest observable distance scale.

(Old) experimental constraints

Because the universe is large compared to the fundamentallength scale

LPlanck =

√G~c3 ≈ 1.6× 10−33cm .

it follows that |Λ| must be very small in fundamental units:

|Λ| . 10−121 .

So let’s just set Λ→ 0?

Quantum contributions to Λ

The vacuum of the Standard Model is highly nontrivial:I ConfinementI Symmetry breakingI Particles acquire masses by bumping into HiggsI . . .

The vacuum carries an energy density, ρvacuum.

Quantum contributions to Λ

In the Einstein equation, the vacuum energy density isindistinguishable from a cosmological constant. We can absorbit into Λ:

Λ = ΛEinstein + 8πGρvacuum .

Einstein could choose to set ΛEinstein → 0.But we cannot set ρvacuum = 0. It is determined by the StandardModel and its ultraviolet completion.

Magnitude of contributions to the vacuum energy

graviton

(a) (b)

I Vacuum fluctuations of each particle contribute(momentum cutoff)4 to Λ

I SUSY cutoff: → 10−64; Planck scale cutoff: → 1I Electroweak symmetry breaking lowers Λ by approximately

(200 GeV)4 ≈ 10−67

I Chiral symmetry breaking, . . .

The cosmological constant problem

I Each known contribution is much larger than 10−121.I Different contributions can cancel against each other or

against ΛEinstein.I But why would they do so to a precision better than

10−121?

Why is the vacuum energy so small?



Recent observations





The Measure Problem

Try solving it

Some ideas, and why they don’t work:

Short- or long-distance modifications of gravity

I Perhaps general relativity should be modified?I We can only modify GR on scales where it has not been

tested: below 1 mm and above astrophysical scales.I If vacuum energy were as large as expected, it would in

particular act on intermediate scales like the solar system.

Violating the equivalence principle

I We have tested GR using ordinary matter, like stars andplanets. Perhaps virtual particles are different? Perhapsthey don’t gravitate?

I But we know experimentally that they do!I Virtual particles contribute different fractions of the mass of

different materials (e.g., to the nuclear electrostatic energyof aluminum and platinum)

I If they did not gravitate, we would have detected thisdifference in tests of the equivalence principle (in thisexample, to precision 10−6)

Degravitating the vacuum

I Perhaps virtual particles gravitate in matter, but not in thevacuum?

I But physics is local.I What distinguishes the neighborhood of a nucleus from the

vacuum?I What about nonperturbative contributions, like scalar

potentials? Why is the energy of the broken vacuum zero?

Initial conditions

I Perhaps there are boundary conditions at the big bangenforcing Λ = 0?

I But this would be a disaster:I When the electroweak symmetry is broken, Λ would drop to−(200 GeV)4 and the universe would immediately crunch.

Gravitational attractor mechanisms

I Perhaps a dynamical process drove Λ to 0 in the earlyuniverse?

I Only gravity can measure Λ and select for the “right” value.I General relativity responds to the total stress tensorI But vacuum energy was negligible in the early universeI E.g. at nucleosynthesis, spacetime was being curved by

matter densities and pressures of order 10−86

I There was no way of measuring Λ to precision 10−121



Recent observations





The Measure Problem

Measuring the cosmological constant

I Supernovae as standard candles→ expansion is accelerating

I Precise spatial flatness (from CMB)→ critical density→ large nonclustering component

I Large Scale Structure: clustering slowing down→ expansion is accelerating

I . . .

is consistent withΛ ≈ 0.4× 10−121

and inconsistent with Λ = 0.

The cosmological constant problem

This result sharpens the cosmological constant problem:

Why is the energy of the vacuum so small, and why is itcomparable to the matter density in the present era?

I Favors theories that predict Λ comparable to the currentmatter density;

I Disfavors theories that would predict Λ = 0.



Recent observations





The Measure Problem

Calling it a duck

Perhaps Λ = 0, and darkenergy is a new form ofmatter that just happens toevolve very slowly(quintessence, . . .)?

“When I see a bird thatwalks like a duckand swims like a duckand quacks like a duck,I call that bird a duck.”

Calling it a duck

Perhaps Λ = 0, and darkenergy is a new form ofmatter that just happens toevolve very slowly(quintessence, . . .)?

“When I see a bird thatwalks like a duckand swims like a duckand quacks like a duck,I call that bird a duck.”

Why “dark energy” is vacuum energy

I Well-tested theories predict huge Λ, in conflict withobservation.

I There is no well-tested, widely accepted solution to thisproblem—in particular, none that predicts Λ = 0.

I It is unwise to interpret an experiment through the lens of abaseless theoretical speculation (such as the prejudicethat Λ = 0).

I If we cannot compute Λ, we should try to measure Λ.I “Dark energy” is

I indistinguishable from ΛI definitely distinct from any other known form of matter

I So it probably is Λ, and we have succeeded in measuringits value.

Not calling it a duck

Wouldn’t it be more exciting ifit was a unicorn?

−→

I Why is this unicorn wearing a duck suit?I Why have we never seen a unicorn without a duck suit?I What happened to the huge duck predicted by our theory?



−→I Why is this unicorn wearing a duck suit?

I Why have we never seen a unicorn without a duck suit?I What happened to the huge duck predicted by our theory?



−→I Why is this unicorn wearing a duck suit?I Why have we never seen a unicorn without a duck suit?

I What happened to the huge duck predicted by our theory?



−→I Why is this unicorn wearing a duck suit?I Why have we never seen a unicorn without a duck suit?I What happened to the huge duck predicted by our theory?

Dynamical dark energy

I Whether Λ is very small, or zero,either way we must explain why it is not huge

I Dynamical dark energy introduces additional complicationsI . . . which would make sense if we were trying to rescue a

compelling theory that predicts Λ = 0 . . .I . . . but we have no such theory.



I Dynamical dark energy introduces additional complications

I . . . which would make sense if we were trying to rescue acompelling theory that predicts Λ = 0 . . .

I . . . but we have no such theory.




compelling theory that predicts Λ = 0 . . .

I . . . but we have no such theory.




compelling theory that predicts Λ = 0 . . .I . . . but we have no such theory.



Recent observations





The Measure Problem

Branes and extra dimensions

Topology and combinatorics

RB & Polchinski (2000)I A six-dimensional manifold contains hundreds of

topological cycles, or “handles”.I Suppose each handle can hold 0 to 9 units of flux, and

there are 500 independent handlesI Then there will be 10500 different configurations.

One theory, many solutions

I String theory: Uniquetheory, no adjustableparameters, manymetastable solutions

I Combine D-branesand their associatedfluxes to tie up 6 extradimensions→

I Huge number ofdifferent choices

I . . . each with its ownlow energy physicsand vacuum energy

I Standard model: A fewadjustable parameters,many metastable solutions

I Combine many copies offundamental ingredients(electron, photon, quarks)→

I Huge number of distinctsolutions (condensedmatter)

I . . . each with its ownmaterial properties(conductivity, speed ofsound, specific weight, etc.)

One theory, many solutions

I String theory: Uniquetheory, no adjustableparameters, manymetastable solutions

I Combine D-branesand their associatedfluxes to tie up 6 extradimensions→

I Huge number ofdifferent choices

I . . . each with its ownlow energy physicsand vacuum energy

I Standard model: A fewadjustable parameters,many metastable solutions

I Combine many copies offundamental ingredients(electron, photon, quarks)→

I Huge number of distinctsolutions (condensedmatter)

I . . . each with its ownmaterial properties(conductivity, speed ofsound, specific weight, etc.)

Many ways to make empty space

Three challenges

To make predictions and testthe landscape of stringtheory, we face threechallenges:

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I Landscape statisticsI Cosmological dynamicsI Measure problem

The prediction of the cosmological constant is sensitive to allthree.



Recent observations





The Measure Problem

The spectrum of Λ

I In each vacuum, Λ receives manydifferent large contributions

I → random variable with valuesbetween about -1 and 1

I With 10500 vacua, Λ has a densespectrum with average spacing oforder 10−500

I About 10379 vacua with |Λ| ∼ 10−121

I But will those special vacua actuallyexist somewhere in the universe?

Λ

0

1

−1

The spectrum of Λ






−1

Λ

0

1

The spectrum of Λ






−1

Λ

0

1

The spectrum of Λ






Λ

0

1

−1

The spectrum of Λ






Λ

0

1

−1



Recent observations





The Measure Problem

Metastability and eternal inflation

I Fluxes can decay spontaneously (Schwinger process)I → landscape vacua are metastableI First order phase transitionI Bubble of new vacuum forms locally.

Metastability and eternal inflation

I New bubble expands to eat up the old vacuumI But for Λ > 0, the old vacuum expands even faster

Guth & Weinberg (1982)I So the old vacuum can decay again somewhere elseI → Eternal inflation

Eternal inflation populates the landscape

I The new vacuum also decays in all possible waysI and so on, as long as Λ > 0I Eventually all vacua will be produced as “pocket universes”I Each vacuum is produced an infinite number of timesI → Multiverse

Our place in the multiverse

I Eternal inflation makes sure that vacua with Λ� 1 arecosmologically produced

I But why do we find ourselves in such a special place in theMultiverse?

I Typical regions have Λ ∼ 1 and admit only structures ofPlanck size, with at most a few quantum states (accordingto the holographic principle). They do not containobservers.

I Because of cosmological horizons, such regions will not beobserved.

Our place in the multiverse

I Eternal inflation makes sure that vacua with Λ� 1 arecosmologically produced

I But why do we find ourselves in such a special place in theMultiverse?

I Typical regions have Λ ∼ 1 and admit only structures ofPlanck size, with at most a few quantum states (accordingto the holographic principle). They do not containobservers.

I Because of cosmological horizons, such regions will not beobserved.

Connecting with standard cosmology

The observable universe fits inside a single pocket:I Vacua can have exponentially long lifetimesI Each pocket is spatially infiniteI Because of cosmological horizons, typical observers see

just a patch of their own pocketI → Low energy physics (including Λ) appears fixed

Collisions with other pockets may be detectable in the CMB

Connecting with standard cosmology

I What we call big bang was actually the decay of our parentvacuum

I Neighboring vacua in the string landscape have vastlydifferent Λ

I → The decay of our parent vacuum released enoughenergy to allow for subsequent nucleosynthesis and otherfeatures of standard cosmology

The string multiverse is special

I This way of solving the cosmological constant problemdoes not work in all theories with many vacua

I In a multiverse arising from an (ad-hoc) one-dimensionalquantum field theory landscape, most observers see amuch larger cosmological constant

I (This is a theory that leads to a multiverse and has beenfalsified!)



Recent observations





The Measure Problem

Probabilities in a large universe

The probability for observing the value I of some observable isproportional to the expected number of times 〈NI〉 this value isobserved in the entire universe.

p1

p2=〈N1〉〈N2〉

.

(Strictly speaking, this is an assumption: We are typical observers.This assumption has been very successful in selecting amongtheories.)

Probabilities in a large universe

The probability for observing the value I of some observable isproportional to the expected number of times 〈NI〉 this value isobserved in the entire universe.

p1

p2=〈N1〉〈N2〉

.

(Strictly speaking, this is an assumption: We are typical observers.This assumption has been very successful in selecting amongtheories.)

The measure problem

1

1

11

1 11

11

1

111

1

1

1

1

1

1

1

11

2

22

2

2

2

2 2

2

2

2

2 2

2

2

22

2

2

2 2 22

22

2

22

22

22

22

22

22

22 22

22

222

2222

22

22 2

2 22 22 2 2 2 222 22 22 222

22 2 2 22

22 2

1

11

1

1

1

1 1

11

1

11

11

1

1

1

1

1

1

11

2

22

2

2

2

2

2 2

222

2

2 22

2

22 2

2

22

21

2

22

2

222

22

2

2

2

1

2

1 1

1 1 11

1 1

I Infinitely many pockets of each vacuumI Each pocket contains infinitely many observers (if any)I Relative probabilities are ill-defined:

p1

p2=〈N1〉〈N2〉

=∞∞

I Need a cutoff to render 〈NI〉’s finite

Holographic approach to the measure problem

Ultimately, the measure should be part of a unique,fundamental description of the multiverse.

The holographic principle is widely expected to be central toany such theory. Different aspects of holography have beenused to motivate different choices of measure:

I Black hole complementarity −→ causal patch cut-off[RB ’06]

I UV/IR relation in AdS/CFT −→ light-cone time cut-off[Garriga & Vilenkin ’08; RB ’09; RB, Freivogel, Leichenauer& Rosenhaus ’10]

Holographic measures

I Complementarity −→ causal patch cut-off

I AdS/CFT −→ light-cone time cut-off

Extensive study of these proposals has yielded the followingencouraging results:

1. They both avoid catastrophic predictions, which plaguedmany older proposals (I won’t show this here)

2. They both give probability distributions for Λ and otherparameters that agree well with observation (which I willshow explicitly for Λ)

3. Despite appearing very different, they are preciselyequivalent (which I will not show) [RB & Yang ’09]






















The quantum xeroxing paradox

I If black hole evaporationis unitary, then globally itwould lead to quantumxeroxing, which conflictswith the linearity ofquantum mechanics

I But no observer can seeboth copies

I Physics need onlydescribe experimentsthat can actually beperformed, so we losenothing by restricting toa causal patch

Causal Patch Cut-off

I Restrict to the causal past of the future endpoint of ageodesic.

I First example of a “local” measure: keep neighborhood ofworldline.

I Roughly, in vacua with Λ > 0, count events inside thecosmological horizon.

I What value of Λ is most likely to be observed, according tothis measure?

Causal Patch Cut-off

I Restrict to the causal past of the future endpoint of ageodesic.

I First example of a “local” measure: keep neighborhood ofworldline.

I Roughly, in vacua with Λ > 0, count events inside thecosmological horizon.

I What value of Λ is most likely to be observed, according tothis measure?

Predicting the cosmological constant

I Consider all observers living around the time tobs after thenucleation of their pocket universe.

I We are a member of this class of observers, so anyconclusions will apply to us, but will be more general in thatthey include observers in very different vacua, withpossibly very different particle physics and cosmology.

I What is the probability distribution over observed Λ?


Landscape statistics: dp̃/dΛ = const for |Λ| � 1, i.e.,most vacua have large Λ

Because of de Sitter expansion, the number of observers insidethe diamond becomes exponentially dilute after tΛ ∼ Λ−1/2:

nobs ∼ exp(−3tobs/tΛ) ,

so there are very few observers that see Λ� t−2obs.

Therefore,

dpd log Λ

∝ dp̃d log Λ

nobs ∝ Λ exp(−√

3Λ tobs)






Therefore,

dpd log Λ

∝ dp̃d log Λ


3Λ tobs)






Therefore,

dpd log Λ

∝ dp̃d log Λ


3Λ tobs)


Therefore, the string landscape + causal patch measure predict

Λ ∼ t−2obs

I Solves the coincidence problem directly.I Agrees better with observation than Λ ∼ t−2

gal [Weinberg ’87](especially if δρ/ρ is also allowed to scan)

I More general: Holds for all observers, whether or not theylive on galaxies [RB & Harnik ’10]

The probability distribution over Λ

!126 !125 !124 !123 !122 !121 !120 !119log( !" )

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Pr

obab

ility

dens

ity

solid line: prediction; vertical bar: observed value[RB, Harnik, Kribs & Perez ’07]

The probability distribution over Λ and Ne

I

II

III

IV

V

log tobslog tc

log tobs

log tL

Geometric effects dominate; no specific anthropic assumptionsrequired→ tΛ ∼ tc ∼ tobs ∼ N̄

[RB, Freivogel, Leichenauer & Rosenhaus ’10]

Documents

The Cosmological Constant Problem and the Multiverse of