A new twist on theNo-Boundary State

Stephen Hawking’s 70th Birthday Conference

January 7, 2012

Thomas Hertog

Institute for Theoretical PhysicsUniversity of Leuven, Belgium

arXiv:0803.1663arXiv:1009.2525arXiv:1111.6090

Stephen’s ”recent” work...

No-boundary cosmology:

Stephen’s ”recent” work...

No-boundary cosmology:

90’s: bottom-up approach → ‘the’ instanton

Stephen’s ”recent” work...

No-boundary cosmology:

90’s: bottom-up approach → ‘the’ instanton

2002: top-down approach → prior on multiverse

Stephen’s ”recent” work...

No-boundary cosmology:

90’s: bottom-up approach → ‘the’ instanton

2002: top-down approach → prior on multiverse

”The history of the universe depends on the questionthat is asked.”

→ local observations: conditional, coarse-grained

Stephen’s ”recent” work...

No-boundary cosmology:

90’s: bottom-up approach → ‘the’ instanton

2002: top-down approach → prior on multiiverse

”The history of the universe depends on the questionthat is asked.”

→ local observations: conditional, coarse-grained

2007: dominated by histories with many efolds

→ eternal inflation

2010: local observations in eternal inflation;

P (CMB1)/P (CMB2) ≈ exp[π(1/V 1ei − 1/V 2

ei)]

Stephen’s ”recent” work...

No-boundary cosmology:

90’s: bottom-up approach → ‘the’ instanton

2002: top-down approach → prior on multiiverse

”The history of the universe depends on the questionthat is asked.”

→ local observations: conditional, coarse-grained

2007: dominated by histories with many efolds

→ eternal inflation

2010: local observations in eternal inflation;

P (CMB1)/P (CMB2) ≈ exp[π(1/V 1ei − 1/V 2

ei)]

Higher order corrections?

→ 2011: dual formulation no-boundary state?

Outline

• No-Boundary state

• Its AdS representation → dual formulation

• Implications for eternal inflation

No-Boundary Wave Function

Ψ[b, h, χ] =∫Cδgδφ exp(−I[g, φ])

”The amplitude of configurations (b, h, χ) on a three-surface Σ is given by the integral over all regularmetrics g and matter fields φ that match (b, h, χ) ontheir only boundary.” [Hartle & Hawking ’83]

Motivation: analogy w/ ground state wave function

Semiclassical Limit

Ψ(b, h, χ) ≈∑e exp[−Ae(b, h, χ)]/~

with

Ae = I(b, h, χ) + ~I(1)(b, h, χ) + · · ·

Extremal geometries generally complex:

I(b, h, χ) = −IR(b, h, χ) + iS(b, h, χ)

WKB Interpretation

Ψ(b, h, χ) ≈ exp[−IR(b, h, χ) + iS(b, h, χ)]/~

The semiclassical wave function predicts Lorentzian,classical evolution in regions of superspace where[Hawking ’84, Grischuk & Rozhansky ’90]

|∇AIR| |∇AS|

The resulting coarse-grained classical histories of theuniverse are given by the integral curves of SL:

pA = ∇AS

and have conserved probabilities

Phistory ∝ exp[−2IR/~]

→ no-boundary measure: prior on multiverse.

Complex Saddle points

ds2 = dτ2 + gij(τ, x)dxidxj, φ(τ, x)

TUNING AND

SP

REGULARITY

CLASSICAL

HISTORY

Hy

Hx

Regularity at SP: gij(0)→ 0, φ(0)→ 0

At final boundary: gij(υ) = b2hij, φ(υ) = χ

Tuning at SP: φ(0) = φ0eiγ, ...

Example

Consider homogeneous/isotropic multiverse:

ds2 = dτ2 + a2(τ)dΩ3, φ(τ)

V (φ) = Λ + 12m

2φ2

Classical evolution requires tuning of φ(0) = φ0eiγ at

the South Pole of the saddle point: ( m2c = 9/4H2)

→ |∇AIR| |∇AS| at large scale factor

→ 1-parameter set of FLRW backgrounds

Saddle point Action

rSP

Hy

HxHx

with Hxr → π/2 for φ0 → 0

I(υ) = 3π2

∫C(0,υ)

dτa[a2(H2 + 2V (φ))− 1]

IR tends to a constant along vertical part

→ probability measure on classical histories.

Inflation

Background histories:

pA = ∇AS

→ no-boundary state predicts inflation

Origin

Background histories:

pA = ∇AS

large φ0 small φ0

Part II: Towards a dual descriptionof the no-boundary state

Complex Saddle points

Lorentzian histories always lie on asymptotically verticalcurves in complex τ -plane:

rSP

Hy

HxHx

Complex Saddle points

Lorentzian histories always lie on asymptotically verticalcurves in complex τ -plane:

rSP

Hy

HxHx

e.g. pure deSitter: a(τ) = 1H sin(Hτ)

horizontal part: ds2 = dτ2 + 1H2 sin2(Hτ)dΩ2

3

vertical part: ds2 = −dy2 + 1H2 cosh2(Hy)dΩ2

3

Complex Saddle points

Lorentzian histories always lie on asymptotically verticalcurves in complex τ -plane:

rSP

Hy

HxHx

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

vertical part: Euclidean ADS

ds2 = −dy2 − 1H2 sinh2(Hy)dΩ2

3

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

With matter: Euclidean ADS domain wall

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

With matter: Euclidean ADS domain wall

• signature complex saddle pt metric varies in τ -plane

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

With matter: Euclidean ADS domain wall

• Domain Wall/Cosmology correspondence in SUGRA[Cvetic ’96; Skenderis, Townsend, Van Proeyen ’07]

→ realized here through universe’s quantum state

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

With matter: Euclidean ADS domain wall

• Domain Wall/Cosmology correspondence in SUGRA[Cvetic ’96; Skenderis, Townsend, Van Proeyen ’07]

V effAdS = −Λ− V

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

Vice versa: starting from Euclidean AdS SUGRAwe predict Lorentzian asymptotically deSitteruniverses.

Representations of Saddle points

a Hx

2

HxSP

Hy

rHx

• Does the AdS/dS connection hold independently ofthe saddle pt approximation?

• What happens along horizontal branch?

Asymptotic Analysis

Asymptotic expansion of metric and fields in smallu ≡ eiτ = e−y+ix [Skenderis,...]:

gij(u,Ω) = −14u2

[hij(Ω) + h(2)ij (Ω)u2

+h(−)ij (Ω)uλ− + h

(3)ij (Ω)u3 + · · ·]

φ(u,Ω) = uλ−(α(Ω) + α1(Ω)u+ · · ·)

+uλ+(β(Ω) + β1(Ω)u+ · · ·)

with λ± ≡ 32[1±

√1− (2m/3)2]

and (arbitrary) ‘boundary values’ (hij, α).

→ AdS/dS connection

Action integral

• Action integral along horizontal part:

Ih =∫hI[g, φ] = +Sct(b, h, χ)− iSct(b, h, χ)

and no finite contribution.

Sct = a0∫ √

h+ a1∫ √

hR(3) + a2∫ √

hφ2

• Action integral along vertical part:

Iv =∫vI[g, φ] = −IRAdS(h, χ)− Sct(b, h, χ)

where IRAdS is finite when a→∞.

→ starting from AdS SUGRA, horizontal branchamounts to adding the usual counterterms.

I(b, h, χ) = −IRAdS(h, χ)− iSct(b, h, χ)

Towards a holographic dual

Ψ(b, h, χ) ≈ exp[+IRAdS(h, χ) + iSct(b, h, χ)]/~

AdS/CFT [Maldacena,Witten,...]:

exp(−IRADS[h, χ]/~) = ZQFT [h, χ] = 〈exp∫d3x√hχO〉

→ ‘dS/CFT dual’ formulation of NBWF:

Ψ(b, h, χ) ≈ 1ZQFT [h,χ,ε]

exp[iSct(b, h, χ)]/~

where UV cutoff ε ∼ 1/b

Remarks

Ψ(b, h, χ) ≈ 1ZQFT [h,χ,ε]

exp[iSct(b, h, χ)]/~

• Z provides measure on configurations (b, h, χ).

• Scale factor evolution as inverse RG flow[Strominger ’02]

• Physical interpretation of counterterms in AdS

• Coarse-graining over UV modes at finite scale factor[see also Vilenkin ’11]

• no-boundary condition of regularity implemented

Part III: Eternal Inflation

Part III: Eternal Inflation

Can the dual formulation of no-boundary state beapplied to regime of eternal inflation only?

Euclidean Eternal Inflation

[Hartle, Hawking & TH, in progress]

Replace inner region of eternal inflation by dual CFTon its boundary:

i

HxHx

Hy

rHxa

2

Euclidean Eternal Inflation

[Hartle, Hawking & TH, in progress]

Replace inner region of eternal inflation by dual CFTon its boundary:

ei

r

• IR CFT with deformation given by φ = φEI.(similar to [Maldacena ’10])

• < O > on inner boundary replaces regularity at SP.

→ remaining saddle point with inner boundary

Euclidean Eternal Inflation

[Hartle, Hawking & TH, in progress]

Replace inner region of eternal inflation by dual CFTon its boundary:

ei

r

|Ψ(b, h, χ)|2 ≈ 1|ZQFT [φEI,he,ε]|2

exp[+2IRAdS/~

Conclusion

• EAdS/dS connection in no-boundary cosmology

• dual description in terms of relevant deformationsof the CFTs that occur in AdS/CFT.

• Reinterpretation of eternal inflation

Conclusion

• EAdS/dS connection in no-boundary cosmology

• dual description in terms of relevant deformationsof the CFTs that occur in AdS/CFT.

• Reinterpretation of eternal inflation

Happy Birthday Stephen!