Accelerated Expansion and Ads/CFT

Home , Eternal inflation

Accelerated Expansion and AdS/CFT

Thomas Hertog Institute for Theoretical Physics University of Leuven, Belgium

with JB Hartle, SW Hawking arXiv:1205...... arXiv:1111.6090 arXiv:0803.1663 Can AdS/CFT be applied to cosmology?

• Time-dependent Lorentzian AdS/CFT

→ ”cosmological” backgrounds Can AdS/CFT be applied to cosmology?

• Time-dependent Lorentzian AdS/CFT

→ ”cosmological” backgrounds

• Analytic continuation from Euclidean AdS

→ wave function of perturbations Wave function of the universe:

• Euclidean AdS/CFT [Horowitz & Maldacena ’04]

R exp(−IADS[h, χ]/~) = ZQF T [h, χ]

• No-boundary State [Hartle & Hawking ’83]

Ψ[h, χ] = exp(−IdS[h, φ]/~)

This talk: connect both ideas. Motivation

• precise formulation of no-boundary state

• singularity resolution

• theoretical foundations of inﬂation

• probabilities in eternal inﬂation No-Boundary State

3 R Ψ[ g, χ] = C δgδφ exp(−IdS[g, φ]/~)

”The amplitude of conﬁgurations (3g, χ) on a three- surface Σ is given by the integral over all regular metrics g and matter ﬁelds φ that match (3g, χ) on their only boundary.” [Hartle & Hawking ’83] Saddle Point Limit

3 3 Ψ[ g, χ] ≈ exp{[−IdS( g, χ)]/~}

REAL

COMPLEX

REGULARITY

3 3 3 IdS( g, χ) = IR( g, χ) − iS( g, χ) WKB Interpretation

3 3 3 Ψ[ g, χ] ≈ exp{[−IR( g, χ) + iS( g, χ)]/~}

Lorentzian space-time evolution emerges if

|∇AIR| |∇AS|

The predicted classical histories are

pA = ∇AS and have conserved probabilities

Phistory ∝ exp[−2IR/~]

No-boundary state → prior on multiverse Complex Saddle points

2 2 i j ds = dτ + gij(τ, x)dx dx , φ(τ, x)

CLASSICAL HISTORY

TUNING AND REGULARITY

SP Hx

iγ Tuning at SP: φ(0) = φ0e Example

Homogeneous/isotropic ensemble: Ψ[b, χ]

2 2 2 ds = dτ + a (τ)dΩ3, φ(τ) 1 2 2 V (φ) = Λ + 2m φ Classical evolution requires tuning:

→ multiverse of FLRW backgrounds Probability measure

SP Hxr Hx

3π R 2 2 I(υ) = 2 C dτa[a (H + 2V (φ)) − 1]

π IR(χ) ≈ − 4V (φ0) Probability measure

SP Hxr Hx

3π R 2 2 I(υ) = 2 C dτa[a (H + 2V (φ)) − 1]

π IR(χ) ≈ − 4V (φ0) Including perturbations:

2 3 2 IR(δζn) ≈ +(/H )n (δζn) Inﬂation

pA = ∇AS

hˆ ≈ mφˆ

No-boundary state predicts inﬂation Was there a Beginning?

large φ0 small φ0 Was there a Beginning?

REAL

COMPLEX

REGULARITY

Saddle points everywhere regular

→ singularities in classical extrapolation no obstacle to asymptotic predictions No-Boundary State: ADS form Complex Saddle points

SP Hxr Hx Complex Saddle points

SP Hxr Hx

1 e.g. a(τ) = H sin(Hτ), φ(τ) = 0 horizontal part: 2 2 1 2 2 ds = dτ + H2 sin (Hτ)dΩ3 vertical part: 2 2 1 2 2 ds = −dy + H2 cosh (Hy)dΩ3 Complex Saddle points

SP Hxr Hx Representations

Diﬀerent representation of same solution:

Hxa SP Hxr Hx Representations

Hxa SP Hxr Hx vertical part: Euclidean ADS

2 2 1 2 2 ds = −dy − H2 sinh (Hy)dΩ3 Representations

Hxa SP Hxr Hx

With matter: Euclidean ADS domain wall

Hy • V acts as eﬀective −V in AdS regime because the signature of the complex saddle point metric varies in the τ-plane,

Veff = −V

• Somewhat reminiscent of Domain Wall/Cosmology correspondence in SUGRA. [Cvetic; Skenderis,Townsend, Van Proeyen]

• Realized here at the level of the wave function of the universe which involves a given complexiﬁed theory. Saddle Point Action

IdS(b, χ) = Iv + Ih

Hxa SP Hxr Hx

• Contribution from vertical part:

R R 3 Iv = v IdS[g, φ] = −IAdS( g,˜ χ˜) + Sct(a, χ˜)

R where IAdS is ﬁnite when y → ∞.

φ = αe−λ−y + βe−λ+y Saddle Point Action

IdS(b, χ) = Iv + Ih

Hxa SP Hxr Hx

• Contribution from horizontal part: R Ih = h I[g, φ] = −Sct(a, χ˜) + iSct(b, χ)

and no ﬁnite contribution. Asymptotic Structure

Expanded in small u ≡ eiτ = e−y+ix,

−1 (2) 2 gij(u, Ω) = 4u2 [hij(Ω) + hij (Ω)u

(−) λ− (3) 3 +hij (Ω)u + hij (Ω)u + ···]

λ− φ(u, Ω) = u (α(Ω) + α1(Ω)u + ···)

λ+ +u (β(Ω) + β1(Ω)u + ···)

3 p 2 with λ± ≡ 2[1 ± 1 − (2m/3) ] and arbitrary ‘boundary values’ (hij, α).

R 3 3 Ih = h I[g, φ] = −Sct(a, g,˜ χ˜) + iSct(b, g,˜ χ)

A universal AdS/dS connection follows directly from an asymptotic analysis Saddle Point Action

Hxa SP Hxr Hx

3 R 3 3 IdS( g, χ) = −IAdS( g,˜ χ˜) + iSct( g, χ) with

3 R p3 R p3 (3) Sct( g, φ) = a0 g + a1 gR + ···

No-Boundary State:

˜ R ˜ ˜ Ψ[b, h, χ] = exp{[+IAdS(h, χ˜) − iSct(b, h, χ)]/~} Two Sets of Saddle points

φ(υ) = φ(υ2) = χ, g˜ij(υ) =g ˜ij(υ2)

Hy Hy

Hxa SP Hxr Hx SP Hxr Hx

COMPLEX φ0 REAL φ0 Accelerated Expansion and AdS/CFT Holographic Cosmology

• No-boundary State:

˜ R ˜ ˜ Ψ[b, h, χ] = exp{[+IAdS(h, χ˜) + iSct(b, h, χ)]/~}

• Euclidean AdS/CFT:

R ˜ ˜ exp(−IAdS[h, χ˜]/~) = ZQF T [h, χ˜]

˜ 1 ˜ Ψ[b, h, χ] = ˜ exp{[iSct(b, h, χ)]/~} ZQF T [h,χ,˜ ]

l with complex source χ˜ and UV cutoﬀ ∼ b Cosmology with AdS Gravity

Using AdS/CFT we evaluate the holographic no-boundary state,

R ˜ ˜ exp(−IADS[h, χ˜]/~) = ZQF T [h, χ˜]

There are two sets of saddle points Cosmology with AdS Gravity

Using AdS/CFT we can evaluate the holographic no-boundary state,

R ˜ ˜ exp(−IADS[h, χ˜]/~) = ZQF T [h, χ˜]

There are two sets of saddle points

A: Real χ˜ → real Euclidean AdS domain walls

SP Hx

˜ R ˜ ˜ Ψ[b, h, χ] ≈ exp{[+IAdS(h, χ˜) − Sct(b, h, χ)]/~} Cosmology with AdS Gravity

Using AdS/CFT we can evaluate the holographic no-boundary state,

R ˜ ˜ exp(−IADS[h, χ˜]/~) = ZQF T [h, χ˜]

There are two sets of saddle points

B: Complex χ˜ → complex Euclidean domain walls

SP Hx

˜ R ˜ ˜ Ψ[b, h, χ] ≈ exp{[+IAdS(h, χ˜) − iSct(b, h, χ)]/~} Saddle points correspond to inﬂationary universes Remarks

˜ R 3 p˜ ZQF T [h, χ˜] = hexp d x hχ˜Oi

• The dependence of Z on the external sources provides a cosmological measure on the space of conﬁgurations (b, h,˜ χ).

• AdS/CFT implements no-boundary condition of regularity in saddle point limit

• Scale factor evolution arises as inverse RG ﬂow

• Physical interpretation of counterterms in AdS What models? What models?

˜ R 3 p˜ ZQF T [h, χ˜] = hexp d x hχ˜Oi

• Lorentzian stability criteria too stringent

• Unitary physics in given classical background

• Ensemble of inﬂationary backgrounds

1 2 2 VAdS(φ) = −Λ − 2m φ

Classical evolution constrains V Singularity Resolution (for Gary) Singularity Resolution (for Gary)

˜ 1 ˜ Ψ[b, h, χ] = ˜ exp{[iSct(b, h, χ)]/~} ZQF T [h,χ,˜ ]

Given an asymptotic structure one can probe the deep interior by taking the RG ﬂow all the way down to the IR.

Singularity described by IR ﬁxed point. Conclusion

The no-boundary proposal and Euclidean AdS/CFT are intimately connected.

• A wave function deﬁned in terms of a gravitational theory with a negative cosmological constant Λ can predict expanding universes with an ‘eﬀective’ positive cosmological constant −Λ.

• The Euclidean AdS/CFT dual provides a more precise, ‘holographic’ formulation of the semiclassical no-boundary state.

Applications:

• Holographic calculation of CMB correlators.

• Get a better handle on eternal inﬂation? Euclidean Eternal Inﬂation

[Hartle, Hawking & TH, in progress] Proposal: replace the inner region of eternal inﬂation by a dual CFT on the threshold surface at υi:

ei i

Hxa Hxr Hx r

• IR CFT with a deformation set by threshold φEI. (similar to [Maldacena ’10])

• < O > on inner boundary replaces regularity condition at origin.