Computing the Wave Function

of the Universe

Penn State, 9 May 2012

Alex Maloney, McGill University

Castro, A. M., to appearCastro, Lashkari & A. M.

Overview

The Problem

Quantum cosmology is confusing:

I How is Unitarity consistent with singularities, inflation, . . . ?

I Is quantum mechanics modified in cosmological settings?

I What are the appropriate observables for eternal inflation?

I What is the meaning and origin of the entropy of acosmological horizon?

Similar questions are answered in the context of black hole physicsby AdS/CFT.

Let us be bold and apply the same techniques to cosmology.

The Wave-function of the Universe

One lesson of AdS/CFT is that the ”wave function of theuniverse” |ψ〉 exists and is computable.

The Hartle-Hawking state

〈h|ψ〉 ∼∫

g |∂M=h

Dg e−S

includes contributions from all geometries. It is the natural“vacuum state” of quantum gravity.

In AdS the radial wave function is a CFT partition function.

Goal: Compute 〈h|ψ〉 in de Sitter space.

de Sitter Space

For three dimensional general relativity with a positivecosmological constant

S [g ] =1

G

∫M

√−g

(R − 2

`2

)the partition function

Z =

∫Dg e−S[g ]

can be computed exactly.

We will be inspired by AdS/CFT but we will not use it.

The Idea

The saddle point approximation is

Z =

∫Dg e−S[g ] =

∑g0

e−kS0+S1+ 1kS2+...

where k = `/G is the coupling. The approximation becomes exactif we can

I Find all classical saddles

I Compute all perturbative corrections around each saddle

We will do both.

The Result

We will compute the wave function of the universe as a function ofthe topology and (conformal) geometry of I+.

This is a version of the Hartle-Hawking computation, where oneanalytically continues from dS to Euclidean AdS space.

The resulting wave function 〈h|ψ〉 is non-normalizable, and peakedwhen the geometry h is singular.

I This comes from a non-perturbative effect

Thus, for pure 2+1 dimensional gravity, de Sitter space is unstable.

The Plan for Today:

• The Wave Function

• The Computation

• The Result

The Wave Function of the Universe

We wish to compute the wave function of the universe 〈h|ψ〉.

Near I+, h→∞ and one can take the WKB approximation of theWheeler-de Witt equation.

Up to local counterterms, the Hartle-Hawking wave functiondepends only on the conformal structure of h

〈h|ψ〉 → e iSct(h)ψ(h)

The dS/CFT conjecture identifies ψ(h) with the partition functionof a Euclidean conformal field theory (Strominger, Maldacena, ...).

We will not use dS/CFT. Instead we will compute ψ(h) directly.

Hartle-Hawking WavefunctionHartle & Hawking propose that 〈h|ψ〉 should be computed byintegrating over smooth, compact Euclidean geometries:

This is difficult to compute.

If we take `→ i`, z → iz the metric of Lorentzian dS becomes thatof Euclidean AdS:

ds2 = `2−dz2 + dx2

z2→ `2

dz2 + dx2

z2

Future infinity of dS becomes the asymptotic boundary of EAdS.

Asymptotics

The Bunch-Davies boundary conditions (ψ ∼ e−iωz → eωz atz → −∞) tell us that wave functions are smooth in the interior ofEAdS.

The Computation

Thus the wave function can be computed by a path integral oversmooth field configurations in Euclidean AdS.

The wave function ψ(h) can be interpreted as the partition of aEuclidean CFT with imaginary central charge, since the analyticcontinuation takes

k =`

G→ i

`

G

Such CFTs are confusing. They do not obey standard unitarity(reflection positivity) conditions.

So instead of trying to identify the CFT we will perform a directbulk computation.

Aside:

This notion of Euclidean continuation is natural if we think ofgravity as a gauge theory

I Analytically continue signature of spacetime withoutcontinuing the local Lorentz group

I Very natural in 3D gravity, Vasiliev theory, . . .

This explains why the AdS/CFT duality between the O(N) modeland Vasiliev theory can be turned into a dS/CFT.

For theories with more complicated degrees of freedom (e.g. RRfields) this is less useful.

The Computation

Wave Function on dS3

The wave function ψ(h) is a function of conformal structure of I+.

A sphere (I+ of global de Sitter) has a unique conformal structure,so ψ is just a number.

But I+ can have more complicated topology:

I Solutions to the classical equations of motion are quotientsdS3/Γ.

I They can have arbitrary topology and conformal structure atfuture infinity.

At early times these universes have Milne-type big bangsingularities.

We wish to compute the topology and conformal modulidependence of ψ(h).

Wave Function on T 2

Today, focus on the case where I+ is a torus.

The quotient dS3/Z

ds2 = `2(−dt2 + cosh2 tdφ2 + sinh2 tdθ2

)with θ ∼ θ + 2π, φ ∼ φ+ 2π has a Milne singularity at t = 0.

This is a de Sitter analogue of the BTZ black hole.

For this geometry, the torus at I+ is square.

More generally the wave function depends on the conformalstructure parameter τ of the torus at I+.

z ∼ z + 1 ∼ z + τ

The Path Integral:

The wave function ψ(τ) is computed by the path integral

ψ(τ) =

∫Dg e−S[g ]

over locally EAdS3 manifolds with T 2 boundary with conformalstructure τ . This includes a sum over topologies

ψ(τ) =∑M

ψM(τ)

In the semiclassical (~→ 0) limit only M which admit a classicalsolution contribute

ψ =∑g0

e−k S(0) + S(1) + k−1S(2)+...

For pure gravity we can find all classical saddles and compute allperturbative corrections.

The Classical Solutions:

The smooth classical solutions are solid tori Mγ , labelled by

elements γ =(

a bc d

)∈ SL(2,Z)

T

X

The Mγ are quotients EAdS3/Z. They are related by modulartransformations

ψγ(τ) = ψ0(γτ), γτ =aτ + b

cτ + d

where ψ0 is the contribution from thermal AdS.

Classical Action:

The wave function is a sum over the modular group

ψ(τ) =∑

γ∈SL(2,Z)

ψ0(γτ)

where ψ0 is the contribution from geometries continuouslyconnected to thermal AdS.

The classical action is obtained by computing the (regularized)volume of EAdS

ψ0(τ) = |q|−2k q = e2πiτ

This becomes a pure phase

ψ0(τ) = |q|2ik = e4πik=τ

upon continuation back to dS.

Quantum Effects:

The one loop term is not pure phase

ψ0(τ) = |q|−2k det∇(1)

√det∇(0) det∇(2)

It is non-trivial even though there are no local degrees of freedom.

This comes from quantizing the space of non-trivialdiffeomorphisms, a la Brown & Henneaux.

ψ0 = |q|−k∏n

1

|1− qn|2

The result is one-loop exact.

The Results

Modular InvarianceWhen we sum over SL(2,Z), the resulting ψ(τ) is invariant underτ → aτ+b

cτ+d . It is a function on H2/SL2(Z).

The Result

ψ(τ) can be written as a convergent series expansion

ψ(τ) ∼ e−πiτ/12

(−6 +

(π6 − 6π)(11 + 24k)

9ζ(3)=τ+ . . .

)

1.000 1.005 1.010 1.015 1.020ImHΤL570 000

575 000

580 000

585 000

590 000

595 000 ΨHΤL¤

Divergence

The divergence at τ = i∞ is non-normalizable∫d2τ

=τ2|ψ(τ)|2 =∞

Peaked at the infinitely stretched torus with τ = i∞.

Thus de Sitter space is unstable; the spatial geometry wants to beinfinity distorted.

I Effect invisible at all orders in perturbation theory around thestandard saddle

I Comes from loop terms around the non-perturbative saddles

Conclusions

The wave function of the universe at I+ is computable as aEuclidean gravity path integral.

The computation is more easily done in Euclidean AdS, rather thanEuclidean dS.

The resulting wave function is peaked when the spatial geometry is(infinitely) inhomogeneous.

I de Sitter space is non-perturbatively unstable.

All known constructions of de Sitter space in string theory arenon-perturbatively unstable.

I We have found the three dimensional version of this instability.