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Alex Maloney, McGill University
I What are the appropriate observables for eternal inflation?
I What is the meaning and origin of the entropy of a cosmological horizon?
Similar questions are answered in the context of black hole physics by 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 the universe” |ψ exists and is computable.
The Hartle-Hawking state
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 positive cosmological constant
S [g ] = 1
∫ Dg e−S[g ]
can be computed exactly.
We will be inspired by AdS/CFT but we will not use it.
The Idea
Z =
e−kS0+S1+ 1 k S2+...
where k = `/G is the coupling. The approximation becomes exact if 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 of the topology and (conformal) geometry of I+.
This is a version of the Hartle-Hawking computation, where one analytically continues from dS to Euclidean AdS space.
The resulting wave function h|ψ is non-normalizable, and peaked when 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 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 the Wheeler-de Witt equation.
Up to local counterterms, the Hartle-Hawking wave function depends only on the conformal structure of h
h|ψ → e iSct(h)ψ(h)
The dS/CFT conjecture identifies ψ(h) with the partition function of a Euclidean conformal field theory (Strominger, Maldacena, ...).
We will not use dS/CFT. Instead we will compute ψ(h) directly.
Hartle-Hawking Wavefunction Hartle & Hawking propose that h|ψ should be computed by integrating over smooth, compact Euclidean geometries:
This is difficult to compute.
If we take `→ i`, z → iz the metric of Lorentzian dS becomes that of Euclidean AdS:
ds2 = `2 −dz2 + dx2
Future infinity of dS becomes the asymptotic boundary of EAdS.
The Bunch-Davies boundary conditions (ψ ∼ e−iωz → eωz at z → −∞) tell us that wave functions are smooth in the interior of EAdS.
The Computation
Thus the wave function can be computed by a path integral over smooth field configurations in Euclidean AdS.
The wave function ψ(h) can be interpreted as the partition of a Euclidean CFT with imaginary central charge, since the analytic continuation takes
k = `
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 direct bulk computation.
This notion of Euclidean continuation is natural if we think of gravity as a gauge theory
I Analytically continue signature of spacetime without continuing the local Lorentz group
I Very natural in 3D gravity, Vasiliev theory, . . .
This explains why the AdS/CFT duality between the O(N) model and Vasiliev theory can be turned into a dS/CFT.
For theories with more complicated degrees of freedom (e.g. RR fields) 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 quotients dS3/Γ.
I They can have arbitrary topology and conformal structure at future infinity.
At early times these universes have Milne-type big bang singularities.
We wish to compute the topology and conformal moduli dependence 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 conformal structure 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 conformal structure τ . This includes a sum over topologies
ψ(τ) = ∑ M
In the semiclassical (~→ 0) limit only M which admit a classical solution contribute
ψ = ∑ g0
e−k S(0) + S(1) + k−1S(2)+...
For pure gravity we can find all classical saddles and compute all perturbative corrections.
The Classical Solutions:
The smooth classical solutions are solid tori Mγ , labelled by
elements γ = (
) ∈ SL(2,Z)
The Mγ are quotients EAdS3/Z. They are related by modular transformations
ψγ(τ) = ψ0(γτ), γτ = aτ + b
cτ + d
Classical Action:
The wave function is a sum over the modular group
ψ(τ) = ∑
where ψ0 is the contribution from geometries continuously connected 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:
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-trivial diffeomorphisms, a la Brown & Henneaux.
ψ0 = |q|−k ∏ n
The Results
Modular Invariance When we sum over SL(2,Z), the resulting ψ(τ) is invariant under τ → aτ+b
cτ+d . It is a function on H2/SL2(Z).
The Result
ψ(τ) ∼ e−πiτ/12
575 000
580 000
585 000
590 000
=τ2 |ψ(τ)|2 =∞
Peaked at the infinitely stretched torus with τ = i∞.
Thus de Sitter space is unstable; the spatial geometry wants to be infinity distorted.
I Effect invisible at all orders in perturbation theory around the standard saddle
I Comes from loop terms around the non-perturbative saddles
The wave function of the universe at I+ is computable as a Euclidean gravity path integral.
The computation is more easily done in Euclidean AdS, rather than Euclidean dS.
The resulting wave function is peaked when the spatial geometry is (infinitely) inhomogeneous.
I de Sitter space is non-perturbatively unstable.