Quantum cosmology { foundations and challenges · Homogeneous, Isotropic Cosmology Quantum...

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Quantum cosmology – foundations and challenges Steffen Gielen Max Planck Institute for Gravitational Physics (Albert-Einstein-Institut) 6 December 2017 – Typeset by Foil T E X

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Page 1: Quantum cosmology { foundations and challenges · Homogeneous, Isotropic Cosmology Quantum cosmology { foundations and challenges The Big Picture of Modern Cosmology We seem to have

Quantum cosmology – foundations and challenges

Steffen GielenMax Planck Institute for Gravitational Physics

(Albert-Einstein-Institut)

6 December 2017

– Typeset by FoilTEX –

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Outline Quantum cosmology – foundations and challenges

Outline

1. Introduction

2. Homogeneous, Isotropic Cosmology

3. Quantum Cosmology?

4. Connection to Quantum Gravity

5. Summary

Steffen Gielen, Max Planck Institute for Gravitational Physics 1/20

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Introduction Quantum cosmology – foundations and challenges

Introduction

Cosmology has made great progress in recent decades, with precise observationssuch as of the cosmic microwave background (CMB) now strongly constrainingany possible theory of the very early universe.

The observed structure of the Universe on largest scales can be described insimple terms: the Universe appears almost exactly homogeneous and isotropicon largest scales, with a specific statistical pattern of small, Gaussian perturbations(inhomogeneities) describing cosmic structure.

The goal of theoretical cosmology is to explain this structure, as well asthe beginning of the Universe: the Big Bang of classical cosmology, wheretemperature, energy density, etc, all diverge, should be replaced by a morecomplete picture presumably involving quantum mechanics.

I will discuss some attempts to obtain such a picture in quantum cosmology.

Steffen Gielen, Max Planck Institute for Gravitational Physics 2/20

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Homogeneous, Isotropic Cosmology Quantum cosmology – foundations and challenges

The Big Picture of Modern Cosmology

We seem to have observed the entire history of the Universe. . .

Steffen Gielen, Max Planck Institute for Gravitational Physics 3/20

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Homogeneous, Isotropic Cosmology Quantum cosmology – foundations and challenges

Homogeneous, Isotropic CosmologyOur Universe appears almost exactly homogeneous and isotropic on its largestscales: It is well described by a three-dimensional geometry of constant curvature.

The Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric is

ds2 = −c2 dt2 + a2(t)

(dr2

1−Kr2+ r2(dθ2 + sin2 θ dϕ2)

).

Steffen Gielen, Max Planck Institute for Gravitational Physics 4/20

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Homogeneous, Isotropic Cosmology Quantum cosmology – foundations and challenges

Scale factor

In the FLRW cosmological model, the expansion of the Universe is described bya single function a(t), the scale factor, that describes how comoving distanceschange under time evolution:

As we go back in time, the scale factor goes to zero within a finite time: this isthe Big Bang singularity.

Steffen Gielen, Max Planck Institute for Gravitational Physics 5/20

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Homogeneous, Isotropic Cosmology Quantum cosmology – foundations and challenges

Friedmann equation

Assuming the Universe is of FLRW form, Einstein’s equations reduce to theFriedmann equation constraining the scale factor a(t),

(a

a

)2

+Kc2

a2=

8πG

3ρ+

Λc2

3

where ρ is the energy density of matter, Λ is the cosmological constant (“darkenergy”), and G is Newton’s constant.

ρ is a sum of the different matter/energy components in the model: in ourUniverse, radiation with ρ ∝ 1

a4, “dust” (nonrelativistic matter, including dark

matter) with ρ ∝ 1a3, inflation, etc.

Any “reasonable” matter will lead to solutions that approach a→ 0 in finite time(in the future or the past, or both).

Steffen Gielen, Max Planck Institute for Gravitational Physics 6/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Quantum Cosmology?

The singularity a → 0, where general relativity breaks down, means that thebeginning of the Universe cannot be explained in conventional cosmology.

This is problematic: cosmology should explain the present state of the Universein terms of “natural” initial conditions; it is not clear how to set these withoutknowing anything about the beginning itself.

One might hope that the singularity a→ 0 can be resolved by quantum physics,similar to, say, the classical singularity in the hydrogen atom.

As an example, study a simple cosmological model, in which the Universe isflat (K = 0), there is no Λ, and the only matter is a free, massless scalar field:

(a

a

)2

=8πG

3ρ =

4πG

3φ2 .

Steffen Gielen, Max Planck Institute for Gravitational Physics 7/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Wheeler-DeWitt equation

Let us proceed as if the Universe was a simple quantum mechanical system.We pass from velocities a and φ to the canonical momenta pa and pφ; theFriedmann equation becomes the constraint

H =p2a

a4− 3

4πG

p2φ

a6

!= 0 .

“Quantising” this in the usual Schrodinger representation of momenta by operatorsand ignoring ordering ambiguities, we write down the Wheeler-DeWitt equation

Hψ(a, φ) ≡ −~2

(a∂

∂a

)2

ψ(a, φ) +3

4πG~2 ∂

2

∂φ2ψ(a, φ) = 0 .

ψ(a, φ) is called the wavefunction of the Universe.

Steffen Gielen, Max Planck Institute for Gravitational Physics 8/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Wheeler-DeWitt equation – challenges

Various technical and conceptual challenges for this approach:

• What is the meaning of ψ(a, φ) (probability interpretation?), where does theinner product come from? Who is the observer?

• What would resolution of the Big Bang singularity a = 0 mean? The probabilityto be exactly at a = 0 is always zero. Is this singularity resolution? Is itimportant whether ψ(a = 0) = 0?

• Out of the different solutions to the Wheeler-DeWitt equation, which oneshould describe our Universe?

• a should be restricted to the positive half of R, so −i~ ∂∂a cannot be self-adjoint.

• . . .

Steffen Gielen, Max Planck Institute for Gravitational Physics 9/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Wheeler-DeWitt equation – challengesSome possible answers (long debate since the 1960’s, and ongoing!):

• Equation of motion has second derivatives, use positive definite inner product

〈ψ|χ〉 ≡ i

∫da

(ψ+

∂χ+

∂φ− ∂ψ

+

∂φχ+ − ψ−∂χ

∂φ+∂ψ−

∂φχ−

)

• Accept only semiclassical states that remain sharply peaked? Many-worldsinterpretation? Consistent/decoherent histories? . . .

• Singularity resolution is only achieved if one can show that the energy densityremains bounded and does not diverge as it does classically.

• Impose boundary conditions selecting a preferred state (e.g., no-boundary)

• Pass from a to α = log(a) which extends over the real axis.

Steffen Gielen, Max Planck Institute for Gravitational Physics 10/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Quantum cosmology – some partial successes

Singularity resolution in the sense of bounded energy density is achieved in loopquantum cosmology [Bojowald, Ashtekar, . . . ], where the Friedmann equation ismodified to, e.g., (

sin(toaa

)to

)2

=8πG

The modification is motivated by loop quantum gravity results about fundamentaldiscreteness of space, but no clear derivation of this form is known yet.

The no-boundary proposal [Hartle, Hawking] is supposed to prescribe a uniquequantum state by

ψ(a, φ) =

∫Dg DΦ e−S[g,Φ]

where the integral is over all (complex) spacetimes without boundary in the pastthat are compatible with the final values a and φ.Unclear whether this is really a definition; requires further input in practice.

Steffen Gielen, Max Planck Institute for Gravitational Physics 11/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Quantum Mechanics of a(t)?

Quantum cosmology, as discussed so far (in terms of minisuperspace models),treats the scale factor a(t) as a quantum degree of freedom that can be consideredseparately from the perturbations around simple homogeneous models.

However, general relativity is nonlinear: perturbations will backreact on a(t).Understanding this backreaction (classically!) is a focus of ongoing research.

More worryingly, quantum effects in quantum cosmology depend on the sizeof the region (“universe”) one studies: indeed,

S =

∫d4x

(√−g R[g]

16πG+ Lm

)=

(∫d3x√h

)∫dt

(−3aa2

8πG+ . . .

)

The arbitrary volume factor∫

d3x√h controls the size of quantum effects (∼ ~/S)

which are hence completely undetermined, and disappear for a large universe.

Steffen Gielen, Max Planck Institute for Gravitational Physics 12/20

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Quantum Cosmology? Quantum cosmology – foundations and challenges

Separate Universe Approach in Cosmology

A useful approximation in classical cosmology for describing perturbations ofa certain range of wavelengths is the separate universe approach: one models theuniverse as consisting of “patches”, each homogeneous and isotropic.

ba

t1

t2

-1

cH

λ

(from [Wands, Malik, Lyth, Liddle 2000]). The “scale factor” or total volume of the(observable) Universe arises from averaging, or summing, over many patches.

Steffen Gielen, Max Planck Institute for Gravitational Physics 13/20

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Connection to Quantum Gravity Quantum cosmology – foundations and challenges

Symmetry reduction or coarse graining?

Symmetry-reduced models can only be an approximation to an effective descriptionobtained by some form of averaging or coarse-graining. This is true already inclassical relativity, let alone in quantum gravity:

(from [Glaser, Loll 2017]). In quantum theory, exact symmetry reduction seems toviolate the uncertainty principle, and so has to be applied before quantisation.

Steffen Gielen, Max Planck Institute for Gravitational Physics 14/20

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Connection to Quantum Gravity Quantum cosmology – foundations and challenges

Towards a local quantum cosmology from quantum gravity(Review: [SG, Sindoni 2016])Quantum gravity is (?) about a local quantisation of geometry.In the group field theory approach one has an “atomistic” picture for space andtime. There is a quantum field creating excitations of geometry,

ϕ†(V, φ)|∅〉 = | 〉•

������������

����

���@

@@

@@@

@@

@������

�����

@@@@@

�������

where |∅〉 is a “no-space” vacuum annihilated by the field ϕ(V, φ).

The total volume of the universe is simply the sum over microscopic volumes:

Vtot(φ) ≡∑V

V NV (φ) =∑V

V ϕ†(V, φ)ϕ(V, φ)

Steffen Gielen, Max Planck Institute for Gravitational Physics 15/20

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Connection to Quantum Gravity Quantum cosmology – foundations and challenges

Towards a local quantum cosmology from quantum gravity

In this approach, the role of the Friedmann equation of cosmology changes– from being an exact equation for a global degree of freedom a(t) to aneffective equation for the total volume of the universe, after one has averagedover microscopic degrees of freedom.

One finds that V (φ) := 〈Vtot(φ)〉 satisfies [Oriti, Sindoni, Wilson-Ewing 2016]

(1

V (φ)

dV (φ)

)2

= 12πG+4voE

V (φ)−

4v2op

V (φ)2

within a class of group field theory models (E, vo are quantum numbers).

For the classical Friedmann equation, one would only have the first term. Theother terms are quantum gravity corrections, somewhat similar to those of loopquantum cosmology: they replace the Big Bang singularity by a “bounce”!

Steffen Gielen, Max Planck Institute for Gravitational Physics 16/20

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Connection to Quantum Gravity Quantum cosmology – foundations and challenges

Big Bounce?

A Big Bounce is a straightforward extension of the Big Bang of classical cosmology:

Steffen Gielen, Max Planck Institute for Gravitational Physics 17/20

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Connection to Quantum Gravity Quantum cosmology – foundations and challenges

Big Bounce?The idea of a “periodic world” goes back to Friedmann’s 1922 paper.

Steffen Gielen, Max Planck Institute for Gravitational Physics 18/20

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Connection to Quantum Gravity Quantum cosmology – foundations and challenges

Adding perturbations

The “local” viewpoint on quantum cosmology makes the incorporation ofinhomogeneities, i.e. perturbations of the FLRW universe, more straightforward.

In standard quantum cosmology, one extends the wavefunction [Halliwell, Hawking

1985]

ψ(a, φ)→ ψ(a, φ;χ~k, . . .

)and studies a Wheeler-DeWitt equation that includes terms up to quadratic orderin perturbations, using Born-Oppenheimer approximations etc.

If quantum cosmology arises from coarse graining quantum gravity, one cango beyond a “scale factor” by computing additional observables beyond the totalvolume

Vtot(φ) =∑V

V ϕ†(V, φ)ϕ(V, φ) .

Statistical pattern of inhomogeneities from quantum geometry! [SG, Oriti 2017]

Steffen Gielen, Max Planck Institute for Gravitational Physics 19/20

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Summary Quantum cosmology – foundations and challenges

Summary

• Standard cosmology, based on general relativity, is incomplete: it cannotaccount for the beginning of the Universe. Basic questions about initialconditions remain open.

• Natural to seek an extension of this framework by quantum theory. Simplemodels in terms of wavefunction of the universe ψ(a, φ) constructedstraightforwardly, but many technical and conceptual challenges. Further(somewhat arbitrary) assumptions needed to define the models.

• Classical cosmology should really arise from coarse graining of classicalrelativity; quantum cosmology from coarse graining of quantum gravity. Localatoms of geometry, rather than wavefunction of the universe.

• First results in the group field theory approach to quantum gravity suggestpossible resolution of Big Bang singularity by a bounce.

Steffen Gielen, Max Planck Institute for Gravitational Physics 20/20

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Thank you!