Quantum Universe. Neil Turok. with thanks to S. Gielen, J. Feldbrugge*, J-L. Lehners, A. Fertig*, L. Sberna*

Transcription

1 Quantum Universe Neil Turok with thanks to S. Gielen, J. Feldbrugge*, J-L. Lehners, A. Fertig*, L. Sberna*

2 Credit: Pablo Carlos Budassi

3 astonishing simplicity: just 5 numbers today Expansion rate: (Temperature) (Age) 67.8±0.9 km s 1 Mpc ± K ±0.038 bn yrs Measurement Error 1%.1%.3% Baryon-entropy ratio 6±.1x % energy Dark matter-baryon ratio 5.4± 0.1 2% Dark energy density 0.69±0.006 x critical 2% Scalar amplitude 4.6±0.006 x % geometry Scalar spectral index (scale invariant = 0) n s -.033± % +m ν 's; but Ω k, 1+ w DE, dn s d ln k, δ 3, δ 4..,r = A gw A s consistent with zero

4 Behind it all is surely an idea so simple, so beautiful, that when we grasp it - in a decade, a century or a millenium - we will all say to each other, how could it have been otherwise? How could we have been so stupid? John A. Wheeler, How Come the Quantum? Ann. N.Y.A.S., 480, (1986).

5 10 26 m dark energy length ( ) galaxies stars (log) length light nuclei planets heavy elements cells molecules complexity m Planck length ( )

6 Nature has found a way to create a huge hierarchy of scales which appears simpler than any current theory. This is a fascinating situation, demanding big new ideas. One of the most conservative is quantum cosmology.

7 The universe must be quantum 1. because gravity must be quantum Feynman Lectures on Gravita2on, Ed. B. Ha9ield, Addison-Wesley (1995) p. 12

8 2. because classical cosmology is singular. cf. hydrogen atom, UV catastrophe. 3. because the vacuum energy has an (apparently) divergent UV quantum contribuaon. It controls the causally accessible volume (UV IR connecaon). 4. because the observed primordial fluctuaaons take the form of a free quantum field.

9 cosmic inflation V (φ) φ

10 quantum fluctuations large scale structure φ

11 fine tuning V (φ) φ

12 the inflationary multiverse Anything that can happen will happen - and it will happen an infinite number of times Guth

13 rethinking quantum cosmology s/ S. Gielen, , (Phys. Rev. LeH. 117 (2016) )

14 quantum gravity U ' U U U '

15 B.S. DeWiK, Phys. Rev. 160, 1967 (p 1140)

16 Many beautiful theoretical works: basic conceptual problems remain unsolved What equation determines the the quantum state? What is the probability of a configuration? This is a tremendous opportunity for theory: observations already provide vital clues.

17 Simplest case: conformal invariant matter T µ µ = 0 4d FRW ds 2 a(t) 2 ( dt 2 + γ ij dx i dx j ); R (3) = 6κ H 3, E 3,S 3 κ < 0,= 0,> 0 Friedmann (!a) 2 = 1 κ a 2 a t as a 0 perfect bounce ds 2 t 2 ( dt 2 + γ ij dx i dx j ) as t 0 unique analytic extension from negative to positive ( or a) t we call this a perfect bounce S. Gielen and N. Turok PRL, 117, (2016)

18 Consider Einstein gravity plus perfect! radiation fluid and M conformally coupled scalar fields χ (e.g. Higgs) It is mathematically convenient to introduce another scalar via a Weyl transformation φ S = ( 1 2 ( φ)2 ( χ)! 2 ) + 1 ( 12 φ 2! χ 2 )R ρ(n) nu µ ϕ µ Invariant under φ Ω 1 φ,!! χ Ω 1 χ, gµν Ω 2 g µν, ρ Ω 4 ρ etc φ may then be used to describe the expansion of the universe (in a Weyl gauge where the metric is fixed).

19 Pick Weyl gauge in which metric is static ds 2 = N(t) 2 dt 2 + γ ij dx i dx j ; ρ r = r = const define action x α = 1 2r (φ,! χ); η αβ = diag( 1,1,...,1) S = m dt N 1!x α!x N(κ x α x +1) 2 α α - relativistic oscillator comoving volume m = 2V 0 r radiaqon density

20 Big crunch-big bang coordinates on space of fields (zero modes) - superspace gravity x µ = at µ, t 2 = 1 H M a > 0 a < 0 x µ = bs µ, s 2 = 1 ds M S ~ (+ b! 2 b 2!φ 2 ) antigravity S ~ (!a 2 + a 2!φ 2 ) gravity

21 Quantization: Hamiltonian x H = N p τ = N dt > 0 Causal (Feynman) propagator x' G F (x, x') = gauge choice ( 2 2m + m 2 (1+κ ) x2 ) 0 t 1; N = τ dn x e i NĤ! x' = i x Ĥ 1 x' Ĥ G (x, x') = iδ (x x') x F 0 0 = dn Dxe i! S x' 0 e.g. κ =0, dτ x i m 2 Dxe 1 0 (more formal but fixes bcs, pos. freq modes) dt(!x2 τ τ ) 0 = i dτ ( ) M +1 2 m 2πiτ e i m 2 ( σ τ +τ ) ; σ = (x x') 2 (usual Feynman propagator in Schwinger representation, in M+1 Minkowski) (*)

22 H M Fourier transform on, saddle point in determines behaviour at large ; a G F (a,a') e iωa, a ~ e iωa', a' This defines the positive and negative frequency modes, from which may be determined by solving (*) via the Wronskian method G F Note: 1) in quantum cosmology, positive frequency expanding! 2) imposing Neumann or Dirichlet bcs at a=0 (which most prior proposals do) prohibits an expanding universe 3) (for perturbation modes) no need for an independent specification of `Bunch-Davies vacuum. Vacuum is consequence of path integral formulation. τ

23 κ > 0 no asymp states - non-normalizable in norm offered below propagator ambiguous (multiple saddle points) κ < 0 Only κ = 0 G F (x,x') ~ e i m 2 κ x 2 as x 2 quantum tunneling to classically allowed solutions in antigravity regime seems quantum consistent path integrals are Gaussian, τ integral is not if m is large, FRW background is heavy : quantum spreading and backreaction are small except around Λ a = 0 when included, among classically allowed universes with asymptotic states, κ = 0 is the most probable

24 emergence of time (à la Wheeler) final expanding a contracting a w/ J. Feldbrugge t P = 1 2 a2 a' 2 +o(m 1 ) initial a' a' t P n irr = o(m 1 ) quantum corrections Cosmological (Einstein-frame) proper time or, in a bounce a > 0 a' < 0 t P = 1 0 a(t) dt t P = 1 2 (a2 + a' 2 ) + o(m 1 ) t P n = +o(m 1 ) quantum fluctuations Leading terms are Just the classical results time is determined by boundary 3-geometries

25 Generic metrics: consider deviations around flat ( κ = 0 ) FRW Anisotropies: ds 2 a(t) 2 ( dt e cλ i dx 2 ); λ = 0 i i i=1 Line element on space of moduli : ds 2 mod = da 2 + a 2 (dh M 2 + de 2 2 ) scalars anisotropy moduli (Note: standard model Higgs (M=1) sufficient to remove BKL chaos)

26 Ordering ambiguities resolved by invariance under coordinate transformations on superspace and redefinitions of lapse N Ω 2 S N, which is equivalent to g µν It follows that The equation Ω 2 g S µν Ĥ =! 2 ( " S +ξ R S +V S ) with ξ = D 2 4( D 1) Ĥ x G F (x,x') = iδ (x x') / g S, V S Ω 2 V S is then invariant Halliwell Moss

27 Under reasonable assumptions, near a = 0 we have ( d 2 da 2 + c' a 2 )Ψ (±) = m 2 Ψ (±), c' =! ς ( M 1) 2( M +2),! ς = conserved anisotropy and scalar momenta This potential occurs in integrable models (Calogero-Sutherland) and, for < a < it is the simplest case of PT-invariant quantum mechanics. Znojil Dorey et al For c' 1 4 the QM Hamiltonian has a real, positive spectrum. So, at least for M>1 and for an open set of anisotropy+scalar momenta, continuation across a=0 seems completely safe.

28 The surprising inverse square potential: classically, m!a2 τ 2 + C ma 2 = m a 2 = C m 2 + τ 2 (t t 0 ) 2 no time delay: invisible! quantum mechanically, there isn t even a phase shift (consequence of scale invariance) Proposals which fix the wavefunction at a=0 violate the correspondence principle and have other undesirable features (superposition of expanding and contracting).

29 Natural quantum propagation of modes a e +ima e +ima e ima X anqgravity excursion e ima

30 General prescription a PosiQve frequency NegaQve frequency Ψ = Ψ (+) + Ψ ( ) analyqc lhp analyqc uhp X

31 We have solved for the scalar and tensor perturbations at linear and nonlinear order, concluding that for a perfect radiation fluid there is precisely zero particle production across a bounce. We also found some surprises S. Gielen and N. Turok PRL, 117, (2016)

32 Our findings: a Incoming positive frequency mode Outgoing positive frequency mode In absence of running or soft masses, particle production vanishes across the singularity. It will be UV-finite when those effects are included.

33 a breaks down due to cosmological blueshift kt > ε breaks down due to wave resonance - shocks form kt < ε 1 perturbation theory valid

34 A quantum measure for the universe The fundamental object in quantum geometrodynamics is the propagator. Let us try to use it to define probabilities. a The scale factor is a natural dynamical variable. But the theory is invariant under PT : a a. Consider the amplitude to go from one copy to the other. x x d M +1 x g S G (x, x) 2 F Normalizable if Λ is positive but only invariant if number of zero modes M=3. In addition to two anisotropy dofs, there must be a scalar (Higgs!)

35 causa sui cosmos physical size time

36 By causa sui I understand that whose essence involves existence, or that whose nature cannot be conceived unless exisqng. Spinoza, Ethics, 1677

37 causal structure big bang big bang Among universes with these asymptotic states, fixing as a function of, one finds is most probable κ κ = 0 Λ and maximising the probability

38 This scenario explains Why the universe is expanding Why it apparently `began in a singularity Why dark energy is essential with a modest addition (one extra scalar) it can also explain the quantum generation of scale-invariant curvature perturbations a more minimal and predictive cosmology