Non-singular quantum cosmology and scale invariant perturbations

th AMT Toulouse November 6, 2007 Patrick Peter Non-singular quantum cosmology and scale invariant perturbations Institut d Astrophysique de Paris GRεCO AMT - Toulouse - 6th November 2007

based upon Tensor perturbations in quantum cosmological backgrounds { JCAP 07, 014 (2005). [hep-th/0509232] } Gravitational wave background in perfect fluid quantum cosmology { PRD 73, 104017 (2006). [gr-qc/0605060] } A non inflationary model with scale invariant cosmological perturbations { PRD 75, 023516 (2007). [hep-th/0610205 } P.P., E. Pinho and N. Pinto Neto AMT - Toulouse - 6th November 2007 2

AMT - Toulouse - 6th November 2007

solves cosmological puzzles uses GR + scalar fields [(semi-)classical] Inflation: can be implemented in high energy theories makes falsifiable predictions is consistent with all known observations Alternative model??? string based ideas (PBB, branes, string gas, ) singularity and initial conditions AMT - Toulouse - 6th November 2007

t From R. Brandenberger, in M. Lemoine et al. Eds., Inflationary cosmology, Lect. Notes Phys. 738 (Springer, Berlin, 2007). t 0 H!1 Scalar field origin? t f (k) Trans-Planckian k t; l(t) = l 0 a(t) a 0 l Pl t R x Hierarchy (amplitude) t i (k) V (ϕ) ϕ 4 10 12 Singularity Validity of GR? t (± ) ; a(t) 0 E inf 10 3 M Pl t i AMT - Toulouse - 6th November 2007 4

A brief history of bouncing cosmology R. C. Tolman, On the Theoretical Requirements for a Periodic Behaviour of the Universe, PRD 38, 1758 (1931) G. Lemaître, L Univers en expansion, Ann. Soc. Sci. Bruxelles (1933) R. Durrer & J. Laukerman, The oscillating Universe: an alternative to inflation, Class. Quantum Grav. 13, 1069 (1996) Penrose: BH formation S 2 > S 1 Quantum nucleation? S 1 S 3 > S 2 > S 1 PBB - Ekpyrotic - Quantum cosmology -... 5 AMT - Toulouse - 6th November 2007

Pre Big Bang scenario: AMT - Toulouse - 6th November 2007 6

Pre Big Bang scenario: In the beginning God created the Heaven and the Earth, and the Earth was without form, and void; and the darkness was upon the face of the deep. And the Breath of God moved upon the face of the water. (Genesis, The Holy Bible). see also M.Gasperini & G. Veneziano, arxiv: hep-th/0703055 AMT - Toulouse - 6th November 2007 6

problem is addressed in the ekpyrotic model. Ekpyrotic scenario: K=0 4 D brane, Quasi BPS K=0 Visible 4 D brane, Quasi BPS Bulk Bulk Hidden where γ is a constant and κ = 8πG = 8π/m 2. Apart Pl from the sign, the potential is the one that leads to the well known power-law inflation[ model if the value of γ lies Sin a given range [24]. The 5 interaction between the two branes results in one (bulk or hidden) brane moving towards the other (visible) boundary until they collide. This impact time is then identified with the Big-Bang[ of standard Fcosmol- ogy. Slightly before that time, the exponential R(4) potential abruptly goes to zero so the boundary brane is led to a d 5 x g R 1 5 (5) M 2 ( ϕ)2 3 2 5 [ e 2ϕ F 2 5! S = d 4 x g 4 4 M 2κ 1 2 ( φ)2 V (φ) singular transition at which the kinetic energy of the bulk 4 brane is converted into radiation. The result is, from this point on, exactly similar to standard [ big bang cosmology, V (ϕ) = V i exp 4 ] with the difference that the flatness problemπγ is claimed to be solved by saying our Universe originated as a(ϕ BPS ϕ i ), state (see however [23]). m Pl ] ],, Visible R orb FIG. 1: Schematic representation of the old ekpyrotic model as a bulk boundary branes in an effective five dimensional theory. Our Universe is to be identified with the visible brane, and a bulk brane is spontaneously nucleated near the hidden brane, moving towards our universe to produce the Big-Bang singularity and primordial perturbations. In the new ekpyrotic scenario, the bulk orb brane is absent and it is the hidden R brane that collides with the visible one, generating the hot Big Bang singularity. Hidden In the old scenario [8], the five dimensional AMT bulk- Toulouse is - 6th November 2007 7 also assumed to contain various fields not described here, whose excitations can lead to the spontaneous nucleation of yet another, much lighter, freely moving, brane. In FIG. 2: Scale factor in the new ekpyrotic scenario. The Universe starts its evolution with a slow contraction phase a ( η) 1+β with β = 0.9 on the figure. The bounce itself is explicitly associated with a singularity which is approached

Initial conditions fixed in the a(η) contracting era Φ+ g = T ij (k) Φ g Φ + d Φ d η AMT - Toulouse - 6th November 2007 8

Initial conditions fixed in the a(η) contracting era Φ+ g = T ij (k) Φ g Φ + d Φ d η??????? AMT - Toulouse - 6th November 2007 8

4D Quantum cosmology Perfect fluid: bounce no horizon problem homogeneity = anthropic solution? flatness = time? monopoles =??? Results: AMT - Toulouse - 6th November 2007 9

Digression: about QM Schrödinger Polar form of the wave function Hamilton-Jacobi quantum potential AMT - Toulouse - 6th November 2007 10

Ontological interpretation (BdB) Trajectories satisfy Properties: strictly equivalent to Copenhagen QM probability distribution (attractor) classical limit well defined state dependent intrinsic reality non local no need for external classical domain! AMT - Toulouse - 6th November 2007 11

The two-slit experiment: AMT - Toulouse - 6th November 2007 12

Quantum Potential in the two-slit experiment AMT - Toulouse - 6th November 2007 13

Trajectories in the two-slit experiment AMT - Toulouse - 6th November 2007 14

Diffraction by a potential AMT - Toulouse - 6th November 2007 15

Quantum cosmology Perfect fluid: Schutz formalism ( 70) Velocity potentials canonical transformation: + rescaling (volume ) + units : simple Hamiltonian: AMT - Toulouse - 6th November 2007 16

Wheeler-De Witt i Ψ T = 1 4 a3(ω 1)/2 a HΨ = 0 [ a (3ω 1)/2 ] Ψ + KaΨ [ a ] [ Technical trick: [ K K = 0 = χ 2a2(1 ω /2 3(1 ω) = i Ψ T = 1 4 2 Ψ χ 2 space defined by χ > 0 constraint (cf. Schrödinger & well) Ψ Ψ χ = Ψ Ψ χ + Gaussian initial wave function AMT - Toulouse - 6th November 2007 17

alternative way of getting the solution: WKB exact superposition: Ψ = e iet ρ(e)ψ E (T )de Gaussian wave packet e (ET 0) 2 Ψ = [ ] ( ) ] ] 1 ( ( ( 4 8T 0 π (T0 2 + T 2 ) 2 exp T 0χ 2 T0 2 + T 2 ) e is(χ,t ) phase S = T χ2 T 2 0 + T 2 + 1 2 arctan T 0 T π 4 Bohmian trajectory ȧ = {a, H} [ ( T ) 2 ] 1 3(1 ω) a = a0 1 + T 0 AMT - Toulouse - 6th November 2007 18

( ) K = 1 K = 0 [ ( ) a(t ) K = +1 Q(T ) quantum potential AMT - Toulouse - 6th November 2007 19

What about the perturbations? Hamiltonian up to 2 nd order H = H (0) + H (2) + factorization of the wave function Ψ = Ψ (0) (a, T ) Ψ (2) [v, T ; a (T )] comes from 0 th order Φ = 3l2 Pl 2 ρ + p ( v a) ( ) ω a d dη ( ) Bardeen (Newton) gravitational potential ds 2 = a 2 (η) { (1 + 2Φ) dη 2 [(1 2Φ) γ ij + h ij ] dx i dx j} conformal time dη = a 3ω 1 dt AMT - Toulouse - 6th November 2007 20

+ canonical transformations: i Ψ (2) ( η = ( d 3 ( x 1 2 δ 2 δv 2 + ω 2 v,iv,i a a ) Ψ (2) (c 2S k2 a ) ( ) Fourier mode v k + a v k = 0 c 2 S = ω 0 vacuum initial conditions v k e ic S kη 2cS k + evolution (matchings and/or numerics) AMT - Toulouse - 6th November 2007 21

10 10 10 8 "e S(x) #m S(x) S(x) s(x) = a 1 2 (1 3ω) v(x) T0 S(x) 10 6 10 4 10 2 ~ k=10-3 -1!10 8-8!10 7-6!10 7-4!10 7-2!10 7 0 2!10 7 4!10 7 6!10 7 8!10 7 1!10 8 x 10 8 10 6 "e v(x) #m v(x) v(x) x T T 0 v(x) 10 4 10 2 10 0 ~ k=10-3 -1!10 8-5!10 7 0 5!10 7 1!10 8 x AMT - Toulouse - 6th November 2007 22

( () ) spectrum P Φ k 3 Φ k 2 A 2 S kn S 1 (k 2 a ) P id. grav. waves: µ + µ = 0 µ h a a µ ini exp( ikη) kη P h k 3 h k 2 A 2 T kn T 12 same dynamics + initial conditions same spectrum n T = n S 1 = 12ω 1 + 3ω CMB normalisation A 2 S = 2.08 10 10 bounce curvature T 0 a 3ω 0 1500l Pl AMT - Toulouse - 6th November 2007 23

Numerical check "=0.01 10 5 10 0 Spectra 10-5 10-10 10-15 P! (x)/ k ~ n S -1, k=10 ~ -7 P h (x)/ k ~ n T, k=10 ~ -7 P! (x)/ k ~ n S -1, k=10 ~ -5 P h (x)/ k ~ n T, k=10 ~ -5 10-20 -100-75 -50-25 0 25 50 75 100 kx ~ AMT - Toulouse - 6th November 2007 24

Full numerical spectrum P! ( ~ k)/ ~ k n S -1, P h ( ~ k)/ ~ k n T and T/S 10 2 10 0 10-2 P! ( ~ k)/ ~ k n S -1 P h ( ~ k)/ ~ k n T T/S 10-4 10-6 10-5 10-4 10-3 10-2 10-1 10 0 10 1 ~ k AMT - Toulouse - 6th November 2007 25

WMAP3 constraint n S = 0.96 ± 0.02 = n S < 8 10 4 predictions spectrum slightly blue power-law + concordance T S = C(T) 10 C (S) 10 = F (Ω, ) A2 T A 2 S w T S 4 10 2 n S 1 AMT - Toulouse - 6th November 2007 26

WMAP3 constraint n S = 0.96 ± 0.02 = n S < 8 10 4 predictions spectrum slightly blue power-law + concordance 0.62 T S = C(T) 10 C (S) 10 = F (Ω, ) A2 T A 2 S w T S 4 10 2 n S 1 AMT - Toulouse - 6th November 2007 26