Bielefeld - 09/23/09. Observing Alternatives to Inflation. Patri Pe r. Institut d Astrophysique de Paris. Bielefeld - 23 rd september 2009

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1 Bielefeld - 09/23/09 Observing Alternatives to Inflation Patri Pe r Institut d Astrophysique de Paris GRεCO Bielefeld - 23 rd september 2009

2 Problems with standard model: Singularity Horizon Flatness Homogeneity Perturbations Dark matter Dark energy / cosmological constant Baryogenesis... Accepted solution = INFLATION Topological defects (monopoles) (Linde s book) Bielefeld - 23 rd september

3 og t Primordial Cosmology Peter & Uzan 2 Primordial Cosmology Patrick Peter Jean-Philippe Uzan oxford graduate texts Problems with standard model: Singularity Horizon Flatness Homogeneity Perturbations Dark matter Dark energy / cosmological constant Baryogenesis... Topological defects (monopoles) Bielefeld - 23 rd september

4 Inflation: solves cosmological puzzles uses GR + scalar fields [(semi-)classical] can be implemented in high energy theories makes falsifiable predictions is consistent with all known observations string based ideas (brane inflation,...) Alternative model??? string based ideas (PBB, other brane models, string gas, ) Quantum gravity / cosmology singularity, initial conditions & homogeneity bounces provide challengers! Bielefeld - 23 rd september 2009

5 t From R. Brandenberger, in M. Lemoine, J. Martin & P. Peter (Eds.), Inflationary cosmology, Lect. Notes Phys. 738 (Springer, Berlin, 2007). t 0 H!1 Scalar field origin? t f (k) t R t i (k) k x Trans-Planckian Hierarchy (amplitude) V (ϕ) ϕ 4 Singularity Validity of GR? t; l(t) = l 0 a(t) a l Pl t (± ) ; a(t) 0 t i E inf 10 3 M Pl Bielefeld - 23 rd september

6 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) A. A. Starobinsky, On one non-singular isotropic cosmological model, Sov. Astron. Lett. 4, 82 (1978) R. Durrer & J. Laukerman, The oscillating Universe: an alternative to inflation, Class. Quantum Grav. 13, 1069 (1996) Penrose: BH formation Quantum nucleation? S 2 > S 1 S 1 S 3 > S 2 > S 1 PBB - Ekpyrotic - Quantum cosmology -... Bielefeld - 23 rd september 2009

7 Pre Big Bang scenario: (cf. M.Gasperini & G. Veneziano, arxiv: hep-th/ ) Bielefeld - 23 rd september

8 Pre Big Bang scenario: (cf. M.Gasperini & G. Veneziano, arxiv: hep-th/ ) a E g S PRE BIG BANG POST BIG BANG PRE BIG BANG POST BIG BANG H 2 Ρe Φ pe Φ t Ρ E g S Η V H E string frame Einstein frame Bielefeld - 23 rd september

9 branes (orbifold fixed planes), one of which to be later identified with our universe, are separated by a finite gap. Both are BPS states [13], i.e., they can be described at low energy by an effective N = 1 supersymmetric model, so that their curvature vanishes. This is how the flatness 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 [ V (ϕ) = V i exp 4 ] πγ (ϕ ϕ i ), (3) m Pl 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 in a given range [24]. S 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 S 4 = d 4 x Big-Bang[ of standard cosmology. Slightly before that time, the exponential R(4) potential abruptly goes to zero so the g boundary brane is led to a 4 singular transition M 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, with the difference that the flatness problem is claimed to be solved by saying our Universe originated as a BPS state (see however [23]). d 5 x [ g 5 R (5) 1 M 5 2 ( ϕ)2 3 e 2ϕ F 2 ], 2 5! [ F 2κ 1 ] 2 ( φ)2 V (φ), [ V (ϕ) = V i exp 4 ] πγ (ϕ ϕ i ), 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 bulk is also assumed to contain various fields not described here, FIG. 2: Scale factor in the new ekpyrotic scenario. The whose excitations can lead to the spontaneous nucleation Universe starts its evolution with a slow contraction phase of yet another, much lighter, freely moving, Bielefeld brane. - In 23 rd september a ( η) 1+β 2009 with β = 0.9 on the figure. The bounce itself 9 the so-called new scenario [9], and its cyclic extension [23], it is the hidden boundary brane that is able to move in the bulk. In both cases, this extra brane, if assumed BPS (as demanded by minimization of the action) is flat, parallel to the boundary branes and initially is explicitly associated with a singularity which is approached by the scalar field kinetic term domination phase, and the expansion then connects to the standard Big-Bang radiation dominated phase.

10 Standard Failures and some solutions Singularity Horizon Flatness Homogeneity Perturbations Others d dt t diffusion t Hubble λ R 1/3 H Merely a non issue in the bounce case! Weyl curvature hypothesis + low initial density + diffusion ( 1+ λ AR 2 H d H a(t) ) t t i dτ a(τ) Ω 1 = 2 ä ȧ 3 accelerated expansion (inflation) or decelerated contraction (bounce) enough time to dissipate any wavelength = vacuum state! see coming slides can be made divergent easily if dark matter/energy, baryogenesis,... ä<0 & ȧ<0 Bielefeld - 23 rd september t i

11 Initial conditions fixed in the contracting era a(η) Φ+ g Φ + d = T ij (k) Φ g Φ d η??????? Bielefeld - 23 rd september

12 S [ ] Self consistent bounce: ( ) dr ds 2 =dt 2 a 2 2 [ (t) 1 Kr 2 + r2 dω 2 One d.o.f. + 4 dimensions G.R. S = d 4 x [ ( R g 6l 2 Pl H 2 = 1 3 ( 1 2 ϕ2 + V ] 2 µϕ µ ϕ V (ϕ) K ) K a 2 Positive spatial curvature N(t) 0 H(t) F. Falciano, M. Lilley & P. P., Phys. Rev. D77, (2008) Bielefeld - 23 rd september t -3 12

13 ( ) ( ) ( Perturbations: ( ) ds 2 = a 2 (η) { (1 + 2Φ)dη 2 [(1 2Φ) γ ij + h ij ]dx i dx j} ( Φ = 3Hu 2a 2 θ u + [k 2 θ θ 3K ( 1 c 2 ) ] u =0 S 15 ( ) θ 1 a [ 10 2 ρ ϕ ρ ϕ + p ϕ ( 1 3K ρ ϕ a 2 ( ) ] P ζ = Ak n S 1 cos 2 ( ω k ph k ) ) + ψ V u (t) ! BST (t) k 3! BST t k or n Bielefeld - 23 rd september

14 Data! l 1 C l 2 Π Wavelength (h 1 Mpc) l No obvious oscillations... Power spectrum today P(k) (h 1 Mpc) CMB SDSS Galaxies Cluster abundance Gravitational lensing Lyman α forest Wave number k (h / Mpc) Bielefeld - 23 rd september

15 l(l+1)c l /2!!!"!#"$ % WMAP1 WMAP3 #=10 3 k * =10-2 Mpc l Bielefeld - 23 rd september

16 ( K-bounce: L = p(x, ϕ) X 1 2 gµν µ ϕ ν ϕ T µν =(ρ + p) u µ u ν pg µν ρ 2X p X p u µ µϕ 2X vanishing spatial curvature possible in 4 dimensions G.R.? ρ (t bounce ) = 0 = p (t bounce ) < 0 H(t) precooling Bounce de Sitter expansion RDE t preheating PPP de Sitter contraction Bielefeld - 23 rd september

17 Model for the bounce phase only: p = p 0 + p X (X X 0 )+p ϕ ϕ + p Xϕ ϕ(x X0) p XX(X X 0 ) p ϕϕϕ "(t) c!! (t) -1 "+p H(t) c!! (t) H(t) "(t) "+p c!! (t) H(t) "+p "(t) #(t) k=10-1 a(t) t Slow #(t) k= "#####" (t) k=10-1 a(t) "#####" (t) k= t 10 4 a(t) "#####" (t) k=10-1 "#####" (t) k= t Fast Oscillations + ζ conserved R. Abramo & P. P., JCAP 09, 001 (2007) Bielefeld - 23 rd september

18 A k FB MB SB k k-mode mixing... Bielefeld - 23 rd september

19 2 d.o.f. + 4 dim G.R. H 2 = l 2 Pl [ ρ + a 3(1+w +) ρ ] a 3(1+w ) ! k / k n th 10 0! k / k n th x k-mode inversion x F. Finelli, P. P. & N. Pinto-Neto, Phys. Rev. D77, (2008) Bielefeld - 23 rd september

20 A specific model: 4D Quantum ( cosmology Perfect fluid: no horizon problem if ω > 1 3 bounce Results: Bielefeld - 23 rd september

21 Digression: about QM Schrödinger Polar form of the wave function Hamilton-Jacobi quantum potential Bielefeld - 23 rd september

22 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! Bielefeld - 23 rd september

23 Quantum cosmology + canonical transformation + rescaling (volume ) + units = a simple Hamiltonian: [ [ [ ] + Technical trick: space defined by Wheeler-De Witt HΨ = 0 K = 0 = χ 2a3(1 ω)/2 3(1 ω) χ > 0 constraint = i Ψ T = 1 2 Ψ 4 χ 2 Ψ Ψ χ = Ψ Ψ χ Bielefeld - 23 rd september 2009

24 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 [ ] ] 1 ( ( 4 8T 0 Ψ = π (T0 2 + T 2 ) 2 exp ( T 0χ 2 ) T0 2 + T 2 e is(χ,t ) phase S = T χ2 T T arctan T 0 T π 4 Bohmian trajectory ȧ = {a, H} a = a 0 [1+ ( T T 0 ) 2 ] 1 3(1 ω) Bielefeld - 23 rd september

25 ( ) ( ) K = 1 K = 0 [ ( ) a(t ) K = +1 Q(T ) quantum potential Bielefeld - 23 rd september

26 What about the perturbations? Hamiltonian up to 2 nd order factorization of the wave function comes from 0 th order Φ = 3l2 Pl ρ + p 2 ω a d ( v dη a) Bardeen (Newton) gravitational potential H = H (0) + H (2) + Ψ = Ψ (0) (a, T ) Ψ (2) [v, T ; a (T )] ( ) ( ) 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 Bielefeld - 23 rd september

27 ( ) ( ) + canonical transformations: i Ψ ( (2) = d 3 ( ( x 1 η 2 δ 2 δv 2 + ω ) 2 v,iv,i a a Ψ (2) ( ) Fourier mode ) v k + (c 2S k2 a v k = 0 a c 2 S = ω 0 vacuum initial conditions v k e ic S kη 2cS k + evolution (matchings and/or numerics) Bielefeld - 23 rd september

28 "e S(x) #m S(x) S(x) s(x) = a 1 2 (1 3ω) v(x) T0 S(x) ~ k= !10 8-8!10 7-6!10 7-4!10 7-2! !10 7 4!10 7 6!10 7 8!10 7 1!10 8 x "e v(x) #m v(x) v(x) x T T 0 v(x) ~ k= !10 8-5! !10 7 1!10 8 x Bielefeld - 23 rd september

29 10 5 "= Spectra 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= kx ~ Bielefeld - 23 rd september

30 Full numerical spectrum P! ( ~ k)/ ~ k n S -1, P h ( ~ k)/ ~ k n T and T/S P! ( ~ k)/ ~ k n S -1 P h ( ~ k)/ ~ k n T T/S ~ k Bielefeld - 23 rd september

31 ( ) ( () ) spectrum P Φ k 3 Φ k 2 A 2 kn S 1 S ) P id. grav. waves: µ + (k 2 a µ = 0 µ h a a µ ini exp( ikη) P h k 3 h k 2 A 2 kη kn T T same dynamics + initial 12 conditions n T = n S 1 = 12ω 1 + 3ω same spectrum CMB normalisation A 2 S = bounce curvature T 0 a 3ω l Pl Bielefeld - 23 rd september

32 WMAP constraint n S =0.96 ± 0.02 = w < 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 n S 1 P. P. & N. Pinto-Neto, Phys. Rev. D78, (2008) Bielefeld - 23 rd september

33 Cosmology without inflation? New solutions to old puzzles No singularity G.R.... monopoles =??? Dark energy... New predictions? Oscillations? Bielefeld - 23 rd september