Cosmological constraints on quantum gravity Jakub Mielczarek - PowerPoint PPT Presentation

Cosmological constraints on quantum gravity Jakub Mielczarek Jagiellonian University, Cracow National Centre for Nuclear Research, Warsaw 4 September, 2014, Trieste Jakub Mielczarek Cosmological constraints on quantum gravity

Jakub Mielczarek Cosmological constraints on quantum gravity

Quantum corrections The holonomy corrections mimic the loop quantization. For the FRW model, the holonomy corrections can be introduced by the following replacement in the classical Hamiltonian: µγ ¯ k → K [ n ] := sin( n ¯ k ) ¯ , n ¯ µγ where ¯ p = a 2 . Furthermore, k is canonically conjugated with ¯ p δ where − 1 / 2 ≤ δ ≤ 0 and n ∈ N . µ ∝ ¯ ¯ Holonomy corrections can be seen as bending (periodification) of the phase space. For the FRW model, the classical phase space Γ G = R × R is deformed onto Γ Q G = R × U (1). Jakub Mielczarek Cosmological constraints on quantum gravity

The holonomy corrections deform the classical Friedmann equation: � � H 2 = 8 π G 1 − ρ ρ , 3 ρ c 3 where the critical energy density ρ c = 8 π G λ 2 γ 2 ∼ ρ Pl . Because ρ ≤ ρ c , the classical Big Bang singularity is replaced by the non-singular Big Bounce 1 . Contracting and expanding branches are causally connected. 1 Bojowald, Ashtekar, Paw� lowski, Singh, ... Jakub Mielczarek Cosmological constraints on quantum gravity

For the model with a massive scalar field, the Big Bounce unavoidably leads to the phase of cosmic inflation 2 . Klein-Gordon equation on the FRW background: φ + 3 H ˙ ¨ φ + m 2 φ = 0 . 2 J. Mielczarek, Phys. Rev. D 81 (2010) 063503, A. Ashtekar and D. Sloan, Phys. Lett. B 694 (2010) 108, J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D 81 (2010) 104049 Jakub Mielczarek Cosmological constraints on quantum gravity

Inhomogeneities - Old story Using the EOM for tensor modes with holonomy corrections derived in Ref. 3 one can compute the following power spectra 4 : 3 M. Bojowald and G. M. Hossain, Phys. Rev. D 77 (2008) 023508 4 J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D 81 (2010) 104049 Jakub Mielczarek Cosmological constraints on quantum gravity

Shape of the power spectrum - intuitive explanation 1 Scale of the bump in the power spectrum k ∗ ∼ λ ∗ . Jakub Mielczarek Cosmological constraints on quantum gravity

Pessimistic plot The present value of λ ∗ is very sensitive on duration of inflationary phase 5 . It makes possibility of observing the suppression due to the bounce almost improbable. There is only very narrow observational window for kinetic bounces with F B ∼ 10 − 13 . 5 J. Mielczarek , M. Kamionka, A. Kurek and M. Szydlowski, “Observational hints on the Big Bounce,” JCAP 1007 (2010) 004. Jakub Mielczarek Cosmological constraints on quantum gravity

The primordial gravitational waves can be used to impose observational constraints on the Big Bounce scenario. It becomes possible thanks to observational limitations on the B-type polarization of the CMB. The Big Bounce is asymmetric 6 : [0 , 1] ∋ F B := V ( φ B ) > 2 . 3 · 10 − 13 . ( ρ = T + V ) ρ c 6 J. Mielczarek, M. Kamionka, A. Kurek and M. Szyd� lowski, JCAP 1007 (2010) 004; J. Grain, A. Barrau, T. Cailleteau and J. Mielczarek, Phys. Rev. D 82 (2010) 123520 Jakub Mielczarek Cosmological constraints on quantum gravity

Inhomogeneities - New story Problems: The procedure of introducing quantum corrections suffers from ambiguities. In general, the algebra of modified constraints is not closed: {C Q I , C Q IJ ( A j i ) C Q J } = g K b , E a K + A IJ . Can we introduce quantum corrections in the anomaly-free manner (i.e. such that A IJ = 0)? We found that, at least for linear inhomogeneities on the flat FRW background, the answer is affirmative: scalar perturbations - T. Cailleteau, J. Mielczarek, A. Barrau, J. Grain, Class. Quantum Grav. 29 (2012) 095010, vector perturbations - J. Mielczarek, T. Cailleteau, A. Barrau and J. Grain, Class. Quant. Grav. 29 (2012) 085009, tensor perturbations - no problem with anomalies. Jakub Mielczarek Cosmological constraints on quantum gravity

Cosmological perturbations Tedious calculations lead us to deformed EOM for the Munhanov variable 7 : ′′ d 2 s ∇ 2 v − z d τ 2 v − c 2 z v = 0 , where the holonomy corrections are introduced through k ) = 1 − 2 ρ 3 µ ¯ c 2 s = Ω = cos(2 γ ¯ where ρ c = 8 π G ∆ γ 2 ∼ ρ Pl . ρ c Similarly for the GWs and scalar matter 8 . Mixed type EOMs: hyperolic - for ρ < ρ c / 2 (Ω > 0), parabolic - for ρ = ρ c / 2 (Ω = 0), elliptic - for ρ > ρ c / 2 (Ω < 0). 7 T. Cailleteau, J. Mielczarek, A. Barrau, J. Grain, Class. Quantum Grav. 29 (2012) 095010 8 T. Cailleteau, A. Barrau, J. Grain and F. Vidotto, Phys. Rev. D 86 (2012) 087301, J. Mielczarek, PhD thesis (2012). Jakub Mielczarek Cosmological constraints on quantum gravity

Algebra of constraints: { D [ N a 1 ] , D [ N a D [ N b 1 ∂ b N a 2 − N b 2 ∂ b N a 2 ] } = 1 ] , � � S Q [ N ] , D [ N a ] − S Q [ N a ∂ a N ] , = � � � � S Q [ N 1 ] , S Q [ N 2 ] g ab ( N 1 ∂ b N 2 − N 2 ∂ b N 1 ) = Ω D , The algebra is closed but deformed with respect to the classical case due to presence of the factor k ) = 1 − 2 ρ 3 µγ ¯ Ω = cos(2¯ ∈ [ − 1 , 1] where ρ c = 8 π G ∆ γ 2 ∼ ρ Pl . ρ c What is the interpretation? Classically, we have � ¯ � N p ∂ a ( δ N 2 − δ N 1 ) { S [ N 1 ] , S [ N 2 ] } = sD , ¯ where s = 1 corresponds to the Lorentzian signature and s = − 1 to the Euclidean one. Jakub Mielczarek Cosmological constraints on quantum gravity

Signature change The sign of Ω reflects a signature of space: Ω > 0 ( ρ < ρ c / 2) - Lorentzian signature, Ω < 0 ( ρ > ρ c / 2) - Euclidean signature 9 Based on this, a new picture of the Big Bounce emerges: In the Planck epoch, space-time becomes a four dimensional Euclidean space. Similarity to the Harte-Hawking no-boundary proposal. Causal connection between the branches is limited. 9 J. Mielczarek, “Signature change in loop quantum cosmology,” Springer Proc. Phys. 157 (2014) 555 [arXiv:1207.4657 [gr-qc]]. Jakub Mielczarek Cosmological constraints on quantum gravity

Silence in LQC The light cones are collapsing onto the time lines for. ρ → ρ c / 2. √ � 1 − 2 ρ The effective speed of light c eff = Ω = ρ c → 0. Communication between different space points is forbidden 10 . The similar behavior is expected at the classical level by virtue of the famous BKL conjecture (Belinsky-Khalatnikov-Lifshitz). 10 J. Mielczarek, “Asymptotic silence in loop quantum cosmology,” AIP Conf. Proc. 1514 (2012) 81 Jakub Mielczarek Cosmological constraints on quantum gravity

Carrollian limit “A slow sort of country ..., ... now, here, you see, it takes all the running you can do to stay in the same place ...” Lewis Carroll Jakub Mielczarek Cosmological constraints on quantum gravity

Silent initial conditions? Is there any natural choice of the initial conditions at the beginning of the Lorentzian phase (silent surface) 11 ? Are correlations between physical quantities at the different points vanishing (white noise)? This situation can be modeled by the following correlation function � G 0 for ξ ≥ r ≥ 0 , G ( r ) = 0 for r > ξ. The corresponding power spectrum can be found straightforwardly: � ∞ P ( k ) = 2 dr G ( r ) r 2 sin( kr ) ≈ 2 G 0 π k 3 π ( k ξ ) 3 , kr 3 0 in the limit k ξ ≪ 1. At large scales, the power spectrum is of the k 3 form. What about the quantum vacuum? 11 J. Mielczarek, L. Linsefors, A.Barrau (in preparation) Jakub Mielczarek Cosmological constraints on quantum gravity

Quantum generation of perturbations Let us focus on the tensor modes (gravitational waves) √ h + , × = 16 π G φ for which we have the following EOM: � d � d 2 H − 1 d Ω d ηφ − Ω ∇ 2 φ = 0 . d η 2 φ + 2 2Ω d η √ By introducing u = z φ with z = a / Ω we get d 2 ′′ d η 2 u − Ω ∇ 2 u − z z u = 0 , � (2 π ) 3 / 2 u k e i k · x takes the form d 3 k which after Fourier transform u = � � d 2 ′′ Ω k 2 − z d η 2 u k + u k = 0 . z Jakub Mielczarek Cosmological constraints on quantum gravity

Characteristic scale for the modes is λ H = a , k H defined such that � � ′′ � � z � � � = � Ω k 2 � � � . H � � z � The modes are super-horizon sized if λ ≫ λ H and sub-horizon sized if λ ≪ λ H . The modes are “frozen” at the super-horizon scales and decaying at the sub-horizon scales. Let us investigate properties of the modes in the case of barotropic fluid p = w ρ for which � 1 � 1 � � 9 � ′′ z ρ c κ a 2 x 2 + 13 w + 9 = 3 − w + x Ω 2 z 2 2 � 11 � � (3 + 30 w + 15 w 2 ) x 2 + 3 + 21 w + 12 w 2 x 3 − , where x = ρ/ρ c . Danger of a divergence at Ω → 0. Jakub Mielczarek Cosmological constraints on quantum gravity