Jurekfest 2019 Faculty of Physics, University of Warsaw, Warszawa, Poland,September 16-20, 2019. -- Conference on the occasion of Jerzy Lewandowski's 60th birthday (Jurekfest). -- Presentación con 24 diapositiva

Mena Marugán, Guillermo A.

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 Primordial Fluctuations in Loop Quantum Cosmology Guillermo A. Mena Marugán IEM-CSICJurekfest20 September 2019 The Universe is approximately homogeneous and isotropic, with cosmological perturbations.  In our era of precision cosmology, observational data can be used to falsify models. Observations might be indicating some possible tensions.  Collections of observations may provide statistical significance.Introduction Introduction  Precision cosmology opens a window to observe genuine QUANTUM COSMOLOGY effects. Perturbations need a gauge invariant  descriptions  (Bardeen, Mukhanov & Sasaki). The passage to quantization asks for a canonical formulation (Langlois, Pinto-Neto, Mena Marugán, Castelló Gomar, Olmedo, Fernández-Méndez, Lewandowski, Dapor, Puchta...). A complete quantum  treatment should include the background (Halliwell & Hawking, Shirai & Wada...).We consider an FLRW cosmology coupled to a scalar field. For simplicity, we assume compact flat (three-torus) spatial topology. We focus the dicussion on SCALAR pertubations. We truncate the action at quadratic  order in perturbations, with the background treated exactly up to that order.ModelWe want to study quantum cosmology modifications to the Mukhanov-Sasaki equations for primordial fluctuations. The FLRW system is described by a scale factor and (the zero-mode of) a homogeneous scalar field:                  We setWe expand the inhomogeneities in a (real) Fourier basis (sines and cosines).Modes are labeled by a wave vector         (with positive first non-vanishing component). The eigenvalue of the Laplacian is Scalar perturbations  are described by the Fourier coefficients of the scalar field, spatial metric (trace and traceless), lapse               and shift The system as a whole is symplectic:  zero modes + perturbations.      (a ,φ).Classical system−ωn2=−n⃗⋅⃗n .n⃗∈ℤ3G=6π2 .( f n⃗ ,± ).(g n⃗ ,± ) ,Constraints: Linear perturbative constraints (Hamiltonian constraint + diffeo constraint)+ Zero-mode of the Hamiltonian constraint.                   Homogeneous lapse                                                                                                                        : momenta.H=N 0 [H 0+∑ H 2n⃗ ,± ]+a−3∑ g n⃗ ,± H 1n⃗ ,±+a−2∑ ωn2 k n⃗ ,± H ↑1n⃗ ,± .H 0=12a3(−a2πa2+πφ2+16π3 a6V (φ)) .πipotentialClassical systemGauge invariance   We change the variables for the perturbations to a new canonical set:  The Mukhanov-Sasaki gauge invariants  Their momenta           which are also gauge invariants. A criterion is needed to fix the contribution of         to them. An Abelianization of the linear perturbative constraints (possible at the truncation order).  Suitable momenta of these, parametrizing possible gauge fixations. v n⃗ ,± .πv n⃗ ,± ,v n⃗ ,±Full system  We extend the canonical transformation to the full system, at the    considered perturbative order.We call                               their momenta, and                    the old perturbative variables. Likewise for        with a flip of sign in the corrections.                     The corrections are QUADRATIC in the perturbations.                                                                   {wqa}≡{a ,φ} , {w pa }w̃qa=wqa+12∑l , n⃗ ,± [X qln⃗ ,± ∂ X pln⃗ ,±∂w pa −∂ X qln⃗ ,±∂w pa X p ln⃗ ,± ].{X qln⃗ ,± , X pln⃗ ,± }w̃ pa ,New Hamiltonian  Since the change of zero modes is quadratic in the perturbations, the new scalar constraint at our truncation order is So, the perturbative contribution to the new scalar constraint is                                                                (up  to gauge).This gives precisely the Mukhanov-Sasaki Hamiltonian.H 0+∑b (wb−w̃b)∂H 0∂wb+∑n⃗ ,± H 2n⃗ ,± at (w̃ a , X̃ ln⃗ ,± ) ,wa−w̃a=∑n⃗ ,± Δ w̃ n⃗ ,±a .H 2n⃗ ,±+∑a Δ w̃ n⃗ ,±a∂H 0∂wa→ H̆ 2n⃗ ,±New Hamiltonian  The total Hamiltonian of the system becomes               Redefined Lagrange  multipliers.  It should include backreaction at the considered perturbative order. The perturbative contribution to the scalar constraint is quadratic in the Mukhanov-Sasaki variables and momenta, and linear in   H= N̄ 0 [H 0+∑n⃗ ,± H̆ 2n⃗ ,± ]+∑n⃗ ,± G n⃗ ,± H̆ 1n⃗ ,±+∑n⃗ ,± K n⃗ ,± H ↑1n⃗ ,± .πφ̃ .Abelianized.Approximation: Quantum geometry effectsare especially relevant in the background.Hybrid quantization Adopt a (loop)  quantum  scheme for zero modes and quantize the perturbations à la Fock. The scalar constraint couples them. We assume:a) Zero modes commute with perturbations after quantization. b) Functions of     act by multiplication. φ̃Fock representation A Fock quantization is fixed in QFT up to unitary equivalence by:The background isometries.The unitarity of the resulting Heisenberg evolution.  The choice of representation does not fix the vacuum: any Fock state is valid.Perturbative constraints  We represent the linear perturbative constraints  (or an integrated version of them) as derivatives (or as translations).  Then, physical states are independent of their momenta (gauge d.o.f.).  Physical states depend only on zero modes and gauge invariants (no gauge fixing).  They still must satisfy the Hamiltonian (or scalar) constraint given by the FLRW and the Mukhanov-Sasaki contributions. .Hamiltonian constraint  This global Hamiltonian constraint can be writtenwhereEven:                                                           Odd:                                         The same                                                         All mode independentΘen⃗ ,±=−[(ϑeωn2+ϑeq)(v n⃗ ,± )2+ϑe(πvn⃗ ,±)2] ,ϑe=ã2, ϑeq=H 0(2)ã2 (19−18 H 0(2)ã2πã2 )+8π3 ã4 (V ' '−4V ) ,H S=12 [πφ̃2−H 0(2)−Θe−Θoπφ̃ ] .H 0(2)=ã2πã2−16π3 ã6 V (φ̃) , Θ=∑n⃗ ,± Θn⃗ ,± .ϑo=−96π3 ã3 V'π ã .Θon⃗ ,±=−ϑo(vn⃗ ,± )2 ,                                               QC: Factor ordering/regularization. Quantum constraint                                 Symmetrization in the linear momentum. It is quadratic in the momentum of the zero mode of the scalar field.  The linear perturbative term goes with the derivative of the potential. .H S=12 [πφ̃2−H 0(2)−Θe−Θoπφ̃ ] .Hamiltonian constraint Consider states for which the dependence on the FLRW geometry and the inhomogeneities         split: The FLRW state is normalized, and evolves in     as:     is an evolution CLOSE to the unperturbed one, with generator            Born-Oppenheimer ansatzΨ=ξ(ã , φ̃)ψ(N , φ̃).ξ(ã , φ̃)=Û (ã , φ̃)χ(ã).(N )Û ̂̃H 0 .φ̃Approximation: Disregard transitions from     to other FLRW states.Taking expectation values in the FLRW geometry, we get a quantum constraint for the Mukhanov-Sasaki field:If we can neglect the first and last terms:                     π̂φ̃2 ψ+2 〈 ̂̃H 0〉ξ π̂φ̃ψ=[〈Θ̂e+12 (Θ̂o ̂̃H 0+ ̂̃H 0 Θ̂o)〉ξ+ 12 〈[π̂φ̃− ̂̃H 0 ,Θ̂o]〉ξ]ψ.π̂φ̃ψ=〈2 Θ̂e+(Θ̂ỗH 0+̂̃H 0Θ̂o)〉ξ4 〈 ̂̃H 0〉ξψ .Schrödinger-like equation for the gauge invariant perturbations Born-Oppenheimer ansatzξMukhanov-Sasaki equationsMoreover, BY ONLY  assuming a direct effective dynamics for the inhomogeneities, we get the modified Mukhanov-Sasaki equations:Conformal time:                                          Recall thatThe expectation values give the quantum corrected mass,  which is mode independent. The effective equations are hyperbolic in the ultraviolet regime.       d ηξ2 v n⃗ ,±=−v n⃗ ,± [ωn2+〈2 ϑ̂eq+(ϑ̂o ̂̃H 0+ ̂̃H 0 ϑ̂o)+ [π̂φ̃− ̂̃H 0 , ϑ̂o]〉ξ2 〈ϑ̂e〉ξ ].〈 ̂̃H 0〉ξ d ηξ=〈ϑ̂e 〉ξ d φ̃ . ϑe=ã2 .Example: LQCWith the standard variables           and   (v ,b)̂̃H 02≈ Ĥ 0(2)= 1(2π)3 ( Ω̂02(2π)3−2∨̂2 V ) ,ϑ̂e=∨̂2/3(2π)2,∨=3(2π)3γ√Δ∣v∣/2,Ω̂0=12γ√Δ∨̂1 /2 [ ŝgn(v) ŝin (b)+ŝin (b) ŝgn(v)]∨̂1 /2 ,Area gapImmirzi parameterMMO prescriptionϑ̂o=12√2πV '∨̂2/3∣Ω̂0∣−1 Λ̂0∣Ω̂0∣−1∨̂2 /3ϑ̂eq=(2π)2̂[ 1∨ ]1/3Ĥ 0(2)(19−18(2π)6Ω̂0−2 Ĥ 0(2))̂[ 1∨ ]1/3+ ∨̂4 /3(2π)4(V ' '−(2π)3 V ) ,ã2(2π)3 ã πã2Λ0(b)≡Ω0(2b) .Neglecting backreactionPossible strategies:Compute the quantum expectation values numerically.    Use an interaction picture around the massless or the de Sitter case.For suitable states, one often adopts the effective LQC description.       Example: LQCInitial conditions on the background within effective LQC:Quantum effects affect modes between the scale of LQC andThe effects may be relevant and compatible with observationsif those modes are entering the Hubble horizon today.  k1=ωn1 /aBk2=ωn2 /a BStandardQuantum   effectsKinetic    Potential≈ (k K−P)Initial conditionsk K−P .For backgrounds where this happens, one gets short-lived inflation.Modes affected by quantum effects do not  first leave the Hubble horizon in the slow-roll regime.Those modes are not in a Bunch-Davies vacuum.  The power spectrum is modulated  by a factor that depends on the Bogoliubov coefficients of the new vacuum state. Vacuum of the perturbations: there are several proposals  (Martín-de Blas & Olmedo, Ashtekar & Gupt...).Initial conditionsConclusions We have studied (scalar) perturbations at quadratic order in the action.At this truncation order, we have found a canonical transformation for the full system leading to Mukhanov-Sasaki gauge invariants.In a hybrid quantization, physical states depend only on the quantum background  and the Mukhanov-Sasaki field.We have derived Mukhanov-Sasaki equations  modified with quantum corrections (beyond homogeneous effective descriptions).In order to extract predictions, it is essential to determine the initial conditions for the background and the vacuum of the perturbations.  Thank you!

Primordial Fluctuations in Loop Quantum Cosmology

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Primordial Fluctuations in Loop Quantum Cosmology

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