Gauge Invariant Perturbations in Quantum Cosmology Guillermo A. - PowerPoint PPT Presentation

Gauge Invariant Perturbations in Quantum Cosmology Guillermo A. Mena Marugán (IEM-CSIC) With Laura Castelló Gomar, Hot Topics in Gen. Relativity & Mercedes Martin-Benito & Gravitation, 14 August 2015

Introduction Introduction Our Universe is approximately homogeneous and isotropic: Background with perturbations . Need of gauge invariant descriptions (Bardeen, Mukhanov-Sasaki). Canonical formulation with constraints (Langlois, Pinto-Nieto) . Quantum treatment including the background (Halliwell-Hawking, Shirai-Wada) . Recently studied in Loop Quantum Cosmology .

Classical system Classical system We consider a FLRW universe with compact flat topology. We include a scalar field subject to a potential (e.g. a mass term). For simplicity, we analyze only SCALAR pertubations .

Classical system: Modes Classical system: Modes 3 ) : (⃗ n ∈ℤ We expand the inhomogeneities in a (real) Fourier basis θ = ( Q ⃗ n , − ) n , + ± iQ ⃗ n , + = √ 2cos ( ⃗ θ ) , n , − = √ 2sin ( ⃗ θ ) n ⋅⃗ n ⋅⃗ n ⋅⃗ ± i ⃗ Q ⃗ Q ⃗ e . √ 2 n 1 ≥ 0. 2 =−⃗ −ω n n ⋅⃗ n . We take The eigenvalue of the Laplacian is Zero modes are treated exactly (at linear perturbative order) in the expansions.

Classical system: Inhomogeneities Classical system: Inhomogeneities Scalar perturbations: metric and field . [ 0 h ij ) } ] 0 h ij + 2 ∑ { n , ± ( t ) ( 3 2 e 2 α 0 h ij + b ⃗ h ij =σ a ⃗ n , ± ( t ) Q ⃗ 2 ( Q ⃗ n ; ± ) ,ij + Q ⃗ , n , ± n , ± ω n k ⃗ n , ± ( t ) N =σ [ N 0 ( t )+ e n , ± ] , 3 α ∑ g ⃗ 2 α ∑ 2 e n , ± ( t ) Q ⃗ N i =σ ( Q ⃗ n , ± ) ;i , 2 ω n 1 2 = G n , ± ] . 3 / 2 [ φ( t )+ ∑ f ⃗ Φ= n , ± ( t ) Q ⃗ σ m = m σ . 2 , ̃ σ( 2 π) 6 π Truncating at quadratic perturbative order in the action: n , ± . H = N 0 [ H 0 + ∑ H 2 n , ± ] + ∑ g ⃗ n , ± + ∑ k ⃗ n , ± ̃ ⃗ ⃗ ⃗ n , ± H 1 H ↑ 1

Classical system: Inhomogeneities Classical system: Inhomogeneities − 3 α H 0 = e 6 α ̃ 2 ( −π α 2 ) , Scalar constraint : 2 +π φ 2 + e 2 φ m 2 +π b ⃗ 2 +π f ⃗ 3 α H 2 n , ± =−π a ⃗ ⃗ 2 + 2 π α ( a ⃗ n , ± π a ⃗ n, ± + 4b ⃗ n , ± π b ⃗ n , ± )− 6 π φ a ⃗ n , ± π f ⃗ 2 e 2 ) +π φ 2 ) − e n, ± n , ± n, ± n , ± 2 ( 2 ( 2 ) 4 α 1 15 2 − 3 ω n 2 f ⃗ 3 ( ω n 2 2 2 a ⃗ 2 2 b ⃗ +π α + 10b ⃗ + 6b ⃗ +ω n 2 a ⃗ 2 a ⃗ n , ± n , ± n , ± n , ± n , ± n , ± 2 ] . n , ± 2 ) + 6 φ a ⃗ 2 [ 3 φ 2 ( 4 α − e 1 6 α ̃ 3 ( 2 ω n n , ± ) + e 2 a ⃗ 2 − 2b ⃗ n , ± + f ⃗ n , ± b ⃗ m 2 a ⃗ n , ± f ⃗ n , ± n , ± n , ± Linear perturbative constraints : 2 n , ± − ω n 3 α H 0 ) a ⃗ n , ± + ( π α ⃗ n , ± =−π α π a ⃗ 2 − 3 π φ 2 + 3 e 4 α ( a ⃗ H 1 n , ± +π φ π f ⃗ 3 e n , ± + b ⃗ n , ± ) n , ± = 1 6 α ̃ ̃ 3 [ −π a ⃗ n, ± ] . 2 φ f ⃗ ⃗ + e m n, ± , H ↑ 1 n , ± +π b ⃗ n , ± +π α ( a ⃗ n, ± + 4 b ⃗ n , ± )+ 3 π φ f ⃗

Gauge invariant perturbations Gauge invariant perturbations Consider the sector of zero modes as describing a fixed background. Look for a transformation of the perturbations -- canonical only with respect to their symplectic structure-- adapted to gauge invariance: a) Find new variables that abelianize the perturbative constraints. 3 α H 0 a ⃗ ̆ ⃗ n , ± = H 1 n , ± − 3 e ⃗ H 1 n, ± . b) Include the gauge-invariant Mukhanov-Sasaki variable. n , ± + π φ α [ f ⃗ n , ± ) ] . n , ± = e π α ( a ⃗ n , ± + b ⃗ v ⃗ c) Complete the transformation with suitable momenta .

Gauge invariant perturbations Gauge invariant perturbations Mukhanov-Sasaki momentum ( removing ambiguities ) : −α [ π f ⃗ n , ± ) ] n , ± + 1 6 α ̃ 2 b ⃗ π φ ( e 2 φ f ⃗ π v ⃗ n , ± = e m n, ± + 3 π φ ̄ − 2 α ( 1 π α ) v ⃗ 2 2 φ+π α + 3 π φ 6 α ̃ − e π φ e m n , ± . (π a ⃗ n , ± , π b ⃗ n , ± ) . The Mukhanov-Sasaki momentum is independent of π b ⃗ n , ± . The perturbative Hamiltonian constraint is independent of π a ⃗ n , ± −π b ⃗ n , ± . The perturbative momentum constraint depends through It is straightforward to complete the transformation: n, ± =− 1 ̃ ̆ ⃗ n , ± = 3 b ⃗ ⃗ π α ( a ⃗ n , ± + b ⃗ n, ± ) . C ↑ 1 n , ± , C 1

Gauge invariant perturbations Gauge invariant perturbations The redefinition of the perturbative Hamiltonian constraint amounts to a redefinition of the lapse at our order of truncation in the action : n , ± , N 0 [ H 0 + ∑ ⃗ n , ± ] + ∑ ⃗ n , ± + ∑ ⃗ H = ̆ n , ± ̆ n , ± ̃ ⃗ ⃗ ⃗ n , ± H 2 n , ± g ⃗ H 1 n , ± k ⃗ H ↑ 1 3 α ∑ ⃗ ̆ N 0 = N 0 + 3 e n , ± g ⃗ n, ± a ⃗ n, ± .

Full system Full system We now include the zero modes as variables of the system, and complete the canonical transformation. We re-write the Legendre term of the action, keeping its canonical form at the considered perturbative order : a ̃ a w p n , ± X p l n , ± V p l ∫ dt [ ∑ a ˙ n , ± ] ≡ ∫ dt [ ∑ a ˙ n , ± ] . a + ∑ l , ⃗ a + ∑ l , ⃗ n , ± ˙ n , ± ˙ ⃗ ⃗ ⃗ ⃗ w q X q l w q w p V q l ̃ a ,w p a , ̃ ( { w q a } = { α , φ } . ) { w q a } → { ̃ a } . Zero modes: Old New w q w p n , ± , X p l { X q l n , ± } → ⃗ ⃗ Inhomogeneities: Old New: n , ± ,V p l n , ± , ̃ n, ± , ̃ { V q l n, ± } = { ( v ⃗ n , ± ) } . n , ± , ̆ n, ± , ̆ n , ± ) , (̄ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ π v ⃗ C 1 C ↑ 1 H 1 H ↑ 1

Full system Full system Using that the change of perturbative variables is linear, it is not difficult to find the new zero modes , which include modifications quadratic in the perturbations . n , ± [ X q l n , ± ] , Expressions: ⃗ n, ± n, ± ⃗ n, ± ∂ X p l ∂ X q l a − 1 2 ∑ l , ⃗ ⃗ ⃗ a = ̃ w q w q a − X p l a ∂ ̃ ∂ ̃ w p w p n , ± [ X q l n , ± ] . n , ± ⃗ n , ± ⃗ n , ± ∂ X p l ∂ X q l a + 1 2 ∑ l , ⃗ a = ̃ ⃗ ⃗ a − w p w p a X p l ∂ ̃ w q ∂ ̃ w q n , ± , X p l { X q l n , ± } → ⃗ ⃗ Old perturbative variables in terms of the new ones.

New Hamiltonian New Hamiltonian Since the change of the zero modes is quadratic in the perturbations , the new scalar constraint at our truncation order is a , X l a )+ ∑ ⃗ n , ± ( w n , ± ) ⇒ ⃗ ⃗ H 0 ( w n , ± H 2 b ) ∂ H 0 a , X l a ,V l n , ± [ ̃ n , ± ) ] , a )+ ∑ b ( w a )+ ∑ ⃗ b − ̃ n, ± ( ̃ ⃗ ⃗ ⃗ H 0 ( ̃ b ( ̃ w w w n , ± H 2 w w ∂ ̃ w a = ∑ ⃗ a − ̃ a n , ± Δ ̃ w w w ⃗ . n , ± The perturbative contribution to the new scalar constraint is: ∂ H 0 n, ± + ∑ a Δ ̃ ̄ ⃗ n , ± = H 2 ⃗ a H 2 w ⃗ a . n , ± ∂ ̃ w

New Hamiltonian New Hamiltonian Carrying out the calculation explicitly, one obtains: n , ± + ( F ↑ 1 n , ± ) ̃ n , ± − 3 e − 3 ̃ α n , ± + 9 n , ± H 0 + ̆ n , ± ̆ n , ± , n , ± = ̆ α ̆ α ̃ ̄ ⃗ ⃗ n , ± + F 2 ⃗ ⃗ ⃗ ⃗ ⃗ − 3 ̃ ⃗ ⃗ H 2 H 2 F 1 H 1 H 1 2 e H ↑ 1 H ↑ 1 π ̃ 2 { [ ω n n , ± ) 2 } . 2 ) ] ( v ⃗ α ( 1 + 15 ̃ − ̃ α φ π ̃ φ 4 m 2 ̃ n , ± = e α ̃ ̆ ⃗ 2 + e − 4 ̃ α π ̃ 2 + ̃ m 2 e 2 ̃ φ 2 − 12 ̃ α − 18 e 6 ̃ φ n , ± ) 2 +(̄ H 2 π v ⃗ π ̃ α π ̃ α F The 's are well determined functions. ̆ n , ± ⃗ H 2 The term is the Mukhanov-Sasaki Hamiltonian. It has no linear contributions of the Mukhanov-Sasaki momentum. π ̃ φ . It is linear in the momentum

New Hamiltonian New Hamiltonian We re-write the total Hamiltonian of the system at our truncation order , redefining the Lagrange multipliers: n , ± + ( F ↑ 1 n, ± ) ̃ − 3 ̃ α n , ± − 3 e n , ± + 9 n , ± ̆ n , ± H 0 + ̆ n, ± = ̆ ̆ ⃗ n , ± + F 2 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ − 3 ̃ α ̃ ⃗ n , ± ⃗ ̄ ⇒ H 2 H 2 F 1 H 1 H 1 2 e H ↑ 1 H ↑ 1 π ̃ α n, ± . N 0 [ H 0 + ∑ ⃗ n , ± ] + ∑ ⃗ n , ± + ∑ ⃗ n , ± ̆ n , ± ̆ n , ± ̆ n, ± ̃ n , ± ̃ H = ̄ ⃗ ⃗ ⃗ H 2 G ⃗ H 1 K ⃗ H ↑ 1

Hybrid quantization Hybrid quantization Approximation : Quantum geometry effects are especially relevant in the background Adopt a quantum cosmology scheme for the zero modes and a Fock quantization for the perturbations. The scalar constraint couples them. We assume: a) The zero modes commute with the perturbations under quantization. b) Functions of act by multiplication. φ ̃

Uniqueness of the Fock description Uniqueness of the Fock description The Fock representation in QFT is fixed (up to unitary equivalence) by: 1) The background isometries ; 2) The demand of a UNITARY evolution. The introduced scaling of the field by the scale factor is essential for unitarity . The proposal selects a UNIQUE canonical pair for the Mukhanov-Sasaki field, precisely the one we chose to fix the ambiguity in the momentum. We can use the massless representation (due to compactness), with its creation and annihilation operators, and the corresponding basis of ⌈ N 〉 . occupancy number states