Gauge Invariant Perturbations and Covariance in Quantum Cosmology - PowerPoint PPT Presentation

Gauge Invariant Perturbations and Covariance in Quantum Cosmology Guillermo A. Mena Marugán (IEM-CSIC) With Laura Castelló Gomar, 2nd APCTP-TUS Workshop, & Mercedes Martin-Benito August 2015

Introduction Introduction Our Universe is approximately homogeneous and isotropic: Background with cosmological perturbations . Need of gauge invariant descriptions (Bardeen, Mukhanov-Sasaki). Perturbations: Canonical formulation with constraints (Langlois, Pinto-Nieto) . Quantum treatment including the background (Halliwell-Hawking, Shirai-Wada) . Hybrid formalism with a Born-Oppenheimer ansatz: Covariance.

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 ⃗ e . Q ⃗ Q ⃗ √ 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 ) , 2 +π φ 2 + e 2 φ m Scalar constraint : 3 α H 2 2 +π b ⃗ 2 +π f ⃗ n , ± =−π a ⃗ 2 ⃗ 2 e + 2 π α ( a ⃗ n , ± π a ⃗ n, ± + 4b ⃗ n , ± π b ⃗ n , ± )− 6 π φ a ⃗ n , ± π f ⃗ 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 2 a ⃗ + 10b ⃗ 2 a ⃗ + 6b ⃗ +ω n 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 n , ± b ⃗ m 2 a ⃗ − 2b ⃗ n , ± f ⃗ 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 are an abelianization of the perturbative constraints. 3 α H 0 a ⃗ ̆ n , ± = H 1 ⃗ n , ± − 3 e ⃗ H 1 n, ± . b) Include the Mukhanov-Sasaki variable, exploiting its gauge invariance. 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 : −α [ π f ⃗ n , ± ) ] + F v ⃗ n , ± + 1 6 α ̃ 2 b ⃗ π φ ( e 2 φ f ⃗ π v ⃗ n, ± = e m n , ± + 3 π φ n , ± . F . There is an ambiguity in a function of the background variables, (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, ± .

Mukhanov-Sasaki momentum Mukhanov-Sasaki momentum We remove the ambiguity in the Mukhanov-Sasaki momentum by any of the following: It equals the time derivative of the Mukhanov-Sasaki variable. The scalar constraint is quadratic in this momentum (no linear terms). It is possible to adopt a Fock quantization with invariance under rigid rotations and unitary evolution ( Cortez, Mena-Marugán, Velhinho ). −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, ± .

Canonical transformation: Inverse Canonical transformation: Inverse n , ± − 1 n , ± = 1 n , ± , n , ± , n , ± , n , ± =−π α ̆ n , ± +π φ ̆ 3 ̃ 3 ̃ ⃗ ⃗ ⃗ n , ± = e −α v ⃗ ⃗ a ⃗ C 1 C ↑ 1 b ⃗ C ↑ 1 f ⃗ C 1 π α ( e π α ) v ⃗ 3 −α n , ± + π φ 2 φ+π φ π α + 3 π φ n , ± =− 1 n , ± + e α ̄ 6 α ̃ π α ̆ ⃗ π a ⃗ π v ⃗ H 1 π α e m n , ± + ( 3 π φ 2 ) ̆ 2 + 1 n , ± − 1 n , ± , 3 π α ̃ 4 α ω n ⃗ ⃗ 2 −π α 3 e C 1 C ↑ 1 π α ( e π α ) v ⃗ 3 n , ± + π φ 2 φ− 2 π φ π α + 3 π φ n , ± + e −α n , ± − 1 6 α ̃ π α ̆ n , ± = 3 ̃ α ̄ ⃗ ⃗ π b ⃗ π v ⃗ H ↑ 1 H 1 π α e m n , ± + 1 n , ± − 4 2 ̆ n , ± , 3 π α ̃ 4 α ω n ⃗ ⃗ 3 e C 1 C ↑ 1 −α ( π α + 3 π φ π α ) v ⃗ 2 6 α ̃ n , ± . 2 φ ̆ α ̄ ⃗ n , ± −π φ ̃ ⃗ π f ⃗ n , ± = e π v ⃗ n , ± + e n , ± − e m C 1 C ↑ 1

Full system Full system We now include the zero modes as variables of the system, and complete the transformation to a canonical one in their presence. We re-write the Legendre term of the action, keeping its canonical form at the considered perturbative order : a w p n , ± X p l a ̃ 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 ( { w q a } = { α , φ } . ) { w q a } → 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 , ± ) , (̄ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ C 1 C ↑ 1 π v ⃗ 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 , ± ] , ⃗ n, ± ⃗ n, ± n, ± ∂ X p l ∂ X q l a − 1 Expressions: 2 ∑ l , ⃗ a = ̃ ⃗ ⃗ a − w q w q 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 = ̃ ⃗ ⃗ w p w p a − a X p l ∂ ̃ ∂ ̃ w q w q n , ± , X p l { X q l n , ± } → ⃗ ⃗ Old perturbative variables in terms of the new.

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 This is the change expected for zero modes treated as time dependent external variables with dynamics generated by H 0 .

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 2 } . 2 ) ] ( v ⃗ α ( 1 + 15 ̃ −̃ α φ π ̃ 4 n, ± = e φ 2 ̃ α π ̃ α ̃ ̆ ⃗ 2 + e − 4 ̃ 2 + ̃ 2 e 2 ̃ 2 − 12 ̃ φ 6 ̃ 2 +(̄ φ α − 18 e n , ± ) π v ⃗ n , ± ) H 2 m m 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