Vacuum state for Loop Quantum Cosmology
Rita Neves, Universidad Complutense de Madrid SFRH/BD/143525/2019
Supervisors: Mercedes Mart´ınBenito, Javier Olmedo
International Loop Quantum Gravity Seminars, March 23rd 2021
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 1/21 - Adiabatic, - Diagonalization of Hamiltonian1 - Minimization of renormalized stress-energy tensor2, uncertainty relations3, - Minimization of smeared quantities4.
1Fahn, et. al, Universe 5, 170 (2019), Elizaga Navascu´es,et. al, Class. Quant. Grav. 36, 185010 (2019) 2Agullo, et. al, PRD 91, 064051 (2015), Handley, et. al, PRD 94, 024041 (2016) 3Danielsson, PRD 66, 023511 (2002), Ashtekar, et. al, Class. Quant. Grav. 34, 035004 (2017) 4de Blas, et. al, JCAP 06, 029 (2016), Elizaga Navascu´es,et. al, Class. Quant. Grav. 38, 035001 (2020), Olbermann, Class. Quant. Grav. 24, 5011 (2007)
Motivation
• Cosmological perturbations in (L)QC: observational window;
• Freedom in choice of vacuum state;
• Natural vacuum?
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 2/21 Motivation
• Cosmological perturbations in (L)QC: observational window;
• Freedom in choice of vacuum state;
• Natural vacuum? - Adiabatic, - Diagonalization of Hamiltonian1 - Minimization of renormalized stress-energy tensor2, uncertainty relations3, - Minimization of smeared quantities4.
1Fahn, et. al, Universe 5, 170 (2019), Elizaga Navascu´es,et. al, Class. Quant. Grav. 36, 185010 (2019) 2Agullo, et. al, PRD 91, 064051 (2015), Handley, et. al, PRD 94, 024041 (2016) 3Danielsson, PRD 66, 023511 (2002), Ashtekar, et. al, Class. Quant. Grav. 34, 035004 (2017) 4de Blas, et. al, JCAP 06, 029 (2016), Elizaga Navascu´es,et. al, Class. Quant. Grav. 38, 035001 (2020), Olbermann, Class. Quant. Grav. 24, 5011 (2007)
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 2/21 Motivation
• Cosmological perturbations in (L)QC: observational window;
• Freedom in choice of vacuum state;
• Natural vacuum? - Adiabatic, - Diagonalization of Hamiltonian1 - Minimization of renormalized stress-energy tensor2, uncertainty relations3, - Minimization of smeared quantities4.
1Fahn, et. al, Universe 5, 170 (2019), Elizaga Navascu´es,et. al, Class. Quant. Grav. 36, 185010 (2019) 2Agullo, et. al, PRD 91, 064051 (2015), Handley, et. al, PRD 94, 024041 (2016) 3Danielsson, PRD 66, 023511 (2002), Ashtekar, et. al, Class. Quant. Grav. 34, 035004 (2017) 4de Blas, et. al, JCAP 06, 029 (2016), Elizaga Navascu´es,et. al, Class. Quant. Grav. 38, 035001 (2020), Olbermann, Class. Quant. Grav. 24, 5011 (2007)
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 2/21 Cosmological perturbations
• Inflaton field φ subject to potential V (φ),
• Scalar and tensor gauge invariant perturbations, minimally coupled: Q, T I ,
• Redefinition: u = aQ, µI = aT I ,
• Fourier modes:
1 Z ~ u(η, ~x) = d3k u (η)eik~x, (2π)3/2 ~k
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 3/21 • in hybrid LQC: 4πG s(t) = − a2 (ρ − 3P ) , s(s) = s(t) + U, 3
U = U [V (φ),V,φ,V,φφ, bckg] .
• Power spectra (z = aφ/H˙ ): k3 |u |2 32k3 |µI |2 P (k) = k , P (k) = k . R 2 2 T 2 2π z η=ηend π a η=ηend
Cosmological perturbations
• Equations of motion u00(η) + k2 + s(s)(η) u (η) = 0, ~k ~k 00 µI (η) + k2 + s(t)(η) µI (η) = 0, k ~k
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 4/21 • Power spectra (z = aφ/H˙ ): k3 |u |2 32k3 |µI |2 P (k) = k , P (k) = k . R 2 2 T 2 2π z η=ηend π a η=ηend
Cosmological perturbations
• Equations of motion u00(η) + k2 + s(s)(η) u (η) = 0, ~k ~k 00 µI (η) + k2 + s(t)(η) µI (η) = 0, k ~k
• in hybrid LQC: 4πG s(t) = − a2 (ρ − 3P ) , s(s) = s(t) + U, 3
U = U [V (φ),V,φ,V,φφ, bckg] .
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 4/21 Cosmological perturbations
• Equations of motion u00(η) + k2 + s(s)(η) u (η) = 0, ~k ~k 00 µI (η) + k2 + s(t)(η) µI (η) = 0, k ~k
• in hybrid LQC: 4πG s(t) = − a2 (ρ − 3P ) , s(s) = s(t) + U, 3
U = U [V (φ),V,φ,V,φφ, bckg] .
• Power spectra (z = aφ/H˙ ): k3 |u |2 32k3 |µI |2 P (k) = k , P (k) = k . R 2 2 T 2 2π z η=ηend π a η=ηend
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 4/21 (n) n− 1 • 2 Wk −−−−→ Wk, at least as O k : k→∞ !2 2 1 W (n)00 3 W (n)0 W (n+2) = k2 + s(η) − k + k , k 2 (n) 4 (n) Wk Wk
• (0) Wk = k.
Cosmological perturbations
Adiabatic states
1 R η • −i Wk(¯η)d¯η ansatz uk(η) = p e : 2Wk(η)
00 0 2 2 2 1 Wk 3 Wk Wk = k + s(η) − + . 2 Wk 4 Wk
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 5/21 Cosmological perturbations
Adiabatic states
1 R η • −i Wk(¯η)d¯η ansatz uk(η) = p e : 2Wk(η)
00 0 2 2 2 1 Wk 3 Wk Wk = k + s(η) − + . 2 Wk 4 Wk
(n) n− 1 • 2 Wk −−−−→ Wk, at least as O k : k→∞ !2 2 1 W (n)00 3 W (n)0 W (n+2) = k2 + s(η) − k + k , k 2 (n) 4 (n) Wk Wk
• (0) Wk = k.
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 5/21 • Candidates for vacuum of perturbations in LQC: - Minimization of regularized energy density, - Exact Hadamard states.
States of Low Energy [Olbermann, Class. Quant. Grav. 24, 5011 (2007)]
• Fewster 2000 [Class. Quant. Grav. 17, 1897–1911 (2000)]: - Renormalized energy density, smeared along time-like curve, is bounded from below as function of state.
• Olbermann 2007: - Particularize for generic FLRW models, smearing function supported on worldline of isotropic observer, - Define procedure to find state that minimizes it mode by mode.
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 6/21 States of Low Energy [Olbermann, Class. Quant. Grav. 24, 5011 (2007)]
• Fewster 2000 [Class. Quant. Grav. 17, 1897–1911 (2000)]: - Renormalized energy density, smeared along time-like curve, is bounded from below as function of state.
• Olbermann 2007: - Particularize for generic FLRW models, smearing function supported on worldline of isotropic observer, - Define procedure to find state that minimizes it mode by mode.
• Candidates for vacuum of perturbations in LQC: - Minimization of regularized energy density, - Exact Hadamard states.
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 6/21 Minimize mode contribution to smeared energy density:
1 Z |π |2 E(T ) = dt f 2(t) Tk + ω2|T |2 , k 2 a6 k k
3 ˙ where πTk = a Tk, f(t) smearing function.
States of Low Energy
A field Tk: ¨ ˙ 2 Tk + 3HTk + ωk(t)Tk = 0,
k2 + s(t) a¨ • scalar or tensor perturbations: ω2(t) = + H2 + . k a2 a
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 7/21 States of Low Energy
A field Tk: ¨ ˙ 2 Tk + 3HTk + ωk(t)Tk = 0,
k2 + s(t) a¨ • scalar or tensor perturbations: ω2(t) = + H2 + . k a2 a
Minimize mode contribution to smeared energy density:
1 Z |π |2 E(T ) = dt f 2(t) Tk + ω2|T |2 , k 2 a6 k k
3 ˙ where πTk = a Tk, f(t) smearing function.
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 7/21 • Defining: 1 Z c (k) := dt f 2(t) |S˙ |2 + ω2|S |2 = E(S ), 1 2 k k k k 1 Z c (k) := dt f 2(t) S˙ 2 + ω2S2 . 2 2 k k k
Procedure
• Arbitrary solution Sk,
• Bogoliubov transformation:
Tk = λ(k)Sk + µ(k)S¯k,
with |λ(k)|2 − |µ(k)|2 = 1, fixing µ(k) ∈ R+,
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 8/21 Procedure
• Arbitrary solution Sk,
• Bogoliubov transformation:
Tk = λ(k)Sk + µ(k)S¯k,
with |λ(k)|2 − |µ(k)|2 = 1, fixing µ(k) ∈ R+,
• Defining: 1 Z c (k) := dt f 2(t) |S˙ |2 + ω2|S |2 = E(S ), 1 2 k k k k 1 Z c (k) := dt f 2(t) S˙ 2 + ω2S2 . 2 2 k k k
Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 8/21 Minimize E(Tk) ⇒ Arg[λ(k)] = π − Arg[c2(k)]:
2 p 2 E(Tk) = (2µ (k) + 1)c1(k) − 2µ(k) µ (k) + 1|c2(k)|.
Minimizing E(Tk) with respect to µ: s c (k) 1 µ(k) = 1 − , p 2 2 2 2 c1(k) − |c2(k)| s c1(k) 1 λ(k) = −e−iArg[c2(k)] + . p 2 2 2 2 c1(k) − |c2(k)|
Procedure