Vacuum state for

Rita Neves, Universidad Complutense de Madrid SFRH/BD/143525/2019

Supervisors: Mercedes Mart´ınBenito, Javier Olmedo

International Loop 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, 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

2  E(Tk) = 2µ (k) + 1 c1(k) + 2 µ(k) Re[λ(k) c2(k)] . | {z } 2µ|λ||c2| cos[Arg(λ)+Arg(c2)]

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 9/21 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

2  E(Tk) = 2µ (k) + 1 c1(k) + 2 µ(k) Re[λ(k) c2(k)] . | {z } 2µ|λ||c2| cos[Arg(λ)+Arg(c2)]

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)|.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 9/21 Procedure

2  E(Tk) = 2µ (k) + 1 c1(k) + 2 µ(k) Re[λ(k) c2(k)] . | {z } 2µ|λ||c2| cos[Arg(λ)+Arg(c2)]

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)|

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 9/21 State of Minimal Energy

The state Tk that is of low energy for any f(t). Only exists in ultrastatic .

Procedure

State of Low Energy associated with f(t)

Tk = λ(k)Sk + µ(k)S¯k, v  u c (k) 1  u 1 µ(k) = t q − ,  2 2 2  2 c1(k) − |c2(k)| v u c (k) 1  −iArg[c2(k)]u 1 λ(k) = −e t q + .  2 2 2 2 c1(k) − |c2(k)|

,→ Dependence on smearing function (through c1 and c2).

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 10/21 Procedure

State of Low Energy associated with f(t)

Tk = λ(k)Sk + µ(k)S¯k, v  u c (k) 1  u 1 µ(k) = t q − ,  2 2 2  2 c1(k) − |c2(k)| v u c (k) 1  −iArg[c2(k)]u 1 λ(k) = −e t q + .  2 2 2 2 c1(k) − |c2(k)|

,→ Dependence on smearing function (through c1 and c2).

State of Minimal Energy

The state Tk that is of low energy for any f(t). Only exists in ultrastatic spacetimes.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 10/21 • de Sitter: singles out Bunch-Davies - support of f → distant past - [massless case: our work]

Properties

In maximally symmetric spacetimes:

• Minkowski: SLE trivially identical to Minkowski vacuum;

- start with Sk = Minkowski vacuum - c2 = 0 trivially ⇒ |λ| = 1, µ = 0 - Tk = Sk (up to phase)

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 11/21 Properties

In maximally symmetric spacetimes:

• Minkowski: SLE trivially identical to Minkowski vacuum;

- start with Sk = Minkowski vacuum - c2 = 0 trivially ⇒ |λ| = 1, µ = 0 - Tk = Sk (up to phase)

• de Sitter: singles out Bunch-Davies - support of f → distant past - [massless case: our work]

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 11/21 Properties (Banerjee, Niedermaier, J. Math. Phys. 61, 103511 (2020))

• Independence of fiducial solution Sk;

• Tk admits UV and IR expansions;

2 • UV aymptotic behavior of |Tk| is independent of f;

• IR symptotic behavior is Minkowski-like for all f;

• As vacuum states of perturbations: agreement with observations for cosmological models with period of kinetic dominance prior to inflation.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 12/21 SLEs in LQC

1 Obtain Sk numerically - give initial conditions (irrelevant) ,→ 0th order adiabatic at bounce: r 1 0 k uk(0) = √ , u (0) = −i 2k k 2

- V (φ) = m2φ2/2

−6 - m = 1.2 × 10 mPl

- φB = 1.225 mPl

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 13/21 SLEs in LQC

1 Obtain Sk numerically - give initial conditions (irrelevant) ,→ 0th order adiabatic at bounce: r 1 0 k uk(0) = √ , u (0) = −i 2k k 2

- V (φ) = m2φ2/2

−6 - m = 1.2 × 10 mPl Planck 2018: predictive posterior plot of free-form Bayesian reconstruction of primordial - φB = 1.225 mPl power spectrum.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 13/21 SLEs in LQC

1 Obtain Sk numerically - give initial conditions (irrelevant) ,→ 0th order adiabatic at bounce: r 1 0 k uk(0) = √ , u (0) = −i 2k k 2

- V (φ) = m2φ2/2

−6 - m = 1.2 × 10 mPl Planck 2018: predictive posterior plot of free-form Bayesian reconstruction of primordial - φB = 1.225 mPl power spectrum. In blue/green: ILLUSTRATIVE

example of PS when varying φB

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 13/21 SLEs in LQC

1 Obtain Sk numerically - give initial conditions (irrelevant) ,→ 0th order adiabatic at bounce: r 1 0 k uk(0) = √ , u (0) = −i 2k k 2

- V (φ) = m2φ2/2

−6 - m = 1.2 × 10 mPl Planck 2018: predictive posterior plot of free-form Bayesian reconstruction of primordial - φB = 1.225 mPl power spectrum. In blue/green: ILLUSTRATIVE

example of PS when varying φB

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 13/21 SLEs in LQC

1 Obtain Sk numerically - give initial conditions (irrelevant) ,→ 0th order adiabatic at bounce: r 1 0 k uk(0) = √ , u (0) = −i 2k k 2

- V (φ) = m2φ2/2

−6 - m = 1.2 × 10 mPl Planck 2018: predictive posterior plot of free-form Bayesian reconstruction of primordial - φB = 1.225 mPl power spectrum. In blue/green: ILLUSTRATIVE

example of PS when varying φB

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 13/21 - here: no need for initial conditions, but need to choose f 2: - support on “whole” evolution (since well before to well after bounce): →results insensitive to shape and support of f 2 as long as it is wide enough - support on expanding branch only (f 2 is sharp (but smooth) step function starting at bounce): →results insensitive to support as long as it is wide enough

SLEs in LQC

2 2 choose f (t): - usually in LQC, 2 strategies for choosing initial conditions of perturbations - at bounce - at asymptotic time well before bounce

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 14/21 SLEs in LQC

2 2 choose f (t): - usually in LQC, 2 strategies for choosing initial conditions of perturbations - at bounce - at asymptotic time well before bounce - here: no need for initial conditions, but need to choose f 2: - support on “whole” evolution (since well before to well after bounce): →results insensitive to shape and support of f 2 as long as it is wide enough - support on expanding branch only (f 2 is sharp (but smooth) step function starting at bounce): →results insensitive to support as long as it is wide enough

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 14/21 - smooth step function: 1

1 − tanh [cot(x)] S(x)= , 2 0 0 x π 1   η − ηi  S π η ≤ η < η + δ,  δ i i 2  f (η)= 1 ηi + δ ≤ η ≤ ηf − δ,   ηf − η  S π ηf − δ < η ≤ ηf . 0 δ ηi ηi+δ ηf-δ ηf

3 compute c1, c2 −→ µ, λ −→ Tk

SLEs in LQC

2 2 choose f (t): - support on “whole” evolution vs expanding branch only

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 15/21 3 compute c1, c2 −→ µ, λ −→ Tk

SLEs in LQC

2 2 choose f (t): - support on “whole” evolution vs expanding branch only - smooth step function: 1

1 − tanh [cot(x)] S(x)= , 2 0 0 x π 1   η − ηi  S π η ≤ η < η + δ,  δ i i 2  f (η)= 1 ηi + δ ≤ η ≤ ηf − δ,   ηf − η  S π ηf − δ < η ≤ ηf . 0 δ ηi ηi+δ ηf-δ ηf

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 15/21 SLEs in LQC

2 2 choose f (t): - support on “whole” evolution vs expanding branch only - smooth step function: 1

1 − tanh [cot(x)] S(x)= , 2 0 0 x π 1   η − ηi  S π η ≤ η < η + δ,  δ i i 2  f (η)= 1 ηi + δ ≤ η ≤ ηf − δ,   ηf − η  S π ηf − δ < η ≤ ηf . 0 δ ηi ηi+δ ηf-δ ηf

3 compute c1, c2 −→ µ, λ −→ Tk

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 15/21 • scalar ∼ tensor at bounce - s(s) = s(t) + U' s(t) - Sk(0) ↔ 0th order adiabatic (Dk = k,Ck = 0)

SLEs in LQC - Results

• Initial conditions: 1 uk(0) = √ , 2Dk r D u0 (0) = k (C − i) , k 2 k

Dk: positive function, Ck: any real function.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 16/21 • scalar ∼ tensor at bounce - s(s) = s(t) + U' s(t) - Sk(0) ↔ 0th order adiabatic (Dk = k,Ck = 0)

SLEs in LQC - Results

• Initial conditions: 1 uk(0) = √ , 2Dk r D u0 (0) = k (C − i) , k 2 k

Dk: positive function, Ck: any real function.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 16/21 SLEs in LQC - Results

• Initial conditions: 1 uk(0) = √ , 2Dk r D u0 (0) = k (C − i) , k 2 k

Dk: positive function, Ck: any real function.

• scalar ∼ tensor at bounce - s(s) = s(t) + U' s(t) - Sk(0) ↔ 0th order adiabatic (Dk = k,Ck = 0)

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 16/21 - minimize smeared energy density, - are exact Hadamard states: computation of stress-energy tensor

• Observationally: close to 2nd order Adiabatic; • Fundamentally different, SLEs:

SLEs in LQC - Results

• Power spectra (z = aφ/H˙ ):

k3 |u |2 P (k) = k , R 2 2 2π z η=ηend 32k3 |µI |2 P (k) = k . T 2 π a η=ηend

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 17/21 - minimize smeared energy density, - are exact Hadamard states: computation of stress-energy tensor

• Observationally: close to 2nd order Adiabatic; • Fundamentally different, SLEs:

SLEs in LQC - Results

• Power spectra (z = aφ/H˙ ):

k3 |u |2 P (k) = k , R 2 2 2π z η=ηend 32k3 |µI |2 P (k) = k . T 2 π a η=ηend

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 17/21 - minimize smeared energy density, - are exact Hadamard states: computation of stress-energy tensor

• Fundamentally different, SLEs:

SLEs in LQC - Results

• Power spectra (z = aφ/H˙ ):

k3 |u |2 P (k) = k , R 2 2 2π z η=ηend 32k3 |µI |2 P (k) = k . T 2 π a η=ηend

• Observationally: close to 2nd order Adiabatic;

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 17/21 SLEs in LQC - Results

• Power spectra (z = aφ/H˙ ):

k3 |u |2 P (k) = k , R 2 2 2π z η=ηend 32k3 |µI |2 P (k) = k . T 2 π a η=ηend

• Observationally: close to 2nd order Adiabatic; • Fundamentally different, SLEs: - minimize smeared energy density, - are exact Hadamard states: computation of stress-energy tensor

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 17/21 SLEs in LQC - Results

• Tensor-to-scalar ratio - scale invariance from smaller k.

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 18/21 SLEs in LQC - Results

• Tensor-to-scalar ratio: • Spectral index: r0.002 ' 0.117 ns ' 0.969

Planck 2018: Y. Akrami et al.(Planck), Astron. Astrophys. 641, A10 (2020)

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 19/21 SLEs in LQC - Results

• Tensor-to-scalar ratio: • Spectral index: r0.002 ' 0.117 ns ' 0.969

Planck 2018: Y. Akrami et al.(Planck), Astron. Astrophys. 641, A10 (2020)

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 19/21 Summary and outlook

• SLEs are good candidates for vacua: - Hadamard states, - Minimize smeared energy density, - IR and UV asymptotic behaviors independent of test function;

• IR and UV agreement with observations when KD precedes inflation;

• In LQC: - qualitative agreement with observations, - insensitive to choice of test function;

• Rigorous statistical analysis: - parameters of the background, - , - V (φ).

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 20/21 Thank you for your attention

Rita Neves, [email protected] ILQGS 23/03/2021 Vacuum state for LQC 21/21