the csl model as a mechanism for the quantum-to …relating the classical quantities with the...

The inflationary paradigm The quantum measurement problem and inflation The CSL inflationary model Comparing to the observational data

The CSL model as a mechanism for the quantum-to-classicaltransition of the primordial perturbations

Gabriel Leon

Departamento de Fısica - Universidad de Buenos Aires

CosmoSur III - June 2015

joint work with: S. Landau (UBA), M.P. Piccirilli (UNLP), L. Kraiselburd(UNLP), G. Bengochea (IAFE), D. Sudarsky (ICN-UNAM)

CosmoSurIII The CSL model and inflation


Successes of the Inflationary Paradigm

Inflation: A phase of accelerated expansion in the early Universe(a > 0) induced by a type of matter satisfying ρ+ 3P < 0. It is usuallymodeled by a scalar field φ called the inflaton.Solves the problems of the Big Bang model (Horizon, Flatness, etc.)The main success...Quantum Fluctuations of the inflaton results in theoretical predictionsof the spectrum of primordial inhomogeneities. [Mukhanov, Chibisov1982]Observational Data in agreement with such predictions



The Data

The temperature anisotropies of the CMB photons emitted by the LastScattering Surface are related to the curvature perturbation Ψ (i.e., thescalar metric perturbation, in the appropriate gauge) by

δT

T0(θ, ϕ) ' 1

3Ψ(ηD, xD) (1)

Moreover, one can use the spherical harmonic functions and writeδTT0

(θ, ϕ) =∑m,l almYlm(θ, ϕ). Thus,

alm =1

3

∫dΩΨ(ηD, xD)Y ?lm(θ, ϕ) (2)

This is the quantity that is measured.The inflationary paradigm is supposed to provide a physical mechanismto relate Ψ with the quantum fluctuations of the matter fields in the earlyUniverse.



Theoretical Predictions

The QFT of inflation can be described by the quantization of theMukhanov-Sasaki variable

v ≡ aδφ+ Ψφ′0a2/a′

where φ(x, η) = φ0(η) + δφ(x, η); a(η) is the scale factor and η is theconformal time.Quantization of v implies quantization of δφ and Ψ, i.e. δφ and Ψ.One proceeds to calculate the 2-point correlation function in the vacuumstate |0〉 :

〈0|Ψ(η, ~x)Ψ(η, ~y)|0〉 (3)

Once 〈0|Ψ(η, ~x)Ψ(η, ~y)|0〉 is known, one proceeds to make theidentification

〈0|Ψ(η, ~x)Ψ(η, ~y)|0〉 = Ψ(η, ~x)Ψ(η, ~y) (4)

For slow roll inflation one defines the slow-roll parametersε ≡ 1−H′/H2 'M2

P /2(∂φV/V )2; δ ≡M2P (∂φφV/V ). The theoretical

prediction is the (scalar) power spectrum

PΨ(k) =1

2π2|Ψk|2k3 ∝

(H2

M2p ε

) ∣∣∣∣k=aH

; ns−1 ≡ d lnP (k)

d ln k= −6ε+2δ

(5)



Something is missing...

We rewrite the expression for the coeficients alm in terms of the Fouriermodes.

alm =4πil

3

∫d3k

(2π)3jl(kRD)Y ∗lm(k)∆(k)Ψk, (6)

with jl(kRD) the spherical Bessel function of order l and RD thecomoving radius of the last scattering surface. Additionally, ∆(k) thetransfer functions.

The classical scalar metric perturbation Ψk corresponds to the primordialcurvature perturbation.

How does the transition Ψk → Ψk takes place?



The Standard Answer

Evidently if one uses 〈0|Ψk|0〉 = Ψk = 0, then alm = 0 for all l,m. Thestandard argument is that it is incorrect to state that alm = 0; the correctstatement is alm = 0, i.e. the average over an “ensemble of universes.”Then, what is the prediction for alm within our universe?

Official Answer: The classic curvature perturbation Ψk that is beingused to calculate alm is related with the quantum fluctuations of thevacuum, this is

Ψk =

√〈0|Ψ2

k |0〉eiαk (7)

thus, the amplitude of the classical quantity Ψk is equal to the quantum

uncertainty√〈0|Ψ2

k |0〉 and αk corresponds to a random phase.

Justification 1: Because it works or Shut up and calculate!

Justification 2: Decoherence, evolution of the vacuum state into asqueezed state, etc.



The fundamental problem

Remarkable: The universe was originally described by a space-timewhich is homogeneous and isotropic, and there is a scalar field (theinflaton) which is in a vacuum state also homogeneous and isotropic(there are some irrelevant deviations of this left from an imperfectinflation at the order of e−80 ), but the universe ended up withinhomogeneities that fit the experimental data.

Question How do we end up in a situation which is not symmetric (thesymmetry being the homogeneity and isotropy) given that there isnothing in the dynamics that breaks such symmetries?

In other words...

How did it happen?

|symmetric〉 =====⇒︸︷︷︸Schrod. Eq.

|non-symmetric〉 (F)



It is clear that the question (F) is closely related to the quantummeasurement problem, which in the cosmological context, appears in anexacerbated manner.

We prefer the Shut up and let me think alternative lifestyle.

Often these issues are resolved as “just philosophy,” with no impactwhatsoever on the theoretical predictions. We will see that suchpreconception is mistaken



The transition and the collapse proposal

Our approach on attempting to answer the question (F) is described as:

By invoking the collapse of the wave function, one could break thesymmetry of the original state.

|symmetric〉 −→ Collapse −→ |non-symmetric〉 (8)

According to standard QM the collapse of the wave function is the resultof a measurement. But it is not clear how to apply this postulate to thecosmological setting?

Phenomenological Model: Some element intrinsic to the systeminduced the collapse of the wave function.

The Continous Spontaneous Localization (CSL) model[Pearle,Ghirardi 1989] was proposed to provide a solution to themeasurement problem. In particular, the “observers” do not play a role atall.

Our goal is to apply the CSL model to the inflationary universe andobtain testable predictions.



CSL model[Ghirardi 1986, Diosi 1987, Pearle 1989]The idea is to modify the Schrodingerequation in such a way that includesthe reduction of the wave function. Inparticular:

For microscopic systems, thewave function must behave likeSchrodinger’s wave function: itmust diffuse, superimpose,interfere, ...For macroscopic objects, thewave function must be always welllocalized in space and behave likea point moving according toNewton’s laws.In a measurement situation, onerecovers quantum probabilitiesand postulate of w.p. reduction



Dynamics of the CSL model

The CSL model proposes the modification

d|ψ〉 =

[−iH − λ

2

(A− 〈A〉

)2]dt+

√λ(A− 〈A〉

)dWt

|ψ〉 (9)

The stochastic behavior is codified inthe noise W (t), formally known as aWiener process, i.e. a process thatsatisfy E(dWt) = 0,E(dWtdWt′) = δ(t− t′)dt2, where Eis an ensemble average over posiblerealizations of Wt.The operator A is known as thecollapse operator. The wavefunction evolves (stochastically) toone of the eigen-states of A.The parameter λ measures the“strength” of the collapse;λ ≡ (m/m0)λCSLLower Bound: λCSL = 10−16s−1.Upper Bound: 1012λCSL



Relating the classical quantities with the quantum objects

Let us recall the expression for alm,

alm =4πil

3

∫d3k

(2π)3jl(kRD)Y ∗lm(k)∆(k)Ψk, (10)

The CSL model evolves the vacuum state |0〉 to a final state |Θ〉 whichdoes not poses the same symmetries as the vacuum state.

How to relate Ψk with Ψk?




First option: Use semi-classical gravity Gab = 8πG〈Tab〉.The geometry is always clasical, only the matter fields are quantized.

Einstein semi-classical equations lead to:

Ψk(η) =H

k2MP

√2

ε〈Θ(η)|πk|Θ(η)〉. (11)

[P. Canate, P. Pearle & D. Sudarsky 2013], where πk is the canonicalconjugated momentum to δφk.

Under some circumstances the model is consistent with observationaldata.

On the other hand, it has been shown that in the semi-classical gravityframework, the tensor-to-scalar ratio is [GL, L. Kraiselburd & S. Landau,2015]

r < 10−9ε2




Second option (our approach) [GL, G. Bengochea 2015]

Ψk = 〈Ψk〉At the beginning of inflation, the quantum state of inflaton is the vacuum|0〉. At the end of inflation the state of each mode of the inflaton is |Θ〉.The CSL model drives the evolution |0〉 → |Θ〉. Thus,

Ψk(τ) = 〈0|Ψk|0〉 = 0 (12)

7−→

Ψk(η) = 〈Θ(η)|Ψk|Θ(η)〉 (13)



CSL inflationary model

Step I:Write the Hamiltonian for the Mukhanov-Sasaki variable

HR,Ik = pR,Ik p?R,Ik +

z′

z

(vR,Ik p?R,Ik + v?R,Ik pR,Ik

)+ k2vR,Ik v?R,Ik , (14)

The indexes R, I denote the real and imaginary parts of the field and itsmomentum .

The canonical conjugated momentum to vk is

pk = v′k − (z′/z)vk, (15)

with z ≡ aφ′0/H and H ≡ a′/aPromote to quantum operators vk and pk with commutator[vR,Ik , pR,Ik′ ] = iδ(k− k′).



CSL inflationary model

Step II:Write the wave functional (it is convenient to work in the momentumrepresentation) Φ[p(η, x)] = ΠkΦk[pRk , p

Ik ] = ΠkΦRk (pRk , η)ΦIk(pIk , η).

The wave functional associated to each model of the real and imaginarypart

ΦR,I(η, pR,Ik ) = exp[−Ak(η)(pR,Ik )2 +Bk(η)pR,Ik + Ck(η)] (16)

evolves according to the CSL-Schrodinger equation, the initial conditionsare Ak(τ) = 1/2k,Bk(τ) = Ck(τ) = 0 corresponding to theBunch-Davies vacuum and τ is the conformal time at the beginning ofinflation.



CSL Inflationary model

Step III: Choose the collapse operator.

Einstein perturbed equations δGab = 8πGδTab lead to

Ψk = 〈Ψk〉 =

√ε

2

H

k2MP〈pk〉 =

√ε

2

H

k2MP[〈pRk 〉+ i〈pIk〉], (17)

The collapse operator is pk.



CSL Inflationary model

Step IV:

Solve the CSL-Schrodinger equation

|ΦR,Ik , η〉 = T exp−∫ η

τ

dη′[iHR,Ik +

1

4λk(W (η′)− 2λkp

R,Ik )2]|ΦR,Ik , τ〉

(18)

We can now compute

Ψk = 〈Ψk〉 =

√ε

2

H

k2MP〈pk〉 =

√ε

2

H

k2MP[〈pRk 〉+ i〈pIk〉], (19)

Additionally, one has

[∆2ΨR,Ik (η)]→ 0 and also 〈ΨR,I

k 〉 6= 0 (20)

as a(η) is increasing.



Modelo CSL inflacionario

Step V:

Obtain the theroetical predictions, in particular, we can calculate thescalar power spectrum defined as:

ΨkΨ?k′ ≡

2π2

k3Ps(k)δ(k− k′) (21)

Once the power spectrum is known, the model yields a prediction for theobservational quantity Cl = (2l + 1)−1∑

m |alm|2.

Cl = 4π

∫ ∞0

dk

kj2l (kRD)∆2(k)Ps(k) (22)



Theoretical Predictions

Our approach predicts

Ps(k) = Askns−1F (λk) (23)

where

As =H2

M2P ε, ns − 1 = −6ε+ 2δ, (24)

If λk = 0 then F (0) = 0, i.e. the space-time is perfectly homogeneousand isotropic Ψk = 〈0|Ψk|0〉 = 0.

The function F (λk) can be made nearly scale invariant if

λk =1

k|τ | ≡λ0

k(25)

The conformal time at the beginning of inflation τ depends on thee-foldings of inflation N ≡ ln(af/ai) and the energy scale asociated toinflation V 1/4.



The Power Spectrum

The standard prediction for the PS is Ps(k) = Askns−1, with ns = 0.96

Our prediction is Ps(k) = Askns−1F (λk), with λk = λ0/k. The value of

λ0 depends on N and V 1/4.We assume EGUT = 1016 GeV and variate NPlot Ps(k) vs. k



Plot of the angular power spectrum (TT) l(l + 1)Cl vs. l



Tensor modes

The prediction for the tensor modes are

Pt(k) = AtkntF (λk) (26)

with

At =H2

M2P

, nt = −2ε (27)

The tensor-to-scalar ratio r

r ≡ PtPs= 16ε (28)



Conclusions

When applying the CSL model to inflation, we obtain a modification forthe power spectrum codified in the function F (λk)

If λk = λ0/k is possible to match the standard prediction under somecircumstances. In particular, λ0 depends on N and V 1/4

On the other hand, it is possible to explore small differences between ourprediction and the standard one by considering λk = [λ0/k]α (mightbehave as a running of the spectral index).

On the other hand, the CSL model is not relativistic; therefore, there isno way to justify, from first principles, the choice of the collapse operatorand the recipe λk = λ0/k

Introducing a collapse mechanism violates the energy conservation,which might induces divergences in the energy-momentum tensor(future work).

In our opinion, the CSL model can be applied, at least in aPhenomenological way, to the inflationary universe. That is, in a way thatis consistent with the observational data. Nevertheless, there is futurework to be done...



Gracias! - Obrigado!


the csl model as a mechanism for the quantum-to …relating the classical quantities with the...

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