cosmological constraints on quantum gravity · 9j. mielczarek, \signature change in loop quantum...

$: Cosmological constraints on quantum gravity · 9J. Mielczarek, \Signature change in loop quantum cosmology," Springer Proc. Phys. 157 (2014) 555 [arXiv:1207.4657 [gr-qc]]. Jakub Mielczarek$
Cosmological constraints on quantum gravity

Jakub Mielczarek

Jagiellonian University, CracowNational Centre for Nuclear Research, Warsaw

4 September, 2014, Trieste

Jakub Mielczarek Cosmological constraints on quantum gravity

Quantum corrections

The holonomy corrections mimic the loop quantization.

For the FRW model, the holonomy corrections can be introducedby the following replacement in the classical Hamiltonian:

k → K[n] :=sin(nµγk)

nµγ,

where k is canonically conjugated with p = a2. Furthermore,µ ∝ pδ where −1/2 ≤ δ ≤ 0 and n ∈ N.

Holonomy corrections can be seen as bending (periodification) ofthe phase space. For the FRW model, the classical phase spaceΓG = R× R is deformed onto ΓQ

G = R× U(1).


The holonomy corrections deform the classical Friedmann equation:

H2 =8πG

3ρ

(1− ρ

ρc

),

where the critical energy density ρc = 38πGλ2γ2 ∼ ρPl. Because

ρ ≤ ρc , the classical Big Bang singularity is replaced by thenon-singular Big Bounce1.

Contracting and expanding branches are causally connected.

1Bojowald, Ashtekar, Paw lowski, Singh, ...Jakub Mielczarek Cosmological constraints on quantum gravity

For the model with a massive scalar field, the Big Bounceunavoidably leads to the phase of cosmic inflation2.

Klein-Gordon equation on the FRW background:

φ+ 3Hφ+ m2φ = 0.

2J. Mielczarek, Phys. Rev. D 81 (2010) 063503, A. Ashtekar and D. Sloan,Phys. Lett. B 694 (2010) 108, J. Mielczarek, T. Cailleteau, J. Grain andA. Barrau, Phys. Rev. D 81 (2010) 104049


Inhomogeneities - Old story

Using the EOM for tensor modes with holonomy correctionsderived in Ref.3 one can compute the following power spectra4:

3M. Bojowald and G. M. Hossain, Phys. Rev. D 77 (2008) 0235084J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D 81

(2010) 104049Jakub Mielczarek Cosmological constraints on quantum gravity

Shape of the power spectrum - intuitive explanation

Scale of the bump in the power spectrum k∗ ∼ 1λ∗

.


Pessimistic plot

The present value of λ∗ is very sensitive on duration of inflationaryphase5. It makes possibility of observing the suppression due to thebounce almost improbable. There is only very narrow observationalwindow for kinetic bounces with FB ∼ 10−13.

5J. Mielczarek , M. Kamionka, A. Kurek and M. Szydlowski, “Observationalhints on the Big Bounce,” JCAP 1007 (2010) 004.


The primordial gravitational waves can be used to imposeobservational constraints on the Big Bounce scenario. It becomespossible thanks to observational limitations on the B-typepolarization of the CMB.

The Big Bounce is asymmetric6:

[0, 1] 3 FB :=V (φB)

ρc> 2.3 · 10−13. (ρ = T + V )

6J. Mielczarek, M. Kamionka, A. Kurek and M. Szyd lowski, JCAP 1007(2010) 004; J. Grain, A. Barrau, T. Cailleteau and J. Mielczarek, Phys. Rev. D82 (2010) 123520


Inhomogeneities - New story

Problems:

The procedure of introducing quantum corrections suffersfrom ambiguities.

In general, the algebra of modified constraints is not closed:

CQI , C

QJ = gK

IJ(Ajb,E

ai )CQ

K +AIJ .

Can we introduce quantum corrections in the anomaly-free manner(i.e. such that AIJ = 0)? We found that, at least for linearinhomogeneities on the flat FRW background, the answer isaffirmative:

scalar perturbations - T. Cailleteau, J. Mielczarek, A. Barrau,J. Grain, Class. Quantum Grav. 29 (2012) 095010,

vector perturbations - J. Mielczarek, T. Cailleteau, A. Barrauand J. Grain, Class. Quant. Grav. 29 (2012) 085009,

tensor perturbations - no problem with anomalies.


Cosmological perturbations

Tedious calculations lead us to deformed EOM for the Munhanovvariable7:

d2

dτ2v − c2

s∇2v − z′′

zv = 0,

where the holonomy corrections are introduced through

c2s = Ω = cos(2γµk) = 1− 2

ρ

ρcwhere ρc =

3

8πG∆γ2∼ ρPl.

Similarly for the GWs and scalar matter8. Mixed type EOMs:

hyperolic - for ρ < ρc/2 (Ω > 0),

parabolic - for ρ = ρc/2 (Ω = 0),

elliptic - for ρ > ρc/2 (Ω < 0).

7T. Cailleteau, J. Mielczarek, A. Barrau, J. Grain, Class. Quantum Grav. 29(2012) 095010

8T. Cailleteau, A. Barrau, J. Grain and F. Vidotto, Phys. Rev. D 86 (2012)087301, J. Mielczarek, PhD thesis (2012).


Algebra of constraints:

D[Na1 ],D[Na

2 ] = D[Nb1 ∂bN

a2 − Nb

2 ∂bNa1 ],

SQ [N],D[Na]

= −SQ [Na∂aN],SQ [N1],SQ [N2]

= ΩD

[gab(N1∂bN2 − N2∂bN1)

],

The algebra is closed but deformed with respect to the classicalcase due to presence of the factor

Ω = cos(2µγk) = 1− 2ρ

ρc∈ [−1, 1] where ρc =

3

8πG∆γ2∼ ρPl.

What is the interpretation? Classically, we have

S [N1], S [N2] = sD

[N

p∂a(δN2 − δN1)

],

where s = 1 corresponds to the Lorentzian signature and s = −1to the Euclidean one.


Signature change

The sign of Ω reflects a signature of space: Ω > 0 (ρ < ρc/2) -Lorentzian signature, Ω < 0 (ρ > ρc/2) - Euclidean signature 9

Based on this, a new picture of the Big Bounce emerges: In thePlanck epoch, space-time becomes a four dimensional Euclideanspace. Similarity to the Harte-Hawking no-boundary proposal.

Causal connection between the branches is limited.9J. Mielczarek, “Signature change in loop quantum cosmology,” Springer

Proc. Phys. 157 (2014) 555 [arXiv:1207.4657 [gr-qc]].Jakub Mielczarek Cosmological constraints on quantum gravity

Silence in LQC

The light cones are collapsing onto the time lines for. ρ→ ρc/2.

The effective speed of light ceff =√

Ω =√

1− 2 ρρc→ 0.

Communication between different space points is forbidden10.

The similar behavior is expected at the classical level by virtue ofthe famous BKL conjecture (Belinsky-Khalatnikov-Lifshitz).

10J. Mielczarek, “Asymptotic silence in loop quantum cosmology,” AIP Conf.Proc. 1514 (2012) 81


Carrollian limit

“A slow sort of country ..., ... now, here, you see, it takes all therunning you can do to stay in the same place ...” Lewis Carroll


Silent initial conditions?

Is there any natural choice of the initial conditions at the beginningof the Lorentzian phase (silent surface)11?

Are correlations between physical quantities at the different pointsvanishing (white noise)? This situation can be modeled by thefollowing correlation function

G (r) =

G0 for ξ ≥ r ≥ 0,0 for r > ξ.

The corresponding power spectrum can be found straightforwardly:

P(k) =2

πk3

∫ ∞0

dr G (r) r2 sin(kr)

kr≈ 2

3

G0

π(kξ)3,

in the limit kξ 1. At large scales, the power spectrum is of thek3 form. What about the quantum vacuum?

11J. Mielczarek, L. Linsefors, A.Barrau (in preparation)Jakub Mielczarek Cosmological constraints on quantum gravity

Quantum generation of perturbations

Let us focus on the tensor modes (gravitational waves)h+,× =

√16πGφ for which we have the following EOM:

d2

dη2φ+ 2

(H− 1

2Ω

dΩ

dη

)d

dηφ− Ω∇2φ = 0.

By introducing u = zφ with z = a/√

Ω we get

d2

dη2u − Ω∇2u − z

′′

zu = 0,

which after Fourier transform u =∫

d3k(2π)3/2 uke

ik·x takes the form

d2

dη2uk +

(Ωk2 − z

′′

z

)uk = 0.


Characteristic scale for the modes is

λH =a

kH,

defined such that ∣∣Ωk2H

∣∣ =

∣∣∣∣∣z′′

z

∣∣∣∣∣ .The modes are super-horizon sized if λ λH and sub-horizonsized if λ λH . The modes are “frozen” at the super-horizonscales and decaying at the sub-horizon scales.

Let us investigate properties of the modes in the case of barotropicfluid p = wρ for which

z′′

z= ρcκa

2 x

Ω2

[1

2

(1

3− w

)+

(9

2+ 13w +

9

2

)x

− (3 + 30w + 15w2)x2 +

(11

3+ 21w + 12w2

)x3

],

where x = ρ/ρc . Danger of a divergence at Ω→ 0.Jakub Mielczarek Cosmological constraints on quantum gravity

Quantization of modes

Let us quantize the Fourier modes uk(η). This follows the standardcanonical procedure. Promoting this quantity to be an operator,one performs the decomposition

uk(η) = i12

(1−sgnΩ)(fk (η)ak + f ∗k (η)a†−k),

where fk (η) is the so-called mode function which satisfies the same

equation as uk(η). The creation (a†k) and annihilation (ak)

operators fulfill the commutation relation [ak, a†q] = δ(3)(k− q).

Two-point correlation function for the φ filed is given by

〈0|φ(x, η)φ(y, η)|0〉 =

∫ ∞0

dk

kPφ(k , η)

sin kr

kr,

where the power spectrum

Pφ(k , η) =k3

2π2

∣∣∣∣ fkz∣∣∣∣2,

and r = |x− y|.Jakub Mielczarek Cosmological constraints on quantum gravity

Vacuum

Vacuum expectation value of the Hamiltonian

H =1

2

∫d3k

[aka−kFk + a†ka

†−kF

∗k +

(2a†kak + δ(3)(0)

)Ek

],

where Fk = (f ′k )2 + ω2k f

2k , Ek = |f ′k |2 + ω2

k |fk |2 and ω2k = Ωk2 − z

′′

zis

〈0|H|0〉 = δ(3)(0)1

2

∫d3kEk .

The ground state (vacuum) can be found by minimizing Ek withthe Wronskian condition fk (f ′k )∗ − f ∗k f

′k = i taken into account.

The energy can be minimized only if ω2k > 0 - interpretation in

terms of particle-like excitations of the filed is possible.


Only for w = −1 (Cosmological constant) evolution of z′′

z acrossthe moment of signature change is regular. No well definedvacuum state in any part of the Euclidean domain forw > 1

6 (−7 +√

17) ≈ −0.48.


Regions of ω2k > 0 for w = −1, where ω2

k = Ωk2 − z′′

z .

In the Euclidean region, the state of vacuum is well defined for thelarge scale modes. This is in opposite to the Lorentzian case.


Vacuum for Ω = −1 with w = −1: |fk |2 =√

32√

2κρc a, which

leads to the power spectrum

Pφ(k) ≡ k3

2π2

∣∣∣∣ fkz∣∣∣∣2 =

√3

4π2√

2κρc

(k

a

)3

∝ k3.

The correlation function is vanishing 〈0|φ(x, η)φ(y, η)|0〉 = 0(white noise).

Vacuum at Ω > 0: |fk |2 = 12k√

Ω, which leads to the power

spectrum

Pφ(k) ≡ k3

2π2

∣∣∣∣ fkz∣∣∣∣2 =

k2

4π2

Ω

a∝ k2.

This is Ω−deformed analogue of the Bunch-Davies vacuum.


Inflationary power spectra for Ω > 0

Using the Ω−deformed Bunch-Davies vacuum the inflationaryspectra can be derived12:

PS(k) = AS

(k

aH

)nS−1

and PT(k) = AT

(k

aH

)nT

,

AS =1

πε

(H

mPl

)2

(1 + 2δH) and nS = 1 + 2η − 6ε+O(δ2H),

AT =16

π

(H

mPl

)2

(1 + δH) and nT = −2ε+O(δ2H).

For the typical inflationary models δH := Vρc∼ 10−12. The

corrections are extremely hard to constrain observationally.

12J. Mielczarek, “Inflationary power spectra with quantum holonomycorrections,” JCAP 1403 (2014) 048


Vacuum for Ω ≈ 0. In case of w = −1, evolution of z′′

z acrossΩ = 0 is regular and in the vicinity of Ω = 0 can beapproximated as follows

z′′

z≈ 1

3κρca

2Ω,

based on which

ω2k ≈ Ω

(k2 − 1

3κρca

2

).

For the super-horizon modes k <√

13κρca2 (while

approaching Ω→ 0−) the vacuum is characterized by thefollowing spectrum

Pφ(k) =

√3√|Ω|

4π2√κρc

(k

a

)3

∝√|Ω|k3.

The power spectrum is vanishing in the limit Ω→ 0−.


Conversion of the power spectrum

The super-horizon k3 power spectrum can be converted to thequasi-flat spectrum in the following sequence:

Pφ ∝ k3 (super) →︸︷︷︸w=1

Pφ ∝ k2 (sub) →︸︷︷︸w=−1

Pφ ≈ const (super)

Exemplary power spectrum:

The oscillations are due to sub-horizon evolution.


Looking beyond the horizon...

The modes relevant from the perspective of testing pre-inflationaryperiod are super-horizon in the present epoch. Within the lineartheory, due to decoupling of the modes, there is no access to theinformation stored at the super-horizon scales.

Is there any hope?


Non-linearities couple modes at the different scales - the IR/UVmixing.

Non-Gaussianity allows us to look beyond the cosmic horizon! Theinformation from the super-horizon scales can be partiallyrecovered.


Non-Gaussian gravitational potential can be parametrised asfollows

Φ = ΦG + fNL(Φ2G − 〈Φ2

G 〉),

where subscript G denotes Gaussianity of a given field and fNL

parametrizes strength of the non-Gaussianity. By decomposinggravitational potential for

Φ = Φsub + Φsuper

in the presence of non-Gaussianity, sub-horizon fluctuations areexplicitly dependent on physics at the super-horizon scales:

〈Φ2sub〉 = 〈Φ2

G ,sub〉(1 + 2fNLΦG ,super)

This simple example shows how the super-horizon modes caninfluence observations performed sub-horizontaly.


Other promising alternatives:

Gravitational phase transitions is the early universe. Secondorder gravitational phase transitions, such as those observedin CDT, may be realized in the very early universe13. Undercertain assumptions such phase transition may lead to nearlyscale invariant spectrum of perturbations14. Gravitationaldefects at the boundaries of domains can be formed.

Dimensional reduction is the early universe. Dimensionalreduction from dS = 4 at large scales to dS = 2 at short scalesmay lead to to nearly scale invariant spectrum of perturbations15. Such dimensional reduction is predicted within variousapproaches to QG, such as CDT, Asymptotic Safety, . . .

13J. Mielczarek,“Big Bang as a critical point,”arXiv:1404.0228 [gr-qc].14J. Magueijo, L. Smolin and C. R. Contaldi,“Holography and the

scale-invariance of density fluctuations,”Class. Quant. Grav. 24 (2007) 369115G. Amelino-Camelia, M. Arzano, G. Gubitosi and J. Magueijo,

“Dimensional reduction in the sky,” Phys. Rev. D 87 (2013) 12, 123532Jakub Mielczarek Cosmological constraints on quantum gravity

From HDA to Poincare algebra

General relativity → Hypersurface deformation algebra

D[Na1 ],D[Na

2 ] = D[Nb1 ∂bN

a2 − Nb

2 ∂bNa1 ],

S [N],D[Na] = −S [Na∂aN],

S [N1],S [N2] = D[gab(N1∂bN2 − N2∂bN1)

].

Special relativity → Poincare algebra

Pµ,Pν = 0,

Mµν ,Pν = ηµρPν − ηνρPµ,Mµν ,Mρσ = ηµρMνσ − ηµσMνρ − ηνρMµσ + ηνσMµρ.


Loop-deformed Poincare algebra

Loop-deformed Poincare algebra16:

[Ji , Jj ] = iεijkJk , [Ji ,Kj ] = iεijkKk , [Ki ,Kj ] = −iΩεijkJk ,

[Ji ,Pj ] = iεijkPk , [Ki ,Pj ] = iδijP0, [Ji ,P0] = 0,

[Ki ,P0] = iΩPi , [Pi ,Pj ] = 0, [Pi ,P0] = 0.

Assuming that Ω = Ω(P0,P2i ) = A(P0)B(P2

i ), the Jacobi identitiesare satisfied if

Ω =P2

0 − αP2

i − α.

The constant of integration α plays a role of the deformationparameter.

16M. Bojowald and G. M. Paily, Phys. Rev. D 87 (2013) 044044,J. Mielczarek, arXiv:1304.2208 [gr-qc]. Essay honorably mentioned in the 2013essay competition of the Gravity Research Foundation.


Combining cosmological and astrophysical constraints

Ω = cos

(πP0

E∗

).

Group velocity:

Deformation of the dispersionrelation → time lags of the GRBphotons. E∗ > 4.7 · 1010 GeV.

Ω =P2

0 − αP2

i − α.

Spectral dimension:

Dimensional reduction fromdS = 4 at the large scales todS = 1 at the short scales.


Summary and outlook

LQG → suppression and oscillations in the spectrum of theprimordial perturbations. However, the relevant features areat super-horizon scales at the moment.

The signature change and the Big Silence open newpossibilities.

Silent initial conditions in the form of a white noise.

Super-horizon vacuum in the Euclidean domain.

Phenomenology available thanks to loop-deformed Poincerealgebra. Complementary methods of testing the samequantum gravitational effects: cosmology and astrophysics.

Studying super-horizon modes through non-Gaussianity.IR/UV mixing and transfer of information from large to shortscales. . .

Gravitational phase transitions and dimensional reduction inthe early universe. Maybe also in LQG. . .


cosmological constraints on quantum gravity · 9j. mielczarek, \signature change in loop quantum...

Documents