Cosmological constraints on quantum gravity Jakub Mielczarek - - PowerPoint PPT Presentation

Cosmological constraints on quantum gravity

Jakub Mielczarek

Jagiellonian University, Cracow National Centre for Nuclear Research, Warsaw

4 September, 2014, Trieste

Jakub Mielczarek Cosmological constraints on quantum gravity

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 introduced by the following replacement in the classical Hamiltonian: ¯ k → K[n] := sin(n¯ µγ¯ k) n¯ µγ , 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) of the phase space. For the FRW model, the classical phase space ΓG = R × R is deformed onto ΓQ

G = R × U(1).

Jakub Mielczarek Cosmological constraints on quantum gravity

The holonomy corrections deform the classical Friedmann equation: H2 = 8πG 3 ρ

• 1 − ρ

ρc

• ,

where the critical energy density ρc =

3 8πGλ2γ2 ∼ ρPl. Because

ρ ≤ ρc, the classical Big Bang singularity is replaced by the non-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 Bounce unavoidably leads to the phase of cosmic inflation2. Klein-Gordon equation on the FRW background: ¨ φ + 3H ˙ φ + 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 and
• A. Barrau, Phys. Rev. D 81 (2010) 104049

Jakub Mielczarek Cosmological constraints on quantum gravity

Inhomogeneities - Old story

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

• 3M. Bojowald and G. M. Hossain, Phys. Rev. D 77 (2008) 023508
• 4J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D 81

(2010) 104049

Jakub Mielczarek Cosmological constraints on quantum gravity

Shape of the power spectrum - intuitive explanation

Scale of the bump in the power spectrum k∗ ∼

1 λ∗ .

Jakub Mielczarek Cosmological constraints on quantum gravity

Pessimistic plot

The present value of λ∗ is very sensitive on duration of inflationary

• phase5. It makes possibility of observing the suppression due to the

bounce almost improbable. There is only very narrow observational window for kinetic bounces with FB ∼ 10−13.

• 5J. Mielczarek , M. Kamionka, A. Kurek and M. Szydlowski, “Observational

hints on the Big Bounce,” JCAP 1007 (2010) 004.

Jakub Mielczarek Cosmological constraints on quantum gravity

The primordial gravitational waves can be used to impose

• bservational constraints on the Big Bounce scenario. It becomes

possible thanks to observational limitations on the B-type polarization of the CMB. The Big Bounce is asymmetric6: [0, 1] ∋ 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. D 82 (2010) 123520

Jakub Mielczarek Cosmological constraints on quantum gravity

Inhomogeneities - New story

Problems: The procedure of introducing quantum corrections suffers from ambiguities. In general, the algebra of modified constraints is not closed: {CQ

I , CQ J } = gK IJ(Aj b, E a i )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 linear inhomogeneities on the flat FRW background, the answer is affirmative: scalar perturbations - T. Cailleteau, J. Mielczarek, A. Barrau,

• J. Grain, Class. Quantum Grav. 29 (2012) 095010,

vector perturbations - J. Mielczarek, T. Cailleteau, A. Barrau and J. Grain, Class. Quant. Grav. 29 (2012) 085009, tensor perturbations - no problem with anomalies.

Jakub Mielczarek Cosmological constraints on quantum gravity

Cosmological perturbations

Tedious calculations lead us to deformed EOM for the Munhanov variable7: d2 dτ 2 v − c2

s ∇2v − z

′′

z v = 0, where the holonomy corrections are introduced through c2

s = Ω = cos(2γ¯

µ¯ k) = 1 − 2 ρ ρc where 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).

Jakub Mielczarek Cosmological constraints on quantum gravity

Algebra of constraints:

{D[Na

1], D[Na 2]}

= D[Nb

1 ∂bNa 2 − Nb 2 ∂bNa 1],

• 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 classical case due to presence of the factor Ω = cos(2¯ µγ¯ k) = 1 − 2 ρ 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 = −1 to the Euclidean one.

Jakub Mielczarek Cosmological constraints on quantum gravity

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 the Planck epoch, space-time becomes a four dimensional Euclidean

• space. 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 ρ

ρc → 0.

Communication between different space points is forbidden10. The similar behavior is expected at the classical level by virtue of the famous BKL conjecture (Belinsky-Khalatnikov-Lifshitz).

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

Jakub Mielczarek Cosmological constraints on quantum gravity

Carrollian limit

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

Jakub Mielczarek Cosmological constraints on quantum gravity

Silent initial conditions?

Is there any natural choice of the initial conditions at the beginning

• f the Lorentzian phase (silent surface)11?

Are correlations between physical quantities at the different points vanishing (white noise)? This situation can be modeled by the following correlation function G(r) = G0 for ξ ≥ r ≥ 0, for r > ξ. The corresponding power spectrum can be found straightforwardly: P(k) = 2 πk3 ∞ 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 the k3 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 dη2 φ + 2

• H − 1

2Ω dΩ dη d dηφ − Ω∇2φ = 0. By introducing u = zφ with z = a/ √ Ω we get d2 dη2 u − Ω∇2u − z

′′

z u = 0, which after Fourier transform u =

• d3k

(2π)3/2 ukeik·x takes the form

d2 dη2 uk +

• Ωk2 − z

′′

z

• uk = 0.

Jakub Mielczarek Cosmological constraints on quantum gravity

Characteristic scale for the modes is λH = a kH , defined such that

• Ωk2

H

• =
• z

′′

z

• .

The modes are super-horizon sized if λ ≫ λH and sub-horizon sized if λ ≪ λH. The modes are “frozen” at the super-horizon scales and decaying at the sub-horizon scales. Let us investigate properties of the modes in the case of barotropic fluid p = wρ for which z

′′

z = ρcκa2 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 standard canonical procedure. Promoting this quantity to be an operator,

• ne performs the decomposition

ˆ uk(η) = i

1 2 (1−sgnΩ)(fk(η)ˆ

ak + f ∗

k (η)ˆ

a†

−k),

where fk(η) is the so-called mode function which satisfies the same equation as uk(η). The creation (ˆ a†

k) and annihilation (ˆ

ak)

• perators fulfill the commutation relation [ˆ

ak, ˆ a†

q] = δ(3)(k − q).

Two-point correlation function for the φ filed is given by 0|ˆ φ(x, η)ˆ φ(y, η)|0 = ∞ dk k Pφ(k, η)sin kr kr , where the power spectrum Pφ(k, η) = k3 2π2

• fk

z

• 2

, and r = |x − y|.

Jakub Mielczarek Cosmological constraints on quantum gravity

Vacuum

Vacuum expectation value of the Hamiltonian ˆ H = 1 2

• d3k
• ˆ

akˆ a−kFk + ˆ a†

kˆ

a†

−kF ∗ k +

• 2ˆ

a†

kˆ

ak + δ(3)(0)

• Ek
• ,

where Fk = (f ′

k)2 + ω2 kf 2 k , Ek = |f ′ k|2 + ω2 k|fk|2 and ω2 k = Ωk2 − z

′′

z

is 0|ˆ H|0 = δ(3)(0)1 2

• d3kEk.

The ground state (vacuum) can be found by minimizing Ek with the Wronskian condition fk(f ′

k)∗ − f ∗ k f

′

k = i taken into account.

The energy can be minimized only if ω2

k > 0 - interpretation in

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

Jakub Mielczarek Cosmological constraints on quantum gravity

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

′′

z across

the moment of signature change is regular. No well defined vacuum state in any part of the Euclidean domain for w > 1

6(−7 +

√ 17) ≈ −0.48.

Jakub Mielczarek Cosmological constraints on quantum gravity

Regions of ω2

k > 0 for w = −1, where ω2 k = Ωk2 − z

′′

z .

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

Jakub Mielczarek Cosmological constraints on quantum gravity

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

√ 3 2√2κρca, which

leads to the power spectrum Pφ(k) ≡ k3 2π2

• fk

z

• 2

= √ 3 4π2√2κρc k a 3 ∝ k3. The correlation function is vanishing 0|ˆ φ(x, η)ˆ φ(y, η)|0 = 0 (white noise). Vacuum at Ω > 0: |fk|2 =

1 2k √ Ω, which leads to the power

spectrum Pφ(k) ≡ k3 2π2

• fk

z

• 2

= k2 4π2 Ω a ∝ k2. This is Ω−deformed analogue of the Bunch-Davies vacuum.

Jakub Mielczarek Cosmological constraints on quantum gravity

Inflationary power spectra for Ω > 0

Using the Ω−deformed Bunch-Davies vacuum the inflationary spectra can be derived12: PS(k) = AS k aH nS−1 and PT(k) = AT k aH nT , AS = 1 πǫ H mPl 2 (1 + 2δH) and nS = 1 + 2η − 6ǫ + O(δ2

H),

AT = 16 π H mPl 2 (1 + δH) and nT = −2ǫ + O(δ2

H).

For the typical inflationary models δH := V

ρc ∼ 10−12. The

corrections are extremely hard to constrain observationally.

• 12J. Mielczarek, “Inflationary power spectra with quantum holonomy

corrections,” JCAP 1403 (2014) 048

Jakub Mielczarek Cosmological constraints on quantum gravity

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

′′

z across

Ω = 0 is regular and in the vicinity of Ω = 0 can be approximated as follows z

′′

z ≈ 1 3κρca2Ω, based on which ω2

k ≈ Ω

• k2 − 1

3κρca2

• .

For the super-horizon modes k <

• 1

3κρca2 (while

approaching Ω → 0−) the vacuum is characterized by the following spectrum Pφ(k) = √ 3

• |Ω|

4π2√κρc k a 3 ∝

• |Ω|k3.

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

Jakub Mielczarek Cosmological constraints on quantum gravity

Conversion of the power spectrum

The super-horizon k3 power spectrum can be converted to the quasi-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.

Jakub Mielczarek Cosmological constraints on quantum gravity

Looking beyond the horizon...

The modes relevant from the perspective of testing pre-inflationary period are super-horizon in the present epoch. Within the linear theory, due to decoupling of the modes, there is no access to the information stored at the super-horizon scales. Is there any hope?

Jakub Mielczarek Cosmological constraints on quantum gravity

Non-linearities couple modes at the different scales - the IR/UV mixing. Non-Gaussianity allows us to look beyond the cosmic horizon! The information from the super-horizon scales can be partially recovered.

Jakub Mielczarek Cosmological constraints on quantum gravity

Non-Gaussian gravitational potential can be parametrised as follows Φ = ΦG + fNL(Φ2

G − Φ2 G),

where subscript G denotes Gaussianity of a given field and fNL parametrizes strength of the non-Gaussianity. By decomposing gravitational potential for Φ = Φsub + Φsuper in the presence of non-Gaussianity, sub-horizon fluctuations are explicitly dependent on physics at the super-horizon scales: Φ2

sub = Φ2 G,sub(1 + 2fNLΦG,super)

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

Jakub Mielczarek Cosmological constraints on quantum gravity

Other promising alternatives: Gravitational phase transitions is the early universe. Second

• rder gravitational phase transitions, such as those observed

in CDT, may be realized in the very early universe13. Under certain assumptions such phase transition may lead to nearly scale invariant spectrum of perturbations14. Gravitational defects at the boundaries of domains can be formed. Dimensional reduction is the early universe. Dimensional reduction from dS = 4 at large scales to dS = 2 at short scales may lead to to nearly scale invariant spectrum of perturbations

• 15. Such dimensional reduction is predicted within various

approaches 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) 3691

• 15G. Amelino-Camelia, M. Arzano, G. Gubitosi and J. Magueijo,

“Dimensional reduction in the sky,” Phys. Rev. D 87 (2013) 12, 123532

Jakub Mielczarek Cosmological constraints on quantum gravity

From HDA to Poincar´ e algebra

General relativity → Hypersurface deformation algebra {D[Na

1], D[Na 2]}

= D[Nb

1 ∂bNa 2 − Nb 2 ∂bNa 1],

{S[N], D[Na]} = −S[Na∂aN], {S[N1], S[N2]} = D

• gab(N1∂bN2 − N2∂bN1)
• .

Special relativity → Poincar´ e algebra {Pµ, Pν} = 0, {Mµν, Pν} = ηµρPν − ηνρPµ, {Mµν, Mρσ} = ηµρMνσ − ηµσMνρ − ηνρMµσ + ηνσMµρ.

Jakub Mielczarek Cosmological constraints on quantum gravity

Loop-deformed Poincar´ e algebra

Loop-deformed Poincar´ e algebra16: [Ji, Jj] = iǫijkJk, [Ji, Kj] = iǫijkKk, [Ki, Kj] = −iΩǫijkJk, [Ji, Pj] = iǫijkPk, [Ki, Pj] = iδijP0, [Ji, P0] = 0, [Ki, P0] = iΩPi, [Pi, Pj] = 0, [Pi, P0] = 0. Assuming that Ω = Ω(P0, P2

i ) = A(P0)B(P2 i ), the Jacobi identities

are satisfied if Ω = P2

0 − α

P2

i − α.

The constant of integration α plays a role of the deformation parameter.

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

essay competition of the Gravity Research Foundation.

Jakub Mielczarek Cosmological constraints on quantum gravity

Combining cosmological and astrophysical constraints

Ω = cos

• πP0

E∗

• .

Group velocity: Deformation of the dispersion relation → time lags of the GRB

• photons. E∗ > 4.7 · 1010 GeV.

Ω = P2

0 − α

P2

i − α.

Spectral dimension: Dimensional reduction from dS = 4 at the large scales to dS = 1 at the short scales.

Jakub Mielczarek Cosmological constraints on quantum gravity

Summary and outlook

LQG → suppression and oscillations in the spectrum of the primordial perturbations. However, the relevant features are at super-horizon scales at the moment. The signature change and the Big Silence open new possibilities. Silent initial conditions in the form of a white noise. Super-horizon vacuum in the Euclidean domain. Phenomenology available thanks to loop-deformed Poincer´ e

• algebra. Complementary methods of testing the same

quantum gravitational effects: cosmology and astrophysics. Studying super-horizon modes through non-Gaussianity. IR/UV mixing and transfer of information from large to short

• scales. . .

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

Jakub Mielczarek Cosmological constraints on quantum gravity