From CDT to Astronomical Observations

Jakub Mielczarek

Jagiellonian University, Cracow Centre de Physique Th´eorique, Marseille

25 May, 2018, Warsaw

Jakub Mielczarek From CDT to Astronomical Observations Causal Dynamical Triangulations

Z i (SHE [gµν ]+SΛ[gµν ]) hout|ini = Dgµνe } → Wick rotation →

Z 1 E E − (S [gµν ]+S [gµν ]) Z = Dgµνe } HE Λ → Discretization →

X 1 − 1 (SE [T ]+SE [T ]) Za = e } HE Λ CT T

Jakub Mielczarek From CDT to Astronomical Observations Two characteristic features

Non-trivial phase structure

The phases are characterized by different scalings → different Hausdorff and spectral dimensions. The spectral dimension is running as a function of diffusion time → Dimensional reduction in phase C.

Jakub Mielczarek From CDT to Astronomical Observations Random walk on a graph

Let us consider diffusion on a graph. The structure of adjacency between the nodes is represented by the links:

The spectral dimension of this graph can be found by determining spectrum (eigenvalues λn) of the Laplace operator ∆ ≡ D − A, where A is an adjacency matrix and D is a degree matrix:

 0 1 1 1   3 0 0 0   1 0 1 1   0 3 0 0  A =   , D =   .  1 1 0 0   0 0 2 0  1 1 0 0 0 0 0 2

Jakub Mielczarek From CDT to Astronomical Observations Spectral dimension

Heat kernel equation: ∂ K(x, y; σ) = −∆ K(x, y; σ), ∂σ x where σ is a diffusion time. Eigenproblem:

∆x φn(x) = λnφn(x). The heat kernel can be decomposed as follows:

X −σλn ∗ K(x, y; σ) = e φn(x)φn(y). n By using the expression for the trace of the heat kernel P(σ) := trK one can introduce

PN −λnσ ∂ log P(σ) i=1 λne dS ≡ −2 = 2σ N . ∂ log σ P −λnσ i=1 e

Jakub Mielczarek From CDT to Astronomical Observations Spectral dimension in CDT

D

DS 2.0 ‡ 4.5

4.0

1.5 3.5 Κ0=2.2, D=0.6

3.0 Κ0=3.6, D=0.6 1.0 C Κ0=4.4, D=0.6 A 2.5

Κ0=4.4, D=2.0 Ê 2.0 ‡ ‡ ‡ Ê 0.5 Ê Ê 1.5 Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê 0 100 200 300 400 500Σ 0.0 Ê 1 2 3 4 Ê 5 k0 Triple point Ê Ê B In the phase C of CDT the spectral dimension can be parametrized as follows dS (σ) = a − b/(c + σ), where σ is the diffusion time. The spectral dimension undergoes reduction from dS ≈ 4 in IR to dS ≈ 2 or dS ≈ 3/2 in UV, depending on location at the phase diagram [ J. Ambjorn, J. Jurkiewicz and R. Loll, Phys. Rev. Lett. 95 (2005) 171301, D. N. Coumbe and J. Jurkiewicz, JHEP 1503 (2015) 151].

Jakub Mielczarek From CDT to Astronomical Observations Reconstruction of the Laplace operator

The spectral dimension is related to the eigenvalues of the Laplace operator. Once again:

∂ log P(σ) d (σ) ≡ −2 , S ∂ log σ where in the continuous limit Z P(σ) = dµeσ4p .

4 The 4p is a 4D momentum space Laplace operator. In the R 2 2 p space we have 4p = −E − p , where p := p~ · p~. Wick rotating back to the Lorentzian case E → iE, the momentum-space wave (Klein-Gordon) equation is

2 2 2 2 4pφ = 0 → (E − p )φ = 0 ⇒ E = p (dispersion relation)

Jakub Mielczarek From CDT to Astronomical Observations Quantum space-time reverse engineering

Deformed dispersion relations → non-trivial diffusion time dependence of spectral dimensions1. We want to do the opposite: The task would be now to recover the form of ∆p from the CDT spectral dimension.

Will focus on the cases with dUV = 2 and dUV = 3/2.

After fixing the IR value dS to be precisely equal to the topological space-time dimension d = 4 and parametrizing UV value by dUV we obtain 4 − d d (σ) = 4 − UV , S 1 + σ/c

such that d(σ → 0) = dUV. The parameter c can be related to the √1 energy scale of the dimensional reduction: E∗ := c .

1D. Benedetti, Phys. Rev. Lett. 102 (2009) 111303; L. Modesto, Class. Quant. Grav. 26 (2009) 242002 Jakub Mielczarek From CDT to Astronomical Observations Following Ref. 2, we simplify the problem by assuming that the departure from classical case is parametrized by the single variable function Ω(p): 2 2 4p = −E − Ω(p) . The corresponding dispersion relation will be E = Ω(p). Furthermore, we assume that the momentum space is flat as in the classical case. Therefore, invariant (with respect to the isometries of a given space-time) measure in the momentum space is dEd3p~ dµ = (2π)4 . Taking the two above assumptions into account we obtain:

∞ 1 4π Z 2 √ 2 −σΩ (p) P(σ) = 2 dp p e . 4πσ (2π) 0

2JM, “From Causal Dynamical Triangulations To Astronomical Observations,” EPL 119 (2017) no.6, 60003. Jakub Mielczarek From CDT to Astronomical Observations As it has been shown in 3, the relation P(Ω) can be converted into the form of the inverse Laplace transform:

γ+iT   1 Z P(σ) 2 p3 = lim 12π5/2 √ eσΩ (p)dσ. 2πi T →∞ γ−iT σ

The form of function P(σ) is predicted by CDT and is given by

1 P(σ) = , 16π2σdUV/2(σ + c)2−dUV/2

∂ log P(σ) (Obtained by integrating dS (σ) ≡ −2 ∂ log σ with the CDT parametrization of dS (σ)). Performing the Laplace transformation we obtain:

3 3 2
p = Ω 1F1 2 − dUV/2, 5/2; −cΩ , where 1F1(a, b; z) is the confluent hypergeometric function. 3T. P. Sotiriou, M. Visser and S. Weinfurtner, Phys. Rev. D 84 (2011) 104018. Jakub Mielczarek From CDT to Astronomical Observations CDT-deformed dispersion relation

Approximations:  3 E∗ p ΩIR(p) ≈ p + (4 − dUV) , 15 E∗   3 p dUV−1 3 d −1 ΩUV(p) ≈ E∗b ∝ p UV . E∗ 3 For dUV = 2 we have ΩUV(p) ∝ p and for dUV = 3/2 the 6 ΩUV(p) ∝ p :

Jakub Mielczarek From CDT to Astronomical Observations Astrophysics

For the particles having energy E  E∗ the IR limit approximation may be applied, leading to the quadratic deviation ∂E ∂Ω(p) 3  E 2 vgr := = ≈ 1 + (4 − dUV) . ∂p ∂p 15 E∗

This expression predicts that for dUV < 4 the group velocity is increasing with the energy → superluminal motion. The effect can be constrained with use of the signals from the high-energy astrophysical sources such as GRBs4.

4G. Amelino-Camelia, et al., Nature 393 (1998) 763 Jakub Mielczarek From CDT to Astronomical Observations In our case 3  E 2 ∆t ' L (4 − dUV) , 15 E∗ where L is the distance to a source. Although a huge value of the energy scale E∗ is expected the multiplication by a sufficiently large distance L may bring the value of ∆t close to the observational window. In particular, the GRB 090510 may be used. Employing the constraint from the Fermi-Large Area Telescope one can find that the energy scale of the dimensional reduction is

p 10 E∗ > 0.7 4 − dUV · 10 GeV at (95%CL) for dUV < 4, p 10 E∗ > 4.8 dUV − 4 · 10 GeV at (95%CL) for dUV > 4.

This constraint is definitely much stronger than any obtained from the accelerator physics experiments (which are reaching energies of the order of 104 GeV).

Jakub Mielczarek From CDT to Astronomical Observations Cosmology

Now we are going to investigate how formation of the primordial inhomogeneities if affected by the effect of dimensional reduction. Here, instead of the IR corrections, the UV part of the dispersion relation may play a crucial role.

Jakub Mielczarek From CDT to Astronomical Observations A short reminder

Primordial perturbations are generated in the quantum process of particle creation from vacuum in the expanding universe.

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

ˆ ∗ ˆ† uˆk(τ) = fk (τ)bk + fk (τ)b−k,

where fk (τ) is the so-called mode function which satisfies the same ˆ† ˆ equation as uk(τ). The creation (bk) and annihilation (bk) † (3) operators fulfill the commutation relation [bˆk, bˆq] = δ (k − q). ˆ 1 Canonical commutation relation between quantum field φ = z uˆ and its conjugated momenta leads to the Wronskian condition

df ∗ df W (f , f 0) ≡ f k − f ∗ k = i. k k k dτ k dτ

Jakub Mielczarek From CDT to Astronomical Observations Power spectrum

Two-point correlation function for tensor modes is given by

Z ∞ dk sin kr h0|φˆ(x, τ)φˆ(y, τ)|0i = Pφ(k, τ) , 0 k kr where k3 |f (τ)|2 P (k, η) = k , φ 2π2 z2 is the power spectrum, |0i is the vacuum state and r = |x − y|. This spectrum is the fundamental observable associated with gravitational wave production.

The mode functions fk (τ) fulfill the same equation as uk:

00 ! d2 z f + k2 − f = 0. dτ 2 k z k

Jakub Mielczarek From CDT to Astronomical Observations The k2 is due to spatial part of the Laplace operator. The CDT modification of the dispersion relation can be, therefore, effectively introduced by the following replacement

k2 → a2Ω2(k/a).

Consequently, our CDT-corrected equation of modes is

00 ! d2 z f + a2Ω2(k/a) − f = 0. dτ 2 k z k

Here, k = pa (a is the scale factor) and the classical limit a2Ω2(k/a) → a2(k/a)2 = k2 is recovered correctly for p → 0. The value of z depends on which kind of perturbations is considered. For the gravitational waves z = zT := a and for the scalar φ˙ perturbations z = zS := a H .

Jakub Mielczarek From CDT to Astronomical Observations 00  2 2 z  In the sub-Hubble limit a Ω (k/a)  z , the vacuum normalization of the fk mode functions leads to 1 |f |2 = . k 2aΩ(k/a)

This is might be viewed as an instantaneous vacuum state. The adiabaticity condition is to be checked for a particular background dynamics. Applying this to the definition of the scalar power spectrum we find that for k/a  E∗

2 3(dUV−2)  E   1 k  dUV−1 3 2 ∗ k |f | EPl E∗ a k nS −1 PS (k) := 2 2 = ∝ k , 2π zS 3π(1 + w)b where n = 1 + 3(dUV−2) . Furthermore, we assume that the S dUV−1 universe is filled with barotropic matter: P = wρ, with w = const.

Jakub Mielczarek From CDT to Astronomical Observations The value of the power spectrum accessible observationally is at the horizon-crossing. In our case, the horizon-crossing condition d −1 00   UV a2Ω2(k/a) = z translates into k ' aE H 3 := k for z ∗ E∗b H H > E∗.

Jakub Mielczarek From CDT to Astronomical Observations In this particular situation the resulting tensor-to-scalar ratio

P (k )  z 2 r := T 0 = 64πG S = 24(1 + w), PS (k0) zT can be directly related to the CMB data. The best current −1 observational bound at the pivot scale k0 = 0.05 Mpc is r < 0.07 (BICEP2/Keck Array and Planck)5, which implicates that w < −0.997 when the perturbations were formed. The universe had to, therefore, be very close to the de Sitter phase when the perturbations were formed. Cosmic inflation is still needed! But, this is in agreement with the CDT assumption, where positive cosmological constant is included from the very beginning.

5P. A. R. Ade et al. [BICEP2 and Planck Collaborations], Phys. Rev. Lett. 114 (2015) 101301 Jakub Mielczarek From CDT to Astronomical Observations The spectral index of scalar perturbations at the horizon-crossing is

d ln PS (k = kH ) 3r(dUV − 2) nS − 1 ≡ ≈ . d ln k (dUV − 1)r − 48

The horizontal blue line corresponds to the value nS − 1 = −0.0384 obtained based on the Planck satellite measurements

Jakub Mielczarek From CDT to Astronomical Observations The scale invariance is still recovered for dUV = 2. However, consistency with the observational data (both the value of the spectral index and the tensor-to-scalar ratio) can be obtained only by taking

dUV > 10.65.

However, this value is in overt contradiction with the values of dUV expected in the C phase of CDT. The scenario considered can be, therefore, ruled out based on the CMB data. Other scenarios, such as those with H < E∗ at the horizon-crossing have to be studied separately, taking into account the evolution of modes in the intermediate energy range. The obtained constraint is consistent with super-diffusion type of behavior, which is, however, not typically predicted within the models of Planck scale physics. One of the exceptions is Relative Locallity [M. Arzano ,T. Trze´sniewski,Phys. Rev. D 89 (2014) no.12, 124024].

Jakub Mielczarek From CDT to Astronomical Observations Going further. . .

Some evidence of this sort of RG lines: J. Ambjorn, A. Grlich, J. Jurkiewicz, A. Kreienbuehl and R. Loll, “Renormalization Group Flow in CDT,” Class. Quant. Grav. 31 (2014) 165003. The issue is discussed in JM,“Big Bang as a critical point,” Adv. High Energy Phys. 2017 (2017) 4015145.

Jakub Mielczarek From CDT to Astronomical Observations The phase B of CDT can be modeled with the use of anisotropic scaling. In Ref. 6 we have proven that the Horava-Lifshitz scaling:

x → bx, t → bz t, leads to the crumpled phase in the (ultralocal) z → 0 limit. Employing the generalized Laplace operator:

z ∆z := ∆ , we have shown that in the z → 0 limit: 1 [∆ ] = δ − = [D ] − [A ] , where 0 ij ij N 0 ij 0 ij

[A0]ij = (1 − δij )w, [D0]ij = δij (N − 1)w,

1 and the weights wij = w = N for i 6= j. 6M. L. Mandrysz and JM, “Ultralocal nature of geometrogenesis,” arXiv:1804.10793 [gr-qc]. Jakub Mielczarek From CDT to Astronomical Observations Ultralocal=Crumpled for finite N

N

disconnected points continuous space → ∞ N

discrete space finite N

z  0 z  1 z

Jakub Mielczarek From CDT to Astronomical Observations By coupling Ising spin matter to the considered graph we show that the process of geometrogensis can be associated with critical behavior. The following Hamiltonian: 1 X E = − w s s , 2 ij i j i,j where wij are the weights associated, has been used. Phase diagram of the system can be reconstructed exhibiting a symmetry broken phase at z ∈ [0, 0.5). At z = 0 the well known Curie-Weiss model is recovered.

TCW Temperature

0 0.0 0.2 0.4 0.6 0.8 1.0 z

Jakub Mielczarek From CDT to Astronomical Observations Gravitational phase transitions in the early universe?

It is not clear whether gravitational phase transitions can generate cosmologically relevant spectrum of primordial perturbations. Numerous conceptual issues. E.g. what is the meaning of d at the transition point? Anyway, measurements of the critical exponents in CDT may tell us something. In particular, by analyzing domains formation we can perform predictions regarding the number of gravitational defects. This can be constrained observationally . Correlation length:

−ν ξ = ξ0 |1 − T /TC |

Relaxation time: −νz τ = τ0 |1 − T /TC |

Jakub Mielczarek From CDT to Astronomical Observations Kibble-Zurek mechanism, domains and defects

Relaxation horizon Z t ξ(T ) τQ ν(z−1)+1 ξc = dt ∝ ξ0 (1 − T /TC ) , 0 τ(T ) τ0

where τQ (quench time) characterizes a speed of the transition. At some point ξc (TF ) = ξ(TF ), which gives us a domain size:   ν τQ νz+1 ξ ≈ ξ0 . τ0 Gravitational defects are formed at the domains boundary. For typical values ξ ≈ lPl. In case there is no inflation, the present size of the domains is

TPl ξtoday ≈ ξ ∼ 1 mm. TCMB

1 An average defect concentration d ∼ mm3 . Jakub Mielczarek From CDT to Astronomical Observations Conclusions

Phenomenology of CDT can be introduced based on non-trivial phase structure and diffusion time dependences of spectral dimensions. A particular scenario has been shown to be in disagreement with observations of the CMB. That is good! Some of the QG predictions are falsifiable.

Other scenarios are to be studied: different ∆p parameterizations, different vacua, curved momentum space, E∗ > H,... → more definitive statements. A possibility of gravitational phase transitions in the early universe (e.g. geometrogenesis) → a new mechanism of generation of primordial perturbations? Critical behavior → non-equilibrium transitions → KZM → domains → gravitational defects Reconstruction of the Laplace operator deirectly from CDT.

Jakub Mielczarek From CDT to Astronomical Observations