Towards reconstructing the quantum effective action of gravity

Benjamin Knorr and Frank Saueressig, 1804.03846

CDT - some results

in recent years, many results accumulated:

phase diagram with 4 different phases

study of spectral dimensions

autocorrelator of 3-volume fluctuations

effect of spatial topology (sphere vs. torus)

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A

B

Cb

CdS

Quadruple point

∆

κ0

1610.05245Ambjorn, Gizbert-Studnicki, Jurkiewicz, Klitgaard, Loll

hep-th/0505113Ambjorn, Jurkiewicz, Loll

CDT - 3-volume and fluctuations

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⟨N3(i)⟩

i

S1xS3 0807.4481

20 40 60 80t

2000

4000

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⟨nt ⟩

S1xT3 1604.08786

<V3(t)>

<ẟV3(t)ẟV3(t’)>

Ambjorn, Goerlich, Jurkiewicz, Loll Ambjorn, Drogosz, Gizbert-Studnicki, Goerlich, Jurkiewicz, Nemeth

CDT - 3-volume and fluctuations

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i

S1xS3 0807.4481

20 40 60 80t

2000

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⟨nt ⟩

S1xT3 1604.08786

<V3(t)>

<ẟV3(t)ẟV3(t’)>

Ambjorn, Goerlich, Jurkiewicz, Loll Ambjorn, Drogosz, Gizbert-Studnicki, Goerlich, Jurkiewicz, Nemeth

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Can we extract information on the effective action of gravity from these correlators?

Effective action - a matching template

idea:

start with ansatz for effective action

calculate correlators measured in CDT

fit to data, extract parameters

if fit unsatisfactory: extend ansatz for effective action

advantages:

EFT spirit, independent of UV completion

access to IR of quantum gravity - nonlocal effects visible at large scales?

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Reconstructing the effective action of QG

inspired by results in Asymptotic Safety and phenomenology, we make the ansatz

at present: only one correlator available, cannot fix non-local term uniquely, for definiteness we consider

effect of this term: gauge-invariant mass term for two-point function

perform semiclassical analysis: quantum fluctuations around classical background geometry

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�QG ⇡ 1

16⇡GN

Zd4x

p�g

�R+ 2⇤� b2

6R 1

⇤2R�

R = R+1

2(3)R

Calculating the correlator

CDT data: expectation value of geometry is sphere/torus - use fixed on-shell metric

use definition of 3-volume:

insert into correlator:

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gµ⌫ = diag(1, a(t)2�ij(x))

�V3(t) =1

2

Zd3x

p� a(t)3 �ij��ij

h�V3(t)�V3(t0)i = 1

4

Z

x

Z

x0h�(x, t)�(x0, t0)i

� = a(t)3�ij��ij

Calculating the correlator II

key ideas:

propagator admits expansion into product eigenfunctions:

use orthogonality of spatial eigenfunctions - only need to know temporal spectrum and eigenfunctions

use ansatz for effective action to calculate spectrum and eigenfunctions

calculate (temporal) propagator

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h�V3(t)�V3(t0)i = 1

4

Z

x

Z

x0h�(x, t)�(x0, t0)i

h�(x, t)�(x0, t0)i =X

n,m

1

�xn + �t

m

�⇤n(x)�n(x

0) ⇤m(t) m(t0)

Comparison to CDT - torus

due to symmetry, correlator only depends on time difference

fitting the data:

fitting lowest eigenvalues gives lattice spacing:

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0.000

0.005

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�t

�2(t

,t+�

t)

GN = 0.14a2CDT

b = 6.93/aCDT

r = 3.09aCDT

Comparison to CDT - torus - remarks

non-zero value of b - evidence for non-local interactions from first principle CDT simulations!

checked higher order curvature terms - completely subdominant (ca. 5 orders of magnitude smaller)

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0.000

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�t

�2(t

,t+�

t)

Outlook: more correlators to investigate

the more correlators, the better we can constrain the effective action

two immediate candidates to extend the present analysis:

higher order three-volume autocorrelations:

three-curvature correlations:

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h�V3(t1) . . . �V3(tn)i

h�(3)R(t)�(3)R(t0)i

Three-point correlator on the torus I

taking derivative of two-point correlator w.r.t. appropriate source gives

make ansatz for illustration:

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h�V3(t1)�V3(t2)�V3(t3)i = �1

8

Z

x1,x2,x3x,y,z

�(3)(x, y, z)G(x, x1)G(y, x2)G(z, x3)

� � 1

2G2

Z

x

⇣�(3)�� + g2�@

2t � + E2�

2⌘

+1

2G3

Z

x

⇣�2(3)�� + g3�

2@2t � + E3�

2⌘

Three-point correlator on the torus II

correlator contains two contributions:

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h�V3(t1)�V3(t2)�V3(t3)i ⇠ a1G(t1, t3)G(t2, t3) + cycl + a2

Z

tG(t, t1)G(t, t2)G(t, t3)

Summary

central idea: constrain effective action by correlators that can be calculated in different approaches

CDT: evidence for non-local interactions which could explain dark energy dynamically

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Summary

central idea: constrain effective action by correlators that can be calculated in different approaches

CDT: evidence for non-local interactions which could explain dark energy dynamically

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QG = EH + minimal non-local extension!