quantum gravity and cosmology - Lagrange...

quantum gravity and cosmology

Daniel F Litim

IHP Trimester 2018Analytics, Interference, and Computation in Cosmology

Analytical Methods (17 - 21 Sep, 2018)

Paris, 18 Sep 2018

“is metric gravity fundamental?”

Daniel F Litim

IHP Trimester 2018Analytics, Interference, and Computation in Cosmology

Analytical Methods (17 - 21 Sep, 2018)

Paris, 18 Sep 2018

standard model

local QFT for fundamental interactionsstrong nuclear forceweak forceelectromagnetic force

dynamics of matterspin 0, 1/2, 1dimensionless couplings and massesperturbatively renormalisable

features

source: PDG (Oct 2015)

triumph of QFT

highlight:asymptotic

freedom

“running couplings”

running couplingsquantum fluctuations modify interactionscouplings depend on energy

predictions into regions where we cannot (yet) make measurements

µ

~ kBfluctuations

energy scale couplings ↵(µ)

QFT provides us with

µd↵

dµ= �(↵)

what’s the trouble with gravity?

gravitationphysics of classical gravity

Einstein’s theory of general relativity

GN = 6.7⇥ 10�11 m3

kg s3

⇤ ⇡ 10�35s�2

Rµ⌫ � 1

2gµ⌫R = �⇤ gµ⌫ + 8⇡GN Tµ⌫

Newton’s coupling

cosmological constant


long distances

short distances


Rµ⌫ � 1

2gµ⌫R = �⇤ gµ⌫ + 8⇡GN Tµ⌫

gravity not tested beyond 1028cm

gravity not tested below 10�2cm

ample space for “new” physics e.g. dark matter, dark energy, extra dimensions, modifications of GR



introduction

• physics of classical gravity

Einstein’s theory GN = 6.7⇧ 10�11 m3

kg s2

• physics of quantum gravity

Planck length ⌃Pl =��GN

c3

⇥1/2 ⌅ 10�33 cm

Planck mass MPl ⌅ 1019GeV

Planck time tPl ⌅ 10�44 s

Planck temperature TPl ⌅ 1032 K

expect quantum modifications at energy scales E ⌅MPl

• hierarchy problem

coupling of Higgs particle to gravity

one-loop level:

either no symmetry breaking

or Higgs particle mass of order MPl

or non-trivial cancelations at loop level (fine-tuning)

• low-scale quantum gravity

particle physics models

fundamental Planck scale = O(1TeV)⇤MPl

Renormalisation group and gravitation – p.2/9

physics of quantum gravity

MPlexpect quantum modifications at energy scales

what’s the trouble with gravity?

perturbatively non-renormalisablew/ and w/o matter

classical singularities (e.g. black holes)geodesically incomplete

loss of predictivity

all known physics

borrowed from Ben Wandelt

all known physics

Standard Model ofParticle Physics

all known physics

General Relativity

all known physics

can the UV cutoff be removed? Q: is metric gravity fundamental?

what means fundamental?

UV fixed point fundamental QFTWilson ’71

running couplingsachieve

under RG evolution




under RG evolution

Gross, Wilzcek ’73Politzer ‘73asymptotic freedom free

UV fixed point

Weinberg ‘79asymptotic safety interacting

UV fixed point



under RG evolution

does metric quantum gravity generate an UV fixed point? Q:


coupling

fermions

gluons

D = 2 + ✏ : ↵ = GN (µ)µD�2

D = 2 + ✏ : ↵ = gGN(µ)µ2�D

D = 4 + ✏ : ↵ = g2YM(µ)µ4�D

Gastmans et al ’78Christensen, Duff ’78

Weinberg ’79Kawai et al ’90

Gawedzki, Kupiainen ’85de Calan et al ’91

Peskin ’80Morris ’04

scalars Brezin, Zinn-Justin ’76Bardeen, Lee, Shrock ’76D = 2 + ✏ : ↵ = gNL(µ)µ

D�2

classes ofgauge-Yukawa theories

D = 4 : several ↵i Litim, Sannino ’14

dimension

gravitons

µ/⇤T

G(µ) ⇡ GN

G(µ) ⇡ ↵⇤µD�2

interacting UV fixed points

coupling

fermions

gluons

D = 2 + ✏ : ↵ = GN (µ)µD�2

D = 2 + ✏ : ↵ = gGN(µ)µ2�D

D = 4 + ✏ : ↵ = g2YM(µ)µ4�D






D�2



dimension

gravitons

µ/⇤T

G(µ) ⇡ GN

G(µ) ⇡ ↵⇤µD�2

classical GR

gravity weakens

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

UV

IR

↵/↵⇤


“antiscreening”

coupling

fermions

gluons

D = 2 + ✏ : ↵ = GN (µ)µD�2

D = 2 + ✏ : ↵ = gGN(µ)µ2�D

D = 4 + ✏ : ↵ = g2YM(µ)µ4�D






D�2



dimension

gravitons

µ/⇤T

G(µ) ⇡ GN

G(µ) ⇡ ↵⇤µD�2

how is this predictive?

relevant, marginal, irrelevant interactions

dangerous UV interactions are softened by quantum fluctuations

predictivity finitely many relevant invariants

UV physics characterised by

coupling

fermions

gluons

D = 2 + ✏ : ↵ = GN (µ)µD�2

D = 2 + ✏ : ↵ = gGN(µ)µ2�D

D = 4 + ✏ : ↵ = g2YM(µ)µ4�D



Gawedzki, Kupiainen ’85 de Calan et al ’91

Peskin ’80


D�2


D = 4 : several ↵i

dimension

gravitons


NEW

NEWLitim, Sannino JHEP ’14

Bond, Litim, EPJC ’16Bond, Litim, PRD ’17

Bond, Litim, Medina-Vasquez, Steudtner, PRD ‘18

supersymmetry D = 4 : several ↵i Bond, Litim, PRL ‘17

fixed points of quantum gravity

4d quantum gravity

running coupling

@tg = (D � 2 + ⌘N ) g

fixed pointsg⇤ = 0g⇤ 6= 0

IRUVlarge anomalous dimension ⌘N = ⌘N (g, all other couplings)

strong coupling effectslarge UV scaling exponents # ⇡ O(1)

g⇤ ⇡ O(1)

4D:

non-perturbative tools mandatory

g(µ) = GN (µ)µD�2t = ln(µ/⇤c)

computational methods

continuum: non-perturbative renormalisation group

lattice: Monte Carlo simulationssimplicial gravitydynamical triangulations

bootstrap search strategyFalls, Litim, Nikolakopoulos, Rahmede ’13, ’14

renormalisation group

• for quantum gravity: “bottom-up”

⇧�

⇧k

⇧ ⇧ ⇧EH

IR

UVpaction

integra

ting⇥out

fixed point

general relativityclassical

• for quantum gravity (Reuter ’96)

kd

dk⇧k[gµ⇥ ; gµ⇥ ] =

1

2Tr

�⇧⇧

(2)k [gµ⇥ ; gµ⇥ ] + Rk

⇥⇥1kdRk

dk

⌅

• effective action

⇧k =1

16⇤Gk

⇤�

g (⇥R + 2⇥k + · · · ) + Smatter,k + Sgf,k + Sghosts,k

Asymptotically safe gravity – p.7/19

Wilson ’71Reuter ’96

Litim ‘03

task: find viable UV fixed point candidates

tool: renormalisation group

“all-in-one” exact, finite, systematic


• functional RG (Wetterich ’93)

kd⇧k

dk=

1

2Tr

⇤�⇤2⇧k[ ]

⇤ ⇤ + Rk

⌃�1

kdRk

dk

�

ren.

12

momentum cutoff Rk

k ddkRk

[k2]

Rk(q2)

q2/k20 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3


kd

dk⇧k[gµ⇤ ; gµ⇤ ] =

1

2Tr

⇥ ⇧

(2)k [gµ⇤ ; gµ⇤ ] + Rk

⌅�1kdRk

dk

⌥


⇧k =1

16⌅Gk

⇧�



optimised choices of regulatorstability & convergence



kd⇧k

dk=

1

2Tr

⌥⇤2⇧k[ ]

⇤ ⇤ + Rk

⇥�1

kdRk

dk

⌅

ren.

12

momentum cutoff Rk

k ddkRk

[k2]

Rk(q2)

q2/k20 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3


kd

dk⇧k[gµ⇤ ; gµ⇤ ] =

1

2Tr

�⇧⇧

(2)k [gµ⇤ ; gµ⇤ ] + Rk

��1kdRk

dk

⇤


⇧k =1

16⌅Gk

⌃⌅

g (�R + 2⇥k + · · · ) + Smatter,k + Sgf,k + Sghosts,k


Litim ’01, ‘02

Wilson ’71, Polchinski ’84, Wetterich ‘92




kd⇧k

dk=

1

2Tr

⇤�⇤2⇧k[ ]

⇤ ⇤ + Rk

⌃�1

kdRk

dk

�

ren.

12

momentum cutoff Rk

k ddkRk

[k2]

Rk(q2)

q2/k20 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3


kd

dk⇧k[gµ⇤ ; gµ⇤ ] =

1

2Tr

⇥ ⇧

(2)k [gµ⇤ ; gµ⇤ ] + Rk

⌅�1kdRk

dk

⌥


⇧k =1

16⌅Gk

⇧�



low energy QCD signatures of confinement

Ising-type critical phenomenaphase transitions, high precision exponents

exact functional flows

• effective equation (Wetterich ’93)

�t⇤k =1

2Tr

⌅⇤⇥2⇤k[⌅]

⇥⌅ ⇥⌅+ Rk

��1

�tRk

⇥12

• Ising universalityquality control: systematic uncertainties (DL, Zappala, 2010)

⌅ ⇥ ⇤

resummed PT 0.0335(25) 0.6304(13) 0.799(11)

�-expansion 0.0360(50) 0.6290(25) 0.814(18)

world average 0.0364(5) 0.6301(4) 0.84(4)

Monte Carlo 0.03627(10) 0.63002(10) 0.832(6)

functional RGs 0.034(5) 0.630(5) 0.82(4)


Litim, Zappala ’10

Pawlowski, Litim, Nedelko, Smekal PRL ’03

Litim ’02Bervillier, Juettner, Litim, ‘07

Wilson ’71, Polchinski ’84, Wetterich ‘92


UV-IR connection



kd

dk k[gµ� ; gµ� ] =

1

2Tr

⌥⇧

(2)k [gµ� ; gµ� ] + Rk

⇥�1kdRk

dk

⌅


k =

� �g

⌃⇥R + 2⇧k

16⇥Gk+ · · ·

⇤+ Smatter,k + Sgf,k + Sghosts,k

• running couplingsprojection of k�k k onto

�g,�

gRn,�

g(Ricci)2 ,�

g(Weyl)2 ,�

g(Riem)2, · · ·

heat kernel techniques, background field method

choice of Rk, stability, analyticity (DL ’01,’02)

• optimisation (DL ’00, ’01, ’02, Pawlowski ’05)

stability⇧ convergence⇧ control of approximations

reduce spurious scheme dependences

improve physics predictions

constructive procedures, analytical flows


Gk = g/k2

�k = � k2Litim ‘03

Einstein-Hilbert gravity

6

0.2 0.5 1.0 2.0 5.0

0.0

0.2

0.4

0.6

0.8

1.0

classical

⇥� quadratic

⇥ quenched

linear ⇤

G(µ)GN

µ/�T

FIG. 1: Crossover of the gravitational coupling G(µ)/GN from classical (dashed line) to fixed point scaling (fulllines) in the Einstein Hilbert theory with n = 3 extra dimensions: classical behavior (black), linear crossover (19)

(blue), quadratic crossover (20) (magenta) and the quenched approximation (18) (red) with �T = �(0)T = �(1)

T =

�(2)T (see text).

dimension cross-over from their classical to fixed point values on a logarithmic energy scale. Note thatboth indices grow with dimension. In particular, the larger ⇥, the faster the cross-over regime in unitsof the RG ‘time’ lnµ. Furthermore, ⇥nG is larger than ⇥G and the ratio ⇥nG/⇥G = 2d/(d + 2) rangesbetween [4/3, 1/2] for d between [4,⌅]. It follows that the cross-over region between IR and UV scalingnarrows with increasing dimension [3, 10].

C. Running Newton’s coupling

For the applications below it is useful to have explicit expressions for the functions g(µ), G(µ), Z(µ),and �(µ) at hand. We introduce three di⇥erent approximations capturing the essential features. Sincethe transition from perturbative to fixed point scaling becomes very narrow with increasing dimension,the scale dependence is well-approximated by an instantaneous jump in the anomalous dimension at theenergy scale µ = �(0)

T . This is the quenched approximation, solely characterized by a transition scale ofthe order of the fundamental Planck scale M⇥, where

Z�1(µ) =

⇤1 for µ < �(0)

T

(�(0)T )d�2/µd�2 for µ ⇥ �(0)

T

�(µ) =

⇤0 for µ < �(0)

T

2� d for µ ⇥ �(0)T

. (18)

In the quenched approximation, continuity in G(µ) follows if the fixed point value g⇥ is related toNewton’s coupling at low energies and the transition scale via g⇥ = GN (�(0)

T )d�2. We will identify thecross-over scale �T with the parameter �(0)

T in (18).

In order to achieve explicit continuous expressions for �(µ) we resolve (16) for g(µ) to find Z(µ)and �(µ) by using mild additional approximations for the ratio ⇥G/⇥nG, see (17). For the additionalapproximation ⇥G/⇥nG = 1, we find an explicit expression for the running coupling (10) from (16) with

Z(µ) = 1 + GNµd�2/g⇥ , �(µ) = (2� d)�

1� 1Z(µ)

⇥. (19)

We have taken the limit g0 ⇤ g⇥ and µ0 ⇤ �T while keeping the gravitational coupling GN = g0/µd�20

fixed to its d-dimensional value. We have also used g⇥ = GN (�(1)T )d�2. Here, the anomalous dimension

is linear in the dimensionless coupling g and reads � = (2�d)g/g⇥. Provided that (2�d)/g⇥ is identified

flow trajectories

• cross-over behaviourintegrated flow with

�gR,

�g

ln(k/MPl)

�

g

⌅D = 4

IR UV

-4 -2 0 2 4-4

-3

-2

-1

0

1

2


Gerwick, Litim, Plehn ‘11



kd

dk k[gµ� ; gµ� ] =

1

2Tr

⌥⇧

(2)k [gµ� ; gµ� ] + Rk

⇥�1kdRk

dk

⌅


k =

� �g

⌃⇥R + 2⇧k

16⇥Gk+ · · ·

⇤+ Smatter,k + Sgf,k + Sghosts,k

• running couplingsprojection of k�k k onto

�g,�

gRn,�

g(Ricci)2 ,�

g(Weyl)2 ,�

g(Riem)2, · · ·

heat kernel techniques, background field method

choice of Rk, stability, analyticity (DL ’01,’02)

• optimisation (DL ’00, ’01, ’02, Pawlowski ’05)

stability⇧ convergence⇧ control of approximations

reduce spurious scheme dependences

improve physics predictions

constructive procedures, analytical flows


Gk = g/k2

�k = � k2

phase diagram

• full flow – vicinity of fixed points4D integrated flow with

�gR,

�g

g

⇥�0.1 0 0.1 0.2 0.3 0.4 0.5

�0.02

�0.01

0

0.01

0.02

0.03


Litim ‘03


background field flow

A

CBD0 1

2-2 - 1

214

- 14

- 18

18

-1-•

12

1

2

3

�

g/g⇤

Einstein-Hilbert gravityLitim , Satz ‘12background field flow

A

CBD0 1

2-2 - 1

214

- 14

- 18

18

-1-•

12

1

2

3


fixed points

A non-trivial UV fixed pointB trivial IR fixed point (class. gravity)C degenerate IR fixed pointD trivial IR fixed point (class. gravity, neg. CC)

Litim , Satz ‘12

Gk = g/k2

�k = � k2

!5 !1!0.5 0 0.1 0.2 0.3 0.4 0.45 0.50

0.1

0.2

0.5

1

2

5.4

10

33.5

0.45 0.5

0

0.015

∞

−∞

Ia

Ib

A

B CD

C

�

gN


Christiansen, Litim , Rodigast, Pawlowski ‘12propagator flow

bootstrapsearch strategy

results

effective action with invariants up to mass dimension D = 2(N � 1)

Ricci scalars �k / f(R)

S =

Zd

4x

pdetgµ⌫

1

16⇡G

[�R+ 2⇤] +N�1X

n=2

�n Rn�k

results



S =

Zd

4x

pdetgµ⌫

1

16⇡G

[�R+ 2⇤] +N�1X

n=2

�n Rn

bootstrap search strategy Falls, Litim, Nikolakopoulos, Rahmede ’13, ’14

1 fix N, compute RG flow2 deduce fixed point and exponents3 increase N to N+1 and start over at 1

�k

results



Falls, Litim, Schroeder, ’18 (in prep.)

up to order N = 2 N = 3 N = 7N = 11 N = 35 N = 71

Codello, Percacci, Rahmede, ’07

Falls, Litim, Nikolakopoulos, Rahmede ’13, ’14, ’16

Bonanno, Contillo,, Percacci, ’10

Souma, ‘99, Reuter, Lauscher ’01, Litim ’03

Reuter, Lauscher ’01

S =

Zd

4x

pdetgµ⌫

1

16⇡G

[�R+ 2⇤] +N�1X

n=2

�n Rn�k

f(R)

5 10 15 20 25 30 350

10

20

30

40

N

UV fixed point

�n(N)

h�ni+ cn

5 10 15 20 25 30 35

-20

0

20

40

60

N

#n(N)

UV scaling exponents

Falls, Litim, Nikolakopoulos, Rahmede 1301.41911410.4815

good

conv

erge

nce

good

conv

erge

nce

f(R)

5 10 15 20 25 30 350

10

20

30

40

N

�n(N)

h�ni+ cn

5 10 15 20 25 30 35

-20

0

20

40

60

N

#n(N)

UV scaling exponents

Falls, Litim, Nikolakopoulos, Rahmede 1301.41911410.4815

good

conv

erge

nce

4

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0 10 20 30 40 50 60 70

0

50

100

150

2 10 18 26 34 42 50 58 66 71

n

10

20

30

40

50

60

70

N=

near-Gaussian strip

“as Gaussian as it gets”

Falls, Litim, Schroeder ’18 (to appear)Ricci scalars

0 2 4 6-5

0

5

10

15

n

#0#1 #2

#3

#4 #5

#6

#7

relevantirrelevantgap

resultsFalls, Litim, Schroeder ’18 (to appear)Ricci scalars

spectrum of eigenvalues

resultsRicci tensors

�k =

Zd

dx

pg[Fk(Rµ⌫R

µ⌫) +R · Zk(Rµ⌫Rµ⌫)]

Falls, King, Litim, Nikolakopoulos, Rahmede, ’18

polynomial expansion up to order N = 21

results

Gaussian

Ù ÙÙ

Ù Ù

Ù Ù

Ù Ù

Ù Ù

Ù Ù

Ù Ù

ÙÙ Ù

Ù ÙÙ

Ú ÚÚ

Ú Ú

Ú Ú

Ú Ú

Ú Ú

Ú Ú

Ú Ú

ÚÚ Ú Ú

Ú

· ··

· ·

· ·

· ·

· ·

· ·

· ·

· · ··

Ê ÊÊ

Ê Ê

Ê Ê

Ê Ê

Ê Ê

Ê Ê

Ê ÊÊ Ê

Ê

≈ ≈≈

≈ ≈

≈ ≈

≈ ≈

≈ ≈

≈ ≈

≈ ≈≈ ≈

ı ıı

ı ı

ı ı

ı ı

ı ı

ı ıı ı

ı

Û ÛÛ

Û Û

Û Û

Û Û

Û Û

Û ÛÛ Û

Ï ÏÏ

Ï Ï

Ï Ï

Ï Ï

Ï ÏÏ Ï

Ï

Ù ÙÙ

Ù Ù

Ù Ù

Ù Ù

Ù ÙÙ Ù

Ú ÚÚ

Ú Ú

Ú Ú

Ú Ú

Ú ÚÚ

· ··

· ·

· ·

· ·

· ·

Ê ÊÊ

Ê Ê

Ê Ê

Ê Ê

Ê

≈ ≈≈

≈ ≈

≈ ≈

≈ ≈

ı ıı

ı ı

ı ı

ı

Û ÛÛ

Û Û

Û Û

Ï ÏÏ

Ï Ï

Ï

Ù ÙÙ

Ù Ù

Ú ÚÚ

Ú

· ·Ê Ê

JnHNL

irrelevant 5

10

15

20

-6-4-20246

N

relevant

0 5 10 15 20

0

10

20

30

n

“as Gaussian as it gets”

Ricci tensors Falls, King, Litim, Nikolakopoulos, Rahmede, ’18

cosmology

-2 -1 0 1 2

-6

-4

-2

0

2

4

6

R

f(R)4E(R)

radius of convergence

de SitterAdS

Ricci scalars

EHrL

ØØØr++r+r-

-2 -1 0 1 2-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

r

Ricci tensors

AdSde Sitter

conclusions

quantum gravity

cosmology

increasing evidence for UV fixed point & asymptotic safetymetric field remains fundamental carrier of gravitational force systematic search strategyUV-IR connection

de Sitter solutions already within quantum domaincheck what happens under RG evolutionasymptotically safe cosmology

thank you!

summary

further directionsinclusion of matter Folkerts Litim, Palwowski ’12

Christiansen, Litim, Pawlowski, Reichert ‘17Falls, Litim, Raghuraman ’10

Falls, Litim ‘12Litim, Nikolakopolous ‘13

quantum black holes, BH thermodynamics

many more …

Hindmarsh, Litim, Rahmede ‘11

extra materialgeodesic completeness of black holes

Penrose diagramimproved Schwarzschildspace-times:

reduced singularity at origin is integrable

test particle trajectories no longer terminate

space-time becomesgeodesically complete

Falls, DL, Raghuraman (`10)

Falls, Litim, Raghuraman (`10)

extra materialconformal scaling and black holes

conformal scaling in QFT

�CFT =d� 1

d

S � (RT )d�1, E � Rd�1T d

Aharony, Banks ’98, Shomer ’07

S ⇠ E⌫

Schwarzschild BH scaling S ⇠ Rd�2/GN E ⇠ Rd�3/GN

S ⇠ E⌫⌫BH =

d� 2

d� 3

conformal scaling

⌫BH 6= ⌫CFT except for d =3

2

conformal scaling in QFT

�CFT =d� 1

d

S � (RT )d�1, E � Rd�1T d

Falls, Litim ’12

Schwarzschild BH scaling S ⇠ Rd�2/GN E ⇠ Rd�3/GN

conformal scaling

S

Rd�1�

�E

Rd�1

⇥�

S

Rd�1�

�E

Rd�1

⇥�

�BH =12

⌫BH 6= ⌫CFT except for d = 2classicalSchwarzschild BHsNOT conformal

0.01 0.1 1 10 100

0.5

0.6

0.7

0.8 �CFT =d� 1

d

�BH =12

R/Rc

Schwarzschild:

conformal:

�BH

�CFT

Falls & DL, 1212.8121 (PRD)

result: asymptotically safe Schwarzschild black holes display conformal scaling

asymptotic safety

extra materialscalar field cosmology

scalar field cosmology(Copeland, Liddle, Wands ’98)

FRW + scalar field, evolution with cosmic time

inflation: dH/dN < 1

scalar field cosmologycosmological fixed points

asymptotic early / late times

scalar field cosmology

classical cosmological fixed points

asymptotically safe cosmology

assumption k = k(N)

consistency condition (Bianchi identity)

(Hindmarsh, DL, Rahmede, ’11)

quantum corrections

asymptotically safe cosmologycosmological fixed points

``freeze-in” ``simultaneous”

quantum gravity and cosmology - Lagrange...

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