quantum gravity and cosmology - lagrange...
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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
gravitationphysics of classical gravity
long distances
short distances
Einstein’s theory of general relativity
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
gravitationphysics of classical gravity
Einstein’s theory of general relativity
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
what means fundamental?
UV fixed point fundamental QFTWilson ’71
running couplingsachieve
under RG evolution
Gross, Wilzcek ’73Politzer ‘73asymptotic freedom free
UV fixed point
Weinberg ‘79asymptotic safety interacting
UV fixed point
UV fixed point fundamental QFTWilson ’71
running couplingsachieve
under RG evolution
does metric quantum gravity generate an UV fixed point? Q:
what means fundamental?
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
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
classical GR
gravity weakens
0.01 0.1 1 10 1000.0
0.2
0.4
0.6
0.8
1.0
UV
IR
↵/↵⇤
interacting UV fixed points
“antiscreening”
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
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
Gastmans et al ’78Christensen, Duff ’78
Weinberg ’79Kawai et al ’90
Gawedzki, Kupiainen ’85 de Calan et al ’91
Peskin ’80
scalars Brezin, Zinn-Justin ’76Bardeen, Lee, Shrock ’76D = 2 + ✏ : ↵ = gNL(µ)µ
D�2
classes ofgauge-Yukawa theories
D = 4 : several ↵i
dimension
gravitons
interacting UV fixed points
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)
tools
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
renormalisation group
• 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
• 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
optimised choices of regulatorstability & convergence
renormalisation group
• 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
• 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.6/14
Litim ’01, ‘02
Wilson ’71, Polchinski ’84, Wetterich ‘92
tool: renormalisation group
renormalisation group
• 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
• 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
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)
Renormalisation group and gravitation – p.7/7
Litim, Zappala ’10
Pawlowski, Litim, Nedelko, Smekal PRL ’03
Litim ’02Bervillier, Juettner, Litim, ‘07
Wilson ’71, Polchinski ’84, Wetterich ‘92
tool: renormalisation group
UV-IR connection
renormalisation group
• for quantum gravity (Reuter ’96)
kd
dk k[gµ� ; gµ� ] =
1
2Tr
⌥⇧
(2)k [gµ� ; gµ� ] + Rk
⇥�1kdRk
dk
⌅
• effective action
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
Renormalisation group and gravitation – p.2/10
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
Asymptotically safe gravity – p.11/19
Gerwick, Litim, Plehn ‘11
renormalisation group
• for quantum gravity (Reuter ’96)
kd
dk k[gµ� ; gµ� ] =
1
2Tr
⌥⇧
(2)k [gµ� ; gµ� ] + Rk
⇥�1kdRk
dk
⌅
• effective action
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
Renormalisation group and gravitation – p.2/10
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
Renormalisation group and gravitation – p.7/10
Litim ‘03
Einstein-Hilbert gravity
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
Einstein-Hilbert gravity
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
Einstein-Hilbert gravity
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
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
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
effective action with invariants up to mass dimension D = 2(N � 1)
Ricci scalars �k / f(R)
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
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irrelevant 5
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-6-4-20246
N
relevant
0 5 10 15 20
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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 cosmology
asymptotically safe cosmologycosmological fixed points
``freeze-in” ``simultaneous”
asymptotically safe cosmology
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