Quantum Einstein Gravity a status report

Transcription

1 p. 1/4 Quantum Einstein Gravity a status report Frank Saueressig Group for Theoretical High Energy Physics (THEP) Institute of Physics M. Reuter and F.S., Sixth Aegean Summer School, Chora, Naxos (Greece), arxiv: ERG 2012 Sept. 3 7, Centre Paul Langevin, France

2 p. 2/4 Outline Lecture I: Fundamental aspects of Quantum Einstein Gravity introduction to Quantum Gravity the concept of Asymptotic Safety a non-gaussian fixed point for gravity predictions from Asymptotic Safety conclusions I Lecture II: Phenomenological applications RG improved cosmology spectral dimension of quantum space-time a prospective IR fixed point for gravity summary: successes and open questions

3 p. 3/4 Asymptotic Safety as viable theory for Quantum Gravity a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters ensures predictivity c) classical limit: RG-trajectories have part where GR is good approximation recover gravitational physics captured by General Relativity: (perihelion shift, gravitational lensing, nucleo-synthesis,...) Quantum Einstein Gravity (QEG)

4 p. 4/4 Einstein-Hilbert-truncation: the phase diagram M. Reuter, F. S., Phys. Rev. D 65 (2002) [hep-th/ ] g Type IIa Type Ia Type IIIa λ 0.25 Type IIIb Type Ib

5 p. 5/4 Addressing the puzzles of Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum 2. singularities in solutions of Einstein equations black hole singularities Big Bang singularity 3. cosmological observations horizon problem: uniform CMB-temperature small positive cosmological constant

6 p. 5/4 Addressing the puzzles of Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum 2. singularities in solutions of Einstein equations black hole singularities Big Bang singularity 3. cosmological observations horizon problem: uniform CMB-temperature small positive cosmological constant 4. Comparison to Monte-Carlo simulations CDT and EDT data for grav. partition function

7 p. 6/4 Addressing the puzzles of Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum 2. singularities in solutions of Einstein equations black hole singularities Big Bang singularity 3. cosmological observations horizon problem: uniform CMB-temperature small positive cosmological constant 4. Comparison to Monte-Carlo simulations CDT and EDT data for grav. partition function

8 p. 7/4 RG-improvement incorporating leading quantum corrections

9 p. 8/4 Classical vs. quantum space-times classical space-times from general relativity S EH 1 = d d x g( R+2Λ) 16πG N Einstein equations R µν = 2 2 d Λg µν solutions are classical space-time metrics g µν : Friedman-Robertson-Walker cosmology Schwarzschild black hole

10 p. 8/4 Classical vs. quantum space-times classical space-times from general relativity S EH 1 = d d x g( R+2Λ) 16πG N Einstein equations R µν = 2 2 d Λg µν solutions are classical space-time metrics g µν : Friedman-Robertson-Walker cosmology Schwarzschild black hole quantum theory: compute observables O DγDCD C O[γ]e S bare[γ,c, C] expectation values for curvatures, two-point correlators,...

11 p. 8/4 Classical vs. quantum space-times classical space-times from general relativity S EH 1 = d d x g( R+2Λ) 16πG N Einstein equations R µν = 2 2 d Λg µν solutions are classical space-time metrics g µν : Friedman-Robertson-Walker cosmology Schwarzschild black hole quantum theory: compute observables O DγDCD C O[γ]e S bare[γ,c, C] expectation values for curvatures, two-point correlators,... Very hard!

12 p. 9/4 Quantum physics from average action Γ k Γ k provides effective description of physics at scale k: capture quantum effects by RG-improvement scheme: transition: classical S EH average action Γ k [g] one-parameter family of effective actions valid at different scales k-dependence captures quantum corrections

13 p. 9/4 Quantum physics from average action Γ k Γ k provides effective description of physics at scale k: capture quantum effects by RG-improvement scheme: transition: classical S EH average action Γ k [g] one-parameter family of effective actions valid at different scales k-dependence captures quantum corrections extracting physics information from Γ k : single-scale problem may allow for cutoff-identification : based on physical intuition: express RG-scale k through physical cutoff ξ modification of classical system by quantum effects

14 p. 10/4 Practical RG-improvement schemes given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions solve classical equations of motion solutions: replace G N G(k(ξ))

15 p. 10/4 Practical RG-improvement schemes given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions solve classical equations of motion solutions: replace G N G(k(ξ)) 2. improved classical equations of motion compute equations of motion from classical action equations of motion: replace G N G(k(ξ)) solve RG-improved equations of motion

16 p. 10/4 Practical RG-improvement schemes given: physically motivated cutoff-identification k = k(ξ) 1. improved classical solutions solve classical equations of motion solutions: replace G N G(k(ξ)) 2. improved classical equations of motion compute equations of motion from classical action equations of motion: replace G N G(k(ξ)) solve RG-improved equations of motion 3. improved average action Γ k : replace G N G(k(ξ)) k 2 R Einstein-Hilbert action f(r)-gravity theory compute modified equations of motion solve modified equations of motion

17 RG-improved cosmology p. 11/4

18 p. 12/4 Motivation for RG-improved cosmology inflation acts as magnifying class for quantum fluctuations non-gaussian fixed point signature imprinted on CMB?

19 p. 13/4 Motivation for RG-improved cosmology [A. Bonanno, M. Reuter, hep-th/ ] [A. Bonanno, M. Reuter, astro-ph/ ] [A. Bonanno, M. Reuter, astro-ph/ ] [M. Reuter, F. Saueressig, hep-th/ ] [A. Bonanno, M. Reuter, arxiv: ] [B.F.L. Ward, arxiv: ] [S. Weinberg, arxiv: ] [A. Bonanno, A. Contillo, R. Percacci, arxiv: ] [M. Hindmarsh, D. Litim, C. Rahmede, arxiv: ] [C. Ahn, C. Kim, E.V. Linder, arxiv: ] [Y. Cai, D. Easson, arxiv: ] [R. Yang, arxiv: ] [A. Contillo, M. Hindmarsh, C. Rahmede, arxiv: ] [S.E. Hong, Y.J. Lee, H. Zoe, arxiv: ] [A. Bonanno, arxiv: ] [M. Hindmarsh, I. Saltas, arxiv: ] [...]

20 p. 14/4 Motivation for RG-improved cosmology [A. Bonanno, M. Reuter, hep-th/ ] [A. Bonanno, M. Reuter, astro-ph/ ] [A. Bonanno, M. Reuter, astro-ph/ ] [M. Reuter, F. Saueressig, hep-th/ ] [A. Bonanno, M. Reuter, arxiv: ] [B.F.L. Ward, arxiv: ] [S. Weinberg, arxiv: ] [A. Bonanno, A. Contillo, R. Percacci, arxiv: ] [M. Hindmarsh, D. Litim, C. Rahmede, arxiv: ] [C. Ahn, C. Kim, E.V. Linder, arxiv: ] [Y. Cai, D. Easson, arxiv: ] [R. Yang, arxiv: ] [A. Contillo, M. Hindmarsh, C. Rahmede, arxiv: ] [S.E. Hong, Y.J. Lee, H. Zoe, arxiv: ] [A. Bonanno, arxiv: ] [M. Hindmarsh, I. Saltas, arxiv: ] [...]

21 p. 15/4 RG-improved Einstein-Hilbert action [A. Bonanno, arxiv: ] objective: realize standard inflation scenario: de Sitter solution with N = 60 e-folds expansion and exits inflation RG-improve the Einstein-Hilbert Lagrangian L EH = 1 16πG ( R+2Λ) linearized RG-flow at non-gaussian fixed point (λ,g) T = (λ,g ) T +2 { [ReCcos(θ t)+imc sin(θ t)]rev +[ImCsin(θ t) ReCcos(θ t)]imv } e θ t cutoff-identification inside Γ k : k 2 R L eff = R 2 +br 2 cos [ θ 2 log(r µ ) ]( R µ ) θ effective action is of f(r)-type at NGFP: agrees with Γ from f(r)-flow equation

22 p. 16/4 solving the RG-improved equation of motion [A. Bonanno, arxiv: ] solve RG-improved equations of motion for Friedman-Robertson-Walker ansatz ds 2 = dt 2 +a(t) 2[ dr 2 +r 2 dω 2 ] 2 Hubble-parameter describing expansion of universe H = ȧ/a de Sitter solution: H = H = const a e Ht

23 p. 16/4 solving the RG-improved equation of motion [A. Bonanno, arxiv: ] solve RG-improved equations of motion for Friedman-Robertson-Walker ansatz ds 2 = dt 2 +a(t) 2[ dr 2 +r 2 dω 2 ] 2 Hubble-parameter describing expansion of universe H = ȧ/a de Sitter solution: H = H = const a e Ht find: discrete set of de Sitter solutions [ ( )] H = µ 12 exp 1 1 θ 2θ tan θ +nπ stability analysis: n = 4 exits after 50 e-folds, n probes stability coefficients of the non-gaussian fixed point

24 p. 16/4 solving the RG-improved equation of motion [A. Bonanno, arxiv: ] solve RG-improved equations of motion for Friedman-Robertson-Walker ansatz ds 2 = dt 2 +a(t) 2[ dr 2 +r 2 dω 2 ] 2 Hubble-parameter describing expansion of universe H = ȧ/a de Sitter solution: H = H = const a e Ht find: discrete set of de Sitter solutions [ ( )] H = µ 12 exp 1 1 θ 2θ tan θ +nπ stability analysis: n = 4 exits after 50 e-folds, n probes stability coefficients of the non-gaussian fixed point possible to obtain quantum gravity driven inflationary phase

25 p. 17/4 fractal-like properties of effective space-time the spectral dimension

26 p. 18/4 Distance depends on length of yard-stick Determined by number N of balls necessary to cover a point-set: N(R) R D Examples: real line: N(R) R 1 D = 1 coast-line of England: D 1.2

27 p. 19/4 Spectral dimension for classical manifolds Heat-equation: diffusion of scalar test particle on manifold with metric g T K g (x,x ;T) = g K g (x,x ;T) define average return probability P g (T) 1 V d d x g(x)k g (x,x;t) = 1 V Tr[exp(T g)] = ( 1 4πT ) d/2 A n T n n=0 asymptotic expansion: space-time dimension seen by diffusion process d = 2 dlnp g(t) dlnt T=0 P g (T): accessible by continuum and Monte Carlo methods!

28 p. 20/4 Spectral dimension of QEG space-times in QEG: metric of manifold is k-dependent = diffusion process with momentum k sees metric g µν k = diffusion equation and return probability will become k-dependent Computation of the spectral dimension: 1. determine k-dependence of (k) = Λ(k)/Λ(k 0 ) (k 0 ) 2. solve the k-dependent heat equation 3. evaluate quantum return probability P g (T) P(T) = d d p (2π) d exp[ p2 F(p 2 )T], F(p 2 ) = Λ(p)/Λ(k 0 ) 4. obtain spectral dimension D s (T) = 2 dlnp g(t) dlnt

29 p. 21/4 Spectral dimension D s of QEG space-times Flow of spectral dimension along a typical RG-trajectory D s k log k classical regime: D s (T) = 4 semi-classical regime: D s (T) = 4/3 NGFP regime: D s (T) = 2

30 p. 22/4 Spectral dimension D s of QEG space-times Flow of spectral dimension along a typical RG-trajectory g P 4 P Ds k P log k 0.0 P Λ P 1 classical regime: D s (T) = 4 semi-classical regime: D s (T) = 4/3 NGFP regime: D s (T) = 2

31 p. 23/4 The spectral dimension puzzle effective QEG space-times [O. Lauscher, M. Reuter 05] classical regime (F(p 2 ) = 1): D s (T) = d NGFP regime (F(p 2 ) p 2 ): D s (T) = d/2 Causal Dynamical Triangulations (d = 4) [J. Ambjorn, J. Jurkiewicz, R. Loll 05] classical regime: D s (T) = 4 short random walks: D s (T) = 2

32 p. 23/4 The spectral dimension puzzle effective QEG space-times [O. Lauscher, M. Reuter 05] classical regime (F(p 2 ) = 1): D s (T) = d NGFP regime (F(p 2 ) p 2 ): D s (T) = d/2 Causal Dynamical Triangulations (d = 4) [J. Ambjorn, J. Jurkiewicz, R. Loll 05] classical regime: D s (T) = 4 short random walks: D s (T) = 2 Causal Dynamical Triangulations (d = 3) [D. Benedetti, J. Henson 09] classical regime: D s (T) = 3 short random walks: D s (T) = 2 Euclidean Dynamical Triangulations (d = 4) [J. Laiho, D. Coumbe 11] classical regime: D s (T) = 4 short random walks: D s (T) = 1.5

33 p. 24/4 Spectral Dimension measured in 3-dimensional CDT [D. Benedetti, J. Henson, Phys. Rev. D 80 (2009) ] D s T T T 20 oscillations (discrete simplex structure) 20 T 500 good data 500 T exponential fall-off (triangulation is compact)

34 p. 25/4 Determining the spectral dimension in CDT [D. Benedetti, J. Henson, Phys. Rev. D 80 (2009) ] D s T T Fit-function I: D s (T) = a b c+t : a = 3.21, b = 46.93, c = D s (T) T=0 = 1.39, D s (T) T= = 3.19

35 p. 26/4 Determining the spectral dimension in CDT [D. Benedetti, J. Henson, Phys. Rev. D 80 (2009) ] D s T T Fit-function II: D s (T) = a+be ct : a = 3.05, b = 1.02, c = 0.02 D s (T) T=0 = 2.03, D s (T) T= = 3.05

36 p. 27/4 The RG-trajectory underlying the CDT-data Matching the spectral dimensions of QEG and CDT: 1. integrate β-functions: g 0,λ 0 g k,λ k 2. substitute RG-trajectory into Ds QEG (T): 3. determine g fit 0,λfit 0 Ds QEG (T) Ds QEG (T;g 0,λ 0 ) by minimizing ( D s ) T=20 ( D QEG s (T;g0 fit,λfit 0 ) DCDT s (T) ) 2

37 p. 27/4 The RG-trajectory underlying the CDT-data Matching the spectral dimensions of QEG and CDT: 1. integrate β-functions: g 0,λ 0 g k,λ k 2. substitute RG-trajectory into Ds QEG (T): 3. determine g fit 0,λfit 0 Ds QEG (T) Ds QEG (T;g 0,λ 0 ) by minimizing ( D s ) T=20 ( D QEG s (T;g0 fit,λfit 0 ) DCDT s (T) ) 2 Best-fit values for CDT-data with N simplices: N g fit 0 λ fit 0 ( D s ) 2 70k k k

38 p. 28/4 Spectral dimension: comparison D s T T D s T T D s T T Ds QEG (T;g0 fit,λfit 0 ) Ds CDT (T) for N = 70k,100k,200k-simplices

39 p. 29/4 Spectral dimension: fit-quality relative error captured by residuals: ɛ DQEG s (T;g0 fit,λfit D QEG s 0 ) DCDT s (T;g fit 0,λfit 0 ) (T) relative error k 100k 200k T

40 p. 30/4 Comparing spectral dimensions in d = 3 D s T Log T CDT and QEG agree with data within 1% accuracy no data-points on the semi-classical and NGFP-plateau resolves puzzle between CDT data and QEG prediction!

41 Robustness of results? p. 31/4

42 p. 32/4 Charting the RG-flow of the R 2 -truncation [O. Lauscher, M. Reuter, hep-th/ ] [S. Rechenberger, F.S., arxiv: ] Includes leading higher-derivative corrections to Einstein-Hilbert truncation Γ grav k [g] = d 4 x g [ 1 ( R+2Λ k )+ 1 ] R 2 16πG k b k

43 p. 32/4 Charting the RG-flow of the R 2 -truncation [O. Lauscher, M. Reuter, hep-th/ ] [S. Rechenberger, F.S., arxiv: ] Includes leading higher-derivative corrections to Einstein-Hilbert truncation Γ grav k [g] = d 4 x g [ 1 ( R+2Λ k )+ 1 ] R 2 16πG k b k b g B Λ A

44 p. 33/4 R 2 -corrections to the spectral dimension D s along the sample trajectories: A B D s t D s t plateau-structure: as in Einstein-Hilbert truncation transitions: manifestly different

45 p. 34/4 Physics encoded in the plateaus plateaus created by universal features of fixed points D s = 4: classical plateau probes eigendirection of GFP towards IR dimensionful couplings are constant classical theory D s = 4/3: semi-classical plateau probes eigendirection of GFP towards UV encodes: universal quantum corrections to General Relativity D s = 2: NGFP-plateau probes: NGFP-scaling encodes: conformal invariance of the theory in UV critical exponents: log-oscillations of D s

46 p. 35/4 modified gravity in the IR a prospective IR-fixed point

47 p. 36/4 The RG trajectory realized in Nature [M. Reuter, H. Weyer, JCAP 0412 (2004) 001] measurement of G N,Λ in classical regime: g T P 1 P 2? <<1 0.5 λ originates at NGFP (quantum regime: G(k) = k 2 g,λ(k) = k 2 λ ) passing extremely close to the GFP long classical GR regime (classical regime: G(k) = const, Λ(k) = const) λ 1/2: strong IR renormalization effects?

48 p. 37/4 IR-completing the trajectory realized in Nature Evidence: gravitational RG-flow has IR-attractive fixed point (g, λ ) = (0, 1/2) caveat: λ = 1/2 is singularity in β-functions IR-fixed point difficult to analyze

49 p. 37/4 IR-completing the trajectory realized in Nature Evidence: gravitational RG-flow has IR-attractive fixed point (g, λ ) = (0, 1/2) caveat: λ = 1/2 is singularity in β-functions IR-fixed point difficult to analyze Indirect evidence for IR-fixed point: bimetric analysis of the two-point correlator [I. Donkin, J. Pawlowski, arxiv: ] diverging correlation length when flow approaches (0, 1/2) [S. Nagy, J. Krizsan, K. Sailer, arxiv: ] study deformed beta-functions where singularities are lifted [D. Litim, A. Satz, arxiv: ]

50 p. 38/4 Charting the RG-flow of the R 2 -truncation [O. Lauscher, M. Reuter, hep-th/ ] [S. Rechenberger, F.S., arxiv: ] Includes leading higher-derivative corrections to Einstein-Hilbert truncation Γ grav k [g] = d 4 x g [ 1 ( R+2Λ k )+ 1 ] R 2 16πG k b k

51 p. 38/4 Charting the RG-flow of the R 2 -truncation [O. Lauscher, M. Reuter, hep-th/ ] [S. Rechenberger, F.S., arxiv: ] Includes leading higher-derivative corrections to Einstein-Hilbert truncation Γ grav k [g] = d 4 x g [ 1 ( R+2Λ k )+ 1 ] R 2 16πG k b k b g B Λ A

52 p. 39/4 The universe is not empty Asymptotic safety in gravity-matter systems

53 p. 40/4 Asymptotically Safe Gravity-Matter systems NGFP occurs in many minimally coupled gravity-matter systems standard model of particle physics SO(10)-GUT models, supersymmetric GUTs [R. Percacci, D. Perini, hep-th/ ]

54 p. 40/4 Asymptotically Safe Gravity-Matter systems NGFP occurs in many minimally coupled gravity-matter systems standard model of particle physics SO(10)-GUT models, supersymmetric GUTs [R. Percacci, D. Perini, hep-th/ ] asymptotically safe gravity-scalar system: NGFP robust with respect to 1-loop counterterm [D. Benedetti, P. Machado, F.S., arxiv: ] estimates for the Higgs-mass 126 m H 176 Gev [M. Shaposhnikov, C. Wetterich, arxiv: ]

55 p. 40/4 Asymptotically Safe Gravity-Matter systems NGFP occurs in many minimally coupled gravity-matter systems standard model of particle physics SO(10)-GUT models, supersymmetric GUTs [R. Percacci, D. Perini, hep-th/ ] asymptotically safe gravity-scalar system: NGFP robust with respect to 1-loop counterterm [D. Benedetti, P. Machado, F.S., arxiv: ] estimates for the Higgs-mass 126 m H 176 Gev [M. Shaposhnikov, C. Wetterich, arxiv: ] non-gaussian fixed point in dilaton-gravity models [G. Narain, R. Percacci, arxiv: ] [G. Narain, C. Rahmede, arxiv: ] conservation of Weyl-invariance under RG-flow [R. Percacci, arxiv: ]

56 p. 41/4 Asymptotically Safe Gravity-Matter systems gravity-induced corrections to QED β-functions [S.P. Robinson, F. Wilczek 05; D.J. Toms 08; D. Ebert, J. Plefka, A. Rodigast 10] symmetry-perserving regulators give vanishing one-loop correction [S. Folkerts, D. Litim, J. Pawlowski 11] asymptotic safety cures Landau pole [J.E. Daum, U. Harst, M. Reuter 10]

57 p. 41/4 Asymptotically Safe Gravity-Matter systems gravity-induced corrections to QED β-functions [S.P. Robinson, F. Wilczek 05; D.J. Toms 08; D. Ebert, J. Plefka, A. Rodigast 10] symmetry-perserving regulators give vanishing one-loop correction [S. Folkerts, D. Litim, J. Pawlowski 11] asymptotic safety cures Landau pole [J.E. Daum, U. Harst, M. Reuter 10] asymptotic safety compatible with light chiral fermions [A. Eichhorn, H. Gies 11]

58 Conclusions p. 42/4

59 p. 43/4 Conclusions Quantum Einstein Gravity constitutes a strong candidate for Quantum Gravity free from unphysical UV-divergences predictive recovers classical General Relativity in the IR unitarity?

60 p. 43/4 Conclusions Quantum Einstein Gravity constitutes a strong candidate for Quantum Gravity free from unphysical UV-divergences predictive recovers classical General Relativity in the IR unitarity? possible to bridge gap between fundamental theory and phenomenology cosmological scenarios physics of black holes comparison to Monte-Carlo simulations (CDT and EDT)

61 p. 43/4 Conclusions Quantum Einstein Gravity constitutes a strong candidate for Quantum Gravity free from unphysical UV-divergences predictive recovers classical General Relativity in the IR unitarity? possible to bridge gap between fundamental theory and phenomenology cosmological scenarios physics of black holes comparison to Monte-Carlo simulations (CDT and EDT)!!! WORK AHEAD!!!