1. IRGAC 2006 Renormalization Group Running Cosmologies – from a Scale Setting to Holographic Dark Energy Branko Guberina Rudjer Boskovic Institute Zagreb, Croatia Ch. I. Cosmological constant – an introitus Ch. II. Cosmological constant renormalization and decoupling – a running scale Ch. III. RG running cosmologies - a scale setting procedure Ch. IV. Holographic dark energy – a setup Ch. V. Conclusions

2. Based on collaboration with Raul Horvat Hrvoje Stefancic Hrvoje Nikolic Ana Babic

3. Ch. I. Cosmological constant - an intro The Einstein equations R μν - ½g μν R + g μν Λ = -8πGT μν The notation follows Ландау-Лифшиц, Теория поля, except for the sign of the Einstein tensor G μν = R μν - ½g μν R (Einstein-Eddington convention used). The signature is η μν = + - - -. The covariant conservation implies the conservation of the sum D μ ( 8πT μν (m) /M 2 P + g μν Λ ) = 0.

4. The energy momentum tensor T μν is given as a functional derivative of the total action S with respect to the metric g μν, δA/δ g μν. T μν must be conserved (invariance of the theory with respect to an arbitrary change of coordinates: Since the metric tensor is covariantly constant this gives Λ = const. However, a number of models with Λ = Λ(t) are around today! The first proposal: M. Bronstein, Physikalische Zeitschrift der Sowietunion, 3 (1933) 73. Strongly critisized by Landau. The same story for G = G(t)! “Such models are not innocent” (A. D. Dolgov, hep-ph/0606230).

5. An argument is the following: one has to derive the energy- momentum tensor T μν by functional differentiation with respect to g μν from the total action S. This would contain the second time derivatives of G. The consistent theory should start from an action S. For example, cf. M. Reuter, Brans - Dicke like theory, PRD 69 (2004)104022

6. Cosmological constant ≠ 0 – unavoidable? Different sources of CC: Zeldovich (1968) – a particle sitting at the bottom of the harmonic oscillator (HO) potential with a minimum energy ½ω should have a nonzero momentum due to the Heisenberg uncertainty principle. Quantum mechanics gives the minimum energy ½ω, indeed. In QFT a quantum field is a collection of infinite number of HOs

7. Other contributions to CC The natural value of the vacuum energy in broken SuGra ρ Λ ~ M P 4 ≥ 10 78 GeV 4. compared to the observed value ρ Λ exp ≈ 10 -47 GeV 4. QCD condensates – nonvanishing and well established experimentally ≠ 0, ≠ 0, with huge values (ρ Λ ) QCD ≈ - 10 45 ρ c. N.B. Inside the proton the gluon condensate gives a mass to the proton m proton = 2m u + m d - ρ G² l ³ proton, l proton ~ 1/m π. ¯

8. Theory versus reality Λ theory /Λ real = = 1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

9. Ch. II. Cosmological constant renormalization and decoupling It is possible to study the cosmological constant without explaining its present value. If treated as parameters in an action S, both G and Λ would become running quantities in any QFT in curved space, cf. N. D. Birell, P.C. W. Davies, “Quantum Fields in Curved Space”, I. L. Buchbinder, S. D. Odintsov, I. L. Shapiro, “Effective Action in Quantum Gravity” Ilya Shapiro, IRGAC 2006 talk. A problem with the identification of the running scale. Normally, in QFT the renormalization scale μ is traded for some physical observable, e. g., a typical energy/momentum of particles in interaction. What is a typical graviton momentum?

10. Quantum field theory in curved space-time as an effective field theory S. Weinberg, Physica (Amsterdam) 96A (1979)327 A very successful Chiral perturbation theory (ChPT) as a low-energy nonlinear realization of QCD (J. Gasser, H. Leutwyler, NPB 250(1985)465). Loop effects make sense in ChPT, since some processes go through loops only.

11. Gravity, with quantum corrections being small up to M P, appears to be, prima facie, even better EFT – expected to govern the effects of Quantum gravity at low energy scale. However ( BAD NEWS ), the possible existence of the singularity in the future, corresponding to the wavelength probed of the order of the size of the universe – deeply in the infrared region – may break the validity of the EFT in the IR limit. In fact QG may lead to a very strong renormalization effects at large distances induced by IR divergences, cf. I. Antoniadis, E. Mottola (1992), N. C. Tsamis, R. P. Woodard (1993, 1995, 1996). E. g. correction to the classical gravity, cf. Donoghue, gr-qc/9607039. Compare to the corrections in nonperturbative quantum Einstein gravity, cf. Reuter, Tbilisi talk, hep-th/0012069

12. The scale μ The meaning of the RGE scales – in the MS scheme the μ dependence in the effective action is compensated by the running of the parameter Λ (as in QED where the μ dependence is compensated by the running charge e(μ). The overall action S which contains a running Λ(μ) is scale independent. The physical interpretation of the RGE scale can be achieved calculating the polarization operator of gravitons arising from particle loops in the linearized gravity, cf. Gorbar-Shapiro, JHEP 02(2003)21 In the physical mass-dependent renormalization scheme an arbitrary parameter μ is usually traded for a Euclidean momentum p. In QCD, for example, one writes an RG equation with respect to the momentum scaling parameter λ, p→ λp, and eliminates the derivatives with respect to μ. The trade is performed by identifying the new scale with the typical average energy (momentum) of the physical process. ¯

13. Einstein-Hilbert Vacuum Action The present age Universe should be well described by the action S The formulation of the theory – rather simple, cf. Shapiro, Sola, PLB 475(2000)236, JHEP 02(2002)006, see also Peccei, Sola, Wetterich, PLB 195(1987)183. - construct a renormalizable gauge theory (e.g., the gauged Higgs lagrangian) in an external gravitation field. - start with the matter action in flat space-time and replace the partial derivatives by the covariant ones, - replace the Minkowski metric η μν by the general one g μν, - replace d 4 x by d 4 x(-g) ½.

14. The vacuum action necessary to ensure the renormalizability of the gauged scalar Lagrangian should contain terms R 2 μνρσ, R 2 μν, R 2, □R, with coefficients a i which are the bare parameters. All divergences can be removed by renormalization of the matter fields, their masses and couplings, the bare parameters a i, G bare, Λ bare, and the non-minimal parameter ξ bare which enters the action via a term (presently negligible) ξφ † φR, where φ is a scalar field.

15. Per definitionem, a vacuum energy density ρ Λ is ρ Λ = Λ/8πG = Λ. The scalar field φ with the potential energy V(φ) contributes to S If φ vac is the value of the field φ(x) which minimizes the potential V(φ), then the lowest state has T μν = g μν V(φ vac ). This is a classical scalar field contribution to the vacuum energy, e.g., for the Higgs field with potential V(φ) = -m² φ † φ + λ(φ † φ)², one has the Higgs condensate contribution (at the classical level) for the cosmological constant ρ Λ Higgs = Λ Higgs /8πG = -m 4 /4λ. ¯

16. Zero-point energy – a renormalization Using the dimensional regularization in d = 4 + ε and MS scheme Bare vs. renormalized Λ + the counterterm → Correct behavior of β(μ) = μ (∂/∂μ)Λ for μ >> M. ¯

17. Decoupling failure Decoupling theorem – Appelquist-Carazzone valid? One expects corrections of the type μ²/M² to be insufficient to suppress the quartic power of the mass. Assume there exist 2 particles, a heavy one with mass M, and a light one with mass m. For μ >> M, m the RGE becomes (Λ = Λ/8πG = ρ Λ ) For m << μ << M one expects the decoupling of a heavy particle with the suppression factor μ²/M² The suppression factor μ²/M² is not sufficient to suppress the contribution of the heavy scalar, since μ² M² >> m 4 ¯ ¯ ¯

18. The zero point energy of a scalar field is (cf. J. Kapusta, Finite-temperature field theory ) and the renormalized and bare CCs are (Λ = Λ/8πG) ¯ ¯ ¯

19. The real scalar field ZPE contribution to the cosmological constant In the μ >> m limit ¯ In the opposite μ << m limit ¯ In the Standard Model ¯

20. A question of the running – a scale? Focus on the effects of running at scales bellow the electron mass (the lightest particle here): The above eq. – the large masses in the μ² term drive the numerical value far out of the range consistent with observations, unless the expression in the brackets vanishes. In a toy model – the SM – one obtains M H ² = 4 ∑ I N i m i 2 – 3m Z 2 – 6m W 2. This leads to a prediction, M H 2 ≈ 550 GeV. ¯ ¯

21. Nucleosynthesis The running of the zero-point energy is ruled by the μ 4 term. Using the expression for the energy density of the radiation during nucleosynthesis ρ R = (π²/30) g * T 4, and making a natural choice μ = T, one obtains [ Λ (μ) – Λ (0) ] (8πGρ R ) -1 = 555/(32 π 4 g * ) With g * = 3.36, the above expression acquires the value 0.053. This value does not disturb nucleosynthesis. Taking μ = 0,02 eV one obtains (8πG) -1 [ Λ (μ) – Λ (0) ] ≈ 10 -48 GeV 4.

22. Experiments vs. theory – an example Cf. Shapiro et al. JCAP 0402 (2004) 006

23. Ch. III. RG running cosmologies – A scale- setting procedure Based on Ana Babic, B. G., Raul Horvat, Hrvoje Stefancic, PRD 71(2005)124041 A class of RGE-based cosmologies – the only running quantities are ρ Λ and G. The RGE improved Einstein equation for these cosmologies R μν - ½g μν R = -8πG(μ)(T μν m + g μν ρ Λ (μ)). (1) The only physical requirement: Eq. (1) should maintain its general covariance with the μ-dependent (and implicitly t-dependent) G(μ) and ρ Λ (μ). This leads to the following equation

24. Assuming the nonvanishing throughout the evaluation of the Universe, one gets This is a master formula. The LHS is a function of the scale factor a, and the RHS is a function of the scale μ, i. e. one has ρ m = f(μ). The inversion gives μ = f -1 (ρ m ). Once the scale setting is completed, one has G(μ) and ρ Λ (μ) as functions of ρ m. The matter energy density – the scale behavior and the Friedmann equation

25. Comments on scale setting procedure Our procedure lacks the first principle connection to quantum gravity. We assumed that ρ m is intrinsically independent of μ. (*) The above equation means: as long as ρ m retains its canonical behavior, the scale μ is univocally fixed and does not even implicitly depend on μ in the above equation. If one allows, e.g., for an interaction between matter and or CC and/or G, then eq. (*) generalizes to Any deviation of ρ m from the canonical form depends on μ. Therefore one is not able to fix a scale without specifying the interactions a priori.

26. Nonperturbative quantum gravity The exact renormalization group approach applied to quantum gravity, cf. M. Reuter, PRD 57(1998)971, IRGAC 2006 talk. The effective average action Γ k [g μν ] – basically a Wilsonian coarse-grained free energy (M. Reuter, C. Wetterich, NPB 417(1994)181; 427((1994)291; 506(1997)483; Berges, Tetradis, Wetterich, Phys. Rep. 363(2002)223.) The momentum scale k is interpreted as an infrared cutoff – for a physical system with a size L, the parameter k ~ 1/L defines an infrared cutoff. The path integral which defines the effective average action Γ k [g μν ] integrates only the quantum fluctuations with the momenta p² > k², thus describing the dynamics of the metrics averaged over the volume (k -1 ) 3. For any scale k there is a Γ k which is an EFT at that scale. Large distance metric fluctuations, p² < k², are not included.

27. Quantum gravity model with an IR fixed point M. Reuter, PRD 57(1998)971, A. Bonano, M. Reuter, PRD 65(2002)043508, PLB 527(2002)9. In this formalism, it is possible to set up the RGE for G(μ) and Λ(μ). The infrared cutoff k plays the role of the general RGE scale μ. In the infrared limit, Λ and G run as follows: λ * and g * are constants and k is an IR cutoff.

28. Inserting the above expressions into the master formula gives The result does not depend explicitly on the topology K of the Universe. A certain level of implicit dependence exists since the expansion of the Universe (and the dependence of ρ m on time) depends on K. The scale-setting procedure unambiguously identifies the scale k in the quantum gravity model with the IR fixed point. The above expression for k leads to Per definitionem, ρ c = 3H 2 /(8πG) Ω Λ = Λ/(8πGρ c ) Ω m = ρ m / ρ c Ω c = -K/(H 2 /a 2 ) Ω Λ + Ω m + Ω c = 0., → Ω Λ = Ω m

29. Using the preceding definitions one obtains an implicit expression for the scale factor of the Universe: Obviously, the expansion of the universe depends on two parameters, w and Ω K 0. Solutions - special choices in the limit t’→ 0 (a→ 0) (cf. BonReuter) (i) Ω K 0 = 0, w arbitrary, one obtains and in agreement with Bonano-Reuter, PLB 527(2002)9.

30. (ii) w = 1/3, Ω K arbitrary. A simple law for the scale factor a/a 0 = H 0 t, and This shows why the Ansatz k = ξ / t functions correctly for cosmologies of any curvature including the radiation only (agreement with Bonano-Reuter PLB 527(2002)9.

31. (iii) w = 0, Ω K ≥ 0. The universe with NR matter and arbitrary curvature - In this case, the scale k can no longer be expressed in the form of ansatz k = ξ/t. The scale k is uniquely defined by choosing the only real solution of the above cubic equation. The product t (a/a 0 ) -3/4 (since k ~ (a/a 0 ) -3/4 ) as a function of t →

32. The product of the cosmic time t and (a/a 0 ) -3/4 versus t for the flat and the open universe with Ω K 0 = 0.02. For the curved universe, the choice of the scale differs from k ~ 1/t.

33. RGE cosmological model from QFT on curved space-time When the RGE scale μ is less than ALL masses in the theory, one may write The application of the scale-setting procedure yields

34. Results The scale μ From the study of cosmologies with the running ρ Λ in QFT in curved space-time, we know that generally C 1 ~ m 2 max, C 2 ~ N b – N f ~1, C 3 ~ 1/m 2 min, etc., and D 1 ~ 1, D 2 ~ 1/m 2 min, etc. The value of C 0 can hardly surpass ρ Λ today.

35. γ 1 and γ 2 are constants of order 1. - a very slow running in the vicinity of μ = m min. - by inserting this scale into the expansion of ρ Λ one arrives at ρ Λ ~ m 2 max m 2 min. In the extreme case we can set m max ~ M Planck, and saturate (ρ Λ ) today with m min ~ m quintessence ~ 10 -33 GeV.

36. Ch. IV. Holographic Dark Energy L = size of the box = radius of a black hole,S BH = the entropy of a black hole, The length scale L provides an IR cutoff which is determined by the UV cutoff Λ. Holographic principle : A field theory overcounts the true dynamical degrees of freedom - therefore extra nonlocal constraints on an effective field theory are necessary. The entropy S scales extensively in an effective quantum field theory for a system of size L with the UV cutoff Λ: Bekenstein: maximum entropy in a box of volume L 3 grows only as the area A of the box. ‘t Hooft, Susskind: 3+1 field theories overcount degrees of freedom. Also, a local quantum field theory cannot correctly describe quantum gravity – too many degrees of freedom in UV. Exit: limit the volume of the system according to

37. A. G. Cohen, D. B. Kaplan, A. E. Nelson, PRL 82 (1999) 4971 An effective field theory satisfying the constraint unavoidably includes many states with the Schwarzschild radius larger than the box size L. Solution: an additional constraint on the IR cutoff 1/L which excludes all states that lie within their Schwarzschild radius. The maximal energy density in the effective QFT is Λ 4, therefore the energy in a given volume ~L 3 should not exceed the energy of a black hole of the same size L, i. e., where M P is the Planck mass.

38. Box of size L Black hole of size L Schwarzschild radius R S < L

39. The IR cutoff L scales as Λ -2 – this bound is more restrictive by far. Near the bound saturation the entropy is given by Cf. a review: R. Bousso, Rev. Mod. Phys. 74(2002)825. Subtle questions concerning black holes and the horizons, cf. T. Padmanabhan, Class. Quantum Grav. 19(2002)5387, T. M. Davis,et al., Class. Quantum Grav. 20(2003)2753, P. C. W. Davies, “The implications of a holographic universe and the nature of physical law”, to appear (priv. comm.).

40. Cosmological constant and holography The usual quantum corrections to the vacuum energy density with no IR cutoff in QFT obviously gives the wrong prediction. Should the fields at the present energy scales fluctuate independently over the entire horizon, or even, over the whole Universe? Cohen et al. proposal: taking the size L to be approximately the size of the present horizon, L ~ H -1, the vacuum energy density Λ 4 is constrained to be of the order of (meV) 4 – the right value of the experimental vacuum energy density. The cosmological constant problem – solved? Not really – one can always add a constant to the quantum corrections. However, if correct, the Cohen et al. bound eliminates the need for fine-tuning.

41. Holographic Dark Energy - A Cosmological Setup A generalized holographic dark energy model – both the cosmological constant (CC) and Newton’s constant G N are scale dependent. A cosmological setup – the renormalization group (RG) evolution of both CC and G N within quantum field theory in curved space. Cohen et al. relation between the UV and IR cutoff results in an upper bound on the zero-point energy density ρ Λ. The largest CC This relation is what we call a generalized dark energy model. A scale μ ~ 1/L represents the IR cutoff. Again, choosing L = H 0 -1 = 10 28 cm, one arrives at the present observed value for the dark energy density ρ Λ = 10 -47 GeV 4.

42. Problems with the choice of the IR cutof: If ρ Λ is considered the energy density of a noninteracting perfect fluid – then the choice μ = L -1 fails to recover the equation of state (EOS) for a dark-energy dominated universe, cf. S. D. Hsu, PLB 594 (2004)13. Even more, choosing L = H -1 always leads to ρ Λ = ρ m for flat space, thus hindering a decelerating era of the universe for redshifts z > 0.5. However, a correct EOS is obtained if one chooses a future event horizon for an IR cutoff, cf. M. Li, PLB 603 (2004) 1.

43. Since the equation (1) was derived using ZPEs, the natural interpretation of dark energy in the above equation is through the variable, or interacting, CC with w = -1. The energy transfer between various components in the universe (where G can also vary with time) is described by a generalized equation of continuity (2) Overdots denote the time derivative; matter is pressureless, w = 0. N. B. ρ Λ is affected not only by matter, but also by a time dependent G. The quantity G N T μν total in (2) is conserved. For G N static, T μν total is conserved. Sourced Friedmann eqs. with holographic dark energy studied by Y.S. Myung, hep-th/0502128.

44. Using the holographic restriction (1) and the generalized equation of continuity (2), to constrain the parameters of the RG evolution in QFT in curved space background, R. Horvat, H. Stefancic, B.G. JCAP 05 (2005) 001. The variation of the CC and G in this model arises solely from particle field fluctuations (no quintessence-like scalar fields). For the RG running scale μ below the lowest mass in the theory, the RG laws read The scale μ cannot be set from first principles. Assuming the convergence of tboth series, and using the studies of cosmologies with running ρ Λ and G, in the formalism of QFT in curved space-time, one generally gets C 1 ~ m 2 max, C 2 ~ N b – N f ~1, C 3 ~ 1/m 2 min etc., and D 1 ~ 1, D 2 ~ 1/m 2 min, etc. m max and m min are the largest and the smallest mass in the theory, respectively. N b and N f are the number of bosonic and fermionic massive degrees of freedom. C 0 = the vacuum ground state, coincides here with the IR limit of CC.

45. The context is set by fixing the matter density law to be a canonical one, i. e. ρ m ~ a -3 – no energy transfer between ρ m and other components. Eq. (2) reduces to N.B. The prime denotes differentiation with respect to the scale μ. Inserting the holographic expession (1) into the above equation (3) Once G N (μ) is known, the IR cutoff μ is fixed. For μ > 0, G’ N 0, i.e., G N (t) increases with increasing cosmic time t. This implies that G N is asymptotically free – the property seen in quantum gravity models at the 1-loop level, cf. Fradkin, Tseytlin, NPB 201 (1982) 469. Asymptotic freedom – of some interest for galaxy dynamics and rotation curves, cf. Shapiro, Sola, Stefancic, JCAP 01 (2005) 012

46. Applying the requrement (1) of the generalized holographic dark energy model to the RG laws for ρ Λ and G N relates C and D : ***************************************************************************************** Let us first discuss the choice G N = const. In this, rather simple case, one obtains with ρ Λ ~ m 2 max μ 2. However, the observational data suggest that μ 0 ~ H 0 today. N.B. The above relation does not imply μ = H. The continuity equation, for G N = const., gives The scale μ cannot be univocally fixed.

47. The equation means a continuous transfer from matter to CC and vice versa (depending on the sign of the interaction term). The dilution of the energy density of matter causes a deviation in the ρ m ~ a -3 behavior – which depends crucially on the choice for μ. The choice μ = H was employed in Shapiro, Sola, Espana-Bonet, Ruiz-Lapuente, PLB 574 (2003)149, JCAP 02 (2004)006. Note: m max ~ M P may be an effective mass describing particles with masses just below the Planck mass.

48. A case ρ Λ = ρ Λ (t), G N = G N (t), ρ m ~ a -3 Inserting the expansions in μ for ρ Λ and G N into the scaling-fixing relation leads to the expression for the scale μ Using the estimates for D n, one obtains

49. Comments on eq (***) 1. The value of μ is marginally acceptable as far as the convergence of the ρ Λ (μ) and G (μ) series is concerned. 2. In addition, from G’ N < 0 one obtains D 1 ≈ C 2 > 0. 3. Equation (***) shows a very slow variation of the scale μ with the scale factor a or the cosmic time t. 4. Once the RG scale crosses below the smallest mass in the theory, it effectively freezes at m min ~ H 0 ~ 10 -33 eV. 5. This is the main result of holography – one finds a hint for possible quintessence-like particles in the spectrum.

50. What holography basically does – it expands the particle spectrum from either side to the extremum – the largest possible particle masses approach the Planck mass, and the smallest possible particle masses are around the lowest mass scale in the universe, H 0. The present value of ρ Λ appears as the product of squared masses of the particles lying on the top and bottom of the spectrum - a natural solution to the coincidence problem.

51. Further studies B. G., R. Horvat, H. Nikolic, PRD 72(2005) The generalized holography dark energy model with scale dependent Λ and G, has been used to set constraints on the RG evolution of both Λ and G. Assuming ρ m scales canonically, it was shown that the continuity equation fixes an IR cutoff, provided a low of variation for either Λ or G is known. Using the RG running CC model, with low energy scaling given by the QFT of particles with masses near the Planck mass, in accordance with holography, amounts to having an IR cutoff scaling as H ½. Such a setup, in which the only undetermined input is the true ground state of the vacuum, yields a transient acceleration. ********************************** B. G., R. Horvat, H. Nikolic, PLB 636(2006)80 – a study of dynamical dark energy (scale dependent G) with a constant ρ Λ.

52. An interesting idea of comprising both the running CC and dark energy coming from a scalar field with a temporary phantom phase obtained with a nonphantom scalar field, which has EOS > -1 by Elizalde, Nojri, Odintsov, Wang, PRD 71 (2005) 103504. Interaction dark matter with dark energy – coincidence problem solution, cf. D. Pavon, W. Zimdahl, PLB 628(2005)206; Spatial curvature & holographic dark energy, cf. W. Zimdahl, D. Pavon, astro-ph/0606555.

53. Conclusions A model with a running CC based on the RG effects in QFT in curved space merged with the concept of holographic dark energy density. Remarkable consequences for the particle spectrum. Consistency with holographic predictions calls for the appearance of the (otherwise redundant) quintessence-like scalar fields. Although the holographic dark energy density in this approach is rather a ‘toy’ model, our order-of- magnitude estimates may indicate a preference for the ‘combined’ dark energy nature.

54. Thanks!

56. Finis enim mundi et omnis creationis homo est.

57. The scale μ? Example 1. Quantum chromodynmics [ μ (∂/∂μ) + β(α)α ∂/∂α … ] Γ ren (p, …, μ) = 0, & [ λ (∂/∂λ) + μ (∂/∂μ) + … ] Γ ren (λp, …, μ) = 0, [- (∂/∂τ) + μ (∂/∂μ) + … ] Γ ren (λp, …, μ) = 0, where τ = ℓn λ, and p→ λp is a momentum scaling. Eliminating μ(∂/∂μ) one obtains the RGE in λ, with the running α (τ). The scale μ is traded for the momentum. ********************************************* Example 2. QFT in curved space-time [ μ (∂/∂μ) + … ] Γ ren (p, …, μ) = 0, & [ τ (∂/∂τ) + μ (∂/∂μ) + … ] Γ ren (λ² g μν, …, μ) = 0, with g μν → λ² g μν, τ = ℓn λ.