Nonlocal gravity. Conceptual aspects and cosmological consequences

Transcription

1 Nonlocal gravity. Conceptual aspects and cosmological consequences Michele Maggiore Sifnos, Sept , 2017

2 based on Jaccard, MM, Mitsou, PRD 2013, MM, PRD 2014, Foffa, MM, Mitsou, PLB 2014, Foffa, MM, Mitsou, IJMPA 2014, Kehagias and MM, JHEP 2014, MM and Mancarella, PRD 2014, Dirian, Foffa, Khosravi, Kunz, MM, JCAP 2014, Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2015, MM PRD 2016, Cusin, Foffa, MM, PRD 2016, Cusin, Foffa, MM, Mancarella, PRD 2016, Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2016, MM (review) Springer, Dirian, Belgacem, Dirian, Foffa, MM, in prep.

3 Summary Conceptual aspects nonlocality in the quantum effective action scenarios for mass scale generation in gravity in the IR FAQ causality degrees of freedoms, ghosts localization and boundary conditions freedom in the choice of nonlocal term

4 Comparison with cosmological data background evolution and dark energy cosmological perturbations Bayesian parameter estimation and model comparison

5 Nonlocality and the quantum effective action At the fundamental level, the action in QFT is local However, the quantum effective action is nonlocal Z e iw [J] D' e is[']+i R J' quantum effective action: W [J] J(x) = h0 '(x) 0i J [J] [ ] W [J] Z J

6 [ ] (x) = J(x) the quantum EA gives the exact eqs of motion for the vev, which include the quantum corrections e i [ ] = Z D' e is[ +'] i R [ ] ' - We are `integrating out' the quantum fluctuations, not the fields It is not a Wilsonian effective action - The regime of validity of the quatum EA is the same as that of the fundamental theory

7 light particles nonlocalities in the quantum effective action these nonolocalities are well understood in the UV. E.g. in QED QED[A µ ]= 1 4 $ Z d 4 x apple F µ 1 e 2 (2) F µ + O(F 4 ) 1 e 2 (2) ' 1 e 2 (µ) 2 log µ 2 Z log it is just the running of the coupling constant in coordinate space 0 dm 2 apple µ 2 + fermionic terms 1 1 m 2 + µ 2 m 2 2 Note: we are not integrating out light particles from the theory!

8 The quantum effective action is especially useful in GR Z e i m[g µ ] = D' e is m[g µ ;'] (vacumm quantum EA. We can also retain the vev's of the matter fields ith the Legendre transform) T µ = 2 p g S m g µ h0 T µ 0i = 2 p g m g µ It gives the exact Einstein eqs including quantum matter loops G µ =8 G h0 in T µ 0 in i Γ = S EH + Γ m is an action that, used at tree level, give the eqs of motion that include the quantum effects

9 The quantum effective action in GR can be computed perturbatively in an expansion in the curvature using heat-kernel techniques Barvinsky-Vilkovisky 1985,1987 = m2 Pl 2 Z d 4 x p gr+ Z d 4 x p g apple Rk R (2)R C µ k W (2)C µ The form factors due to a matter field of mass m s are known in closed form Gorbar-Shapiro m 2 s For m s <<E k(2) ' k(µ) k log µ 2 + c c m 4 s For m s >>E k(2) ' O(2/m 2 s)

10 However, these corrections are only relevant in the UV (ie for quantum gravity) and not in the IR (cosmology): R log( 2/m 2 s)r m 2 PlR unless R m 2 Pl unavoidable, since these are one-loop corrections, and we pay a factor 1/m Pl 2 For application to cosmology, we rather need some strong IR effect

11 Dynamical mass generation in gravity in the IR? The techniques for computing the quantum EA are well understood in the UV, but much less in the IR infrared divergences of massless fields in ds lead to dynamical mass generation, m 2 dyn / H 2p Rajaraman 2010,... the graviton propagator has exactly the same IR divergences Antoniadis and Mottola 1986,... quantum stability of ds. Decades of controversies...

12 Euclidean lattice gravity suggests dynamical generation of a mass m, and a running of G N G(k 2 )=G N "1+ m 2 k O m 2 k 2 1 # ν 1/3 Hamber 1999,.., 2017 non-perturbative functional RG techniques find interesting fixed-point structure in the IR, and strong-gravity effects Wetterich 2017 the dynamical emergences of a mass scale in the IR in gravity is a meaningful working hypothesis

13 Gauge-invariant (or diff-invariant) mass terms can be obtained with nonlocal operators eg massive electrodynamics Dvali 2006 = 1 Z d 4 x F µ F µ m 2 1 F µ 4 2 F µ in the µ A µ =0 1 4 m2 F µ 1 2 F µ = 1 2 m2 A µ A µ a nonlocal but gauge-inv photon mass

14 Numerical results on the gluon propagator from lattice QCD and OPE are reproduced by adding to the quantum effective action a term m 2 g 2 Tr Z d 4 xf µ 1 2 F µ F µ = F a µ T a, 2 ab = D ac µ D µ,cb, D ab µ = µ gf abc A c µ (Boucaud et al 2001,Capri et al 2005,Dudal et al 2008) it is a nonlocal but gauge invariant mass term for the gluons, generated dynamically by strong IR effects

15 Our approach: we will postulate some nonlocal effective action, which depends on a mass scale, and is supposed to catch IR effects in GR phenomenological approach. Identify a non-local modification of GR that works well attempt at a more fundamental understanding

16 Our prototype model will be MM and M.Mancarella 2014 RR = Z d 4 x p g apple m 2 Pl 2 R R 4 RR 2 2 R ``RR model'' Λ RR generated dynamically, analog to Λ QCD g µ (x) =e 2 (x) µ! R = 62 + O( 2 ) is a mass term for the conformal mode! m H 0! RR mev 4 RR m2 m 2 Pl Λ RR is the fundamental scale. No ultralight particle provides a solution to the naturalness problem

17 FAQ 1. Is the theory causal? in a fundamental QFT nonlocality è acausality. E.g. Z Z dx 0 (x 0 )(2 1 )(x 0 )= dx 0 dx 00 (x 0 )G(x 0 ; x 00 ) (x 00 ) (x) (x) Z = dx 0 [G(x; x 0 )+G(x 0 ; x)] (x 0 ) the quantum EA generates the eqs of motion of h0 g µ 0i h0 out g µ 0 in i h0 in g µ 0 in i Feynman path integral, acausal eqs Schwinger-Keldish path integral, nonlocal but causal eqs

18 2. What is the domain of validity of the effective theory? Are you integrating out massless particles? The eqs e i m[g µ ] = Z D' e is m[g µ ;'] h0 T µ 0i = 2 p g m g µ are exact. It is not a Wilsonian approach of course, the issue is to compute the EA, but if one can, it is valid in the whole regime of the fundamental action

19 3. What are the d.o.f. of your theory? Have you done a Hamiltonian analysis? You cannot read the dof from the vacuum quantum effective action! You need the fundamental theory Example: Polyakov quantum EA in D=2 is an example of the fact that we can get non-perturbative informations on the quantum EA since it can be computed exacly, it is useful to clarify conceptual aspects that create confusion in the literature (dof, ghosts,propagating vs non-propagating fields, etc)

20 consider gravity + N conformal matter fields in D=2 conformal anomaly: h0 T a a 0i = N 24 R this result is exact. Only the 1-loop term contributes. In D=2: g µ = e 2 µ R = 22 = N 12 2 [ ] [ = 0] = N 24 Z d 2 xe 2 2 in D=2, Γ[0]=0. This is the exact quantum EA

21 we can covariantize [ ]= N 24 Z d 2 xe 2 2 using p g = e 2 R = 22 we get = N Z 48 N = 96 d 2 x p g R Z d 2 x p gr2 1 R Polyakov quantum effective action

22 More generally, if we also quantize gravity Z S = d 2 x p g(appler )+S m = = Z N d 2 x p Z ḡ d 2 x p gr 1 Z 2 R d 2 x p g apple N b + N R e2 Apparently, for N>25, σ is a ghost! But in fact the theory is healthy and there are no quanta of σ in the physical spectrum, once we impose the physical state conditions in the fundamental theory (David 1988, Distler-Kawai 1989, Polchinski 1989)

23 linearizing the RR model over flat space: D µ (k) = i 2k 2 ( µ + µ µ ) + 1 i i 6 k 2 k 2 m 2 µ, a scalar ghost?? No! The degrees of freedom cannot be read from the quadratic part of a quantum EA! The idea is that this should be the quantum EA of usual Einstein gravity plus matter, which includes massless fields (there are at least the graviton and the photon).

24 4. Is nonlocal gravity equivalent to a scalar-tensor theory? The RR model can be put into local form introducing two auxiliary fields U = 2 1 R, S = 2 1 U then the eqs of motion read G µ = m2 6 K µ (U, S)+8 GT µ 2U = R 2S = U 4 RR = 1 12 m2 m 2 Pl are U and S associated to real quanta? Then U would be a ghost

25 consider again = N Z 48 N = 96 d 2 x p g R Z d 2 x p gr2 1 R we could localize it introducing again By definition U = 2 1 R 2U = R ie 2U =22 The most general solution is U =2 + U hom however, only U=2σ gives back the original action U is fixed in terms of the metric. There are no creation/ annihilation operators associated to U

26 Similarly, consider L = = 1 4 F µ F µ 1 2 m2 A µ A µ 1 4 F µ 1 m 2 2! F µ we could localize it, preserving gauge-invariance, introducing U µ = 2 1 F µ however, there is no new dof associated to U µν The initial conditions on the original fields (metric,a µ ) fix in principle the initial conditions on the auxiliary fields

27 5. If we had a derivation of the RR model from a fundamental action, the inital conditions on the auxiliary fields would be fixed in terms of the initial conditions on the metric. However, we do not have a derivation. So, how do you choose in practice the initial conditions on U,S? in cosmology, at the background level initial conditions at early times are taken to be homogeneous: a priori the space of theories is parametrized by {U(t 0 ), U(t 0 ),S(t 0 ), Ṡ(t 0)} however, for the RR model the cosmological solutions are an attractor in this space : 3 stable directions, one marginal parameter U=u 0 For the Δ 4 model, 4 stable directions

28 at the level of cosmological perturbations, what are the initial conditions on the auxiliary fields? recall that for the Polyakov action in this case, writing U(t, x) U = 2 1 R =2 =Ū(t)+ U(t, x) Ū(t in )= (t in ) U(t in, x) = (t in, x), U(t in, x) = (t in, x) in general, δu, δv must vanish if the metric perturbation vanish (they are not independent dof!)

29 for the RR model we set U(t in, x) = c U (t in, x), U 0 (t in, x) =c U 0 (t in, x), V (t in, x) = c V (t in, x), V 0 (t in, x) =c V 0 (t in, x), (V=H 02 S) and vary c U,c V at a level -10<c U,c V <10 result: very little dependence of the final results Belgacem, Dirian, Foffa, MM, in prep = [ ( )- ( )]/ ( ) =

30 6. How much freedom we have in the choice of the nonlocal term? We have explored several possibilities massive photon: can be described replacing a first guess for a massive deformation of GR could be (Dvali 2006) (Arkani-Hamed, Dimopoulos, Dvali and Gabadadze 2002) however, we lose energy-momentum conservation

31 to preserve energy-momentum conservation: S µ = S T µ (r µs + r S µ ) G µ m 2 ( 1 G µ ) T =8 GT µ (Jaccard,MM, Mitsou, 2013) however, instabilities in the cosmological evolution (Foffa,MM, Mitsou, 2013) G µ m 2 (g µ 1 R) T =8 GT µ (MM 2013) ``RT model'' works very well if started from RD and fits well the data however, cosmological perturbations are unstable if we start from desitter inflation (Belgacem, Cusin, MM and M.Mancarella, 2017, in prep.)

32 a related model: (MM and M.Mancarella, 2014) RR = m2 Pl 2 Z d 4 x p g appler m2 6 R R stable cosmological evolution and stable perturbations in all phases fits very well the data ``RR model'' a more general model = m2 Pl 2 Z d 4 x p g apple R µ 1 R R µ 2C µ C µ µ 3 R µ R µ stability requires µ 2 =µ 3 =0 (Cusin, Foffa, MM and M.Mancarella, 2016)

33 another useful variant Z = m2 Pl d 4 x p 4 2 (`Δ 4 model') not at all easy to construct a nonlocal model that works g apple R 1 6 m2 R 1 4 R 4 = R µ r µ r 2 3 R gµ r µ Rr (Cusin, Foffa, MM and M.Mancarella, 2016) (Belgacem, Foffa, Dirian and MM, in prep.) (viable background evolution, stable cosmological perturbations during RD, MD and also inflation) the models that work correspond to dynamical mass generation for the conformal mode, rather than for the graviton

34 Cosmological background evolution the nonlocal term acts as an effective DE density! define w DE from DE(x)/ wde(z) RR model with u 0 = x z x=ln a DE(x)/ wde(z) Δ 4 model x z a phantom DE equation of state!

35 Cosmological perturbations well-behaved? YES Dirian, Foffa, Khosravi, Kunz, MM JCAP 2014 this step is already non-trivial, see e.g. DGP or bigravity consistent with data? YES comparison with ΛCDM implement the perturbations in a Boltzmann code compute likelihood, χ 2, perform parameter estimation Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2015, 2016 Dirian, 2017

36 Boltzmann code analysis and comparison with data We test the non-local models against Planck 2015 TT, TE, EE and lensing data, isotropic and anisotropic BAO data, JLA supernovae, local H_0 measurements, growth rate data and we perform Bayesian parameter estimation and model comparison we modified the CLASS code and use Montepython MCMC we vary! b = b h 2 0,! c = c h 2 0, H 0, A s, n s, z re In ΛCDM, Ω Λ is a derived parameter, fixed by the flatness condition. Similarly, in our model the mass parameter m 2 is a derived parameter, fixed again from Ω tot =1 we have the same free parameters as in ΛCDM

37 A crucial point: neutrino masses (Dirian 2017) Oscillation and β-decay experiments give 0.06 ev Σ ν m ν 6.6 ev The Planck baseline analysis from ΛCDM fixes Σ ν m ν = 0.06 ev If we Σ ν m ν vary, in ΛCDM we hit the lower bound CMB Planck data, interpreted with ΛCDM, only give an upper bound Σ ν m ν < 0.23 ev Planck coll., Ade et al 2015 The RR and Δ 4 nonlocal models predict a non-zero value of Σ ν m ν the interval 0.06 ev Σ ν m ν 6.6 ev. It is wrong to keep fixed Σ ν m ν = 0.06 ev in the nonlocal models

38 Parameter estimation. Results the most interesting predictions are on H 0 and Σ ν m ν from CMB+BAO+SNae: prediction for neutrino masses (consistent with the limits 0.06 ev Σ ν m ν 6.6 ev) nonlocal models consistent with the value of H 0 obtained by local measurements H 0 =73.02 ± 1.79 (Riess et al ) discrepancy 3.1σ for ΛCDM, 2.0σ for RR, 1.5σ for Δ 4

39 the predictions for Σ ν m ν and H 0 are correlated with CMB+BAO+SNa, RR and ΛCDM have comparable χ 2, while Δ 4 is disfavored currently running chains including also H 0

40 Conclusion: at the phenomenological level, these non-local models are quite satisfying solar system tests OK generates dynamically a dark energy cosmological perturbations work well comparison with CMB,SNe,BAO with modified Boltzmann code ok pass tests of structure formation higher value of H0 Same number of parameters as ΛCDM, and competitive with ΛCDM from the point of view of fitting the data

41 Next step: understanding from first principles where such nonlocal term comes from

42 Thank you!

43 bkup slides

44 growth rate and structure formation

45 Absence of vdvz discontinuity the propagator is continuous for m=0 D µ (k) = i 2k 2 ( µ + µ µ ) + 1 i i 6 k 2 k 2 m 2 µ, write the eqs of motion of the non-local theory in spherical symmetry: for mr <<1: low-mass expansion A. Kehagias and MM 2014 ds 2 = A(r)dt 2 + B(r)dr 2 + r 2 (d 2 +sin 2 d 2 ) for r>>r S : Newtonian limit (perturbation over Minowski) match the solutions for r S << r << m -1 (this fixes all coefficients)

46 result: for r>>r s for r s <<r<< m -1 : m! 0 A(r) = 1 the limit is smooth! r S r B(r) = 1+ r S r A(r) ' 1 r S r apple 1+ 1 (1 cos mr) 3 apple 1 1 (1 cos mr mr sin mr) 3 1+ m2 r 2 6 By comparison, in massive gravity the same computation gives A(r) =1 4 3 vdvz discontinuity r S r 1 r S 12m 4 r 5 breakdown of linearity below r V =(r s /m^4) 1/5

47 consider Background evolution RR = m2 Pl 2 Z d 4 x p g appler m2 6 R R localize using U = 1 R, S = 1 U in FRW we have 3 variables: H(t), U(t), W(t)=H^2(t)S(t). define x=ln a(t), h(x)=h(x)/h 0, γ=(m/3h 0 ) 2 ζ(x)=h'(x)/h(x)

48 h 2 (x) = M e 3x + R e 4x + Y (U, U 0,W,W 0 ) U 00 +(3+ )U 0 = 6(2 + ) W (1 )W 0 2( )W = U there is an effective DE term, with DE (x) = 0 Y (x) 0 =3H 2 0 /(8 G) define w DE from the model has the same number of parameters as ΛCDM, with Ω Λ γ (plus the inital conditions, parametrized by u 0, c U, c W )

49 the perturbations are well-behaved and differ from ΛCDM at a few percent level = [1+µ(a; k)] GR = [1+ (a; k)]( ) GR m S z z DE-related deviations at z=0.5 of order 4% consistent with data: (Ade et al., Planck XV, 2015) µ Planck Planck+BSH Planck+WL Planck+BAO/RSD Planck+WL+BAO/RSD

50 effective Newton constant (RR model) Geff ê G z growth index: ΛCDM d log M (a; k) d ln a =[ M ] (z;k) g our model k= z

51 linear power spectrum 1.07 matter power spectrum compared to ΛCDM PHkL ê PLHkL ê MpcD 10 4 DE clusters but its linear power spectrum is small compared to that of matter ê hl 3 D Matter DE ê MpcD

52 sufficiently close to ΛCDM to be consistent with existing data, but distinctive prediction that can be clearly tested in the near future phantom DE eq of state: w(0)= (RR) + a full prediction for w(z) DES Δw=0.03 EUCLID Δw=0.01 linear structure formation µ(a) =µ s a s! µ s = 0.09, s = 2 Forecast for EUCLID, Δµ=0.01 non-linear structure formation: 10% more massive halos lensing: deviations at a few % Barreira, Li, Hellwing, Baugh, Pascoli 2014