Nonlocal gravity. Conceptual aspects and cosmological consequences

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1 Nonlocal gravity. Conceptual aspects and cosmological consequences Michele Maggiore Moriond 2018

2 based on recent works with Enis Belgacem, Giulia Cusin, Yves Dirian, Stefano Foffa, Martin Kunz, Michele Mancarella, Valeria Pettorino see Belgacem, Dirian, Foffa, MM, , JCAP for a recent detailed review

3 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

4 [ ] (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

5 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!

6 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

7 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)

8 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

9 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...

10 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

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

12 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

13 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

14 Our prototype model is MM and M.Mancarella 2014 RR = m2 Pl 2 Z d 4 x p g apple R 1 6 m2 R R ``RR model'' m generated dynamically, replaces Λ g µ (x) =e 2 (x) µ! R = 62 + O( 2 ) is a mass term for the conformal mode!

15 Cosmological background evolution the nonlocal term acts as an effective DE density! DE(x)/ wde(z) x ln a z define w DE from a phantom DE equation of state!

16 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 Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2015, 2016 Dirian, 2017 Belgacem, Dirian, Foffa, MM 2017 implement the perturbations in a Boltzmann code compute likelihood, χ 2, perform parameter estimation

17 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

18 overall χ 2 statistically equivalent to ΛCDM higher value of H 0 prediction for neutrino masses forecasts for surveys with the features expected from Euclid, DESI, SKA are under way (Casas, Dirian, Maierotto, Kunz, MM, Pettorino)

19 GW propagation h 00 A +2H[1 ( )] h 0 A + k 2 ha =0 c GW =c ok with GW the ``GW luminosity distance" is different from the standard (electromagnetic) luminosity distance Z z d gw em dz 0 L (z) =dl (z)exp 1+z 0 (z 0 ) 0

20 d L gw /dl em z at ET and LISA this propagation effect dominates over that from the dark energy EoS! (after taking into account parameter estimation)

21 dl/dl z non-local model dl/dl wcdm, w= z

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

23 Thank you!

24 bkup slides

25 growth rate and structure formation 0.50 f(z) 8 (z) z z

26 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)

27 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

28 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)

29 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 )

30 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

31 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

32 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

33 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

34 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

35 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

36 A closer example: = 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