Nonlocal gravity. Conceptual aspects and

cosmological consequences

Michele Maggiore

Sifnos, Sept. 18-23, 2017

based on Jaccard, MM, Mitsou, PRD 2013, 1305.3034 MM, PRD 2014, 1307.3898 Foffa, MM, Mitsou, PLB 2014, 1311.3421 Foffa, MM, Mitsou, IJMPA 2014, 1311.3435 Kehagias and MM, JHEP 2014, 1401.8289 MM and Mancarella, PRD 2014, 1402.0448 Dirian, Foffa, Khosravi, Kunz, MM, JCAP 2014, 1403.6068 Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2015, 1411.7692 MM PRD 2016, 1603.01515 Cusin, Foffa, MM, PRD 2016, 1512.06373 Cusin, Foffa, MM, Mancarella, PRD 2016, 1602.01078 Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2016, 1602.03558 MM (review) Springer, 1606.08784 Dirian, 1704.04075 Belgacem, Dirian, Foffa, MM, in prep.

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

•  Comparison with cosmological data

– background evolution and dark energy

– cosmological perturbations

– Bayesian parameter estimation and model comparison

Nonlocality and the quantum effective action At the fundamental level, the action in QFT is local However, the quantum effective action is nonlocal

quantum effective action:

eiW [J] ⌘Z

D' eiS[']+iRJ'

�W [J ]

�J(x)= h0|'(x)|0iJ ⌘ �[J ]

�[�] ⌘ W [J ]�Z

�J

- 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

��[�]

��(x)= �J(x)

the quantum EA gives the exact eqs of motion for the vev, which include the quantum corrections

ei�[�] =

ZD' eiS[�+']�i

R ��[�]�� '

•  light particles nonlocalities in the quantum effective action these nonolocalities are well understood in the UV. E.g. in QED

it is just the running of the coupling constant in coordinate space Note: we are not integrating out light particles from the theory!

1

e2(2)' 1

e2(µ)� �0 log

✓�2

µ2

◆

log

✓�2

µ2

◆⌘

Z 1

0dm2

1

m2+ µ2

� 1

m2 �2

�

$

�QED[Aµ] = �1

4

Zd

4x

Fµ⌫

1

e

2(2)F

µ⌫ +O(F 4)

�+ fermionic terms

The quantum effective action is especially useful in GR It gives the exact Einstein eqs including quantum matter loops Γ = SEH+ Γm is an action that, used at tree level, give the eqs of motion that include the quantum effects

Tµ⌫ =2p�g

�Sm

�gµ⌫

Gµ⌫ = 8⇡G h0in|Tµ⌫ |0ini

h0|Tµ⌫ |0i =2p�g

��m

�gµ⌫

(vacumm quantum EA. We can also retain the vev's of the matter fields ith the Legendre transform)

ei�m[gµ⌫ ] =

ZD' eiSm[gµ⌫ ;']

The quantum effective action in GR can be computed perturbatively in an expansion in the curvature using heat-kernel techniques The form factors due to a matter field of mass ms are known in closed form For ms<<E For ms>>E

� =m

2Pl

2

Zd

4x

p�g R+

Zd

4x

p�g

RkR(2)R+

1

2Cµ⌫⇢�kW (2)Cµ⌫⇢�

�

k(2) ' k(µ)� �k log

✓�2

µ2

◆+ c1

m2s

2+ c2

m4s

22+ . . .

Gorbar-Shapiro 2003

Barvinsky-Vilkovisky 1985,1987

k(2) ' O(2/m2s)

However, these corrections are only relevant in the UV (ie for quantum gravity) and not in the IR (cosmology): unavoidable, since these are one-loop corrections, and we pay a factor 1/mPl

2

For application to cosmology, we rather need some strong IR effect

R log(�2/m2s)R ⌧ m2

PlR R ⇠ m2Plunless

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, Rajaraman 2010,....

the graviton propagator has exactly the same IR divergences Antoniadis and Mottola 1986,....

•  quantum stability of dS. Decades of controversies....

m2dyn / H2

p�

•  Euclidean lattice gravity suggests dynamical generation of a mass m, and a running of GN

•  non-perturbative functional RG techniques find interesting fixed-point structure in the IR, and strong-gravity effects

the dynamical emergences of a mass scale in the IR in gravity is a meaningful working hypothesis

G(k2) = GN

"1 +

✓m2

k2

◆ 12⌫

+O✓m2

k2

◆ 1⌫

#

Hamber 1999, .., 2017

Wetterich 2017

ν≈1/3

Gauge-invariant (or diff-invariant) mass terms can be obtained with nonlocal operators

eg massive electrodynamics Dvali 2006

in the gauge a nonlocal but gauge-inv photon mass

� = �1

4

Zd

4x

✓Fµ⌫F

µ⌫ �m

2�Fµ⌫

1

2F

µ⌫

◆

@µAµ = 0

1

4m2

�Fµ⌫1

2Fµ⌫ =

1

2m2

�AµAµ

•  Numerical results on the gluon propagator from lattice QCD and OPE are reproduced by adding to the quantum effective action a term

(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

m

2g

2Tr

Zd

4xFµ⌫

1

2F

µ⌫

Fµ⌫ = F aµ⌫T

a , 2ab = Dacµ Dµ,cb , Dab

µ = �ab@µ � gfabcAcµ

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

Our prototype model will be MM and M.Mancarella 2014

•  ΛRR generated dynamically, analog to ΛQCD

• 

is a mass term for the conformal mode! •  ΛRR is the fundamental scale. No ultralight particle provides a solution to the naturalness problem

g

µ⌫

(x) = e

2�(x)⌘

µ⌫

! R = �62� +O(�2)

``RR model'' �RR =

Zd

4x

p�g

m

2Pl

2R�R

⇤4RR

22R

�

⇤4RR ⇠ m2m2

Pl m ⇠ H0 ! ⇤RR ⇠ meV

FAQ 1. Is the theory causal? in a fundamental QFT nonlocality è acausality. E.g. the quantum EA generates the eqs of motion of

�

��(x)

Zdx

0�(x0)(2�1

�)(x0) =�

��(x)

Zdx

0dx

00�(x0)G(x0;x00)�(x00)

=

Zdx

0[G(x;x0) +G(x0;x)]�(x0)

h0|gµ⌫ |0i

h0out

|gµ⌫ |0ini Feynman path integral, acausal eqs

h0in|gµ⌫ |0ini Schwinger-Keldish path integral, nonlocal but causal eqs

2. What is the domain of validity of the effective theory?

Are you integrating out massless particles?

The eqs

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

h0|Tµ⌫ |0i =2p�g

��m

�gµ⌫

ei�m[gµ⌫ ] =

ZD' eiSm[gµ⌫ ;']

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)

•  consider gravity + N conformal matter fields in D=2

conformal anomaly: this result is exact. Only the 1-loop term contributes. In D=2:

in D=2, Γ[0]=0. This is the exact quantum EA

h0|T aa |0i =

N

24⇡R

gµ⌫ = e2�⌘µ⌫

��

��= � N

12⇡2�

R = �22�

�[�]� �[� = 0] = � N

24⇡

Zd

2x e

2��2�

we can covariantize using we get

Polyakov quantum effective action

p�g = e2�

� =N

48⇡

Zd

2x

p�g �R

= � N

96⇡

Zd

2x

p�g R2�1

R

�[�] = � N

24⇡

Zd

2x e

2��2�

R = �22�

More generally, if we also quantize gravity

S =

Zd

2x

p�g(R� �) + Sm

� = �N � 25

96⇡

Zd

2x

p�g R

1

2R� �

Zd

2x

p�g

=

Zd

2x

p�g

N � 25

24⇡g

ab@a�@b� +

N � 25

24⇡R� � �e

2�

�

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)

•  linearizing the RR model over flat space: 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).

Dµ⌫⇢�(k) =i

2k2(⌘µ⇢⌘⌫� + ⌘µ�⌘⌫⇢ � ⌘µ⌫⌘⇢�)

+1

6

✓i

k2� i

k2 �m2

◆⌘µ⌫⌘⇢� ,

a scalar ghost??

4. Is nonlocal gravity equivalent to a scalar-tensor theory?

The RR model can be put into local form introducing two auxiliary fields then the eqs of motion read are U and S associated to real quanta? Then U would be a ghost

U = �2�1R , S = �2�1U

Gµ⌫ =m2

6Kµ⌫(U, S) + 8⇡GTµ⌫

2U = �R

2S = �U

⇤4RR =

1

12m2m2

Pl

consider again

we could localize it introducing again

By definition ie The most general solution is 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

� =N

48⇡

Zd

2x

p�g �R

= � N

96⇡

Zd

2x

p�g R2�1

R

U = �2�1R

2U = �R 2U = 22�

U = 2� + Uhom

Similarly, consider

we could localize it, preserving gauge-invariance, introducing however, there is no new dof associated to Uµν

The initial conditions on the original fields (metric,Aµ) fix in principle the initial conditions on the auxiliary fields

L = �1

4Fµ⌫F

µ⌫ � 1

2m2

�AµAµ

= �1

4Fµ⌫

1�

m2�

2

!Fµ⌫

Uµ⌫ = �2�1Fµ⌫

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

however, for the RR model the cosmological solutions are an attractor in this space : 3 stable directions, one marginal parameter U=u0 For the Δ4 model, 4 stable directions

{U(t0), U(t0), S(t0), S(t0)}

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 in general, δU, δV must vanish if the metric perturbation vanish (they are not independent dof!)

U = �2�1R = 2�U(t,x) = U(t) + �U(t,x)

�U(tin,x) = ��(tin,x) , �U(tin,x) = ��(tin,x)

U(tin) = �(tin)

for the RR model we set (V=H0

2S) and vary cU,cV at a level -10<cU,cV <10 result: very little dependence of the final results

U(tin,x) = cU�(tin,x) , U 0(tin,x) = cU�0(tin,x) ,

V (tin,x) = cV �(tin,x) , V 0(tin,x) = cV �0(tin,x) ,

-15 -10 -5 0-0.00002

-0.00001

0

0.00001

0.00002

�� �

[�(��)-�(�)]/�(�) �=�

Belgacem, Dirian, Foffa, MM, in prep

-15 -10 -5 0 5-0.002

-0.001

0.000

0.001

0.002

�� �

��

�=�

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

however, we lose energy-momentum conservation

(Dvali 2006)

(Arkani-Hamed, Dimopoulos, Dvali and Gabadadze 2002)

•  to preserve energy-momentum conservation:

however, instabilities in the cosmological evolution •  works very well if started from RD and fits well the data however, cosmological perturbations are unstable if we start from deSitter inflation

Gµ⌫ �m2(⇤�1Gµ⌫)T = 8⇡GTµ⌫

(Jaccard,MM, Mitsou, 2013)

(Foffa,MM, Mitsou, 2013)

Gµ⌫ �m2(gµ⌫⇤�1R)T = 8⇡GTµ⌫ (MM 2013)

Sµ⌫ = STµ⌫ +

1

2(rµS⌫ +r⌫Sµ)

(Belgacem, Cusin, MM and M.Mancarella, 2017, in prep.)

``RT model''

•  a related model:

–  stable cosmological evolution and stable perturbations in all phases

–  fits very well the data

•  a more general model

stability requires µ2=µ3=0

``RR model''

�RR =m

2Pl

2

Zd

4x

p�g

R� m

2

6R

1

22R

�(MM and M.Mancarella, 2014)

� =m

2Pl

2

Zd

4x

p�g

R� µ1R

1

22R� µ2C

µ⌫⇢� 1

22Cµ⌫⇢� � µ3R

µ⌫ 1

22Rµ⌫

�

(Cusin, Foffa, MM and M.Mancarella, 2016)

•  another useful variant (`Δ4 model')

•  not at all easy to construct a nonlocal model that works (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

��4 =m

2Pl

2

Zd

4x

p�g

R� 1

6m

2R

1

�4R

�

�4 = 22 + 2Rµ⌫rµr⌫ � 2

3R2+

1

3gµ⌫rµRr⌫

(Cusin, Foffa, MM and M.Mancarella, 2016) (Belgacem, Foffa, Dirian and MM, in prep.)

Cosmological background evolution •  the nonlocal term acts as an effective DE density! define wDE from

-8 -6 -4 -2 00.0

0.2

0.4

0.6

0.8

1.0

1.2

x

� DE(x)/�

0

0 2 4 6 8 10-1.20

-1.15

-1.10

-1.05

-1.00

zwDE(z)

RR model with u0=0

-8 -6 -4 -2 00.0

0.2

0.4

0.6

0.8

x

� DE(x)/�

0

x=ln a

0 2 4 6 8 10-1.4

-1.3

-1.2

-1.1

-1.0

-0.9

-0.8

z

wDE(z)

Δ4 model

a phantom DE equation of state !

Cosmological perturbations •  well-behaved? YES

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, χ2, perform parameter estimation

Dirian, Foffa, Khosravi, Kunz, MM JCAP 2014

Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2015, 2016 Dirian, 2017

•  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

In ΛCDM, ΩΛ is a derived parameter, fixed by the flatness condition. Similarly, in our model the mass parameter m2 is a derived parameter, fixed again from Ωtot=1

we have the same free parameters as in ΛCDM

!b = ⌦bh20, !c = ⌦ch

20, H0, As, ns, zre

Boltzmann code analysis and comparison with data

•  A crucial point: neutrino masses (Dirian 2017)

Oscillation and β-decay experiments give 0.06 eV≤ Σν mν ≤ 6.6 eV The Planck baseline analysis from ΛCDM fixes Σν mν = 0.06 eV If we Σν mν vary, in ΛCDM we hit the lower bound CMB Planck data, interpreted with ΛCDM, only give an upper bound Σν mν < 0.23 eV The RR and Δ4 nonlocal models predict a non-zero value of Σν mν the interval 0.06 eV≤ Σν mν ≤ 6.6 eV. It is wrong to keep fixed Σν mν = 0.06 eV in the nonlocal models

Planck coll., Ade et al 2015

•  Parameter estimation. Results the most interesting predictions are on H0 and Σν mν from CMB+BAO+SNae:

•  prediction for neutrino masses (consistent with the limits 0.06 eV≤ Σν mν ≤ 6.6 eV)

•  nonlocal models consistent with the value of H0 obtained by local measurements H0 =73.02 ± 1.79 (Riess et al 1604.01424)

discrepancy 3.1σ for ΛCDM, 2.0σ for RR, 1.5σ for Δ4

•  the predictions for Σν mν and H0 are correlated

•  with CMB+BAO+SNa, RR and ΛCDM have comparable χ2, while Δ4 is disfavored •  currently running chains including also H0

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

Next step: understanding from first principles where such non-

local term comes from

Thank you!

bkup slides

growth rate and structure formation

Absence of vDVZ discontinuity

•  the propagator is continuous for m=0

•  write the eqs of motion of the non-local theory in spherical symmetry:

for mr <<1: low-mass expansion for r>>rS: Newtonian limit (perturbation over Minowski) match the solutions for rS<< r << m-1 (this fixes all coefficients)

A. Kehagias and MM 2014

ds2 = �A(r)dt2 +B(r)dr2 + r2(d✓2 + sin2 ✓ d�2)

Dµ⌫⇢�(k) =i

2k2(⌘µ⇢⌘⌫� + ⌘µ�⌘⌫⇢ � ⌘µ⌫⌘⇢�)

+1

6

✓i

k2� i

k2 �m2

◆⌘µ⌫⌘⇢� ,

•  result: for r>>rs

the limit is smooth ! By comparison, in massive gravity the same computation gives

A(r) = 1� rSr

1 +

1

3

(1� cosmr)

�

B(r) = 1 +

rSr

1� 1

3

(1� cosmr �mr sinmr)

�

A(r) ' 1� rSr

✓1 +

m2r2

6

◆

m ! 0

A(r) = 1� 4

3

rSr

⇣1� rS

12m4r5

⌘

vDVZ discontinuity breakdown of linearity below rV=(rs/m^4)1/5

for rs<<r<< m-1:

Background evolution

•  consider

localize using 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)/H0 , γ=(m/3H0)2 ζ(x)=h'(x)/h(x)

U = �⇤�1R , S = �⇤�1U

�RR =m

2Pl

2

Zd

4x

p�g

R� m

2

6R

1

22R

�

•  there is an effective DE term, with

•  define wDE from •  the model has the same number of parameters as ΛCDM, with ΩΛ ↔ γ (plus the inital conditions, parametrized by u0, cU, cW)

⇢0 = 3H20/(8⇡G)

⇢DE(x) = ⇢0�Y (x)

h

2(x) = ⌦M

e

�3x + ⌦R

e

�4x + �Y (U,U 0,W,W

0)

U

00 + (3 + ⇣)U 0 = 6(2 + ⇣)

W

00 + 3(1� ⇣)W 0 � 2(⇣ 0 + 3⇣ � ⇣

2)W = U

•  the perturbations are well-behaved and differ from ΛCDM at a few percent level = [1 + µ(a; k)] GR

� � = [1 + ⌃(a; k)]( � �)GR

0.0 0.5 1.0 1.5 2.00.00

0.02

0.04

0.06

0.08

0.10

z

m

•  deviations at z=0.5 of order 4% •  consistent with data: (Ade et al., Planck XV, 2015)

0.00 0.25 0.50 0.75 1.00

⌃0 � 1

�0.8

0.0

0.8

1.6µ

0�

1

DE-related

Planck

Planck+BSH

Planck+WL

Planck+BAO/RSD

Planck+WL+BAO/RSD

0.0 0.5 1.0 1.5 2.00.00

0.02

0.04

0.06

0.08

0.10

zS

0.0 0.5 1.0 1.5 2.0

0.530

0.535

0.540

0.545

0.550

0.555

z

g

k=5

growth index: ΛCDM

our model

effective Newton constant (RR model)

d log �M (a; k)

d ln a= [⌦M ]

�(z;k)

0.0 0.5 1.0 1.5 2.01.00

1.01

1.02

1.03

1.04

1.05

1.06

z

GeffêG

•  linear power spectrum

0.001 0.01 0.1 1 101.04

1.045

1.05

1.055

1.06

1.065

1.07

k @h êMpcDPHkLêP LHkL

0.001 0.01 0.1 1 1010-4

0.01

1

100

104

k @h êMpcD

PHkL@HM

pcêhL3D

Matter

DE

DE clusters but its linear power spectrum is small compared to that of matter

matter power spectrum compared to ΛCDM

•  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)= - 1.14 (RR) + a full prediction for w(z) •  DES Δw=0.03 •  EUCLID Δw=0.01

–  linear structure formation

•  Forecast for EUCLID, Δµ=0.01

–  non-linear structure formation: 10% more massive halos

–  lensing: deviations at a few %

µ(a) = µsas ! µs = 0.09, s = 2

Barreira, Li, Hellwing, Baugh, Pascoli 2014