Non-local infrared modifications of gravity and dark energy

Michele Maggiore

Los Cabos, Jan. 2014

based on

M. Jaccard, MM and E. Mitsou, 1305.3034, PR D88 (2013) MM, arXiv: 1307.3898 S. Foffa, MM and E. Mitsou, 1311.3421, 1311.3435

A. Kehagias and MM, in preparation Y. Dirian, S. Foffa, N. Khosravi, M. Kunz, MM, in preparation

we will only give a summary of the main results. Glad to discuss the details during this week!

•  The model:

where we use the fact that any symmetric tensor can be written as

where

Gµ⌫ � d� 1

2dm2

�gµ⌫⇤�1R

�T= 8⇡GTµ⌫

our initial motivation:

introduce a mass m and deform GR in the infrared in a way which is fully covariant and does not require an external reference metric

some sources of inspiration: •  Proca theory

is equivalent to

•  degravitation

S =Rd

4x

⇥� 1

4Fµ⌫Fµ⌫ � 1

2m2�AµA

µ � jµAµ⇤

@µFµ⌫ �m2�A

⌫ = j⌫ !n m2

� @⌫A⌫ = 0

(⇤�m2�)A

µ = 0⇣1� m2

�

⇤⌘@µFµ⌫ = j⌫

(Dvali 2006; Dvali, Hofmann, Khoury 2007; review Hinterbichler 2012)

✓1� m2

⇤

◆Gµ⌫ = 8⇡GTµ⌫

Arkani-Hamed, Dimopoulos, Dvali and Gabadadze 2002

Cosmological consequences

•  consider

•  specialize to FRW in d=3. Define

•  in FRW Sµ= (S0,0) , so we have 3 variables: H(t), U(t), S0(t)

•  a final massaging of the eqs.

and their final form is

•  there is an effective DE term, with

•  define wDE from then:

•  at the level of background evolution, the model has the same number of parameters as ΛCDM, with ΩΛ ↔ γ.

•  results.

•  Fixing γ = 0.05.. (m=0.67 H0) we reproduce ΩDE=0.68

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gYHxL

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x

WiHxL

•  having fixed γ we get a pure prediction for the EOS:

fit w(x)=w0+(1-a) wa in the region -1< x <0

best fit values: w0= -1.042, wa=-0.020

on the phantom side !

(general consequence of together with ρ>0 and dρ/dt>0)

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-1.0

-0.5

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0.5

xwHxL

•  Planck+WP+BAO gives w0 = -1.04 , wa<1.32

•  since we predict |wa|<<1, we can also compare with the Planck prediction for constant w

Planck+WP+SNLS: w0 = -1.13 (95% c.l.)

Planck+WP+Union2.1 w0 = -1.09 ± 0.17 (95% c.l.)

+0.72 -0.69

+0.13 -0.14

Consistency issues •  no issue of ghost-induced vacuum decay. The non-local

eqs must be understood as an effectve classical eq for the in-in quantum expectation values, and not as the eq of motion of a fundamental non-local QFT

•  no vDVZ discontinuity. Solar system tests passed •  No strong classical non-linearities at short distances.

Corrections to the Schwarzschild solutions are 1+O(m2r2)

•  cosmological perturbations are being worked out

Conclusions

•  we have an interesting IR modification of GR

•  and a testable prediction for the dark energy EOS

Thank you!

A locality / gauge-invariance duality for massive gauge fields

•  Example: Proca theory for massive photons

•  non-local formulation (Dvali 2006; Dvali, Hofmann, Khoury 2007)

Stueckelberg trick:

add one field and gain a gauge symmetry

S =Rd

4x

⇥� 1

4Fµ⌫Fµ⌫ � 1

2m2�AµA

µ � jµAµ⇤

m2� @⌫A

⌫ = 0@µFµ⌫ �m2

�A⌫ = j⌫ !

n(⇤�m2

�)Aµ = 0

Aµ ! Aµ + 1m�

@µ'

Aµ ! Aµ � @µ✓ , ' ! '+m�✓

S =Rd

4x

⇥� 1

4Fµ⌫Fµ⌫ � 1

2m2�AµA

µ � 12@µ'@

µ'�m�A

µ@µ'� jµA

µ⇤

@µFµ⌫ = m2

�A⌫ +m�@

⌫'+ j⌫ ,

⇤'+m�@µAµ = 0 .

If we choose the unitary gauge φ=0 we get back to the original formulation of Proca theory (and loose the gauge sym because of gauge fixing).

Instead, keep the gauge sym explicit and integrate out φ using its own equation of motion:

'(x) = �m�⇤�1(@µAµ)

we have explicit gauge invariance for the massive theory, at the price non-locality

•  a sort of duality between explicit gauge-invariance and explicit locality

•  we can fix the gauge and the non-local term disappears (and we are back to Proca eqs.)

•  with hindsight, the Stueckelberg trick was not needed

Substituting in the eq of motion for Aν :

or

(⇤�m2�)A

⌫ =⇣1� m2

�

⇤⌘@⌫@µAµ + j⌫

@µAµ = 0

⇣1� m2

�

⇤⌘@µFµ⌫ = j⌫

the original idea: write the linearize Fierz-Pauli theory in non-local form to preserve linearized diff, and covariantize

SFP = 12

Rd

4x

⇥hµ⌫Eµ⌫,⇢�

h⇢� �m

2(hµ⌫hµ⌫ � h

2)⇤

hµ⌫ ! hµ⌫ + 1m (@µA⌫ + @⌫Aµ) ,Stueckelberg trick:

restores inv under

eq of motion for Aν:

α undetermined (peculiar of FP)

Dvali, Hofmann, Khoury (2007)

•  in QED, we found that a massive deformation of the theory is obtained replacing

•  for gravity, a first guess for a massive deformation of GR could be

however this is not correct since

We would lose energy-momentum conservation.

6 d.o.f: massive spin 2 + 1 scalar ghost linearized Ricci scalar

The field N (which is basically the longitudinal mode of Aµ ) acts as a Lagrange multiplier.

At the linearized level it kills the ghost.

However, if we covariantize the theory, it imposes the condition R=0, which is not present in GR ⇒ a fully covariant vDVZ discontinuity ! (Porrati 2002)

•  to understand the properties of the theory we linearize over Minkowski space: in d spatial dimensions

the corresponding Lagrangian is (only formally! see later)

we add a gauge fixing and we get the propagator

•  no vDVZ discontinuity! •  For m=O(H0), solar system test easily passed. Corrections are O(m2/k2) =10-30 for k=(1 a.u)-1. •  massless graviton + extra contribution to

one massles scalar + one massive scalar ghost ?

The ghost a ghost has in principle two type of effects:

•  at the classical level, it can give rise to runaway solution. for a ghost with m=O(H0) this can even be welcome for

explaining the acceleration of the Universe

•  at the quantum level, it produces a vacuum decay rate, Γ for a ghost interacting gravitationally with local interactions:

!

"

g

"

!

Non-local QFT or classical effective equations?

•  we have directly in the EoM (rather than in the solution). This EoM cannot come from the variation of a Lagrangian. E.g.

•  we can repalce after the variation, as a formal trick to get the EoM from a Lagrangian.

However, any connection to the QFT described by this Lagrangian is lost.

⇤�1ret

�

��(x)

Zdx

0�(x0)(⇤�1

�)(x0) =�

��(x)

Zdx

0dx

00�(x0)G(x0

, x

00)�(x00)

=

Zdx

0[G(x, x0) +G(x0, x)]�(x0)

⇤�1 ! ⇤�1ret

Deser-Waldron 2007, Barvinski 2012

Taking

as the quantum Lagrangian of our problem introduces spurious propagating degrees of freedom. In

the apparent ghost-like pole in the propagator already comes with a retarded prescription. It is not a Feynman propagator and does not describe a propagating dof!

Our non-local EoM is not the classical EoM of a non-local QFT!

EoMs involving emerge from a classical or a quantum averaging of a more fundamental (local) QFT

•  classically, when separating long and short wavelength and integrating out the short wave-length (e.g cosmological perturbation theory, or GWs)

•  in QFT, when computing the effective action that includes the effect of radiative corrections. This provides effective non-local field eqs for

•  the in-in matrix elements satisfy non-local and retarded equations

⇤�1ret

h0|�|0i, h0|gµ⌫ |0i

Jordan 1986, Calzetta-Hu 1987

•  So, we interpret our non-local eqs as a classical, effective equation, derived from a more fundamental local theory by a classical or quantum averaging

•  any problem of quantum vacuum stability can only be addressed in this fundamental theory

•  in any case, the apparent ghost that we found has nothing to do with quantum vacuum decay in our model.

It can however trigger classical cosmological instabilities

A fake ghost in massless GR

S

(2)EH =

1

2

Zd

d+1xhµ⌫Eµ⌫,⇢�

h⇢�

hµ⌫ = hTTµ⌫ +

1

2(@µ✏⌫ + @⌫✏µ) +

1

d⌘µ⌫s

S

(2)EH =

1

2

Zd

d+1x

h

TTµ⌫ ⇤(hµ⌫)TT � d� 1

d

s⇤s

�

It looks as if there are many more propagating d.o.f Furthermore s seems a ghost !

Sint =

2

Zd

d+1xhµ⌫T

µ⌫ =

2

Zd

d+1x

h

TTµ⌫ (T

µ⌫)TT +1

d

sT

�

⇤hTTµ⌫ = �

2TTTµ⌫ , ⇤s =

2(d� 1)T

•  the origin of the problem is that s is a non-local function of hµν :

•  example:

it looks as if we have generated a dynamical dof! However, the solution of the homogeneous eq are spurious!

the same happens for s: s is non-radiative, and we must discard the solutions of the homogeneous eq

•  at the quantum level, no annihilation/creation operators associated to it; s cannot be put on the external lines (otherwise, the vacuum in GR would decay!)

s =

✓⌘µ⌫ � 1

⇤@µ@⌫

◆hµ⌫ = Pµ⌫hµ⌫

r2� = ⇢

� ⌘ ⇤�1� ⇤� = r�2⇢ ⌘ ⇢

⇤s = 0

•  the same happens in our non-local theory. The extra term in

is just a mass term for s ! However, it remains a non-radiative field, as in GR, and we must discard the plane-wave solutions of

again, no propagating dof associated to s, and no issue of quantum vacuum decay !

L2 =1

2hµ⌫Eµ⌫,⇢�h⇢� � d� 1

2dm2(Pµ⌫hµ⌫)

2

=1

2

hTTµ⌫ ⇤(hµ⌫)TT � d� 1

ds(⇤+m2)s

�

(⇤+m2)s = 2(d�1)T ,

•  the local formulation introduces a spurious degrees of freedom, given by Uhom(x). We must be aware that, given a definition of

the original non-local model, Uhom(x) is fixed (so also the initial conditions on U are fixed)

•  in flat space, this spurious degree of freedom gives rise to plane waves, but there are no creation/annihilation operators associated to it

(it is another way to see that the apparent ghost is non-dynamical)

•  the stability problem is different in the original non-local formulation and in the local one

•  a degravitation mechanism

the localformulation of the model is invariant under

it is a sort of degravitation mechanism where a cosmological constant can be traded for the initial condition u0. However, this transformation connect different non-local theories

Gµ⌫ +d� 1

2dm2(Ugµ⌫)

T = 8⇡GTµ⌫ ,

�⇤U = R

gµ⌫ ! gµ⌫ , U ! U + c⇤ , Tµ⌫ ! Tµ⌫ + ⇤8⇡Ggµ⌫

with c⇤ ⌘ [2d/(d� 1)](⇤/m2)

A more general class of models

•  when we write we must define In FRW me must invert

even after choosing the retarded Green's function we have the freedom of specifying the solutions of

the possible definitions of the non-local model are in one-to-one correspondence with the initial conditions on U, S0

�gµ⌫⇤�1

retR�T

⇤�1ret and (...)T

⇤U = 0, DS0 = 0

�⇤U = R, �⇤ = @20 + dH@0

DS0 = U , D = @20 + dH@0 � dH2

•  the initial conditions on U,S0 are fixed by the definition of the non-local operators. In general, the solution of is

we chose G=Gret but we must also specify fhom In FRW, and we define

(with t* in RD), i.e. we choose Uhom=0. Then

Later we will study the most general definition

⇤f = j

f(x) = (⇤�1

j)(x) ⌘ f

hom

(x) +

Zd

d+1

x

0p

�g(x0)G(x;x0)j(x0)

⇤f = �a�d@0(ad@0f)

Deser-Waldron 2007

U(t⇤) = U(t⇤) = 0, S0(t⇤) = S0(t⇤) = 0

U(t) ⌘ �(⇤�1retR)(t) ⌘ �

Z t

t⇤

dt01

ad(t0)

Z t0

t⇤

dt00 ad(t00)R(t00)

•  the space of initial conditions: recall that

where

in the early Universe we can neglect the contribution of Y to ζ and write ζ(x)≃ ζ0 (-2 in RD, -3/2 in MD)

trade {U,U',Y,Y'}(xin) for {u0,u1,a1,a2}

u0= marginal direction u1,a1,a2 =decaying directions

Y (x) = � 2(2 + ⇣0)⇣0(3 + ⇣0)(1 + ⇣0)

+6(2 + ⇣0)

3 + ⇣0x+ u0 �

6(2 + ⇣0)u1

2⇣20 + 3⇣0 � 3e

�(3+⇣0)x

+a1e↵+x + a2e

↵�x ,

•  we can forget about u1, a1,a2 and the most general modification of the model amounts to including u0.

This means that we now define

inserting this into

we get a cosmological constant!!

in d=3,

⇤ = [(d� 1)/2d]m2u0

⌦⇤ = �u0

U(t) ⌘ �(⇤�1retR)(t) ⌘ u0 �

Z t

t⇤

dt01

ad(t0)

Z t0

t⇤

dt00 ad(t00)R(t00)

•  effect of u0 on the evolution:

(w0,wa) shift toward their ΛCDM values (-1,0), but w0 is always on the phantom side

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u0=4 u0=400

u0= -10 for u0 < uc ≃ -12, no solution with γ Y(0) =0.68

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•  the results for this class of models can be summarized by:

•  w0 always on the phantom side: -1.33 < w0 < -1 •  prediction of a precise relation wa = wa (w0)

•  EUCLID nominal target: Δw0=0.01, Δwa=0.1

¯

-1.14-1.12-1.10-1.08-1.06-1.04-1.02-1.00-0.20

-0.15

-0.10

-0.05

0.00

w0

wa

future directions

•  understanding what fundamental theory is behind these effective classical eqs

•  cosmological perturbations

•  structure formation