Lorentz violation in gravity

w/ B. Audren, E. Barausse, M. Ivanov, J. Lesgourgues, O. Pujolàs, S. Sibiryakov, K. Yagi, N. Yunes

Diego Blas

1412.4828 [gr-qc] Review (w/ E. Lim)

Gravitational theories landspace

General Relativity

Minimal, Unitary, Lorentz invariant,Massless, Metric, EFT, Local, 4D

Finely tuned dark sector

‘pheno' scale

Exte

nsio

n 2

Loren

tz vio

lation

Incomplete at high E (QG) or high-densities (BH)

✓ quantum gravity (Hořava gravity, CDT,…)✓ new ideas for black holes✓ Natural alternative to ΛCDM and DM✓ Only known Higgs mechanism for massive gravitons

benefits of violating LI in gravity:

Lorentz Violation Valley

Lorentz invariance (LI) is a key ingredient ofParticle Physics, Gravity, Dark Sector

is it necessary? which are the bounds?

✓ quantum gravity (Hořava gravity, CDT,…)✓ new ideas for black holes✓ Natural alternative to ΛCDM and DM✓ Only known Higgs mechanism for massive gravitons

benefits of violating LI in gravity:

Lorentz Violation Valley

Lorentz invariance (LI) is a key ingredient ofParticle Physics, Gravity, Dark Sector

is it necessary? which are the bounds?

. 10�20

I will only discuss the deviations in the gravity sector!

⇠ 0

Einstein-aether

There is a preferred frame at each point of the space-time set by a dynamical unit vector - aether

Jacobson, Mattingly, 2000

uµ

S = �M2P

2

⇤d4x⇥�g

�R + Kµ�

⇤⇥⇤µu⇤⇤�u⇥ + l(uµuµ � 1)⇥

Kµ�⇤⇥ � c1g

µ�g⇤⇥ + c2�µ⇤��

⇥ + c3�µ⇥ ��

⇤ + c4uµu�g⇤⇥

Lagrange multiplier:enforces unit norm

Space-time filled by a preferred time directionAssociated to a time-like unit vector uµ

Generic: Einstein-æther

Scalar-vector

uµuµ = 1

Hypersurface orthogonal:Khronometric

x

µ

' = '0' = '1

uµ ⌘ @µ'p@↵'@↵'

khronon

Jacobson, Mattingly 01

The preferred frame case

tx

' 7! f(')

DB, Pujolas, Sibiryakov 09Horava 09

massless spin 2 graviton extra massless scalar with

,

,

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

uµuµ = 1

Jacobson, Mattingly 01

c2t =1

1� �!2 = c2t k

2

!2 = c2� k2

Ingredients: uµ , gµ⌫

Khronometric

' = t+ �

extra vector polarizations

Einstein-æther (generic ): extra term uµ

c2� =� + �

↵

uµ ⌘ @µ'p@↵'@↵'

uµ = uµ + �uµ

�rµu⌫rµu⌫

Gravitational Lagrangian (IR)

' 7! f(')

DB, Pujolas, Sibiryakov 09

massless spin 2 graviton extra massless scalar with

,

,

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

uµuµ = 1

Jacobson, Mattingly 01

c2t =1

1� �!2 = c2t k

2

!2 = c2� k2

Ingredients: uµ , gµ⌫

Khronometric

' = t+ �

extra vector polarizations

Einstein-æther (generic ): extra term uµ

c2� =� + �

↵

uµ ⌘ @µ'p@↵'@↵'

uµ = uµ + �uµ

�rµu⌫rµu⌫

Gravitational Lagrangian (IR)

' 7! f(')

DB, Pujolas, Sibiryakov 09

+1

Md�4?

Od>4(uµ, gµ⌫)UV:

EFT with cut-off ⇤IR ⇠ p

↵MP

M? . ⇤IRwith

massless spin 2 graviton extra massless scalar with

,

,

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

uµuµ = 1

Jacobson, Mattingly 01

c2t =1

1� �!2 = c2t k

2

!2 = c2� k2

Ingredients: uµ , gµ⌫

Khronometric

' = t+ �

extra vector polarizations

Einstein-æther (generic ): extra term uµ

c2� =� + �

↵

uµ ⌘ @µ'p@↵'@↵'

uµ = uµ + �uµ

�rµu⌫rµu⌫

Gravitational Lagrangian (IR)

' 7! f(')

DB, Pujolas, Sibiryakov 09Possible UV completion:

Hořava gravity

+1

Md�4?

Od>4(uµ, gµ⌫)UV:

EFT with cut-off ⇤IR ⇠ p

↵MP

M? . ⇤IRwith

�R

L = M2PN

p�⇣KijK

ij � (1� �0)(�ijKij)2

| {z }@20

�(1� �0)(3)R+ ↵0 aiai...+

�2(3)R

M4?

⌘

Unitary gauge: ' = t

UV� = � mpp

8⇡M2PR

⇥1� e�M?R

cos (M?R)

⇤

Kij ⇠@�ij@t

⇠ ! �ij

Renormalizable UV: Hořava Gravity

ds2 = N

2dt2 � �ij(dxi +N

idt)(dxj +N

jdt),ai ⌘ @ilogN,

� = � mpp

8⇡M20R

h1 + ↵ e�R/�

i

Finite andfinite derivatives!! M�1

? & µm ?

DB, Lim 14

(3)Rijkl ⇠ k2�ij,

stability, no ghostsno gravitational Cherenkov

Theoretical

IR phenomenology I (Kh)

massless spin 2 graviton extra massless scalar with

,

,

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

c2t =1

1� �!2 = c2t k

2

!2 = c2� k2' = t+ � c2� =� + �

↵

Gravitational Lagrangian (KH IR)

' 7! f(')

stability, no ghostsno gravitational Cherenkov

0 < ↵ < 2c2� > 0, c2t > 0

c2t � 1, c2� � 1

Theoretical,

IR phenomenology: TH & Solar System (Kh)DB, Pujolas, Sibiryakov 10

stability, no ghostsno gravitational Cherenkov

0 < ↵ < 2c2� > 0, c2t > 0

c2t � 1, c2� � 1

Theoretical,

�

��!v

�!r

�!v ⇠ 10�2 ↵PPN1 = �4(↵� 2�)

↵PPN2 =

(↵� 2�)(↵� �� 3�)

2(�+ �). 10�7

. 10�4

h00 = �2GNM

r

✓1� (↵PPN

1 � ↵

PPN2 )v2

2� ↵

PPN2

2

(xiv

i)2

r

2

◆

h0i =↵

PPN1

2GN

m

r

v

i

Solar system (PN gravity)

↵ = 2� identical to GR in the Solar System!

GN ⌘ 1

8⇡M2P (1� ↵/2)

Will 05

IR phenomenology: TH & Solar System (Kh)DB, Pujolas, Sibiryakov 10

Matter forces are not modifiedGravitation modified (couplingbetween gravitons and )

Violation of strong equivalence principle (SEP) (Nordtvedt effect)

effectively, for strong gravity regimesthis produces a coupling matter-æther for point particles!

g p

the orbital equations depend on uµvµ

Spp = �m

Zds f(uµv

µ)Spp = �m

Zds

uµ vµ

staruµ

Strongly gravitating bodies

tFoster 07

Yagi, DB, Yunes, Barausse 13

SEP violation : dipolar radiation expected

⌃i ⌘Z

d

3x⇢x

i

SEP violated: the conserved momentum does not correspond to

The dipole mode can be seen in interferometers or in the evolution of binaries

(similar phenomenon in scalar-tensor)

,

Eardley, Will 70s

h ⇠ G

c3d

dt

⌃i

r⇠ G

c3Pi

r

h ⇠ G

c3

˙Pi

r

Pi = m1vi1 + m2v

i2

Damour, Esposito-Farese 92

Dipolar radiation

+O(1/c4)GR

0.0 0.2 0.4 0.6 0.8 1.0-10

-5

0

5

10

b

l 0 0.040.020

0.1

0.05

StabilityêCherenkovBinary pulsarsBBN

�

�

Combined constraints from WD-NS and NS-NS systemsPSR J1141-6545, PSR J0348+0432, PSR J0737-3039

Gravitational Radiation Constraints (Kh)

Yagi, DB, Yunes, Barausse 13

(Solar system constraints)

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

Lm = LLI(SM,DM,⇤, gµ⌫) + SM LLV (SM, gµ⌫ , uµ)

+DM LLV (DM, gµ⌫ , uµ) + DE LLV (DE, gµ⌫ , uµ)

LV in late cosmology

Natural dark energy ΘCDM DB, Sibiryakov 11

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

Lm = LLI(SM,DM,⇤, gµ⌫) + SM LLV (SM, gµ⌫ , uµ)

+DM LLV (DM, gµ⌫ , uµ) + DE LLV (DE, gµ⌫ , uµ)

LV in late cosmology

Natural dark energy ΘCDM DB, Sibiryakov 11

LV effects from the coupling to :

gravitons & DM gravitate differently: no equivalence principle and enhanced collapse!

(i) background modifies the inertial mass and with (ii) new interaction from

uµ = uµ + �uµ

uµ

�uµ

COSMOLOGÍA

2003

WMAP mejoró la precisión de las observaciones del CMB

COBE(7 degree resolution)

WMAP(0.25 degree resolution)

COBE(resolución de 7 grados)

WMAP(resolución de .25 grados)

(analogía con resolución en mapas terrestres)

Fondo de radiación de microondas

La anisotropía tiene una estructura granular.

La escala característica del “grano” es 300.000 años luz,

el tamaño del Universo observable en la época de desacoplamiento

H2 =8⇡

3Gc⇢m Gc 6= GN

L�GR = LEH +M2P

p�g

⇣� (rµuµ)

2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

Lm = LLI(SM,DM,⇤, gµ⌫) + SM LLV (SM, gµ⌫ , uµ)

+DM LLV (DM, gµ⌫ , uµ) + DE LLV (DE, gµ⌫ , uµ)

LV in late cosmology

Natural dark energy ΘCDM DB, Sibiryakov 11

Faster Jeans instability: DM dom, subhorizon

Anisotropic stress �� = O(�)

ds2 = a(t)2⇥(1 + 2�)dt2 � �ij(1� 2 )dxidxj

⇤

�00 + 2H�0 = �k2�

a2k2�

a2=

3H2(1 + �/2 + 3�/2)

2(1� ↵/2)� =

3GN

2GcH2�

� ⇠ t�1+q

1+24GNGc

Kobayashi, Urakawa, Yamaguchi 10DM = 0

LV Perturbations: Pure gravity

;

�� + k2c2s�� � �✓4k2

3

◆

k2 ⇠ GN

Gc�� ceffs

Shift of the peaks, change of zero point of oscillation and amplitude

http://class-code.net

E.g. Cosmic Microwave BackgroundAudren, Blas, Lesgourgues, Sibiryakov 13

9

scales, the peaks are further suppressed by Silk damp-ing. Indeed, due to the shift in the phase of oscillations,they correspond to smaller physical scales at recombina-tion, that are more a↵ected by di↵usion damping. TheDoppler e↵ect, which depends on ⇥

�

at recombination,is also modified. Finally, a prominent feature clearly vis-ible on the plot is the significant enhancement of theISW contribution in the range 10 < l < 100, i.e. be-tween the regions usually a↵ected by the early ISW e↵ect(100 < l < 200) and the late ISW e↵ect (2 < l < 10).The ISW e↵ect is proportional to the time derivative ofthe gravitational potential. In the ⇤CDM model, the po-tential varies only during the epochs of radiation and ⇤domination. However, in the enhanced gravity model, thegrowth of density perturbations entails a slow increase ofthe gravitational potential also during the matter domi-nated era, enhancing the ISW e↵ect on a wide range ofscales.

All in all, we conclude that the enhanced gravity modelproduces significant modifications in the spectrum ofCMB anisotropies. The pattern of these modificationsis quite specific, and apparently not degenerate with thee↵ects of standard cosmological parameters.

Shear model : We recall that in this model, the � fieldgenerates some anisotropic stress and contributes to theshear of the perturbed metric, as described by Eq. (43).The presence of shear tends to smooth out metric per-turbations on scales smaller than the sound horizon asso-ciated with the sound speed of the � field. Note that forparameters of the particular shear model studied here,the field � is superluminal16 (c

�

=p

3) and the suppres-sion appears already on super-Hubble scales.

This is indeed observed on the top panel of Fig. 1,where the gravitational potential is clearly smaller (inabsolute value) compared to ⇤CDM. It also exhibits no-table wiggles caused by oscillations in the �-field [4]. Thefact that c

�

is much larger than the photon-baryon soundspeed explains the shift between the phase of the oscil-lations seen in and in ⇥

�

. The suppression of shiftsthe zero-point of the oscillations in the photon tempera-ture ⇥

�

. On the other hand, we do not see any shift inthe positions of the peaks, which is compatible with theprevious discussion: the self-gravity of radiation is notmodified in this model.

For fixed initial conditions ⇥�

(⌧0), the amplitude ofthe acoustic oscillations depends crucially on boostinge↵ects, imprinted around the time of Hubble crossing,and caused by the three gravitational driving terms onthe right-hand side of Eq. (44). In the shear model, theamplitude of acoustic oscillations is damped as a conse-quence of smaller metric fluctuations and reduced grav-itational boosting. This translates into an overall sup-

16 As pointed above, this does not present any inconsistencies intheories without Lorentz invariance.

0

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

7e-10

8e-10

9e-10

1e-09

10 100 1000

l(l+1

)Cl

ΛCDMenhanced gravity

0

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

7e-10

8e-10

9e-10

1e-09

10 100 1000

l(l+1

)Cl

l

ΛCDMshear

FIG. 2: Temperature anisotropy spectrum (solid) and itsdecomposition in terms of Sachs–Wolfe (dotted), Doppler(dashed) and Integrated Sachs–Wolfe (dot-dashed) contribu-tions. For clarity, we do not show cross correlations betweenthese contributions. Thick black lines represent the ⇤CDMmodel, while thin blue lines are used for the two ⇥CDM ref-erence models.

pression of the SW e↵ect visible in the bottom panel ofFig. 1.

On Fig. 2, we see that the Doppler and ISW contri-butions to the total temperature spectrum C

`

are alsolower in the shear model than in ⇤CDM. The net resultis a uniform suppression of all peaks. One may expectthat this e↵ect could be compensated, at least partially,by a rescaling of the initial amplitude of perturbations.This suggests that pure shear models might be less con-strained than enhanced gravity ones.

Our shear model is similar to the one studied in [11],where it was claimed that the dominant e↵ect on theCMB comes through the ISW. Our analysis demonstratesthat the changes in the SW and Doppler contributionsare equally important for this model.

9

scales, the peaks are further suppressed by Silk damp-ing. Indeed, due to the shift in the phase of oscillations,they correspond to smaller physical scales at recombina-tion, that are more a↵ected by di↵usion damping. TheDoppler e↵ect, which depends on ⇥

�

at recombination,is also modified. Finally, a prominent feature clearly vis-ible on the plot is the significant enhancement of theISW contribution in the range 10 < l < 100, i.e. be-tween the regions usually a↵ected by the early ISW e↵ect(100 < l < 200) and the late ISW e↵ect (2 < l < 10).The ISW e↵ect is proportional to the time derivative ofthe gravitational potential. In the ⇤CDM model, the po-tential varies only during the epochs of radiation and ⇤domination. However, in the enhanced gravity model, thegrowth of density perturbations entails a slow increase ofthe gravitational potential also during the matter domi-nated era, enhancing the ISW e↵ect on a wide range ofscales.

All in all, we conclude that the enhanced gravity modelproduces significant modifications in the spectrum ofCMB anisotropies. The pattern of these modificationsis quite specific, and apparently not degenerate with thee↵ects of standard cosmological parameters.

Shear model : We recall that in this model, the � fieldgenerates some anisotropic stress and contributes to theshear of the perturbed metric, as described by Eq. (43).The presence of shear tends to smooth out metric per-turbations on scales smaller than the sound horizon asso-ciated with the sound speed of the � field. Note that forparameters of the particular shear model studied here,the field � is superluminal16 (c

�

=p

3) and the suppres-sion appears already on super-Hubble scales.

This is indeed observed on the top panel of Fig. 1,where the gravitational potential is clearly smaller (inabsolute value) compared to ⇤CDM. It also exhibits no-table wiggles caused by oscillations in the �-field [4]. Thefact that c

�

is much larger than the photon-baryon soundspeed explains the shift between the phase of the oscil-lations seen in and in ⇥

�

. The suppression of shiftsthe zero-point of the oscillations in the photon tempera-ture ⇥

�

. On the other hand, we do not see any shift inthe positions of the peaks, which is compatible with theprevious discussion: the self-gravity of radiation is notmodified in this model.

For fixed initial conditions ⇥�

(⌧0), the amplitude ofthe acoustic oscillations depends crucially on boostinge↵ects, imprinted around the time of Hubble crossing,and caused by the three gravitational driving terms onthe right-hand side of Eq. (44). In the shear model, theamplitude of acoustic oscillations is damped as a conse-quence of smaller metric fluctuations and reduced grav-itational boosting. This translates into an overall sup-

16 As pointed above, this does not present any inconsistencies intheories without Lorentz invariance.

0

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

7e-10

8e-10

9e-10

1e-09

10 100 1000

l(l+1

)Cl

ΛCDMenhanced gravity

0

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

7e-10

8e-10

9e-10

1e-09

10 100 1000

l(l+1

)Cl

l

ΛCDMshear

FIG. 2: Temperature anisotropy spectrum (solid) and itsdecomposition in terms of Sachs–Wolfe (dotted), Doppler(dashed) and Integrated Sachs–Wolfe (dot-dashed) contribu-tions. For clarity, we do not show cross correlations betweenthese contributions. Thick black lines represent the ⇤CDMmodel, while thin blue lines are used for the two ⇥CDM ref-erence models.

pression of the SW e↵ect visible in the bottom panel ofFig. 1.

On Fig. 2, we see that the Doppler and ISW contri-butions to the total temperature spectrum C

`

are alsolower in the shear model than in ⇤CDM. The net resultis a uniform suppression of all peaks. One may expectthat this e↵ect could be compensated, at least partially,by a rescaling of the initial amplitude of perturbations.This suggests that pure shear models might be less con-strained than enhanced gravity ones.

Our shear model is similar to the one studied in [11],where it was claimed that the dominant e↵ect on theCMB comes through the ISW. Our analysis demonstratesthat the changes in the SW and Doppler contributionsare equally important for this model.

6

use the Poisson equation to define the total density per-turbation,

�⇢[tot] = � k2�

4⇡GN

a2, (29)

where � is the perturbation of the Newtonian potentialrelated to ⌘ and h by the standard formulas [31]. Thenthe power spectrum is defined as the ratio of this densityperturbation to the apparent background density of DMand baryons. The latter di↵ers from the true densityentering the Friedmann equation due to the di↵erencebetween G

cos

and GN

. Taking this into account, oneobtains the final formula for the power spectrum [10],

P (k) =

✓G

N

Gcos

◆2 h|�⇢[tot](~k)|2i⇢[dm] + ⇢[b]

. (30)

The definition (29) has a limited applicability: it is mean-ingful only well inside the Hubble horizon and for neg-ligible shear. However, both conditions are satisfied forthe range of wavenumbers where the power spectrum isactually measured. Taking the gravitational potential asthe basis for the definition of LPS is motivated by thefact that this quantity is directly probed by the lensingsurveys. The amount of galaxies and clusters is also be-lieved to trace linearly the underlying gravitational fieldat large scales (k < 0.1hMpc�1), though contamina-tion by a scale-dependent bias cannot be excluded (cf.[38]). A dedicated study of the clustering dynamics inthe model at hand is required to definitely resolve thisissue, which is beyond the scope of this paper.

A cleaner observable, insensitive to the above ambi-guities, is the spectrum of CMB anisotropies. The tem-perature spectrum computed using Class in the modelwith the parameters (27) is shown in the lower panelof Fig. 1. The main modification comes from the verystrong ISW e↵ect induced by the accelerated growth ofperturbations (26). It extends over a wide range of mul-tipoles from l ⇠ k

Y,0⌧0 (⇡ 25 for the chosen parameters)up to l ⇠ 1000. Modifications in the Sachs–Wolfe andDoppler e↵ects are small, consistent with the decouplingof DM from photon-baryon plasma at the epoch of recom-bination [39]. In particular, the positions of the acousticpeaks are not changed, which distinguishes the ⇤LVDMimpact on CMB from that of the modified Poisson equa-tion (i), helping to break the degeneracy between them.

V. COMPARISON WITH DATA

The e↵ects discussed above are constrained by the dataon the CMB anisotropies and LPS. In this way the cos-mological observations can be used to put bounds on LVin dark matter and gravity. For this purpose we use inthis work the CMB data from the Planck 2013 release[12] combined with the galaxy power spectrum from theWiggleZ redshift survey [13]. To compute the predictions

Y10 log10(c�)

0

1.32

2.37

log 1

0(c2 �

)

0 0.0322 0.058Y

-4.44 -3.43 -2.41log10(�)

0

0.0322

0.058

Y

0 1.32 2.37log10(c

2�)

FIG. 2: Marginalized one-dimensional posterior distributionand two-dimensional probability contours (at the 68% and95% CL) of the ⇤LVDM parameters for Einstein-aether (up-per panel) and khronometric (lower panel) cases. Only thesubspace of parameters responsible for Lorentz violation isshown.

for observables at various values of the model parame-ters, we modify the Boltzmann code Class [37]. Theparameter space is explored using the Monte Carlo codeMonte Python [40]. We consider both Einstein-aetherand khronometric models focusing only on the scalarsector of perturbations. We separately study the casesY 6= 0 (Lorentz violation in dark matter) and Y ⌘ 0(dark matter is Lorentz invariant).For non-zero Y the fit includes nine cosmological pa-

↵ = 2�

Planck, SPT, WiggleZ

Cosmological Constraints (Kh)

http://montepython.net/

Audren, Blas, Ivanov, Lesgourgues, Sibiryakov 14

c2� =� + �

↵

Y ⌘ CDM

< 55

< 10�3

< 0.029

First bound on LI of DM!!

↵PPN1 ↵PPN

2. 10�4 . 10�7

Tests in the gravitational sector (long distance)

Cosmological constraints (background and perturbations): growth rate + anisotropic stressEffects on the CMB and matter power spectrum

↵ = 2�

GW (strong fields): SEP, Dipolar radiation Solar system tests:

Exploring Lorentz violation yields a rich phenomenology with strong theoretical motivations (from QG to DM & DE)Lorentz violation modifies gravity at every scale

(extra massless d.o.f. )

DM . O(.01)

' = t+ �

Conclusions

�, � . O(.01)

�, � . O(.01)

Peculiar (finite) short-range gravitational potentials M�1? & µm ?

Open issues

?why are the CMB and aligned?

Effects in the primordial universe

Constraints on

Is Horava gravity renormalizable? Landau poles?

1412.4828 [gr-qc]Review (w/ E. Lim)

can LV be so big in the gravity sector given the LI in the SM?

uµ

Are singularities present?

M?

Constraints from large-scale structure…