Diego Blas, COSMO’12 Beijing

Natural Dark Energyfrom Lorentz Breaking

w/ B. Audren, M. Ivanov, J. Lesgourgues, S. Sibiryakov

JCAP 1107 (2011) 026 [arXiv:1104.3579]arXiv:1302.xxxx

Diego Blas

Diego Blas, COSMO’12 Beijing

Natural Dark Energyfrom Lorentz Breaking

w/ B. Audren, M. Ivanov, J. Lesgourgues, S. Sibiryakov

JCAP 1107 (2011) 026 [arXiv:1104.3579]arXiv:1302.xxxx

Diego Blas

(a variation on LB and cosmology)

Aspects of LB theories of gravity

Better UV properties than GR

New ideas for black hole physics

Potentially useful for AdS/CFT (condensed matter)

Models of modified gravity at low energies

Any new ideas for cosmology? Late time, inflation,...

Big effort to understand gravity at large distances!

DES

Euclid

SNAP

HETDEX

WFIRST

JEDI

PlanckSDSS

Dark energy missions

....

EUCLID

SPT

WFIRST

...

Breaking Lorentz InvarianceEinstein-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�⇤⇥ � c1gµ�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-aether

Scalar-vector

uµuµ = 1

Hypersurface orthogonal:Khronometric

x

µ

' ='0'

= '1

uµ ⌘@µ'p

@↵'@↵'

Khronon

Gravitational Lagrangian (low energies)

L�GR = LEH +p�g

⇣� (rµuµ)2 + ↵ (u⌫r⌫uµ)2 + �rµu⌫r⌫uµ

⌘

Ingredients:

↵ = 2�

↵ ⇠ � . 10�2

Solar system + GW:

uµ , gµ⌫

Einstein-Aether:

low energy Lagrangian of Hořava gravity

Khronometric:

...+

Solar system + GW:

c1p�grµu⌫rµu⌫

Expressed as if uµ ⌘@µ'p

@↵'@↵'

c1 = 0E.g. scalar pert. around FRW

↵ = �(3�+ �)

↵ ⇠ � . 10�2,

,

⇤IR ⇠p↵MP

Matter part: +SM Fields + DM + DE

Lm = LLI(SM,DM,DE, gµ⌫) + SM LLB(SM, gµ⌫ , uµ)+DM LLB(DM, gµ⌫ , uµ) + DE LLB(DE, gµ⌫ , uµ)

SM . 10�20

DM ,DE?

Particle physics

Matter Lagrangian (low energies)

In the following SM = 0 DM = 0

LB and Dark Energy

DE

Gµ⌫ =1

M2PTmµ⌫ +

1

M2PT fluidµ⌫ +

1

M2PT aetherµ⌫ + ⇤ gµ⌫

Background: Homogeneous and isotropic (preferred foliation aligned with CMB frame)

✓ȧ

a

◆2=

8⇡Gc3

⇢m

Friedmann equations almost not modified!

Gc =1

8⇡M2P [1 + 3�/2 + �/2]

From BBN ( abundance)4He Gc = GN +O(.01)

ds2 = gµ⌫dxµdx⌫ = dt2 � a(t)2dxidxi

uµ = (u0(t), 0, 0, 0)

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

= vµ , ⇢(t)

Homogeneous Cosmology

GN =1

8⇡M2P (1� ↵/2)

Quintessence model: ΘCDM

Add a field with an exact shift symmetry⇥ 7! ⇥+ C

Derivatively coupled! Very simple Lagrangian

L⇥ =(@⌫⇥)2

2+ µ2u⌫@⌫⇥

Stable under radiative corrections:breaks ⇥ 7! �⇥

Assuming that LI is broken (EAther or Khronon):

It is technically natural to assume a small ! µ

Friedman equation

d

dt

�a3⇥̇+ µ2a3

�= 0

Scalar equation Naturally small!

NB: If ⇢mat = 0 Minkowski is a solution!(tachyonic instability)

⇢⇥ = µ4/2

! = �1

H2 =8⇡Gc3

⇥̇2

2+ ⇢m

!

=8⇡Gc3

✓µ4

2+ ⇢m

◆

ΘCDM: Homogeneous cosmology

⇥̇ = �µ2 + Ca(t)3

Stiff matter at BBN C ⇡ 0

✦ Small mass protected by shift symmetry� 7! �+ C

- Phenomenology very similar to ΛCDM

Non-perturbative potential

L = @µ�@µ�+ ⇤4 (1 + cos(�/f))

⇤4 ⇠ H20M2P f & MP

Non-perturbative gravitational effects breakthe invariance: appearance of operators

PseudoGoldstones & DE

�2

M2PR

gapped mode

slow mode (clusters)

expected enhancement of structure formation at large scales

c� ⇠ k/kc ⌧ 1

!2+ = k2c + (c

2� + 1)k

2

k ⌧ kc

✴Spectrum k � kc

✴Spectrum

!2� = c2�k

2, !2⇠ = k2,

' = t+ �

⌘ µ2/(MPp↵) ⇠ H0/

p↵

⌘ µ2/(MPp↵) ⇠ H0/

p↵

c2� ⌘� + �

↵

!2� =c2� k

4

k2c

ΘCDM: Perturbations⇥ = ⇥̄+ ⇠,

ΘCDM: Perturbations II✴ LB enhancement structure: DM dom, subhorizon

✴ Anisotropic stress

�� = O(�)

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

⇤

�00 + 2H�0 = �k2�

a2

k2�

a2=

3H2(1 + �/2 + 3�/2)

2(1� ↵/2) � =3GN2Gc

H2�

� ⇠ ⌧�1+q

1+24GNGc

Khronometric+ Solar system constraints

Einstein-Aether:

GNGc

� 1 = ↵+ � + 3�2

+O(2)

GNGc

� 1 = 3(� + �)2

+O(2) > 0GNGc

� 1 = O(2)

Cosmological perturbations

h�(k)�(k0)i ⌘ �(3)(k + k0)P (k)k3

k (h Mpc�1)

↵ = 2�

↵ ⇠ � . 10�2

↵ ⇠ � . 10�1⇤CDM

⇥CDM

⇥CDM

P(k)

[h�1Mpc)

3]

http://class-code.net

0.001 0.01 0.1 1

5001000

50001¥ 104

5¥ 1041¥ 105

More structure and shift

http://class-code.nethttp://class-code.net

Cosmological perturbations

h�(k)�(k0)i ⌘ �(3)(k + k0)P (k)k3

k (h Mpc�1)

↵ = 2�

↵ ⇠ � . 10�2

↵ ⇠ � . 10�1⇤CDM

⇥CDM

⇥CDM

P(k)

[h�1Mpc)

3]

http://class-code.net

0.001 0.01 0.1 1

5001000

50001¥ 104

5¥ 1041¥ 105

Linear

More structure and shift

http://class-code.nethttp://class-code.net

9

10�3 10�2 10�1 100

k [Mpc�1]

101

102

103

104

105P(

k,z=

0)[M

pc3 ]

SDSS DR7 (Reid et al. 2010)LyA (McDonald et al. 2006)ACT CMB Lensing (Das et al. 2011)ACT Clusters (Sehgal et al. 2011)CCCP II (Vikhlinin et al. 2009)BCG Weak lensing(Tinker et al. 2011)ACT+WMAP spectrum (this work)

1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023Mass scale M [Msolar]

10�6

10�5

10�4

10�3

10�2

10�1

100

101

102

Mas

sVa

rianc

e� M

/M

SDSS DR7 (Reid et al. 2010)LyA (McDonald et al. 2006)ACT CMB Lensing (Das et al. 2011)ACT Clusters (Sehgal et al. 2011)CCCP II (Vikhlinin et al. 2009)BCG Weak lensing (Tinker et al. 2011)ACT+WMAP spectrum (this work)

Fig. 5.— The reconstructed matter power spectrum: the stars show the power spectrum from combining ACT and WMAP data (toppanel). The solid and dashed lines show the nonlinear and linear power spectra respectively from the best-fit ACT ⇤CDM model withspectral index of ns = 0.96 computed using CAMB and HALOFIT (Smith et al. 2003). The data points between 0.02 < k < 0.19 Mpc�1

show the SDSS DR7 LRG sample, and have been deconvolved from their window functions, with a bias factor of 1.18 applied to the data.This has been rescaled from the Reid et al. (2010) value of 1.3, as we are explicitly using the Hubble constant measurement from Riess et al.(2011) to make a change of units from h�1Mpc to Mpc. The constraints from CMB lensing (Das et al. 2011), from cluster measurementsfrom ACT (Sehgal et al. 2011), CCCP (Vikhlinin et al. 2009) and BCG halos (Tinker et al. 2011), and the power spectrum constraintsfrom measurements of the Lyman–↵ forest (McDonald et al. 2006) are indicated. The CCCP and BCG masses are converted to solar massunits by multiplying them by the best-fit value of the Hubble constant, h = 0.738 from Riess et al. (2011). The bottom panel shows thesame data plotted on axes where we relate the power spectrum to a mass variance, �M/M, and illustrates how the range in wavenumber k(measured in Mpc�1) corresponds to range in mass scale of over 10 orders of magnitude. Note that large masses correspond to large scalesand hence small values of k. This highlights the consistency of power spectrum measurements by an array of cosmological probes over alarge range of scales.

Hlozek et alt., 11

http://inspirehep.net/author/Hlozek%2C%20Renee?recid=901279&ln=enhttp://inspirehep.net/author/Hlozek%2C%20Renee?recid=901279&ln=en

-5

0

5

10

15

20

25

30

35

40

1e-05 0.0001 0.001 0.01 0.1 1

P k (M

pc/h

)3

k (h/Mpc)

enhanced gravity (Pk/Pk ΛCDM - 1)shear 20 x (Pk/Pk ΛCDM - 1)

(EA)

http://class-code.net

Signature of ΘCDM

� ⇠ 10�1

GNGc

� 1 ⇠ 10�1

GNGc

� 1 < 10�4

http://class-code.nethttp://class-code.net

Cosmic Microwave Background

�̈� + k2c2s�� � �

✓4k2

3

◆k2 ⇠ GN

Gc�� ceffs

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

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

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

(EA) � ⇠ 1

http://class-code.net

GNGc

� 1 ⇠ 10�1 GNGc

� 1 < 10�4

http://class-code.nethttp://class-code.net

lhttp://class-code.net

⇤CDM

l(l+1)C

l/2⇡

↵ = 2�↵ ⇠ � . 10�2⇥CDM

↵ ⇠ � . 10�1⇥CDM

10 100 1000

2.¥ 10-10

4.¥ 10-10

6.¥ 10-10

8.¥ 10-10

1.¥ 10-9

1.2¥ 10-9 Peaks shiftand amplitude (SW)

ISW

⇥l(k, ⌘0) ⇠ [⇥0(k, ⌘⇤) + (k, ⌘⇤)]jl[k(⌘0 � ⌘⇤)]

+3⇥1(k, ⌘⇤)⇣jl�1[k(⌘0 � ⌘⇤)]� (l+1)jl[k(⌘0�⌘⇤)]k(⌘0�⌘⇤)

⌘

+R ⌘00 d⌘ e

�⌧h ̇(k, ⌘)� �̇(k, ⌘)

ijl[k(⌘0 � ⌘)] ISW

Doppler

SW

⇥� ⌘ ��/4

http://class-code.nethttp://class-code.net

Fig. 2. Left: Power spectrum of CMB temperature anisotropies, showing data from WMAP5 [19],the 2003 flight of BOOMERANG [20], CBI [21] and the full ACBAR dataset [22]. The red line isthe best-fit LCDM model to the data. Right: Matter power spectrum, showing data from the SDSS2006 data release and the best-fit LCDM curve; the inset shows the imprint (in Fourier space) of theCMB acoustic peaks, known as the baryon acoustic oscillations (from [2]).

dynamical dark energy fields have also been considered. For the simplest option of vacuumenergy, the observed value of the cosmological constant is overwhelmingly smaller than theprediction of current particle physics. In particular,

ρΛ,obs = Λ ∼ H20M2p ∼ (10−33 eV)2(1019 GeV)2 ∼ (10−3 eV)4, (1.1.6)

whereasρΛ,theory ∼ M4fundamental ! 1 TeV4 " ρΛ,obs . (1.1.7)

In addition, the Λ value needs to be strongly fine-tuned to be of the same order of magnitudetoday as the current matter density, i.e.,

ρΛ ∼ ρ(m)0 ⇒ ΩΛ ∼ Ω(m) , (1.1.8)

otherwise galaxies and then life could not emerge in the universe. The question is how this“coincidence” arises at late times, given that

ρΛ = constant , while ρ(m) ∝ (1 + z)3 . (1.1.9)

No convincing or natural explanation has yet been proposed. String theory provides a tanta-lising possibility in the form of the “landscape” of vacua [23,24]. There appears to be a vastnumber of vacua admitted by string theory, with a broad range of energies above and belowzero. The idea is that our observable region of the universe corresponds to a particular smallpositive vacuum energy, whereas other regions with greatly different vacuum energies will lookentirely different. This multitude of regions forms in some sense a “multiverse”. This is aninteresting idea, but it is highly speculative, and it is not clear how much of it will survive thefurther development of string theory and cosmology.

7

– 101 –

Fig. 32.— The nine-year WMAP TT angular power spectrum. The WMAP data are inblack, with error bars, the best fit model is the red curve, and the smoothed binned cosmic

variance curve is the shaded region. The first three acoustic peaks are well-determined.

WMAP9

Other effects

Anisotropic stress

0.1

10-2

10-3

10-4

10-5

0.001 0.01

(φ-ψ

)/φ

k (h Mpc-1)

abc

�� � �(�̇+ 2H�)

O(�)

� ⇠ � . 10�3

� ⇠ � . 10�2

� ⇠ � . 10�4

2.1 2.26 2.41

100 !b= 2.22+0.0412�0.044

0.112 0.121 0.13

!cdm= 0.121+0.00273�0.00289

0.112

0.121

0.13

0.919 0.955 0.991

ns= 0.955+0.0105�0.0113

0.919

0.955

0.991

2.3 2.59 2.87

10+9As= 2.56+0.0908�0.0917

2.3

2.59

2.87

0.64 0.681 0.722

h= 0.678+0.0128�0.0135

0.64

0.681

0.722

6.05 9.78 13.5

zreio= 10.1+1.18�1.16

6.05

9.78

13.5

0 0.0648 0.117

�= 0.0152+0.00103�0.0152

0

0.0648

0.117

0.0001 0.0115 0.0206

�plus�= 0.00478+0.000552�0.00468

0.0001

0.0115

0.0206

0 6.98 12.6

SPTSZ= 4.89+2.19�2.42

0

6.98

12.6

10.7 19.7 28.6

SPTPS= 20.5+2.74�2.72

10.7

19.7

28.6

-2.1 5.56 13.2

SPTCL= 4.97+2.26�2.21

2.1 2.26 2.41-2.1

5.56

13.2

0.112 0.121 0.13 0.919 0.955 0.991 2.3 2.59 2.87 0.64 0.681 0.722 6.05 9.78 13.5 0 0.0648 0.1170.0001 0.0115 0.0206 0 6.98 12.6 10.7 19.7 28.6

↵ = 2�

Bounds

Khronometric

� = 0.0152 +0.001�0.0152

� + � = 0.00478+0.00055�0.00468

WMAP7, SPT, WiggleZ

http://montepython.net/

http://montepython.nethttp://montepython.net

Breaking Lorentz invariance in the dark sector as an alternative to GR/ΛCDM with better UV properties.

ΘCDM model for dark energy

Same background evolution. Different perturbations. growth of structure enhanced

anisotropic stress at linear scales: effects on the CMB and matter power spectrum

w = �1

Conclusions

Comparison with data (LB bounds at percent level!)

Other bounds: weak lensing, polarization (EA)?...

⇢⇥ = µ4/2 UV insensitive