From Dark Energy to Dark Force

Luca AmendolaINAF/Osservatorio Astronomico di

Roma

• Dark energy-dark matter interactions• Non-linear observational effects of DE• Modified gravity

Outline

What do we know about cosmicexpansion ?

Nucleosynthesis (z~109)

CMB (z~1000)

Standard candles (z~1)

Perturbations (z~0-1000)

DE

Four hypotheses Four hypotheses on dark energy on dark energy

A) Lambda

B) scalar field

C) modified gravity

D) non-linear effect

Scalar fieldScalar field

VV ''

pw

Vp

V

s ta te o f eq .

)(2

1

)(2

1

2

2

• It is more general• Scalars are predicted by fundamental theories

Compton wavelength = Hubble length

10

1

3 3

3 0 0 0

1 0

HM pcm

eVmV()

Observational requirements:A) Evolve slowly B) Light mass

VV '

An ultra-An ultra-lightlight scalar field scalar field

DM

FIf

Abu

ndan

ce

MassL.A. & R. Barbieri 2005

Evolution of background

zzyxz

yyxxyy

yxyxx

energyradiationz

energypotentialy

energykineticx K

)331(2

1'

)333(2

1'

)333(2

1'

222

22

222

2

2

,2

AAeV , Potential Energy Dark

)(1 222 zyxm Flat space:

0'3

04

03

))(2

1(

3

8 22

VH

H

H

VH

radrad

mm

m

Tracking vs. attractors

In a phase space, tracking is a curve,attractor is a point

Ωγ

ΩK

ΩP

AV

7.022 yx

The coupling

• But beside the potential there can be also a coupling…

0

0

;)(

;)(

T

T m

;)(;)(

;)(;)(

m

mm

CTT

CTT

Dark energy as scalar gravity

Einstein frameEinstein frame Jordan frame Jordan frame

TRR

CTT

CTT

mLRL

m

mm

m

82

1

),(

,)(;)(

,)(;)(

;;;,

,

,

;)(

;)(

)(

)(2

18

)2

1(

0

0

)()(

fL

LLT

RRL

T

T

mLRfL

R

R

R

m

m

geg f ˆ'2

Dark energy as scalar gravity T

(m)= CT(m)

T= -CT(m)

coupled conservation laws :

3

)'(3

mmm

m

CH

CVH

C

m

Cm

emm

ea

0

30

)(

First basic property:

C2/G = scalar-to-tensor ratio

An extra gravityAn extra gravity

0)/13

41(4')2

'1(''

222

2

kkk HkmG

H

H

)3

41( 2* rmeGG

Newtonian limit: the scalar interaction generates an attractive extra-gravity

Yukawa term

Local tests of gravity: λ<1 a.u.

Only on baryons and on sublunar scales

Adelberger et al. 2002

)1()( / rbar eGrG

α

λ

0 0 1.0bar

Astrophysical tests of gravity: λ<1 Mpc

Distribution of dark matter and baryons in galaxies and clusters (rotation curves, virial theorem, X-ray clusters,…)

Gradwohl & Frieman 1992

α

λ

)1()( / rdm eGrG

5.1dm

Cosmological tests of gravity: λ>1/H0

gravitational growth of structures: CMB, large scale structure

)1()1()( /ij

rijij GeGrG

)3

41( 2*

iGG

Since αb=βb2<0.001, baryonsbaryons must be very weaklyvery weakly

coupled

Since αc=βc2<1.5, dark matterdark matter can be strongly strongly

coupled

T(cdm)= CT(cdm)

T= -CT(cdm)

T(bar)= 0

T(rad)= 0

A species-dependent interaction

Dark energy and the equivalence principle

cdm

baryon

cdm

G*=G(1+4β2/3)

G

G

baryon

G

Cem 0Cem 0

bm bm

A 3D phase space

zzyxz

yyxxyy

zyxyxyxx

densityradiationz

energypotentialy

energykineticx K

)331(2

1'

)333(2

1'

)1()333(2

1'

222

22

222222

2

2

,2

Phase spaces

© A. Pasqui

Ωrad

ΩK

ΩP

Two qualitatively different cases:weak coupling strong coupling

rad mat

field

rad mat

field

No No couplingcoupling

couplingcoupling

MDE = /9

a ~ tp

p = 6/(42+9)

= 0

a ~ tp

p = 2/3

MDE:

kinetic phase, indep. of potential!

MDE:

toda

y

Weak coupling: density trends

The equation of state w=p/depends on during MDE and on during tracking:

we = 4 past value (decelerated)w = present value (accelerated)

Deceleration and accelerationAssume

V =

toda

y

rad mat

field

Dominated bykinetic energy

β

Dominated bypotential energy

α

cl)

WMAP and the coupling

Planck:

Scalar force 100 times weaker than gravity

strong coupling

Dark energy

•Acceleration has to begin at z<1 •Perturbations stop growing in an accelerated universe•The present value of Ωm depends on the initial conditions

Strongly coupled dark energy

•Acceleration begins at z > 1•Perturbations grow fast in an accelerated universe•The present value of Ωm does not depend on the initial conditions

A Strong coupling and the coincidence problem…

< 1

> 1to

day

2

2

)(4

1 84

M

0M

Weak:

Strong: AeV

High redshift supernovae at z > 1

L.A., M. Gasperini & F. Piazza: 2002 MNRAS,2004 JCAP

Dream of a global attractor

zzyxz

yyxxyy

zyxyxyxx

densityradiationz

energypotentialy

energykineticx K

)331(2

1'

)333(2

1'

)1()333(2

1'

222

22

222222

2

2

,2

7.022 yx

Stationary models

couplingsl

ope

bar

stationary

3

)/(33

2

2

)(4

1844

a

w

aa

b

eff

M

large βany μ baryon baryon

epoch !epoch !

baryonbaryon

densitydensity

is theis the

controllingcontrolling

factorfactor

Does it work ?

constraints from SN,constraints on omegaconstraints from ISW

Does it work ? No !No !

naak 22 4

L. A. & D. Tocchini-Valentini 2002

Second try

X

X

matter

Xpw

Xpp

X

geLUXpR

L

,

,

;;

21

2

2/

),(),(2

Generalized coupled scalar field Lagrangian

Under which condition one gets a stationary attractor Ω, w constant?

Theorem

)( XeXgp

A stationary attractor is obtained if and only if

Piazza & Tsujikawa 2004L.A., M. Quartin, I. Waga, S. Tsujikawa 2006

For instance :

dark energy with exp. pot.

tachyon field

dilatonic ghost condensate

eVXp 0 Y

Vg 01

2/10 )21( XeeVp

Y

Yg

21

eXXp 2 Yg 1

XeY

Perturbations on Stationary attractors

0)3

41(

2

3')'

'2(''

,

2

X

m pH

H

New perturbation equation in the Newtonian limit

which can be written using only the observable quantities w,Ω

0)6

1)(1(2

3')91(

2

1''

2

eff

effeff w

ww

L.A., S. Tsujikawa, M. Sami, 20051

2

Analytical solution

)4(2

12

211

m

am

Therefore we have an analytical solution for the growth oflinear perturbations on any stationary attractor:

In ordinary scalar field cosmology, m lies between 0 and 1. Now itcan be larger than 1, negative or complex !

Two interesting regions: phantom (p;X<0) and non-phantom (p;X>0)

Phantom damping

contour plot of Re(m)

Theorem 1: a phantom fieldon a stationary attractor alwaysproduces a damping of the perturbations: Re(m)<0.

0)3

41(

2

3')'

'2(''

,

2

X

m pH

H

Does it work?

Theorem 2: the gravitational potential is constant (i.e. no ISW) for

2.1,2.07.0

For

3/)342(

s

s

w

w

22 4 ak

Poisson equation

Still quite off the SN constraints !!

A No-Go theorem

• Take a general p(X,U)• Require a sequence of decel. matter era followed by acceleration

Theorem: no function p(X,U) expandable in a finite polynomial can achievea standard sequence matter+scaling acceleration !

END OF THE SCALING DREAM ???

L.A., M. Quartin, I. Waga, S. Tsujikawa 2006

Background expansion

Linear perturbations

What’s next ?

Non-linearity

1) N-Body simulations2) Higher-order perturbation theory

Interactions• Two effects: DM mass is varying, G is different for baryons and DM

22 r

Gm

r

eGmH vv b

Cc

bb

mb mc

22

*

)2(r

Gm

r

emGvHv b

Cc

cc

N-body recipe

• Flag particles either as CDM (Flag particles either as CDM (cc) or baryons () or baryons (bb) in ) in proportions according to present valueproportions according to present value

• Give identical initial conditionsGive identical initial conditions• Evolve them according the their Newtonian equation: Evolve them according the their Newtonian equation:

at each step we calculate two gravitational potentials at each step we calculate two gravitational potentials and evolve the and evolve the cc particle mass particle mass

• Reach a predetermined varianceReach a predetermined variance• Evaluate clustering separately for Evaluate clustering separately for cc and and bb particles particles• Modified Adaptive Refinement Tree code (Kravtsov et Modified Adaptive Refinement Tree code (Kravtsov et

al. 1997, Mainini et al, Maccio’ et al. 2003)al. 1997, Mainini et al, Maccio’ et al. 2003)

Collab. with S. Bonometto, A. Maccio’, C. Quercellini, R. MaininiPRD69, 2004

N-body simulations

© A. Maccio’

Λ β=0.15 β=0.25

N-body simulations

© A. Maccio’

β=0.15 β=0.25

N-body simulations: halo profiles

β dependent behaviour towards the halo center.

Higher β: smaller rc

2

1

)(:

cc

c

cr

r

r

r

r

rNFW

A scalar gravity friction

22

*

)2(r

Gm

r

emGvHv b

Cc

cc

• The extra friction term drives the halo steepeningThe extra friction term drives the halo steepening• How to invert its effect ? How to invert its effect ? • Which cosmology ?Which cosmology ?

Linear Newtonian perturbations

2

3)/'1('

0'

vHHv

v

022

3

3

S

A field initially Gaussian remains Gaussian:the skewness S3 is zero

Non-linearity:Non-linearity:Higher order perturbation theoryHigher order perturbation theory

Non-linear Newtonian perturbations

7

3 422

3

3

S

A field initially Gaussian develops a non-Gaussianity:the skewness S3 is a constant value

2

3)()/'1('

0)1('

vvvHHv

v

Independent of Ω, of eq. of state, etc.: S3 is a probeof gravitational instability, not of cosmology

(Peebles 1981)

Non-linear scalar-Newtonian perturbations

)6.01(7

3 4 2

22

3

3

S

the skewness S3 is a constant

2

3)()

'1(' :B a ry o n s

)3

41(

2

3)()'2

'1(' :C D M

0)1('

2

vvvH

Hv

vvvH

Hv

v

therefore S3 is also a probeof dark energy interaction

(L.A. & C. Quercellini, PRL 2004)

Skewness as a test of DE coupling

Sloan DSS:Predicted error on S3 less

than 10%

7/3 43S

Modified 3D gravityModified 3D gravity

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

μνστμνmatter R,RR,φ,f+L+Rgxd 4

)1)(12(3

)2(21

4

nn

nw

L+R+Rgxd

asympt

mattern

Simplest case:

Higher order gravity !

Turner, Carroll, Capozziello, Odintsov…

L.A., S. Capozziello, F. Occhionero, 1992

Modified N-dim gravityModified N-dim gravity

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

ymatter,R,Fdx=+L+Rggdyxd mattern

n 4

4444 ...

matterL+Rφfgxd 4

Simplest case:

Aspects of the same Aspects of the same physicsphysics

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

matterL+Rφ,fgxd 4

Extra-dim. Degrees of freedom

Higher order gravity

Coupled scalar field

Scalar-tensor gravity

The simplest caseThe simplest case

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

matterL+

R

μ+Rgxd 4

is equivalent to coupled dark energy

RφL+R'+Lg'xd matter,Rφ4

But with strong coupling !

2/1=β

2

33

2

3)'(3

mmm

m

H

VH

R+1/R modelR+1/R model

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

rad mat

field

rad mat

fieldMDE

toda

y

9/1=Ωφ

2/1=β

a= t 1/2

R+RR+Rnn model model

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

2/1=β

9/1=Ωφ

L.A., S. Tsujikawa, D. Polarski 2006

a=t 1 /2

Distance to last scatteringDistance to last scatteringin R+Rin R+Rnn model model

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

2/1=β

9/1=Ωφ

decz

zH

dz=zr

)(

a=t 1 /2

General f(R, Ricci, General f(R, Ricci, Riemann)Riemann)

A) Lambda

B) scalar field

C) modified gravity

D) non linear effect

mμνστμνστ

nμνμν RRβ+RRα+Rgxd 4

a=t 1 /2

we find again the same past behavior:

so probably most of these models are ruled out.

Anti-gravity has many side-effects…