Bologna 2007

of gravity

Luca Amendola

INAF/Osservatorio Astronomico di Roma

The dark side

Bologna 2007

Observations are converging…

…to an unexpected universe

Bologna 2007

The dark energy problem

GTRgR 82

1

gravity matter

1tot 3.0cluster

1 XX

X wp

Solution: modify either the Matter sector

or the Gravity sector

)( gF )(8 GT

...but remember :

DE

MG

Bologna 2007

Modified matter

Problem:All the matter particles we know possess an effective interaction range that is

much smaller the cosmological ones

the effective pressure is always positive !

Solution:add new forms of matter with strong interaction/self-interaction

the effective pressure can be large and negative

Dark Energy=scalar fields, generalized perfect fluids etc

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Can we detect traces of modified gravity atbackground linear level ?non-linear

{ }

Modified gravity

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What is gravity ?

A universal force in 4D mediated by a massless tensor field

What is modified gravity ?

What is modified gravity ?

A non-universal force in nD mediated by (possibly massive) tensor, vector and scalar fields

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Cosmology and modified gravity

in laboratory

in the solar system

at astrophysical scales

at cosmological scales

} very limited time/space/energy scales;only baryons

complicated by non-linear/non-gravitational effects

unlimited scales; mostly linear processes;baryons, dark matter, dark energy !

Bologna 2007

L = crossover scale:

• 5D gravity dominates at low energy/late times/large scales

• 4D gravity recovered at high energy/early times/small scales

5D Minkowski bulk:

infinite volume extra dimension

gravity leakage

2

1

1

rVLr

rVLr

brane

Simplest MG (I): DGP

RgxdLRgxdS 4)5()5(5

(Dvali, Gabadadze, Porrati 2000)(Dvali, Gabadadze, Porrati 2000)

3

82 G

L

HH

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f(R) models are simple and self-contained (no need of potentials) easy to produce acceleration (first inflationary model) high-energy corrections to gravity likely to introduce higher-order terms particular case of scalar-tensor and extra-dimensional theory

matterL+Rfgxd 4eg higher order corrections ...324 RR+Rgxd

Let’s start with one of the simplest MG model: f(R)

Simplest MG (II): f(R)

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f(R) is popular....

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Is this already ruled out by

local gravity? matterL+Rfgxd )(4

is a scalar-tensor theory with Brans-Dickeparameter ω=0 or

a coupled dark energy model with coupling β=1/2

''

1

'

4'

''

1

)1()3

41(

22

/2*

ff

fRf

fm

eGeGG rrm

(on a local minimum)

α

λ

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Dark Fog

The trouble with f(R): it’s Fourth Order Gravity (FOG) !

Higher order equations introduce: new solutions (acceleration ?)

new instabilities (is the universe stable?)

''3])126('[2

1'3 22 fRHfHHfHf m

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The simplest caseThe simplest case

matterL+

R

μRgxd

44

2/1=β

03

0)'(3

mm H

VH

In Einstein Frame

2

33

2

3)'(3

mmm

m

H

VH

Turner, Carroll, Capozzielloetc. 2003

)'(g 2 gf

'log

'

')'(

2

f

f

ffRV

Bologna 2007

R-1/R model :R-1/R model : the the φφMDEMDE

rad mat

field

rad mat

fieldMDE

toda

y

9/1=Ωφ

2/1=β

a= t 1/2Caution:Plots in theEinstein frame!

)( 3

8

2

33

2

3)'(3

2

m

mmm

m

H

H

VH

2/3t=ainstead of !!

In Jordan frame:

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Sound horizon in R+RSound horizon in R+Rnn model model

dec

dec

z

z

s

zH

dz

zH

dzc

0 )(/

)(

2/1t=a

3/1=weff

!!2a...and by the way

L.A., D. Polarski, S. Tsujikawa, PRL 98, 131302, astro-ph/0603173

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cl)

WMAP and the coupling

Planck:

L.A., C. Quercellini et al. 2003

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Classification of f(R) solutions

,...)21

)1(2(

,...)1

3(

)0,5,4(

)0,0,1(

)0,0,1(

)2,1,0(

6

5

4

3

2

1

m

mP

m

mP

P

P

P

P

deSitter acceleration, w = -1

General acceleration, any w0

)1(2

)107(1

0

0

2

0

2

m

m

m

m

m

m

m

mm

For all f(R) theories, define the characteristic curve:

The problem is: can we go from matter to acceleration?

wrong matter era (t1/2)

good matter era (t2/3) for m≥0

fRfr

fRfrm

/'

'/'')(

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The m,r plane

The dynamics becomes 1-dimensional !

The qualitative behavior of any f(R) model can beunderstood by looking at the geometrical properties of the

m,r plot

m(r) curve

crit. line

acceleration

matter era

deSitter

L.A., D. Polarski, S. Tsujikawa, PRD, astro-ph/0612180

f

Rfr

f

Rfrm

'

'

'')(

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The power of the m(r) method

REJECTED

REJECTED

REJECTED

REJECTED

1/0)( ReRRf

REJECTED

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The triangle of viable trajectories

There exist only two kinds of cosmologically viable trajectories

baRRf )()(

pp

p

RRRf 1

1

1 )()(naRRRf )(

Notice that in the triangle m>0

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Constraints on viable trajectories

Cosmological constraints

Constraint from thematter expansion:CMB peak requires

m(past)<0.1

Constraint from accelerated expansion:SN require

m(present)<0.1

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Viable trajectories are cool !

Viable trajectories have a very peculiar effective equation of state

Define wDE implicitely as

daawaw

aaHH

DE

wmm

11

)ˆ1(3320

2

)(logˆ

])1([

We find )1(

3

9

1

,

2

2Rm

DE f

HR

Hw

Theorem: diverges if grows in the past, i.e. if Rf , 0)( Rm

Corollary: all viable f(R) cosmologies possess a divergent

DEw

DEw

baRRf )()( L.A., S. Tsujikawa, 2007

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Phantom crossing

Conclusions: for all viable f(R) models

there is a phantom crossing of there is a singularity of both occur typically at low z when 1m

DEwDEw

phantom DE

standard DE

baRRf )()(

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Crossing/singularity as signatures of

modified gravity The same phenomenon occurs for

DGP (Alam et al 2005) Scalar,Vector, Tensor models (Libanov et al. 2007)

and in the Riess et al. (2004) dataset !!

Nesseris Perivolaropoulos 2005

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...but don’t forget theLocal Gravity Constraints...

However, if we apply naively the LGC at the present epoch.

mmfm 1''01

!!10'

'')( 58

0

000

f

fRrm

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relaxing the Local Gravity Constraints ?

However, the mass depends on the local field configuration

623 1010'

'')(

s

sss f

fRRm

depending on the experiment: laboratory, solar system, galaxy

see eg. Nojiri & Odintsov 2003; Brookfield et al. 2006Navarro & Van Acoyelen 2006; Faraoni 2006; Bean et al. 2006;Chiba et al. 2006; Hu, Sawicky 2007;....

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LGC+Cosmology

Take for instance the ΛCDM clone

baRRf )()(

Applying the criteria of LGC and Cosmology

23101 ba

i.e. ΛCDM to an incredible precision

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However. . . perturbations

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MG at the linear level

At the linear perturbation level and sub-horizon scales, a modified gravity model will

mmakQGak ),(4 22 modify Poisson’s equation

induce an anisotropic stress

modify the growth of perturbations

)])(21()21[( 222222 dzdydxdtads

),( ak

0),(4')'

1('' kkk akGQH

H

Bologna 2007

MG at the linear level

scalar-tensor models

2

2

2

2

0,

*

'

')(

'32

)'(2)(

FF

Fa

FF

FF

FG

GaQ

cav

0),(

1),(

ak

akQ

standard gravity

DGP

13

2)(

21;3

11)(

a

wHraQ DEc

f(R)

Ra

km

Ra

km

a

Ra

km

Ra

km

FG

GaQ

cav2

2

2

2

2

2

2

2

0,

*

21

)(,

31

41)(

Lue et al. 2004; Koyama et al. 2006

Bean et al. 2006Hu et al. 2006Tsujikawa 2007

coupled Gauss-Bonnet see L. A., C. Charmousis, S. Davis 2006...)(

...)(

a

aQ

Boisseau et al. 2000Acquaviva et al. 2004Schimd et al. 2004

Bologna 2007

Observer

Dark matter halos

Background sources

Radial distances depend on

geometry of Universe

Probing gravity with weak lensing

Statistical measure of shear pattern, ~1% distortion

Foreground mass distribution depends on growth/distribution of structure

Bologna 2007

Probing gravity with weak lensing

In General Relativity, lensing is causedby the “lensing potential”

and this is related to the matter perturbationsvia Poisson’s equation. Therefore the lensing signal depends on thetwo modified gravity functions

and in the growth function

in the WL power spectrum

Bologna 2007

Growth of fluctuations

A good fit to the linear growth of fluctuations is

where

LCDM

DE

DGP

ST

we parametrizeInstead of

Peebles 1980Lahav et al. 1991Wang et al. 1999Bernardeau et al. 2002L.A. 2004Linder 2006

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Weak lensing measures Dark Gravity

DGP Phenomenological DE

Weak lensing tomography over half sky

LCDM

DGP

L.A., M. Kunz, D. Sapone arXiv:0704.2421

Bologna 2007

Weak lensing measures Dark Gravity

scalar-tensor model

Weak lensing tomography over half sky V. Acquaviva, L.A., C. Baccigalupi, in prep.

Bologna 2007

Weak lensing measures Dark Gravity

Marginalizing over modified gravity parameters

FOM

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Non-linearity

N-Body simulations

Higher-order perturbation theory

Maccio’ et al. 2004Jain et al. 2006....

Kamionkowski et al. 2000Gaztanaga et al. 2003Freese et al. 2002Makler et al. 2004Lue et al. 2004L.A. & C. Quercellini 2004....

Bologna 2007

N-body simulations in MG

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

22 r

Gm

r

eGmHvv b

Cc

bb

mb mc

22

*

)2(r

Gm

r

emGvHv b

Cc

cc

Dark energy/dark matter coupling

Bologna 2007

N-body simulations

A. Maccio’, L.A.,C. Quercellini, S. Bonometto, R. Mainini 2004

Λ β=0.15 β=0.25

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N-body simulations

β=0.15 β=0.25

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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

Bologna 2007

Conclusions: the teachings of DE

There is much more than meets the eyes in the Universe Two solutions to the DE mismatch: either add “dark energy” or “dark gravity” The high precision data of present and near-future observations allow to test for dark energy/gravity It is crucial to combine background and perturbations Weak Lensing is a good bet...(to be continued)

Bologna 2007

Bologna 2007

An ultra-An ultra-lightlight scalar field scalar field

DM

FIf

Abu

ndan

ce

MassL.A. & R. Barbieri 2005

Adopting a PNGBpotential

eVmm 3030 10/

Hubble size Galactic size

Bologna 2007

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

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An extra gravityAn extra gravity

0)/13

41(4')2

'1(''

22

2

kkkkm

GH

H

)3

41( 2* rmeGG

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

Yukawa term

in real space

Bologna 2007

Understanding Understanding dark energydark energy

Let him who seeks continue seeking until he finds. When he finds, he will become troubled. When he becomes troubled, he will be astonished

Coptic Gospel of Thomas

Bologna 2007

Bologna 2007

An ultra-An ultra-lightlight scalar field scalar field

DM

FIf

Abu

ndan

ce

MassL.A. & R. Barbieri 2005

Adopting a PGBpotential

eVmm 3030 10/

Hubble size Galactic size