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of gravity

Luca Amendola

INAF/Osservatorio Astronomico di Roma

The dark side

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Observations are converging…

…to an unexpected universe

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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 !

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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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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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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

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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

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

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A recipe to modify gravity

Can we find f(R) models that work?

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MG in the background (JF)

321

23

22

1

16

'6

'

'

xxx=ΩH

Rx

Hf

fx

Hf

fx

m

An autonomous dynamical system

f

Rfr

f

Rfrm

'

'

'')(

)2(2)/(

'

)42()/(

'

31'

3332

313

13232

312

312

1231

xxxxm

xxx

xxxxxm

xxx

xxxxxx

characteristic function

r

prrmeRRf

r

rnrmRRRf

nmRRf

mRRf

qRp

n

n

)()(

1)()(

1)(

0)(

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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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A theorem on viable models

Theorem: 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 )()(

L.A., S. Tsujikawa, 2007

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Local Gravity Constraints are very tight

Depending 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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Two free functions

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

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

),( ak

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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. 2004L.A., Kunz &Sapone 2007

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Parametrized MG: Growth of fluctuationsas a measure of modified gravity

we parametrizeInstead of

LCDMDE

DGPST

is an indication of modified gravity/matter

0),(4')'

1('' kkk akGQH

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

Di Porto & L.A. 2007

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Present constraints on gamma

Viel et al. 2004,2006; McDonald et al. 2004; Tegmark et al. 2004

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Present constraints on gamma

C. Di Porto & L.A. 2007

DGP

LCDM

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Two MG observables

Correlation of galaxy positions:galaxy clustering

Correlation of galaxy ellipticities:galaxy weak lensing

22 ),(),( zkPbzkP mattgal

2)(),( zkPellipt

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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

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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 two modified gravity functions

and in the growth function

in the WL power spectrum

{

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Forecasts for Weak Lensing

L.A., M. Kunz, D. Sapone JCAP 2007

Marginalization over the modified gravity parameters

does not spoil errors on standard parameters

)1/()( 0 zzwwzw a

zz 01)(

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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

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Weak lensing measures Dark Gravity

scalar-tensor model

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

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Non-linearity in WL

Weak lensing tomography over half sky

=1000,3000,10000

log

max

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Non-linearity in BAO

Matarrese & Pietroni 2007

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Conclusions: the teachings of DE

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/gravityNew MG parameters: γ,Σ A general reconstruction of the first order metric requires galaxy correlation and galaxy shear Let EUCLID fly...

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Current Observational Status: CFHTLS

First resultsFrom CFHT Legacy Survey with Megacam

(w=constant and other priors assumed)

Weak Lensing

Type IaSuper-novae

Hoekstra et al. 2005Semboloni et al. 2005

Astier et al. 2005