Portsmouth 2008
Can we detect traces of modified gravity atbackground linear level ?non-linear
{ }
Modified gravity
Portsmouth 2008
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
Portsmouth 2008
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 !
Portsmouth 2008
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
Portsmouth 2008
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)
Portsmouth 2008
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
Portsmouth 2008
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
Portsmouth 2008
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
Portsmouth 2008
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)(
Portsmouth 2008
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
/'
'/'')(
Portsmouth 2008
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
Portsmouth 2008
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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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
Portsmouth 2008
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
Portsmouth 2008
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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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
Portsmouth 2008
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
{
Portsmouth 2008
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)(
Portsmouth 2008
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
Portsmouth 2008
Weak lensing measures Dark Gravity
scalar-tensor model
Weak lensing tomography over half sky V. Acquaviva, L.A., C. Baccigalupi, in prep.
Portsmouth 2008
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...