Transcript The nature of Dark Energy

The dark side
of gravity
Luca Amendola
INAF/Osservatorio Astronomico di Roma
Portsmouth 2008
Observations are converging…
…to an unexpected universe
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Modified gravity
Can we detect traces of modified gravity at
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{
background
linear
non-linear
}
level ?
What is modified gravity ?
What is gravity ?
A universal force in 4D mediated by a massless tensor field
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
}
very limited time/space/energy scales;
only baryons
at astrophysical scales
complicated by non-linear/nongravitational effects
at cosmological scales
unlimited scales; mostly linear processes;
baryons, dark matter, dark energy !
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Simplest MG (I): DGP
(Dvali, Gabadadze, Porrati 2000)
S   d 5 x  g ( 5) R ( 5)  L  d 4 x  g R
H2 
H 8G


L
3
brane
L = crossover scale:
1
r  L  V 
r
1
r  L  V  2
r
5D Minkowski
bulk:
infinite volume
extra dimension
gravity
leakage
• 5D gravity dominates at low energy/late times/large scales
• 4D gravity recovered at high
energy/early times/small scales
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Simplest MG (II): f(R)
Let’s start with one of the simplest MG model: f(R)
 dx
4
eg higher order corrections
g  f R + Lmatter 
4
d
x



g R + R 2  R 3  ...
 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 higherorder terms
 particular case of scalar-tensor and extra-dimensional theory
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Is this already ruled out by local gravity?
4
d
x

g  f ( R ) + Lmatter 
is a scalar-tensor theory with Brans-Dicke
parameter ω=0 or
a coupled dark energy model with coupling β=1/2
4 2  m r
G  G (1   e )  G (1  e  r /  )
3
1 Rf '4 f
1
2
m


2
f ''
f'
f ''
*
α
(on a local minimum)
λ
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The simplest case
4
d
x



μ4
g
 R  R + Lmatter 



In Einstein Frame
3



  3H  V ( )' 
 m



  3H  V ( )' 20
 m  3H m  0 3 

 m  3H m  
 m
2
β = 1/ 2
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Turner, Carroll, Capozziello
etc. 2003
gˆ   ( f ' ) 2 g 
fR  f '
f '2
  log f '
V ( )' 
R-1/R model : the φMDE
  3H  V ( )' 
 m  3H m  
H2 
3
 m
2
3 
 m
2
8
(  m   )
3
rad
mat
field
β = 1/ 2
Ωφ = 1 / 9
1/2
In Jordan frame: a= t
instead of
a = t 2/3
rad
MDE field
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!!
mat
Caution:
Plots in the
Einstein frame!
Sound horizon in R+Rn model
1/ 2
a=t
weff = 1/ 3



z dec
cs dz
/
H ( z)
z dec

0
dz
H ( z)
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L.A., D. Polarski, S. Tsujikawa, PRL 98, 131302,
astro-ph/0603173
A recipe to modify gravity
Can we find f(R) models that work?
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MG in the background (JF)
An autonomous dynamical system
f '
x1  
Hf '
f
x2  
6 f 'H 2
R
x3 
6H 2
Ωm = 1  x1  x2  x3
x'1  1  x3  3x2  x1  x1 x3
2
x1 x3
x '2 
 x2 (2 x3  4  x1 )
m( x2 / x3 )
x1 x3
x'3  
 2 x3 ( x3  2)
m( x2 / x3 )
Rf ' '
f'
Rf '
r
f
m( r ) 
characteristic function
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f ( R)  R    m  0
f ( R )  R  n  m   n  1
1 r
r
p
 m( r )   r 
r
f ( R )  R  R  n  m( r )   n
f ( R)  R p e qR
Classification of f(R) solutions
For all f(R) theories, define the characteristic curve:
m(r )  Rf ' ' / f '
r  Rf ' / f
P1  (0,1,2)
m  0
P2  ( 1,0,0)
m  2
P3  (1,0,0)
m  0
P4  ( 4,5,0)
deSitter acceleration, w = -1
wrong matter era (t1/2)
m  0
3m
m(7  10m)
,...)
m  1 
1 m
2(1  m) 2
2(1  m)
P6  (
,...)  m  0
1  2m
P5  (
good matter era (t2/3) for m≥0
General acceleration, any w
The problem is: can we go from matter to acceleration?
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The m,r plane
The qualitative behavior of any f(R) model can be
understood by looking at the geometrical properties of the
m,r plot
matter era
deSitter
m(r) curve
Rf ' '
f'
Rf '
r
f
m( r ) 
acceleration
crit. line
The dynamics becomes 1-dimensional !
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L.A., D. Polarski, S. Tsujikawa, PRD, astro-ph/0612180
The power of the m(r) method
f ( R)  R  0e  R / 1
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The triangle of viable trajectories
There exist only two kinds of cosmologically viable trajectories
f ( R)  R
f ( R)  R  aR n
f ( R)  ( R a  ) b
Notice that in the triangle m>0
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p
p 1
( R  )
1
1 p
A theorem on viable models
Theorem: for all viable f(R) models
 there is a phantom crossing of wDE
 there is a singularity of wDE
 both occur typically at low z when m  1
standard DE
phantom DE
f ( R)  ( R a  ) b
L.A., S. Tsujikawa, 2007
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Local Gravity Constraints are very tight
Depending on the local field configuration
Rs f s ' '
m( Rs ) 
 10  23  10 6
fs '
depending on the experiment: laboratory, solar system, galaxy
see eg. Nojiri & Odintsov 2003; Brookfield et al. 2006
Navarro & 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
f ( R)  ( R a  ) b
Applying the criteria of
LGC and Cosmology
a  b  1  10
23
i.e. ΛCDM to an incredible precision
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However. . . perturbations
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Two free functions
ds  a [(1  2 )dt  (1  2 )(dx  dy  dz )]
2
2
2
2
2
2
At the linear perturbation level and sub-horizon scales, a modified gravity model will
 modify Poisson’s equation
k 2  4Ga2Q(k , a)  m m
 induce an anisotropic stress
 
 (k , a) 

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MG at the linear level
 standard gravity
 scalar-tensor models
Q(k , a)  1
 (k , a)  0
G * 2( F  F '2 )
Q(a) 
FGcav, 0 2 F  3F '2
Boisseau et al. 2000
Acquaviva et al. 2004
Schimd et al. 2004
L.A., Kunz &Sapone 2007
F '2
 (a) 
F  F '2
 f(R)
 DGP
 coupled Gauss-Bonnet
Q(a) 
G*
FGcav,0
k2
a2R ,
k2
1  3m 2
a R
1  4m
k2
a2R
 (a) 
k2
1  2m 2
a R
m
1
;   1  2 Hrc wDE
3
2
 (a) 
3  1
Bean et al. 2006
Hu et al. 2006
Tsujikawa 2007
Q(a)  1 
Q (a )  ...
 (a )  ...
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Lue et al. 2004;
Koyama et al. 2006
see L. A., C. Charmousis,
S. Davis 2006
Parametrized MG: Growth of fluctuations
as a measure of modified gravity
H'
 k ' '(1  ) k '4GQ (k , a)  k  0
H
Instead of
good fit
Peebles 1980
Lahav et al. 1991
Wang et al. 1999
Bernardeau 2002
L.A. 2004
Linder 2006
we parametrize
LCDM
DE
Di Porto
& L.A.
2007
DGP
ST
is an indication of modified gravity/matter
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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
LCDM
DGP
C. Di Porto & L.A. 2007
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Two MG observables
Pgal (k , z)  b 2 Pmatt (k , z)   2
Correlation of galaxy positions:
galaxy clustering
Correlation of galaxy ellipticities:
galaxy weak lensing
Pellipt (k , z )  (  ) 2
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Probing gravity with weak lensing
Statistical measure of shear pattern, ~1% distortion
Dark matter halos
Background
sources
Observer
Radial distances depend on
geometry of Universe
Foreground mass distribution depends on
growth/distribution of structure
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Probing gravity with weak lensing
In General Relativity, lensing is caused
by the “lensing potential”
and this is related to the matter perturbations
via Poisson’s equation.
Therefore the lensing signal depends on
two modified gravity functions
in the WL power spectrum
and in the growth function
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{
Forecasts for Weak Lensing
Marginalization over the modified gravity parameters
does not spoil errors on standard parameters
( z)  1  0 z
w( z)  w0  wa z /(1  z)
L.A., M. Kunz, D. Sapone JCAP 2007
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Weak lensing measures Dark Gravity
DGP
Phenomenological DE
DGP
LCDM
Weak lensing tomography over half sky
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L.A., M. Kunz, D. Sapone
arXiv:0704.2421
Weak lensing measures Dark Gravity
scalar-tensor model
Weak lensing tomography over half sky
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V. Acquaviva, L.A., C.
Baccigalupi, in prep.
Non-linearity in WL
 max
=1000,3000,10000
log 
Weak lensing tomography over half sky
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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/gravity
New 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
Weak
Hoekstra et al. 2005
Semboloni et al. 2005
Lensing
First results
From CFHT
Legacy
Survey with
Megacam
Type Ia
Supernovae
(w=constant
and other
priors
assumed)
Astier et al. 2005
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