Transcript The nature of Dark Energy

The dark side of
gravity
Luca Amendola
Munich 2008
INAF/Osservatorio Astronomico di Roma
Why DE/MG is interesting
g
How to observe it
Munich 2008
Observations are converging…
…to an unexpected universe
Munich 2008
Classifying the unknown
Standard cosmology:
GR gravitational equations + FRW metric
a) change the equations
i.e. add new matter field (DE) or modify gravity (MG)
b) change the metric
i.e. inhomogeneous non-linear effects, void models, etc
Munich 2008
Modified gravity
Which are the effects of modified gravity at
Munich 2008
{
background
linear
non-linear
}
level ?
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 !
Munich 2008
How to hide modified gravity
(in the solar system)
L.A., C. Charmousis, S. Davis,
PRD 2008, arXiv 0801.4339
Generalized Brans-DickeGauss-Bonnet Lagrangian
Solution in static spherical
symmetry in a linearized
PPN metric with
   /U  1 
Conclusion:
there are solutions which
look “Einsteinian” but are not…
Munich 2008
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
Munich 2008
Simplest MG (II): f(R)
The simplest MG in 4D: f(R)
 dx
4
eg higher order corrections
g  f R + Lmatter 
4
d
x


 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
Munich 2008

g R + R 2  R 3  ...
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
2
m
f ''
*
α
λ
Munich 2008
Adelberger et al. 2005
The fourfold way out of local gravity
4 2 m r
G  G(1   e )
3
*
m , 
{
depend on time
depend on space
depend on local density
depend on species
Munich 2008
Sound horizon in R+R - n model
 dx


μ4

gR
+ Lmatter 

R


4
Turner, Carroll, Capozziello etc. 2003
at

1/ 2


z dec
in the Matter Era !
cs dz
/
H ( z)
z dec

0
dz
H ( z)
Munich 2008
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?
Munich 2008
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
Munich 2008
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
MG in the background
Ωγ
ΩK
Munich 2008
ΩP
Classification of f(R) solutions
For all f(R) theories:
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  (
Munich 2008
good matter era (t2/3) for m≥0
General acceleration, any w
The power of the m(r) method
f ( R)  R  0e  R / 1
Munich 2008
The triangle of viable trajectories
cosmologically viable trajectories
m(r )  Rf ' ' / f '
r   Rf ' / f
f ( R)  R
f ( R)  R  aR n
f ( R)  ( R a  ) b
Notice that in the triangle m>0
L.A., D. Polarski,
S. Tsujikawa 2007 PRD
astro-ph/0612180
Munich 2008
p
p 1
( R  )
1
1 p
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; Mota et al. 2006;....
Munich 2008
c
LGC+Cosmology
Take for instance the ΛCDM clone
f ( R)  ( R a  ) b
Applying the criteria of
LGC and background cosmology
a  b  1  10
23
i.e. ΛCDM to an incredible precision
Munich 2008
What background hides
perturbations reveal
The background expansion
only probes H(z)
The (linear) perturbations probe
first-order quantities
ds 2  a 2 [(1  2 )dt 2  (1  2 )(dx 2  dy 2  dz 2 )]
Full metric reconstruction
at first order requires 3 functions
H ( z )  (k , z )  (k , z )
Munich 2008
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) 

(most of what follows in collaboration with M. Kunz, D. Sapone)
Munich 2008
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 )  ...
Munich 2008
Lue et al. 2004;
Koyama et al. 2006
see L. A., C. Charmousis,
S. Davis 2006
Reconstruction of the metric
Correlation of galaxy positions:
galaxy clustering
Correlation of galaxy ellipticities:
galaxy weak lensing
Pgal (k , z,  )  (1   2 ) 2 b 2 2 (k , z )

'
b
Pellipt (k , z)  (  ) 2
Munich 2008
Peculiar velocities
Correlation of galaxy velocities:
galaxy peculiar field
v  x
 z  r 
H 0 x
Pz  (1   2 ) Pr
Pz  (1  2 ) Pr
redshift distortion parameter
'

b
=0.70±0.
Munich 2008
2
Guzzo et al. 2008
The Euclid theorem
Observables:
Conservation equations:
b  P(k , z , transv)
'
 P(k , z , rad ) / P(k , z , transv)  1
b
z
Pellipt (k , z )   dz ' K ( z ' )(  ) 2
0
5 unknown variables:
 '  3 '

Ha
k2
 '   

Ha
b(k , z ),  (k , z ),  (k , z ), (k , z ),  (k , z )
We can measure 3 combinations and we have 2 theoretical relations…
Theorem: lensing+galaxy clustering allows to measure all
(total matter) perturbation variables at first order without
assuming any particular gravity theory
Munich 2008
The Euclid theorem
Observables:
Conservation equations:
b  P(k , z , transv)
'
 P(k , z , rad ) / P(k , z , transv)  1
b
z
Pellipt (k , z )   dz ' K ( z ' )(  ) 2
0
5 unknown variables:
 '  3 '

Ha
k2
 '   

Ha
b(k , z ),  (k , z ),  (k , z ), (k , z ),  (k , z )
We can measure 3 combinations and we have 2 theoretical relations…
Theorem: lensing+galaxy clustering allows to measure all
(total matter) perturbation variables at first order without
assuming any particular gravity theory
Munich 2008
The Euclid theorem
b(k , z ),  (k , z ),  (k , z ), (k , z ),  (k , z )
From these we can estimate deviations from Einstein’s gravity:
k   4Ga Q(k , a) 
2
2
 
 (k , a) 

Munich 2008
Euclid
A geometrical probe of the universe proposed for
Cosmic Vision
All-sky optical
imaging for
gravitational
lensing
=
+
All-sky near-IR
spectra to
H=22 for BAO
Munich 2008
Weak lensing
Euclid forecast
Present constraints
  0.4
DGP
d log
 
d log a
  0.02
LCDM
Weak lensing tomography over half sky
Munich 2008
L.A., M. Kunz, D. Sapone
arXiv:0704.2421
DiPorto & L.A. 2007
Power spectrum
Galaxy clustering at 0<z<2 over half sky
Munich 2008
....if you know the bias to 1%
Non-linearity in BAO
Matarrese & Pietroni 2007
Munich 2008
Poster advertisement
Cosmic parallax
t  10 yrs
  0.1 1as
Garcia-Bellido & Haugbolle 2008
See poster by Miguel Quartin…
LTB void model
Quercellini, Quartin & LA,
arXiv 0809.3675
Munich 2008
Conclusions
 Two solutions to the DE mismatch: either add “dark
energy” or “dark gravity”
 High-precision next generation cosmological
observations are the best tool to test for modifications of
gravity
 It is crucial to combine background and perturbations
 A full reconstruction to first order requires imaging and
spectroscopy: Euclid
Munich 2008
The bright side of
Munich
Luca Amendola
INAF/Osservatorio
Astronomico di Roma
Munich 2008
Weak lensing measures Dark Gravity
scalar-tensor model
Weak lensing tomography over half sky
Munich 2008
V. Acquaviva, L.A., C.
Baccigalupi, in prep.
Non-linearity in WL
 max
=1000,3000,10000
log 
Weak lensing tomography over half sky
Munich 2008
Non-linearity in BAO
Matarrese & Pietroni 2007
Munich 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/gravity
New MG parameters: γ,Σ
 A general reconstruction of the first order metric
requires galaxy correlation and galaxy shear
 Let EUCLID fly...
Munich 2008
References
Basics:
CMB:
Bias:
WMAP:
N-body:
L.A. , Phys. Rev. D62, 043511, 2000;
L.A. , Phys. Rev. Lett. 86,196,2001;
L.A. & D. Tocchini-Valentini, PRD66, 043528, 2002
astro-ph/0303228, Phys Rev 2003
A. Maccio’ et al. 2004
Dilatonic dark energy: L.A., M. Gasperini, D. Tocchini-Valentini, C.
Ungarelli, Phys. Rev. D67, 043512, 2003
Munich 2008
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
Munich 2008