Transcript The nature of Dark Energy - INAF

The dark side
of gravity
Luca Amendola
INAF/Osservatorio Astronomico di Roma
Bologna 2007
Observations are converging…
…to an unexpected universe
Bologna 2007
The dark energy problem
F ( g  ) 
R  
1
2
g  R  8 GT 
gravity
 8GT ( )
matter
 tot  1
 cluster  0 . 3
Solution: modify either the Matter sector
DE
or the Gravity sector
MG
...but remember :
pX
X
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 w X  1
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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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 x g
5
(5)
H 
2
R
H
L
(5)

 L d x  g R
4
8G

3
brane
L = crossover scale:
r  L  V 
1
r
r  L  V 
1
r
5D Minkowski
bulk:
infinite volume
extra dimension
gravity
leakage
2
• 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  + L matter

dx
4

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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
f(R) is popular....
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Is this already ruled out by local gravity?

dx
4
g
f
( R ) + L matter

is a scalar-tensor theory with Brans-Dicke
parameter ω=0 or
a coupled dark energy model with coupling β=1/2
G  G (1 
*
4
 e
2
 m r
)  G (1   e
r /
)
3
m
2


1
f ''

Rf ' 4 f
f'
2

α
1
f ''
(on a local minimum)
λ
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Dark Fog
The trouble with f(R): it’s Fourth Order Gravity (FOG) !
3 f 'H
2
 m 
1
2
[ f ' ( 6 H  12 H )  f ]  3 H R f ' '
2
Higher order equations introduce:
new solutions (acceleration ?)
new instabilities (is the universe stable?)
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The simplest case

dx
4
4

μ
g
 R  R + L matter

In Einstein Frame
3



  3 H   V ( )' 
 m



  3 H   V ( )'  20
 m  3 H  m  0 3 

 m  3H m  
  m
2
β = 1/ 2
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



Turner, Carroll, Capozziello
etc. 2003
2
gˆ   ( f ' ) g 
V ( )' 
fR  f '
f'
  log f '
2
R-1/R model : the φMDE
3
  3 H   V ( )' 
2
3
 m  3 H  m  
H
2

8
3
2
 m
  m
( m   )
rad
mat
field
β = 1/ 2
Ωφ = 1 / 9
In Jordan frame:
a= t
instead of a = t
rad
1/2
2/3
MDE field
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!!
mat
Caution:
Plots in the
Einstein frame!
Sound horizon in R+Rn model
a=t
1/ 2
w eff = 1 / 3

 

z dec
c s dz
H (z)
z dec
/

0
dz
H (z)
...and by the way
  a !!
2
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L.A., D. Polarski, S. Tsujikawa, PRL 98, 131302,
astro-ph/0603173
WMAP and the coupling 
cl)
Planck: 
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L.A., C. Quercellini et al. 2003
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 )
m  0
P5  (
P6  (
3m
1 m
m  1
,...)
2 (1  m )
1  2m
,...)
deSitter acceleration, w = -1
wrong matter era (t1/2)
m ( 7  10 m )
2 (1  m )
good matter era (t2/3) for m≥0
2
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 ' '
m (r ) 
f'
r  
acceleration
Rf '
f
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  0 e
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 R / 1
The triangle of viable trajectories
There exist only two kinds of cosmologically viable trajectories
p
f ( R)  R  aR
f ( R)  ( R  )
a
f ( R)  R
n
b
Notice that in the triangle m>0
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p 1
1
( R  )
1 p
Constraints on viable trajectories
Cosmological constraints
Constraint from
accelerated expansion:
SN require
Constraint from the
matter expansion:
CMB peak requires
m(present)<0.1
m(past)<0.1
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Viable trajectories are cool !
Viable trajectories have a very peculiar effective equation of state
Define wDE implicitely as
H
2
 H 0 [ m a
Theorem: w DE
3
 (1   m ) a
 3 (1  wˆ )
]
1
1
wˆ  (log a )  w DE a da
w DE  
We find
2
diverges if f , R
R  3H
1
9H
2
2
(1   m f , R )
grows in the past, i.e. if m ( R )  0
Corollary: all viable f(R) cosmologies possess a divergent w DE
f ( R)  ( R  )
a
b
L.A., S. Tsujikawa, 2007
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Phantom crossing
Conclusions: for all viable f(R) models
 there is a phantom crossing of w DE
 there is a singularity of w DE
 both occur typically at low z when  m  1
standard DE
phantom DE
f ( R)  ( R  )
a
b
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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 !!
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Nesseris
Perivolaropoulos 2005
...but don’t forget the
Local Gravity Constraints...
However, if we apply naively the LGC at the present epoch.
m
1


m(r0 ) 
f 0 ' '  1 mm
R0 f 0 ' '
 10
58
!!
f0 '
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relaxing the Local Gravity Constraints ?
However, the mass depends on the local field configuration
m( Rs ) 
Rs f s ' '
 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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MG at the linear level
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
k   4Ga Q(k , a)  m m
2
 modify Poisson’s equation
2
 (k , a) 
 induce an anisotropic stress
 modify the growth of perturbations
 k ' ' (1 
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H'
H
 

) k ' 4  GQ ( k , a )  k  0
MG at the linear level
 standard gravity
 scalar-tensor models
Q(k , a)  1
 (k , a)  0
Q(a) 
 (a) 
 f(R)
Q(a ) 
G
G
2( F  F ' )
*
2
FGcav, 0 2 F  3F '
F'
2
F  F'
2
k
1  4m
*
FGcav, 0
2
m
2
a R,
2
k
1  3m
2
a R
 DGP
Q(a )  1 
 (a) 
 coupled Gauss-Bonnet
Boisseau et al. 2000
Acquaviva et al. 2004
Schimd et al. 2004
2
1
3
;
 (a) 
k
2
2
a R
2
k
1  2m 2
a R
Bean et al. 2006
Hu et al. 2006
Tsujikawa 2007
  1  2 Hrc wDE
2
Lue et al. 2004;
Koyama et al. 2006
3  1
Q ( a )  ...
 ( a )  ...
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see L. A., C. Charmousis,
S. Davis 2006
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 the
two modified gravity functions
in the WL power spectrum
and in the growth function
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Growth of fluctuations
A good fit to the linear growth of fluctuations is
where
LCDM
DE
DGP
ST
Instead of
we parametrize
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Peebles 1980
Lahav et al. 1991
Wang et al. 1999
Bernardeau et al. 2002
L.A. 2004
Linder 2006
Weak lensing measures Dark Gravity
DGP
Phenomenological DE
DGP
LCDM
Weak lensing tomography over half sky
Bologna 2007
L.A., M. Kunz, D. Sapone
arXiv:0704.2421
Weak lensing measures Dark Gravity
scalar-tensor model
Weak lensing tomography over half sky
Bologna 2007
V. Acquaviva, L.A., C.
Baccigalupi, in prep.
Weak lensing measures Dark Gravity
Marginalizing over modified gravity parameters
FOM
Bologna 2007
Non-linearity
N-Body simulations
Maccio’ et al. 2004
Jain et al. 2006
....
Higher-order
perturbation theory
Bologna 2007
Kamionkowski et al. 2000
Gaztanaga et al. 2003
Freese et al. 2002
Makler et al. 2004
Lue et al. 2004
L.A. & C. Quercellini 2004
....
N-body simulations in MG
Dark energy/dark matter coupling
Two effects: DM mass is varying, G is different for baryons and DM
mb
mc
*
v c   H ( v c  2   ) 
v b   Hv b 
Gmc e
r
G mc e
r
C
2
Bologna 2007

C
2
r
Gmb
r

Gmb
2
2
N-body simulations
Λ
A. Maccio’, L.A.,C. Quercellini, S.
Bonometto, R. Mainini 2004
β=0.15
Bologna 2007
β=0.25
N-body simulations
β=0.25
β=0.15
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N-body simulations: halo profiles
β-dependent behaviour towards
the halo center.
NFW :
 (r )
 cr

c
r 
r 
1  
rc 
rc 
2
Higher β: smaller rc
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-light scalar field
Hubble size
fI 
F
Abundance
Adopting a PNGB
potential
Galactic size
 DM
m30  m / 10
30
eV
Mass
Bologna 2007
L.A. & R. Barbieri 2005
Dark energy as scalar gravity
Jordan frame
Einstein frame
L   f ( ) R  Lm ( m )
L   R  Lm ( , m )

T( m ) ;  0

T( m ) ;  CT( m ) ,

T( ) ;  CT( m ) ,

R 
1
2
g   e
2f'
gˆ  

T( ) ;  0
1 

L, R ( R    R ) 
2
  R  8T


1

8T    ( L  L, R ) 
2

;

  ( L, R );  f ;
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An extra gravity
Newtonian limit: the scalar interaction generates an
attractive extra-gravity
4


 k ' ' (1 
 2   ) k ' 4  G (1 
)  k  0
2
2
H
3 1  m / k
2
H'
in real space
G  G (1 
*
4
 e
2
3
Yukawa term
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 m r
)
Understanding dark
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-light scalar field
Hubble size
fI 
F
Abundance
Adopting a PGB
potential
Galactic size
 DM
m30  m / 10
30
eV
Mass
Bologna 2007
L.A. & R. Barbieri 2005