Transcript The nature of Dark Energy

The dark side of
gravity
Luca Amendola
INAF/Osservatorio Astronomico di Roma
SAIT 2008
Why DE is interesting
g
How to observe it
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Observations are converging…
…to an unexpected universe
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The dark energy problem
F ( g  )  R 
1
g  R  8GT  8GT ( )
2
gravity
matter
 tot  1

0.3
cluster
Solution: modify either the Matter sector
DE
or the Gravity sector
MG
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Modified
Modified gravity
gravity
Can we detect traces of modified gravity at
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{
background
linear
non-linear
}
level ?
Whatisis modified
modified gravity
?
What
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
gravity
Cosmology
andmodified
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
times/small scales
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2008
SimplestMG
MG
(II):
f(R)
Simplest
(II):
f(R)
The simplest modification of Einstein’s gravity: f(R)
f
xg
R
+
L 
d
4
matter
eg higher order corrections


4
2
3
d
x
g
R
+
R

R

...

 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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MGalready
alreadyruled
ruledout
out by
bylocal
local gravity
IsIsthis
gravity??
Yukawa correction to Newton’s potential
4 2
m
r

r/
G

G
(
1
 e  )
G
(
1


e
)
3
α
1 Rf
'
4
f
1
2
m
  2 
f''
f'
f''
*
(on a local minimum)
λ
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From
Energyto
todark
Darkforce
Force
From Dark
dark energy
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
L.A., D. Polarski,
S. Tsujikawa,
PRL 98, 131302,
astro-ph/0603173
recipefor
to modify
gravity
AArecipe
modified
gravity
Can we find f(R) models that work?
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The
plane
Them,r
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  0 e  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
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p
p 1
( R  )
1
1 p
A theorem on phantom crossing
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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DE observations checklist
What do we need to test DE?
1) observations at z~1
2) observations involving background and perturbations
3) independent, complementary probes
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Why z~1
probes at z=0 and z=1000
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Why background+perturbations
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
H ( z )  (k , z )  (k , z )
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Why independent probes
a) systematics: because we need to test systematics like
evolutionary effects in SN Ia, biasing effects in the baryon
acoustic oscillations, intrinsic alignments in lensing, etc.
b) theory: because in all generality we need to measure two
free functions at pert. level (in addition to H(z) at background
level)
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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
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
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Present constraints on gamma
s ≡ d log δ /d log a
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. Phys.Rev. 2007
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Observables
Correlation of galaxy positions:
galaxy clustering
Pgal (k , z)  b2 Pmatt (k , z)  b2  2
Correlation of galaxy ellipticities:
galaxy weak lensing
Pellipt (k , z )  (  ) 2
Correlation of galaxy velocities:
galaxy peculiar field
' 2
Ppec.vel . (k , z )  ( )
b
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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
Guzzo et al. 2008
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The EUCLID theorem
Correlation of galaxy positions:
galaxy clustering
Pgal (k , z)  b2 Pmatt (k , z)  b2  2
Correlation of galaxy ellipticities:
galaxy weak lensing
Pellipt (k , z )  (  ) 2
Correlation of galaxy velocities:
galaxy peculiar field
' 2
Ppec.vel . (k , z )  ( )
b
THE EUCLID THEOREM:
reconstructing
Ψ ,Φ,b
in k,z requires
An imaging tomographic survey Pellipt (k , z )
and a spectroscopic survey:
Pgal (k , z ), Ppec.vel . (k , z )
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DUNE x SPACE = EUCLID
- a satellite mission that merges DUNE and SPACE proposals
- one of 4 mission selected by ESA Cosmic Vision in 2007
- final selection 2011
- launch 2015-2020, 5 years duration, half-sky
- an imaging mission in several bands optical+NIR (>1 billion galaxies)
- a spectroscopic survey: 500.000.000 galaxy redshifts
- a pan-european collaboration
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Requirements for Weak Lensing
Statistical Requirements:
• a 20,000 deg2 survey at high galactic latitude (|b|>30 deg)
• sample of at least 35 galaxies/amin2 usable for weak lensing (SNR[Sext]>7,
FWHM>1.2FWHM[PSF]) with a median z~1 and an rms shear error per galaxy of
=0.35 (or equivalent combination)
• a PSF FWHM smaller than 0.23’’ to be competitive with ground based surveys
• photometric redshifts to derive 3 redshift bins over the survey area (from ground
based observations)
Systematics Requirements:
• Survey scanned in compact regions <20 on a side, with 10% overlap between
adjacent stripes
• a precision in the measurement of the shear after deconvolution of the PSF better
than about 0.1%. This can be achieved with a PSF with a FWHM of 0.23’’, an
ellipticity |e|<6% with an uncertainty after calibration of |e|<0.1%.
• good image quality: low cosmic ray levels, reduced stray light, linear and stable
CCDs, achromatic optics
• Photometric redshifts with precision ∆z<0.1 in a subset of the survey to place limits
on Review
the intrinsic correlations of CNES
galaxy
shapes (from ground based observations)
CPS
Paris
Science from DUNE/EUCLID
• Primary goal: Cosmology with WL and SNe
–
Measurement of the evolution of the dark energy equation of state (w,w’) from z=0 to ~1
–
Statistics of the dark matter distribution (power spectrum, high order correlation functions)
–
Reconstruction of the primordial power spectrum (constraints on inflation)
• Cross-correlation with CMB (Planck)
–
Search for correlations of Galaxy shear with ISW effect, SZ effect, CMB lensing
–
Search for DE spatial fluctuations on large scales
• Study of Dark Matter Haloes:
–
Mass-selected halo catalogues (about 80,000 haloes) with multi- follow-up (X-ray, SZ, op
 halo mass calibration
–
Strong lensing: probe the inner profiles of haloes
• Galaxy formation:
–
Galaxy bias with galaxy-galaxy and shear-galaxy correlation functions
–
Galaxy clustering with high resolution morphology
• Core Collapse supernovae:
–
constraints on the history of star formation up to z~1
• Fundamental tests:
– Test of gravitational instability paradigm
CPS Review
CNES Paris
– Dark Energy clustering
Weak Lensing Power Spectrum Tomography
The power of WL
DUNE baseline: 20,000 deg2, 35 galaxies/amin2, ground-based
photometry for photo-z’s, 3 year WL survey
WL power spectrum for each z-bin
z>1
z<1
CPS Review
CNES Paris
Redshift bins from photo-z’s
The power of WL
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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
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
Σ0
LCDM
Weak lensing tomography over half sky
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L.A., M. Kunz, D. Sapone
arXiv:0704.2421
Non-linearity in WL
ell_max=1000,3000,10000
Weak lensing tomography over half sky
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Non-linearity in BAO
Matarrese & Pietroni 2007
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Clustering measures Dark Gravity
Galaxy clustering at 0<z<2 over half sky
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....if you know the bias to 1%
Combining P(k) with WL
Weak lensing/ BAO over half sky
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Conclusions
 Two solutions to the DE mismatch: either add “dark
energy” or “dark gravity”
 The high precision data of present and near-future allow
to test for dark energy/gravity
 It is crucial to combine background and perturbations
 A full reconstruction to first order requires imaging and
spectroscopy
 Let EUCLID fly...
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