From Dark Energy to Dark Force
Luca AmendolaINAF/Osservatorio Astronomico di
Roma
• Dark energy-dark matter interactions• Non-linear observational effects of DE• Modified gravity
Outline
What do we know about cosmicexpansion ?
Nucleosynthesis (z~109)
CMB (z~1000)
Standard candles (z~1)
Perturbations (z~0-1000)
DE
Four hypotheses Four hypotheses on dark energy on dark energy
A) Lambda
B) scalar field
C) modified gravity
D) non-linear effect
Scalar fieldScalar field
VV ''
pw
Vp
V
s ta te o f eq .
)(2
1
)(2
1
2
2
• It is more general• Scalars are predicted by fundamental theories
Compton wavelength = Hubble length
10
1
3 3
3 0 0 0
1 0
HM pcm
eVmV()
Observational requirements:A) Evolve slowly B) Light mass
VV '
An ultra-An ultra-lightlight scalar field scalar field
DM
FIf
Abu
ndan
ce
MassL.A. & R. Barbieri 2005
Evolution of background
zzyxz
yyxxyy
yxyxx
energyradiationz
energypotentialy
energykineticx K
)331(2
1'
)333(2
1'
)333(2
1'
222
22
222
2
2
,2
AAeV , Potential Energy Dark
)(1 222 zyxm Flat space:
0'3
04
03
))(2
1(
3
8 22
VH
H
H
VH
radrad
mm
m
Tracking vs. attractors
In a phase space, tracking is a curve,attractor is a point
Ωγ
ΩK
ΩP
AV
7.022 yx
The coupling
• But beside the potential there can be also a coupling…
0
0
;)(
;)(
T
T m
;)(;)(
;)(;)(
m
mm
CTT
CTT
Dark energy as scalar gravity
Einstein frameEinstein frame Jordan frame Jordan frame
TRR
CTT
CTT
mLRL
m
mm
m
82
1
),(
,)(;)(
,)(;)(
;;;,
,
,
;)(
;)(
)(
)(2
18
)2
1(
0
0
)()(
fL
LLT
RRL
T
T
mLRfL
R
R
R
m
m
geg f ˆ'2
Dark energy as scalar gravity T
(m)= CT(m)
T= -CT(m)
coupled conservation laws :
3
)'(3
mmm
m
CH
CVH
C
m
Cm
emm
ea
0
30
)(
First basic property:
C2/G = scalar-to-tensor ratio
An extra gravityAn extra gravity
0)/13
41(4')2
'1(''
222
2
kkk HkmG
H
H
)3
41( 2* rmeGG
Newtonian limit: the scalar interaction generates an attractive extra-gravity
Yukawa term
Local tests of gravity: λ<1 a.u.
Only on baryons and on sublunar scales
Adelberger et al. 2002
)1()( / rbar eGrG
α
λ
0 0 1.0bar
Astrophysical tests of gravity: λ<1 Mpc
Distribution of dark matter and baryons in galaxies and clusters (rotation curves, virial theorem, X-ray clusters,…)
Gradwohl & Frieman 1992
α
λ
)1()( / rdm eGrG
5.1dm
Cosmological tests of gravity: λ>1/H0
gravitational growth of structures: CMB, large scale structure
)1()1()( /ij
rijij GeGrG
)3
41( 2*
iGG
Since αb=βb2<0.001, baryonsbaryons must be very weaklyvery weakly
coupled
Since αc=βc2<1.5, dark matterdark matter can be strongly strongly
coupled
T(cdm)= CT(cdm)
T= -CT(cdm)
T(bar)= 0
T(rad)= 0
A species-dependent interaction
Dark energy and the equivalence principle
cdm
baryon
cdm
G*=G(1+4β2/3)
G
G
baryon
G
Cem 0Cem 0
bm bm
A 3D phase space
zzyxz
yyxxyy
zyxyxyxx
densityradiationz
energypotentialy
energykineticx K
)331(2
1'
)333(2
1'
)1()333(2
1'
222
22
222222
2
2
,2
Phase spaces
© A. Pasqui
Ωrad
ΩK
ΩP
Two qualitatively different cases:weak coupling strong coupling
rad mat
field
rad mat
field
No No couplingcoupling
couplingcoupling
MDE = /9
a ~ tp
p = 6/(42+9)
= 0
a ~ tp
p = 2/3
MDE:
kinetic phase, indep. of potential!
MDE:
toda
y
Weak coupling: density trends
The equation of state w=p/depends on during MDE and on during tracking:
we = 4 past value (decelerated)w = present value (accelerated)
Deceleration and accelerationAssume
V =
toda
y
rad mat
field
Dominated bykinetic energy
β
Dominated bypotential energy
α
cl)
WMAP and the coupling
Planck:
Scalar force 100 times weaker than gravity
strong coupling
Dark energy
•Acceleration has to begin at z<1 •Perturbations stop growing in an accelerated universe•The present value of Ωm depends on the initial conditions
Strongly coupled dark energy
•Acceleration begins at z > 1•Perturbations grow fast in an accelerated universe•The present value of Ωm does not depend on the initial conditions
A Strong coupling and the coincidence problem…
< 1
> 1to
day
2
2
)(4
1 84
M
0M
Weak:
Strong: AeV
High redshift supernovae at z > 1
L.A., M. Gasperini & F. Piazza: 2002 MNRAS,2004 JCAP
Dream of a global attractor
zzyxz
yyxxyy
zyxyxyxx
densityradiationz
energypotentialy
energykineticx K
)331(2
1'
)333(2
1'
)1()333(2
1'
222
22
222222
2
2
,2
7.022 yx
Stationary models
couplingsl
ope
bar
stationary
3
)/(33
2
2
)(4
1844
a
w
aa
b
eff
M
large βany μ baryon baryon
epoch !epoch !
baryonbaryon
densitydensity
is theis the
controllingcontrolling
factorfactor
Does it work ?
constraints from SN,constraints on omegaconstraints from ISW
Does it work ? No !No !
naak 22 4
L. A. & D. Tocchini-Valentini 2002
Second try
X
X
matter
Xpw
Xpp
X
geLUXpR
L
,
,
;;
21
2
2/
),(),(2
Generalized coupled scalar field Lagrangian
Under which condition one gets a stationary attractor Ω, w constant?
Theorem
)( XeXgp
A stationary attractor is obtained if and only if
Piazza & Tsujikawa 2004L.A., M. Quartin, I. Waga, S. Tsujikawa 2006
For instance :
dark energy with exp. pot.
tachyon field
dilatonic ghost condensate
eVXp 0 Y
Vg 01
2/10 )21( XeeVp
Y
Yg
21
eXXp 2 Yg 1
XeY
Perturbations on Stationary attractors
0)3
41(
2
3')'
'2(''
,
2
X
m pH
H
New perturbation equation in the Newtonian limit
which can be written using only the observable quantities w,Ω
0)6
1)(1(2
3')91(
2
1''
2
eff
effeff w
ww
L.A., S. Tsujikawa, M. Sami, 20051
2
Analytical solution
)4(2
12
211
m
am
Therefore we have an analytical solution for the growth oflinear perturbations on any stationary attractor:
In ordinary scalar field cosmology, m lies between 0 and 1. Now itcan be larger than 1, negative or complex !
Two interesting regions: phantom (p;X<0) and non-phantom (p;X>0)
Phantom damping
contour plot of Re(m)
Theorem 1: a phantom fieldon a stationary attractor alwaysproduces a damping of the perturbations: Re(m)<0.
0)3
41(
2
3')'
'2(''
,
2
X
m pH
H
Does it work?
Theorem 2: the gravitational potential is constant (i.e. no ISW) for
2.1,2.07.0
For
3/)342(
s
s
w
w
22 4 ak
Poisson equation
Still quite off the SN constraints !!
A No-Go theorem
• Take a general p(X,U)• Require a sequence of decel. matter era followed by acceleration
Theorem: no function p(X,U) expandable in a finite polynomial can achievea standard sequence matter+scaling acceleration !
END OF THE SCALING DREAM ???
L.A., M. Quartin, I. Waga, S. Tsujikawa 2006
Background expansion
Linear perturbations
What’s next ?
Non-linearity
1) N-Body simulations2) Higher-order perturbation theory
Interactions• Two effects: DM mass is varying, G is different for baryons and DM
22 r
Gm
r
eGmH vv b
Cc
bb
mb mc
22
*
)2(r
Gm
r
emGvHv b
Cc
cc
N-body recipe
• Flag particles either as CDM (Flag particles either as CDM (cc) or baryons () or baryons (bb) in ) in proportions according to present valueproportions according to present value
• Give identical initial conditionsGive identical initial conditions• Evolve them according the their Newtonian equation: Evolve them according the their Newtonian equation:
at each step we calculate two gravitational potentials at each step we calculate two gravitational potentials and evolve the and evolve the cc particle mass particle mass
• Reach a predetermined varianceReach a predetermined variance• Evaluate clustering separately for Evaluate clustering separately for cc and and bb particles particles• Modified Adaptive Refinement Tree code (Kravtsov et Modified Adaptive Refinement Tree code (Kravtsov et
al. 1997, Mainini et al, Maccio’ et al. 2003)al. 1997, Mainini et al, Maccio’ et al. 2003)
Collab. with S. Bonometto, A. Maccio’, C. Quercellini, R. MaininiPRD69, 2004
N-body simulations
© A. Maccio’
Λ β=0.15 β=0.25
N-body simulations
© A. Maccio’
β=0.15 β=0.25
N-body simulations: halo profiles
β dependent behaviour towards the halo center.
Higher β: smaller rc
2
1
)(:
cc
c
cr
r
r
r
r
rNFW
A scalar gravity friction
22
*
)2(r
Gm
r
emGvHv b
Cc
cc
• The extra friction term drives the halo steepeningThe extra friction term drives the halo steepening• How to invert its effect ? How to invert its effect ? • Which cosmology ?Which cosmology ?
Linear Newtonian perturbations
2
3)/'1('
0'
vHHv
v
022
3
3
S
A field initially Gaussian remains Gaussian:the skewness S3 is zero
Non-linearity:Non-linearity:Higher order perturbation theoryHigher order perturbation theory
Non-linear Newtonian perturbations
7
3 422
3
3
S
A field initially Gaussian develops a non-Gaussianity:the skewness S3 is a constant value
2
3)()/'1('
0)1('
vvvHHv
v
Independent of Ω, of eq. of state, etc.: S3 is a probeof gravitational instability, not of cosmology
(Peebles 1981)
Non-linear scalar-Newtonian perturbations
)6.01(7
3 4 2
22
3
3
S
the skewness S3 is a constant
2
3)()
'1(' :B a ry o n s
)3
41(
2
3)()'2
'1(' :C D M
0)1('
2
vvvH
Hv
vvvH
Hv
v
therefore S3 is also a probeof dark energy interaction
(L.A. & C. Quercellini, PRL 2004)
Skewness as a test of DE coupling
Sloan DSS:Predicted error on S3 less
than 10%
7/3 43S
Modified 3D gravityModified 3D gravity
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
μνστμνmatter R,RR,φ,f+L+Rgxd 4
)1)(12(3
)2(21
4
nn
nw
L+R+Rgxd
asympt
mattern
Simplest case:
Higher order gravity !
Turner, Carroll, Capozziello, Odintsov…
L.A., S. Capozziello, F. Occhionero, 1992
Modified N-dim gravityModified N-dim gravity
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
ymatter,R,Fdx=+L+Rggdyxd mattern
n 4
4444 ...
matterL+Rφfgxd 4
Simplest case:
Aspects of the same Aspects of the same physicsphysics
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
matterL+Rφ,fgxd 4
Extra-dim. Degrees of freedom
Higher order gravity
Coupled scalar field
Scalar-tensor gravity
The simplest caseThe simplest case
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
matterL+
R
μ+Rgxd 4
is equivalent to coupled dark energy
RφL+R'+Lg'xd matter,Rφ4
But with strong coupling !
2/1=β
2
33
2
3)'(3
mmm
m
H
VH
R+1/R modelR+1/R model
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
rad mat
field
rad mat
fieldMDE
toda
y
9/1=Ωφ
2/1=β
a= t 1/2
R+RR+Rnn model model
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
2/1=β
9/1=Ωφ
L.A., S. Tsujikawa, D. Polarski 2006
a=t 1 /2
Distance to last scatteringDistance to last scatteringin R+Rin R+Rnn model model
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
2/1=β
9/1=Ωφ
decz
zH
dz=zr
)(
a=t 1 /2
General f(R, Ricci, General f(R, Ricci, Riemann)Riemann)
A) Lambda
B) scalar field
C) modified gravity
D) non linear effect
mμνστμνστ
nμνμν RRβ+RRα+Rgxd 4
a=t 1 /2
we find again the same past behavior:
so probably most of these models are ruled out.
Anti-gravity has many side-effects…
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