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Physical Constraints on Physical Constraints on Gauss-Bonnet Dark Energy Gauss-Bonnet Dark Energy
CosmologiesCosmologies
Ishwaree NeupaneIshwaree Neupane University of Canterbury, NZUniversity of Canterbury, NZ
DARK 2007, Sydney September 25, 2007
Recently, there has been a renewal of Recently, there has been a renewal of Interest in scenarios that propose Interest in scenarios that propose alternatives or corrections to alternatives or corrections to Einstein’s gravity. Einstein’s gravity.
The proposals are of differing origin as The proposals are of differing origin as well as motivations: some are based on well as motivations: some are based on multi -dimensional theories, others on multi -dimensional theories, others on scalar-curvature couplingsscalar-curvature couplings. .
RRSRSTS
RL GBeff 222
222)(Im
8
1)(Re
8
1
)T(T
T 3
)S(S
S
2
1
22
2
) (
1Re
Im
,2
Re
radiuscationcompactifieT
axionarpseudoscalS
dilatonstringeg
Ss
Gauss-Bonnet Gravity: Motivations
Gauss-Bonnet gravity is motivated by the stability and naturalness of the models, uniqueness of a Lagrangian in higher dimensions, the low-energy effective string actions (heterotic string),
Dark energy from stringy gravityDark energy from stringy gravityOne-loop corrected (heterotic) superstring actionOne-loop corrected (heterotic) superstring action
2
GB22
24 R )()()(
2)(
2),(
2
fV
RgxdSg
a Brans-Dicke-like a Brans-Dicke-like runway dilatonrunway dilaton
modulusmodulus
22223
2
4)(24)(
GBabcdabcd
abab RRRRRRHHH
aaa
Gauss-Gauss-BonnetBonnet
curvature curvature densitydensity
.....)()/(
00 eff
......)/cosh(3
2)2ln()( 0
In a known example of string In a known example of string compactificationcompactification
No good reason to omit the scalar-No good reason to omit the scalar-curvature couplings curvature couplings
apart from complicationapart from complication
How can current observations constrain such models?
The simplest Example: A fixed modulus & The simplest Example: A fixed modulus & no Gauss-Bonnet coupling no Gauss-Bonnet coupling
This simplifies the theory a lotThis simplifies the theory a lot
2
24 )(
2)(
2
VR
gxdS
DefineDefine
2
22
22
/
)/(
)/(2
HH
HVy
Hx
EOMsEOMs xy ,3 sufficiently sufficiently
simplesimple
qw3
2
3
1
3
21
Effective Equation of StateEffective Equation of State
x=0 and y=3 is a de Sitter fixed point : Lambda-CDM
Too Many choicesToo Many choices
...2
1)( 22
0 mVV
2 ,)( 04
V
0
4 cos)( CV
Quadratic Quadratic
Inverse power-Inverse power-lawlaw
Axion Axion potentialpotential
eVV)/(
00)(
Exponential Exponential potentialpotential
The issue may not be simply to achieve the dark The issue may not be simply to achieve the dark
energy equation of stateenergy equation of state 1DEwFor the model to work the scalar field must relax its For the model to work the scalar field must relax its potential energy after inflation down to a sufficiently potential energy after inflation down to a sufficiently low value: close to the observed of dark energylow value: close to the observed of dark energy
Gauss-Bonnet driven effective dark energy Gauss-Bonnet driven effective dark energy
abcdabcd
abab
GB RRRRRRHHHaaa
4)(24)( 2222
3
2
2
GB2
24 R )(
8
1)(
2
1)(
2
fV
RgxdSgrav
)/( 0'4
)/(10
0
0
3
)-2(1)V(
,e )(
eVf
fff
Number of e-folds Number of e-folds primarily primarily
depends on the field valuedepends on the field value
constta )](ln[
2/ ,31 20
2 GB gravity may be a GB gravity may be a
solution to the dark energy solution to the dark energy problem, but a large scalar problem, but a large scalar
coupling strength is coupling strength is requiredrequired
GB term is topological in 4D, and, if coupled, no Ghost for Minkowski background. Cosmology requires FRW, Inflation non-constant scalar coupling
Crossing the barrier of cosmological constant
0/2 0 )V( eV
10 ,8 ,6 ,5 ,40
Equation of state parameter for the potential From top to bottom
Dynamics may be well behaved, butDynamics may be well behaved, but
An exact solutionAn exact solution: :
)()3(........)(8
1)()( 22 HRfV GB
LetLet
)( 0
2 )()( NeuuHf
consttaN )](ln[
Ansatz
Scalar spectral indexScalar spectral index
HHsn 24 1
CMB
+LSS
Nature of the dark energy
Tegmark et al. 2004
Is
1DEw
)(2
1 2.
VL
Null dominant Null dominant energy condition : energy condition : energy doesn’t energy doesn’t propagate outside propagate outside the light conethe light cone
A model withA model with
1wGauss-Bonnet corrections: No Gauss-Bonnet corrections: No need to introduce a wrong sign need to introduce a wrong sign kinetic term kinetic term
11 DEDE ww
preferred?
A couple of remarks:A couple of remarks:
DE
DEDE
pw
q
H
Hweff 3
2
3
1
3
21
2
.
DEw
effDE ww 1.
effw
01.0
.
H
2.
does not depend on the equation of state of other fluid components, while definitely does
Dark energy or cosmological constant problem is a cosmological problem: Almost every model of scalar gravity behaves as Einstein’s GR for
PP mm effeVV /0,
/0 , )(
The Simplest Potentials
Perhaps too naïve: The slopes of the potentials considered in a post inflation scenario are too large to allow the required number of e-folds of inflation
The above choices hold some validity as a post-inflation approximation
Dashed lines (SNe IA plus CMBR shift parameter) Shaded regions (including Baryon Acoustic Oscillation scale)
Koivisto & Mota hep-th/0609155
mattergrav SSS
)( )(),( 442sradmmm AgxdASS
d
AdQ
)(ln
A non-minimally coupled scalar field
5.4 ,104 5222
d
dQmQm PlPl
Local GR constraints on Q and its derivatives (Damour et al. 1993, Esposito-Farese 2003)
Within solar system and laboratories distances: is less than years
GdtdG /)/(1210
d
AdQ
)(ln
5-
.
10 .5~ ,8.0 Q
H
For the validity of weak equivalence principle
Damour et al. gr-qc/0204094 (PRL)
Crossing of w = -1Crossing of w = -1??
1wIn the absence of GB-scalar coupling, a crossing between non-In the absence of GB-scalar coupling, a crossing between non-phantom and phantom cosmology is unlikelyphantom and phantom cosmology is unlikely.. 1w
A smooth progression to
10
2 )( , )()( efeHV
10
,3/2
,9
80
1w
Ghost and Superluminal modesGhost and Superluminal modes One may also consider a metric spacetime under quantum One may also consider a metric spacetime under quantum
effect: perturbed metric about a FRW backgroundeffect: perturbed metric about a FRW background
H
A gauge invariant quantity:so-called a comoving perturbation
2
23
a
D(t)C(t)- adtSlinear
)(
)(2
tC
tDCk
Speed of propagation No-ghost and stability conditions:
10 2 kC
0)( ),( tDtC
)1](3 )1(2[
])1(4[1
22
..2
2
x
fcR
.... )( ..., 00, eVVeff
right) (left to 2/3 3, 8, ,12 and 3/2
Propagation speed of a scalar mode
HfGB
Propagation speed for a tensor mode
right) (left to 2/3 3, 8, ,12 and 3/2
.... )( ..., 00, eVVeff Hf
f.
..
2T
1
1c
mGH~
42 HfGB
0 01.0~/
~H
G
dtGd
Observing the effects of a GB coupling
m
m
is the matter density contrast
The growth of matter fluctuations
GBGB f
f
HGG
231
~
11210)(/|| yrttGGG nucleonownownucleonow
)1(~ ~)(
(0.1) ~
/
'
Oef
OH
Pm
Growth of matter perturbations isGrowth of matter perturbations is
GB
m
GB
q4
311
-0.6q ,26.0 m
1.051.0
With the inputs the observational limit on growth factor
implies that 2.0|| GB on large cosmological scales
SummarySummary
1DEw
Gauss-Bonnet modification of Einstein’s gravity can easily account for an accelerated expansion with quintessence, cosmological constant or phantom equation-of-state
The scalar-curvature coupling can also trigger onset of a late dark energy domination with
The model to be compatible with astrophysical observations, the GB dark energy density fraction should not exceed 15%.
The solar system constraints, due to a small fractional anisotropic stress 5
210121
f
f
HGB
can be more stronger