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