Extra Dimensions, Cosmology and the end of Space and Time
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Transcript Extra Dimensions, Cosmology and the end of Space and Time
Physical Constraints on
Gauss-Bonnet Dark Energy
Cosmologies
Ishwaree Neupane
University of Canterbury, NZ
DARK 2007, Sydney September 25, 2007
Recently, there has been a renewal of
Interest in scenarios that propose
alternatives or corrections to
Einstein’s gravity.
The proposals are of differing origin as
well as motivations: some are based on
multi -dimensional theories, others on
scalar-curvature couplings.
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),
Leff
1
2 2
R
S S
(S S ) 2
Re S
3
T T
1
1
2 2
2
(Re
S
)
R
(Im
S
)
R
GB
R
2
8
(T T ) 8
2
g s2
e , stringdilaton
Im S pseudoscalar axion
Re T e
2
1
(compactification radius) 2
Dark energy from stringy gravity
One-loop corrected (heterotic) superstring action
R
2
2
S g d x g 2 V ( , ) ( ) ( ) 2 f ( ) ( ) R GB
2
2
2
4
modulus
a Brans-Dicke-like
runway dilaton
(a)
2
3
a
a 24H
2
2
( H H 2 ) R 2 4R ab Rab R abcd Rabcd RGB
Gauss-Bonnet
curvature density
In a known example of string compactification
f ( ) f 0
( / 0 )
e
2
..... ( ) ln(2) cosh( / 0 ) ......
3
No good reason to omit the scalar-curvature couplings
apart from complication
How can current observations
constrain such models?
The simplest Example: A fixed modulus &
no Gauss-Bonnet coupling
This simplifies the theory a lot
R
2
S d x g 2 V ( ) ( )
2
2
Define
x
2
( / H )
2
y 2 (V / H 2 )
EOMs
H/ H
2
4
2
2
1 2
w 1
q
3
3 3
Effective Equation of State
y 3 , x
sufficiently simple
x=0 and y=3 is a de Sitter fixed point : Lambda-CDM
Too Many choices
Quadratic
Exponential potential
1 2 2
V ( ) V0 m ...
2
V ( ) V0
Inverse power-law
Axion potential
V ( ) C cos
0
4
e
( / 0 )
4 0
V ( ) , 2
The issue may not be simply to achieve the dark
energy equation of state
w 1
DE
For the model to work the scalar field must relax its
potential energy after inflation down to a sufficiently
low value: close to the observed of dark energy
Gauss-Bonnet driven effective dark energy
1
1
R
2
S grav d 4 x g 2 V ( ) ( ) 2 f ( ) R GB
2
8
2
(a)
2
3
a
a 24H
2
2
( H H 2 ) RGB
R 2 4R ab Rab R abcd Rabcd
GB term is topological in 4D, and, if coupled, no Ghost for Minkowski background.
Cosmology requires FRW, Inflation non-constant scalar coupling
ln[a(t )] const
Number of e-folds primarily
depends on the field value
f ( ) f 0 f1 e ( / 0 ) ,
V( )
GB gravity may be a solution to the
dark energy problem, but a large
scalar coupling strength is required
2(1- )
3 4 f
'
V0 e ( / 0 )
1 3 , 202 / 2
Crossing the barrier of cosmological constant
Equation of state parameter for the potential
From top to bottom
0 4, 5, 6, 8, 10
V( ) V0 e 2 /0
Dynamics may be well behaved, but
An exact solution:
Let
1
2
2
( ) V ( ) f ( ) RGB ........ (3 ) H ( )
8
Ansatz
N ( )
f ( ) H u( ) u0e
2
N ln[a(t )] const
Scalar spectral index
ns 1 4 H 2 H
Nature of the dark energy
Is
CMB wDE 1
+
preferred?
Null dominant energy
condition : energy
LSS
doesn’t propagate
outside the light cone
A model with
. 2
w 1
1
L V ( )
2
Tegmark et al. 2004
Gauss-Bonnet corrections: No need to
introduce a wrong sign kinetic term
wDE 1 wDE 1
A couple of remarks:
wDE weff
1.
w DE
.
p DE
weff 1
DE
2 H
1 2
q
2
3H
3 3
wDE does not depend on the equation of state of other
fluid components, while weff
definitely does
2.
Dark energy or cosmological constant problem is
a cosmological problem: Almost every model of
scalar gravity behaves as Einstein’s GR for
.
H
0.0 1
The Simplest Potentials
V ( ) V0 e
/ mP
,
/ mP
f , f 0 e
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
A non-minimally coupled scalar field
S S grav S matter
S m S A 2 ( ), m d 4 x g A 4 ( ) ( m rad s )
d ln A( )
Q
d
Local GR constraints on Q and its derivatives
(Damour et al. 1993, Esposito-Farese 2003)
2
2
mPl
Q2
dQ
4 10 , mPl
4.5
d
5
Within solar system and laboratories distances: (dG / dt) / G
12 years
is less than
10
d ln A( )
Q
d
For the validity of weak equivalence principle
.
0.8, Q ~
5.10-5
H
Damour et al. gr-qc/0204094 (PRL)
Crossing of w = -1?
In the absence of GB-scalar coupling, a crossing between nonphantom w 1 and phantom cosmology w 1 is unlikely.
A smooth progression to
V ( ) H 2 ( ) 0 1e ,
9,
2 / 3,
0 108
w 1
f ( ) e
Ghost and Superluminal modes
One may also consider a metric spacetime under quantum
effect: perturbed metric about a FRW background
A gauge invariant quantity:
so-called a comoving perturbation
S linear
H
D(t)
2
dt a - C(t) 2
a
3
No-ghost and stability conditions:
C (t ), D(t ) 0
0 Ck2 1
Speed of
propagation
C k2
D (t )
C (t )
Propagation speed of a scalar mode
..
[4 (1 ) f ]
2
cR 1
[2 (1 ) x 2 3 2 ](1 )
2
f, f 0 e ..., V ( ) V0 e ....
2 / 3 and 12, 8, 3, 2/3 (left to right)
GB f H
Propagation speed for a tensor mode
2 / 3 and 12, 8, 3, 2/3 (left to right)
f , f 0 e
..., V ( ) V0 e
....
..
c T2
1 f
.
1 f H
Observing the effects of a GB coupling
The growth of matter fluctuations
GB f H
~
2H 4G m
f
~
G G 1 3GB
2
H
f
GB
m
m
is the matter
density contrast
12
| Gnow Gnucleo | / Gnow (t now t nucleo ) 10
~
dG / dt
~ 0.01 H 0
G
'
yr
1
~ O(0.1)
H
f ( ) ~ e / mP
~ O(1)
Growth of matter perturbations is
3 m
1 q1
GB
4
GB
With the inputs m 0.26, q -0.6 the observational
limit on growth factor
0.51 0.1
implies that | GB | 0.2 on large cosmological scales
Summary
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 wDE 1
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
f
anisotropic stress
1
can be more stronger
5
2GB 1
10
H 2
f