Carsten van de Bruck University of Sheffield

Saturday, 3 July 2010
The dilaton and modified
gravity
Carsten van de Bruck
University of Sheffield
Work in collaboration with P. Brax, A. Davis and D. Shaw
arXiv:1005.3735

Saturday, 3 July 2010
Scalar Fields and Modified Gravity
Scalar fields are common in particle physics (although not a single
one has been observed).
Usually those fields couple to matter.
Properties are often difficult to reconcile with cosmology/gravity
experiments.
Examples: a) Dilaton in string theory, which couples to all matter
forms and is rather light. b) Moduli fields in supersymmetry/gravity
and/or string theory.
Need mechanism to hide those type of fields from current
experiments, otherwise ruled out!

Saturday, 3 July 2010
Here we consider two mechanisms
to “hide” the scalar field(s):
Damour-Polyakov (DP) mechanism
(Damour&Polyakov, Nucl.Phys.B 423, 532 (1994))
The Chameleon Mechanism
(Khoury&Weltman, Phys.Rev.D 69, 044026 (2004))

Saturday, 3 July 2010
Chameleon mechanism
(for h(Φ)=1):
Chameleon effect is based on interactions of field with ambient matter
and nonlinear self-interactions.
Essential for chameleon mechanism is the thin-shell solutions to the
field equation! 80
!!! [M]
60
40
20
0 40 80 120 160
r [M -1 0
]

Saturday, 3 July 2010
Chameleon mechanism:
The force (given by the gradient of the scalar field) is suppressed
outside the body.
Not all bodies have this thin shell! Field can behave as a standard
massive scalar field and force can be long-ranged.
Field follows an effective potential, which consists of the bare
potential and a part which depends on the ambient matter
density:
Veff(φ) =V (φ)+ρmatterA(φ)

The coupling, usually given by
β =
∂ ln A(φ)
∂φ
is in the case of a thin shell given by
β 2 eff = β 2 |φ∞ − φc|
βΦN (R)
This results in constraints on the theory!
Saturday, 3 July 2010

Not all scalar-tensor theories are chameleon
theories!! Thin-shell solution essential!! Whether
or not a thin-shell solution exists, depends on
the potential V and the coupling function A!
What about the string-theory dilaton? We
require ßeff to be small. Consider a theory for
which
Saturday, 3 July 2010
A(φ) =e βφ
V (φ) =V0e −φ

The Damour-Polyakov mechanism
This mechanism is operating in theories in which
the coupling function A(ϕ) has a minimum. The
coupling
β =
∂ ln A(φ)
∂φ
is zero at this point. Cosmologically, the scalar
field is driven to the minimum of A(Φ) if the bare
potential vanishes! Damour and Polyakov (1994)
applied this mechanism to the dilaton, ignoring the
influence of V(Φ).
Saturday, 3 July 2010

Saturday, 3 July 2010
In our work we took the influence of the
potential into account and promoted the dilaton to be
a dark energy scalar field.
Damour, Piazza & Veneziano (Phys.Rev.D 66,
046007 (2002)) considered the dilaton as a dark
energy field too, but in their case the minimum of
A(Φ) was at Φ=∞! Here, we consider the minimum of
A(Φ) at finite value: very different constraints on the
theory and also cosmological behaviour is very
different!

Due to the non-canonical nature of the scalar
field (k(Φ) is not 1!), the coupling to matter is
not β, but
α = β(φ)2
=
k(φ) 2
Cosmologically, if field is near the minimum:
α =0.04...0.33
depending on λ. The field is long-ranged
(cosmological scales)!
Saturday, 3 July 2010
β(φ) 2
λ −2 +3β(φ) 2

Saturday, 3 July 2010
Note that
β(φmin) =
V (φmin)
4V (φmin)+A(φmin)ρm
therefore β → 0 for ρ →∞
The coupling α therefore tends to zero in
large density regions. We need it to be
smaller than 10 -5 in the solar system (the
strongest local constraint, from Cassini
experiment). One can indeed show that β is
driven to small values, provided A2 is large.

Value of A 2 × 10 5
Saturday, 3 July 2010
10 2
10 1
10 0
10 1
10 0
Allowed Parameter Space for Environmentally Dependent Dilaton
Lower bound on A 2 × 10 5
from necessary condition
Unshaded region violates Cassini bound
10 1
Value of s
Allowed Region
10 2

This translates into the allowed cosmological
interaction range
Cosmological Range of
Fifth Force: (in Mpc)
cos
Saturday, 3 July 2010
2.5
2
1.5
1
0.5
0
10 0
Allowed Values of the Cosmological Fifth Force Range: cos
Ruled out by local tests
10 1
Value of s
Allowed Region
10 2

Saturday, 3 July 2010
Interaction range important for structure formation
(on scales 0.5-2 Mpc). No modification of gravity
well above that scale.
Background evolution very similar to ΛCDM.
Deviations from w=-1 very small (not checked but
expect situation to be similar as in chameleon
cosmology).
Coupling rather small (0.04...0.33), but could be
significant, even if additional force is below gravity!

Observables are fgal=dln δg/dln a, and
the functions Σκm and ΣκI and the indicator
EG:
k 2 (Φ + Ψ) =−8πGa 2 ¯ρΣκmDGRδi
H −1 k 2 ( ˙ Φ + ˙ Ψ)=−8πGa 2 ¯ρΣκI(fGR − 1)DGRδi
EG = k2 (Ψ + Φ)
−3H 2 0 a−1 θ
and θ = − dδg
dlna
covered in talk by Brax; Brax, vdB, Davis, Shaw, JCAP 1004, 032 (2010) and for EG see
Zhang, Ligouri, Bean & Dodelson, Phys.Rev.Lett. 99, 141302 (2007).
Saturday, 3 July 2010

GR
Relative Modified Gravity Parameter: E /E 1
G G
Saturday, 3 July 2010
0.07
0.06
0.05
0.04
0.03
0.02
0.01
10 1
0
z=0
z=1
z=2
Modified Gravity Parameter, E G , for s = 10
10 0
10 1
Spatial Scale: (A 2
1/2 H0 ) 1 k
10 2
10 3

GR
Relative Modified Gravity Parameter: E /E 1
G G
Saturday, 3 July 2010
4.5
4
3.5
3
2.5
2
1.5
1
0.5
x 103
10 1
0
z=0
z=1
z=2
Modified Gravity Parameter, E G , for s = 1
10 0
10 1
Spatial Scale: (A 2
1/2 H0 ) 1 k
10 2
10 3

ISW Slip Parameter, I m
Saturday, 3 July 2010
1.2
1
0.8
0.6
0.4
0.2
0
10 1
0.2
z=0
z=1
z=2
ISW Slip Parameter, I m , for s = 10
10 0
10 1
Spatial Scale: (A 2
1/2 H0 ) 1 k
10 2
10 3

ISW Slip Parameter, I m
Saturday, 3 July 2010
1.005
1
0.995
0.99
0.985
0.98
0.975
10 1
0.97
ISW Slip Parameter, I m , for s = 1
10 0
10 1
Spatial Scale: (A 2
1/2 H0 ) 1 k
10 2
z=0
z=1
z=2
10 3

Weak Lensing Slip Parameter, m
Saturday, 3 July 2010
1.15
1.1
1.05
10 1
1
Weak Lensing Slip Parameter, m , for s = 10
z=0
z=1
z=2
10 0
10 1
Spatial Scale: (A 2
1/2 H0 ) 1 k
10 2
10 3

Weak Lensing Slip Parameter, m
Saturday, 3 July 2010
1.007
1.006
1.005
1.004
1.003
1.002
1.001
10 1
1
z=0
z=1
z=2
Weak Lensing Slip Parameter, m , for s = 1
10 0
10 1
Spatial Scale: (A 2
1/2 H0 ) 1 k
10 2
10 3

Saturday, 3 July 2010
Conclusions
Considered dilaton action in strong coupling regime, assuming minimum in
coupling function A(Φ) (motivated by work of Damour & Polyakov) in order to
“hide” the field.
Promoted field to be responsible for dark energy: field displaced from
minimum of effective coupling function.
Coupling therefore not zero, but can be small locally (depending on
parameter).
As a side remark: if A(Φ) would not have minimum, chameleon mechanism
cannot come to rescue.
Constraints on parameter (λ, A2) given from local constraints.
Cosmological interaction range below ∼2 Mpc, constraints are currently weak,
but hopefully soon much better constraints available. Leads to more stringent
constraints on strong coupling limit of string theory. For this, a more detailed
calculation of the non-linear regime is necessary.