Domain wall solution in $F(R)$ gravity and variation of the fine structure constant

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Domain wall solution in F (R) gravity and variation of the fine
structure constant
Kazuharu Bamba1,∗, Shin’ichi Nojiri1,2,† and Sergei D. Odintsov3,‡,§
1Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan
2Department of Physics, Nagoya University, Nagoya 464-8602, Japan
3Institucio` Catalana de Recerca i Estudis Avanc¸ats
(ICREA) and Institut de Ciencies de l’Espai (IEEC-CSIC),
Campus UAB, Facultat de Ciencies, Torre C5-Par-2a pl,
E-08193 Bellaterra (Barcelona), Spain
Abstract
We construct a domain wall solution in F (R) gravity. We first compare a scalar field theory hav-
ing a runaway type potential with a corresponding scalar field theory obtained through a conformal
transformation of F (R) gravity and illustrate a behavior of F (R) as a function of R. Furthermore,
we reconstruct a static domain wall solution in a scalar field theory. We also reconstruct an explicit
F (R) gravity model in which a static domain wall solution can be realized. Moreover, we show
that there could exist an effective (gravitational) domain wall in the framework of F (R) gravity. In
addition, it is demonstrated that a logarithmic non-minimal gravitational coupling of the electro-
magnetic theory in F (R) gravity may produce time-variation of the fine structure constant which
may increase with decrease of the curvature. We also present cosmological consequences of the
coupling of the electromagnetic field to a scalar field as well as the scalar curvature and discuss
the relation between variation of the fine structure constant and the breaking of the conformal
invariance of the electromagnetic field.
PACS numbers: 04.50.Kd, 95.36.+x, 98.80.Cq
∗ E-mail address: bamba@kmi.nagoya-u.ac.jp
† E-mail address: nojiri@phys.nagoya-u.ac.jp
‡ E-mail address: odintsov@ieec.uab.es
§ Also at Tomsk State Pedagogical University.
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I. INTRODUCTION
According to recent cosmological observations, e.g., Supernovae Ia (SNe Ia) [1], cosmic mi-
crowave background (CMB) radiation [2, 3], large scale structure (LSS) [4], baryon acoustic
oscillations (BAO) [5], and weak lensing [6], it has been implied that the current expansion
of the universe is accelerating. Studies on the late time cosmic acceleration are classified
into the representative two categories. One is to introduce dark energy such as cosmological
constant in the framework of general relativity (for a recent review, see, e.g., [7]). The other
is to modify the gravitational theory, for instance, F (R) gravity, where F (R) is an arbitrary
function of the scalar curvature R (for recent reviews on F (R) gravity, see, e.g., [8–10]).
Recently, not only temporal [11, 12] but also spatial [13] variations of the fine structure
constant αEM have been suggested. To account for the spatial variation of αEM, the signature
of a domain wall produced in the spontaneous symmetry breaking involving a dilaton-like
scalar field coupled to electromagnetism has been considered in Ref. [14]. Furthermore, in
Ref. [15] it has been shown that a runaway domain wall, which is formed by a runaway type
potential of a scalar field [16], can explain both the time variation by its potential and the
spatial one by its formation simultaneously.
On the other hand, a domain wall solution in the framework of F (R) gravity has not
been investigated in detail yet. In particular, it is interesting to reconstruct an F (R) gravity
model in which a domain wall solution can be realized. It is known that F (R) gravity can
be written as a corresponding scalar field theory through a conformal transformation to the
Einstein frame. In this paper, as one approach, by comparing a scalar field theory having
a runaway type potential considered in Ref. [15] with a corresponding scalar field theory
to which F (R) gravity is transformed through a conformal transformation, we examine
a behavior of F (R) as a function of R. It is demonstrated that the deviation of F (R)
gravity from general relativity increases as the curvature becomes large and it asymptotically
becomes constant in the high curvature regime. As another approach, we reconstruct an
explicit F (R) gravity model in which a static domain wall solution can be realized. First,
by using a procedure proposed in Ref. [17], we reconstruct a static domain wall solution
in a scalar field theory. Next, in a similar configuration, we reconstruct an explicit form
of F (R) with forming a static domain wall solution. Moreover, by applying the method
of reconstruction of F (R) gravity in Ref. [18], we show that there could exist an effective
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(gravitational) domain wall in the framework of F (R) gravity. In addition, we discuss an
issue of a connection between F (R) gravity and variation of the fine structure constant
by exploring non-minimal Maxwell-F (R) gravity. Furthermore, we present cosmological
consequences of the coupling of the electromagnetic field to a scalar field as well as the
scalar curvature. We also study the relation between variation of the fine structure constant
and the breaking of the conformal invariance of the electromagnetic field. We use units of
kB = c = ~ = 1 and denote the gravitational constant 8piG by κ2 ≡ 8pi/MPl2 with the
Planck mass of MPl = G−1/2 = 1.2 × 1019GeV. Moreover, in terms of electromagnetism we
adopt Heaviside-Lorentz units.
The paper is organized as follows. In Sec. II, we describe F (R) gravity and a correspond-
ing scalar field theory by using a conformal transformation of F (R) gravity to the Einstein
frame. Furthermore, we compare a scalar field theory having a runaway type potential with
a corresponding scalar field theory and study a behavior of F (R) as a function of R. In
Sec. III, we reconstruct a static domain wall solution in a scalar field theory. In Sec. IV, we
also reconstruct an explicit F (R) gravity model in which a static domain wall solution can
be realized. In Sec. V, we demonstrate that there could exist an effective (gravitational)
domain wall in F (R) gravity. In Sec. VI, we consider non-minimal Maxwell-F (R) gravity
and examine a relation between F (R) gravity and variation of the fine structure constant.
In addition, we investigate cosmological consequences of the coupling of the electromagnetic
field to a scalar field as well as the scalar curvature in Sec. VII. Finally, conclusions are given
in Sec. VIII.
II. COMPARISON OF F (R) GRAVITY WITH A SCALAR FIELD THEORY HAV-
ING A RUNAWAY TYPE POTENTIAL
A. F (R) gravity and a corresponding scalar field theory
The action of F (R) gravity with matter is written as
S =
∫
d4x
√
−gF (R)2κ2 +
∫
d4xLM (gµν ,ΨM) , (2.1)
where g is the determinant of the metric tensor gµν and LM is the matter Lagrangian.
We make a conformal transformation to the Einstein frame:
g˜µν = Ω2gµν , (2.2)
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where
Ω2 ≡ F,R , (2.3)
F,R ≡
dF (R)
dR . (2.4)
Here, a tilde represents quantities in the Einstein frame. We introduce a new scalar field φ,
defined by
φ ≡
√
3
2
1
κ lnF,R . (2.5)
The relation between R and R˜ is expressed as
R = e1/
√
3κφ
[
R˜ +
√
3�˜ (κφ)− 12 g˜
µν∂µ (κφ) ∂ν (κφ)
]
, (2.6)
where
�˜ (κφ) = 1√−g˜ ∂µ
[
√
−g˜g˜µν∂ν (κφ)
]
. (2.7)
The action in the Einstein frame is given by [19]
SE =
∫
d4x
√
−g˜
(
R˜
2κ2 −
1
2 g˜
µν∂µφ∂νφ− V (φ)
)
+
∫
d4xLM
(
(F,R)−1 (φ)g˜µν ,ΨM
)
, (2.8)
where
V (φ) = F,RR˜− F
2κ2 (F,R)2
. (2.9)
B. Runaway domain wall and a varying fine structure constant αEM
In Ref. [15], the following action describing a runaway domain wall and a space-time
varying fine structure constant αEM has been proposed:
SE =
∫
d4x
√
−g˜
(
R˜
2κ2 −
1
2 g˜
µν∂µφ∂νφ− V (φ)
)
+
∫
d4x
√
−g˜
(
−14B(φ)g˜
µαg˜νβFµνFαβ
)
+ Smatter , (2.10)
where
V (φ) = M
2p+4
(φ2 + σ2)p , (2.11)
B(φ) = e−ξκφ . (2.12)
Fµν = ∂µAν − ∂νAµ . (2.13)
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Here, Fµν is the electromagnetic field-strength tensor and Aµ is the U(1) gauge field. Smatter
is the action for other ordinary matters (radiation, non-relativistic matter and baryon).
Moreover, V (φ) is a scalar field potential of runaway type, M is a mass scale, p(> 1) is a
constant assumed to be larger than unity, σ(< φ) is a constant assumed to be smaller than
the value of φ. It is known that although there is no minima in the potential V (φ), the
discrete symmetry φ↔ −φ can be broken dynamically and consequently a domain wall can
be formed. Furthermore, B(φ) is a coupling function of φ to the electromagnetic kinetic
term and ξ is a constant. The spatio-temporal variations of αEM come from the variation of
B(φ) in terms of space and time because αEM(φ) = α(0)EM/B(φ), where α
(0)
EM = e2/ (4pi) with e
being the charge of the electron [26], is the bare fine structure constant, and ξ is a constant.
We note that since the electromagnetic fields have the conformal invariance, the conformal
transformation in Eq. (2.2) does not generate the non-trivial coupling of the scalar filed φ
with the electromagnetic fields.
By equating Eq. (2.9) to Eq. (2.11), we obtain
F (R˜) = e
√
2/3κφ
[
R˜− 2κ2 M
2p+4
(φ2 + σ2)p e
√
2/3κφ
]
. (2.14)
We analyze Eq. (2.14) by using Eq. (2.5) and F,R = e
√
2/3κφ. We describe F (R) as
F (R) = R + f(R) . (2.15)
By combining Eq. (2.14) with Eq. (2.15), we find
F (R˜) = R˜ + f(R˜)
=
(
1 + df(R˜)
dR˜
)




R˜− 2κ2 M
2p+4
{
[3/ (2κ2)]
[
ln
(
1 + df(R˜)/dR˜
)]2
+ σ2
}p
(
1 + df(R˜)
dR˜
)




.(2.16)
We consider the case in which
∣
∣
∣
df(R˜)/dR˜
∣
∣
∣
≪ 1. Using Eq. (2.16) and taking the term in
terms of df(R˜)/dR˜ up to the first order, we acquire the following approximate differential
equation:
df(R˜)
dR˜
+ 1
R˜− 4β
f(R˜) + 2β
R˜− 4β
= 0 , (2.17)
with
β ≡ κ2M4
(M
σ
)2p
. (2.18)
We solve Eq. (2.17) numerically.
5