Hybrid metric-Palatini brane system

0.29 MB·

by  Qi-Ming Fu

.
Hybrid metric-Palatini brane system
Qi-Ming Fu , Li Zhao , Yu-Xiao Liu
∗ † ‡
Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China
ItisknownthatthemetricandPalatiniformalisms ofgravitytheorieshavetheirowninteresting
features but also suffer from some different drawbacks. Recently, a novel gravity theory called
hybridmetric-Palatinigravitywasputforwardtocureorimprovetheirindividualdeficiencies. The
action of this gravity theory is a hybrid combination of the usual Einstein-Hilbert action and a
f(R) term constructed by the Palatini formalism. Interestingly, it seems that the existence of a
light andlong-range scalar fieldin thisgravity maymodify thecosmological and galactic dynamics
withoutconflictingwiththelaboratoryandSolarSystemtests. Inthispaperwefocusonthetensor
6 perturbation of thick branes in this novel gravity theory. We consider two models as examples,
1 namely, the thick branes constructed by a background scalar field and by pure gravity. The thick
0 branesin bothmodels havenoinnerstructure. However, thegraviton zeromode inthefirst model
2 hasinner structurewhen the parameter in this model is larger than its critical value. Wefind that
n the effective four-dimensional gravity can be reproduced on the brane for both models. Moreover,
a thestability of both branesystems against the tensor perturbation can also beensured.
J
5 PACSnumbers: 04.50.Kd,04.50.-h,11.27.+d
2

] I. INTRODUCTION likeC-theories. Thenewfeatureofsuchhybridgravityis
c that it predicts the existence of a light long-range scalar
q
field, which can be used to explain the late-time cosmic
- General relativity is a successful gravitational theory
r acceleration [9].
g at the scale of the Solar System. However, it does not
[ work well at largerscales. Thus, many modified theories Consideringthecharacteristicsoflightandlong-range,

1 of gravity have been put forward to describe cosmolog- thereisapossibilitythatthisscalarfieldmaymodifythe
cosmological and galactic dynamics [9–12] without con-
v ical behaviors such as cosmic acceleration and galactic
flicting with the laboratory and Solar System tests. In
6 dynamics [1–5]. In general, there are three kinds of for-
4 malisms for modified gravity theories, namely, the met- Ref. [11], the authors analyzed the criteria for obtain-
5 ingcosmicaccelerationandobtainedseveralcosmological
ric formalism, Palatini formalism (matters do not cou-
6 solutions, which describe both accelerating and deceler-
ple with the priori metric-independent connection), and
0 ating Universes, depending on the form of the effective
metric-affine formalism (matters couple with the metric
.
1 and priori metric-independent connection) [1]. They all scalar potential. The virial theorem was studied in the
0 context of the galaxy cluster, where the mass dispersion
have their own interesting properties and, at the same
6 relationwasmodifiedbyatermrelatedtothenewscalar
time, suffer from different drawbacks. Recently, the so-
1
field predicted by the hybrid metric-Palatini theory of
called C-theories were proposed in Refs. [6–8], which
:
v establishes a bridge between the first and second for- gravity[12]. Stabilityofthe EinsteinstaticUniversewas
i alsoanalyzedinRef. [13]andalargeclassofstablesolu-
X malisms in order to find ways to cure or improve their
tionswerefound. InRef. [14],theauthorsconsideredthe
individual deficiencies. In C-theories, the Levi-Civita
r
a connection Γˆ of the metric gˆ is conformally related to possibilitythatwormholesolutionsmaybesupportedby
µν
this hybrid metric-Palatini gravity according to the null
the spacetimemetricg ,namely,gˆ = ( )g ,where
µν µν µν
C R energy conditions at the throat, and found some specific
is an arbitrary function of the Ricci curvature scalar
C = [g,Γˆ]=gµν [Γˆ] only. examples. In Ref. [15], the authors showed that the ini-
µν
R R R tialvalueproblemcanbewell-formulatedandwell-posed.
Alternatively, another novel modified gravity was pre- Moreover,the dynamics of linear perturbation and ther-
sented in Ref. [9], whose action is a hybrid combination modynamics in hybrid metric-Palatini gravity were also
oftheusualEinstein-Hilbertactionandaf( )termcon- investigated in Refs. [16] and [17], respectively. For a
R
structedby the Palatiniformalism. It has a dynamically detailed introduction, see the recent review [18].
equivalentscalar-tensorrepresentationlikethepuremet-
On the other hand, it has been extensively considered
ric and pure Palatini cases [9–12]. It also shares proper-
in the past two decades that our four-dimensional world
ties ofboth the metric formalismandPalatiniformalism
might be just a brane embedded in a higher-dimensional
spacetime. This idea provides new insights into solving
some long existing problems, such as the gauge hierar-
chy and cosmologicalconstant problems [19–24]. Dating
∗[email protected]
†[email protected],correspondingauthor back to the original Kaluza-Klein (KK) theory, the ex-
‡[email protected],correspondingauthor tra dimension is compactedinto a circle with the Planck
2
scale radius. This makes detecting the extra dimension where κ2 = 8πG with G the five-dimensional New-
5 5
hopeless. While in brane scenarios, the sizes of extra di- tonian gravitational constant and we have set c = 1.
mensionscanbetheorderofsubmillimeter[21]orinfinite S is the standard matter action, R = gMNR is
m MN
[23]. theEinstein-HilbertRicciscalarconstructedbythemet-
In the Randall-Sundrum-II brane scenario [23], the ric, and =gMN MN is the Palatini curvature, where
R R
thicknessofthebraneisneglected. Inmorerealisticthick MN is defined in terms of a torsionless independent
brane models the original singular brane is replaced by Rconnection, Γˆ, as
a smooth domain wall generated by matter fields. The
thick brane models have been extensively studied in the ΓˆP ΓˆP +ΓˆP ΓˆQ ΓˆP ΓˆQ .(2)
context of higher-dimensional gravity theories [25–40]. RMN≡(cid:16) MN,P − MP,N PQ MN − MQ PN(cid:17)
The linearization of a brane system is one of the most
Introducing an auxiliary scalar field φ, the action (1)
important issues in brane models [41–56]. First, it is a
can be deformed as
key procedure to investigate stability of the brane solu-
tion. Second,toreproducethe effectivefour-dimensional 1
S = d5x√ g R+φ V (φ) +S , (3)
gravity, we need also to study the linear perturbation 2κ2 Z − h R− 1 i m
of the brane system. Third, the linear perturbation will
result in the interaction of matter fields with the KK where
gravitons, which can be tested by experiments.
df( )
In the previous investigations about a brane system,
φ F( )= R , V (φ) F( ) f( ). (4)
1
the metric [51–55] and Palatini formalisms [56, 57] were ≡ R d ≡R R − R
R
individually considered. Therefore, it is interesting to
The field equations can be obtained by varying the
study the properties of a brane system in a gravity the-
action (3) with respect to the metric g , the scalar
ory containing both formalisms. For example, how does MN
the hybrid of the two formalisms affect the properties field φ, and the independent connection ΓˆPMN:
of the brane solutions and stability of the linear per-
1
turbation? This motivates us to investigate the hybrid R +φ (R+φ V )g =κ2T , (5a)
MN MN 1 MN MN
metric-Palatini brane system. In this paper, inspired by R − 2 R−
thescalar-tensorrepresentationofhybridmetric-Palatini V1φ = 0, (5b)
R−
gravity, we will consider two models: the thick branes ˆ √ gφgMN = 0, (5c)
P
constructed by a background scalar field (model A) and ∇ −
(cid:0) (cid:1)
bypuregravitationalsystem(modelB)inhybridmetric- where V dV1(φ), the matter stress-energy tensor is
Palatini gravity. In Refs. [58–65], some brane mod- 1φ ≡ dφ
els have been constructed for pure gravitationalsystems definedasusualTMN ≡−√2gδ(√δg−MgNLm),and∇ˆP iscom-
without matter fields. This scenario is the same as pro- patible with the connection−ΓˆP .
MN
ducinganexpandinguniversefromf(R)gravitywithout
The solution of Eq. (5c) implies that the indepen-
introducing an extra scalar field (inflation without infla-
dentconnectionistheLevi-Civitaconnectionofametric
ton). Inordertostudytheissuesofstabilityofthetensor q =φ2/3g . Then, the relation between and
MN MN MN
perturbation and localizationof the gravitonzero mode, R
R is
MN
we will investigate the linearization of these two brane
models. 4 1
= R + ∂ φ∂ φ φ
In this work, we focus on brane in hybrid metric- RMN MN 3φ2 M N − φ(cid:16)∇M∇N
Palatini gravity. Therefore, in Sec. II we briefly in- 1
troduce the hybrid metric-Palatini model and find thick + gMN(cid:3)φ , (6)
3
(cid:17)
brane solutions for model A and model B. Stability of
the brane system and localization of the graviton zero where (cid:3) gKL . Using the relation (6), one can
K L
mode are analyzed in Sec. III. Section IV comes with obtain a s≡calar-t∇enso∇r representation [9–11]:
the conclusion.
1 4
S = d5x√ g (1+φ)R+ ∂Kφ∂ φ
2κ2 Z − h 3φ K
II. THE HYBRID METRIC-PALATINI BRANE V (φ) +S . (7)
1 m
MODELS AND SOLUTIONS − i
Now, it is clear that the free choice of the form of the
Now,letusstartwiththeactionofthefive-dimensional f( ) is transformed to the potential V (φ) of a scalar
1
R
brane model in hybrid metric-Palatini gravity [9]. profileφ. Inspiredbythescalar-tensorrepresentation,we
consider two models: model A for the brane constructed
1 by a matter scalar field χ and model B for the brane
S = d5x√ g R+f( ) +S (g,χ), (1)
2κ2 Z − h R i m constructedbythepuregravitywithoutanymatterfield.
3
A. Model A: with matter Inordertoavoidtheghostproblem,weshouldensurethe
positive definiteness of the coefficient of R in the action
The action of the matter part with a scalar field is (7),soweshouldtakea>0. Nowitcanbe checkedthat
the system supports the following solutions:
1
S = d5x√ g gMN∂ χ∂ χ V (χ) . (8)
m Z − h− 2 M N − 2 i χ(y)=tanh(ky) 1 3(5a+3)cosh(2ky)+5a+9
r6
(cid:16) (cid:17)
Substituting Eq. (6) and Eq. (5b) in Eq. (5a), the gravi-
10a+9 15a+9
tational field equation can be written as +i E iky,
r 3 10a+9
h (cid:16) (cid:17)
4 15a+9
(1+φ)RMN + ∂Mφ∂Nφ M Nφ gMN(cid:3)φ F iky, , (15a)
3φ −(cid:16)∇ ∇ − (cid:17) − (cid:16) 10a+9(cid:17)i
1 4 1
g (1+φ)R+ ∂Kφ∂ φ V (φ) =κ2T . V (y) = k2sech4(ky) (12 8a)cosh(2ky)
−2 MNh 3φ K − 1 i MN 1 −2 h −
(9) + 3(a+1)cosh(4ky)+49a+9 , (15b)
i
Considering Eq. (5b) and the trace of Eq. (6), one finds V (y)= 5k2sech2(ky) a sech2(ky)+5a+3 , (15c)
that the scalar field φ is governed by the second-order 2 2 h− i
evolution equation
wherewehavechosenb=1andκ=1forsimplicity,and
8φ(cid:3)φ 4∂ φ∂Kφ φ2 5V (φ) 3(1+φ)V the functions E andF are two kinds ofelliptic integrals.
K 1 1φ
− − − The shapes of this brane solution and the energy den-
(cid:2) +2φ2κ2T =0(cid:3). (10)
sity are shown in Fig. 1. Obviously, the energy density
peaks at the origin of the extra dimension, which rep-
Our discussion will be limited to the static flat brane
resents a single brane. It is not difficult to analyze the
scenario, for which the metric is given by
structure of the five-dimensional spacetime at y = ,
±∞
ds2 =e2A(y)η dxµdxν +dy2, (11) where the curvature R = 20k2 < 0. This means that
µν −
the spacetime is asymptotic AdS.
with y the extra dimension. Meanwhile, the scalar field, Wecanalsoobtaintheexpressionoff( )fromEq.(4)
R
φ = φ(y), is independent of the brane coordinates. For
the system (9)-(11), the Einstein field equations and f( )= 2a a 2+ 26ak2 +12k2. (16)
equation of motion of the scalar field φ are read as R 3 R− 120k2R 3
3(A +4A2)(1+φ)+7Aφ +V Then, the complete Lagrangian for gravity can be ex-
′′ ′ ′ ′ 1
pressed as
+2κ2V +φ = 0, (12a)
2 ′′
12(A′′+A′2)(1+φ)+4A′φ′+V1+4φ′′ =R+ 2a a 2+ 26ak2 +12k2. (17)
+2κ2V2+3κ2χ′2 4φ′2/φ = 0, (12b) L 3 R− 120k2R 3
−
32A′φ′2φ−φ2 5φ′V1−3(1+φ)V′1 −4φ′3
κ2φ(cid:0) φ2 10V +3χ2 +(cid:1)8φ φφ = 0, (12c) B. Model B: without matter
′ 2 ′ ′′ ′
−
(cid:0) (cid:1)
where a prime stands for the derivative with respect to
We can also construct a brane from the scalar profile
the extra-dimensional coordinate y.
φ(y)withoutintroducingthe matter fieldχ(y). Thenwe
Theequationofmotionofthematterfieldisdescribed
can obtain the field equations of the whole system just
by the following equation:
by omitting the terms about the matter field χ(y) from
Eqs. (12):
dV (χ)
2
4Aχ +χ = . (13)
′ ′ ′′
dχ 3(1+φ)(A′′+4A′2)+7A′φ′+V1(φ)+φ′′ = 0, (18a)
Equations (12)-(13) describe the whole system. There 12φ(1+φ)(A′′+A′2)+4A′φ′φ+φV1(φ)
are five variables, namely, A(y), φ(y), χ(y), V1(φ), and +4φφ′′ 4φ′2 = 0, (18b)
−
V (χ). However, there are only three independent equa-
2 8φ +32Aφ 5φV (φ)+3φ(1+φ)V = 0. (18c)
′′ ′ ′ 1 1φ
tions. So, one needs assume two conditions to solve this −
system. Now,thereareonlythreevariables,namely, A(y), φ(y),
In this model, we consider the following configuration and V (φ), but only two independent equations. So, we
1
for the scalar field φ(y) and warp factor A(y): just need one condition to solve this system.
Subtracting (18a) from (18b) yields
φ(y) = a tanh2(ky), (14a)
A(y) = b ln[sech(ky)]. (14b) 9φ(1+φ)A +3φφ 3φAφ 4φ2 =0. (19)
′′ ′′ ′ ′ ′
− −
4
ã2AHyL ΦHyL
ΧHyL
1.0 10
10
0.8 8
0.6 6 5
0.4 4 y
-10 -5 5 10
0.2 2
-5
y y
-10 -5 5 10 -10 -5 5 10 -10
(a)e2A(y) (b)φ(y) (c)χ(y)
V2HyL ΡHyL
V1HyL
100 120
y 100
-10 -5 5 10 80
80
60
-100 60
40
40
-150
20 20
-200 y y
-10 -5 5 10 -10 -5 5 10
(d)V1(y) (e)V2(y) (f)ρ(y)
FIG. 1: Plots of the brane solution (14)-(15) and energy density for model A. The parameters are set to b = 1, k =1, a = 1
for thin lines, a=5 for red dashed thick lines, a=10 for black thick lines.
It is easy to check that this equation yields a thin brane influence on the localization of fermions [66] from the
solution, i.e., φ(y) = φ and A(y) = αy , where both case of odd scalar kink solutions in other brane models
1
− | |
φ and α are constants. In this paper, we mainly focus [26, 67].
1
on thick brane solution, so we suppose
ΦHyL
A(y)=b ln[sech(ky)], (20) V1HyL
0.0
y
where b is a positive parameter in order to localize the -0.2 -6 -4 -2-2 2 4 6
graviton zero mode on the brane (see Sec. III). In order -0.4 -4
tokeeptheZ2 symmetryoftheextradimension,weonly -0.6 -6
lookfor evenfunctionsolutionforthe scalarφ(y). Thus, -8
-0.8
the initial condition for φ(y) can be assumed as y -10
-6 -4 -2 2 4 6 -12
φ(0)=φ , φ(0)=0, (21) (a)φ(y) (b)V1(y)
0 ′
from which and Eq. (19), we can get
FIG. 2: Plots of the scalar profile φ(y) and scalar potential
φ′′(0)=3(1+φ0)k2b. (22) V1(y) for model B. The parameters are set to b = 1, k = 1.
The black thick, red dashed thin, blue thin lines correspond
In order to ensure the positive definiteness of the coeffi- to φ =−0.9, −0.8, −0.7 respectively.
0
cientofRintheaction(7),werequire1+φ(y)>0,from
which one has 1+φ >0 and so φ (0)>0.
0 ′′
Consideringtheasymptoticbehaviourofthewarpfac-
tor A(y ) bk y , one can obtain the asymp- III. LOCALIZATION OF GRAVITY
→ ±∞ → − | |
totic behavior of the scalar profile from Eq. (19):
c Sincewealreadyhavetwobranemodels(modelAand
1
φ(y ) c , (23)
→±∞ → (ec2 +e bky )3 → 1 model B), we will consider stability of these two models
− | |
under the tensor perturbation of the spacetime metric
where c and c are integral constants related to φ(0) andthelocalizationofthegravitonzeromode,whichare
1 2
andφ(0). Thenumericalsolutionsofthescalarfieldand two important issues in brane theory. Generally speak-
′
scalar potential V (y) are plotted in Fig. 2, from which ing,the four-dimensionalmasslessgravitonshouldbe lo-
1
one can see that c increases with φ(0). Note that the calized on the brane in order to reproduce the familiar
1
even parity of the scalar φ(y) here would have different four-dimensional Newtonian potential. We will analyze
5
these issues in the following context. Equation(29)istheequationofmotionfortheKKmode
Since the scalar, vector, and tensor fluctuations are H(z), and it can be factorized as the supersymmetric
decoupled from each other, we can write the spacetime formL LH(z)=m2H(z)with L=( d +∂zf)andL =
† dz f †
metric under the tensor fluctuation as ( d + ∂zf). The Hermitian and positive definite of the
−dz f
ds2 =e2A(y)(η +h )dxµdxν +dy2, (24) operator L L ensure that m2 > 0. Thus, there is no
µν µν †
tachyonic KK mode.
where h represents the tensor fluctuation and it
µν By setting m = 0 in Eq. (29), the graviton zero mode
is transverse-traceless, i.e., ηµβ∂ h = 0 and h
ηµνh = 0. The field equationβofµtνhe tensor pertu≡r- can be solved as
µν
bation reads H (z)=N f 1(z)=N 1+φ e3A/2, (32)
0 0 − 0
φ p
h′µ′ν + 4A′+ 1+′φ h′µν +e−2A(cid:3)(4)hµν =0, (25) whereN0isanormalizationconstant. Thenormalization
(cid:16) (cid:17)
of the zero mode is expressed as
where (cid:3)(4) = ηµν∂ ∂ stands for the four-dimensional
µ ν
D’Alembertian operator. By making a coordinate trans- H2dz = H2 e Ady =N2 1+φ e2Ady =1. (33)
formation dy =eAdz, Eq. (25) can be rewritten as Z 0 Z 0 − 0 Z
(cid:0) (cid:1)
∂2h + 3∂ A+ ∂zφ ∂ h +(cid:3)(4)h =0. (26) For model A with the other parameters set to b = 1,
z µν (cid:16) z 1+φ(cid:17) z µν µν k =1, and κ=1, N can be calculated as N = 3 .
0 0 6+2a
q
After making the KK decomposition h = For arbitrary positive parameters, it can be shown that
µν
ε (x)f(z)H(z), we canget the following two equations: the integral in (33) is finite. So, the graviton zero
µν
mode in model A can be localized on the brane. Fig-
(cid:3)(4)εµν(x)=m2εµν(x), (27a) ure 3 shows the shapes of the effective potential of the
∂2H(z) 3∂ A+∂ ln f2(1+φ) ∂ H(z) gravitationalfluctuationandthenonnormalizedgraviton
− z −(cid:16) z z (cid:0) (cid:1)(cid:17) z zero mode. It can be seen that the shape of the effec-
∂2f ∂ A ∂ f ∂ φ ∂ f tive potential changes from volcano-like well to double
z +3 z z + z z H(z)=m2H(z).
−(cid:16) f f 1+φ f (cid:17) well with increasing a. From Eq. (31), we can obtain
U (0)= 4a2 18a+27/2. It is obviousthat the shape
(27b) ′′
− −
of the effective potential is volcano-like for U (0) > 0
′′
where Eq. (27a) is the Klein-Gordon equation for the and double well for U (0)< 0. The critical value of the
′′
four-dimensional massless (m = 0) or massive (m = 0) parametera is a = 3 √15 3 since we only need pos-
graviton. In order to get a Schr¨odinger-like equatio6 n of itivea. Thus,thec effe4c(cid:0)tivep−oten(cid:1)tialhasasinglewelland
the KK mode H(z), its first-order derivation should be a double well for 0<a<a and a>a , respectively.
c c
vanishing. Thus, the function f(z) can be solved from It can also be seen that the graviton zero mode is lo-
3∂zA+∂zln f2(1+φ) =0 as calized gradually far away from the origin of the extra
(cid:0) (cid:1) dimension (see Fig. 3(b)) because the shape of the effec-
e 3A/2
f(z)= − . (28) tive potential changes from volcano like to double well,
√1+φ see Fig. 3(a). This character does not mean a double
brane but a single brane because the energy density still
Then, Eq. (27b) can be rewritten as
peaksattheoriginoftheextradimension(seeFig.1(f)).
∂2+U(z) H(z) = m2H(z), (29) Therefore,evenalthoughthebranehasnointernalstruc-
− z
(cid:0) (cid:1) ture, the effective potential has a double well and the
where the effective potential for the KK mode is given
gravitonzero mode has a split for large a. This is a new
by
resultofthis modelthatdifferentfromthe previousones
(∂ f)2 ∂2f in the literatures.
U(z) = 2 z z Figure4showstheshapesoftheeffectivepotentialand
f2 − f
graviton zero mode in model B. From Eqs. (20), (21),
3 9 (∂ φ)2
= ∂2A+ (∂ A)2 z (22), and (31), one can get
2 z 4 z − 4(1+φ)2
+3∂zA ∂zφ+∂z2φ, (30) U(0)= φ′′(0) 3bk2 = 3bk2 3bk2 =0.
2(1+φ) 2(φ0+1) − 2 2 − 2
which can be rewritten in coordinate y as Thus, the shape of the effective potential of the gravita-
tionalfluctuationisalwaysadoublewellforanypositive
3 15 φ2
U(z(y)) = e2A A + A2 ′ parameters b and k. From the asymptotic behaviour of
′′ ′
(cid:16)2 4 − 4(φ+1)2 the warpfactor A(y ) bk y and scalar profile
→±∞ →− | |
+ 4A′φ′+φ′′ . (31) φ(y → ±∞) → c1, it is easy to check that the corre-
2(φ+1) sponding graviton zero mode for model B can also be
(cid:17)
6
H0HyL The brane in model A was constructed by a background
UHyL 1.5 scalar field χ. This brane system can be solved analyt-
8 ically. On the other hand, inspired by the scalar-tensor
6 1.0 representationofthisgravity,weconsideredthe possibil-
4 ity of the thick brane constructedby pure gravity,which
0.5
2 is called model B. We obtained a set of numerical solu-
y y tions for the brane system in this model. Then, we de-
-3 -2 -1 1 2 3
-2 -4 -2 2 4 rived the field equation of the tensor perturbation (25).
(a)U(y) (b)H0(y) AftertheKKdecomposition,weobtainedaSchr¨odinger-
like equation of the KK modes H(z), which is the equa-
tion of motion of the graviton along the extra dimen-
FIG. 3: Plots of the effective potential U(y) of the gravita- sion. This equationcanbe factorizedas a supersymmet-
tionalfluctuationandthegravitonzeromodeH(y)formodel ricform,whichensuresthestabilityofthebranesystem.
A.Theparametersaresettob=1,k=1,κ=1,anda=0.1
Furthermore, we also gave the condition that avoids the
for thin lines, a = 5 for red dashed thick lines, a = 10 for
ghost gravitons.
black thick lines.
In order to produce the four-dimensional Newtonian
potential, we analyzed the graviton zero modes in both
H HyL
0
models. The graviton zero mode in model A splits from
UHyL
0.30
one peak to two peaks with the parameter a increasing;
10 0.25
however, the brane does not split. This means that the
5 0.20
y 0.15 graviton zero mode is localized gradually far away from
-1.5-1.0-0.5-5 0.5 1.0 1.5 0.10 the origin of the extra dimension with increasing a. The
-10 0.05 reason is that the shape of the effective potential of the
y
-15 -2 -1 1 2 gravitational fluctuation changes from volcano like (0 <
a<a ) to double well (a>a ). The gravitonzero mode
(a)U(y) (b)H0(y) c c
in model B is localized around the origin of the extra
dimension and becomes thinner with the parameter k
FIG. 4: Plots of the effective potential of the gravitational increasing. The shape of the effective potential of the
fluctuation and the graviton zero mode of model B. The pa- gravitational fluctuation is always a double well. The
rametersaresettob=3,φ =−0.9,andk=1forthinlines, graviton zero modes in both models are localized on the
0
k=2 for red dashed thick lines, k=3 for black thick lines. branes. So, we can obtain the familiar four-dimensional
Newtonian potential for both models.
The localizationof various matter fields on the branes
normalizable: H2(z)dz < . Therefore, the graviton
0 ∞ in the two models (especially the one constructed by
zero mode canRbe localized on the brane. pure gravity) is an important and interesting question.
So,wecanconcludethatthe branecanbeconstructed
It leaves for our future work.
by the background scalar field or by pure gravity, and
the Newtonianpotentialonthebranecanbereproduced
on both models since the graviton zero mode can be lo-
calized on the brane. Acknowledgments
This workwassupportedbythe NationalNaturalSci-
IV. CONCLUSIONS AND DISCUSSION ence Foundation of China (Grants No. 10905027, No.
11375075, and No. 11522541), and the Fundamental
In this paper, we investigated two thick brane models Research Funds for the Central Universities (Grant No.
(modelAandmodelB)inhybridmetric-Palatinigravity. lzujbky-2015-jl01).
[1] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 [6] L.Amendola,K.Enqvist,andT.S.Koivisto,Phys.Rev.
(2010). D 83, 044016 (2011).
[2] A.DeFeliceandS.Tsujikawa,LivingRev.Relativity13, [7] T. S. Koivisto, Phys. Rev. D 83, 101501 (2011); T. S.
3 (2010). Koivisto, Phys. Rev.D 84, 121502 (2011).
[3] S.CapozzielloandM.DeLaurentis,Phys.Rep.509,167 [8] T. S. Koivisto, D. F. Mota, and M. Sandstad, Novel as-
(2011). pects of C-theories in Cosmology, arXiv:1305.4754.
[4] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, [9] T.Harko,T.S.Koivisto,F.S.N.Lobo,andG.J.Olmo,
Phys.Rep.513, 1 (2012). Phys. Rev.D 85, 084016 (2012).
[5] S.NojiriandS.D.Odintsov,Phys.Rep.505,59(2011). [10] S.Capozziello, T.Harko,F.S.N.Lobo,andG.J.Olmo,
7
Int.J. Mod. Phys.D 22, 1342006 (2013). Emerging Gauge Symmetry and de Sitter Bounce,
[11] S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo, arXiv:1511.03681.
and G. J. Olmo, JCAP 1304, 011 (2013). [41] A.KehagiasandK.Sfetsos,Phys.Lett.B472,39(2000).
[12] S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo, [42] J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778
and G. J. Olmo, JCAP 1307, 024 (2013). (2000).
[13] C. G. Boehmer, F. S. N. Lobo, and N. Tamanini, Phys. [43] M. Giovannini, Phys.Rev. D 64, 064023 (2001).
Rev.D 88, 104019 (2013). [44] D. Bazeia, C. Furtado, and A. R. Gomes, J. Cosmol.
[14] S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo, Astropart. Phys.0402, 002 (2004).
and G. J. Olmo, Phys.Rev. D 86, 127504 (2012). [45] M. Cvetic and M. Robnik, Phys. Rev. D 77, 124003
[15] S.Capozziello, T.Harko,F.S.N.Lobo,G.J.Olmo,and (2008).
S.Vignolo,Int.J.Geom.Meth.Mod.Phys.11,1450042 [46] A. Herrera-Aguilar, D. Malagon-Morejon, and
(2014). R. R.Mora-Luna, JHEP 1011, 015 (2010).
[16] N.A. Lima, Phys. Rev.D 89, 083527 (2014). [47] W. T. Cruz, A. R. Gomes, and C. A. S. Almeida, Euro-
[17] T.AziziandN.Borhani,Astrophys.SpaceSci.357,146 phys.Lett. 96, 31001 (2011).
(2015). [48] K.Yang,Y.X.Liu,Y.Zhong,X.L.Du,andS.W.Wei,
[18] S. Capozziello, T. Harko, T. S. Koivisto, F. S. N. Lobo, Phys. Rev.D 86, 127502 (2012).
and G. J. Olmo, Universe1, 199 (2015). [49] V.SantosandC.A.S.Almeida,Phys.Lett.B718,1114
[19] I.Antoniadis, Phys. Lett. B 246, 377 (1990). (2013).
[20] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, [50] A.AhmedandB.Grzadkowski,JHEP1301,177(2013).
Phys.Lett. B 429, 263 (1998). [51] K. Yang, X. L. Du, and Y. X. Liu, Phys. Rev. D 88,
[21] I.Antoniadis, N.Arkani-Hamed,S.Dimopoulos, and G. 124037 (2013).
R.Dvali, Phys.Lett. B 436, 257 (1998). [52] W. T. Cruz, L. J. S. Sousa, R. V. Maluf, and C. A. S.
[22] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 Almeida, Phys.Lett. B 730, 314 (2014).
(1999). [53] N. Barbosa-Cendejas, A. Herrera-Aguilar, U. Nuca-
[23] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 mendi, I. Quiros, and K. Kanakoglou, Gen. Rel. Grav.
(1999). 46, 1631 (2014).
[24] R. Maartens and K. Koyama, Living Rev. Rel. 13, 5 [54] M. Higuchi and S. Nojiri, Gen. Rel. Grav. 46, 1822
(2010). (2014).
[25] W.D.GoldbergerandM.B.Wise, Phys.Rev.Lett.83, [55] D. Bazeia, A. Lobao, L. Losano, R. Menezes, and G. J.
4922 (1999). Olmo, Phys.Rev. D 91, 124006 (2015).
[26] M. Gremm, Phys.Lett. B 478, 434 (2000). [56] B. M. Gu, B. Guo, H. Yu, and Y. X. Liu, Phys. Rev. D
[27] M. Gremm, Phys.Rev. D 62, 044017 (2000). 92, 024011 (2015).
[28] O.DeWolfe,D.Z.Freedman,S.S.Gubser,andA.Karch, [57] D. Bazeia, L. Losano, R. Menezes, G. J. Olmo, and D.
Phys.Rev.D 62, 046008 (2000). Rubiera-Garcia, Eur. Phys. J. C 75, 569 (2015).
[29] C. Csaki, J. Erlich, T. J. Hollowood, and Y. Shirman, [58] V. Dzhunushaliev, V. Folomeev, B. Kleihaus, and J.
Nucl.Phys. B 581, 309 (2000). Kunz, JHEP 1004, 130 (2010).
[30] T.GherghettaandA.Pomarol, Nucl.Phys.B586,141 [59] H.Liu,H.Lu,andZ.L.Wang,JHEP1202, 083(2012).
(2000). [60] Y. Zhong and Y. X. Liu, Pure geometric thick
[31] N. Arkani-Hamed, M. Porrati, and L. Randall, JHEP f(R)-branes: stability and localization of gravity,
0108, 017 (2001). arXiv:1507.00630.
[32] A.Campos, Phys. Rev.Lett. 88, 141602 (2002). [61] M. Parry, S. Pichler, and D. Deeg, JCAP 0504, 014
[33] S. Kobayashi, K. Koyama, and J. Soda, Phys. Rev. D (2005).
65, 064014 (2002). [62] K.A.BronnikovandS.G.Rubin,Grav.Cosmol.13,191
[34] A.Z. Wang, Phys.Rev.D 66, 024024 (2002). (2007).
[35] C. Charmousis, S. C. Davis, and J. F. Dufaux, JHEP [63] N. Deruelle, M. Sasaki, and Y. Sendouda, Prog. Theor.
0312, 029 (2003). Phys. 119, 237 (2008).
[36] D.Bazeia and A.R. Gomes, JHEP 0405, 012 (2004). [64] A. Balcerzak and M. P. Dabrowski, Phys. Rev. D 77,
[37] Y.X.Liu,L.Zhao,X.H.Zhang,andY.S.Duan, Nucl. 023524 (2008).
Phys.B 785, 234 (2007). [65] A. Herrera-Aguilar, D. Malagon-Morejon, and R. R.
[38] V. Dzhunushaliev, V. Folomeev, and M. Minamitsuji, Mora-Luna, JHEP 1011, 015 (2010).
Phys.Rev.D 79, 024001 (2009). [66] Y.X.Liu,Z.G.Xu,F.W.Chen,andS.W.Wei, Phys.
[39] Y. X. Liu, Y. Zhong, Z. H. Zhao, and H. T. Li, JHEP Rev. D 89, 086001 (2014).
1106, 135 (2011). [67] M.CveticandH.H.Soleng,Phys.Rep.282,159(1997).
[40] W. J. Geng and H. Lu, Einstein-Vector Gravity,