Table Of ContentCosmic evolution of scalar fields with multiple vacua: generalized DBI and
quintessence
Changjun Gao∗
Key Laboratory for Computational Astrophysics, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China and
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
You-Gen Shen†
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
(Dated: June9, 2016)
We find a method to rewrite the equations of motion of scalar fields, generalized DBI field and
6
quintessence, in the autonomous form forarbitrary scalar potentials. With the aid of this method,
1
we explore the cosmic evolution of generalized DBI field and quintessence with the potential of
0
multiple vacua. Then we find that the scalars are always frozen in the false or true vacuum in
2
the end. Compared to the evolution of quintessence, the generalized DBI field has more times of
n oscillationsaroundthevacuumofthepotential. Thereasonforthispointisthat,withtheincreasing
u ofspeedφ˙,thefrictiontermofgeneralizedDBIfieldisgreatlydecreased. ThusthegeneralizedDBI
J field acquires more times of oscillations.
8
PACSnumbers: 98.80.Es,98.80.Cq
]
c
q
I. INTRODUCTION of this article is to report that we have found a way.
-
r The DBI field describes the dynamics of D-branes
g
evolving in a higher-dimensional warped spacetime. A
[ Scalarfieldsplayanimportantroleinboththeoretical
novel aspect of this field is the existence of a speed limit
physics and modern cosmology for their simple but non-
5 on the field space, resulting from causality restrictions
trivialdynamics. Intheoreticalphysics,they arepresent
v on the motion of the branes in the bulk spacetime. The
0 intheJordan-Brans-DicketheoryasJordan-Brans-Dicke
speedlimitisenforcedbythenon-canonicalkineticterms
6 scalar[1];inKaluza-Kleincompactificationtheoryasthe
in the DBI field. This is different from the quintessence
9 radion [2]; in the Standard Model of particle physics as
whose speed φ could be arbitrarily large. From this
6 the Higgs boson [3]; in the low-energy limit of the su- ∇
0 point of view, quintessence and DBI field are the coun-
perstring theory as the dilaton [4], tachyon [5] and DBI
. terpart of Newtonian and Special Relativity mechanics,
1 (Dirac-Born-Infeld) field [6]. In cosmology, scalar fields
respectively. ThecosmicevolutionofDBIfieldhavebeen
0 are employed to model the inflaton [7], the quintessence
5 [8–14] (for a recent review of quintessence, see Refs. [15] studied in Refs. [29]. These researches only apply to
1 some special forms of potentials. Thus, to find a general
and references therein), the k-essence [16, 17], the phan-
: method applying to arbitrary DBI potential constitutes
v tom [18], in order to drive the inflation of the early Uni-
the second purpose of this article.
i verse or to speed up the expansion of the late Universe.
X
We note that most of the scalar fields are hypothetical
r
a particles. ButtheHiggsBosonhasuniquelybeendiscov-
ered by experiments. II. GENERALIZED DBI FIELD
Quintessence is a canonical scalar field which is as-
sumed to be minimally coupled to gravity. Compared We consider a D3-brane with tension T evolving in a
to other scalar fields, quintessence turns out to be the 5-dimensional spacetime. The dynamics of the mobile
simplest scenario which is free of ghosts and instability D3-brane is described by the DBI action. The D3-brane
problems. The dynamics of quintessence in the presence isfreetomoveonthe internalcompactCalabi-Yauman-
of matters has been studied in great detail for many dif- ifold. The generalized DBI action can be written as fol-
ferent potentials [12–14, 19–28]. However, for a general lows [6]
quintessencepotential,theequationsofmotionarerather
involved. To ourknowledge,onehavenotyetfindagen- 1
S = d4x√ g T (ψ)W (ψ) 1+ ∂ ψ∂µψ
eral method to write the equations of motion in the au- − " s T (ψ) µ
tonomousformforarbitrary potential. Thusthepurpose Z
T (ψ)+V (ψ)]+S . (1)
m
−
Here T(ψ) is the warped tension of the brane and S
m
∗Electronicaddress: [email protected] is the action for matters localized in the D3-brane. The
†Electronicaddress: [email protected] potential W(ψ) could arise under the condition that the
2
brane is a non-BPS one [5] or there are multiple coinci- constrainedtobe smallerthanthe speedoflight. Thisis
dent branes [30]. The potential V(ψ) is related to the remarkably different from the usual quintessence which
brane’s interactions with bulk fields or other branes. could have arbitrary large speed in the sense of classical
Inordertosimplifyourderivations,wedefinethevari- mechanics.
able φ as follows Observing Eqs. (6) and Eq. (7), we could absorb the
constant κ2/3 (κ2 =8π) into W,V and ρ , respectively,
m
1
∂ ψ =∂ φ. (2)
µ µ
T (ψ)
3
W W , (9)
Then above actiopn can be written as −→ · κ2
3
S = d4x√ g T (φ)W (φ) 1+∂µφ∂µφ V −→V · κ2 , (10)
−
Z−T(φ)+Vh(φ)]+Sm . p (3) ρm −→ρm· κ32 . (11)
Withoutthelossofphysicalsignificance,wecouldabsorb
Then above equations of motion turns out to be
the term T(φ) into W(φ) and V(φ), respectively. Then
we find the action is simply H2 =Wγ+V +ρ , (12)
m
2H˙/3+H2 =W/γ+V ω ρ , (13)
m m
−
S = d4x√ g W (φ) 1+∂µφ∂µφ+V (φ) +Sm(4.) φ¨+ 1 3Hφ˙ + 1 V,φ + 1 W,φ =0. (14)
Z − h p i γ2 · γ3 · W γ2 · W
We shall investigate the cosmic evolution of the DBI
Given the scalar potential W(φ), V(φ) and the equation
field in the background of spatially flat Friedmann-
ofstateω ,weareleftwiththreevariables,a(t),ρ and
Robertson-Walker Universe m m
the DBI field φ(t). Then we have three variables and
ds2 = dt2+a(t)2 dr2+r2dΩ2 , (5) three independent differential equations. So the system
− of equations is closed.
wherea(t)isthe cosmicscale(cid:0)factor. Wem(cid:1)odelthe mat- It is rather difficult to find the analytic solutions to
ter sourcespresentinthe Universe asperfect fluids. The the equations of motion (12-14). Hence in order to solve
perfect fluids can be baryonic matter, relativistic mat- them numerically, we had better rewrite them in the au-
ter and dark energy. We assume there is no interaction tonomous form. To this end, we introduce the following
between the generalized DBI scalar field and the matter dimensionless quantities
fields, other than by gravity. Then the Einstein equa-
V
tions and the equation of motion of the scalar field are X φ˙ , U ,φ , (15)
given ≡ ≡ W32
√W W
3H2 =κ2(Wγ+V +ρ ) , Y , Q ,φ , (16)
m ≡ H ≡ W32
2H˙ +3H2 = κ2( W/γ V +ω ρ ) , (6)
− − − m m √V V,φ
Z , S , (17)
and ≡ H ≡ V 32
N lna. (18)
φ¨+ 1 3Hφ˙ + 1 V,φ + 1 W,φ =0, (7) ≡
γ2 · γ3 · W γ2 · W We see U, Q, S are the functions of DBI field, φ. So
they can be expressed as the function of Y/Z. Now we
respectively. Here H a˙/a denotes the Hubble pa-
≡ have only three variables, namely, X, Y, Z and it is
rameter and dot is the derivative with respect to cos-
sufficient for us to derive the corresponding three inde-
mic time, t. ρ and ω are the energy density and
m m
pendent differential equations. We remember that the
the equation of state for the matter sources. We have
system of equations of X, Y Z are an autonomous sys-
ω = 1, 0, 1/3, +1forthecosmologicalconstant,dust
m
− temifandonlyifU, Q, S areexpressedasthe functions
matter, relativistic matter and stiff matter, respectively.
of X, Y, Z. For simplicity, we assume
In this paper, we shall consider the case of dust matter,
ω =0. V and W denote the derivative with respect
m ,φ ,φ V
to φ. γ is defined by =φ2 . (19)
W
1
γ = , (8) We emphasize that one could in general assume V/W =
1 φ˙2 F(φ) with F(φ) other simple functions, for example,
− F(φ) = φn, eαφ, lnφ and so on. Then we could obtain
q
which has the physical meaning of the generalized φ = φ (V/W) = φ (Z/Y) from this equation. Similarly,
Lorentz factor. It is apparent the speed of scalar φ˙ is U, Q, S could be expressed as the function of Z/Y.
3
Now with the aid of this assumption, we are able to
write the Eqs. (12-14) in the autonomous form and deal 3.5
with any scalar potential, V(φ) in the calculations. In
this article, we shall focus on the scalar potential V(φ) V3
with the expression of
2.5
ξ1 ξ2
V
V = V + 1 (φ a )(φ a )(φ a ) 2
0 φ6 − 1 − 2 − 3
(φ a )(φ a )(φ a ) , (20) 1.5 σ1
4 5 6
· − − −
1
whereV anda areallpositiveconstants. Thereasonfor σ2
i i
this choice of potential is that what we want to study is 0.5 σ3
φ
a potential with multiple vacua. Scalar fields with mul-
tiple vacua are very interesting because they have both 0
1 1.2 1.4 1.6 1.8 2
theoreticalorigininstringlandscape[31]andtheoretical
study in Coleman-De Luccia tunnelling [32].
FIG.1: TheDBIpotentialV(φ)withrespecttoφ. Thereare
We note that the case of the well-known AdS throat,
W(φ) = λ/φ4 has been included in the desired one. As two local maximum (ξ1 =1.11, ξ2 = 1.56), one real vacuum
(σ3 =1.86) and two false vacuum (σ1 =1.03, σ2 =1.31) for
an example, we put V = 2, V = 104, a = 1, a =
0 1 1 2 thepotential. Physically,thescalar fieldwouldrolldownthe
11/10, a3 = 9/8, a4 = 39/25, a5 = 157/100, a6 = 2 in potential and then passes through the first (σ1) and the sec-
the following discussions. ond (σ2) false vacuum. Finally, it arrives at the real vacuum
InFig.1,weplotthepotentialV(φ)withrespecttoφ. (σ3).
.
There are two local maximum (ξ = 1.11, ξ = 1.56),
1 2
one real vacuum (σ = 1.863) and two false vacuum
3
(σ = 1.029, σ = 1.305) for the potential. Physically,
1 2
the scalar field would roll down the potential and then where U, Q, S are the functions of Y/Z and
passes through the first (σ ) and the second (σ ) false
1 2
vacuum. Finally, it arrivesat the real vacuum (σ ). The 1
3 γ = , (22)
detail of the trajectory is closely related to the initial √1 X2
velocity, φ˙i (with the initial value, φi fixed). When the H˙ −1 γ2 3
initialspeedφ˙ is smallenough,the scalarwouldacquire = 3Y2 − 1 γY2 Z2 . (23)
i H2 · 2γ − 2 − −
dampedoscillationsduetotheHubblefrictioninthefirst
(cid:0) (cid:1)
falsevacuumandfinallyitisfrozeninthisvacuum. How-
The Friedmann equation becomes the constraint equa-
ever,withtheincreasingofinitialspeed,thescalarcould
tion
cross the first local maximum (ξ ) and finally is frozen
1
in the second false vacuum (σ2). With even much larger 1= Y2 +Z2+ ρm . (24)
initial velocity, the scalar could cross the two local max- √1 X2 H2
imum (ξ and ξ ) and finally dwells on the real vacuum −
1 2
(σ ). Since the speed of the scalar is constrained to be The equation of state w of the DBI scalar field is
3
smaller than the speed of light, the scalar can not climb
the hill with arbitrary height. Due to the Hubble fric- 1/γ Z2/Y2
w − − . (25)
tion, we expect the fate of the scalar is to dwell on the ≡ γ+Z2/Y2
realvacuum. Inwhatfollows,weshallshowthesespoints
numerically. In Table I, we present the properties of the five fixed
points for the scalar field. The points (a, c, e) corre-
Using the dimensionless variables defined in (15-18),
spond to the false vacua (σ , σ ) and real vacuum (σ ).
1 2 3
the equations of motion (12-14) can be written in the
The three points are stable spirals. On these epoches,
following autonomous form
the scalar field behaves as a damping oscillator with the
equation of state of firstly behaving as the dust mat-
ter and then oscillating approaching 1 (see Fig. (5)).
dX 1 1 1
−
= 3X UY QY , The points (b, d,) correspondto the two localmaximum
dN −γ2 · − γ3 · − γ2 ·
(ξ , ξ ) andthey aresaddlepoints. Onthese points, the
1 2
dY = 1QXY2 Y H˙ , speedofthescalarfieldexactlyvanishesandtheDBIfield
dN 2 − H2 acquires the equation of state of cosmologicalconstant.
dZ 1 H˙ In Fig. (2-4), we plot the evolution of the speed, φ˙ of
= SXZ2 Z , (21) thegeneralizedDBIfieldwithφ. Wefixtheinitialvalues
dN 2 − H2
4
Name X Y Z φ Stability Ω w
φ φ
(a) 0 0.70 0.72 σ1 =1.03 Stable spiral (attractor) 1 −1
(b) 0 0.67 0.74 ξ1 =1.11 Saddle point 1 −1
(c) 0 0.61 0.79 σ2 =1.31 Stable spiral (attractor) 1 −1
(d) 0 0.54 0.84 ξ2 =1.56 Saddle point 1 −1
(e) 0 0.47 0.88 σ3 =1.86 Stable spiral (attractor) 1 −1
TABLE I:Properties of thecritical points for thescalar potential given by Eq.(20).
of φ˙, φ and ρ /H2. By this way, the initial values of Y
m
and Z are determined. Fig. (2) shows that when the
initialspeedφ˙i is smallenough,the scalarwouldacquire X1
dampedoscillatingduetotheHubblefrictioninthefalse
0.8
vacuum(σ )andfinallyitisfrozeninthisvacuum. With
1
the increasing of initial speed, the scalarcrossesthe first 0.6
local maximum (ξ ) and finally is frozen in the second
1 0.4
false vacuum (σ ) (see Fig. (3)). With even much larger
2
initialvelocity,thescalarcrossesthetwolocalmaximum 0.2
(ξ and ξ ) and finally dwells on the real vacuum (σ )
1 2 3 0
(see Fig. (4)).
In Fig. (5), we plot the evolution of the equation of –0.2
state w of the generalized DBI field corresponding to
–0.4
Fig. (4). We see the generalizedDBI field previouslybe-
φ
havesasthe dustmatterandthenoscillatingapproaches –0.6
1afterasufficientlongtime. Thereasonforoscillating 1.02 1.03 1.04 1.05 1.06 1.07 1.08
−
ofω canbeunderstoodasfollows. Eq.(25)tellsuswhen
the speed of DBI field vanishes, the equation of state is
FIG. 2: Evolution of the speed (X = φ˙) of generalized DBI
1. Fig. (4) shows there are many times for the vanish-
−ing of speed during the damped oscillating. Every time field with φ. The point (φ = σ1 = 1.03, X = 0) is a stable
spiralandthusanattractor. Inthiscase,thegeneralizedDBI
the generalized DBI field acquires vanishing velocity, its
fieldbehavesasadampingoscillatorinthefalsevacuum(σ1)
equation of state is 1.
and finally is frozen.
−
.
III. QUINTESSENCE
Inthissection,weshallpresentthemethodfordealing
The Einstein equations and the equation of motion of
with arbitrary quintessence potential. As an example,
the quintessence are given by
we would explore the cosmic evolution of quintessence
field with multiple vacua. To this end, let’s focus on the
potential as follows 1
3H2 =κ2 φ˙2+V +ρ ,
m
2
V =V eb(φ−a1)(φ−a2)(φ−a3)(φ−a4)(φ−a5)(φ−a6) , (26) (cid:18) (cid:19)
0 1
2H˙ +3H2 = κ2 φ˙2 V +ω ρ , (27)
m m
where V0, b, ai are positive constants. As an example, − (cid:18)2 − (cid:19)
we consider, V = 1, b = 1, a = 1/2, a = 3/5, a =
0 1 2 3
1, a =17/10, a =2, a =5/2.
4 5 6 and
InFig.6,weplotthepotentialV(φ)withrespecttoφ.
There are two local maximum (ξ = 0.813, ξ = 1.862),
1 2
one real vacuum (σ3 = 2.342) and two false vacuum φ¨+3Hφ˙+V,φ =0, (28)
(σ = 0.545, σ = 1.355) for the potential. Physically,
1 2
the scalar field would roll down the potential and then
respectively. Here ρ and ω are the energy density
damped oscillates between these vacua. Given an initial m m
finite velocity φ˙ and field value φ , the fate of the scalar and the equation of state for the matter sources. In this
0
section,wealsoconsiderthecaseofdustmatter,ω =0.
is expectedto dwellonthe one ofthe vacuumdue to the m
Hubble friction. In what follows, we shall show theses ObservingEqs.(27)andEq.(28),wecouldabsorbthe
points numerically. constantκ2/3(κ2 =8π) into φ2,V and ρ , respectively,
m
5
X1 w0
–0.2
0.5
–0.4
0
–0.6
–0.5 –0.8
φ –1
–1
1.25 1.3 1.35 1.4 1.45 1.5 1 2 3 4 5 6 N
FIG. 3: Evolution of the speed (X = φ˙) of generalized DBI FIG.5: Theevolutionoftheequationofstateforthegeneral-
field with φ. The point (φ = σ2 = 1.31, X = 0) is a stable izedDBIscalar field. Itbehavesasthedustmatterathigher
spiralandthusanattractor. Inthiscase,thegeneralizedDBI redshiftsandoscillating approaches−1atthelowerredshifts
fieldbehavesasadampingoscillator inthefalsevacuum(σ2) .
and finally is frozen.
.
Then above equations of motion turns out to be
H2 =φ˙2+V +ρ , (32)
m
X1 2H˙/3+H2 = φ˙2+V ω ρ , (33)
m m
− −
1
φ¨+3Hφ˙ + V =0. (34)
,φ
2
0.5
GiventhescalarpotentialV(φ)andtheequationofstate
ω , we are left with three variables, a(t),ρ and the
m m
0 quintessence φ(t). Then we have three variables and
three independent differential equations. So the system
of equations is closed.
–0.5 The same as the DBI case,it is rather involvedto find
theanalyticsolutionstotheequationsofmotion(32-34).
Hence in order to solve them numerically, we had better
φ
–1 rewrite them in the autonomous form. To our knowl-
1.78 1.8 1.821.841.861.88 1.9 1.921.941.96 edge, one have only explored some special form of the
quintessencepotential,namely,theexponentialpotential
[12, 20], power-law type potential [9, 13], the Albrecht
FIG. 4: Evolution of the speed (X = φ˙) of generalized DBI andSkordispotential [33] anso on. For arbitrary poten-
field with φ. The point (φ = σ2 = 1.86, X = 0) is a stable tial, one have not yet find a generalmethod to write the
spiralandthusanattractor. Inthiscase,thegeneralizedDBI
equations of motion in the autonomous form. In what
field behavesas adampingoscillator in thereal vacuum(σ3)
follows, we shall propose a method that can be used to
and finally is frozen.
deal with arbitrary potentials.
.
To this end, we introduce the following dimensionless
quantities
φ˙ V
,φ
X , Q =Q(Z) , (35)
6 ≡ H ≡ V
φ2 φ2 , (29)
−→ · κ2 √V
Y , N lna, (36)
3 ≡ H ≡
V V , (30)
−→ · κ2 Z φ. (37)
≡
3
ρm −→ρm· κ2 . (31) With the aid of these definitions, we can write the
6
equations of motion in the autonomous form with ar-
bitrary potentials: 1.2
dX H˙
= 3X Y2Q X ,
dN − − − · H2 1.1
dY 1 H˙
= XYQ Y , ξ1 ξ2
dN 2 − H2
dZ V1
= X , (38) σ1
dN
σ2
with
0.9
H˙ 3
= 3X2 (1+ω ) 1 X2 Y2 . (39) σ3
H2 − − 2 m − − φ
0.8
(cid:0) (cid:1)
WenotethatQisthefunctionofZ. Therefore,Eqs.(38)
0.5 1 1.5 2 2.5
is indeed an autonomous system of equations. In Table
II,wepresentthepropertiesofthefivefixedpointsforthe
quintessence. The points (a,c,e)correspondtothe false FIG. 6: The quintessence potential V(φ) with respect to
vacua (σ , σ ) and real vacuum (σ ). The three points φ. There are two local maximum (ξ1 = 0.813, ξ2 =
1 2 3
are stable spirals. On these epoches, the quintessence 1.862), one real vacuum (σ3 = 2.342) and two false vacuum
behavesasadampingoscillatorwiththeequationofstate (σ1 = 0.545, σ2 = 1.355) for the potential. Physically, the
quintessence would firstly damped oscillates between these
offirstlybehavingasthedustmatterandthenoscillating
vacuaandfinallydwellononeofthevacuumduetotheHub-
approaching 1. Thepoints(b,d,)correspondtothetwo
− ble friction.
local maximum (ξ , ξ ) and they are saddle points. On
1 2 .
thesepoints,thespeedofthescalarfieldexactlyvanishes
and the quintessence acquires the equation of state of
cosmologicalconstant.
In Fig. (7-9), we plot the evolution of the rescaled
speed, X of the quintessence with φ. The figures
1
show that, with the increasing of initial speed, the
quintessenceisfrozeninthefirstfalsevacuum,thesecond
false vacuum and the real vacuum, respectively. Com-
0.5
pared to evolution of DBI field, we see from Figs.(2-4)
that the DBI field has more times of oscillations than
quintessence. How to understand this point? The equa-
X0
tions of motion tell us the friction term due to Hubble σ1
(Attractor)
expansionis3Hφ˙ and3Hφ˙ 1 φ˙2 forquintessenceand
−
–0.5
DBIfield,respectively. The(cid:16)nwitht(cid:17)heincreasingofspeed
φ˙, the friction term of DBI field is greatly decreased. So
the DBI field acquires more times of oscillations. φ
–1
0 0.5 1 1.5 2 2.5 3 3.5
IV. CONCLUSION AND DISCUSSION
FIG. 7: Evolution of the rescaled speed (X) of quintessence
Ingeneral,theequationsofmotionforgeneralizedDBI
withφ. Thepoint(φ=σ1, X =0)isastablespiralandthus
fieldandquintessencearerathercomplicated. Soonere-
an attractor. The quintessenceis finally frozen at φ=σ1.
sorttothemethodofphaseanalysisbywritingtheequa-
.
tions of motion in the autonomous form. Many special
form of potentials have been studied for DBI field [29]
and quintessence [12–14, 19–28]. However, the general
method for dealing with arbitrary potentials have not initialspeed,bothgeneralizedDBIfieldandquintessence
yet been proposed. Thus the outcome of this article is are successively frozen, in the first false vacuum, the
that we have found the method. second false vacuum and the real vacuum, respectively.
Different from the potentials studied in Refs. [12– Compared to the evolution of quintessence, the general-
14, 19–29], we investigate the cosmic evolution of the izedDBI fieldhas moretimes ofoscillations. The reason
generalizedDBIfieldandquintessencewiththepotential for this point is that, with the increasing of speed φ˙, the
of multiple vacua. We find that, with the increasing of frictiontermofgeneralizedDBIfieldisgreatlydecreased.
7
Name X Y Z φ Stability Ω w
φ φ
(a) 0 1 0.545 σ1=0.545 Stable spiral (attractor) 1 −1
(b) 0 1 0.813 ξ1=0.813 Saddle point 1 −1
(c) 0 1 1.355 σ2=1.355 Stable spiral (attractor) 1 −1
(d) 0 1 1.862 ξ2=1.862 Saddle point 1 −1
(e) 0 1 2.342 σ3=2.342 Stable spiral (attractor) 1 −1
TABLE II:Properties of thecritical points for the quintessencepotential given by Eq.(26).
ThusthegeneralizedDBIfieldacquiresmoretimesofos-
cillations.
1
The conclusion in this paper may be trivial, but the
proof and the method are not. As an example, one
couldstudytheColeman-DeLucciatunnellingusingthis
0.5
method. After making the Wick rotation, t = iτ
−
in Eqs. (21) and Eqs. (38), we are able to study the
X0 (Attractor)σ2 Coleman-De Luccia tunnelling numerically but exactly.
–0.5
φ
–1
Acknowledgments
0 1 2 3
FIG. 8: Evolution of the rescaled speed (X) of quintessence We thank one of the referees for pointing out some
withφ. Thepoint(φ=σ2, X =0)isastablespiralandthus important typos. This work is supported by the Chi-
an attractor. The quintessenceis finally frozen at φ=σ2. neseMoST863programundergrant2012AA121701,the
. NSFC under grant 11373030, 10973014, 11373020 and
11465012.
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