Quintessence and phantom emerging from the split-complex field and the
split-quaternion field
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
arXiv:1501.03220v5 [gr-qc] 30 Dec 2015
Xuelei Chen†
Key Laboratory for Computational Astrophysics, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China and
Center of High Energy Physics, Peking University, Beijing 100871, China
You-Gen Shen‡
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
(Dated: December 31, 2015)
Motivated by the mathematic theory of split-complex numbers (or hyperbolic numbers, also perplex numbers) and the split-quaternion numbers (or coquaternion numbers), we define the notion of
split-complex scalar field and the split-quaternion scalar field. Then we explore the cosmic evolution
of these scalar fields in the background of spatially flat Friedmann-Robertson-Walker Universe. We
find that both the quintessence field and the phantom field could naturally emerge in these scalar
fields. Introducing the metric of field space, these theories fall into a subclass of the multi-field
theories which have been extensively studied in inflationary cosmology.
PACS numbers: 98.80.Jk, 04.40.Nr, 04.50.+h, 11.25.Mj
I.
INTRODUCTION
More and more accurate and convincing astronomical
observations [1–3] indicate that dark energy dominates
our current Universe. Although a host of observationally
viable dark energy models have been proposed, the nature of dark energy is still undetermined. The Einstein
cosmological constant is in many respects the most economical solution to the problem of dark energy. But it is
confronted with two fundamental problems: the fine tuning problem and the coincidence problem (see e.g. [4]).
So one turn to the study of other options for dark energy,
for example, quintessence [5–7], quintom [8], k-essence
[9], Chaplygin gas [10], holographic dark energy [11] and
so on.
One find that in many models of quintessence field,
there exist the so-called tracker solutions. In these solutions, the quintessence field always has a energy density
closely tracks (but is smaller than) that of the radiation
until the matter-radiation equality. By this way, the coincidence problem is solved [5–7]. Phantom energy is
introduced into the study of cosmic evolution by Caldwell [12, 13]. The Lagrangian of phantom takes the form
Lφ = − 21 ∇µ φ∇µ φ + V (φ), with φ the phantom field
and V (φ) the phantom potential. By changing the sign
of kinetic term in quintessence field by hand, this form
of Lagrangian could be obtained. One then find that the
energy density of phantom actually increases with cosmic
time. So the fate of the Universe is a Big Rip.
Now we want to ask: can we unify quintessence and
phantom in a natural way? The answer is yes. We
find that the quintessence and phantom can naturally
emerge in the theory of split-complex scalar field, the
split-quaternion scalar field and the split-octonion scalar
field. This finding is motivated by the mathematic theory
of split-complex numbers (or hyperbolic numbers, also
perplex numbers), the split-quaternion numbers (or coquaternion numbers) and the split-octonion numbers [14]
(for references, see also the Wikipedia 1 ).
We note that our application of split-complex number is not just a mathematical tool but rather it
has physical information too. Actually, the theory of
split-complex number has been applied in gravitational
fields [15], quantum group [16], quantum mechanics
[17], quintessence cosmology [18] and so on. What is
more, with the help of split-complex variables, a pseudocomplex field theory [19] and a pseudo-complex General Relativity [20] are recently presented. For the splitcomplex scalar field, if one require it obeys the symmetry
of invariance under hyperbolic rotation, the split-complex
field would restore to the Hessence field [21]. On the other
hand, if the symmetry is allowed to be broken, the split-
∗ Electronic
address: gaocj@bao.ac.cn
address: xuelei@cosmology.bao.ac.cn
‡ Electronic address: ygshen@center.shao.ac.cn
† Electronic
1
http : //en.wikipedia.org/wiki/Split − complex number.
2
complex scalar field would turn out to be the quintom
field [8]. Coleman found that for conventional complex
field, there exists Q-Balls solutions [22] due to the conserved charge. We find there is also conserved charge for
split-complex field. So we expect Q-Balls-like solutions
exist which makes the split-complex field more physical.
Finally, we find the split-complex scalar field could be reformulated as one specific model of the multi-filed theory
which attracts a lot of effort in the study of inflationary
universe [23–27] in recent years. In all, the split scalar
fields have rich physics.
The paper is organized as follows. In section II, we
define the notion of split-complex scalar field and show
quintessence and phantom could emerge from this field.
In section III, we investigate the cosmic evolution of
the split-complex field and show that the detail of dynamics is closely related to the initial conditions on the
quintessence and phantom. In section IV, we investigate the cosmic evolution of the split-quaternion field.
In section V, the linear perturbations of these fields
in the background of Friedmann-Robertson-Walker Universe are present. Conclusions and discussions are given
in section VI. Throughout this paper, we adopt the system of units in which G = c = ~ = 1 and the metric
signature (−, +, +, +).
II.
A.
What is split-complex scalar field
The mathematic theory of split-complex numbers (or
hyperbolic numbers, also perplex numbers) can be found
in Ref. [14] or the Wikipedia 2 . Motivated by the theory
of these numbers, we define the split-complex scalar field
Φ as follows
(1)
where φ and ψ are two real scalar fields. The quantity j
is similar to the imaginary unit i except that [14]
j 2 = +1 .
(2)
Choosing j 2 = −1 results in the conventional complex
scalar field. It is this change of sign which distinguishes
the split-complex scalar field from the ordinary complex
one. The quantity j here is not a real number but an
independent quantity. Namely, it is not equal to ±1.
Just as for ordinary complex field, one can define the
notion of a split-complex conjugate as follows
Φ∗ = φ − jψ .
2
http : //en.wikipedia.org/wiki/Split − complex number.
ΦΦ∗ = φ2 − ψ 2 .
(3)
(4)
This quadratic form is split into positive and negative
parts, in contrast to the positive definite form of the ordinary complex scalar field.
Similar to the ordinary complex field which can be
written in the form of Euler’s formula
Φ = φeiθ = φ cos θ + iφ sin θ ,
(5)
the split-complex field has the Euler’s formula as follows
Φ = φejθ = φ cosh θ + jφ sinh θ .
(6)
This can be derived from a power series expansion using
the fact that cosh has only even powers while that for
sinh has odd powers. It follows that ΦΦ∗ = φ2 .
B.
Quintessence and phantom from split-complex
field
We shall consider the theory of a massive, splitcomplex, self-interacting scalar field with the Lagrangian
density as follows
L =
SPLIT-COMPLEX SCALAR FIELD
Φ = φ + jψ ,
Then the modulus of a split-complex scalar field is given
by the isotropic quadratic form
2
1
1
∇µ Φ∇µ Φ∗ + λ2 ΦΦ∗ + m2 /λ2 ,
2
4
(7)
with m the mass of the scalar and λ a coupling constant. Here the potential is the same as minimally coupled Higgs potential which has been given experimental
evidence. Of course, one may consider other potentials,
i.e, exponential potential, power-law potential and so on.
However, if one demand the theory obeys the symmetry
of hyperbolic rotation (see subsection C below), the potential should be constrained to be the function of ΦΦ∗ ,
namely V (ΦΦ∗ ). On the other hand, if the symmetry is
broken, the potential could be V (Φ + Φ∗ , Φ − Φ∗ ). In
the next, we will see the former is exactly the Hessence
field [21] and the latter the quintom field [8]. So the splitcomplex scalar field procedure is not just a mathematical
re-formulation of the Hessence or Quintom, but has the
physical meaning of internal symmetry.
Substituting Eq. (1) into Eq. (7), we obtain
L =
1
1
∇µ φ∇µ φ − ∇µ ψ∇µ ψ
2
2
2
1
+ λ2 φ2 − ψ 2 + m2 /λ2 .
4
(8)
If we take the field Φ in the Lagrangian Eq. (7) as the
ordinary complex scalar field, Φ = φ+iψ, the Lagrangian
takes the form
1
1
L = ∇µ φ∇µ φ + ∇µ ψ∇µ ψ
2
2
2
1
(9)
+ λ2 φ2 + ψ 2 + m2 /λ2 .
4
3
It is apparent there is sign of difference before the terms
1
µ
2
2
2 ∇µ ψ∇ ψ and ψ . This is due to the fact that i = −1
2
and j = 1. One can recognize that Lagrangian, Eq. (8)
is nothing but the Hessence field proposed by Wei et al
in Ref. [21]. φ and ψ plays the role of quintessence and
phantom, respectively. The difference of Hessence field
from the quintom field [8]
1
1
∇µ φ∇µ φ − ∇µ ψ∇µ ψ + V (φ, ψ) , (10)
2
2
is that the form of the scalar potential is greatly constrained by V (φ2 − ψ 2 ) in Hessence. We point out that
there is a remarkable difference in our motivation from
the Hessence. Ref. [21] propose the Lagrangian density
of Hessence as follows
1
1
,
L = ∇µ Φ∇µ Φ + ∇µ Φ∗ ∇µ Φ∗ + V Φ2 + Φ∗2 (11)
2
2
with Φ the conventional complex scalar field. However,
our Lagrangian density Eq. (7) is different from Eq. (11)
not only on the Lagrangian expression but also on the
physical meaning of scalar field Φ. In Fig. 1 we plot the
scalar potential V ∝ (ΦΦ∗ + M 2 )2 with the quintessence
field φ and the phantom field ψ. The potential has the
vanishing absolute vacuum energy on the hyperbola φ2 −
ψ 2 + M 2 = 0.
L =
with α a constant. Expressed the split-complex scalar
field in terms of two real fields φ and ψ, the hyperbolic
rotation corresponds to the O(1, 1) transformations
φ → φ cosh α + ψ sinh α ,
ψ → φ sinh α + ψ cosh α .
(13)
(14)
Then the Noëther theorem tells us this symmetry leads
to a conserved charge which is given by the formula
Z
1
d3 x (Φ∗ ∂0 Φ − Φ∂0 Φ∗ ) ,
(15)
Q=
2j
in the background of four dimensional Minkowski spacetime. Here ∂0 represents the derivative with the time.
Coleman found that for conventional complex field, there
would exist Q-Balls solutions [22] due to the conserved
charge. Since there is also conserved charge for splitcomplex field, we expect Q-Balls-like solutions also exist
which makes the split-complex field more physical.
D.
the stability
Introduce the metric tensor ηIJ in the field space ΦI =
(φ, ψ),
1 0
,
(16)
ηIJ = η IJ =
0 −1
where I, J = 1, 2. Eq. (8) can be expressed as
2
L =
V
0
–4
–4
–2
–2
phi 0
2
2
4
0psi
4
symmetry
The theory of Lagrangian Eq. (7) has the symmetry
that it is invariant after a hyperbolic rotation
Φ → Φejα ,
(17)
With this form, the theory belongs to the general multiple scalar field theory considered in Ref. [23–27]. The
equation of motion and the energy momentum are give
by
(18)
∇µ ∇µ ΦI − λ2 ΦJ ΦJ + m2 /λ2 ΦI = 0 ,
and
FIG. 1: The sketch of the potential V ∝ (ΦΦ∗ +M 2 )2 with the
quintessence field φ and the phantom field ψ. The potential
has the vanishing absolute vacuum energy on the hyperbola
φ2 − ψ 2 + M 2 = 0.
.
C.
2
1
1
∇µ ΦI ∇µ ΦI + λ2 ΦI ΦI + m2 /λ2 .
2
4
(12)
Tµν = −∇µ ΦI ∇µ ΦI + gµν L .
(19)
From the equation of motion, we see both the
quintessence φ and the phantom ψ acquire a vanishing
effective mass m0 :
q
m0 = λ ΦJ ΦJ + m2 /λ2 |(ΦJ ΦJ +m2 /λ2 ) = 0 , (20)
at the global minimum of the potential. In this case, not
only quintessence but also phantom is stable in dynamics.
This is different from the usual phantom field with the
Lagrangian
1
L = − ∇µ ψ∇µ ψ + V (ψ) ,
2
(21)
4
and the equation of motion
∇µ ∇µ ψ + m20 ψ = 0 ,
(22)
where m20 ≡ V,ψψ |ψ=ψ0 . The value of ψ = ψ0 corresponds to the global minimum of the potential. In
this case, the phantom field acquires an imaginary mass
which would lead to the classically and quantum instability. Then why is there such a difference in the mass?
The reason are as follows. The usual phantom scalar potential has the form V = 12 m20 ψ 2 at the global minimum.
However, for the Lagrangian Eq. (8), the phantom potential takes the form V = − 21 m20 ψ 2 at the global minimum.
Thus it is very important that the potential of the splitcomplex scalar is constrained to be V (φ2 − ψ 2 ) in order
to avoid the problem of instability .
E.
conformal invariant split-complex field
Recently, Kallosh and Linde [28] proposed a simple
conformally invariant two-field model of dS/AdS space.
The model consists of two real scalar fields, φ and ψ,
which has the SO(1, 1) symmetry:
LKL =
1
1
1
φ2 − ψ 2 R
∇µ φ∇µ φ − ∇µ ψ∇µ ψ +
2
2
12
1
2
2 2
− λ φ −ψ
.
(23)
4
Here R is the Ricci scalar and λ a coupling constant. This
theory is locally conformal invariant under the following
transformations,
g̃µν = e−2σ(x) gµν ,
φ̃ = eσ(x) φ , ψ̃ = eσ(x) ψ . (24)
The global SO(1, 1) symmetry is a boost between these
two fields. Using the concept of our split-complex scalar
field ΦI , the theory is equivalent to
L =
2
1
1
1
∇µ ΦI ∇µ ΦI + RΦI ΦI − λ ΦI ΦI . (25)
2
12
4
In other words, the two scalar fields considered in
Ref. [28] is exactly the conformal invariant split-complex
field.
III.
matter and dark energy. We assume there is no interaction between the split-complex scalar field and other
matter fields, other than by gravity. Then the Einstein
equations and the equation of motion of the scalar fields
are given by
1 2 1 2
3H 2 = κ2
φ̇ − ψ̇ + V + ρr + ρm ,(27)
2
2
1
1 2 1 2
φ̇ − ψ̇ − V + ρr , (28)
2Ḣ + 3H 2 = −κ2
2
2
3
and
where a(t) is the cosmic scale factor. We model all other
matter sources present in the Universe as perfect fluids.
These matter sources can be baryonic matter, relativistic
(29)
ψ̈ + 3H ψ̇ − V,ψ = 0 ,
(30)
respectively. Here H ≡ ȧ/a denotes the Hubble parameter and dot is the derivative with respect to cosmic time,
t. ρm and ρr are the energy density of dark matter and
relativistic matter. V,φ and V,ψ denote the derivative
with respect to φ and ψ, respectively. Introduce the following dimensionless quantities
κ φ̇
κ ψ̇
X≡√
, Y ≡ √
,
H
6
6H
√
√
κ ρm
κ V
, U≡ √
,
Z≡ √
6 H
3 H
2 V,φ
2 V,ψ
λφ ≡ − √
, λψ ≡ − √
,
κV
6
6 κV
N ≡ ln a ,
(31)
then the equations of motion can be written in the following autonomous form
dX
dN
dY
dN
dZ
dN
dU
dN
3
Ḣ
= −3X − λφ Z 2 − X 2 ,
2
H
3
Ḣ
= −3Y − λψ Z 2 − Y 2 ,
2
H
3
Ḣ
= − Z (λφ X + λψ Y ) − Z 2 ,
2
H
3
Ḣ
= − U −U 2 ,
2
H
dλφ
=
dN
DYNAMICS OF SPLIT-COMPLEX SCALAR
FIELD
In this section, we investigate the cosmic evolution of
the split-complex field in the background of spatially flat
Friedmann-Robertson-Walker (FRW) Universe
(26)
ds2 = −dt2 + a (t)2 dr2 + r2 dΩ2 ,
φ̈ + 3H φ̇ + V,φ = 0 ,
dλψ
=
dN
3X λ2ψ − λ2φ
r
2 + 2 1 − s2 λ2φ − λ2ψ
3
+ λφ (Xλφ − Y λψ ) ,
2
3Y λ2φ − λ2ψ
r
2 + 2 1 − s2 λ2φ − λ2ψ
3
+ λψ (Xλφ − Y λψ ) ,
2
together with a constraint equation
X2 − Y 2 + Z2 + U 2 +
κ 2 ρr
=1.
3H 2
(32)
(33)
5
Here
s=
√
6 mκ
,
4 λ
(34)
Z 1
Attractor
and
Ḣ
1
= −2 − X 2 + Y 2 + 2Z 2 + U 2 .
(35)
2
H
2
The equation of state w and the fraction of the energy
density for the split-complex scalar field are
X2 − Y 2 − Z2
,
X2 − Y 2 + Z2
≡ X2 − Y 2 + Z2 ,
0.8
0.6
0.4
w ≡
ΩΦ
(36)
As an example, we consider s = 1. Physically, the
quintessence φ would roll down the potential and the
phantom ψ roll up the potential. Therefore we can safely
assume X > 0, Y > 0, λφ > 0, λψ < 0. We investigate the
cosmology model with the following values of parameters:
Ωk0 = 0, Ωm0 = 0.27, Ωr0 = 8.1 · 10−5, ΩX0 = 0.73, which
are consistent with current observations [29].
In Fig. 2, we plot the phase plane for the evolution of
the split-complex scalar with a range of different initial
conditions. The point (0, 0, 0) corresponds to the radiation or matter dominated epoches. The point (0, 0,
0) is unstable and the point (0, 0, 1) is stable and thus
an attractor. These trajectories show that the Universe
always ends at the split-complex scalar potential dominated epoch.
In Fig. 3, we plot the evolution of density fractions for
radiation, dark matter and the split-complex scalar field.
This shows that the split-complex scalar field can mimic
the dark energy very well.
In Fig. 4, we plot the evolution of the equation of state
for the split-complex scalar. It behaves as the stiff matter
at higher redshifts and a cosmological constant for the
lower redshifts. It can cross the phantom divide around
the value of N = −5 (redshift 3000).
IV.
DYNAMICS OF SPLIT-QUATERNION
SCALAR FIELD
A.
0.2
8
6
http : //en.wikipedia.org/wiki/Split − quaternion.
2
0
2
4
Y
6
8
FIG. 2: The phase plane for the evolution of the split-complex
scalar with a range of different initial conditions. The point
(0, 0, 0) corresponds to the radiation or matter dominated
epoches. The point (0, 0, 0) is unstable and the point (0, 0, 1)
is stable and thus an attractor. These trajectories show that
the Universe always ends at the split-complex scalar potential
dominated epoch.
1
0.8
0.6
Ω
0.4
0.2
–20
In this subsection, we define the notion of splitquaternion scalar field and the split-octonion field. We
find they actually consists of two quintessence fields and
two phantom fields. The number of quintessence is exactly the same as that of the phantom. To this end, we
start from the theory of split-quaternion number q (or
coquaternion) which is given by [30]3
3
4
0
split-quaternion scalar field
q = u + ix + jy + kz ,
X
–15
–10 N
–5
0
5
FIG. 3: The evolution of density fractions for radiation (circled line), dark matter (crossed line) and the split-complex
scalar field(solid line)
(37)
where u, x, y, z are real numbers. The quantity i, j, k here
are not real numbers but independent quantities. The
6
Here Λ is a positive constant. It is apparent the theory
consists of two quintessence field φ1 , φ2 and two phantom
field ψ1 , ψ2 . Introduce the metric tensor ηIJ in the space
of field ΦI = (φ1 , φ2 , ψ1 , ψ2 ),
1
w
ηIJ = η IJ
–1
1
0
=
0
0

0
1
0
0
0
0
−1
0

0
0 
,
0 
−1
(44)
where I, J = 1, 2, 3, 4. Eq. (43) can be expressed as
2
1
6Λ
1 2
I
µ I
L = ∇µ ΦI ∇ Φ + λ ΦI Φ + 2
.
2
4
κ
–1.00005
–14
–11
–8
–5
N
–2
1
4
FIG. 4: The evolution of the equation of state for the splitcomplex scalar. It behaves as the stiff matter at higher redshifts and a cosmological constant for the lower redshifts. The
split-complex scalar can cross the phantom divide around the
value of N = −5 (redshift 3000)
.
With this form, the theory also belongs to the general multiple scalar field theory considered in Ref. [23–
27]. Furthermore, we could define the split-octonion
field (motivated by the split-octonion4 ) by introducing the metric tensor in the space of field ΦI =
(φ1 , φ2 , φ3 , φ4 , ψ1 , ψ2 , ψ3 , ψ4 ),
ηIJ = η IJ
(38)
1
0

0

0
=
0
0

0
0
0
1
0
0
0
0
0
0

products of these elements are [30]
ij = k = −ji,
jk = −i = −kj,
ki = j = −ik,
i2 = −1,
j 2 = +1,
k 2 = +1,
(45)
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
−1
0
0
0
0
0
0
0
0
−1
0

0
0 

0 

0 
 ,
0 
0 

0 
−1
(46)
where I, J = 1, 2, 3, 4, 5, 6, 7, 8.
The Lagrangian
Eq. (45) can also describe a massive, self-interacting splitoctonion scalar field.
and hence ijk = 1. A split-quaternion has a conjugate
q ∗ = u − ix − jy − kz ,
(39)
qq∗ = u2 + x2 − y 2 − z 2 .
(40)
Φ = φ1 + iφ2 + jψ1 + kψ2 .
(41)
and multiplicative modulus
This quadratic form is split into positive and negative
parts, in contrast to the positive definite form on the
algebra of quaternions.
We now define the split-quaternion scalar field Φ as
B.
In this section, we study the dynamics of the splitquaternion scalar field in the background of spatially flat
FRW Universe. For simplicity, we only consider the cosmic evolution of the split-quaternion scalar field.
The Einstein equations and the equation of motion of
the scalar fields take the form
2
1
6Λ
1
,
∇µ Φ∇µ Φ∗ + λ2 ΦΦ∗ + 2
2
4
κ
(42)
can be written as
1
1
L = ∇µ φ1 ∇µ φ1 + ∇µ φ2 ∇µ φ2
2
2
1
1
− ∇µ ψ1 ∇µ ψ1 − ∇µ ψ2 ∇µ ψ2
2
2
2
1 2
6Λ
2
2
2
2
+ λ φ1 + φ2 − ψ1 − ψ2 + 2
. (43)
4
κ
1
Φ̇I Φ̇I + V ,
2
1
Φ̇I Φ̇I − V ,
2Ḣ + 3H 2 = −κ2
2
3H 2 = κ2
Then the Lagrangian
L =
dynamics
(47)
and
Φ̈I + 3H Φ̇I + V,ΦI = 0 ,
4
http : //en.wikipedia.org/wiki/Split − octonion.
(48)
7
respectively.
Introduce the following dimensionless quantities
κ φ̇1
3 V,φ1
X≡ √
, λφ1 ≡ √
,
H
6
6κ V
3 V,φ2
κ φ̇2
, λφ2 ≡ √
,
Y ≡ √
6H
6κ V
3 V,ψ1
κ ψ̇1
, λψ1 ≡ − √
,
Z≡ √
6H
6κ V
κ ψ̇2
3 V,ψ2
U≡ √
, λψ2 ≡ − √
,
H
6
6κ V
N ≡ ln a ,
(49)
then the equations of motion can be written in the following autonomous form
dX
dN
dY
dN
dZ
dN
dU
dN
dλφ1
dN
= −3X − XB − λφ1 A ,
= −3Y − Y B − λφ2 A ,
= −3Z − ZB − λψ1 A ,
= −3U − U B − λψ2 A ,
= −Xλ2φ1 − Y λφ1 λφ2 − Zλφ1 λψ1 − U λφ1 λψ2
+
1+
XC
√
,
1 − ΛC
dλφ2
= −Xλφ1 λφ2 − Y λ2φ2 − Zλφ2 λψ1 − U λφ2 λψ2
dN
YC
√
+
,
1 + 1 − ΛC
dλψ1
= Xλφ1 λψ1 + Y λφ2 λψ1 + Zλ2ψ1 + U λψ2 λψ1
dN
ZC
√
,
+
1 + 1 − ΛC
dλψ2
= Xλφ1 λψ2 + Y λφ2 λψ2 + Zλψ1 λψ2 + U λ2ψ2
dN
UC
√
+
,
1 + 1 − ΛC
A ≡ 1 − X2 − Y 2 + Z2 + U 2 ,
B ≡ −3 X 2 + Y 2 − Z 2 − U 2 ,
C ≡ λ2φ1 + λ2φ2 − λ2ψ1 − λ2ψ2 ,
(50)
together with a constraint equation
X2 + Y 2 − Z2 − U 2 +
κ2 V
=1.
3H 2
(51)
The equation of state w of the split-quaternion scalar
field is
w ≡ −1 −
2
2 Ḣ
= −1 − B .
3 H2
3
(52)
We note that the coupling constant λ is not present
in above equations. So there is only one free parameter,
Λ. As an example, we consider Λ = 1. Physically, the
quintessence φ1 , φ2 would roll down the potential and the
phantom ψ1 , ψ2 roll up the potential. Therefore we shall
assume λφ1 > 0, λφ2 > 0, λψ1 < 0, λψ2 < 0, X < 0, Y <
0, Z > 0, U > 0.
In Fig. 5 and Fig. 6, we plot the evolution of the
scaled velocities X, Y, Z, U for the quintessence fields
φ1 , φ2 and the phantom fields ψ1 , ψ2 , respectively,
with respect to the equation of state w. They show
that if the kinetic energy of quintessence dominates
over that of phantom initially, X 2 + Y 2 > Z 2 + U 2
(X(0) = −1.22, Y (0) = −0.95, Z(0) = 0.99, U (0) =
0.70, λφ1 (0) = 1/20, λφ2 (0) = 1/20, λψ1 (0) =
−1/20, λψ2 (0) = −1/20), the equation of state of the
split-quaternion field would evolve from +1 to −1 with
the point (X, Y, Z, U, w) = (0, 0, 0, 0, −1) as the attractor (Fig. 5). On the other hand, if the kinetic energy of phantom dominates over that of quintessence initially, X 2 + Y 2 < Z 2 + U 2 (X(0) = −0.95, Y (0) =
−0.78, Z(0) = 1.2, U (0) = 1.0, λφ1 (0) = 1/20, λφ2 (0) =
1/20, λψ1 (0) = −1/20, λψ2 (0) = −1/20), the equation
of state of the split-quaternion field would evolve from
minus infinity to −1 with the point (X, Y, Z, U, w) =
(0, 0, 0, 0, −1) as the attractor (Fig. 6). Corresponding
to Fig. 5 and Fig. 6, we plot the equation of state in
Fig. 7 and Fig. 8, respectively.
In Fig. 9, we plot the evolution of the scaled velocities
X, Y, Z, U for the quintessence fields φ1 , φ2 , the phantom fields ψ1 , ψ2 , respectively, with respect to the equation of state w. The initial values are put by X(0) =
−1, Y (0) = −1.1, Z(0) = 1, U (0) = 1.1, λφ1 (0) =
1/20, λφ2 (0) = 1/20, λψ1 (0) = −1/27, λψ2 (0) = −1/20.
The figure shows that the equation of state of the splitquaternion field could cross the phantom divide with the
point (X, Y, Z, U, w) = (0, 0, 0, 0, −1) as the attractor.
In Fig. 10, we plot evolution of the scaled velocities X, Y, Z, U for the quintessence fields φ1 , φ2 , the
phantom fields ψ1 , ψ2 , respectively, with respect to the
equation of state w. The initial values are put by
X(0) = −0.8, Y (0) = −0.99, Z(0) = 0.805, U (0) =
0.99, λφ1 (0) = 1/15, λφ2 (0) = 1/15, λψ1 (0) =
−1/20, λψ2 (0) = −1/20. The figure shows that the
equation of state of the split-quaternion field cold cross
the phantom divide with the point (X, Y, Z, U, w) =
(0, 0, 0, 0, −1) as the attractor. In Fig. (11) and Fig. (12),
we plot the evolution of their equation of state, respectively.
In all, the detail of dynamics of the split-quaternion
scalar field is closely related to the initial conditions on
the fields.
C.
multi-field theory
In the proceeding sections, we have studied the cosmic evolution of the spilt-complex field and the split-
8
1
0.5
0
U
1
U
Z
0.5
Z
Attractor
0
Attractor
Y
–0.5
Y
–0.5
–1
X
X
–1
–1 –0.8 –0.6 –0.4 –0.2
0
w
–2.6 –2.4 –2.2 –2 –1.8 –1.6 –1.4 –1.2 –1
w
0.2 0.4 0.6 0.8
FIG. 5: Evolution of the scaled velocities X, Y, Z, U for the
quintessence fields φ1 , φ2 and the phantom fields ψ1 , ψ2 , respectively, with respect to the equation of state w. The initial values are put by X(0) = −1.22, Y (0) = −0.95, Z(0) =
0.99, U (0) = 0.70, λφ1 (0) = 1/20, λφ2 (0) = 1/20, λψ1 (0) =
−1/20, λψ2 (0) = −1/20. In other words, we set the same
strength of fields but different scaled velocities. Here and
in the afterwards, (· · ·)(0) ≡ (· · ·)(N = 0). The figure
shows that if the kinetic energy of quintessence dominates
over that of phantom initially (with the same strength of
fields), X 2 + Y 2 > Z 2 + U 2 , the equation of state of the
split-quaternion field would evolve from +1 to −1 with the
point (X, Y, Z, U, w) = (0, 0, 0, 0, −1) as the attractor.
.
FIG. 6: Evolution of the scaled velocities X, Y, Z, U for the
quintessence fields φ1 , φ2 , the phantom fields ψ1 , ψ2 , respectively, with respect to the equation of state w. The initial
values are put by X(0) = −0.95, Y (0) = −0.78, Z(0) =
1.2, U (0) = 1.0, λφ1 (0) = 1/20, λφ2 (0) = 1/20, λψ1 (0) =
−1/20, λψ2 (0) = −1/20. In other words, we set the same
strength of the fields but different scaled velocities. The figure shows that if the kinetic energy of phantom dominates
over that of quintessence initially, X 2 + Y 2 < Z 2 + U 2 (with
the same strength of fields), the equation of state of the splitquaternion field would evolve from minus infinity to −1 with
the point (X, Y, Z, U, w) = (0, 0, 0, 0, −1) as the attractor.
V.
quaternion field with the Lagrangian as follows, L =
1
I µ
2 ∂µ Φ ∂ ΦI + V , Eq. (17) and (45). In this subsection,
we extend the expression of Lagrangian and show that
they actually belong to the multi-field theories studied
in the inflationary cosmology [23–27]. To this end, we
define
1
X IJ ≡ − ∂µ ΦI ∂ µ ΦJ .
2
(53)
In the spirit of k-inflation [31], the very general Lagrangian is of the form
L =K X
IJ
, Φ
I
,
(54)
where I = 1, · · ·, N labels the multiple fields. Here ΦI
can be split-complex field (N = 2), split-quaternion field
(N=4), split-octonion field (N=8) and so on. The field
indices I, J are raised and lowered with the metric tensor
ηIJ and η IJ of the field space. If η IJ is replaced with the
most general metric g̃IJ (ΦK ), it is just the extensively
studied multi-field theory in inflationary cosmology [23–
27].
LINEAR PERTURBATIONS
Up to now, we have explored the cosmic evolution
of split-complex field and the split-quaternion field, respectively. In this section, we focus on the linear perturbations for the split-complex field. The extension of
the result to split-quaternion and split-octonion field is
straightforward.
We start from the action as follows (following the convention and notation in Ref. [23])
Z
√
S = d4 x −gP̃ (Y, ΦI ) ,
(55)
where
Y = ηIJ X
IJ
b ΦK
+
2
X 2 − XIJ XJI ,
(56)
where ΦK is the split-complex field. ηIJ is defined by
Eq. (16). When b = 0, the Lagrangian, Eq. (17) is included as a particular case of P̃ (X, ΦK ).
In order to study the evolution of linear perturbations in the background of FRW Universe, we expand
the above action to second order, including both metric
and scalar field perturbations. In the uniform curvature
gauge, the perturbed split-complex field takes the form
ΦI (x, t) = ΦI0 (t) + QI (x, t) ,
(57)
9
U
1
10
0.5
Z
5
w0
0
Attractor
–5
–0.5
Y
–10
–1
X
–0.5
0
0.5
N
1
1.5
2
FIG. 7: The evolution of the equation of state for the splitquaternion scalar when the kinetic energy of quintessence
dominates over that of phantom (with the same strength of
fields). It behaves as the stiff matter at higher redshifts and
a cosmological constant for the lower redshifts.
.
w
–1
–2.5
–2
–1.5
–1
–0.5
w
0
0.5
1
FIG. 9: Evolution of the scaled velocities X, Y, Z, U for the
quintessence fields φ1 , φ2 , the phantom fields ψ1 , ψ2 , respectively, with respect to the equation of state w. The initial values are put by X(0) = −1, Y (0) = −1.1, Z(0) =
1, U (0) = 1.1, λφ1 (0) = 1/20, λφ2 (0) = 1/20, λψ1 (0) =
−1/27, λψ2 (0) = −1/20. The figure shows that the equation
of state of the split-quaternion field could cross the phantom
divide with the point (X, Y, Z, U, w) = (0, 0, 0, 0, −1) as the
attractor.
.
The second order action can be expressed as
Z
1
3
3
S(2) =
dtd xa
P̃,Y ηIJ + P̃,Y Y Φ̇I Φ̇J Q̇I Q̇J
2
1
− 2 P̃,Y [(1 + bX)ηIJ − bXIJ ] ∂i QI ∂ i QJ
a
–1.2
–1.4
−M̄IJ QI QJ + 2P̃,Y J Φ̇I QJ Q̇I ,
–1.6
(58)
–1.8
–2
with the effective squared mass matrix
0.2
0.4
0.6
0.8
N
1
1.2
1.4
FIG. 8: The evolution of the equation of state for the splitquaternion scalar when the kinetic energy of phantom dominates over that of quintessence (with the same strength of
fields). The equation of state is always smaller than unit one.
.
where QI denotes the field perturbations. In the following, we will usually drop the subscript “0” on ΦI0 and
simply identify ΦI as the fields in FRW Universe unless
otherwise stated.
X P̃,Y
(P̃,Y J Φ̇I + P̃,Y I Φ̇J )
H
3
X P̃,Y
1
+
(1 − 2 )Φ̇I Φ̇J
2H 2
cad
·
1
1 a3 2
1+ 2
Φ̇I Φ̇J . (59)
P̃
− 3
a 2H ,Y
cad
M̄IJ = −P̃,IJ +
Here P̃,Y and P̃,J denote the derivative of P̃ with respect
to Y and ΦJ . Dot denotes the derivative with respect to
cosmic time t. a and H are the scale factor and Hubble
parameter of the Universe, respectively. cad is the sound
speed for adiabatic perturbations defined by
c2ad ≡
P̃,Y
P̃,Y
.
+ 2X P̃,Y Y
(60)
10
1.5
U
1
1
Z
0.5
w0
0
Attractor
–1
–0.5
Y
–2
–1
X
–1.5 –1.015 –1.01 –1.005
–3 –3
–1 –0.995 –0.99
w
FIG. 10: Evolution of the scaled velocities X, Y, Z, U for the
quintessence fields φ1 , φ2 , the phantom fields ψ1 , ψ2 , respectively, with respect to the equation of state w. The initial
values are put by X(0) = −0.8, Y (0) = −0.99, Z(0) =
0.805, U (0) = 0.99, λφ1 (0) = 1/15, λφ2 (0) = 1/15, λψ1 (0) =
−1/20, λψ2 (0) = −1/20. The figure shows that the equation
of state of the split-quaternion field cold cross the phantom
divide with the point (X, Y, Z, U, w) = (0, 0, 0, 0, −1) as the
attractor.
.
The sound speed squared of entropy perturbations is defined by
c2en ≡ 1 + bX .
M̄IJ
·
.
(64)
The equation of motion is given by
Q̈I −
1
∂i ∂ i QI + M̄IJ QJ = 0 .
a2
(65)
In the gauge of Q2 = 0, we have
1
Q̈1 − 2 ∂i ∂ i Q1 + λ2 φ2 Q1 = 0 .
a
0
–0.99
(62)
where
a3
Φ̇I Φ̇J
H
–0.5
FIG. 11: The evolution of the equation of state for the splitcomplex scalar corresponding Fig. (9). It behaves as the stiff
matter at higher redshifts and a cosmological constant for
the lower redshifts. The split-complex scalar can cross the
phantom divide around the value of N = −1.5
.
–1.01
and the second order action is simply
Z
1
1
S(2) =
dtd3 xa3 Q̇I Q̇I − 2 ∂i QI ∂ i QI − M̄IJ QI QJ ,
2
a
(63)
–1
w –1
cad = cen = 1 ,
1
= λ ΦI ΦJ − 3
a
–1.5
N
(61)
For Lagrangian, Eq. (17), we have
2
–2
–2.5
0
0.5
The squared mass term is positive. Thus the perturbation to quintessence Q1 is stable. On the other hand, in
1.5
N
2
2.5
3
FIG. 12: The evolution of the equation of state for the splitcomplex scalar corresponding Fig. (10). It behaves as the
phantom matter at higher redshifts and a cosmological constant for the lower redshifts. The split-complex scalar can
cross the phantom divide around the value of N = 0.3
.
the gauge of Q1 = 0, we have
Q̈2 −
(66)
1
1
∂i ∂ i Q2 − λ2 ψ 2 Q2 = 0 .
a2
(67)
In this case, the squared mass term is negative and the
perturbation to phantom Q2 is unstable.
In order to achieve a scale invariant power spectrum,
we now decompose the perturbations into adiabatic and
11
entropy part as follows, QI = Qσ eIσ + Qs eIs , where
Φ̇I
,
eIσ = eI1 = q
P,X JK Φ̇J Φ̇K
Refs. [32] and [33] have ever shown that above equations
lead to exact scale invariant power spectrum.
(68)
VI.
I
and eIs is the unit
R vector orthogonal to eσ . Use the conformal time τ = dt/a(t) and define the canonically normalized fields
a
a
vσ ≡
Qσ ,
vs ≡
Qs ,
(69)
cad
cen
one derive the equations of motion for vσ and vs
α′′
z′
′′
′
2
2
2 2
vs + ξvσ + cen k −
+ a µs vs − ξvσ = 0 ,
α
z
′
′′
(zξ)
z
vσ −
vs = 0 .
(70)
vσ′′ − ξvs′ + c2ad k 2 −
z
z
Here the prime denotes the derivative with respect to τ
and
i
h
a
ξ ≡
(1 + c2ad )P̃,s − c2ad σ̇ 2 P̃,Y s ,
σ̇ P̃,Y cad
1 P̃,s2
P̃,ss
P̃,Y s P̃,s
− 2
+2
,
2
2
2cad X P̃,Y
P̃,Y
P̃,Y
q
q
aσ̇
z ≡
P̃,Y ,
α ≡ a P̃,Y ,
cad H
µ2s ≡ −
(71)
with
σ̇ ≡
√
2X ,
P̃,Y s ≡ P̃,Y I eIs
P̃,s ≡ P̃,I eIs
q
P̃,Y cen ,
P̃,ss ≡ P̃,IJ eIs eJs P̃,Y c2en .
q
P̃,Y cen ,
(72)
In the limit of weak coupling, ξ ≃ 0, small′′ effective mass,
µs ≃ 0 and slow-roll, Ḣ ≃ 0, one have z /z = 2/τ 2 and
′′
α /α = 2/τ 2 . So the equations of motion turn out to be
2
′′
2
2
vs + cen k − 2 vs = 0 ,
τ
2
′′
2 2
vσ + cad k − 2 vσ = 0 .
(73)
τ
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CONCLUSION AND DISCUSSION
In conclusion, motivated by the mathematic theory
of split-complex numbers, the split-quaternion numbers
and split-octonion numbers, we have proposed the notion
of split-complex scalar field, split-quaternion scalar field
and the split-octonion scalar field. We note that our proposal of split scalar fields is not just a mathematical tool
but rather it has physical information.
In the first place, the well-known quintessence and
phantom fields could naturally emerge in these fields. So
in order to construct a phantom field, one need not resort to quintessence by changing the sign of its kinetic
term by hand. On the other hand, the conventional
complex scalar field usually can not generate a phantom field by the decomposition, Φ = φ + iψ except for
a non-canonical form of Lagrangian [21]. Secondly, if
one require the split-complex scalar field obeys the symmetry of invariance under hyperbolic rotation, the splitcomplex field would restore to the Hessence field [21].
But if the symmetry of invariance is allowed to be broken, the split-complex scalar field would turn out to be
the quintom field [8]. So compared to Hessence and quintom, they have rich physics. Thirdly, we find there is a
conserved charge for the split-complex field. Therefore,
the Coleman’s Q-Balls-like solutions are expected to exist in the split-complex field which makes the field even
more physical. Finally, by introducing the metric of field
space, these split scalar fields fall into a subclass of the
multi-field theories which are being studied in inflationary cosmology [23–27]. Also, following the usual procure
of quantization of the complex scalar field [34], one could
carry out the quantization of split-complex scalar field.
Acknowledgments
This work is supported by the Chinese MoST 863 program under grant 2012AA121701, the NSFC under grant
11373030, 10973014, 11373020 and 11465012.
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