Prog. Theor. Exp. Phys. 2014, 113E01 (13 pages)
DOI: 10.1093/ptep/ptu147
Time variation of the equation of state
for dark energy
Tetsuya Hara1,∗ , Ryohei Sakata2 , Yusuke Muromachi1 , and Yutaka Itoh1
1
Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan
Altech Corporation, Queen’s Tower C 18F, Yokohama, 220-6218, Japan
∗
E-mail: [email protected]
2
Received September 10, 2014; Revised October 1, 2014; Accepted October 1, 2014; Published November 17, 2014
...............................................................................
The time variation of the equation of state (w Q ) for the dark energy is analyzed by the current
values of the parameters Q , w Q and their time derivatives. In the future, detailed features of the
dark energy could be observed, so we have considered the second derivative of w Q for two types
of potential: One is an inverse power-law type (V = M 4+α /Q α ) and the other is an exponential
one (V = M 4 exp (β M/Q)). The first derivative dw Q /da and the second derivative d 2 w Q /da 2
for both potentials are derived. The first derivative is estimated by the observed two parameters = w Q + 1 and Q , with the tracker approximation for Q. In the limit → 0, the first
derivative is null and, under the tracker approximation, the second derivative also becomes null.
The evolution of forward and/or backward time variation could be analyzed from some fixed time
point. If the potential is known, the evolution will be estimated from values Q and Q̇ at this point,
because the equation for the scalar field is the second derivative equation. For the inverse power
potential, if we do not adopt the tracker approximations, the observed first and second derivatives with and Q must be utilized to determine the two parameters of the potential, M and α.
For the exponential potential, the second derivative is estimated by the observed parameters ,
Q , and dw Q /da, because the parameter for this potential is essentially one, β. If the parameter
number is n for the potential form, it will be necessary for n + 2 independent observations to
determine the potential, Q and Q̇, for the evolution of the scalar field.
...............................................................................
Subject Index
1.
E64, E65
Introduction
Even though it is almost ten and several years since the detection of the acceleration of the universe,
the dark energy is not well understood [1]. We do not yet know whether it is the cosmological constant
or not [2–4]. Then a search is undertaken to observe the variation of the equation of state (w = w Q )
for the dark energy. Many works have been done on the study of dark energy in the form of a slowly
rolling scalar field and time variation of the equation of state for the dark energy. Usually, a parameter
is taken to denote the variation of w as [1,5–11]
w(a) = w0 + wa (1 − a),
(1)
where a, w0 , and wa are the scale factor (a = 1 at present), the current value of w(a), and the first
derivative of w(a) by wa = −dw/da, respectively.
Even now, not very much is known about the values of w0 and wa ; we extend the parameter space as
w(a) = w0 + wa (1 − a) + 12 wa2 (1 − a)2 ,
© The Author(s) 2014. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),
which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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T. Hara et al.
including the second derivative of w(a) as wa2 = d 2 w/da 2 . Although it may be hard to observe the
parameter wa2 , it must be a good clue to understanding the features of dark energy in the future.
One of the new ingredients of this work in comparison with past works is the inclusion of the second
derivative for the parameter space, where w Q and dw Q /da were mainly considered up to now [12].
We follow the single scalar field formalism of Ref. [13] and take the potential of two types as
V = M 4+α /Q α and V = M 4 exp(β M/Q). Although there are many scalar potentials motivated by
particle physics [14,15], the inverse power-law type is one of the most commonly investigated; it was
chosen by Peebles and Ratra [2,3] for the time variable “cosmological constant”. It is also examined
in Ref. [13] as one of the leading candidates for the “tracker solution”, which is the attractor-like solution to the “coincidence problem”, converging a very wide range of initial conditions to a common
evolutionary track of w Q (t).
Steinhardt et al. [13] also studied the exponential of the inverse power-law model V (Q) =
M 4 exp(1/Q) as a more generic potential with a mixture of inverse power-laws, through changing the
effective power index by varying Q (initially Q is small with a high power index; later it is large with a
low power index). Wang et al. [16] have analyzed the tracker field quintessence models with potentials
V (Q) ∝ Q −α , exp(M p /Q), exp(M p /Q) − 1, exp(β M p /Q), and exp(γ M p /Q) − 1, with observational data from the SNIa, CMB, and BAO. They have found that models with potentials exp(M p /Q),
exp(M p /Q) − 1, and exp(γ M p /Q) − 1 are not supported by the data. So we examine the inverse
power-law and the exponential one for the single scalar field model of dark energy.
The goal of this paper is to explore the dark energy under the quintessence model in a single scalar
field by assuming the potential. We have tried to increase the parameter space to explore the features
of dark energy, by adding the second derivative. We examine our trial under typical potentials. To
determine the potential form we must observe the evolution of the universe. If the parameter number
is n for the potential form, it will be necessary to make n + 2 independent observations to determine
the potential, Q and Q̇, at some time for the time variation of the scalar field.
In Sect. 2, the equation of state for the scalar field and parameters to describe the potentials are
presented. The first derivatives of w Q for two types of potentials are calculated in Sect. 3. In Sect. 4,
the second derivatives are presented, and the detailed calculations are shown in Appendix A. The
results and discussion are presented in Sect. 5.
2. Equation of state
2.1. Scalar field
For the dark energy, we consider the scalar field Q(x, t), where the action for this field in the
gravitational field is described by [13]
1 μν
1
4 √
R + g ∂μ Q∂ν Q − V (Q) + S M ,
S = d x −g −
(3)
16π G
2
where S M is the action of the matter field and G is the gravitational constant, occasionally putting
G = 1. Neglecting the coordinate dependence, the equation for Q(t) becomes
Q̈ + 3H Q̇ + V = 0,
(4)
where H is the Hubble parameter and V is the derivative of V by Q. As κ = 8π/3, H satisfies the
following equation:
2
ȧ
= κ(ρ B + ρ Q ) = κρc ,
(5)
H2 =
a
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where ρ B , ρ Q , and ρc are the energy density of the background and scalar fields and the critical
density of the universe. The energy density and pressure for the scalar field are written as
ρ Q = 12 Q̇ 2 + V,
(6)
p Q = 12 Q̇ 2 − V,
(7)
and
respectively. Then the parameter w Q for the equation of state is described by
wQ ≡
2.2.
1 2
2 Q̇
1 2
2 Q̇
pQ
=
ρQ
−V
+V
.
(8)
Time variation of w Q
It is assumed that the current value of w Q is slightly different from a negative unity by (> 0) as
w Q = −1 + .
(9)
By using Eq. (8), Q̇ 2 is written as
Q̇ 2 =
2V
,
2−
(10)
which is also given by using the density parameter Q = ρ Q /ρc as
Q̇ 2 = 2(ρc Q − V ).
(11)
Combining Eqs. (10) and (11), V is given by
V = ρc Q
.
1−
2
(12)
From Eqs. (11) and (12), Q̇ is given as
Q̇ =
(ρc Q ).
(13)
If we determine the potential, the parameters to describe the evolution of the scalar field are the
values of Q and Q̇ at some fixed time, because Eq. (4) is the second derivative equation. Then the
evolution or backward variation could be estimated from this fixed point. In the following we take
this fixed time as it is at present and estimate backward the accelerating behavior in the near past.
As ρc is given by observation through the Hubble parameter H , Q̇ is determined by Q and ,
which also determine the value of V . If we adopt the form and parameter of the potential, the value
of V could be used to estimate the value of Q. In reality, the evolution of H in Eq. (4) depends on the
background densities, which include radiation density. The effect of radiation density can be ignored
in the near past (z ≤ 103 ) and so is not considered in this work.
In the following, we investigate the power inverse potential V = M 4 (M/Q)α ; (α > 0) and the
exponential potential V = M 4 exp(β M/Q); (β > 0), respectively.
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2.2.1. V = M 4 (M/Q)α
The parameters of the potential are M and α. When we take M = M∗ , V becomes
α
4 M∗
V = M∗
.
Q
From Eq. (12), as M∗4 (M∗ /Q)α = ρc Q (1 −
Q=
2 ),
(14)
Q is given as
M∗4+α
ρc Q 1 −
1/α
2
.
(15)
If we take Q = Q 0 Mpl at present, Mpl being the Planck mass, M∗ becomes
M∗ = Mpl
Q α0
1/(4+α)
ρc
1−
.
4 Q
2
Mpl
(16)
Then Q 0 , Q , , and α determine the parameter M∗ , which means that the parameters determining
the accelerating behavior are Q 0 , Q , , and α.
4 is described by the observed value N as
The difference in the observed value ρc and the value Mpl
4
× 10−N ;
ρc = Mpl
(17)
as N 122 [4], M∗ becomes
M∗ = Mpl ×
Q α0 Q
1−
2
1/(4+α)
× 10−N /(4+α) .
(18)
The problem is how to estimate Q 0 and α.
2.2.2. V = M 4 exp(β M/Q)
If we put Mpl = β M, V is written as
V =
Mpl
β
4
exp
Mpl
.
Q
(19)
In essence, as β is combined with M, the parameter of this potential is β. From Eq. (12), exp(Mpl /Q)
is given by
Mpl
β 4
=
.
(20)
exp
ρc Q 1 −
Q
Mpl
2
Then Q is estimated as
Q=
Mpl
ln β 4 ρc Q 1 −
2
4
/Mpl
.
(21)
If we take Q = Q 0 Mpl as it is at present, Q 0 determines the parameter β as
β=
1/4
4
Mpl
ρc Q (1 − /2)
exp Q −1
/4
.
0
(22)
In this potential, the parameters determining the accelerating behavior are Q 0 , Q , and . The
problem of how to estimate Q 0 is also left.
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3.
T. Hara et al.
First derivative of w Q
To investigate the variation of w Q , we calculate dw Q /da using Eqs. (4): (6), and (7),
1 d
dw Q
=
da
ȧ dt
pQ
ρQ
=
1 ṗ Q ρ Q − p Q ρ̇ Q
2
ȧ
ρQ
=
1 Q̇(
Q̈
+
V
Q̇(
Q̈
−
V
)ρ
−
p
)
Q
Q
2
a Hρ Q
=
Q̇ (−3H Q̇ − 2V )ρ Q − p Q (−3H Q̇)
2
a Hρ Q
Q̇ 3H Q̇( p Q − ρ Q ) − 2V ρ Q
2
a Hρ Q
2V Q̇
V
=
−3H Q̇ − ρ Q .
2
V
a Hρ Q
=
(23)
In the limit → 0 where Q̇ = ρc Q → 0, dw Q /da becomes null. To investigate further, we
must consider the potential form.
3.1.
V = M 4+α /Q α
As V /V = −α/Q, Eq. (23) becomes
αρ Q
− 3H Q̇
Q
2V Q̇
3H Q̇ Q
=
α−
.
a H Qρ Q
ρQ
dw Q
2V Q̇
=
2
da
a Hρ Q
(24)
To investigate the signature of dw Q /da, we must estimate the following term:
3H Q̇ Q
3H =
(ρc Q )Q 0 Mpl =
ρQ
ρc Q
24π × Q0.
Q
(25)
To estimate Q 0 , we consider the tracker approximation that w Q is almost constant as
−
Q
V
= 1,
24π(1 + w Q ) V
(26)
which is given by Eq. (9) of Ref. [13]. If we adopt this approximation, Q/Mpl becomes
Qα =
Q
× α.
24π We approximate the current value Q 0 as Q 0 = (1 + ε) × Q α . Then, from Eq. (25),
becomes (1 + ε)α. If ε > 0, dw Q /da < 0, and vice versa.
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(27)
24π Q
× Q0
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From Eq. (24), Q is derived as
Q=
then Q 0 is given by
Q0 = α
2αρ Q V Q̇
,
2 dwq + 6V Q̇ 2
H aρ Q
da
3 Q
(1 − /2)
.
dw Q
2π
+
6(1
−
/2)
da
(28)
(29)
If dw Q /da is observed, Q 0 /α will be determined by the observed values Q , , and dw Q /da.
dw
For a positive value of Q 0 , daQ must be greater than −6(1 − /2). Assuming that dw Q /da 1
and 0.1, it is estimated that 0 Q 0 ∞.
3.2.
V =
Mpl 4
Mpl
exp
β
Q
As V /V = −Mpl /Q 2 , Eq. (23) becomes
dw Q
3H Q̇ 2
3 Q̇ 2
2V
2V
V
V Q̇
−
−
=
Q̇ =
−
−
da
a Hρ Q
ρQ
V
aρ Q
ρQ
VH
⎡
⎤
4 × 10−Nc
M
Q
M
pl
2V ⎣ 3ρc Q
pl
⎦
+ 2√
=
−
aρ Q
ρQ
Q
8π/3Mpl × 10−Nc /2
Q
2V
−3 +
=
√
aρ Q
(Q/Mpl )2 8π/3
√
3 Q 2V
2
=
1 − / Q (Q/Mpl ) 24π .
aρ Q (Q/Mpl )2
8π
If we adopt the tracker approximation in Eq. (26), Q becomes
Q 1/4
Qβ =
.
24π (30)
(31)
We approximate the current value Q 0 as Q 0 = (1 + ε) × Q β . Then from Eq. (30), the above value
within [ ] becomes −ε. If ε > 0, dw Q /da < 0, and vice versa.
From Eq. (30), Q is determined as
⎛
⎞1/2
2Vρ
Q̇
Q
⎠ .
Q=⎝
(32)
dw Q
2
H (aρ Q da + 6V Q̇ 2 )
Then Q 0 is estimated by the observable parameters Q , , and dw Q /da as
⎞1/2
⎛
3
2(1
−
/2)
Q
⎠ .
Q0 = ⎝
dw
8π
a Q + 6(1 − /2)
(33)
da
Q 0 does not depend on the potential parameter β, which is determined by Eq. (22).
dw
For the real value of Q 0 , daQ must be greater than −6(1 − /2). Assuming that dw Q /da 1
and 0.1, it is estimated that 0 Q 0 ∞.
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T. Hara et al.
The second derivative of w Q
From Eq. (23), the second derivative of w Q is given by
d 2wQ
1 2
2
=
ρ
−
p
ρ̈
)
ȧρ
−
(
p
˙
ρ
−
p
ρ̇
)
äρ
+
2
ȧρ
ρ̇
(
p̈
.
Q Q
Q Q
Q Q
Q Q
Q Q
Q
Q
4
da 2
ȧ 3 ρ Q
(34)
The time derivatives of p Q and ρ Q are written as
ṗ Q = −3H Q̇ 2 − 2V Q̇,
ä
2
p̈ Q = −3 + 21H − 2V
Q̇ 2 + 12H Q̇V + 2(V )2 ,
a
ä
2
2
ρ̇ Q = −3H Q̇ , ρ̈ Q = −3 + 21H Q̇ 2 + 6H Q̇V ,
a
(35)
(36)
where ä/a = −4π G(ρc + p Q ). Using these equations, the detailed calculation of Eq. (34) is
described in Appendix A.
From Eq. (A21), d 2 w/da 2 becomes
d 2wQ
3 Q
1−
=
da 2
4π a 2
2
V
6π V
2
× −Mpl
+
(1 − ) (6 + Q ) − Q /3 Mpl
V
Q
V
2
V
8π 2
Mpl
+
(7 − 6) .
(37)
+ 1−
2
V
Q
From this equation, the value of d 2 w Q /da 2 can be estimated. In the limit → 0, the signature
of d 2 w Q /da 2 is positive under the condition V /V = 0. In principle, when the first derivative is
observed, Q 0 is estimated through Eqs. (29) or (33). Then the second derivative is estimated by the
observed parameters , Q , and Q 0 .
From this equation, we estimate d 2 w Q /da 2 for each potential in the following.
4.1.
V = M 4+α /Q α
As the following relations are derived,
V α(α + 1)
=
,
V
Q2
α
V
=− ,
V
Q
V
V
2
=
α2
,
Q2
we put them in Eq. (37) and get
d 2wQ
3 Q
1−
=
da 2
4π a 2
2
6π α
α(α + 1)
−
× −
(1 − ) (6 + Q ) − Q /3
(Q/Mpl )2
Q
Q/Mpl
2
α
8π +
(7 − 6) .
+ 1−
2
Q/Mpl
Q
(38)
If dw Q /da is observed, Q/α will be determined by Eq. (29). If d 2 w Q /da 2 is observed, one can
estimate the value of α from the above equation.
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When we take the tracker approximation as Q 0 (1 + ε) × Q α (1 + ε) × Q /(24π ) × α
of Eq. (27), the part within [ ] of the above equation becomes
(1 + 1/α)
1
24π (1
−
)(6
+
−
−
)
−
/3
Q
Q
Q
(1 + ε)2
2(1 + ε)
1
1
+ (7 − 6) .
(39)
1−
+
(1 + ε)2
2
3
In the limit → 0 where Q α → ∞, d 2 w Q /da 2 becomes null in this approximation.
4.2.
V = (Mpl /β)4 exp(Mpl /Q)
As the following relations are derived,
2
2
2
Mpl
Mpl
2Mpl
Mpl
V V
V
=
=
−
+
,
,
=
,
V
Q3
Q4 V
Q2
V
Q4
we put them in Eq. (37) and get
Mpl 3
Mpl 4
d 2wQ
3 Q
− 2
1−
=
+
da 2
4π a 2
2
Q
Q
Mpl 2
6π −
(1 − )(6 + Q ) − Q /3
Q
Q
Mpl 4 8π + 1−
+
(7 − 6) .
2
Q
Q
(40)
When we take the tracker approximation as Q 0 (1 + ε) × Q β = (1 + ε) × ( Q /(24π ))1/4
in Eq. (31), the part within [ ] of the above equation becomes
24π 24π −1/4
1
2
1
+
(1 − )(6 + Q ) − Q /3
−
−
3
4
2
Q
(1 + ε)
Q
(1 + ε)
2(1 + ε)
1
1
+ (7 − 6) .
(41)
1−
+
4
(1 + ε)
2
3
In the limit → 0 where Q β → ∞, d 2 w Q /da 2 becomes null.
5.
Conclusions and discussion
It is important to know the variation of the equation of state w of the background field for the
investigation of the expansion of the universe. It is known that ä is described by the following
equation:
4π G
(1 + 3w)aρ.
ä = −
3
At present, backward observation of the large-scale structure of the universe has been undertaken to
estimate w Q at the age (1 + z) [7]. For the moment, the values of w(a = a0 ) and dw/da have been
pursued:
dw
da.
(42)
w(a) = w(a = a0 ) +
da
Observation of the second derivative of w could be expected in the future:
1 d 2w
dw
da +
(da)2 + · · · ,
da
2 da 2
so we estimate the second derivative of w with a for two typical potentials in this work.
w(a) = w(a0 = 1) +
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The first derivative dw Q /da and the second derivative d 2 w Q /da 2 for the power inverse and
exponential potentials are calculated. The first derivative is estimated by the two observed parameters = w Q + 1 and Q , assuming parameters Q 0 . In the limit → 0, the first derivative is null
and, under the tracker approximation, the second derivative also becomes null. For the inverse power
potential V = M∗4+α /Q α , the observed first and second derivatives are used to determine the potential parameters M∗ and α. For the exponential potential V = M 4 exp (β M/Q), the second derivative
is calculated by the observed parameters , Q , and dw Q /da, where the potential parameter is
essentially one, β.
The evolution of forward and/or backward time variation could be analyzed from some fixed time
point. If the form of the potential is known, the evolution will be calculated with values Q and Q̇ at
a fixed time, because the equation for the scalar field is the second derivative equation. Q̇ could be
derived from and Q . If dw Q /da is observed, Q/α could be estimated for the inverse power-law
potential and Q could be estimated for the exponential potential. To estimate the parameter α, one
must observe the second derivative d 2 w Q /da 2 .
After the parameters of the potential are determined, the time variation of the dark energy, such
as d 3 w Q /da 3 , d 4 w Q /da 4 , and so on, can be calculated. If they are predicted values, it will be
understood that the dark energy can be described by quintessence with a single scalar field.
We have increased the parameter space for dark energy such as w and dw/da by adding the second
derivative d 2 w Q /da 2 . Then we have tried to derive the second derivatives for well investigated potentials, suggested by the observations [16]. Intimate relations between the second derivative and the
observables have been derived. If we do not adopt some assumptions, such as the tracker approximation and/or matter-dominant stage, it will be necessary to observe Q , w Q , dw Q /da, and d 2 w Q /da 2
for the inverse power potential V = M 4 (M/Q)α , where the form parameters are two M and α. When
other forms of potential with parameter n are taken, it is necessary to make independent n + 2 observations for the determination of Q, Q̇, and form parameter n of the potential. Such observations
would be values of Q , w, dw/da . . ., and d n w/da n .
If we assume a matter-dominant approximation, we could estimate α and M∗ from the attractor
solution [17], which is outlined in Appendix B. If < 0, we must consider utterly different models
such as phantoms, quintom, or k-essence [18–20].
Appendix A. Derivation of Eq. (34)
By using Eqs. (35) and (36), the terms within Eq. (34) become
!
p̈ Q ρ Q − p Q ρ̈ Q = − 2V Q̇ 2 + 6H Q̇V + 2(V )2 12 Q̇ 2
ä
+ −6 + 42H 2 − 2V Q̇ 2 + 18H Q̇V + 2(V )2 V
a
ṗ Q ρ Q − p Q ρ̇ Q = Q̇[3H Q̇( p Q − ρ Q ) − 2V ρ Q ].
Next, we try to calculate the second term within [ ] of Eq. (34). It is given as [17]
ρ
ä
c
= 4π G − − p Q ,
a
3
(A1)
(A2)
(A3)
where we neglect the radiation pressure. By using the relation of p Q = ρ Q − 2V = ρc Q − 2V and
Eq. (12), we get
p Q = ρc Q (−1 + ).
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(A4)
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Putting this relation into Eq. (A3), it becomes
ä
1
= 4π Gρc Q (1 − ) − Q .
a
3
(A5)
As Q̇ 2 = ρ Q , the first derivatives of p Q and ρ Q become
ṗ Q = −3Hρ Q − 2V Q̇,
ρ̇ Q = −3Hρ Q .
(A6)
Using H 2 = (8π G/3)ρc , the term on the right-hand side ( ) of the second term in the [ ] of Eq. (34)
becomes
! 2
2
+ 2ȧρ Q ρ̇ Q = 4π Gaρc Q (1 − ) − 13 Q ρ Q
+ 2H aρ Q (−3Hρ Q )
äρ Q
!
2
ρc Q (1 − ) − 13 Q − 4 .
(A7)
= 4π Gaρ Q
Taking p Q − ρ Q = −2V and Eq. (A2), the term on the left-hand side ( ) of the second term is
√
V
√
24π G +
Q .
(A8)
ṗ Q ρ Q − p Q ρ̇ Q = −2Vρ Q ρc
V
From Eqs. (A7) and (A8), the second term in the [ ] of Eq. (34) becomes
1−
+ 2ȧρ Q ρ̇ Q ) = −
Q
2
√
V ×
24π G +
Q V
1
× Q (1 − ) − Q − 4 .
3
1/2
( ṗ Q ρ Q −
2
p Q ρ̇ Q )(äρ Q
5ρ
8π Gaρ Q
c
(A9)
Next, we try to calculate the first term within [ ] of Eq. (34). Equation (A1) can be changed to
2 1 2 2
H Q̇ V V
V Q̇ 2
+6
+2
Q̇ V
p̈ Q ρ Q − p ρ̈ Q = −2
V V
V V
V
2
2 H2
V Q̇ 2
H Q̇ V V
ä 1
+ 42
−2
+ 18
+2
+
−6
(A10)
V 3.
aV
V
V
V
V V
V
Here we calculate the elements separately in the above equation as
Q̇ 2
ρQ =
=
,
V
ρ Q (1 − /2)
1 − /2
√
√
(8π G/3)ρc ρc Q H Q̇
(8π G/3)
=
=
,
V
ρc Q (1 − /2)
Q (1 − /2)
1 2 2 1
2 1 3
2
Q̇ V = ρ Q ρ 1 −
= ρQ 1 −
,
2
2
2
2
2
!
!
4π Gρc Q (1 − ) − Q /3
ä 1
4π G Q (1 − ) − Q /3
=
=
,
aV
ρc Q (1 − /2)
Q
(1 − /2)
H2
(8π G/3)ρc
8π G
1
=
=
.
V
ρc Q (1 − /2)
3 Q (1 − /2)
10/13
(A11)
(A12)
(A13)
(A14)
(A15)
PTEP 2014, 113E01
T. Hara et al.
Then Eq. (A10) can be described as
1 3 2
p̈ Q ρ Q − p Q ρ̈ Q = ρ Q 1 −
2
2
√
√
8π G/3 36 8π G/3(1 − /2) V V
4
1−
+ 6 ×
−2 −
+
2
V
V
Q Q
2
2
4
2
V
1−
+ 2 1−
+
2
2
V
2 −24π G Q (1 − ) − Q /3 + 112π G .
+
(A16)
Q
2 = a Hρ 2 = a √(8π/3)ρ ρ 2 , the first term within [ ] of Eq. (34) becomes
As ȧρ Q
c Q
Q
1 5
2
2
( p̈ Q ρ Q − p Q ρ̈ Q )ȧρ Q = aρ Q (8π G/3)ρc 1 −
2
2
√
√
8π G/3 36 8π G/3(1 − /2) V 4 V
+ 6 ×
−
+
V
V
Q Q
2
2
4
2
V
+
1−
+ 2 1−
2
2
V
2 −24π G Q (1 − ) − Q /3 + 112π G . (A17)
+
Q
Then Eq. (34) can be written as
8π G 1/2 5
2
ρc ρ Q a 1 −
3
2
√
2 24π G
× ([ ] in Eq. (A17)) +
({ } in Eq. (A9)) .
Q 2
1
1
d 2wQ
= 3 4 ×
2
da
2
ȧ ρ Q
The term within [ ] is calculated as
6 8π G
V
1
4 V + 2
(1 − )(6 + Q ) − Q
−
V
3 Q
3
V
2 2
2
V
16π G
+
1−
+
(14 − 12).
2
V
Q
The outside factor of [ ] is
3 Q
16π G a 2
2 .
1−
2
(A18)
(A19)
(A20)
Using Eqs. (A19) and (A20), d 2 w/da 2 becomes
d 2wQ
6π 0.5 V
3 Q
2 V
−4Mpl
+4
1−
=
(1 − ) (6 + Q ) − Q /3 Mpl
2
2
da
16πa
2
V
Q
V
2
V
32π 2
Mpl
+
(7 − 6) .
(A21)
+ 4 1−
2
V
Q
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T. Hara et al.
Appendix B. Matter-dominant and attractor-solution approximation
For the era ρ M ≥ ρ Q , we can use Eq. (4) as the matter-dominant approximation where a ∝ t 2/3 . For
the inverse power potential, it becomes
Q̈ +
2
Q̇ − α M 4+α Q −α−1 = 0,
t
which has the attractor solution as [17]
1/(2+α)
α(2 + α)2 M 4+α t 2
Q=
.
2(4 + α)
(B1)
(B2)
If this solution was used, the parameters α and M would be derived by and the time tc at which
ρM = ρQ .
For this solution, it is derived as
1 2
α
Q̇ /V =
,
(B3)
2
4+α
then
"
α
2
α
α
wQ =
−1
+1 =−
= −1 +
.
(B4)
4+α
4+α
2+α
2+α
As =
α
2+α ,
α could be estimated by as
α=
2
.
1−
(B5)
If a t 2/3 and ≤ 0.1 could be approximated until the current t0 , it could be approximated as
tc = t0 /( Q /(1 − Q ))1/2 . The parameter M is derived by the relation ρ M = ρ Q at tc as
2 4+α t 2 −α/(2+α)
1
1
c
4+α α(2 + α) M
·
=M
2 6π Gtc2
2(4 + α)
1+α
1/(2+α)
2 (2 + α)2−α
−2α/(2+α)
=
M 2(4+α)/(2+α) tc
,
α α (4 + α)2
where ρ M = 1/(6π Gt 2 ) and ρ Q = Q̇ 2 /2 + V are used. So M∗ is determined by α and tc as
1/(2(4+α))
−2/(4+α)
M∗ = (3 · 23 π )−(2+α) α α (2 + α)−2+α (4 + α)2
× tc
.
(B6)
(B7)
(B8)
Then the observed values t0 , Q , and could determine the potential parameters α and M∗ under
the matter-dominant and attractor-solution approximation.
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