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International Journal of Modern Physics E
c© World Scientific Publishing Company
Dark energy and neutrino model in SUSY
– Remarks on active and sterile neutrinos mixing –
Ryo Takahashi∗
Graduate School of Science and Technology, Niigata University, 950-2181 Niigata, Japan
takahasi@muse.sc.niigata-u.ac.jp
Morimitsu Tanimoto
Department of Physics, Niigata University, 950-2181 Niigata, Japan
tanimoto@muse.sc.niigata-u.ac.jp
We consider a Mass Varying Neutrinos (MaVaNs) model in supersymmetric theory.
The model includes effects of supersymmetry breaking transmitted by the gravitational
interaction from the hidden sector, in which supersymmetry was broken, to the dark
energy sector. Then evolutions of the neutrino mass and the equation of state parameter
of the dark energy are presented in the model. It is remarked that only the mass of a
sterile neutrino is variable in the case of the vanishing mixing between the left-handed
and a sterile neutrino on cosmological time scale. The finite mixing makes the mass of
the left-handed neutrino variable.
1. Introduction
Cosmological observations have provided the strong evidence that the Universe
is flat and its energy density is dominated by the dark energy component whose
negative pressure causes the cosmic expansion to accelerate.1 In order to clarify the
origin of the dark energy, one has tried to understand the connection of the dark
energy with particle physics.
In the Mass Varying Neutrinos (MaVaNs) scenario proposed by Fardon, Nelson
and Weiner, relic neutrinos could form a negative pressure fluid and cause the
present cosmic acceleration.2 In the model, an unknown scalar field, which is called
“acceleron”, is introduced and neutrinos are assumed to interact through a new
scalar force. The acceleron sits at the instantaneous minimum of its potential, and
the cosmic expansion only modulates this minimum through changes in the number
density of neutrinos. Therefore, the neutrino mass is given by the acceleron, in
other words, it depends on the number density of neutrinos and changes with the
expansion of the Universe. The equation of state parameterw and the energy density
of the dark energy also evolve with the neutrino mass. Those evolutions depend
∗talked at the International Workshop on Neutrino Masses and Mixings, University of Shizuoka,
Shizuoka, Japan, December 17-19, 2006
http://arxiv.org/abs/0704.0186v1
October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka
2 Ryo Takahashi
on the form of a scalar potential and the relation between the acceleron and the
neutrino mass strongly. Some examples of the potential have been considered.3
The idea of the variable neutrino mass was considered at first in a model of
neutrino dark matter and was discussed for neutrino clouds.4 Interacting dark en-
ergy scalar with neutrinos was considered in the model of a sterile neutrino.5 The
coupling to the left-handed neutrino and its implication on the neutrino mass limit
from baryogenesis was discussed.6 In the context of the MaVaNs scenario, there
have been a lot of works. 7,8,9,10,11,12
In this talk, we present a MaVaNs model including the supersymmetry breaking
effect mediated by the gravity. Then we show evolutions of the neutrino mass and
the equation of state parameter in the model.
2. MaVaNs Model in Supersymmetric Theory
We discuss the Mass Varying Neutrinos scenario in supersymmetric theory and
present a model.
We assume a chiral superfield A in dark sector. A is assumed to be a singlet
under the gauge group of the standard model. It is difficult to construct a viable
MaVaNs model without fine-tunings in some parameters when one assumes one
chiral superfield in dark sector, which couples to only the left-handed lepton doublet
superfield. 8 Therefore, we assume that the superfield A couples to both the left-
handed lepton doublet superfield L and the right-handed neutrino superfield R. For
simplicity, we consider the MaVaNs scenario in one generation of neutrinos.a
In such framework, we suppose the following superpotential,
AA+mDLA+MDLR+
RR, (1)
where λ is a coupling constant of O(1) and MA, MD, MR and mD are mass pa-
rameters.b The scalar and the spinor component of A are represented by φ and ψ,
respectively. The scalar component corresponds to the acceleron which cause the
present cosmic acceleration. The spinor component is a sterile neutrino. The third
term of the right-hand side in Eq. (1) is derived from the Yukawa coupling such as
yLAH with y < H >= mD, where H is the Higgs doublet.
In the MaVaNs scenario, the dark energy is assumed to be composed of the
neutrinos and the scalar potential for the acceleron. Therefore, the energy density
of the dark energy is given as
ρDE = ρν + V (φ). (2)
Since only the acceleron potential contributes to the dark energy, we assume the
vanishing vacuum expectation values of sleptons, and thus we find the following
aThree generations model of this scenario has presented in non supersymmetric theory.9
bOther supersymmetric model so called “hybrid model” has been proposed.10
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 3
effective scalar potential,
V (φ) =
|φ|4 +M2A|φ|
2 +m2D|φ|
2. (3)
We can write down a lagrangian density from Eq. (1),
L = λφψψ +MAψψ +mDνLψ +MDνLνR +MRνRνR + h.c.. (4)
It is noticed that the lepton number conservation in the dark sector is violated
because this lagrangian includes both MAψψ and mDνLψ. After integrating out
the right-handed neutrino, the effective neutrino mass matrix is given by
mD MA + λφ
, (5)
in the basis of (νL, ψ), where c ≡ −M
D/MR and we assume λφ≪ MD ≪MR. The
first term of the (1, 1) element of this matrix corresponds to the usual term given
by the seesaw mechanism in the absence of the acceleron. We obtain masses of the
left-handed and a sterile neutrino as follows,
mνL =
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
, (6)
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
. (7)
It is remarked that only the mass of a sterile neutrino is variable in the case of
the vanishing mixing (mD = 0) between the left-handed and a sterile neutrino on
cosmological time scale. The finite mixing (mD 6= 0) makes the mass of the left-
handed neutrino variable. We will consider these two cases of mD = 0 and mD 6= 0
later.
In the MaVaNs scenario, there are two constraints on the scalar potential. The
first one comes from cosmological observations. It is that the magnitude of the
present dark energy density is about 0.74ρc. ρc is the critical density. Thus, the
first constraint turns to
V (φ0) = 0.74ρc − ρ
ν , (8)
where “0” means the present value.
The second one is the stationary condition:
∂V (φ)
= 0. (9)
In this scenario, the neutrino mass is represented by a function of the acceleron;
mν = f(φ). Since the energy density of the neutrino varies on cosmological times
scale, the vacuum expectation value of the acceleron also varies. This property
makes the neutrino mass variable. If ∂mν/∂φ 6= 0, Eq. (9) is equivalent to
∂V (φ(mν))
= 0. (10)
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4 Ryo Takahashi
Eq. (10) is rewritten by using the cosmic temperature T :
∂V (φ)
= −T 3
∂F (ξ)
, (11)
where ξ ≡ mν/T , ρν = T
4F (ξ) and
F (ξ) ≡
y2 + ξ2
ey + 1
. (12)
We can get the time evolution of the neutrino mass from Eq. (11). Since the sta-
tionary condition should be always satisfied in the evolution of the Universe, this
one at the present epoch is the second constraint on the scalar potential:
∂V (φ)
mν=m0ν
= −T 3
∂F (ξ)
mν=m0ν ,T=T0
. (13)
In addition to two constraints for the potential, we also have two relations between
the vacuum expectation value of the acceleron and the neutrino masses at the
present epoch:
m0νL =
c+MA + λ < φ >
[c− (MA + λ < φ >0)]2 + 4m
, (14)
m0ψ =
c+MA + λ < φ >
[c− (MA + λ < φ >0)]2 + 4m
. (15)
Next, let us consider the dynamics of the acceleron field. In order that the
acceleron does not vary significantly on distance of inter-neutrino spacing, the ac-
celeron mass at the present epoch must be less than O(10−4eV) 2. Here and below,
we fix the present acceleron mass as
m0φ = 10
−4 eV. (16)
Once we adjust parameters which satisfy five equations (8) and (13)∼(16), we can
have evolutions of the neutrino masses by using the Eq. (11).
The dark energy is characterized by the evolution of the equation of state pa-
rameter w. The equation of state is derived from the energy conservation law and
the stationary condition Eq. (11):
w + 1 =
[4− h(ξ)]ρν
, (17)
where
h(ξ) ≡
∂F (ξ)
F (ξ)
. (18)
It seems that w in this scenario depend on the neutrino mass and the cosmic
temperature. This means that w varies with the evolution of the Universe unlike
the cosmological constant.
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 5
In the last of this section, we comment on the hydrodynamical stability of the
dark energy in the MaVaNs scenario. The speed of sound squared in the neutrino-
acceleron fluid is given by
c2s =
ẇρDE + wρ̇DE
, (19)
where pDE is the pressure of the dark energy. Recently, it was argued that when
neutrinos are non-relativistic, this speed of sound squared becomes negative in this
scenario.11 The emergence of an imaginary speed of sound means that the MaVaNs
scenario with non-relativistic neutrinos is unstable, and thus the fluid in this sce-
nario cannot acts as the dark energy. However, finite temperature effects provide a
positive contribution to the speed of sound squared and avoid this instability. 12
Then, a model should satisfy the following condition,
5aT 2
25aT 20 (z + 1)
> 0, (20)
where z is the redshift parameter, z ≡ (T/T0)− 1, and
≃ 6.47. (21)
The first and the second term of left hand side in Eq. (20) are negative and positive
contributions to the speed of sound squared, respectively. We find that a model
which leads to small ∂mν/∂z is favored. A model with a small power-law scalar
potential; V (φ) = Λ4(φ/φ0)k, k ≪ 1, and a constant dominant neutrino mass;
mν = C + f(φ), f(φ) ≪ C, leads to small ∂mν/∂z.
c Actually, some models have
been presented.9
3. Effect of supersymmetry breaking
Let us consider effect of supersymmetry breaking in the dark sector. We assume
a superfield X , which breaks supersymmetry, in the hidden sector, and the chiral
superfield A in the dark sector is assumed to interact with the hidden sector only
through the gravity. This framework is shown graphically in Fig. 1. Once supersym-
metry is broken at TeV scale, its effect is transmitted to the dark sector through
the following operators:
X† +X
A†A, (22)
where Mpℓ is the Planck mass. Then, the scale of soft terms FX(TeV
2)/Mpℓ ∼
O(10−3-10−2eV) is expected. In the “acceleressence” scenario, this scale is identi-
fied with the dark energy scale.14 We consider only one superfield which breaks
cA model with the masses of the left-handed neutrinos given by the see-saw mechanism is unstable
even if it has a small power-law scalar potential.13
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6 Ryo Takahashi
Fig. 1. The illustration of interactions among three sectors. The dark sector couples to the left-
handed neutrino through a new scalar force in the MaVaNs scenario. The dark sector is also
assumed to be related with the hidden sector only through the gravity.
supersymmetry for simplicity. If one extends the hidden sector, one can consider a
different mediation mechanism between the standard model and the hidden sector
from one between the dark and the hidden sector.
In this framework, taking supersymmetry breaking effect into account, the scalar
potential is given by
V (φ) =
|φ|4 −
(φ3 + h.c.) +M2A|φ|
2 +m2D|φ|
2 −m2|φ|2 + V0, (23)
where κ and m are supersymmetry breaking parameters, and V0 is a constant
determined by the condition that the cosmological constant is vanishing at the true
minimum of the acceleron potential.
We consider two types of the neutrino mass matrix in this scalar potential. They
are the cases of the vanishing and the finite mixing between the left-handed and a
sterile neutrino.
3.1. Case of the Vanishing Mixing
When the mixing between the left-handed and a sterile neutrino is vanishing,mD =
0 in the neutrino mass matrix (5). Then we have the masses of the left-handed and
a sterile neutrino as
mνL = c, (24)
mψ = MA + λ < φ > . (25)
In this case, we find that only the mass of a sterile neutrino is variable on cosmo-
logical time scale due to the second term of the right hand side in Eq. (25).
Let us adjust parameters which satisfy Eqs. (8) and (13)∼(16). In Eq. (8), the
scalar potential Eq. (23) is used. Putting typical values for four parameters by hand
as follows:
λ = 1, mD = 0, m
= 2× 10−2 eV, m0ψ = 10
−2 eV, (26)
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 7
we have
< φ >0≃ −1.31× 10−5 eV, c = 2× 10−2 eV, MA ≃ 10
−2 eV,
m ≃ 10−2 eV, κ ≃ 4.34× 10−3 eV. (27)
We need fine-tuning between MA and m in order to satisfy the constraint on the
present accerelon mass of Eq. (16).
We show evolutions of the mass of a sterile neutrino and the equation of state
parameter in Figs. 2, 3 and 4. The behavior of the mass of a neutrino near the
present epoch is shown in Fig. 3. We find that the mass of a sterile neutrino have
varied slowly in this epoch. This means that the first term of the left hand side in
Eq. (20), which is a negative contribution to the speed of sound squared, is tiny.
We can also check the positive speed of sound squared in a numerical calculation.
Therefore, the neutrino-acceleron fluid is hydrodynamically stable and acts as the
dark energy.
3.2. Case of the Finite Mixing
Next, we consider the case of the finite mixing between the left-handed and a sterile
neutrino (mD 6= 0). In this case, the left-handed and a sterile neutrino mass are
given by
mνL =
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
, (28)
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
. (29)
We find that both masses of the left-handed and a sterile neutrino are variable on
cosmological time scale due to the term of the acceleron dependence.
Taking typical values for four parameters as
λ = 1, mD = 10
−3 eV, m0νL = 2× 10
−2 eV, m0ψ = 10
−2 eV, (30)
we have
< φ >0≃ −1.31× 10−5 eV, c ≃ 1.99× 10−2 eV, MA ≃ 1.01× 10
−2 eV,
m ≃ 1.02× 10−2 eV, κ ≃ 4.34× 10−3 eV. (31)
where we required that the mixing between the active and a sterile neutrino is
tiny. In our model, the small present value of the acceleron is needed to satisfy the
constraints on the scalar potential in Eqs. (8) and (13).
Values of parameters in (31) are almost same as the case of the vanishing mixing
(27). However, the mass of the left-handed neutrino is variable unlike the vanishing
mixing case. The time evolution of the left-handed neutrino mass is shown in Fig. 5.
The mixing does not affect the evolution of a sterile neutrino mass and the equation
of state parameter, which are shown in Figs. 6, 7. Since the variation in the mass of
the left-handed neutrino is not vanishing but extremely small, the model can also
avoid the instability of speed of sound.
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8 Ryo Takahashi
Finally, we comment on the smallness of the evolution of the neutrino mass at
the present epoch. In our model, the mass of the left-handed and a sterile neutrino
include the constant part. A variable part is a function of the acceleron. In the
present epoch, the constant part dominates the neutrino mass because the present
value of the acceleron should be small. This smallness of the value of the acceleron
is required from the cosmological observation and the stationary condition in Eqs.
(8) and (13).
4. Summary
We presented a supersymmetric MaVaNs model including effects of the supersym-
metry breaking mediated by the gravity. Evolutions of the neutrino mass and the
equation of state parameter have been calculated in the model. Our model has
a chiral superfield in the dark sector, whose scalar component causes the present
cosmic acceleration, and the right-handed neutrino superfield. In our framework,
supersymmetry is broken in the hidden sector at TeV scale and the effect is assumed
to be transmitted to the dark sector only through the gravity. Then, the scale of
soft parameters of O(10−3-10−2)(eV) is expected.
We considered two types of model. One is the case of the vanishing mixing
between the left-handed and a sterile neutrino. Another one is the finite mixing
case. In the case of the vanishing mixing, only the mass of a sterile neutrino had
varied on cosmological time scale. In the epoch of 0 ≤ z ≤ 20, the sterile neutrino
mass had varied slowly. This means that the speed of sound squared in the neutrino
acceleron fluid is positive, and thus this fluid can act as the dark energy. In the
finite mixing case, the mass of the left-handed neutrino had also varied. However,
the variation is extremely small and the effect of the mixing does not almost affect
the evolution of the sterile neutrino mass and the equation of state parameter.
Therefore, this model can also avoid the instability.
References
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(2005); P. McDonald, U. Seljak, R. Cen, P. Bode, and J. P. Ostriker, Mon. Not. Roy.
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2. R. Fardon, A. E. Nelson and N. Weiner, JCAP. 0410, 005 (2004).
3. R. D. Peccei, Phys. Rev. D 71, 023527 (2005).
4. M. Kawasaki, H. Murayama and T. Yanagida, Mod. Phys. Lett. A 7, 563 (1992); G. J.
http://arxiv.org/abs/astro-ph/0603449
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Stephenson, T. Goldman and B. H. J. McKellar, Int. J. Mod. Phys. A 13, 2765 (1998),
Mod. Phys. Lett. A 12, 2391 (1997).
5. P. Q. Hung, hep-ph/0010126.
6. P. Gu, X-L. Wang and X-Min. Zhang, Phys. Rev. D 68, 087301 (2003).
7. D. B. Kaplan, A. E. Nelson, N. Weiner, Phy. Rev. Lett. 93, 091801 (2004); V. Barger,
D. Marfatia and K. Whisnant, Phys. Rev. D73, 013005 (2006); P-H. Gu, X-J. Bi, B.
Feng, B-L. Young and X. Zhang, hep-ph/0512076; X-J. Bi, P. Gu, X-L. Wang and
X-Min. Zhang, Phys. Rev. D 69, 113007 (2004); P. Gu and X-J. Bi, Phys. Rev. D 70,
063511 (2004); P. Q. Hung and H. Päs, Mod. Phys. Lett. A 20, 1209 (2005); V. Barger,
P. Huber and D. Marfatia, Phys. Rev. Lett. 95, 211802 (2005); M. Cirelli and M. C.
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H. Li and X-Min. Zhang, Phys. Rev. D72 123523, (2005); A. W. Brookfield, C. van de
Bruck, D. F. Mota and D. Tocchini-Valentini, Phys. Rev. Lett. 96, 061301 (2006); R.
Horvat, JCAP 0601, 015 (2006); R. Barbieri, L. J. Hall, S. J. Oliver and A. Strumia,
Phys. Lett. B 625, 189 (2005); N. Weiner and K. Zurek, Phys. Rev. D 74, 023517
(2006); H. Li, B. Feng, J-Q. Xia and X-Min. Zhang, Phys. Rev. D 73, 103503 (2006);
A. W. Brookfield, C. van de Bruck, D. F. Mota and D. Tocchini-Valentini, Phys. Rev.
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R. Takahashi and M. Tanimoto, Phys. Rev. D 74, 055002 (2006); A. Ringwald and L.
Schrempp, JCAP. 0610, 012 (2006); R. Takahashi and M. Tanimoto, hep-ph/0610347;
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http://arxiv.org/abs/hep-ph/0010126
http://arxiv.org/abs/hep-ph/0512076
http://arxiv.org/abs/hep-ph/0511027
http://arxiv.org/abs/hep-ph/0610347
http://arxiv.org/abs/astro-ph/0611353
http://arxiv.org/abs/hep-ph/0611284
http://arxiv.org/abs/astro-ph/0611912
http://arxiv.org/abs/astro-ph/0606034
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10 Ryo Takahashi
Fig. 2. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 2000)
Fig. 3. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 20)
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 11
Fig. 4. Evolution of w (0 ≤ z ≤ 50)
Fig. 5. Evolution of the mass of the left-handed neutrino (0 ≤ z ≤ 2000)
October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka
12 Ryo Takahashi
Fig. 6. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 2000)
Fig. 7. Evolution of w (0 ≤ z ≤ 50)
	Introduction
	MaVaNs Model in Supersymmetric Theory
	Effect of supersymmetry breaking
	Case of the Vanishing Mixing
	Case of the Finite Mixing
	Summary

  We consider a Mass Varying Neutrinos (MaVaNs) model in supersymmetric theory.
The model includes effects of supersymmetry breaking transmitted by the
gravitational interaction from the hidden sector, in which supersymmetry was
broken, to the dark energy sector. Then evolutions of the neutrino mass and the
equation of state parameter of the dark energy are presented in the model. It
is remarked that only the mass of a sterile neutrino is variable in the case of
the vanishing mixing between the left-handed and a sterile neutrino on
cosmological time scale. The finite mixing makes the mass of the left-handed
neutrino variable.

Introduction
Cosmological observations have provided the strong evidence that the Universe
is flat and its energy density is dominated by the dark energy component whose
negative pressure causes the cosmic expansion to accelerate.1 In order to clarify the
origin of the dark energy, one has tried to understand the connection of the dark
energy with particle physics.
In the Mass Varying Neutrinos (MaVaNs) scenario proposed by Fardon, Nelson
and Weiner, relic neutrinos could form a negative pressure fluid and cause the
present cosmic acceleration.2 In the model, an unknown scalar field, which is called
“acceleron”, is introduced and neutrinos are assumed to interact through a new
scalar force. The acceleron sits at the instantaneous minimum of its potential, and
the cosmic expansion only modulates this minimum through changes in the number
density of neutrinos. Therefore, the neutrino mass is given by the acceleron, in
other words, it depends on the number density of neutrinos and changes with the
expansion of the Universe. The equation of state parameterw and the energy density
of the dark energy also evolve with the neutrino mass. Those evolutions depend
∗talked at the International Workshop on Neutrino Masses and Mixings, University of Shizuoka,
Shizuoka, Japan, December 17-19, 2006
http://arxiv.org/abs/0704.0186v1
October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka
2 Ryo Takahashi
on the form of a scalar potential and the relation between the acceleron and the
neutrino mass strongly. Some examples of the potential have been considered.3
The idea of the variable neutrino mass was considered at first in a model of
neutrino dark matter and was discussed for neutrino clouds.4 Interacting dark en-
ergy scalar with neutrinos was considered in the model of a sterile neutrino.5 The
coupling to the left-handed neutrino and its implication on the neutrino mass limit
from baryogenesis was discussed.6 In the context of the MaVaNs scenario, there
have been a lot of works. 7,8,9,10,11,12
In this talk, we present a MaVaNs model including the supersymmetry breaking
effect mediated by the gravity. Then we show evolutions of the neutrino mass and
the equation of state parameter in the model.
2. MaVaNs Model in Supersymmetric Theory
We discuss the Mass Varying Neutrinos scenario in supersymmetric theory and
present a model.
We assume a chiral superfield A in dark sector. A is assumed to be a singlet
under the gauge group of the standard model. It is difficult to construct a viable
MaVaNs model without fine-tunings in some parameters when one assumes one
chiral superfield in dark sector, which couples to only the left-handed lepton doublet
superfield. 8 Therefore, we assume that the superfield A couples to both the left-
handed lepton doublet superfield L and the right-handed neutrino superfield R. For
simplicity, we consider the MaVaNs scenario in one generation of neutrinos.a
In such framework, we suppose the following superpotential,
AA+mDLA+MDLR+
RR, (1)
where λ is a coupling constant of O(1) and MA, MD, MR and mD are mass pa-
rameters.b The scalar and the spinor component of A are represented by φ and ψ,
respectively. The scalar component corresponds to the acceleron which cause the
present cosmic acceleration. The spinor component is a sterile neutrino. The third
term of the right-hand side in Eq. (1) is derived from the Yukawa coupling such as
yLAH with y < H >= mD, where H is the Higgs doublet.
In the MaVaNs scenario, the dark energy is assumed to be composed of the
neutrinos and the scalar potential for the acceleron. Therefore, the energy density
of the dark energy is given as
ρDE = ρν + V (φ). (2)
Since only the acceleron potential contributes to the dark energy, we assume the
vanishing vacuum expectation values of sleptons, and thus we find the following
aThree generations model of this scenario has presented in non supersymmetric theory.9
bOther supersymmetric model so called “hybrid model” has been proposed.10
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 3
effective scalar potential,
V (φ) =
|φ|4 +M2A|φ|
2 +m2D|φ|
2. (3)
We can write down a lagrangian density from Eq. (1),
L = λφψψ +MAψψ +mDνLψ +MDνLνR +MRνRνR + h.c.. (4)
It is noticed that the lepton number conservation in the dark sector is violated
because this lagrangian includes both MAψψ and mDνLψ. After integrating out
the right-handed neutrino, the effective neutrino mass matrix is given by
mD MA + λφ
, (5)
in the basis of (νL, ψ), where c ≡ −M
D/MR and we assume λφ≪ MD ≪MR. The
first term of the (1, 1) element of this matrix corresponds to the usual term given
by the seesaw mechanism in the absence of the acceleron. We obtain masses of the
left-handed and a sterile neutrino as follows,
mνL =
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
, (6)
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
. (7)
It is remarked that only the mass of a sterile neutrino is variable in the case of
the vanishing mixing (mD = 0) between the left-handed and a sterile neutrino on
cosmological time scale. The finite mixing (mD 6= 0) makes the mass of the left-
handed neutrino variable. We will consider these two cases of mD = 0 and mD 6= 0
later.
In the MaVaNs scenario, there are two constraints on the scalar potential. The
first one comes from cosmological observations. It is that the magnitude of the
present dark energy density is about 0.74ρc. ρc is the critical density. Thus, the
first constraint turns to
V (φ0) = 0.74ρc − ρ
ν , (8)
where “0” means the present value.
The second one is the stationary condition:
∂V (φ)
= 0. (9)
In this scenario, the neutrino mass is represented by a function of the acceleron;
mν = f(φ). Since the energy density of the neutrino varies on cosmological times
scale, the vacuum expectation value of the acceleron also varies. This property
makes the neutrino mass variable. If ∂mν/∂φ 6= 0, Eq. (9) is equivalent to
∂V (φ(mν))
= 0. (10)
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4 Ryo Takahashi
Eq. (10) is rewritten by using the cosmic temperature T :
∂V (φ)
= −T 3
∂F (ξ)
, (11)
where ξ ≡ mν/T , ρν = T
4F (ξ) and
F (ξ) ≡
y2 + ξ2
ey + 1
. (12)
We can get the time evolution of the neutrino mass from Eq. (11). Since the sta-
tionary condition should be always satisfied in the evolution of the Universe, this
one at the present epoch is the second constraint on the scalar potential:
∂V (φ)
mν=m0ν
= −T 3
∂F (ξ)
mν=m0ν ,T=T0
. (13)
In addition to two constraints for the potential, we also have two relations between
the vacuum expectation value of the acceleron and the neutrino masses at the
present epoch:
m0νL =
c+MA + λ < φ >
[c− (MA + λ < φ >0)]2 + 4m
, (14)
m0ψ =
c+MA + λ < φ >
[c− (MA + λ < φ >0)]2 + 4m
. (15)
Next, let us consider the dynamics of the acceleron field. In order that the
acceleron does not vary significantly on distance of inter-neutrino spacing, the ac-
celeron mass at the present epoch must be less than O(10−4eV) 2. Here and below,
we fix the present acceleron mass as
m0φ = 10
−4 eV. (16)
Once we adjust parameters which satisfy five equations (8) and (13)∼(16), we can
have evolutions of the neutrino masses by using the Eq. (11).
The dark energy is characterized by the evolution of the equation of state pa-
rameter w. The equation of state is derived from the energy conservation law and
the stationary condition Eq. (11):
w + 1 =
[4− h(ξ)]ρν
, (17)
where
h(ξ) ≡
∂F (ξ)
F (ξ)
. (18)
It seems that w in this scenario depend on the neutrino mass and the cosmic
temperature. This means that w varies with the evolution of the Universe unlike
the cosmological constant.
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 5
In the last of this section, we comment on the hydrodynamical stability of the
dark energy in the MaVaNs scenario. The speed of sound squared in the neutrino-
acceleron fluid is given by
c2s =
ẇρDE + wρ̇DE
, (19)
where pDE is the pressure of the dark energy. Recently, it was argued that when
neutrinos are non-relativistic, this speed of sound squared becomes negative in this
scenario.11 The emergence of an imaginary speed of sound means that the MaVaNs
scenario with non-relativistic neutrinos is unstable, and thus the fluid in this sce-
nario cannot acts as the dark energy. However, finite temperature effects provide a
positive contribution to the speed of sound squared and avoid this instability. 12
Then, a model should satisfy the following condition,
5aT 2
25aT 20 (z + 1)
> 0, (20)
where z is the redshift parameter, z ≡ (T/T0)− 1, and
≃ 6.47. (21)
The first and the second term of left hand side in Eq. (20) are negative and positive
contributions to the speed of sound squared, respectively. We find that a model
which leads to small ∂mν/∂z is favored. A model with a small power-law scalar
potential; V (φ) = Λ4(φ/φ0)k, k ≪ 1, and a constant dominant neutrino mass;
mν = C + f(φ), f(φ) ≪ C, leads to small ∂mν/∂z.
c Actually, some models have
been presented.9
3. Effect of supersymmetry breaking
Let us consider effect of supersymmetry breaking in the dark sector. We assume
a superfield X , which breaks supersymmetry, in the hidden sector, and the chiral
superfield A in the dark sector is assumed to interact with the hidden sector only
through the gravity. This framework is shown graphically in Fig. 1. Once supersym-
metry is broken at TeV scale, its effect is transmitted to the dark sector through
the following operators:
X† +X
A†A, (22)
where Mpℓ is the Planck mass. Then, the scale of soft terms FX(TeV
2)/Mpℓ ∼
O(10−3-10−2eV) is expected. In the “acceleressence” scenario, this scale is identi-
fied with the dark energy scale.14 We consider only one superfield which breaks
cA model with the masses of the left-handed neutrinos given by the see-saw mechanism is unstable
even if it has a small power-law scalar potential.13
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6 Ryo Takahashi
Fig. 1. The illustration of interactions among three sectors. The dark sector couples to the left-
handed neutrino through a new scalar force in the MaVaNs scenario. The dark sector is also
assumed to be related with the hidden sector only through the gravity.
supersymmetry for simplicity. If one extends the hidden sector, one can consider a
different mediation mechanism between the standard model and the hidden sector
from one between the dark and the hidden sector.
In this framework, taking supersymmetry breaking effect into account, the scalar
potential is given by
V (φ) =
|φ|4 −
(φ3 + h.c.) +M2A|φ|
2 +m2D|φ|
2 −m2|φ|2 + V0, (23)
where κ and m are supersymmetry breaking parameters, and V0 is a constant
determined by the condition that the cosmological constant is vanishing at the true
minimum of the acceleron potential.
We consider two types of the neutrino mass matrix in this scalar potential. They
are the cases of the vanishing and the finite mixing between the left-handed and a
sterile neutrino.
3.1. Case of the Vanishing Mixing
When the mixing between the left-handed and a sterile neutrino is vanishing,mD =
0 in the neutrino mass matrix (5). Then we have the masses of the left-handed and
a sterile neutrino as
mνL = c, (24)
mψ = MA + λ < φ > . (25)
In this case, we find that only the mass of a sterile neutrino is variable on cosmo-
logical time scale due to the second term of the right hand side in Eq. (25).
Let us adjust parameters which satisfy Eqs. (8) and (13)∼(16). In Eq. (8), the
scalar potential Eq. (23) is used. Putting typical values for four parameters by hand
as follows:
λ = 1, mD = 0, m
= 2× 10−2 eV, m0ψ = 10
−2 eV, (26)
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 7
we have
< φ >0≃ −1.31× 10−5 eV, c = 2× 10−2 eV, MA ≃ 10
−2 eV,
m ≃ 10−2 eV, κ ≃ 4.34× 10−3 eV. (27)
We need fine-tuning between MA and m in order to satisfy the constraint on the
present accerelon mass of Eq. (16).
We show evolutions of the mass of a sterile neutrino and the equation of state
parameter in Figs. 2, 3 and 4. The behavior of the mass of a neutrino near the
present epoch is shown in Fig. 3. We find that the mass of a sterile neutrino have
varied slowly in this epoch. This means that the first term of the left hand side in
Eq. (20), which is a negative contribution to the speed of sound squared, is tiny.
We can also check the positive speed of sound squared in a numerical calculation.
Therefore, the neutrino-acceleron fluid is hydrodynamically stable and acts as the
dark energy.
3.2. Case of the Finite Mixing
Next, we consider the case of the finite mixing between the left-handed and a sterile
neutrino (mD 6= 0). In this case, the left-handed and a sterile neutrino mass are
given by
mνL =
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
, (28)
c+MA + λ < φ >
[c− (MA + λ < φ >)]2 + 4m
. (29)
We find that both masses of the left-handed and a sterile neutrino are variable on
cosmological time scale due to the term of the acceleron dependence.
Taking typical values for four parameters as
λ = 1, mD = 10
−3 eV, m0νL = 2× 10
−2 eV, m0ψ = 10
−2 eV, (30)
we have
< φ >0≃ −1.31× 10−5 eV, c ≃ 1.99× 10−2 eV, MA ≃ 1.01× 10
−2 eV,
m ≃ 1.02× 10−2 eV, κ ≃ 4.34× 10−3 eV. (31)
where we required that the mixing between the active and a sterile neutrino is
tiny. In our model, the small present value of the acceleron is needed to satisfy the
constraints on the scalar potential in Eqs. (8) and (13).
Values of parameters in (31) are almost same as the case of the vanishing mixing
(27). However, the mass of the left-handed neutrino is variable unlike the vanishing
mixing case. The time evolution of the left-handed neutrino mass is shown in Fig. 5.
The mixing does not affect the evolution of a sterile neutrino mass and the equation
of state parameter, which are shown in Figs. 6, 7. Since the variation in the mass of
the left-handed neutrino is not vanishing but extremely small, the model can also
avoid the instability of speed of sound.
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8 Ryo Takahashi
Finally, we comment on the smallness of the evolution of the neutrino mass at
the present epoch. In our model, the mass of the left-handed and a sterile neutrino
include the constant part. A variable part is a function of the acceleron. In the
present epoch, the constant part dominates the neutrino mass because the present
value of the acceleron should be small. This smallness of the value of the acceleron
is required from the cosmological observation and the stationary condition in Eqs.
(8) and (13).
4. Summary
We presented a supersymmetric MaVaNs model including effects of the supersym-
metry breaking mediated by the gravity. Evolutions of the neutrino mass and the
equation of state parameter have been calculated in the model. Our model has
a chiral superfield in the dark sector, whose scalar component causes the present
cosmic acceleration, and the right-handed neutrino superfield. In our framework,
supersymmetry is broken in the hidden sector at TeV scale and the effect is assumed
to be transmitted to the dark sector only through the gravity. Then, the scale of
soft parameters of O(10−3-10−2)(eV) is expected.
We considered two types of model. One is the case of the vanishing mixing
between the left-handed and a sterile neutrino. Another one is the finite mixing
case. In the case of the vanishing mixing, only the mass of a sterile neutrino had
varied on cosmological time scale. In the epoch of 0 ≤ z ≤ 20, the sterile neutrino
mass had varied slowly. This means that the speed of sound squared in the neutrino
acceleron fluid is positive, and thus this fluid can act as the dark energy. In the
finite mixing case, the mass of the left-handed neutrino had also varied. However,
the variation is extremely small and the effect of the mixing does not almost affect
the evolution of the sterile neutrino mass and the equation of state parameter.
Therefore, this model can also avoid the instability.
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Fig. 2. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 2000)
Fig. 3. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 20)
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Dark energy and neutrino model in SUSY– Remarks on active and sterile neutrinos mixing – 11
Fig. 4. Evolution of w (0 ≤ z ≤ 50)
Fig. 5. Evolution of the mass of the left-handed neutrino (0 ≤ z ≤ 2000)
October 31, 2018 9:21 WSPC/INSTRUCTION FILE shizuoka
12 Ryo Takahashi
Fig. 6. Evolution of the mass of a sterile neutrino (0 ≤ z ≤ 2000)
Fig. 7. Evolution of w (0 ≤ z ≤ 50)
	Introduction
	MaVaNs Model in Supersymmetric Theory
	Effect of supersymmetry breaking
	Case of the Vanishing Mixing
	Case of the Finite Mixing
	Summary
