Dark energy interacting with neutrinos and dark matter: a
  phenomenological theory
Dark energy interacting with neutrinos and dark matter: a
phenomenological theory
G. M. Kremer∗
Departamento de F́ısica, Universidade Federal do Paraná
Caixa Postal 19044, 81531-990 Curitiba, Brazil
October 29, 2018
Abstract
A model for a flat homogeneous and isotropic Universe composed of dark energy, dark
matter, neutrinos, radiation and baryons is analyzed. The fields of dark matter and neutrinos
are supposed to interact with the dark energy. The dark energy is considered to obey either
the van der Waals or the Chaplygin equations of state. The ratio between the pressure and the
energy density of the neutrinos varies with the red-shift simulating massive and non-relativistic
neutrinos at small red-shifts and non-massive relativistic neutrinos at high red-shifts. The
model can reproduce the expected red-shift behaviors of the deceleration parameter and of the
density parameters of each constituent.
The recent astronomical measurements of type-IA supernovae [1, 2, 3, 4] and the analysis of
the power spectrum of the CMBR [5, 6, 7, 8, 9] provided strong evidence for a present accelerated
expansion of the Universe [3, 10, 11, 12, 13, 14]; the nature of the responsible entity, called dark
energy, still remains unknown. Furthermore, the measurements of the rotation curves of spiral
galaxies [15] as well as other astronomical experiments suggest that the luminous matter represents
only a small amount of the massive particles of the Universe, and that the more significant amount
is related to dark matter. That offered a new setting for cosmological models with dark energy
and dark matter and in these contexts many interesting phenomenological models appear in the
literature analyzing the interaction of neutrinos [16, 17, 18] and dark matter [19, 20, 21, 22, 23, 24]
with dark energy. With respect to dark energy some exotic equations of state were proposed in the
literature and among others we quote the van der Waals [25, 26, 27, 28, 29] and the Chaplygin [30,
31, 32, 33] equations of state.
In the present work a very simple cosmological model – for a homogeneous, isotropic and flat
Universe composed by dark matter, dark energy, baryons, radiation and neutrinos – is investigated
where the dark energy is modeled either by the van der Waals or the Chaplygin equations of state
and interact with neutrinos and dark matter. Units have been chosen so that 8πG/3 = c = 1,
whereas the metric tensor has signature (+,−,−,−).
Let a homogeneous, isotropic and spatially flat Universe be characterized by the Robertson
Walker metric ds2 = dt2 − a(t)2δijdxidxj , where a(t) denotes the cosmic scale factor. The sources
of the gravitational field are related to a mixture of five constituents described by the fields of dark
energy, dark matter, baryons, neutrinos and radiation. The components of the energy-momentum
tensor of the sources is written as
(T µν) = diag(ρ,−p,−p,−p), (1)
where ρ and p denote the total energy density and pressure of the sources, respectively. In terms
of the energy densities and pressures of the constituents it follows
ρ = ρb + ρdm + ρr + ρν + ρde, p = pb + pdm + pr + pν + pde. (2)
∗kremer@fisica.ufpr.br
http://arxiv.org/abs/0704.0371v1
Above the indexes (b, dm, r, ν, de) refer to the baryons, dark matter, radiation, neutrinos and dark
energy, respectively.
The conservation law of the energy-momentum tensor T µν ;ν = 0 leads to the evolution equation
for the total energy density of the sources, namely
ρ̇+ 3
(ρ+ p) = 0, (3)
where the dot refers to a differentiation with respect to time.
The baryons and radiation are considered as non-interacting fields so that the evolution equa-
tions for their energy densities read
ρ̇b + 3
ρb = 0, ρ̇r + 4
ρr = 0, (4)
once the baryons represent a pressureless fluid, i.e., pb = 0, and the radiation pressure is given in
terms of its energy density by pr = ρr/3.
According to a model proposed by Wetterich [19] the evolution equation for the energy density
of a pressureless (pdm = 0) dark matter field which interacts with a scalar field φ is given by
ρ̇dm + 3
ρdm = βρdmφ̇. (5)
Here the scalar field plays the role of the dark energy and β is a constant which couples the fields
of dark matter and dark energy.
For interacting neutrinos with dark energy it is supposed that the evolution equation of the
energy density is given by (see [17, 18])
ρ̇ν + 3
(ρν + pν) = α(ρν − 3pν)φ̇. (6)
The coefficient α is connected with the mass of the neutrinos and for more details one is referred
to [17, 18] and to the references therein. Here α will be consider a phenomenological coefficient
that couples the dark energy field with the neutrinos. Note that if pν = ρν/3, there is no coupling
between the fields of dark energy and neutrinos. Moreover, it is also important to note that the
neutrinos in the past must behave as massless particles where the relationship between the pressure
and the energy density is pν = ρν/3. Due to the coupling of the neutrinos with the scalar field
they become massive and non-relativistic. For these reasons a barotropic equation of state for the
neutrinos is proposed where the ratio between the pressure and the energy density wν = pν/ρν ,
given in terms of the red-shift z, reads
K3(1/z)
K2(1/z)
K3(1/z)
K2(1/z)
. (7)
Above K2(1/z) and K3(1/z) are modified Bessel functions of second kind. For small values of z,
wν tends to the non-relativistic limit equal to 2/3, whereas for large values of z, wν tends to the
relativistic limit equal to 1/3. It is noteworthy that for red-shifts z ≈ 10 this ratio reaches the value
wν ≈ 1/3 and the coupling between the neutrinos and the dark energy is negligible. The expression
given in (7) is motivated by the equation of the specific heat of a relativistic gas (see e.g. [34]).
The evolution equation for the energy density of the dark energy field is obtained from equations
(2) through (6), yielding
ρ̇de + 3
(ρde + pde) = −αφ̇(ρν − 3pν)− βρdmφ̇. (8)
The energy density and pressure of the dark energy are connected with the scalar field by φ̇ =√
ρde + pde. Since the purpose of this work is to develop a phenomenological theory, it is assumed
0 2 4 6 8 10
vw Ωb
Figure 1: Density parameters as functions of red-shift: van der Waals fluid (solid lines) and Chap-
lygin fluid (dashed lines).
that the dark energy field behaves either as a van der Waals or a Chaplygin fluid with an equation
of state given by [28, 29, 30, 31, 32, 33]
pvw =
8wvwρvw
3− ρvw
− 3ρ2vw, pch = −
, (9)
where wvw and A are positive free parameters in the van der Waals and Chaplygin equations of
state, respectively.
For the determination of the time evolution of the energy densities one has to close the system
of differential equations by introducing the Friedmann equation
= ρ. (10)
From now on the red-shift will be used as a variable instead of time thanks to the following
relationships
ρ(1 + z)
. (11)
Equations (4) can be easily integrated leading to the well-known dependence of the energy
densities of the baryons and radiation with the red-shift
ρr(z) = ρr(0)(1 + z)
4, ρb(z) = ρb(0)(1 + z)
3, (12)
whereas equations (5), (6) and (8) become a system of coupled differential equations for the energy
densities ρdm, ρν and ρde, namely,
(1 + z)ρ′dm − 3ρdm
(ρde + pde)/ρ
= −βρdm, (13)
(1 + z)ρ′ν − 3(ρν + pν)
(ρde + pde)/ρ
= −α(ρν − 3pν), (14)
(1 + z)ρ′de − 3(ρde + pde)
(ρde + pde)/ρ
= βρdm + α(ρν − 3pν). (15)
In the above equations the prime refers to a differentiation with respect to the red-shift.
In order to solve the coupled system of differential equations (13) – (15) one has to specify initial
values for the energy densities at z = 0. The following initial values for the density parameters
Ωi(z) = ρi(z)/ρ(z) taken from the literature (see [35] for a review) were chosen: Ωde(0) = 0.72,
0 500 1000 1500 2000 2500 3000
Figure 2: Density parameters as functions of red-shift for a van der Waals fluid as dark matter.
Ωdm(0) = 0.229916, Ωb(0) = 5 × 10−2, Ωr(0) = 5 × 10−5, Ων(0) = 3.4 × 10−5. Moreover, one has
to specify values for the coupling parameters α and β and for the parameters wvw and A which
appear in the van der Waals and Chaplygin equations of state (9). One way to fix the two last
parameters is through the use of the value of the deceleration parameter q = 1/2+ 3p/2ρ at z = 0.
Indeed, by considering q(0) = −0.55 it follows wvw = 0.33851 and A = 0.50403. For the coupling
parameters two sets of values were chosen, namely, (a) α = 5 × 10−5 and β = −5 × 10−5 for the
van der Waals equation of state and (b) α = 10−1 and β = −10−2 for the Chaplygin equation of
state. Its is also important to note that by increasing the value of the coupling parameter α (and/
or β) the transfer of energy between the dark energy and neutrinos (and/or dark matter) becomes
more efficient.
In Fig. 1 the density parameters are plotted as functions of the red-shift for values in the range
0 ≤ z ≤ 10. The straight lines refer to the case where the van der Waals equation of state is used
to describe the dark energy field whereas the dashed lines correspond to the Chaplygin equation
of state. The two density parameters that represent the dark energy field are denoted by Ωvw and
Ωch. One can infer from this figure that the dark energy density parameter tends to zero for high
red-shifts when the van der Waals equation of state is used, whereas it tends to a constant value
for the Chaplygin equation of state. While for high red-shifts the van der Waals equation of state
simulates a cosmological constant with pvw = −ρvw, the pressure of the Chaplygin fluid vanishes
indicating that it becomes another component of the dark matter field (see also the behavior of
the pressures indicated in Fig. 4). It is also important to note that the density parameters of the
baryons and of the dark matter increase more with the red-shift for the van der Waals equation of
state, since there is an accentuated decrease in the density parameter of the dark energy for this
case. Note that the density parameters of the radiation and neutrinos are very small in this range
of the red-shift and are not represented in this figure.
The behavior of the density parameters for the cases of the van der Waals and Chaplygin
equations of state are shown in Figs. 2 and 3, respectively, for red-shifts in the range 0 ≤ z ≤ 3000.
One can conclude from these figures, as expected, that the density parameters of the neutrinos
and radiation increase with the red-shift whereas those of the baryons and dark matter decrease.
Furthermore, the equality between the “matter” and “radiation” fields occurs when z ≈ 3000 for
the case where the dark matter field is modeled as a van der Waals fluid and z ≈ 4200 for the case of
a Chaplygin fluid. This can be easily understood, since in the latter case the dark energy becomes
dark matter for high red-shifts contributing for the density parameter of the “matter” field.
In Fig. 4 are plotted the deceleration parameter and the ratio between the pressure and the
energy density for both cases, the large frame corresponding to the van de Waals fluid whereas the
small frame to the Chaplygin fluid. For both cases the deceleration parameter at z = 0 is equal to
q(0) = −0.55, since this value was fixed in order to find the parameters wvw and A in the equations
of state (9). The transition from the decelerated to the accelerated phase of the Universe occurs at
zT = 0.73 and zT = 0.53 for the van der Waals and Chaplygin equations of state, respectively. It
0 500 1000 1500 2000 2500 3000
 + Ων
Figure 3: Density parameters as functions of red-shift for a Chaplygin fluid as dark matter.
-0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.751.5 2
-0.5 0 0.5 1 1.51.5 2
Figure 4: Deceleration parameter and ratio between the pressure and the energy density as functions
of red-shift: large frame (van der Waals), small frame (Chaplygin).
0 2 4 6 8 10
with interactions
without interactions
Figure 5: Density parameters as functions of red-shift for a Chaplygin fluid with and without
interactions.
is interesting to note that while the Chaplygin equation of state simulates a cosmological constant
with pch = −ρch for negative red-shifts which implies an accelerated phase of the Universe in the
future, the van der Waals equation of state leads to a positive pressure and brings the Universe to
another decelerated phase in the future. It is noteworthy to call attention that for positive values
of the red-shift, the solution of the coupled differential equations (13) through (15) predicts that
the van der Waals fluid behaves close to a cosmological constant with pvw ≈ −ρvw. This behavior
does not lead to a new transition from a decelerated to an accelerated phase in the very early
Universe, since the energy density of the radiation field increases so that the radiation pressure
becomes larger than that of the van der Waals fluid. For high red-shifts the Universe first becomes
dominated by the baryon and dark matter fields and for higher red-shifts by the radiation field.
This model does not attempt to model the inflationary period, where the inflaton field dominates
a short rapid evolution of the Universe.
As final remarks we call attention to the fact that one expects that the coupling between dark
energy, dark matter and neutrinos should be weak so that the parameters α and β are restricted to
small values. The difference between the parameters adopted for the van der Waals and Chaplygin
equations of state is due to stability conditions of the non-linear coupled system of differential
equations (13) – (15), the van der Waals equation of state being more unstable for large values
of these parameters than the Chaplygin equation of state. In Fig. 5 we have plotted the density
parameters as functions of the red-shift for the case where a Chaplygin equation of state is used as
dark energy. One can infer from this figure that the decay of the dark energy density parameter
and the increase of the dark matter density parameter with the red-shift are more pronounced when
there exists a coupling between the fields. The density parameter of the baryons remains unchanged
since the baryons are uncoupled.
As final comment it is important to note that even without couplings between the fields of
dark energy, dark matter and neutrinos, this phenomenological model – with the equations of state
of van der Waals and Chaplyging as dark energy – can describe satisfactorily the evolution of a
Universe whose constituents are dark energy, dark matter, baryons, neutrinos and radiation.
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