Interacting diffusive unified dark energy and dark matter from scalar fields
Eur. Phys. J. C
Interacting diffusive unified dark energy and dark matter from scalar fields
David Benisty 0
E. I. Guendelman 0
0 Department of Physics, Ben Gurion University of the Negev , 84105 Beersheba , Israel
Here we generalize ideas of unified dark matterdark energy in the context of two measure theories and of dynamical space time theories. In two measure theories one uses metric independent volume elements and this allows one to construct unified dark matterdark energy, where the cosmological constant appears as an integration constant associated with the equation of motion of the measure fields. The dynamical spacetime theories generalize the two measure theories by introducing a vector field whose equation of motion guarantees the conservation of a certain Energy Momentum tensor, which may be related, but in general is not the same as the gravitational Energy Momentum tensor. We propose two formulations of this idea: (I) by demanding that this vector field be the gradient of a scalar, (II) by considering the dynamical space field appearing in another part of the action. Then the dynamical space time theory becomes a theory of Diffusive Unified dark energy and dark matter. These generalizations produce nonconserved energy momentum tensors instead of conserved energy momentum tensors which leads at the end to a formulation of interacting DEDM dust models in the form of a diffusive type interacting Unified dark energy and dark matter scenario. We solved analytically the theories for perturbative solution and asymptotic solution, and we show that the CDM is a fixed point of these theories at large times. Also a preliminary argument as regards the good behavior of the theory at the quantum level is proposed for both theories. The best explanation, and fitting with data for the accelerated expansion of our universe, is the CDM model, which tells us that our universe contains 68% of dark energy, and 27% of dark matter. This model present two big questions: The

cosmological constant problem [1–3] – why there is a large
disagreement between the vacuum expectation value of the
energy momentum tensor which comes from quantum field
theory and the observable value of dark energy density – and
the coincidence problem [4] – why the observable values of
dark energy and dark matter densities in the late universe are
of the same order of magnitude.
In order to solve this problem, many approaches emerged
[5–8]. One interesting suggestion was a diffusive exchange
of energy between dark energy and dark matter made by
Calogero [9,10], Haba et al. [11], and Szydlowski and
Stachowski [12], with some solution to cosmic problems. The
basic notion is that the diffusion equation (or more exactly,
the Fokker–Planck equation [13,14], which describes the
time evolution of the probability density function of the
velocity of a particle under the influence of random forces),
implies a nonconserved stress energy tensor T μν , which has
some current source j μ:
where σ is the diffusion coefficient of the fluid. This
generalization is Lorentz invariant and fit for curved spacetime.
The current j μ is a timelike covariantly conserved vector
field and its conservation tells us that the number of particles
in this fluid is constant.
However, in the gravitational equations, the Einstein
tensor is proportional to the conserved stress energy tensor
∇μT(μGν) = 0, which we labeled with “G” [15,16]. So
Calogero came up with what he called φCDM model, which
achieves a conserved total energy momentum tensor
appearing in the right hand side of Einstein’s equation. But for the
dark energy and dust stress tensors there is some source
current for those tensors (however, the sum is conserved):
As Calogero mentioned [9], the diffusion model introduced
in his paper lacks an action principle formulation. Therefore
we develop from a generalization of two measure theories
[17–28] a “diffusive energy theory” which can produce on
one hand a nonconserved stress energy tensor (T(μχν)), as in
(1), and on the other hand the conserved stress energy tensor
(T(μGν)) that we know from the right hand side of Einstein’s
equation. As we will see, this suggested theory is
asymptotically different from the φCDM model, and more close in this
limit to the standard CDM.
a dynamical time (here represented by the dynamical time
χ 0). Some cosmological solutions of (5) have been studied
in [30], in the context of spatially flat radiationlike solutions,
and considering gauge field equations in curved space time.
For a related approach where a set of dynamical
spacetime coordinates are introduced, not only in the measure of
integration, but also in the lagrangian, see [31].
2 Two measure theories and dynamical time theories
There have been theoretical approaches to gravity theories,
where a fundamental constraint is implemented; like in two
measure theories where one works, in addition to the regular
measure of integration in the action √−g, with yet another
measure, which is also a density and which is also a total
derivative. In this case, one can use for constructing this
measure four scalar fields ϕa , where a = 1, 2, 3, 4. Then we can
define the density = εαβγ δεabcd ∂αϕa ∂β ϕb∂γ ϕc∂δϕd , and
we can write an action that uses both of these densities:
S =
−gL2.
As a consequence of the variation with respect to the scalar
fields ϕa , assuming that L1 and L2 are independent of the
scalar fields ϕa , we obtain
where Aaα = εαβγ δεabcd ∂β ϕb∂γ ϕc∂δϕd . Since det[ Aaα] ∼
3 as one easily sees, for = 0, (4) implies that L1 = M =
Const. This result can expressed as covariant conservation
of a stress energy momentum of the form T μν
( ) = L1gμν ,
and using the second order formalism, where the covariant
derivative of gμν is zero, we obtain ∇μT(μν) = 0. This implies
∂αL1 = 0. This suggests generalizing the idea of the two
measure theory, by imposing the covariant conservation of a
more nontrivial kind of the energy momentum tensor, which
we denote T(μχν) [29]. Therefore, we consider an action of the
form
d4x √−g R
where χμ;ν = ∂ν χμ − μλν χλ. If we assume T(μχν) to be
independent of χμ and having μλν being defined as the Christoffel
connection coefficients, then the variation with respect to χμ
gives a covariant conservation: ∇μT(μχν) = 0.
Notice the fact that the energy density is the canonically
conjugated variable to χ 0, which is what we expect from
3 Diffusive energy theory from action principle
Let us consider a four dimensional case, where there is a
coupling between the scalar field χ and the stress energy
momentum tensor T(μχν):
where, μ; ν are covariant derivatives of the scalar field. When
λ
μν is being defined as the Christoffel connection
coefficients, the variation with respect to χ gives a covariant
conservation of a current f μ:
which is the source of the stress energy momentum tensor.
This corresponds to the “dynamical spacetime” theory (5),
where the dynamical spacetime 4vector χμ is replaced by a
gradient of a scalar field χ . In the “dynamical space theory”
we obtain four equations of motion, by the variation of χμ,
which correspond to covariant conservation of the energy
momentum tensor, ∇μT(μχν) = 0. By changing the four
vector to a gradient of a scalar ∂μχ at the end, what we do is
to change the conservation of the energy momentum
tensor to the asymptotic conservation of the energy momentum
tensor (7) which corresponds to a conservation of a current
∇ν f ν = 0. In an expanding universe, the current fμ gets
diluted, so then we recover asymptotically a covariant
conservation law for T μν again. Equation (7) has a close
cor(χ)
respondence with the one obtained in a “diffusion scenario”
for DE–DM exchange [9,10].
This stress energy tensor is substantially different from
the stress energy tensor we all know, which is defined as
8πc4G T(μGν) = Rμν − 21 gμν R. In this case, the stress energy
momentum tensor T(μχν) is not conserved (but there is some
conserved current f ν , which is the source to this stress energy
momentum tensor nonconservation), here there is some
conserved stress energy tensor T(μGν), which comes from variation
of the action according to the metric:
The lagrangian LM could be the modified term χ,μ;ν T(μχν),
but as we will see, it could be added to more action terms.
Using different expressions for T(μχν), which depend on other
variables, will give the connection between the scalar field χ
and those other variables.
Notice that for the theory the shift symmetry holds, and
χ → χ + Cχ ; T(μχν) → T μν
(χ) + gμν CT
will not change any equation of motion. When Cχ , CT are
some arbitrary constants. This means that if the matter is
coupled through its energy momentum tensor as in (9), the
process of redefinition of the energy momentum tensor will
not affect the equations of motion. Of course, a redefinition
of such a type of energy momentum tensor is exactly what
is done in the process of normal ordering in quantum field
theory, for example.
4 Diffusive energy theory without high derivatives
Another model that does not involve high derivatives is
obtained, by keeping χμ as a 4vector, which is not
gradient, but we introduce the vector field χμ in another part of
the action:
where A is another scalar field. Then from variation with
respect to χμ we obtain
as (7), where the source is
But in contrast to (7), where fμ appears as an integration
function, here f μ appears as a function of the dynamical
fields. From the variation with respect to A, we indeed see
that the current χ μ + ∂μ A is conserved, which means again
μν
with (7) that ∇μ∇ν T(χ) = 0, but it does not tell us that all of
the equations of motion are the same. Nevertheless,
asymptotically, for the late universe, the two theories coincide.
To start with, we discuss a toy model in one dimension
describing a system that allows the nonconservation of a
certain energy function, which increases or decreases linearly
with time, while there is another energy which is conserved.
It is of interest to compare with a mechanism that produces
nonconserved energy momentum tensors which leads to a
formulation of interacting DE–DM models; however, there
are crucial differences.
5 A mechanical system with a constant power and
diffusive properties
In order to see the applications of the ideas, we start with
a simple action of one dimensional particle in a potential
V (x ). We introduce a coupling between the total energy of
the particle 21 m x˙2 + V (x ) and the second derivative of some
dynamical variable B:
S =
In order to see the applications of the ideas, we start with
a simple action of a one dimensional particle in a potential
V (x ). We introduce a coupling between the total energy of
the particle 21 m x˙2 + V (x ) and the second derivative of some
dynamical variable B:
S =
The equation of motion according to the dynamical variable
B shows that the second derivative of the total energy is zero.
In other words, the total energy of the particle is linear in time:
m x˙2 + V (x ) = E (t ) = Pt + E0
where P is a constant power which is given to the particle
or taken from it, and E0 is the total energy of the particle at
time equal to zero.
From the equation of motion according to coordinate x
we get a close connection between the dynamical variable B
and the coordinate of the particle:
which with Eq. (15) gives
B¨˙ 2V (x ) P
B¨ = √2m(E (t ) − V (x )) − 2(E (t ) − V (x ))
To get a feeling of these kinds of theories, let us look at the
case of a harmonic oscillator V (x ) = 21 k x 2. First of all, we
see from Eq. (5) and the condition that the right hand side
be positive; since the left hand side obviously is positive, we
see that there is a boundary time τ = − EP0 , for which for
P > 0 we get t > τ , and for P < 0 there is a maximal time
t < τ . Let us consider the case that the power P is positive.
The equations of motion in that case will not oscillate, but
they will grow exponentially until the “Pt” term present in
Eq. (15) dominates, when x 2 ∝ t . This is very similar to
Brownian motion.
The momenta for this toy model are
πB = ∂∂LB˙ − ddt ∂∂LB¨ = − ddt E (t ),
∂L
B = ∂ B¨ = E (t ).
Using the Hamiltonian formalism (with second order
derivative [32,33]) we see that the hamiltonian of the system is
Since the action in not explicitly dependent on time, the
hamiltonian is conserved. So even if the total energy of the
particle is not conserved, we have the conserved
hamiltonian (21). This notion is equivalent to a nonconserved stress
energy tensor T(μχν), in addition to the conserved stress energy
T(μGν), which appears in the Einstein equation.
Notice that this hamiltonian is not necessarily bounded
from below. However, there are only mild instabilities in
the solutions. For example, for the case V (x ) = 0, we get
1
x˙ ∝ B˙ ∝ t 2 . In the case of a harmonic oscillator, where
V (x ) = 21 k x 2, there is an even milder behavior at large times:
1
x ∝ t 2 , which resembles a diffusive behavior, or Brownian
motion. This behavior is a mild kind of instability, since no
exponential growth appears, only power law growth. The
related model in cosmology, as we will see, because of the
coupling to an expanding spacetime shows dumped
perturbations, shows a trend towards a fixed point solution, where
it coincides with the standard CDM model. This is because
whatever potential instabilities the model may have in a flat
background, the expanding space (most notably the de Sitter
space) has the counter property of red shifting any
perturbation; this effect overcomes and cancels these rather soft
instabilities (power law instabilities that may exist for the
solution in flat space) as we will see in Sect. 7. The
exponential expansion is known to counter all kind of unstable
behaviors, for example, it goes against the gravitational
instability and a big enough cosmological constant can prevent
galaxy formation; our case is much simpler than that, but
the basic reason is the same. In this context it is important
to notice that in an expanding universe a noncovariant
conservation of an energy momentum tensor, which may imply
that some energy density is increasing in the locally inertial
frame, does not mean a corresponding increase of the energy
density in the comoving cosmological frame. For example, a
noncovariant conservation of the dust component of the
universe, in the examples we study, will produce a still
decreasing dust density, although there is a positive flow of energy
in the inertial frame. The result of this flow of energy in the
local inertial frame is going to be just that the dust energy
density decreases a bit slower that the conventional dust in
the comoving frame.
Independently of this, we will see how it is possible to
construct theories with positive Euclidean action that describe
diffusive DE–DM unification.
6 Gravity, “kessence” and diffusive behavior Our starting point is the following nonconventional gravityscalarfield action, which will produce a diffusive type of interacting DE–DM theory:
d4x √−g R +
with the following explanations for the different terms: R is
the Ricci scalar which appears in the Einstein–Hilbert action.
L(φ, X ) is the generalcoordinate invariant Lagrangian of a
single scalar field φ, which can be of an arbitrary generic
“kessence” type: some function of a scalar field φ and the
combination X = ∂μφ∂μφ [34–36]):
N =1
As we will see, this last action will produce a diffusive
interμν
action between DE–DM type theory. For the ansatz of T(χ)
we choose to use some tensor which is proportional to the
metric, with a proportionality function (φ, X ):
T(μχν) = gμν (φ, X ) ⇒ S(χ) =
From the variation of the scalar field χ we get = 0,
whose solution will be interpreted as a dynamically generated
cosmological constant with diffusive source.
We take the simple example for this generalized theory,
and for the functions L, we take the first order of the Taylor
expansion from (23), or L = = X ( A1 = 1, A2 = A3 =
· · · = 0). From the variation according to the scalar field we
get a conserved current j;μμ = 0:
For a cosmological solution we take into account only the
change as a function of time φ = φ (t ). From that we see
that the ‘0’ component of the current jα is nonzero. The last
variation we should take is according to the metric (using
the identities in Appendix A), which gives a conserved stress
energy tensor:
,σ ) + j μφ,ν − χ ,μ ,ν − χ ,ν ,μ.
For cosmological solutions the interpretation of the dark
energy is by a term proportional to the metric − + χ ,σ ,σ ,
and dark matter dust by the ‘00’ component of the tensor
j μφ,ν − χ ,μ ,ν − χ ,ν ,μ. Let us see the solution for the
Friedman–Robertson–Walker metric:
ds2 = −dt 2 + a2(t )
The basic combination becomes L = = X = ∂μφ∂μφ =
−φ˙ 2. Notice that there are high derivative equations, but all
such types of equations correspond to conservation laws. For
example, we see that the variation of the scalar field (24) will
give ddt (2φ˙ φ¨a3) = 0, which can be integrated to
which can be integrated again to give
The conserved current from Eq. (25) gives the relation
C3
2φ˙ ( χ + 1) = a3 ,
which can also be integrated to give
which provides the solution for the scalar field χ . From (26)
we get the terms for the DE–DM densities:
and the pressure of DE: pde = −ρde and DM: pdm = 0.
This leads to the Friedman equations with (32) and (33) as
source, and there are a few approximations that we want to
discuss. The first one is the asymptotic solution.
7 Asymptotic solution and stability of the theory
We can solve asymptotically and by the way show the basic
stability of the theory (which should eliminate any concerns
related to the formal unboundedness of the action). First we
solve for χ˙ ; see Eq. (31). We see that the leading term is
the fraction a13 a3dt . For an asymptotically de Sitter space,
where a(t ) ≈ a0 exp (H0t ), we see that there is a unique
asymptotic value:
This is in accordance with our expectations that the
expansion of the universe will stabilize the solutions, indeed (30)
is basically equivalent to the equation of a particle rolling
down a linear potential plus additional negligible terms as
a(t ) goes to infinity; the fixed point solution is of course that
of constant velocity, when friction × velocity = force = 1;
since friction = 3H, we obtain Eq. (34).
With this information we can check what the asymptotic
value of DE is, from (28), (29) and (32). We see that in this
limit, the nonconstant part of φ˙ 2 is canceled by 2χ˙ φ˙ φ¨, and
then asymptotically
with the same analysis for DM density we obtain
The Friedman equation provides a relation between C1 and
H0 (the asymptotic value of Hubble constant) which is H 2
0 =
8π G C1. For negative C2 we have decaying dark energy, the
3
last term of the contribution for dark energy density is
positive (and the opposite). This behavior, where C2 < 0, has
a chance of explaining the coincidence problem, because
unlike the standard CDM model, where the dark energy
is exactly constant, and the dark matter decreases like a−3,
in our case, dark energy can slowly decrease, instead of being
constant, and dark matter also decreases, but not as fast as
a−3.
As suggested, this behavior can be understood by the
observation that in an expanding universe a noncovariant
conservation of an energy momentum tensor, which may
imply that some energy density is increasing in the locally
inertial frame, does not mean a corresponding increase of the
energy density in the comoving cosmological frame, here in
particular the noncovariant conservation of the dust
component of the universe will produce a still decreasing dust
density, although, for C2 < 0, there will be a positive flow
of energy in the inertial frame to the dust component, but the
result of this flow of energy in the local inertial frame will
be just that the dust energy density will decrease a bit more
slowly than conventional dust (but still it decreases).
This is yet another example where potential instabilities
are softened or in this case eliminated by the expansion of the
universe. It is well known in the case of the Jeans gravitational
instability, which is much softer in the expanding universe
and also in other situations [37].
Another application of this mechanism could be to use
it to explain the particle production, “taking vacuum energy
and converting it into particles” as expected from the
inflation reheating epoch. Maybe this can be combined with a
mechanism that creates standard model particles.
As we see, the expansion of the universe stabilizes the
solutions, such that for large times all of them become
indistinguishable from CDM, which appears as an attractor fixed
point of our theory, showing a basic stability of the solutions
at large times. Choosing C1 as positive is necessary, because
of the demand that the terms with √C1 will not be imaginary.
But for the other constants of integration, there is only the
condition C3√C1 > 32CH20 , which gives a positive dust density
at large times.
8 C2 = 0 solution
Another special case is when C2 = 0. That means that the
dark energy of this universe is constant: we have φ˙ 2 = C1
and φ¨ = 0. The equation of motions for the dark energy
and dust (32) and (33) are independent on the scalar field
χ , and therefore the density of dust in that universe behaves
as C3√C1 . This solution says there is no interaction between
a3
dark energy and dark matter. This is precisely the solution of
the two measure theory [38–40], with the action
d4x √−g R +
which provides a unified picture of DE–DM. For more about
two measure theory and related models and solutions for DE–
DM, see the discussion in Appendix C. The FRWM for both
theories gives the solution
For this trivial case, C2 = 0, there is no diffusion effect
between dark matter and dark energy. The current f μ, which
is the source of the stress energy tensor T(μχν) (see (7)) is zero,
and both stress energy tensors are conserved. This is
equivalent to CDM. The exact solution of the case of constant
dark energy and dust, using (38) and (39) is [41]
a0(t ) =
where α = 23 √C1. From comparing to the CDM solution,
we can see how the observables values are related to the
constant of integration that come from the solution of the
theory:
= H ;
m =
C1√C3
where H is the Hubble constant for the late universe. For
exploring the nontrivial diffusive effect for C2 = 0, we use
perturbation theory.
9 Perturbative solution
The conclusion from this correspondence is that the diffusion
between dark energy and dark matter dust at the late universe
is very small, since that is the effect of the C2 term, and
therefore we can estimate the solution by perturbation theory.
So we obtain two dimensionless terms, which are dependent
on time and scale factor, and they tell us the “diffusion rate”:
where the integration is between two close times t0 and t .
For C2 = 0, both λ1 and λ2 are equal to zero, and there is
no dissipative effect, which as we saw, gives us the CDM
model. For any nonzero λ1,2 1, the stress energy tensor
T(μχν) is not conserved, and there is a little diffusion effect.
The use of defining these two dimensionless terms is
evident when C2 is small enough for using perturbation
theory. By using λ1 we can write the scalar field term as
φ˙ 2 = C1(1 + λ). The definition for λ2 is from the
assumption that the leading term in (33), whose scale √C1C3, is
much bigger than the other term χ˙ φ˙ φ¨ (with the χ˙ C2
component, using (28)). The total contributions for the densities, in
the context of perturbation theory at the first order, are
We can see from those terms that in the deviation from the
unperturbed standard solution, the behavior of dark energy
and dust are opposite – for increasing dark energy (for
example the components are C2 < 0; C1, C3, C4 > 0), the dark
matter amount (a3ρdm ) gets lower. Or in the case of
decreasing dark energy, the amounts of dark matter increases (and
C1, C2, C3, C4 > 0).
10 Equation of motion and solutions for diffusive
energy without higher derivatives
For the second class of theories we proposed in (10)–(12),
we can write the diffusive energy action, without high
derivatives:
1
S = 16π G
σ
+ 2
√−g R +
√−g
and, as before, the stress energy tensor T(μχν) = gμν . From
the variation with respect to the vector field χμ:
jα = 2(χ;λλ + 1)φ,α; j;α = 0,
α
as (7) and (25). Finally for the stress energy tensor, which
comes from variation with respect to the metric we obtain
1
,λ − 2σ
+ j μφ,ν − χ ,μ ,ν − χ ,ν ,μ
Both theories, (22) and (46), give rise to similar final
equations of motion, besides the variation according to the metric,
which asymptotically for large times behave in the same way.
The new terms − 21σ ,μ ,μ and σ1 ,μ ,μ are negligible at
the late universe, since they go as a16 . For the early universe
those terms may be very important, which we will study in
future publications.
The modified model of diffusion gives rise to the simpler
model when σ goes to infinity. Since, in this case, the extra
σ2 √−g(χμ + ∂μ A)2 term forces χμ = −∂μ A (because
(χμ + ∂μ A)2 = 0 and and we are also disregarding light like
solutions for (χμ + ∂μ A), which do not appear relevant to
cosmology), i.e. the χμ is a gradient of a scalar. Therefore
the theory of dynamical time (5) with a source (46) becomes
a diffusive action with high derivatives (6) and (22).
11 Some preliminary ideas on quantization and the
boundedness of the Euclidean action
Let us take the action (46); by integration by parts of
χμ;ν T(μχν), and throwing away total derivatives, we obtain the
action
√−g R +
We notice that there are no derivatives acting on the χμ field
at this action, and therefor χμ is a Lagrange multiplier. It
is legitimate to solve χμ from its equation of motions, and
insert the result back into the action. The equation of motion
according to the χμ variation is
√−g R +
Considering the functional integral quantization for this
theory will give a few integrations over the field variables. The
functional integral over the scalar A gives rise to a delta
function that enforces the covariant conservation of the current
∇μT(μχν) = f ν . The Euclidean functional integral will be
Z =
This partition function excludes the Hilbert–Einstein action
term, which has its own problems, which are not particularly
of interest to this paper. In the full theory we need to include
the integration over all the Euclidean geometries.
We can see that the integration measure is positive
definite, and the argument of the integrals in the exponents
are negative definite in a Euclidean signature spacetime
sign[+, +, +, +], following the Hawking approach [42].
The terms fμ f μ and φ,μφ,μ are positive definite, and by
choosing the proper sign of σ , the action is positive definite,
and the partition function is convergent. The original theory
we formulated in (22) is equivalent to (46) when σ goes to
minus infinity.
Therefore, this proof is valid for both theories. However,
the simple model (22) has to be regularized by first taking
A complete set of solutions of these differential equations
(in the form of Friedman equations) is very complicated, but
one phenomenological solution for this theory predicts a DE–
DM ratio similar to the observed one [11]. Both approaches
(which are described in this paper and in Calogero’s theory)
become very similar when the time derivative of the scalar
field is low χ˙ C2 1. In that case, the dark energy density
(32) becomes
The dark matter dust will reduce to the term (33):
and for those equations it implies diffusion between dark
energy and dark matter dust, like Calogero has found. In this
model they assumed that the dark energy and the dust are not
separately conserved.
We can see that our asymptotic solution does not fit with
Calogero’s model, for general C2. As opposed to Eq. (2), in
our asymptotic solution (35)–(36) the dark energy density
becomes constant, providing a behavior much closer to the
standard CDM model. The main reason for this
nonequivalence of those theories is the role of the χ˙ field, which has the
effect of making the exchange between dark matter and dark
energy less symmetric than in the φCMD model. In our case,
χ˙ makes the decay of DE much lower than in φCDM, and it
keeps the DM evolution still decreasing as CDM (a−3).
finite and negative σ , and then letting the σ go to minus
infinity. This is a preliminary approach, because in the quantum
theory, there are many issues concerning how one goes from
the Hamiltonian formulation to the path integral formulation,
etc. But we see that the quantum theory has a chance to be
well defined.
12 Diffusive dark energy and dust by Calogero
The solution for the Calogero suggestion we presented at the
beginning, see (1) and (2), leads to the following
interdependence between the densities of dark matter and dark energy
and the scale parameter:
13 Discussion, conclusions and prospects
In this paper we have generalized the TMT and the dynamical
space time theory, which imposes the covariant conservation
of an energy momentum tensor. We demand that the
dynamical spacetime 4vector χμ, which appears in the dynamical
spacetime theory, be a gradient ∂μχ . We do not obtain the
covariant conservation of the energy momentum tensor that is
introduced in the action. Instead we obtain current
conservation. The current is the divergence of this energy momentum
tensor. This current, which drives the nonconservation of
the energy momentum tensor, is dissipated in the case of an
expanding universe. So we get asymptotic conservation of
this energy momentum tensor. Because the fourdivergence
of the covariant divergence of both the dark matter and dark
energy is zero, we can make contact with the dissipative
models of [9,10]. This might give a deeper motivation for these
models and allow for the construction of new models.
This energy tensor is not the gravitational energy tensor
which appears in the right hand side of the Einstein
tensor, in the gravity equations, but the noncovariant
conservation of the energy momentum tensor that appears in the
action induces an energy momentum transfer between the
dark energy and dark matter components, of the gravitational
energy momentum tensor, in a way that resembles the ideas in
[11]. But one did not provide any action principle to support
their ideas. Although the mechanism is similar, our
formulation and theirs are not equivalent.
From the asymptotic solution we see that when C2 < 0,
unlike the standard CDM model, where the dark energy is
exactly constant, and the dark matter decreases like a−3, in
our case, dark energy can slowly decrease, instead of being
constant, and dark matter also decreases, but not as fast as
a−3. This special property is different in the φCMD model,
where the exchange between DE and DM is much stronger
in the asymptotic limit.
This behavior, where C2 < 0, has a chance of explaining
the coincidence problem, because unlike the standard CMD
model, where the dark energy is exactly constant, and the
dark matter decreases like a−3, in our case, dark energy can
slowly decrease, instead of being constant, and dark matter
also decreases, but not as fast as a−3. This behavior can be
understood by the observation that in an expanding universe
a noncovariant conservation of an energy momentum
tensor, which may imply that some energy density is increasing
in the locally inertial frame, does not mean a corresponding
increase of the energy density in the comoving
cosmological frame. Here in particular the noncovariant conservation
of the dust component of the universe will produce a still
decreasing dust density, although, for C2 < 0, there will be
a positive flow of energy in the inertial frame to the dust
component, but the result of this flow of energy in the local
inertial frame will be just that the dust energy density will
decrease a bit more slowly than the conventional dust (but it
still decreases).
We have seen that in perturbation theory, the behavior of
dark energy and dust are different – for increasing dark energy
(for example the components are C2 < 0; C1, C3, C4 > 0),
the dark matter amount (a3ρdm ) grows lower. Or in the case of
decreasing dark energy, the amounts of dark matter increase
(and all the constants of integration are positive).
For another suggestion for diffusive energy action, which
does not produce high derivative equations, we have kept the
χμ field as a 4vector (not the gradient of a scalar), but now
χμ appears in another term at the action, in addition to a scalar
field A. The equations of motion produce again a diffusive
energy equation, but with the additional contribution of two
terms that are negligible for the late universe.
A preliminary argument about the good behavior of the
theory at the quantum level is also proposed for both
theories. Some additional investigations concerning the quantum
theory could be developed by using the WDW equation, in
the minisuper space approximation.
Also in the future we will study not only the asymptotic
behavior, but the full numerical solution of the dark energy
and dark matter components, starting from the early universe,
for all the theories we suggested.
Acknowledgements We are very grateful to Professor Zbigniew Haba
for interesting comments and encouragement. This study was funded
by Foundational Questions Institute.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
14 Appendix A: Identities
1 ν σ μ σ
= 4 [gμτ (δαν δλσ + δλδα ) + gτ ν (δαμδλσ + δλ δα )
−2 ∂
−g
15 Appendix B
An equivalent expression for (7), when T(μχν) is formulated as
a perfect fluid in FRWM space, is
when C2 = 0, the stress energy tensor is conserved, and
there is no diffusive effect. For late times, where the scale
parameter goes to infinity, we see that the diffusive effect
vanishes.
16 Appendix C
TMTs also have many points of similarity with ‘Lagrange
multiplier gravity (LMG)’ [43,44]. The Lagrange multiplier
field in LMG enforces the condition that a certain function
be zero. In the TMT this is equivalent to the constraint that
requires some lagrangian to be constant. The two measure
models presented here are different from the LMG models
of [43,44], and they provide us with an arbitrary constant of
integration for the value of a given lagrangian, this constant of
integration, if nonzero, can generate spontaneous symmetry
breaking of scale invariance, which is present in the theory
for example. Recently a lot of interest has been attracted
by the socalled “mimetic” dark matter model proposed in
[45,46]. The latter employs a special covariant isolation of
the conformal degree of freedom in Einstein gravity, whose
dynamics mimics cold dark matter as a pressureless “dust”.
Important questions concerning the stability of “mimetic”
gravity are studied in Refs. [47,48] where also one
formulates a generalized tensor–vector–scalar “mimetic” gravity,
which avoids those problems. In [49] the idea is applied to
inflationary scenarios.
Most versions of the mimetic gravity, except for [47]
appear to be equivalent to a special kind of Lagrange
multiplier theory or TMT models that were known before, where
we have the simple constraint that the kinetic term of a scalar
field be constant. This of course gives identical results to the
very special TMT where the lagrangian that couples to the
new measure is the kinetic term of this scalar field.
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