#### The cosmological nature of the dark Universe

Eur. Phys. J. C
The cosmological nature of the dark Universe
Christian Henke 0
0 Department of Mathematics, University of Technology at Clausthal , Erzstrasse 1, 38678 Clausthal-Zellerfeld , Germany
This paper deals with the cancellation mechanism, which identifies the energy density of space-time expansion in an empty universe with the zero-point energy density and avoids the scale discrepancy with the observed energy density (cosmological constant problem). Using an intrinsic degree of freedom which describes the coupling of a variable cosmological term Λ with non-relativistic matter and radiation, the following consequences are demonstrated by coupling only a small contribution of Λ with nonrelativistic matter. First, the standard model of cosmology with a positive cosmological constant can be generalised such that the missing mass problem of dark matter is solved by an overall negative variable cosmological term. Second, the model under consideration is compatible with constraints from the standard model of particle physics. Third, an equation of state parameter of dark matter is derived which agrees with observations of rotational curves of galaxies. Moreover, the creation and annihilation process of dark matter is presented.
1 Introduction
In the recent paper [
1
], the author has demonstrated that the
variable cosmological term Λ(a) = Λ0 + Λ1a−r , r = 4 − ,
= 9.151 × 10−122 solves the fine-tuning of the
cosmological constant problem (see [
2–6
]) and generates the missing
mass of dark matter which constitutes 26% of the
matterenergy density. The remaining matter-energy constraints of
the universe such as 69% dark energy and 5% ordinary
matter as well as the initial singularity are also satisfied by the
theory.
Unfortunately, the age of the universe without
contributions from radiation which is given by
d x
0.05/x + 0.69x 2 + 0.26/xr−2 = 10.6×109years,
disagrees with the accepted value of 13.80 × 109 years. Here,
H0 = 67.74 km/(s Mpc) denotes the present-day Hubble
constant. The age of 13.80 billion years can be obtained in the
case r ≈ 3. In order to retain the accepted age of the universe,
provided that the fine-tuning property of the cosmological
constant problem and the generation of the missing mass of
dark matter are preserved, an intrinsic degree of freedom for
a parameterised coupling of the cosmological term with
nonrelativistic matter and radiation is used. It turns out that in our
universe, only a very small fraction of the cosmological term
couples with non-relativistic matter. Interestingly, this is the
same result as derived from the observational constraints for
the baryon and antibaryon pair annihilation [
7
]. As another
consequence, an equation of state parameter of dark matter
is found which agrees with observations of rotational curves
of galaxies [
8
].
The remainder of the paper is organised as follows. In
section 2 we review the Friedmann equations where the
cosmological term Λ is a function of the scale factor a. Section 3
is devoted to the dark sector and states the basic ideas behind
the derivation of the cancellation mechanism of the
cosmological constant problem from [
1
]. Then, it is demonstrated
that the attractive force of dark matter is a consequence of a
positive energy density of the cosmological term. Finally, we
show that our solution of the cosmological constant problem
explains the already mentioned cosmological observations
and how dark matter is created/destroyed by the interaction
of non-relativistic matter/radiation.
2 A time-dependent cosmological term
Let us start by recalling the Robertson–Walker space-time
line element (see [
9
] for notational conventions)
ds2 = −c2dt 2 + a(t )2
aa˙ 22 − 13 Λ + ak2 = κ3c2 ρ ,
3 aa¨ − Λ = − κ2 (ρc2 + 3 p).
which reduces Einstein’s field equations to the Friedmann
equations for the scaling factor a(t )
Here, dΩ2 = dθ 2 + sin2 θ dφ2 and k denotes the curvature
parameter of unit length−2. Moreover, the variable
cosmological term is defined by
Λ(a) = Λ0 + Λ1a−r ,
Λ0, r > 0.
It is convenient to include the Λ-term in the
energy-momentum tensor of the right-hand side. Therefore, we define
an effective density and pressure field which includes
nonrelativistic matter and radiation
ρeff = ρm + ρr + ρΛ, peff = pm + pr + pΛ,
where ρΛ = Λ(a)/κc2 and pΛ = −Λ(a)/κ denote the
background fields.
The zero covariant divergence of the energy-momentum
tensor gives
d
da
(ρeffa3) + 3 pce2ff a2 = 0, peff = 0,
and leads with pm = 0 and pr = c2ρr /3 to
a13 dda (ρm a3) + a14 dda (ρr a4) = − κ1c2 dda Λ(a),
which is satisfied by the parametrised coupling
a13 dda (ρm a3) = −α κ1c2 dda
1 d
a4 da
1 d
(ρr a4) = −(1 − α) κc2 da
Λ(a),
Integrating both equations lead to
Λ(a), α ∈ R.
αr Λ1 1
ρm = ρm,0a−3 + κc2 3 − r
(1 − α)r Λ1
ρr = ρr,0a−4 + κc2
a−r , r = 3,
1
4 − r
a−r , r = 4.
(1)
(2)
(4)
(5)
(6)
(7)
(8)
(9)
3 The dark sector
In this section we review the basic ideas of the cancellation
mechanism between the quantum zero-point energy and the
energy density of the cosmological constant in an empty
universe (cf. [
1
]). First, using a metric which is independent
of the scale factor, the second Friedmann equation can be
identified with an Euler-Lagrange equation. Therefore, the
related Lagrangian and the first Friedmann equation leads
to the cancellation mechanism between the energy density
of space-time expansion and the energy density of the
cosmological term. Identifying the energy density of space-time
expansion with the quantum zero-point energy, we get
Λ1 r
ρzpe = κc2 r − 4
a−r ,
r − 4
Λ1 = 2πrl2p , r = 4,
and l p denotes the Planck length. As the consequence, there
is no scale discrepancy between the total energy densities
and the fine-tuning problem of the cosmological constant
problem is solved.
Now, we investigate the parameter range for r such that
an initial singularity is guaranteed. Using the settings
Hence, depending on the exponent r, the presence of the
variable cosmological term can completely change the dynamical
behavior of ρm and ρr .
In order to analyse the gravitational nature of the
Λdepending components of ρeff, the acceleration behaviour
is discussed by the term
where
4 a Λ(α)α3 dα
κc2a4
,
(10)
(11)
we have to discuss the inequality
By considering the equation
κc2
− 2
a−r , 3 = r = 4,
(13)
from Eq. (2). As usual, the Λ0-depending term acts repulsive
and is labeled with dark energy. Since
it follows from the considered parameter range (12) that the
Λ1-depending term has an attractive effect and can be
identified as dark matter. Notice that the attractive nature of dark
matter is a direct consequence of a positive Λ1-depending
density.
From (8) and (9) we can conclude that dark matter doesn’t
interact with ordinary matter and radiation (ρm,0 and ρr,0
are independent of Λ1). The interaction takes place between
the three components of dark matter (self-interacting dark
matter) which can be classified by their equation of state
parameter w = −1, w = 0 and w = 1/3.
4 Cosmological constraints
It remains to consider some observational constraints for the
present-day composition of the universe. Using the usual
settings ρcrit = 3H02/κc4,
kc2
Ωk = − a02 H02 ,
Ωr = ρρcr,r0it a0−4,
Ωm = ρρmcr,i0t a0−3,
Λ0c2
ΩΛ0 = 3H02 ,
where a0 = a(t0) denotes the present-day scale factor, the
dark matter contribution is taken into account by
Ωdm =
ρΛ1 + ρmΛ + ρrΛ .
ρcrit
Multiply (1) by c2a2/H02a02, we get
d x 2
dτ
= Ωk +
Ωm
x
Ωr 2 Ωdm ,
+ x 2 + ΩΛ0 x + xr−2
(14)
(15)
(16)
where x = a/a0 and τ = H0t. This leads to the present-day
constraint
1 = Ωk + Ωm + Ωr + ΩΛ0 + Ωdm.
(17)
To relate the last equation with observations (cf. [
10
]), we
consider
(Ωk , Ωm , Ωr , ΩΛ0 , Ωdm ) = (0, 0.05, 5 × 10−5, 0.69, 0.26)
and H0 = 67.74 s kMmpc . The setting ΩΛ0 = 0.69 leads to a
cancellation mechanism (10) which automatically cancel 121
decimal places without fine-tuning. Moreover, using Ωdm =
0.26 and
=
2πl2p H02Ωdm
c2
it follows
3 r (3 − r ) = r (4 − α) − 12,
where r0 and r1 are independent of . By substituting the
decomposition (19) in (18), a hierarchy of equations for the
terms multiplied by the same power of follows and yields
the solution to the unknowns r0 and r1. Namely, we obtain
As already mentioned in the introduction a small coupling
between the non-relativistic matter and the dynamical part of
the cosmological term is necessary for an universe which is
13.80 billion years old.
Namely, using (20) with 0 < α < 10−5 we get by
numerical integration
1
H0 0
1
d x
Ωxm + Ωx2r + ΩΛ0 x 2 + xr−2
Ωdm
= 13.80 × 109years.
Therefore, the remaining parameter of the cosmological term
is Λ1 = −2.03 × 1068. Hence, the overall cosmological term
Λ(a) is negative! (cf. [
11
] and Fig. 1).
Now, some consequences of the value 0 < α < 10−5 are
investigated. The low production (ρmΛ > 0) with respect to
α of non-relativistic (dark) matter particles by the coupling
process between non-relativistic matter and the
cosmological term is in agreement to the results of [
7
]: the limitation
of baryon/antibaryon pair annihilations, the isotropy of the
10−5
10−4
α
−0.5
microwave background and Eq. (5) lead to a scale factor
which is nearly indistinguishable to the scale factor in the
matter dominated epoch. Hence, the term dda (ρm a3) is only
a small perturbation of Eq. (5), which is equivalent to the
case 0 < α 1.
Interestingly, the amount of annihilation of dark matter
corresponds to the amount of the zero-point energy (cf. [
1
])
ρrΛ = −(1 − α)ρzpe ≈ −ρzpe.
Finally, the total equation of state parameter of dark matter
is given by
13 ρrΛ − ρΛ1
wdm = c2 ρΛ1 + ρmΛ + ρrΛ
α
= 4 − α + O( ) ≤ 2.5 × 10−6,
r
= 3 − 1
which agrees with the equation of state parameter wdm ∈
(1.3 × 10−8, 1.5 × 10−7) from observed rotational curves of
galaxies (see [
8
]) (Fig. 2).
In contrast to the standard ΛCDM model with 6
parameters, the equation of state parameter wdm extends this model
to the ΛwDM model from [
11
]. The above constraint for wdm
increases the Hubble constant by a maximum of 0.007% and
decreases Ωdm by a maximum of 0.02% (cf. [11, Figure 7])
which is within the specified accuracies of the parameters.
Moreover, the derived equation of state parameter wdm meets
the constraint −0.000896 < wdm < 0.00238 from cosmic
microwave background observations [
11
].
The structure formation of the universe for the ΛwDM
model was investigated in [
12
]. The main result is that the
clustering scale of the large-scale structure is independent of
the equation of state parameter.
In this paper, the variable cosmological term Λ(a) = Λ0 +
Λ1a−r , r > 0 has been applied and it has been confirmed
that the total energy density of an empty Friedmann universe
is related to the cosmological term such that the fine-tuning
problem was avoided by setting Λ1 = r − 4/2πrl2p, r = 4.
As a consequence, the dynamical part of the cosmological
term generates the attractive force of dark matter.
Moreover, it has been demonstrated that the accepted age
of our universe requires that only a small fraction of the
cosmological term couples with non-relativistic matter. More
precisely, the cosmological term creates/destroys dark
matter with the interaction of non-relativistic matter/radiation.
Similar to black holes, the decrease of the density by
radiation is caused by the vacuum energy density. This could be
a further argument that dark matter consists of black holes
and that ρdm represents the spatially averaged densities of
black holes. On the other hand, the results of this paper
cannot exclude that dark matter is made of particles. The range
3 < r < 3.0000075 is compatible with constraints from
baryon/antibaryon pair annihilation [
7
].
Furthermore, the parameter range of r produces a non-zero
equation of state parameter of dark matter wdm ≤ 2.5 × 10−6
which agrees with observational data of rotational curves
of galaxies [
8
], generates the missing mass of dark matter,
describes the present-day composition of the universe and
realises a negative cosmological term. Living in an universe
with a negative cosmological term will completely change
our understanding of cosmology. This could have
important consequences for holographic correspondence-theories
which are mainly formulated on space-times with a negative
cosmological constant (cf. [
13
]).
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Funded by SCOAP3.
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