1 Introduction
Recent astro-physical observations show that the Universe consists of 4 % ordinary matter,
23 % dark matter, 73 % dark energy .
The existence of dark matter was pointed out firstly by Jan Oort (1930) and Frizt
Zwicky (1933) based on the studies of the rotation curves of galaxies and galactic clusters.
The main candidates for dark matter are MACHOs (Massive Astrophysical Compact Halo
Ob jects) and WIMPs (Weakly Interacting Massive Particles).
Dark energy is an unknown form of energy with negative pressure. Nowadays it causes
the accelerating expansion of the Universe.
The existence of dark energy has been pointed out directly by two independent groups
based on Supernovae (SNe) type Ia observations [1] and also indirectly been suggested by
independent studies based on fluctuations of the 3K relic radiation [2], large scale structure
[3], age estimates of globular clusters, old high red-shift ob jects [4], as well as by the X-ray
data from galaxy clusters [5].
It is widely known that the main distinction between the pressure-less CDM and dark
energy is that the former agglomerates in small scales whereas dark energy is a smooth
component in the Universe. Such properties seems to be directly linked to the equation
of the state of both components. Recently, the idea of a unified description for CDM and
dark energy scenarios has received much attention. For example, Wetterich [6] suggested
that dark matter might consist of quintessence lumps, Kasuya [7] showed that spintessence
type scenarios are generally unstable to formation of Q balls which behave as pressure less
matter.
In this paper, based on the vector model of gravitational field we also introduce a united
description for dark matter and dark energy. This description does not require the anti-
gravitation properties for dark energy. It also points out the truncation of dark matter
halos and a lower limit for the average mass of dark matter and dark energy particles. The
coincident problem between the density of dark energy and one of matter is a clear fact.

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Journal of Science of Hanoi National University of Education
Natural sciences, Volume 52, Number 4, 2007, pp. 55- 63
AN APPROACH TO DARK MATTER AND DARK
ENERGY
Vo Van On
Department of Physics, University of Natural
Sciences, Vietnam National University - Ho Chi Minh City
Abstract. In this paper, we show an approach to the dark matter and dark energy.
This approach does not demand dark energy with the anti-gravitational property.
It also points out a lower limit of the average mass of the particles of cosmological
energy (ordinary matter, dark matter and dark energy particles) m 54eV.
Keyword(s): Dark matter, Dark energy, Cosmological energy
1 Introduction
Recent astro-physical observations show that the Universe consists of 4 % ordinary matter,
23 % dark matter, 73 % dark energy .
The existence of dark matter was pointed out firstly by Jan Oort (1930) and Frizt
Zwicky (1933) based on the studies of the rotation curves of galaxies and galactic clusters.
The main candidates for dark matter are MACHOs (Massive Astrophysical Compact Halo
Objects) and WIMPs (Weakly Interacting Massive Particles).
Dark energy is an unknown form of energy with negative pressure. Nowadays it causes
the accelerating expansion of the Universe.
The existence of dark energy has been pointed out directly by two independent groups
based on Supernovae (SNe) type Ia observations [1] and also indirectly been suggested by
independent studies based on fluctuations of the 3K relic radiation [2], large scale structure
[3], age estimates of globular clusters, old high red-shift objects [4], as well as by the X-ray
data from galaxy clusters [5].
It is widely known that the main distinction between the pressure-less CDM and dark
energy is that the former agglomerates in small scales whereas dark energy is a smooth
component in the Universe. Such properties seems to be directly linked to the equation
of the state of both components. Recently, the idea of a unified description for CDM and
dark energy scenarios has received much attention. For example, Wetterich [6] suggested
that dark matter might consist of quintessence lumps, Kasuya [7] showed that spintessence
type scenarios are generally unstable to formation of Q balls which behave as pressure less
matter.
In this paper, based on the vector model of gravitational field we also introduce a united
description for dark matter and dark energy. This description does not require the anti-
55
VO VAN ON
gravitation properties for dark energy. It also points out the truncation of dark matter
halos and a lower limit for the average mass of dark matter and dark energy particles. The
coincident problem between the density of dark energy and one of matter is a clear fact.
2 An approach to the Dark matter and dark energy prob-
lems
From result in the paper [8], we have known that the density of the cosmological energy
(ordinary matter, dark matter and dark energy) dilutes in the form ρ ∝ R−2 in the vector
model of the gravitational field. Thus, based on the dilution of the density, the cosmological
energy is like dust matter (ρm ∝ R−3) than the radiation energy (ρR ∝ R−4). Because of
this fact, we assume that the classical Bolzmann distribution can be used to describe the
distribution of the cosmological energy around galaxies and galactic clusters.
We consider a galaxy with the gravitational mass Mg, it is in a sea of the cosmological
energy. We investigate the gravitational field at a point A in this sea.
Call N0 is the density of cosmological energy particles at a very distance point from the
galaxy when ϕg = 0.
Call ϕg is the gravitational potential at A. When we assume that the classical Boltzmann
distribution can be applied to the cosmological energy, we have the density of particles at
A
N = N0 exp
(
−mgϕg
kT
)
(2.1)
Here mg is the gravitational mass of a particle. Thus density of gravitational mass at A is
ρg = Nmg = mgN0 exp
(
−mgϕg
kT
)
(2.2)
Here mg is the gravitational mass of particle, T is the absolute temperature of the particle
gas. At a remote distanct from the galaxy, the kinetic energy of particles is much larger
than its potential energy. We suppose that
mgϕg kT (2.3)
We have
exp
(
−mgϕg
kT
)
≈ 1− mgϕg
kT
(2.4)
From (2.2)
ρg = mgN0
(
1− mgϕg
kT
)
= mgN0 −
m2gN0ϕg
kT
(2.5)
We recall the 3rd equation of the system of non-relativistic equations [9]
∇−→E g = −ρg
εg
(2.6)
56
AN APPROACH TO DARK MATTER AND DARK ENERGY
Notice that
−→
Dg = εg
−→
E g [9], G = 1/4piεg and
−→
E g = −∇ϕg due to −→A g = 0, thus
∇2ϕg = ρg
εg
(2.7)
Substituting (2.5) into (2.7), we have
∇2ϕg = mgN0
εg
− m
2
gN0
εgkT
ϕg (2.8)
We rewrite (2.8) in the following form
∇2ϕg = a− b2ϕg (2.9)
Here
a ≡ mgN0
εg
(2.10)
b2 ≡ m
2
gN0
εgkT
(2.11)
Assuming that the cosmological energy particles distributes symmetrically around the
galaxy Mg, we can rewrite (2.9) in the following form:
1
r
d2
dr2
(rϕg) = −b2ϕg + a (2.12)
We seek the general root of equation (2.12). The general root of homogeneous equation
(2.12)
1
r
d2
dr2
(rϕg) = −b2ϕg (2.13)
is
ϕg0 =
1
r
(
C1e
−ibr + C2e
+ibr
)
(2.14)
A special root of inhomogeneous equation (2.12) is
ϕg1 =
1
r
(
e−ibr + e+ibr
)
+
a
b2
(2.15)
Thus, the general root of inhomogeneous equation (2.12) is
ϕg = ϕg0 + ϕg1 =
1
r
(
C1e
−ibr + C2e
+ibr
)
+
1
r
(
e−ibr + e+ibr
)
+
a
b2
(2.16)
Due to ϕg and a/b
2
are real, so C1 = C2 and they are real or C1 = −C2 and they are purely
imaginary. We require that when r → 0, we obtain Newtonian limit for the gravitational
potential, i.e.
ϕg → ϕg = −GMg
r
+ constant (2.17)
57
VO VAN ON
When r → 0, ϕg in (2.16) becomes
ϕg =
1
r
(C1 +C2) +
2
r
+
a
b2
(2.18)
In the case when C1 = C2, from (2.17) and (2.18), we obtain
C1 + 1 = C2 + 1 = −GMg
2
(2.19)
and a/b2 = constant. In the case when C1 = −C2, we don't obtain the classical Newtonian
limit. We shall discuss this case in a different paper. Thus the general root of (2.12) is
ϕg = −GMg
r
(
e−ibr + eibr
)
+
a
b2
= −GMg
r
cos br +
a
b2
(2.20)
We also obtain the gravitational field around galaxy Mg when the cosmological energy
presents itself as follows
Eg = −gradϕg = −GMgb
r
sin br − GMg
r2
cos br (2.21)
Finally, the gravitational force acts on a star mg1 which moves in this gravitational field
as follows
Fg = mg1Eg = −GMgbmg1
r
sin br − GMgmg1
r2
cos br (2.22)
We rewrite (2.22) in the following form
Fg = FV + FN (2.23)
With
FV ≡ −GMgbmg1
r
sin br (the vacuum force) (2.24)
FN ≡ −GMgmg1
r2
cos br (the Newtonian force) (2.25)
We consider now the correlation between FV and FN when r varies from 0 to ∞.
1. Region 1 (Newtonian region): when br 1 we have sin br ≈ 0, cos br ≈ 1 so
FV FN we see that
Fg = FN = −GMgmg1
r2
(2.26)
We return the Newtonian limit.
2. Region 2 (Region of dark matter): when br = pi2 ± ε we have sin br ≈ 1, cos br ≈ 0
so FV FN Therefore
Fg = FV = −GMgmg1b
r
(2.27)
If we investigate now the motion of a star in this region, we have
mi1
v2
r
=
GMgmg1b
r
(2.28)
58
AN APPROACH TO DARK MATTER AND DARK ENERGY
H¼nh 1: The dependence of star velocity on the distance r from the center of the galaxies.
Because mi1 ∼= mg1 therefore
v2 = GMgb (2.29)
i.e. v is independent of r. When taking into account of the term sinbr, we have
v2 = GMgb sin br or
v = (GMgb sin br)
1/2 with
pi
2
− ε < br < pi
2
+ ε (2.30)
We express the results on the figure (1).
3. Region 3 (Region of Dark Energy): when | cos br| > sin br or br > pi. FN changes sign
and becomes repulsive force and |FN | > FV or both FN and FV become repulsive
forces. A star or other galaxy into this region would be repulsed away and accelerated.
Perhaps acceleration of the Universe on large distances occurs when galaxies are in
this region.
4. Region 4 (large attractive region): when br is very large. FV and FN change signs
again and become attractive forces. We show these regions on the figure (2).
3 Discussions
Now we evaluate the value of b. We examine the rotation curve of the Milky Way . The
Sun which is in the region of dark matter has the rotation velocity around the Milky Way
59
VO VAN ON
H¼nh 2: Space regions around galaxy when r increases.
60
AN APPROACH TO DARK MATTER AND DARK ENERGY
about 200 km/s, the Milky Way's mass is about 1011Msun. From (2.29), we have
b =
v2
GMg
(3.1)
where v = 2× 105 m/s and Mg ∼ 1011 × 2× 1030 kg. We get
b ∼ 3× 10−21 m−1 (3.2)
We also find that radii of planets in the solar system are in the Newtonian region. Indeed,
when we choose rmax = rPluto = 5500.10
9 m we obtain
brPluto = 3× 10−21 × 5500 × 109 = 165 × 10−10 << 1 (3.3)
We have also known that most masses of galaxies are about 1011 Msun, their velocities in
rotation curves are about 150 km/s → 300 km/s, therefore value of b in (3.2) is true.
We evaluate the first dark matter region of galaxies. From Fig.(2) and formulas (2.30)
we find that the region of dark matter starts at the distance so that FV FN and ends
as brmax = pi or rmax ∼ 1.04 × 1021 m ∼ 33.8 kpc.
We evaluate the mean mass of the cosmological energy particles in this approach. It
is known that particles with integral spins obey the Bose - Einstein statistics , particles
with half- integral spins obey the Fermi-Dirac statistics. However, when the particle gas
satisfies the non- degeneracy condition [10]
n0h
3
(2pimkT )3/2
1 (3.4)
These two statistics lead to the classical Bolzmann statistics. Here n0 is the particle
density,h = 6.63×10−34 J.s is the Planck constant,k = 1.38×10−23 J.K−1 is the Bolzmann
constant,m is the particle mass,T is the absolute temperature of the particle gas. We can
substitute n0 = ρ/m with ρ is the mass density of particles. (3.4) becomes
ρh3
(2pikT )3/2m5/2
1 (3.5)
But we have ρ0 = mgN0 ≤ ρ (because ρ0 is the density at the points which are very
distance from the galaxy). Therefore
ρ0h
3
(2pikT )3/2m5/2
1 (3.6)
We also recall (2.11)
b2 =
m2gN0
εgkT
(3.7)
=
mgρ0
εgkT
(3.8)
61
VO VAN ON
Therefore
kT =
mgρ0
b2εg
=
mgρ04piG
b2
(3.9)
Substitute (3.9) into (3.6), we have:
h3b3
(8pi2G)3/2ρ
1/2
0 m
4
g
1 (3.10)
or
(hb)3/4
(8pi2G)3/8ρ
1/8
0
mg (3.11)
If we choose ρ0 ∼ 10−29 g/cm3 ∼ 10−26kg/m3, we have:
mg 1.3× 10−34 kg ∼ 54 eV (3.12)
We also remark that b can have different values in clusters of galaxies and superclusters
of galaxies by (2.11) and (2.29).
A remarkable point of this approach is that it does not demand Dark Energy with anti-
gravitational properties.
4 Conclusion
In this paper, we have introduced an approach to dark matter and dark energy. we have
obtained a modified expression for the gravitational force and have found a lower limit of
the average mass of the particles of cosmological energy.
Acknowledgments
We thank Prof. Nguyen Ngoc Giao for his helpful discussions guidance.
MËT CCH TIP CN VT CHT TÈI V NNG L×ÑNG TÈI
Vã V«n În
1
Bë Mæn Vªt Lþ Lþ Thuy¸t- Khoa Vªt lþ ,
¤i håc Khoa håc Tü nhi¶n, ¤i håc Quèc gia Hç Ch½ Minh
Tâm tt nëi dung
Trong b i b¡o n y, chóng tæi ÷a v o mët ti¸p cªn v· vªt ch§t tèi v n«ng l÷ñng
tèi. Ti¸p cªn n y khæng y¶u c¦u n«ng l÷ñng tèi vîi t½nh ch§t ph£n h§p d¨n. Chóng
tæi công d¨n ra mët cæng thùc c£i ti¸n cho lüc h§p d¨n cõa Newton, chi ra mët giîi
h¤n d÷îi ch khèi l÷ñng trinh b¼nh cõa c¡c h¤t n«ng l÷ñng vô trö m 54eV.
1
vovanon1963@gmail.com
62
AN APPROACH TO DARK MATTER AND DARK ENERGY
T i li»u
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(1999).
[2] P. de Bernardis et al.,Nature 404,(2000)955; L. Knox and L. Page, PLR 85,(2000)1366.
[3] R. G. Carlberg et al.,Astroph. J 462,(1996)32.
[4] J. Dunlop et al.,Nature 381,(1996)581;
[5] G. Steigman and J. E. Felten,Space Sci. Rev. 74,(1995)245;
WMAP Collaboration,astro-ph/0302207-09,13-15,22-25.
[6] C. Wetterich,Phys. Rev. D 65,(2002)123512.
[7] S. Kasuya,Phys. Lett. B 515,(2001)121.
[8] Vo Van On, Comm. in Phys. 17 (2007)13-17.
[9] Vo Van On,Journal of Science and Technology Development, Vietnam National Uni-
versity - Ho chi Minh City,9, No.4 (2006)5-11.
[10] B. Yavorsky and A. Detlaf, Handbook of Physics, Translated from the Russian by
Nicholas Weinstein, Mir Publishers, Moscow, 1975.
63