Vol.3, No.9, 743-749 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.39099
Copyright © 2011 SciRes. OPEN ACCESS
The basic blocks of the universe matter: Boltzmann
fundamental particle and energy quanta of dark matter
and dark energy
Murad Shibli1*, Sohail Anwar2
1College Requirement Unit, Abu Dhabi Polytechnic, Institute of Applied Technology, Abu Dhabi, United Arab Emirates;
*Corresponding Author: murad.alshibli@iat.ac.ae
2Pennsylvania State University, Altoona College, Altoona, USA.
Received 30 November 2010; revised 15 February 2011; accepted 10 March 2011.
ABSTRACT
Recent astronomical NASA observations indi-
cates that visible matter contributes only to
about 4% of the universe total energy density,
meanwhile, dark matter and dark energy con-
tributes to 26% and 70% of the universe total
energy, respectively, with an average density
close to 10–26 kg/m3. This paper proposes an
equation of state of dark energy and dark matter
as one unified entity. This equation is derived
based on the ideal gas equation, Boltzmann
constant, Einstein energy-mass principle and
based on the assumption that dark energy and
dark matter behave as a perfect fluid. This
analysis presents what could be the most fun-
damental particle and quanta of dark matter and
dark energy. Considering NASA’s Cosmic Mi-
crowave Background Explorer (CMB) which es-
timated that the sky has an average temperature
close to 2.7251 Kelvin, then the equivalent mass
and energy of the proposed fundamental particle
is determined. It is found that this candidate
particle has an equivalent mass of 4.2141 × 10–40
Kg which is equivalent to 3.7674 × 10–23 J. Sur-
prisingly, this value has the same order of
Boltzmann constant KB = 1.38 × 10–23 J/K. This
candidate particle could be the most funda-
mental and lightest particle in Nature and serves
as the basic block of matter (quarks and gluons).
Moreover, assuming a uniform space dark en-
ergy/dark matter density, then the critical tem-
perature at which the dark matter has a unity
entity per volume is determined as 34.983 × 1012
K. Analytically, it proposes that at this trillion
temperature scale, the dark matter particles uni-
fied into a new quark-hydron particle. Finally,
tentative experimental verification can be con-
ducted using the Relativistic Heavy Ion Collider
(RHIC).
Keywords: Dark Energy; Dark Matter; Equation of
State; Boltzmann Constant; Boltzamnn Particles;
Einstine’s Cosmological Constant
1. INTRODUCTION
Recent astronomical observations by the Supernova
Cosmology Project, the High-z Supernova Search Team
and cosmic microwave background (CMB) have pro-
vided strong evidence that our universe is not only ex-
panding, but also expanding at an accelerating rate [1-8].
It was only in 1998 when dark energy proposed for the
first time, after two groups of astronomers made a survey
of exploding stars, or supernovas Ia, in a number of dis-
tant galaxies [1,3]. These researchers found that the su-
pernovas were dimmer than they should have been, and
that meant they were farther away than they should have
been. The only way for that to happen, the astronomers
realized, was if the expansion of the universe had sped
up at some time in the past, as well as accounting for a
significant portion of a missing component in the uni-
verse [9,10]. The only explanation is that there is a kind
of force that has a strong negative pressure and acting
outward in opposition to gravitational force at large
scales which was proposed for the first time by Einstein
in his General Relativity and given the name the cosmo-
logical constant Lambda [2]. This force is given the
name Dark Energy, since it is transparent and cannot be
observed or detected directly. The fourth law of thermo-
dynamics is proposed by the author to account for the
dark energy [11].
These cosmological observations strongly suggest that
the universe is dominated by a smoothly homogenous
M. Shibli et al. / Natural Science 3 (2011) 743-749
Copyright © 2011 SciRes. OPEN ACCESS
744
distributed dark energy component [12-20]. The quantity
and composition of matter and energy in the universe is a
fundamental and important issue in cosmology and
physics. Based on the Lambda-Cold Dark Matter Model
(Lambda-CDM 2006), dark energy contributes about
70% of the critical density and has a negative pressure.
The cold dark matter contributes 25%, Hydrogen, He-
lium and stars contributes 5% and, finally the radiation
contributes 5 × 10–5. The measurements of the Wilkinson
Microwave Anisotropy Probe (WMAP) satellite indicate
the universe geometry is very close to flat [21].
Using the Doppler Shift phenomena, scientists can
learn much about the motions of galaxies. They know
that galaxies rotate because, when viewed edge-on, the
light from one side of the galaxy is blue shifted and the
light from the other side is red shifted. One side is mov-
ing toward the Earth, the other is moving away. They can
also determine the speed at which the galaxy is rotating
from how far the light is shifted. Knowing how fast the
galaxy is rotating, they can then figure out the mass of
the galaxy mathematically. According to Newton’s laws,
the rotation speed satisfies vGMr, where M is the
mass within radius r, and G is the Universal Gravitation
constant. But as scientists look closer at the speeds of
galactic rotation, they find something strange. The indi-
vidual stars in a galaxy should act like the planets in our
solar system—the farther away from the center, the
slower they should move. But the Doppler Shift reveals
that the stars in many galaxies do not slow down at far-
ther distances. On the contrary, the stars move at flat
speeds (see Figures 1 and 2) that should rip the galaxy
apart because there is not enough measured mass to sup-
ply the gravity needed to hold the galaxy together. These
high rotational speeds suggest that the galaxy contains
more mass than was calculated. Scientists theorize that,
if the galaxy was surrounded by a halo of unseen matter,
the galaxy could remain stable at such high rotational
speeds.
Much of the evidence for dark matter comes from the
study of the motions of galaxies. Many of these appear to
be fairly uniform, by the virial theorem the total kinetic
energy should be half the total gravitational binding en-
ergy of the galaxies. Experimentally, however, the total
kinetic energy is found to be much greater: in particular,
assuming the gravitational mass is due to only the visible
matter of the galaxy, stars far from the center of galaxies
have much higher velocities than predicted by the virial
theorem.
Galactic rotation curves, which illustrate the velocity
of rotation versus the distance from the galactic center,
cannot be explained by only the visible matter. Assuming
that the visible material makes up only a small part of the
cluster is the most straightforward way of accounting for
Figure 1. Rotation following Kepler’s 3rd law is shown above
as planet-like or differential rotation. Notice that the orbital
speeds falls off as you go to greater radii within the Galaxy.
This is called a Keplerian rotation curve.
Figure 2. The observed rotation curve for the our galaxy Milky
Way. To determine the rotation curve of the Galaxy, stars are
not used due to interstellar extinction. Instead, 21-cm maps of
neutral hydrogen are used. When this is done, one finds that the
rotation curve of the Galaxy stays flat out to large distances,
instead of falling off as in the figure above. This means that the
mass of the Galaxy increases with increasing distance from the
center.
this. Galaxies show signs of being composed largely of a
roughly spherically symmetric, centrally concentrated
halo of dark matter with the visible matter concentrated
in a disc at the center.
Accordingly, dark matter can be defined as the matter
of unknown composition that does not emit or reflect
M. Shibli et al. / Natural Science 3 (2011) 743-749
Copyright © 2011 SciRes. OPEN ACCESS
745
enough electromagnetic radiation to be observed directly,
but its presence can be inferred from gravitational effects
on visible matter like galaxies and stars [22-25]. Ac-
cording to present observations of structures larger than
galaxy-sized as well as Big Bang cosmology, dark matter
accounts for the vast majority of mass in the observable
universe (22%). The observed phenomena consistent
with dark matter observations include the rotational
speeds of galaxies, orbital velocities of galaxies in clu-
sters, gravitational lensing of background objects by ga-
laxy clusters, (Figure 3) and the temperature distribution
of hot gas in galaxies and clusters of galaxies.
This paper introduces a proposed equation of state of
dark energy and dark matter as one unified entity (Sec-
tion 2). Such an equation is derived based on the as-
sumption that dark energy and dark matter behave as a
perfect fluid and using the ideal gas equation, Boltzmann
constant and the energy-mass principle of Einstein.
Moreover, this paper suggests what could be the most
fundamental particle and quanta of dark matter and dark
energy and its characteristics (Section 3). Moreover,
based on NASA’s Cosmic Microwave Background Ex-
plorer (CMB) which estimated that the sky has an aver-
age temperature close to 2.7251 Kelvin, then the equiva-
lent mass and energy of fundamental particle of the dark
matter/dark energy is determined with an equivalent
mass of 40
4.2141 10
Kg which is equivalent to
23
3.767410 J
. Since this value has the same order of
Boltzmann constant 23
1.38 10
B
K
 J/K.
Furthermore, dark matter particle could be the most
fundamental and lightest particle in Nature and serves
the basic block of matter (quarks and gluons). Moreover,
assuming a uniform space dark energy/dark matter den-
sity, then the critical temperature at which the dark mat-
ter has a unity entity per volume is determined as
12
34.98310 K. At this temperature Boltzmann particles
are melt (unified) to generate quarks which are consi-
dered the basic blocks of physical matter (Section 4). Fi-
nally, conclusions are discussed.
2. PRELIMINARY: THE EQUATION OF
STATE OF DARK ENERGY AND DARK
MATTER: THE UNIFIED ENTITY
This equation relates the pressure P, temperature T and
the volume V of a substance behaves as an ideal gas [26],
that is
PV mRT (1)
As it can be seen easily that Eq.1 represents the en-
ergy associated with an ideal gas at given pressure P,
temperature T and the volume V, that is
PV mRT E (2)
Note that both sides of the equation has the units of
Figure 3. Gravitational lenses (Hubble Space Telescope,
NASA).
energy (work done by pressure P). Assume now that dark
energy behaves like an ideal gas with a negative pressure
(P) that causes the universe to expand with a total vol-
ume V, then by dividing both side of the equation of state
(5) by V, then
mE
PRT
VV
(3)
Defining the mass density as m
m
V
and energy
density as E
E
V
, Eq.3 yields to
mE
PRT
(4)
Now by taking the ratio between the mass density and
energy density then
E
m
RT
(5)
It can be concluded that the ratio between the mass
density and energy density are proportional to the pro-
duct of the temperature T and dark energy-dark matter
constant R (known as Universal gas constant). It is worth
to mention that NASA’s Cosmic Microwave Background
Explorer (CMB) in 1992 estimated that the sky has a
temperature close to 2.7251 Kelvin. Moreover, the Wil-
kinson Microwave Anisotropy Probe (WMAP) in 2003
has made a map of the temperature fluctuations of the
CMB with more accuracy [27-29].
The Boltzmann constant
B
K
is a physical constant
that relates temperature to microscopic energy.
B
A
RN
, where NA is the Avogadro Number.
23
1.38 10
B
K
 J/K The numerical value of
B
K
measures the conversion factor for mapping from this
microscopic energy E to the macroscopically-derived
temperature scale.
M. Shibli et al. / Natural Science 3 (2011) 743-749
Copyright © 2011 SciRes. OPEN ACCESS
746
The ideal gas law can now be expressed in terms of
Boltzmann constant such that
B
PVNK T (6)
where N is the actual number of entities (particles). Now
dividing both sides of (10) by volume to get the energy
density then
B
NB E
N
PKTKT
V

 (7)
This shows that the ratio between the energy density
and the entities density is proportional to the absolute
temperature times the Boltzmann constant.
The Boltzmann constant
B
K
is a physical constant
that relates temperature to energy.
B
A
RN where
A
N is the Avogadro Number [27].
23
1.3806505 10
B
K

J/K. The numerical value of
B
K
measures the conversion factor for mapping from
this characteristic microscopic energy E to the macro-
scopically-derived temperature scale.
The ideal gas law can now be expressed in terms of
Boltzmann constant such that
B
PVNK TE (8)
where N is the actual number of molecules. Now divid-
ing both sides of (8) by the volume to get the energy
density then
B
NB E
N
PKTKT
V
 (9)
By taking the ration between the energy density
E
and number of molecules density N/V, one gets
E
B
N
KT
(10)
or
E
N
B
K
T
(11)
This shows that the ratio between the energy density
and the molecular density is proportional to the absolute
temperature times the Boltzmann constant. The simula-
tion results demonstrate such a model.
3. PROPOSED DARK MATTER
PARTICLE CANDIDATE
A quark-gluon plasma (QGP) or quark soup is a phase
of quantum chromodynamics (QCD) which exists at ex-
tremely high temperature and/or density. This phase con-
sists of (almost) free quarks and gluons, which are se-
veral of the basic building blocks of matter. Recent
analyses from the Relativistic Heavy Ion Collider
(RHIC), a 2.4-mile-circumference (atom smasher) at the
US Department of Energy’s (DOE) Brookhaven Na-
tional Laboratory, establish that collisions of gold ions
traveling at nearly the speed of light have created matter
at a temperature of about 4 trillion degrees Celsius—the
hottest temperature ever reached in a laboratory, about
250,000 times hotter than the center of the Sun [30]. This
temperature, based upon measurements by the PHENIX
collaboration at RHIC, is higher than the temperature
needed to melt protons and neutrons into a plasma of
quarks and gluons. These new temperature measure-
ments, combined with other observations analyzed over
nine years of operations by RHIC’s four experimental
collaborations of BRAHMS, PHENIX, PHOBOS, and
STAR indicate that RHIC’s gold-gold collisions produce
a freely flowing liquid composed of quarks and gluons.
Such a substance, often referred to as quark-gluon
plasma, or QGP, filled the universe a few microseconds
after it came into existence 13.7 billion years ago. At
RHIC, this liquid appears, and the quoted temperature is
reached, in less time than it takes light to travel across a
single proton. Search for the axions is investigated in
work [30,31]. The axion is a proposed candidate particle
for dark energy.
The Hadron Epoch covers the time from 10–6 seconds
to 1 second after the Big Bang as shown in Figure 4. The
temperature during this epoch is estimated to decrease
from 1013 K to 1010 K. At 10–6 seconds Electrons and
positrons annihilate each other during the hadron epoch.
At 10–5 seconds, the temperature of the Universe is ap-
proximately 1013 K. Quarks combine to form protons and
neutrons. The lowering temperature allows quark/anti-
quark pairs to combine into mesons. After this period
quarks and anti-quarks can no longer exist as free parti-
cles. At 10–4 seconds the temperature of the Universe is
approximately 1010 (10 million) Kelvin. The existence of
antimatter is cancelled out, as lepton/anti-lepton pairs are
annihilated by existing photons. Neutrinos break free and
exist on their own.
Now consider the estimations which show that values
of universe dark energy density (=1.2622 × 10–26 kg/m3 =
6.8023 GeV), universe critical density (=1.8069 × 10–26
kg/m3 = 9.7378 GeV), universe matter density (=
0.54207 × 10–26 kg/m3 = 2.9213 GeV), and universe radi-
ation density (= 2.73 × 10–31 kg/m3 = 1.4558 MeV).
In this proposed paper and based on astronomical ob-
servations that the average density of dark matter and
dark energy is approximately 26
10 Kg/m3 and based on
previous published work [12] that the density of dark
matter is 26
0.54 10
Kg/m3 which is equivalent to
10
4.8277 10
J/m3. Now benefiting from (7) at CMB
temperature T = 2.73 K, then
12 3
12.8110entities m
N
 (12)
Since 12 3
12.8110entitiesm
N
 is corresponding
to 26
0.54 10
Kg/m3, then each entity has a mass of
M. Shibli et al. / Natural Science 3 (2011) 743-749
Copyright © 2011 SciRes. OPEN ACCESS
747
26 1240
0.541012.81104.214110 Kg

. The equiva-
lent energy of this particle is 23
3.767410 J
. For the
purpose of comparison, the mass of the most fundamen-
tal particles is listed in Table 1.
Figure 4. Quark-hadron transition and related temperature
and time occurrence after big bang.
Table 1. Masses of different quarks and particles are known
[32,33].
Symbol Description Value
me Electron mass 511 keV
mμ Muon mass 105.7 MeV
mτ Tau mass 1.78 GeV
mu Up quark mass 1.9 MeV
md Down quark mass 4.4 MeV
ms Strange quark mass 87 MeV
mc Charm quark mass 1.32 GeV
mt Top quark mass 172.7 GeV
mb Bottom quark mass 4.24 GeV
Furthermore, considering the lowest temperature in
nature at Boomerang nebula which is 1 Kelvin, then the
dark matter should be exactly equivalent to Boltzmann
constant. As it can be seen, the mass of the electron is
much heavier than this candidate particle by 2.159 Bil-
lion times. Furthermore,
N
is unity when the tem-
perature T is equal to 12
34.98310 K . This tempera-
ture value is called the critical temperature.
As introduced before, it is estimated that at 100 mi-
croseconds after the Big Bang [34] the temperature was
10 TK. At 3 - 5 TK proton-antiproton reactions occur. If
the density of dark matter/dark energy is uniform, ho-
mogeneous and constant through the universe, and since
the density is at the same order of the proton-nitron, then
it is very possible that dark energy/dark matter is con-
verted into quarks at this critical temperature as fol-
lowed.
Proposed dark matter candidate particle is charac-
terized in the following table:
Boltzmann Particle, B
mB 4.2141 × 10–40 kg
2.5386 × 10–13 u
3.7674 × 10–23 J
mBc2
0.022695
A comparison with the most known particles is shown
below
Electron, e
me 9.10638215(45) × 10–31 kg
5.4857990943(23) × 10–4 u
8.18710438(41) × 10–14 J mec2
0.510998910(13) Mev
Muon, μ
mμ 1.88353130(11) × 10–28 kg
0.1134289256(29) u
1.692833510(95) × 10–11 J mμc2
105.6583668(38) Mev
Proton, p
mp 1.672621637(83) × 10–27 kg
1.00727646677(10) u
1.503277359(75) × 10–10 J mpc2
938.272013(23) Mev
Neutron, n
mn 1.674927211(84) × 10–27 kg
1.00866491597(43) u
1.505349505(75) × 10–10 J mnc2
939.565346(23) Mev
Deuteron, d
md 3.34358320(17) × 10–27 kg
2.013553212724(78) u
3.00506272(15) × 10–10 J mdc2
1875.612793(47) Mev
M. Shibli et al. / Natural Science 3 (2011) 743-749
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748
Triton, t
mt 5.00735588(25) × 10–27 kg
3.0155007134(25) u
4.50038703(22) × 10–10 J
mtc2
2808.920906(70) Mev
Helion, h
mh 5.00641192(25) × 10–27 kg
3.0149322473(26) u
4.49953864(22) × 10–10 J mhc2
2808.391383(70) Mev
Alpha Particle, α
mα 6.64465620(33) × 10–27 kg
mαc2 4.001506179127(62) u
5.97191917(30) × 10–10 J
3727.379109(93) Mev
4. UTILIZATION OF DARK MATTER
AND PROPOSED EXPERIMENT
There density are affected by the space temperature
then utilization of its energy can be achieved at high tem-
peratures such as Fermi melting point of quarks into
quarkgluon plasma (0.5 – 1.2 × 1012 K) or higher. Hence
N
is unity when the temperature T is equal to 34.983
×1012 K, then this temperature value is called the critical
temperature. In other words at this temperature the
12 3
12.8110entities m
N
 of dark matter particles
are unified (melted) to form quarks which work as the
basic blocks of matter. Cooling quarks to 1013 K then
quarks combine to form protons and neutrons.
heat
e
Buuu ddde

(13)
B is after the dark matter candidate particle.
Similar to Relativistic Heavy Ion Collider (RHIC),
here it is proposed to conduct an experiment at 4 Trillion
Kelvin to generate quark-gluon plasma as explained in
Section 3.
The following two tables, Tables 2 and 3 show some a
comparison with some physical phenomena temperatures
so as to compare with the critical temperature at which
dark matter particles unified into quarks.
5. CONCLUSIONS
A proposed equation of state of dark energy and dark
matter as one unified entity is introduced such that dark
energy and dark matter are not distinct. On the contrary,
both dark energy and dark matter represent one unified
entity. Such an equation is derived based on the assump-
tion that dark energy and dark matter behave as a perfect
fluid and using the ideal gas equation, Boltzmann con-
stant and the energy-mass principle of Einstein. This
principle agrees with the recent observations of NASA
Table 2. The temperature order between 1 and 10 K.
1 K At the Boomerang nebula the coldest natural environment
1.5 K Melting point of overbound helium
2.19 KLambda point of overbound superfluid helium
2.725 KCosmic microwave background
4.1 K Superconductivity point of mercury
4.22 KBoiling point of bound helium
5.19 KCritical temperature of helium
7.2 K Superconductivity point of lead
9.3 K Superconductivity point of niobium
Table 3. The temperature order TK (1012 K).
0.5 - 1.2 TKFermi melting point of quarks into quark-gluon plasma
3 - 5 TK In proton-antiproton reactions
Z0 Electronuclear excitations
10 TK 100 microseconds after the Big Bang
300 - 900 TKAt proton-nickel conversions in the Tevatron’s Main
Injector
that dark energy and dark matter has close density values
and in the range of 10–26 kg/m3.
Additionally, in this paper presents it is presented what
could be the most fundamental particle and quanta of
dark matter and dark energy and its characteristics. It is
found that this candidate particle has an equivalent mass
of 40
4.2141 10
Kg which is equivalent to
23
3.767410 J
. This value has the same order of
Boltzmann constant 23
1.3810J K
B
K
 . Benefiting
from CMB temperature T = 2.73 K, then each cubic me-
ter of space contains 12 3
12.8110entitiesm of Bolt-
zmann particles. As it can be seen, the mass of the elec-
tron is much heavier than this candidate particle by 2.159
Billion times. It could be the most fundamental and
lightest particle in nature and serves as the basic blocks
of matter (quarks and gluons).
Moreover, the critical temperature at which the dark
matter has a unity entity per unit volume is determined as
12
34.98310 K. Considering the lowest temperature in
nature at Boomerang nebula which is 1 Kelvin, then the
dark matter should be exactly equivalent to Boltzmann
constant. An experiment (similar to Relativistic Heavy
Ion Collider (RHIC)) is proposed at the critical tempera-
ture to form quarks.
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