The basic blocks of the universe matter: Boltzmann fundamental particle and energy quanta of dark matter and dark energy

doi:10.4236/ns.2011.39099

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Vol.3, No.9, 743-749 (2011) Natural Science

http://dx.doi.org/10.4236/ns.2011.39099

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: [email protected]

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

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

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

3.767410 J



. Since this value has the same order of

Boltzmann constant 23

1.38 10

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

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

PRT



 (3)

Defining the mass density as m



 and energy

density as E





, Eq.3 yields to

PRT





 (4)

Now by taking the ratio between the mass density and

energy density then



 (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

is a physical constant

that relates temperature to microscopic energy.



, where NA is the Avogadro Number.

1.38 10



 J/K The numerical value of

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

746

The ideal gas law can now be expressed in terms of

Boltzmann constant such that

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

NB E

PKTKT



 (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

is a physical constant

that relates temperature to energy.

RN where

N is the Avogadro Number [27].

1.3806505 10

K



J/K. The numerical value of

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

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

NB E

PKTKT



 (9)

By taking the ration between the energy density



and number of molecules density N/V, one gets



 (10)



 (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

4.8277 10



 J/m3. Now benefiting from (7) at CMB

temperature T = 2.73 K, then

12 3

12.8110entities m



 (12)

Since 12 3

12.8110entitiesm



 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

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,



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

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



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



 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

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

3.767410 J



. This value has the same order of

Boltzmann constant 23

1.3810J K

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

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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