1. Introduction

“Hubble volume” can be considered as a key tool in cosmology and unification. In this paper an attempt is made to understand the basic unified concepts of the four fundamental cosmological interactions. This is a new approach and particle physics and cosmology can be studied in a cohesive mode.

1.1. Basic Assumptions in Particle Cosmology

With reference to the Mach’s principle[1-6] and the Hubble volume, if “Hubble mass” is the product of cosmic critical density and the Hubble volume[7-9], then it can be assumed that,

1) Within the Hubble volume, each and every point in free space is influenced by the Hubble mass.

2) Within the Hubble volume, the Hubble mass plays a vital role in understanding the properties of electromagnetic and nuclear interactions.

3) Within the Hubble volume, Hubble mass plays a key role in understanding the geometry of the universe.

With reference to the Avogadro number[10] and from unification point of view, the utmost fundamental question is: How to understand the origin of “mass” of elementary particles? In this connection it can be assumed that,

1) “Molar electron mass” can be considered as the rest mass of a new heavy charged elementary particle.

2) Atomic gravitational constant is Avogadro number times the classical gravitational constant.

2. Key Concepts in Particle Cosmology

Concept-1: In atomic and nuclear physics, atomic gravitational constant is Avogadro number times the classical gravitational constant .

(1)

This idea may come under the subject classification of “strong gravity” and is not in the main stream physics. K.P. Sinha, C. Sivaram, Abdus Salam, E. Recami and colleagues developed the subject in a unified gravitational approach [11-15]. It is reasonable to say that - since the atomic gravitational constant is times the classical gravitational constant, atoms are themselves arranged in a systematic manner and generate the “gram mole”.

Concept -2: The key conceptual link that connects the gravitational and non-gravitational forces is - the classical force limit

(2)

It can be considered as the upper limit of the string tension. In its inverse form it appears in Einstein's theory of gravitation[6] as It has multiple applications in Black hole physics and Planck scale physics[16]. It has to be measured either from the experiments or from the cosmic and astronomical observations.

Concept -3: Ratio of ‘classical force limit ’ and ‘ weak force magnitude’ is where is a large number close to the Avogadro number.

(3)

Thus the proposed weak force magnitude is newton. Considering this , Higgs fermion and boson masses can be fitted. In this connection please see our earlier published papers[17-21] and application-9 of this paper.

Concept-4: In the expanding cosmic Hubble volume, can be considered as the gravitational or electromagnetic interaction range.

Concept-5: In the expanding cosmic Hubble volume, characteristic cosmic Hubble mass is the product of the cosmic critical density and the Hubble volume. If the critical density is and characteristic Hubble radius is mass of the cosmic Hubble volume is

(4)

Concept-6: There exists a charged heavy massive elementary particle in such a way that, inverse of the fine structure ratio is equal to the natural logarithm of the sum of number of positively and negatively charged in the Hubble volume. If the number of positively charged particles is and the number of negatively charged particles is also then

(5)

From experiments and from the current observations[22,23,24], magnitude of the Hubble constant is, Km/sec/Mpc. Thus

(6)

If is the Avogadro number and is the rest mass of electron, surprisingly it is noticed that, and this is close to the above estimation of Thus it can be suggested that,

(7)

In this way, Avogadro number can be coupled with the cosmic, atomic and particle physics. Then with reference to the obtained cosmic Hubble mass is and thus the obtained Hubble’s constant is Km/sec/Mpc. Note that large dimensionless constants and compound physical constants reflect an intrinsic property of nature [25,26]. Whether to consider them or discard them depends on the physical interpretations, logics, experiments, observations and our choice of scientific interest. In most of the critical cases, ‘time’ only will decide the issue. The mystery can be resolved only with further research, analysis, discussions and encouragement.

Concept -7: For any observable charged particle, there exist two kinds of masses and their mass ratio is Let this number be First kind of mass seems to be the ‘gravitational or observed’ mass and the second kind of mass seems to be the ‘electromagnetic’ mass. This idea can be applied to proton and electron.

This number is obtained in the following way. In the Planck scale, similar to the Planck mass, with reference to the elementary charge, a new mass unit can be constructed in the following way.

(8)

(9)

Here is the elementary charge. How to interpret this mass unit? Is it a primordial massive charged particle? If two such oppositely charged particles annihilate, a large amount of energy can be released. This may be the root cause of cosmic energy reservoir. Such pairs may be the chief constituents of black holes. In certain time interval with a well defined quantum rules they annihilate and release a large amount of energy in the form of photons. In the Hubble volume, with its pair annihilation, “origin of the CMBR” can be understood. Clearly speaking, gravitational and electromagnetic force ratio of

(10)

It can be interpreted that, if is the observable or gravitational mass of, then is the electromagnetic mass of. With reference to the electron rest mass,

(11)

Concept-8: If is the quantum of the gravitational angular momentum, then the electromagnetic quantum can be expressed as Thus the ratio,

(12)

where is very close to the weak mixing angle

Concept-9: In modified quark SUSY[17,18], if is the mass of quark fermion and is the mass of quark boson, then

(13)

and represents the effective quark fermion mass. The number can be fitted with the following empirical relation

(14)

With this idea super symmetry can be observed in the low and high energy strong interactions and can also be observed in the electroweak interactions[17,18].

3. To Fit the Rest Masses of Electron, Proton and Neutron

If is the light charged elementary particle and is the heavy charged elementary particle to be detected or observed, it is possible to represent the relation in the following form.

(15)

(16)

(17)

In this way the origin of the electron rest mass can be understood. It is noticed that,

(18)

and is roughly 5 times greater than the nucleon rest energy. If ,

An attempt is made to fit the rest masses of proton and neutron in the following way.

(19)

(20)

(21)

In support of these relations an attempt is made to implement the number in fitting the nuclear binding energy constants and other areas of physics like strong interaction range, potential energy of electron in hydrogen atom, electroweak physics etc.

3.1. To Fit the Nuclear Binding Energy Constants

The semi-empirical mass formula (SEMF) is used to approximate the mass and various other properties of an atomic nucleus[27,28,29]. As the name suggests, it is based partly on theory and partly on empirical measurements. Based on the ‘least squares fit’, volume energy coefficient is MeV, surface energy coefficient is MeV, coulombic energy coefficient is MeV, asymmetric energy coefficient is = 23.21 MeV and pairing energy coefficient is MeV. The semi empirical mass formula is

(22)

It is noticed that,

(23)

Asymmetric energy constant be

(24)

Pairing energy constant be

(25)

Let the maximum nuclear binding energy per nucleon be

(26)

Coulombic energy constant be

(27)

Table 1. SEMF binding energy with the proposed energy coefficients

Surface energy constant be

(28)

Volume energy constant be

(29)

In table-1 within the range of to nuclear binding energy is calculated and compared with the measured binding energy[30]. Column-3 represents the calculated binding energy and column-4 represents the measured binding energy.

3.2. Proton-nucleon Stability Relation

It is noticed that

(31)

where is the stable mass number of This is a direct relation. Assuming the proton number in general, for all atoms, lower stability can be fitted directly with the following relation[27].

(32)

Stable super heavy elements can be predicted with this relation. In between to obtained is lower compared to the actual It is noticed that, upper stability in light and medium atoms up to can be fitted with the following relation.

(33)

From this relation for ,obtained upper Note that, for ,actual stable where is the fine structure ratio. This seems to be a nice and interesting coincidence. In between 0.00615 and 0.0080, for light and medium atoms up toor mean stability can be fitted with the following relation.

(34)

Surprisingly it is noticed that, in this relation, Thus up to or mean stability can be expressed as

(35)

4. To Fit the Rms Radius of Proton

Let be the rms radius of proton. Define two radii and as follows.

(36)

(37)

It is noticed that,

(38)

Thus,

(39)

This can be compared with the 2010 CODATA recommended rms radius of proton fm. Recent work on the spectrum of muonic hydrogen (an exotic atom consisting of a proton and a negative muon) indicates a significantly lower value for the proton charge radius, fm and the reason for this discrepancy is not clear. This is 10 times more precise than all the previous determinations[31,32]. Thus from proton rest mass and rms radius,

(40)

(41)

Here the most interesting thing is that, is very close to the Bohr radius of Hydrogen atom. It is very interesting to note that, with ionic radii of atoms can be fitted very easily as

(42)

where is the ionic radius of mass number If nm, if nm and if nm. Their corresponding recommended radii are 0.076 nm, 0.102 nm and 0.138 nm respectively[31,32].

5. To Fit the Characteristic Potential Radius of Nucleus 1.4 fm

It is noticed that, gram mole is a black hole where the operating gravitational constant is but not . That means for the simplest case of gram mole of electrons or gram mole of protons, there exist number of electrons or number of protons. Let it follows the concept of Schwarzschild radius. It can be expressed in the following way. Let us define two radii and as follows.

(43)

(44)

(45)

(46)

For the above two cases, the characteristic mean distance in between electrons or in between protons, can be obtained as

(47)

(48)

It is noticed that,

(49)

This can be compared with the characteristic alpha scattering experimental radius [31] of nucleus fm. Based on the Yukawa’s Pion exchange model nuclear interaction range is 1.4 fm. Thus if is the charged pion rest mass,

(50)

6. Applications of the Proposed Assumptions and Concepts

6.1. Part-1: Applications in Particle and Nuclear Physics

6.1.1. Application-1: The Characteristic Nuclear Charge Radius

If Km/sec/Mpc, is the characteristic radius of nucleus, it is noticed that,

(51)

where is the proton rest mass. This can be compared with the characteristic charge radius of the nucleus and the strong interaction range.

6.1.2. Application-2: Scattering Distance between Electron and the Nucleus

If fm is the minimum scattering distance between electron and the nucleus, it is noticed that,

(52)

Here is the molar electron mass. Here it is very interesting to consider the role of the Schwarzschild radius of the ‘electron mass’.

6.1.3. Aplication-3: To fit the charged lepton rest masses

Muon and tau rest masses can be fitted in the following way[33]. Let be the characteristic nuclear unit size. The key relation seems to be

(53)

Considering the ratio of the volumes and , let

(54)

Now muon and tau masses can be fitted with the following relation[17,18].

(55)

where x = 0,1 and 2. At x = 0, At x = 1, MeV and can be compared with the rest mass of muon (105.66 MeV). At x = 2, MeV and can be compared with the rest mass of tau (1777.0 MeV). x = 0,1 and 2 can be considered as the 3 characteristic vibrating modes.

6.1.4. Aplication-4: Electromagnetic and Strong Interaction Ranges

For electron, starting from, its characteristic interaction ending range can be expressed as

(56)

Similarly, for proton, its characteristic interaction starting range can be expressed as

(57)

6.1.5. Application-5: Ratio of Electromagnetic and Strong Interaction Ending Range

Ratio of electromagnetic ending interaction range and strong interaction ending range can be expressed as

(58)

Thus if

(59)

Interesting observation is

(60)

This can be considered as the mean strong interaction range and is close to the proton rms radius!

6.1.6. Application-6

For any elementary particle of charge electromagnetic mass and characteristic radius, it can be assumed as

(61)

This idea can be applied to proton as well as electron. Electron’s characteristic radius is

(62)

Similarly proton’s characteristic radius is

(63)

6.1.7. Application-7: Potential Energy of Electron in Hydrogen Atom

Let be the potential energy of electron in the Hydrogen atom. It is noticed that,

(64)

where is the Bohr radius[34,35]. With 99.6822% this is matching with eV. After simplification it takes the following form.

(65)

Thus the Bohr radius can be expressed as

(66)

Electron’s orbit radii can be expressed as

(67)

where is the radius of orbit and Thus in Hydrogen atom, potential energy of electron in orbit can be expressed as

(68)

Note that, from the atomic theory it is well established that, total number of electrons in a shell of principal quantum number is Thus on comparison, it can suggested that, is the potential energy of electrons and potential energy of one electron is equal to

6.1.8. Application-8: Magnetic Moments of the Nucleon

If magnetic moment of electron can be expressed as[36,37]

(69)

It can be suggested that electron’s magnetic moment is due to the electromagnetic interaction range. Similarly magnetic moment of proton is due to the strong interaction ending range.

(70)

If proton and neutron are the two quantum states of the nucleon, by considering the mean strong interaction range magnetic moment of neutron can be fitted as

(71)

6.1.9. Application-9: To Correlate the Charged Higgs Fermion Mass and the Electron Mass

If is the charged Higgs fermion, it is noticed that,

(72)

Thus,

(73)

From relation (52),

(74)

If Higgs fermion and Higgs boson mass ratio is 2.2627, then obtained Higgs boson mass is 45576.27 MeV and the most surprising thing is that, Higgs boson pair generates the neutral boson of rest energy 91152.53 MeV. Estimated top quark rest energy[17,18] is 182160 MeV and its corresponding boson is 80505.6 MeV. Thus the surprising thing is that, susy boson of the top quark seems to be the electroweak boson. Another interesting idea is that boson and Higgs boson generate a neutral boson of mass 126 GeV. It can be suggested that, boson pair generates a neutral boson of rest energy 161 GeV.

6.2. Part-2 : Applications in Cosmology

6.2.1. Application-10: To fit the Hubble’s Constant

Combining the relations (51) and (52) and if Km/sec/Mpc, it is noticed that,

(75)

Surprisingly this ratio is close to unity! How to interpret this ratio? From this relation it can be suggested that, along with the cosmic variable, in the presently believed atomic and nuclear physical constants, on the cosmological time scale, there exists one variable physical quantity. ‘Rate of change’ in its magnitude may be a measure of the present cosmic acceleration. Thus independent of the cosmic red shift and CMBR observations, from the atomic and nuclear physics, cosmic acceleration can be verified. Based on the above coincidence, magnitude of the present Hubble’s constant can be expressed as

(76)

6.2.2. Application-11: Pair Creation of M_c within the Hubble Volume and the CMBR Temperature

Pair particles creation and annihilation is a characteristic phenomena in `free space’, and is the basic idea of quantum fluctuations of the vacuum. In the expanding universe, from relation (8) by considering the proposed charged and its pair annihilation as characteristic cosmic phenomena, origin of the isotropic CMB radiation can be addressed. At any time it can be suggested that

(77)

where is the cosmic mass at time Please note that, at present

(78)

Qualitatively and quantitatively this can be compared with the present CMBR temperature . But it has to be discussed in depth. It seems to be a direct consequence of the Mach's principle.

6.2.3. Application-12: A Quantitative Approach to Understand the CMBR Radiation

It is noticed that, there exists a very simple relation in between the cosmic critical density, matter density and the thermal energy density. It can be expressed in the following way. At any time

(79)

where is the matter density and is the thermal energy density expressed in or Considering the Planck - Coulomb scale, at the beginning if

(80)

(81)

Thus at any time

(82)

(83)

(84)

In this way, observed matter density and the thermal energy density can be studied in a unified manner. The observed CMB anisotropy can be related with the inter galactic matter density fluctuations.

6.3. Present Matter Density of the Universe

From (76) at present if Km/sec/Mpc,

(85)

where and Based on the average mass-to-light ratio for any galaxy[6]

(86)

where for any galaxy, and the number .

Note that elliptical galaxies probably comprise about 60% of the galaxies in the universe and spiral galaxies thought to make up about 20% percent of the galaxies in the universe. Almost 80% of the galaxies are in the form of elliptical and spiral galaxies. For spiral galaxies, and for elliptical galaxies, For our galaxy inner part, Thus the average is very close to 8 to 9 and its corresponding matter density is close to and can be compared with the above proposed magnitude of

6.4. Present Thermal Energy Density of the Universe

At present if Km/sec/Mpc,

(87)

and thus

(88)

At present if

(89)

where is the radiation energy density constant, then obtained CMBR temperature is, This is accurately fitting with the observed CMBR temperature[24] , Thus in this way, the present value of the Hubble’s constant and the present CMBR temperature can be co-related with the following trial-error relation.

(90)

7. Discussion & Conclusions

String theory or QCD is not in a position to address the basics of cosmic structure[38]. In understanding the basic concepts of unification or theory of every thing, role of dark energy and dark matter is insignificant. Even though string theory was introduced for understanding the basics of strong interaction, its success seems to be a dilemma because of its higher dimensions and the non-coupling of the nuclear and Planck scale. Based on the proposed relations and applications, Hubble volume or Hubble mass, can be considered as a key tool in unification as well as cosmology. From relations (51,52,75), if it is possible to identify the atomic cosmological physical variable, then by observing the rate of change in its magnitude (on the cosmological time scale), the “future” cosmic acceleration can be verified and thus the cosmic geometry can be confirmed from atomic and nuclear physics! Without the advancement of nano- technology or fermi-technology this may not be possible. Not only that, independent of the cosmic red shift and CMBR observations “future” cosmic acceleration can be checked in this new direction.

Considering the proposed relations and concepts it is possible to say that there exists a strong relation between cosmic Hubble mass, Avogadro number and unification..

ACKNOWLEDGEMENTS

The first author is indebted to professor K. V. Krishna Murthy, Chairman, Institute of Scientific Research on Vedas (I-SERVE), Hyderabad, India and Shri K. V. R. S. Murthy, former scientist IICT (CSIR) Govt. of India, Director, Research and Development, I-SERVE, for their valuable guidance and great support in developing this subject.