Mathematical Overview of Hypersphere World-Universe Model

The Hypersphere World-Universe Model (WUM) provides a mathematical framework that allows calculating the primary cosmological parameters of the World which are in good agreement with the most recent measurements and observations. WUM explains the experimental data accumulated in the field of Cosmology and Astroparticle Physics over the last decades: the age of the World and critical energy density; the gravitational parameter and Hubble’s parameter; temperatures of the cosmic microwave background radiation and the peak of the far-infrared background radiation; the concentration of intergalactic plasma and time delay of Fast Radio Bursts. Additionally, the model predicts masses of dark matter particles, photons, and neutrinos; proposes new types of particle interactions (Super Weak and Extremely Weak); shows inter-connectivity of primary cosmological parameters of the World. WUM proposes to introduce a new fundamental parameter Q in the CODATA internationally recommended values. This paper is the summary of the mathematical results obtained in [1]-[4].


Introduction
Hypersphere World-Universe Model (WUM) views the World as a 3-dimensional hypersphere that expands along the fourth spatial dimension in the Universe.A hypersphere is an example of a 3-manifold which locally behaves like regular euclidean 3-dimensional space: just as a sphere looks like a plane to small enough observers.WUM is based on Maxwell's equations (ME) that form the foundation of Electromagnetism and Gravitoelectromagnetism.According to ME, there exist two measurable physical characteristics: energy density and energy flux density.
WUM makes reasonable assumptions in the main areas of cosmology.The remarkable agreement of the calculated values of the primary cosmological parameters with the observational data gives us considerable confidence in the model.
The principal idea of WUM is that the energy density of the World W ρ equals to the critical energy density cr ρ necessary for 3-manifold at any cos- mological time.cr ρ can be found by considering a sphere of radius M R and enclosed mass M, with a small test mass m on the periphery of the sphere.Mass M can be calculated by multiplication of cr ρ by the volume of the sphere.The equation for cr ρ can be found from the escape speed calculation for test mass m: where G is the gravitational constant, H is Hubble's parameter, and c is the gravitoelectrodynamic constant that is identical to the electrodynamic constant c in Maxwell's equations.
WUM introduces a fundamental dimensionless time-varying parameter Q that is the measure of the curvature of the Hypersphere.Q can be calculated from the average value of the gravitational constant and in present epoch equals to (see Section 2): WUM develops a mathematical framework that allows for direct calculation of a number of cosmological parameters through Q.The precision of such parameters increases by orders of magnitude (see Section 2).Below we will use the following fundamental constants: • Basic unit of length 0 2π a a = , 0 a being the classical electron radius; • Planck constant h; • Basic unit of energy 0 hc E a = that is the basic gravitoelectrodynamic charge; • Basic unit of energy density 0 4 hc a ρ = ; • Basic unit or surface energy density 0 0 3 hc a a σ ρ = = ; • Basic unit of mass 0 h m ac = ; • Basic unit of frequency 0 c a ν = ; • Fine-structure constant α .

Primary Cosmological Parameters
Equation (1.1) can be rewritten as where g µ is the gravitomagnetic parameter and M ρ is the energy density of the Medium.Hubble's parameter H can be expressed: = , where R is the Hubble's radius and is the radius of the Hypersphere in WUM.Introducing the dimensionless parameter Q: we can rewrite (2.1) where MO ρ is the energy density of Macroobjects of the World.Assuming that we can find the equation for the critical energy density: and for the gravitational constant: We can calculate the value of G based on the value of H.

Gravitation
In frames of WUM the parameter G can be calculated based on the value of the energy density of the Medium M ρ [2]: where a dimension-transposing parameter P equals to: Then the Newton's law of universal gravitation can be rewritten in the following way: where we introduced the measurable parameter of the Medium M ρ instead of the phenomenological coefficient G ; and gravitoelectromagnetic charges 3) have a dimension of "Area", which is equivalent to "Energy", with the constant that equals to the basic unit of surface energy density 0 σ .
Following the approach developed in [2] we can find the gravitomagnetic parameter of the Medium M µ : and the impedance of the Medium M Z : where τ is a cosmological time.These parameters are analogous to the per- meability 0 µ and impedance of electromagnetic field , where 0 ε is the permittivity of electromagnetic field and

Intergalactic Plasma
In where e is the elementary charge.Since the formula calculating the potential energy of interaction of protons and electrons contains the same parameter pe k : ( ) where we assume that e ω is proportional to Energy densities of protons and electrons are then proportional to 1 Q − , similar to the critical energy density We substitute ( ) ) and calculate concentration of protons and electrons:  6] corresponding to the WMAP measurement of the baryon density [7].The Mediums' plasma density (4.3) is in good agreement with the estimated value [6].
From Equation (4.2) we obtain the value of the lowest frequency pl ν : Photons with energy smaller than ph pl E h ν = cannot propagate in plasma, thus pl hν is the smallest amount of energy a photon may possess.Following the authors of [8] we can call this amount of energy the rest energy of photons that equals to The above value is in good agreement with the value  This value is in good agreement with experimentally found value of 0.049 0.013 ± [9].The results obtained in [6] [8] and [9] prove assumption ( ) where B k is the Boltzmann constant and MBR T is MBR temperature.We can now calculate the value of MBR T : 1 4 Thus calculated value of MBR T is in excellent agreement with experimentally measured value of 2.72548 0.00057 K ± [10] and proves assumption (4.7).

Fast Radio Bursts
Fast Radio Burst (FRB) is a high-energy astrophysical phenomenon manifested as a transient radio pulse lasting only a few milliseconds.These are bright, unresolved, broadband, millisecond flashes found in parts of the sky outside the Milky Way.The component frequencies of each burst are delayed by different amounts of time depending on the wavelength.This delay is described by a value referred to as a Dispersion Measure (DM) which is the total column density of free electrons between the observer and the source of FRB.Fast radio bursts have DMs which are: much larger than expected for a source inside the Milky Way [11]; and consistent with propagation through ionized plasma [12].In this Section we calculate a time delay of FRB based on the characteristics of the Intergalactic Plasma discussed in [4] (see Section 4).
Consider a photon with initial frequency emit ν and energy emit E emitted at time emit τ when the radius of the hypersphere World in the fourth spatial dimension was emit R .The photon is continuously losing kinetic energy as it moves from galaxy to the Earth until time obsv τ when the radius is The observer will measure obsv ν and energy obsv E and calculate a redshift: Recall that emit ( ) Let's calculate photons' traveling time ph t from a galaxy to the Earth taking into account that the rest energy of photons ph E is much smaller than the energy of photons where ph t ∆ is photons' time delay relative to the light-travel time LTT t that equals to: All observed FRBs have redshifts 1 z < .It means that we can use the Hubble's law: Photons' rest energy squared at radius r between emit R and 0 R equals to (3.5): According to WUM, photons' energy E γ on the way from galaxy to an ob- server can be expressed by the following equation: which reduces to emit E at (5.5) and to obsv E at 0 r R = .Placing the values of the parameters (5.5), (5.6), (5.7) into (5.4),we have for photons' time delay: where . Taking 0.492 z = [12] we get the calculated value of photons' time delay which is in good agreement with experimentally measured value [12] 2 2.438 1 GHz It is worth to note that in our calculations there is no need in the dispersion measure.

Neutrinos
It is now established that there are three different types of neutrino: electronic e ν , muonic µ ν , and tauonic τ ν , and their antiparticles.Neutrino oscillations imply that neutrinos have non-zero masses [13] Their concentrations n v are then proportional to and energy densities of neutrinos are proportional to Experimental results obtained by M. Sanchez [16] show nos respectively.Significantly more accurate result was obtained by P. Kaus, et al. [17] for the ratio of the mass splitting: Let's assume that muonic neutrino's mass indeed equals to From equation (6.5) it then follows that Then the squared values of the muonic and tauonic neutrino masses fall into ranges (6.3) and (6.4): Let's assume that electronic neutrino mass equals to is also in a good agreement with the value of 2 0.06 eV c discussed in literature [18].
Considering that all elementary particles, including neutrinos, are fully characterized by their four-momentum , , , , we obtain the following neutrino energy densities i ν ρ in accordance with theoretical calculations made by L. D. Landau and E. M. Lifshitz [19]: where F p is Fermi momentum, Let's take the following value for Fermi momentum p F : where is the extrapolated value of F p at the Beginning when 1 Q = .Using (6.13), we obtain neutrinos relative energy densities i ν Ω in the Medium in terms of the critical energy density cr ρ : ( ) where ( ) It's commonly accepted that concentrations of all types of neutrinos are equal.This assumption allows us to calculate the total neutrinos relative energy density in the Medium: which is remarkably close to its value calculated in (6.20).
The assumptions made in (6.6), (6.9), (6.17) and (6.22) are further supported by the excellent numerical agreement of calculated and measured value of Fine-structure constant α discussed in Section 11.

Cosmic Far-Infrared Background
The cosmic Far-Infrared Background (FIRB), which was announced in January 1998, is part of the Cosmic Infrared Background, with wavelengths near 100 microns that is the peak power wavelength of the black body radiation at temperature 29 K.In this Section we introduce Bose-Einstein Condensate (BEC) drops of dineutrinos whose mass is about Planck mass, and their temperature is around 29 K.These drops are responsible for the FIRB [15].
According to [20] [21] [22], the size of large cosmic grains G D is roughly equal to the length F L : and their mass G m is close to the Planck mass ( ) 2) The density of grains G ρ is about: According to WUM, Planck mass P M equals to [15]   1 2 0 Note that the value of P M is increasing with cosmological time, and is pro- portional to 1 2   τ .Then, The received energy will increase the grain's temperature G T , until equili- brium is achieved: power received equals to the power irradiated by the surface of a grain in accordance with the Stefan-Boltzmann law where SB σ is Stefan-Boltzmann constant: With Nikola Tesla's principle at heart-There is no energy in matter other than that received from the environment-we apply the World equation [23] to a grain: where 0 σ is a basic unit of surface energy density: We then calculate the grain's stationary temperature G T to be This result is in an excellent agreement with experimentally measured value of 29 K [24]- [35] and proves the assumptions (7.1), (7.2) and (7.9).
Cosmic FIRB radiation is not a black body radiation.Otherwise, its energy density FIRB ρ at temperature G T would be too high and equal to the energy density of the Medium of the World: ( )

Bose-Einstein Condensate
New cosmological models employing the Bose-Einstein Condensates (BEC) have been actively discussed in literature in recent years [36]- [50].The transition to BEC occurs below a critical temperature c T , which for a uniform three-di- mensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by ( ) where X n is the particle concentration, X m is the mass per boson, ζ is the Riemann zeta function: ( ) and the value of the mass X m : X m is about 10 orders of magnitude larger than the rest mass of photon's (see which is 10π times smaller than the energy density of MBR and dineutrinos: The ratio between FIRB and MBR corresponds to the value of 3.4% calculated by E. L. Wright [51].

Multicomponent Dark Matter
Dark Matter (DM) is among the most important open problems in both cosmology and particle physics.Dark Matter problem can be, in principle, achieved through extended theories of gravity, as it is discussed, for example, in [52].
There pr e e = which we took to match the Quark Model.ELOPs break into two preons at nuclear densities or at high temperatures.In particle physics, preons are postulated to be "point-like" particles, conceived to be subcomponents of quarks and leptons [53].
WUM postulates that masses of DMP are proportional to 0 m multiplied by different exponents of α and can be expressed with the following formulae: CDM particles (neutralinos and WIMPs): that one of outstanding puzzles in particle physics and cosmology relates to so-called cosmic coincidence: the ratio of dark matter density in the World to baryonic matter density in the Medium of the World 5 ≅ [55]  [56].
Neutralinos, WIMPs, and sterile neutrinos are Majorana fermions, which partake in the annihilation interaction with strength equals to 2 α − , 1 α − , and 2 α respectively (see Section 10).The signatures of DMP annihilation with ex- pected masses of 1.3 TeV, 9.6 GeV, 70 MeV, 340 keV, and 3.7 keV are found in spectra of the diffuse gamma-ray background and the emission of various macroobjects in the World [23].
The assumptions made in (8.3) and (8.6) are further supported by the excellent numerical agreement of calculated and measured value of fine-structure constant α discussed in Section 11.

Macroobject Cores Built up from Fermionic Dark Matter
In this section, we discuss the possibility of all macroobject cores consisting of DMP introduced in Section 9.The first phase of stellar evolution in the history of the World may be dark stars, powered by Dark Matter heating rather than fusion.Neutralinos and WIMPs, which are their own antiparticles, can annihilate and provide an important heat source for the stars and planets in the World.
In our view, all macroobjects of the World (including galaxy clusters, galaxies, star clusters, extrasolar systems, and planets) possess the following properties: • Macroobject cores are made up of DMP; • Macroobjects consist of all particles under consideration, in the same proportion as they exist in the World's Medium; • Macroobjects contain other particles, including DM and baryonic matter, in shells surrounding the cores.Taking into account the main principle of the World-Universe Model (all physical parameters can be expressed in terms of , Q α , small integer numbers, and π ) we modify the published theory of Fermionic Compact Stars (FCS) developed by G. Narain, et al. [57] as follows.We take a scaling solution for a free Fermi gas consisting of fermions with mass f m in accordance with following equations: Maximum mass: Minimum radius: Maximum density: max ; ; 2π taken by G. Narain, et al. [57]).Then diameter of FCS is proportional to the fermion Compton length Cf L .We use π 6 as the value of 1 A (instead of 1 0.384 A = taken by G. Narain, et al. [57]).Then 3 A will equal to   Table 1 summarizes the parameter values for FCS made up of various fermions: where which is the maximum possible density of any macroobject in the World.
A Compact Star made up of heavier particles, WIMPs and neutralinos, could in principle have a much higher density.In order for such a star to remain stable and not exceed the nuclear density, WIMPs and neutralinos must partake in an annihilation interaction whose strength equals to Scaling solution for interacting WIMPs can also be described with equations

M
. It makes strongly interacting WIMPs good candidates for stellar and planetary cores of extrasolar systems with Red stars [23].
When the mass of a FCS made up of WIMPs is much smaller than the maximum mass, the scaling solution yields the following equation for parameters 1 A and 2 A : )-one order of magnitude.It makes WIMPs good candidates for brown dwarf cores too [23].
Scaling solution for interacting neutralinos can be described with the same equations (10.1), (10.2), (10.In this case, the maximum mass and minimum radius increase about four orders of magnitude each and the maximum density equals to the nuclear density.Note that parameters of a FCS made up of strongly interacting neutralinos are identical to those of neutron stars. Practically constant value of FCS minimum radius takes place in the huge range of masses ( )

Grand Unified Theory
At the very Beginning (Q = 1) all extrapolated fundamental interactions of the World-strong, electromagnetic, weak, Super Weak and Extremely Weak (proposed in WUM), and gravitational-had the same cross-section of The gravitational coupling parameter G α is similarly decreasing: The weak coupling parameter W α is decreasing as follows:

Conclusions
WUM holds that there exist relations between all Q-dependent parameters: Newtonian parameter of gravitation and Hubble's parameter; critical energy density and Fermi coupling parameter; temperatures of the microwave background radiation and far-infrared background radiation peak.The calculated values of these parameters are in good agreement with the latest results of their measurements.
Today, Fermi coupling parameter F G is known with the highest precision masses m and M ( Cm L and CM L are Compton length of mass m and M respectively).The gravitoelectromagnetic charges in (3. measuring the value of Hubble's parameter anywhere in the World and taking its inverse value allows us to calculate the absolute Age of the World.The Hubble's parameter is then the most important characteristic of the World, as it defines the Worlds' Age.While in our Model Hubble's parameter H has a clear physical meaning, the gravitational parameter the Newton's law of universal gravitation.The second important characteristic of the World is the gravitomagnetic parameter M µ .Taking its inverse value, we can find the absolute radius of curva- ture of the World in the fourth spatial dimension.We emphasize that the above two parameters ( M Z and M µ ) are principally different physical characteristics of the Medium that are connected through the gravitoelectrodynamic constant c .It means that Time is not a physical dimension and is absolutely different entity than Space.Time is a factor of the World.It follows that Gravity, Space and Time itself can be introduced only for a World filled with Matter consisting of elementary particles which take part in simple interactions at a microscopic level.The collective result of their interactions can be observed at a macroscopic level.Gravity, Space and Time are then emergent phenomena [3].
, et al. found that the mean diffuse intergalactic plasma density is 8].It is more relevant to call ph E the minimum energy of pho- tons which can pass through the Intergalactic plasma.
the energy density of protons in the Medium.The relative energy density of protons p Ω is then the ratio of p cr ρ ρ :

( 4 .
2).According to WUM, the black body spectrum of Microwave Background Radiation (MBR) is due to thermodynamic equilibrium of photons with low density intergalactic plasma consisting of protons and electrons. of electrons in the Medium.We assume that the energy density of MBR MBR ρ equals to twice the value of e ρ : (ages of the World at the moments of emitting and observing).A light-travel time distance to a galaxy LTT d equals to for solar and atmospheric neutri- The sum of the calculated neutrino masses 2 0.053eV c m ν ≅ ∑ (6.10) principal ideas of WUM holds that energy densities of Medium particles are proportional to proton energy density in the World's Medium[2]: 21) which depends on the Fine-structure constant α .We take the value of ν mass 1 P B M and radius 2 F B L is receiving energy from the Me- dium of the World as the result of dineutrinos Bose-Einstein Condensation (see Section 8) at the following rate: of the FIRB radiation is the sum of the contributions of all individual grains.Comparing Equations (7.11) and (4.8), we can find the relation between the grains' temperature and the temperature of the MBR: energy density e Ω in terms of the critical energy densi-

( 4 . 5 )
) and is in the range of neutrinos masses (seeSection 6).The calculated values of mass and concentration of dineutrinos satisfy the conditions for their Bose-Einstein condensation.Consequently, BEC drops whose masses are about Planck mass can be created.The stability of such drops is provided by the detailed equilibrium between the energy absorption from the Medium of the World (provided by dineutrinos as a result of their Bose-Einstein condensation) and re-emission of this energy in FIRB at the stationary tempera-

3 e m and consists of two preons with mass 1
are three prominent hypotheses on nonbaryonic DM, namely Hot Dark Matter (HDM), Warm Dark Matter (WDM), and Cold Dark Matter (CDM).A neutralino with mass N m in 2 100 10000 GeV c ⇔ range is the leading CDM candidate.Light DMP is heavier than WDM and HDM but lighter than neutralinos are DM candidates too.Subsequently, we will refer to the light DMP asWIMPs.Their mass WIMP m good WDM candidate.In our opinion, a tauonic neutrino is a good HDM candidate.In addition to fermions discussed above, we offer another type of DMP-bosons, consisting of two fermions each.There exist two types of DM bosons which we called DIRACs and ELOPs[23].DIRACs are magnetic dipoles with mass 0 m , consisting of two Dirac monopoles with mass about can only exist at nuclear densities or at high temperatures.In our opinion, Dirac monopoles are the smallest building blocks of constituent quarks and hadrons (mesons and baryons).The second boson is the ELOP (named by analogy to an ELectron-nortisOP dipole).ELOP weighs 2

5 )
These values fall into the ranges estimated in literature.The role of those particles in macroobject cores built up from fermionic dark matter will be discussed in Section 10.Our Model holds that the energy densities of all types of DMP are proportional to the proton energy density p ρ in the World's Medium (see(4.6)).In all, there are 5 different types of DMP.Then the total energy density of DMP is which is close to the measured DM energy density: of neutron stars equals to the nuclear density:

( 10 . 1 )
, (10.2),(10.3)and the following values of 1 and minimum radius increase about two orders of magnitude each and the maximum density equals to the nuclear density.Note that parameters of a FCS made up of strongly interacting WIMPs are identical to those of neutron stars.In accordance with the paper by G. Narain, et al.[57], the most attractive feature of the strongly interacting Fermi gas of WIMPs is practically constant value of FCS minimum radius in the large range of masses eight orders of magnitude smaller than max WIMP

4 ≅
with the value of 91 used by G. Narain, et al.[57].Minimum mass and maximum radius take on the following values: three orders of magnitude, and the range of FCS core radii ( 3) and the following values of * 1

M
is more than seventeen orders of magnitude smaller than max NM.It makes strongly interacting neutralinos good candidates for stellar and planetary cores of extrasolar systems with Main-sequence stars[23].When the mass of a FCS made up of neutralinos is much smaller than the maximum mass, the scaling solution yields the following equation for paramemaximum radius take on the following values:

1 U
by the Unified coupling constant: α = .The extrapo- lated energy density of the World was four orders of magnitude smaller than the nuclear energy density[1].The average energy density of the World has since been decreasing in time

3 )
The difference in the strong and the electromagnetic interactions is not in the coupling parameters but in the strength of these interactions depending on the particles involved: electrons with charge e and monopoles with charge 2 e µ α = in electromagnetic and strong interactions respectively.The super weak coupling parameter SW α and the extremely weak coupling parameter EW α proposed in WUM are decreasing as follows:

5 )
According to WUM, the coupling strength of super-weak interaction is 10 ~10 − times weaker than that of weak interaction.The possibility of such ratio of interactions was discussed in the developed theoretical models explaining CP and Strangeness violation[58] [59] [60][61].Super-weak and Extremely-weak interactions provide an important clue to Physics beyond the Model.

1 )
Based on its average value, we can calculate and significantly increase the precision of all Q-dependent parameters.We propose to introduce Q as a new fundamental parameter tracked by CODATA, and use its value in calculation of all Q-dependent parameters.
based on the value of H, but the accuracy of its measurements is very poor.We have obtained the value of Q in (1.2) based on the Equation (2.6), and value of G that is measured with much better accuracy.Then we can calculate the value of Conversely, we can find the value of the Hubble's parameter based on the value of the gravitational parameter.H and G are interchangeable!Knowing value of one, it is possible to calculate the other.According to (2.2) we can find the value of dimensionless parameter Q our Model, the World consists of stable massive elementary particles with lifetimes longer than the age of the World.Protons with mass p m and energy

Table 1 .
Parameter values for FCS made up of various fermions.

Energy Density of Dineutrinos, FIRB and the World Our
Model holds that the energy densities of all types of Dark Matter particles (DMP) are proportional to the proton energy density in the World's Medium.In all, there are 5 different types of DMP (seeSection 9).Then the total energy density of Dark Matter (DM) We took energy density of dineutrinos νν Ω and FIRB FIRB Ω (see Section 8):