_{1}

^{*}

Recent discussions attributed fermion mass to an analogue of Weyl curvature which occurred by hypothesis when closed, spin
strings swept out closed world tubes. A new degree of freedom and corresponding curvature class were attributed to “second order tubes” that were swept out by initially introduced closed tubes, etc. Curvature classes were associated by hypothesis with composite masses
where d
enoted a mass-less spin
field and where a
and
respectively denoted an LH quark and an RH anti-lepton that were characterized by opposite I
_{3} values and shared a common generation. The resulting model accounted for known quark masses and predicted a new quark of mass 30 GeV/c
^{2}. The composite masses
form a symmetry, the preservation of which is equivalent to the conservation of electrical charge and string scale. SUGRA interactions that preserve the proposed symmetry can therefore be precisely defined. In this context, gauge transformations that establish the proposed curvature classes also associate with a second realization of the originally generated symmetry, the preservation of which is equivalent to the conservation of string length and of the curvature
from which the postulated model generates admissible increments of large scale expansion. The latter symmetry is associated by hypothesis with the large scale structure of the observable universe, thereby motivating a theoretical approximation of the total number of galaxies. This result parallels the approximation that is indicated by observation.

Previous papers have reviewed the Weyl geometry in which the parallel displacement of a vector around a closed curve admits an increment of vector magnitude. It was observed that the Weyl model associated increments of vector magnitude l with curvature states

[

where S represented the length of the circulating string. This analogue of Weyl curvature was called W-curva- ture and was designated “W.” By hypothesis the W-curvature state (3) was regarded as an admissible physical state if and only if it satisfied a principle of gauge invariance

The value n = 1 was regarded as describing the closed nature of the first world tube and n = 2 was regarded as enforcing the closed nature of the postulated “second order” closed tube that was swept out by the circulation of the first order tube (that corresponding to n = 1). Additional oscillations of the closed string and closed world tube were not described.

Finally a generalization

of (5) resulted in a world tube of nth order. The compounded world tube (7) was regarded as having been established by the circulation of an (n-1)^{th} order tube as this tube swept out a closed tube. Again the curvature states (7) were regarded as physical states if and only if they satisfied a principle of gauge invariance

The value n = 1 enforced the closed nature of the first order world tube, n = 2 enforced the closed nature of the postulated “second order” closed tube which was swept out by the circulation of the first order tube, etc. The value n = 6 enforced the closed nature of the n^{th} order world tube. Again, additional oscillations of the closed string, of the closed world tube etc. were not described.

Each of the W-curvature states (7) that was established by a principle of gauge invariance (8) was associated, by hypothesis, with a stable or stationary, spin-2h state

where _{3} value that was opposite that of _{3})-neutral) [

The proposed model was calibrated by associating the gauge invariant W-curvature state

[

Specifically, an expression that is algebraically equivalent to expression (10) was obtained by dividing both sides of (10) by “6.” The result was

Interpretation of this unfamiliar mass was deferred until after the massive states described by expressions (12), (13), (14) and (15) had been interpreted. Continuing, both sides of (11) were divided by “5” to produce

The theoretical mass represented by expression (12) motivated the association of (12) with the I_{3} values and generation that correspond to _{4} is that associated with the spin-2h composite^{2} and the observed mass of the anti-tauon ^{2} [_{3} = −1/2 partners in a heavy fermion generation.

To observe a fourth state both sides of (12) were divided by “4” to produce

The theoretical mass of expression (13) motivated the association of (13) with the I_{3} values and generation that correspond to ^{2} [^{2}). The charmed quark and the muon’s neutrino were regarded as I_{3} = +1/2 partners in a moderately heavy generation.

To introduce a fifth state, both sides of (13) were divided by “3” to produce

The mass of expression (14) motivated the association of (14) with the I_{3} values and generation that correspond to the pair consisting of ^{2} [_{3} = 0 partners in (what is in the proposed model regarded as) a light generation.

Finally, to observe a sixth state, both sides of (14) were divided by “2” to produce

The mass of expression (15) motivated the association of (15) with the I_{3} values and generation that correspond to that of the spin-2h state_{3} value of _{3} value that is obtained when the I_{3} value of the up quark and that of the down quark are averaged; and the I_{3} value of _{3} value that is obtained when the I_{3} value of the anti-electron and that of the anti-electron’s neutrino are averaged. Thus the spin-2h state _{3} value of zero and with the light generation. The observed mass of ^{2} [

To interpret the mass of the composite that is described by expression (11), it was first observed that the left-handed muon _{L} was interpreted as an unobserved, left-handed quark that is characterized by I_{3} = −1/2 and 7_{L} was regarded as a member of a moderately heavy generation. Finally, since the mass of ^{2}.

To further elaborate upon the relevant aspects of string theory, it is necessary to review the theory of super-gravitation.

To discuss super-gravitational interactions it is necessary to review the theory of pure super-gravity. Osp(1/4)- pure super-gravity on M_{4}

L

is regarded as dual to the string background AdS_{7}XS^{4} [

where

and transform under Osp(1/4) as

The covariant derivative is

and the curvature tensor is derived from

[

String theory as proposed in Section 1 produces the six spin-2h curvature states _{3} values are those that characterize the left handed fermions in the states_{3} conjugate classes

Since string theory as proposed in Section 1 has now incorporated position-independent SUGRA connections (e.g. the connections in the

where

are identified by hypothesis as admissible physical states if and only if they are established by gauge invariance:

A large scale model is founded upon the postulated invariance of

If established boundary conditions are adopted, the geometry of the proposed model motivates a theoretical approximation of the total number of galaxies. Let us now consider this model.

The proposed model is calibrated in terms of a large-scale boundary condition that is based upon observation. Specifically if a constant of proportionality

then the phase transitions that are described by the expression (23) become

The selection of base 10 is founded upon the observed boundary condition that galaxies are typically separated by a distance that is about ten times the diameter of the typical galaxy; that galactic clusters are typically separated by a distance that is about ten times the diameter of the typical galactic cluster etc.

The typical galactic cluster, which corresponds by hypothesis to n = 2, will be modeled in terms of local but globally typical boundary conditions. Observations of such conditions establish that the typical galaxy is about 10^{5} light years (ly) in diameter, and that five galaxies populate the typical basic cluster. The local cluster is approximated as a volume that is enclosed by an abstract sphere and that is measured in terms of a number of galaxies; i.e. as a volume that is associated with the equation

where R_{2} is the average radius of a typical local cluster in terms of a number of galaxies. Rounding to three digits, it is concluded that, typically,_{2} = 1.06, the Equation (26), and the boundary condition of homogeneity that is established by observation, and rounding to three digits, one determines the approximate radius R_{3}, in terms of a number of galaxies, of a typical n = 3 state: R_{3} = (2(10)1.06 = 21.2). Utilizing R_{3} = 21.2 and rounding to three digits one determines the approximate value R_{4} in terms of a number of galaxies of a typical n = 4 state: R_{4} = (2(10)21.2 = 424). Utilizing the value R_{4} = 424 and rounding to three digits, one determines the radial value R_{5} of a typical n = 5 state: R_{5} = (2(10)424 = 8480). Finally, utilizing the value R_{5} = 8480, and rounding to three digits, one determines the value R_{6} of a typical n = 6 state: R_{6} = (2(10)8480 = 169,600); or (rounding to three digits) 170,000.

It is assumed that each radius is established by a class of phase transitions which produces a space-like blister that is initially vacuous in its interior and is thinly bounded by a mass energy distribution. By hypothesis each blister is ultimately converted into a 3-dimensional distribution (as the spherical mass-energy distribution is inwardly dispersed). By hypothesis, each filled-in blister becomes an element of a larger spherical distribution (initially vacuous interior) that is generated by the next class of phase transitions. Near the event horizon spherical distributions of mass-energy are, by hypothesis, not yet inwardly dispersed. It is therefore assumed that the final radius that is calculated above is the radius of a spherical distribution that involves the entire galactic hierarchy. The area of this spherical distribution is now calculated in terms of a number of galaxies.

[

Previous discussions that attribute mass to a string analogue of Weyl curvature are reviewed and adopted as a foundation for proposed large scale considerations. A string analogue of Weyl curvature is again described as emerging when closed spin-2h strings sweep out closed world tubes. A second degree of freedom and corresponding class of world tubes are described as occurring when world tubes themselves sweep out closed tubes etc. The proposed hypothesis is described as paralleling Wheeler’s ideal, which attributes mass to space-time curvature [^{2}.

Postulated curvature classes are associated, by hypothesis, with composite masses

notes a mass-less spin-2h field and where

ton that are characterized by opposite I_{3} values and share a common generation. The composite masses

form a symmetry, the preservation of which is equivalent to the conservation of electrical charge and of string scale. SUGRA interactions that preserve the relevant symmetry can therefore be precisely defined. In this context, moreover, gauge transformations that establish the curvature classes associate, by hypothesis, with a second realization of the originally generated symmetry, the preservation of which, by hypothesis, is equivalent to the

conservation of string length and the conservation of the curvature

model generates admissible increments of large scale expansion. The latter symmetry is associated by hypothesis with the large scale structure of the observable universe, thereby motivating a theoretical approximation of the total number of galaxies. This result parallels the approximation that is indicated by observation.

Figures are designed by R. James Towe.

JosephTowe, (2015) Outline of a String Cosmology. Journal of Modern Physics,06,1856-1863. doi: 10.4236/jmp.2015.613190