Section III

III.THE NEW COSMOLOGY WITH MATTER ADDED

This section will couple the apex and epoch fields to matter, thus creating an interaction LD. I define matter as members of the following group: three generations of quarks and leptons, gauge vector-bosons (gluons, photons, weak bosons), the Higgs quartet, and gravitational fields—the standard model. There is no dark matter, no dark energy, no dark anything in NC. New particles or forces can be included in the future. Gravitational fields will be discussed in section IV.
There will be an important constraint on the coupling—eliminate the cosmological pathologies of general relativity. The cyclic cosmology of section II eliminates the one-sided pathology, but one cannot have a big bang in each cycle—no infinities. Thus, the temperature of cosmic thermal radiation must have a finite limit during each cycle (about \(200\,K\)). In that case, one would expect quarks and gluons to be eternally confined (except possibly in some core-collapse supernovae) and some symmetries to be eternally broken.
NC will be the story of the first generation of composites (proton, neutron), the electron, the electron-neutrino, and the photon. The other constituents will be either virtual, rare, or ephemeral. This section will illustrate the coupling of the apex and epoch fields to photons and (charged and neutral) fermions.
Unless noted, physics is assumed to be described in the rest-frame of the apex field in all that follows. The rest-frame of the cosmic microwave background should coincide with the rest-frame of the apex field.
The epoch field coupling is defined through use of the epoch parameter,\(E({\phi _{ep}})\). The Lagrangian density for matter (LDM) contains many universal constant parameters, e.g., masses and coupling constants. These constants have (known) present-day values (corresponding to the summer epoch). I propose that the NC version of the LDM is generated by substituting NC values of the constants in place of the present-day constants

\[{\rm{NC constant = (1 – K(E)) * present – value + K(E) * winter – value,}}\tag{3.1}\]

where K is a function to be determined.
The winter value of each constant is to be determined, but only a few will differ from the summer value. Specifically, the strong and electromagnetic interactions will be the same in all seasons. The gravitational fields will be slightly different in winter, and the weak interactions will be significantly different in winter.
The Higgs quartet plays a puzzling role in NC. Are they independent (massive) epoch fields or should the epoch field modify the parameters of the quartet? In either case, the quartet would be made stable. I have chosen to evade this question by decoupling the quartet from the apex and epoch fields. Where necessary, the epoch field will over-ride the quartet to make masses or coupling constants have seasonal values.

In what follows here, recall that the NC labels scalar, vector, etc., are mushy. The apex field is a “tensor”, so a coupling tensor can be generated. I propose the following two connector “tensors”,\({\Delta _{\mu \nu }}\) and \({\sigma _{\mu \nu }}\). \[ \begin{array}{l} \Delta_{\mu \nu} \equiv \mathcal{P} g_{\mu \nu} + v_\mu^{\ \nu}(apex) W, \\ v_j^{\ k}(apex) = A \delta_j^{\ k} \ \text{(only non-vanishing elements)}, \\ \mathcal{P} = [J(A,E(\phi_{ep}))]^{-1/2}, \\ W = (\mathcal{P} – 1)/A, \\ \sigma^{\mu \lambda} \Delta_{\lambda \nu} = \delta_\nu^{\ \mu}, \end{array}\tag{3.2} \] where \(\delta _\nu ^\mu = 1{\rm{ if }}\mu {\rm{ = }}\nu {\rm{ else = 0}}\). \(J(A,E)\) is an arbitrary real function of \(A\) and E, but there are some constraints. \(J( – A,E) = J(A,E)\) \(J(0,0) = 1\) and \(J(A,E) > 0\) for all \(A\) and E attainable during each cycle. As a result, \(W\) is well behaved at mid-summer when \(A = 0\). J will be defined in subsection F below. Eq. (3.2) insures that \(\Delta _0^0 = \mathcal{P}, \quad \Delta _k^k = 1 = \sigma _k^k, \quad \sigma _0^0 = \mathcal{P}^{-1}\) , and the constraints will eliminate any pathology. \(\mathcal{P}\) is a scalar since E and A are scalars, and it is a function of fields without any explicit dependence on \({x^0}\). Thus, \({\Delta _{\mu \nu }}\) and \({\sigma _{\mu \nu }}\) can be inserted (anywhere and in any manner) into the LDM.
The apex field can be thought of as a master clock. One can create an LDM for a field of interest. Using Eq. (2.1), the equation of evolution for the field results, but the evolution is relative to \({x^0}\) (the apex clock). There is another clock. We measure evolution (physical time) by using devices (clocks) that operate according to physical laws of matter. There must be a relationship (transformation) between physical time (\(\tau \)) and \({x^0}\)—subject to important constraints. The transformed physical time equations must have the same speed limit for all constituents (and for all \({x^0}\))—the universal constant \(c = 2.99799x{10^8}{\rm{ M}}{{\rm{S}}^{ – 1}}\). Furthermore, the transformed equation must be the same familiar version except for corrections. The corrections must be so tiny that they could only be measured by an experiment that ran for a (extremely) long physical time.
It is useful to introduce a notation that distinguishes between apex and physical clocks: \[ \begin{array}{l} \partial’_0 = \partial / \partial x^0 \; ; \; \partial’_k = \partial / \partial x^k, \\ \partial_0 = \partial / \partial c\tau \; ; \; \partial_k = \partial / \partial x^k . \end{array} \tag{3.3} \] The procedure is clear enough. The first step is to form the “bare” primed LDM—just the familiar LDM with primes for the fields and their derivatives (and Dirac matrixes). Next, form a “dressed” primed LDM by inserting delta, sigma, and theta tensors (shown below) to form an interaction with the apex field. Next, obtain the primed equation of motion from the Euler-Lagrange equation (by variation of the primed field of interest). Finally, use a set of transformations to the unprimed equation of motion using the delta, sigma, and theta tensors.
The method of dressing and transforming may seem arbitrary. However, the desired result is that the unprimed equation of motion will look exactly like the familiar equation except for (possibly) a tiny correction and \(c\tau \) in place of \({x^0}\). I have chosen to use \(\tau \) in place of the customary t as a reminder. What we think of time has a hidden structure.
The ratio of the physical clock evolution to the apex clock evolution is given as a power series in \(c\tau {L_{ax}}^{ – 1}\), \[ d(c\tau)/dx^0 \equiv f = \mathcal{P} – (\partial’_0 \ln \mathcal{P}) \, c\tau + \cdots . \tag{3.4} \] The leading terms in Eq. (3.4) are a first order differential equation for \(\tau ({x^0})\), and there will be an arbitrary constant of integration that can be chosen so that \(\tau = 0\) corresponds to now for an observer. The solution of Eq. (3.4), \[ c\tau(x^0) = \mathcal{P}(x^0)^{-1} \int_0^{x^0} d{x^0}’ \, \mathcal{P}({x^0}’)^2, \tag{3.5} \] fixes now as corresponding to \({x^0} = 0 = \tau \). Thus, the present-day value of the apex field \({A_{now}} = A({x^0} = 0)\). \({A_{now}}\) will be determined in Appendix C, but it is not a universal constant. In what follows, the tiny corrections will be of order \({L_{ax}}^{ – 1}\) and smaller corrections will be ignored. The final transformation of derivatives follows: \[ \begin{array}{l} {\partial _0}^\prime = f{\partial _0} \quad ; \quad {\partial _k}^\prime = {\partial _k}, \\ {\partial _\mu }^\prime = \Sigma _\mu ^\lambda {\partial _\lambda }, \\ \Theta _\mu ^\lambda \Sigma _\lambda ^\nu = \delta _\mu ^\nu , \end{array}\tag{3.6} \] where \(\Sigma _0^0 = f\), \(\Sigma _k^k = 1 = \Theta _k^k\), and \(\Theta _0^0 = {f^{ – 1}}\). Note that \({\partial _0}\Theta _0^0 = 0\) (order \({L_{ax}}^{ – 2}\)). \(\Sigma \) is not quite a Lorenz invariant tensor because the speed of light in the apex clock system is not a constant. The transformation will make light-speed seem constant but possibly with tiny deviations. A (theoretically perfect) Michelson interferometer (with unequal-length arms) will not have a null result, but over a long time it will measure Hubble’s parameter. In NC, Lorenz invariance is slightly broken. It is possible to add higher power terms in Eq. (3.4) to marginalize any mischief generated by multiples of the leading term—no evaporating fermions.

The units used for electromagnetism (EM) are MKSQ, where electric charge (Q) is measured in coulombs. The four-potential (\({A_\mu }\)) has units \(KM{S^{ – 1}}{Q^{ – 1}}\), and the vacuum-permittivity has units \({Q^2}{S^2}{K^{ – 1}}{M^{ – 3}}\) ( \({\varepsilon _0} = 8.854x{10^{ – 12}}\)). The dressed primed LD for a fermion that could have electrical charge is chosen as follows: \[ {\mathcal{N}_F}^{-1} \mathcal{L}_F = \tfrac{i}{2} \left( \bar{\psi}’ \, \Gamma^\mu \partial_\mu’ \psi’ – \mathrm{c.c.} \right) – \kappa A_\mu’ \, \bar{\psi}’ \, \Gamma^\mu \psi’ – \mathcal{L}_F^{-1} \bar{\psi}’ \, \psi’, \tag{3.7} \] where \(\kappa = {\rm{q/}}\hbar \), q is the electric charge of the fermion, \({L_F}^{ – 1} = {m_F}c/\hbar \), and \(\mathcal{N}_F\) is a constant normalization factor. The transformations are governed by minimal EM coupling as follows: \[ \begin{array}{l} \psi’ = \psi, \\ A_\mu’ = \Sigma_\mu^{\ \lambda} A_\lambda . \end{array} \tag{3.8} \] The usual (constant) Dirac matrices are used as follows: \[ \begin{aligned} \{ \gamma^\mu , \gamma^\nu \} &= 2 g^{\mu \nu}, \\ \bar \psi &= \psi^\dagger \gamma^0, \\ \Gamma^\mu &= \Theta_\lambda^\mu \gamma^\lambda. \end{aligned} \tag{3.9}\] Next, the primed equation of motion follows from a variation (\(\delta \bar \psi ‘\)), \[ 0 = i \Gamma^\mu \partial_\mu’ \psi’ + \tfrac{i}{2} (\partial_\mu’ \Gamma^\mu) \psi’ – \kappa A_\mu’ \, \Gamma^\mu \psi’ – L^{-1} \psi’. \tag{3.10} \] The unprimed equation results by using Eq. (3.2) to (3.10) and then taking \(\tau \to 0\) to obtain the now equation, \[0 = i{\gamma ^\mu }{\partial _\mu }\psi – \kappa {A_\mu }{\gamma ^\mu }\psi – {L^{ – 1}}\psi ,\tag{3.11}\] which is the familiar Dirac equation without explicit reference to the apex field. There is a conserved current (\({\partial _\mu }\bar \psi {\gamma ^\mu }\psi = 0\)), and a source term for primed EM, \[ \partial \mathcal{L}_F / \partial A_\mu’ = – \kappa \mathcal{N}_F \Theta_\lambda^\mu (\bar \psi \, \gamma^\lambda \psi). \tag{3.12} \] Eq. (3.12) will be useful in the following subsection.

The dressed primed LD for EM is \[ \begin{aligned} \mathcal{N}_{EM}^{-1} \mathcal{L}_{EM} &= \sigma_\varepsilon^\alpha \sigma_\delta^\beta {F_{\alpha \beta}}^\prime \, \sigma_\gamma^\varepsilon \sigma_\lambda^\delta {F^{\gamma \lambda}}^\prime, \\ F_{\mu \nu}^\prime &= {\partial_\mu}^\prime {A_\nu}^\prime – {\partial_\nu}^\prime {A_\mu}^\prime \end{aligned} \tag{3.13} \] where \( \mathcal{N}_{EM} = -\frac{\varepsilon_0 c}{4} \) . The primed Euler-Lagrange equation for photons and charged fermions has the following form, \[ 0 = (\partial \mathcal{L}_F / \partial A_\nu’) – \partial_\mu’ \left[ \partial \mathcal{L}_{EM} / \partial (\partial_\mu’ A_\nu’) \right]. \tag{3.14} \] Next, obtain the primed equation from Eq. (3.14): \[ {\partial_\mu}^\prime \big[ \sigma_\varepsilon^\mu \sigma_\delta^\nu \sigma_\gamma^\varepsilon \sigma_\lambda^\delta {F^{\gamma \lambda}}^\prime \big] = \kappa \mathcal{N}_{FEM} \, \Theta_\lambda^\nu \, \bar\psi \gamma^\lambda \psi \tag{3.15}\] where \( \mathcal{N}_{FEM} = -\frac{1}{4} \mathcal{N}_F \mathcal{N}_{EM} \) Since \({\partial _k} \mathcal{P} = 0\), only time-like derivatives of \(\Delta ,{\rm{ }}\sigma ,{\rm{ }}\Sigma {\rm{, and }}\Theta \)will survive. Note that \(\Delta ,{\rm{ }}\sigma ,{\rm{ }}\Sigma {\rm{, and }}\Theta \) are diagonal. Convert Eq. (3.15) to the unprimed version using the transformations above, \[\begin{array}{l}{F_{\mu \nu }}^\prime = \Sigma _\mu ^\alpha \Sigma _\nu ^\beta {F_{\alpha \beta }},\\{F_{\mu \nu }} = {\partial _\mu }{A_\nu } – {\partial _\nu }{A_\mu }.\end{array}\tag{3.16}\] \(\) Eq. (3.15) can be rewritten, \[\Sigma _\kappa ^\nu \Sigma _\mu ^\chi {\partial _\chi }[\sigma _\varepsilon ^\mu \sigma _\delta ^\kappa \sigma _\gamma ^\varepsilon \sigma _\lambda ^\delta \Sigma _\alpha ^\gamma \Sigma _\beta ^\lambda {F^{\alpha \beta }}] = {\mu _0}{J^\nu },\tag{3.17}\] where \({\mu _0}{J^\nu } = \kappa \mathcal{N}_{FEM}\,\bar \psi \,{\gamma ^\nu }\,\psi\) and \({\partial _\mu }{J^\mu } = 0\). Eq. (3.17) can be rewritten as well, \[A_{\alpha \beta }^{\chi \nu }{\partial _\chi }{F^{\alpha \beta }} + B_{\alpha \beta }^\nu {F^{\alpha \beta }} = {\mu _0}{J^\nu }.\tag{3.18}\] As \(\tau \to 0\), \(\Sigma_\mu^\nu \to \Delta_\mu^\nu\), and \(\partial_0 \Sigma_0^0 \to 0\). Thus, \(A_{\alpha \beta}^{\chi \nu} \to \delta_\alpha^\chi \delta_\beta^\nu\), \(B_{\alpha \beta}^\nu \to -2[\partial_0 \ln(\mathcal{P})] \delta_\alpha^0 \delta_\beta^\nu\), and the now equation results, \[\partial_\mu F^{\mu \nu} – 2[\partial_0 \ln(\mathcal{P})] F^{0\nu} = \mu_0 J^\nu. \tag{3.19}\] Unlike the fermion equation, Eq. (3.11), the EM Eq. (3.19) has an explicit reference to the apex field and \({x^0}\). However, for the electric field of a point charge (a stationary charged fermion) the \(\partial_0 \ln(\mathcal{P})\) term of Eq. (3.19) vanishes identically, and \({\partial _\mu }{F^{\mu 0}} = {\mu _0}{J^0}\). Thus, the energy level of a stationary state of the charged fermion equation will be independent of \({x^0}\)—a universal constant. This fact will be important in what follows.
The relative size of the \(\partial_0 \ln(\mathcal{P})\) term of Eq. (3.19) for typical radiation calculations will be of order \(\lambda /{L_{ax}}\), where \(\lambda \) is the wavelength of the radiation. For example, if \(\lambda = 1\) meter (low temperature thermal radiation), the \(\partial_0 \ln(\mathcal{P})\) term would be of order \({10^{ – 26}}\)—safely ignored. The only case where the \(\partial_0 \ln(\mathcal{P})\) term can be significant is when free photons travel for eons from source to observer. Thermal radiation is created at the same time in the current cosmology (GRLCDM). In NC, thermal photons and some portions of background x-rays and gamma rays are relic in nature. These relic photons could have traveled for hundreds of billion years—from the preceding winter or summer.

An observer today of distant objects will observe shifted spectrums. That shift has two components—a cosmological shift toward red and a doppler shift. The doppler component is due to the velocity of the source relative to the observer, and it is generally unknown and neglected. This subsection focuses only on the cosmological component.
Consider a (polarized and monochromatic) electromagnetic wave-train moving through empty space in the \({x^1}\) direction in the first half of red summer. The vector potential will have one component, \({A_2}({x^1},\tau )\), and Eq. (3.19) will take the following form, \[0 = \partial_0^2 A_2 – \partial_1^2 A_2 – 2(\partial_0 A_2)[\partial_0 \ln(\mathcal{P})]. \tag{3.20}\] Assume that the wave-train has a length of N+1 crests, and that the vector potential has a form (accurate near the center of the train), ${A_2} = \cos (2\pi {\lambda ^{ – 1}}\xi )$, and $\xi = {x^1} – c\tau $. The center of the train corresponds to $\xi = 0$. In the limit $\xi \to 0$, Eq. (3.20) requires the following constraint, \[\partial_0 \ln(\lambda) = – \partial_0 \ln(\mathcal{P}). \tag{3.21}\] The train was created at a certain time in the past (then) and observed in the observer’s now. Eq. (3.21) causes a connection between then and now. \[\lambda_{now}/\lambda_{then} = \mathcal{P}_{then}/\mathcal{P}_{now}, \tag{3.22}\] where \({\lambda _{now}}\) is the \(\lambda \) of \({A_2}\) in Eq. (3.20). During the first half of red summer \({\partial _0}\mathcal{P} < 0\), and ${\lambda _{now}} > {\lambda _{then}}$. Now for an observer can be at any value of \({x^0}\) (or \(\mathcal{P}\)), and one can define a red shift parameter (z) for a distant source in the usual fashion, \[1 + z = {\lambda _{observer}}/{\lambda _{source}}{\rm{ }}{\rm{.}}\tag{3.23}\] The red shift can be connected directly with the apex field, \[ 1 + z = {\mathcal{P}_{source}}/{\mathcal{P}_{observer}}.\tag{3.24} \] Consider two observers (both in the rest-frame of the apex field) widely separated in space but observing the same wave-train from the same source. Each observer can measure the wavelength at the center of the train. The velocity of the center of the train can be established by measuring the physical time lag (\(\Delta \tau \)) for the center to travel a known distance (\(\Delta {x^1}\)). The velocity of the center is independent of the wavelength, so \(\Delta {x^1}/\Delta \tau = c\) for both observers. If observer1 is closer to the source than observer 2 then \({z_1} < {z_2}\). If a monochromatic wave-train is N+1 crests long, then the length (and volume) of the train will be proportional to ${\mathcal{P}^{ - 1}}$. The electromagnetic energy density is proportional to ${\mathcal{P}^2}$, so the total energy of the photons within the volume is proportional to ${\mathcal{P}}$. The total number of photons within the volume is constant since photons cannot be created or destroyed in flight through a vacuum. Therefore, the energy of a single photon in flight is determined, \[ {E_\gamma } = {E_{source}}\mathcal{P}/{\mathcal{P}_{source}}{\rm{ }}.\tag{3.25} \] The free photon will always be observed to move with velocity c (the universal constant), and it can directly gain energy from or lose energy to the apex field in flight.
An observer today looking into the past at a distant object will observe a red shifted spectrum—i.e., the observer is in the red branch. An observer in early blue summer looking into the past will see blue shifted spectrums. Blue and red people will use the same physics to explain experiments—the only tangible difference will be cosmological in nature. During the last half of red summer an observer looking into the past will see blue shifted spectrums for nearer objects and red shifts for more distant objects. The corresponding blue situation will be the opposite.

What is the relationship between the apex clock and the physical clock? The evolution of cosmology is governed by the apex clock while our lives march by the physical clock. Eq. (3.4) gives the appropriate relation, \[ d(c\tau)/dx^0 = \mathcal{P}, \tag{3.26} \] where \(\tau \to 0\) (the physical clock is being continuously reset). Since observers are interested in the evolution of distant physical systems, it is necessary to use physical time to describe the evolution of the universe. In NCNG the universe is not expanding, so the distance to an observed object will always be \(c\Delta \tau \). The duration of physical time between two values of the apex field is \[\int_{{\tau _1}}^{{\tau _2}} {d\tau = } \int_{{A_1}}^{{A_2}} {dA} (dc\tau /d{x^0}){(dA/d{x^0})^{ – 1}}{c^{ – 1}}.\tag{3.27}\] As an example, the number of years between now and the middle of the current summer, \[\Delta \tau = {L_{ax}}{c^{ – 1}}\int_0^{{A_{now}}} {dA} {[J(A,0)]^{ – 1/2}}{[1 – {G_{ax}}(A)]^{ – 1/2}},\tag{3.28}\] can be evaluated once the universal constants in Eq. (3.28) have been established. That task will be relegated to Appendix C.

I have chosen the following form for the J function, \[\begin{array}{l}J(A,E) = N(A)(1 – E) + M(A){E^2},\\N(A) = 1 – A\,{\rm{tanh(}}{\Gamma _{cl}}A),\\M(A) = (1 – {\kappa _{cl}})\exp [{\beta _{cl}}(1 – {A^2})],\end{array}\tag{3.29}\] where \(1 > > {\kappa _{cl}} > 0\), \({\beta _{cl}} > > 1\), and \({\Gamma _{cl}} > > 1\) are universal clock constants. This arcane form for J results from various requirements for a consistent cosmology.
Eq. (3.29) satisfies the criteria of Eq. (3.2), and it is well-behaved over the whole cycle and continuous except for tiny slices during bubble eras. J has the simplest form \(J = 1 – \left| A \right|\) during summer (E=0) except for tiny regions near \({A^2} = 1\) and \(A = 0\).
For reasons that will become clear in the section about reverse nucleosynthesis, it is necessary that each winter season needs to last about \(33{L_{ax}}{c^{ – 1}}\) years. That constraint requires the following relation, \[33 \sim 2\int_1^{{A_W}} {dA} {[J(A,1)]^{ – 1/2}}{[1 + \alpha _W^{ax} – {G_{ax}}(A)]^{ – 1/2}},\tag{3.30}\] where \({G_{ax}}({A_W}) = 1 + \alpha _W^{ax}\). The integrand of Eq. (3.30) has two factors. The G factor is determined by observations that have nothing to do with winter. If the J factor is of order unity, the integral will be too small. The only way to satisfy Eq. (3.30) is if J can be very small in winter—hence the exp beta term of J.
In red spring after the bubble era, there is a maximum value of z. The size of \({z_{\max }}\) is influenced by the M term of Eq. (3.29). Using \({{\mathop{\rm E}\nolimits} ^2}\) instead of E and the kappa factor keeps \({z_{\max }}\) in the proper range—larger than the highest observed z, but not too big (\({z_{\max }} \sim 15\)).
I have gone into extra detail here to illustrate a recurring theme in NCNG. There are new universal constants and new functions that seem as if they could have any form or value, but consistency and observability will lead one to the best choice.
Every aspect of NCNG can be verified by observations in principle, but there is a proviso—if one has a big enough telescope with low enough noise. Unlike GRLCDM, NCNG has no dark side since z is always in a cool range. Furthermore, the cool range of z eliminates the first pathology of GR—infinite z at the origin of time.

Eq. (3.26) shows that a bubble wall moves at light speed only when \(J = 1\), e.g., when \(A = 0\). In general, \(\left| v_{wall} \right| / \left| v_{photon} \right| = \mathcal{P}\). On the summer side of the wall, Eq. (3.29) will generate \(\mathcal{P} \gg 1\), and on the winter side \(\mathcal{P} > 1\). In (red or blue) spring, inside is summer and a photon inside will move much slower than the receding wall that surrounds it. Such a photon will never cross back into winter during the bubble era. On the winter side, photons are moving slightly slower than the wall, and the wall can overtake photons. Thus, any spring photon will only cross a bubble wall once.
The situation in (red or blue) fall is similar. Inside is winter and a photon inside will move slightly slower than the receding wall that surrounds it, and the photons on the summer side can be overtaken by the much faster moving wall. There are no observers in winter, so this case is moot. Any photon (existing before the spring bubble era) that arrives at a summer observer’s telescope will have crossed only one bubble wall.
What happens when a photon crosses a bubble wall? That question depends on the forces that can interact with a photon traveling in a vacuum—there are two. If the bubble wall is moving through a gravitational field, there could be a discontinuity across the wall. If so, the photon could be deflected crossing the wall. That possibility will be eliminated in section VII(B). The second “force” is the interaction of a photon with the apex field via Eq. (3.19). In that case, there would be an insignificant change in the cosmological red shift of the photon across the wall. Thus, each relic photon making the necessary single wall crossing will not be affected. The bubble era is transparent for relic photons.

Now that \(P\) has been given form, The Hubble parameter (\({H_0}\)) can be formulated in red summer. The definition of \({H_0}\) is \[{H_0} = – {(dz/d\tau )_{now}},\tag{3.31}\] where now corresponds to \(E = 0\), and \(J = 1 – A\). Eq. (3.24) gives z as a function of \(A\), \[z = – 1 + {[(1 – {A_0})/(1 – A)]^{1/2}},\tag{3.32}\] where \({A_0}\) is the present-day value of \(A\). Eq. (3.31) can be rewritten, \[{H_0} = – c{[(dz/dA)(dA/d{x^0}){(dc\tau /d{x^0})^{ – 1}}]_{A = {A_0}}}.\tag{3.33}\] Eq. (3.26) and Eq. (2.24) give the final form, \[{H_0} = 0.5c{L_{ax}}^{ – 1}{(1 – {A_0})^{ – 1/2}}{[1 – {G_{ax}}({A_0})]^{1/2}}.\tag{3.34}\] The most accurate value (in obsolete units) is \({H_0} = 73.8{\rm{ (km/sec) per mega – parsec}}\) from [1] and [2]. A more useful version is \({H_0} = c{L_H}^{ – 1}\), where \({L_H} = {\rm{13}}{\rm{.26}}x{10^9}\)LY. An important relationship results, \[{L_{ax}} = 0.5{L_H}{(1 – {A_0})^{ – 1/2}}{[1 – {G_{ax}}({A_0})]^{1/2}}.\tag{3.35}\] Since \({L_{ax}}\) is a universal constant (see Appendix Z) and \({G_{ax}}\) is formed from universal constants, it follows that \({L_H} \Rightarrow {A_0}\). This is important because all cosmological red shift information springs from a knowledge of \({A_0}\).

Cosmic microwave background-radiation (CMB) has been observed and identified as black body thermal radiation. CMB accurately matches the Planck distribution, \(B(\nu ,T)\), giving a present-day temperature of \(2.7250K\). The physical expansion of the universe in GR allows the use of thermodynamic relations to establish how the characteristic temperature of the radiation (T) changes as the universe expands—an avenue not available in NC. In NC the thermal photons in flight can lose or gain energy via Eq. (3.25). Each photon will have the following change between two physical times, \[ \nu / \nu’ = \mathcal{P}(\tau) / \mathcal{P}(\tau’) \equiv \xi , \tag{3.36} \] where \(\xi \) can be treated as a constant in what follows. The Planck distribution has the following constraints, \[ \begin{array}{l} B(\nu ,T) = 2h\nu^3 c^{-2} \,[\exp(h\nu / kT) – 1]^{-1}, \\ \int_0^\infty d\nu \, B(\nu ,T) = \pi^{-1} \sigma T^4, \\ \int_0^\infty d\nu’ \, B(\nu’,T’) = \pi^{-1} \sigma (T’)^4. \end{array} \tag{3.37} \] If one substitutes Eq. (3.36) into the first of the integrals of Eq, (3.37), one obtains the following, \[ \begin{array}{l} T’ = T \, \xi^{-1}, \\ T(\tau’) / T(\tau) = \mathcal{P}(\tau’) / \mathcal{P}(\tau), \end{array}\tag{3.38} \] where \(\mathcal{P}(\tau)\) is established using the techniques of subsections 3(E) and 3(F) above.