APPENDIX C: MODEL HISTORIES OF THE APEX FIELD AND PHYSICAL TIME
A model history is a tabulation of three variables of interest versus the Apex field over a (red branch) range of interest using the universal constants of Appendix Z. The variables are z (cosmological red shift relative to now), the epoch parameter E, and the elapsed physical time relative to an edge of the range. Appendix Z is a work in progress. New observations and deeper research will require periodic updates of the model and thus change these histories.
The equations that govern the variables are the following set. \(E(A)\) is given by Eq. (2.12) through Eq. (2.17). The relation of physical time and the Apex field is as follows:
\[
\begin{array}{l}
G_{ax}(A) = d_1^{ax}A^2 + d_2^{ax}A^4 + d_3^{ax}A^6,\\[6pt]
\dfrac{dA}{cd\tau} = -L_{ax}^{-1}\,\mathcal{F}(A)\,J\!\left(A,E(\phi_{ep}(A))\right)^{1/2},\\[6pt]
\mathcal{F}(A) = \left[\,1 + \xi(A) – G_{ax}(A)\,\right]^{1/2},\\[6pt]
\xi(A) = \dfrac{\alpha_W^{ax}}{2}\left\{ 1 + \tanh\!\left[2A_{ep}\Gamma_{ep}(A – A_{ep})\right]\right\},
\end{array}
\tag{C.1}
\]
where J is given by Eq. (3.29). The cosmologic red shift is zero at earth (now), and z is defined as follows:
\[\begin{array}{l}z = J{(A,E({\phi _{ep}}(A)))^{ – 1/2}}J{({A_0},E({\phi _{ep}}({A_0})))^{1/2}} – 1,\\{L_{ax}} = 0.5{L_H}{(1 – {A_0})^{ – 1/2}}{[1 – {G_{ax}}({A_0})]^{1/2}},\\{L_H} = {\rm{13}}{\rm{.26}}x{10^9}{\rm{ light – years,}}\end{array}\tag{C.2}\]
where \({A_0}\) is the value of A today (not a universal constant). Use the lower members of Eq. (C.2) to calculate \({A_0}\), then use the upper member to obtain z.
Each history requires (the same) knowledge of some of the universal constants of Appendix Z—five apex constants, five epoch constants, three clock constants, and \({x_{Bera}}\)—a total of fourteen. If there was a set of precise standard-candles (for a wide range of z), the fourteen constants could be determined from observations, but that is not the case. However, there are consistency constraints to be imposed. Furthermore, the effect of a variation of each constant on the history can be quantified.
Several constraints are obvious. All histories will have a maximum value of z—necessarily larger than the most distant observations to date. All histories must have a (blue to red) winter lasting at least 300 billion years—in accordance with the reverse nucleosynthesis model. There are also some constraints that control what can be seen for z near the maximum.
The maximum value of z is largely dependent on \({\beta _{cl}}\). I have chosen the values of Appendix Z so that \({z_{\max }} \sim 15\) (a spec for the JWST). The constants of this document yield \({z_{\max }} = 14.9213\). The most recent (oldest) observation for this edition is galaxy JADES-GS-z13-0 [18]. That value is \({z_{\max }} = 13.2\), the current candidate. Since JWST can go to higher z, one should expect that future observations will clarify if JWST can establish \({z_{\max }}\). Note that obtaining a value for z from JWST data is arcane—possibly leading to controversy.
There is another complication in NCNG. There are two different distances for \(z = 13.2\) which bracket the maximum. Which distance is JADES-GS-z13-0? Using the constants of this edition, the oldest distance occurs about 496,000 years after local gravity is turned on, but 378,000 years before fusion is turned on (Fig. C5). The older version of JADES-GS-z13-0 would consist of only Kelvin contraction stars and gas, The more recent version of JADES-GS-z13-0 would occur 418,000 years after the older version (after fusion is turned on). One should expect that the newer version would have greater luminosity than the older version—more time to convert gas into stars and hotter fusion stars compared to Kelvin contraction stars. I believe that JADES-GS-z13-0 is the older, dimmer version. This object is erroneously identified as a supermassive dark star [18], but it is actually a normal galaxy in very early red summer.
Many of the astronomical observations (during my lifetime) have been interpreted through the filter of the suspect GRLCDM theory—suspect conclusions requiring re-examination. Some of the rungs in the ladder of distance will survive and some will require a new expression. The goal will be a coherent model of distance versus cosmological red shift (\(d(c\tau )/dz\)) derived from an ultimate version of Appendix Z. The first step is outlined below using the version of the standard model of Appendix Z in this document.
The apex field (A) varies between 1.043716 and -1.043716 between midwinter extremes. Thus, the value of A for a point within a cycle is a long decimal—inconvenient for an axis of a figure. One can use Eq. (C.2) to obtain the now parameter, \({A_0} = 0.8150217\). Then, I have chosen the following definition:
\[A = {A_0} – \Delta A\,\alpha ,\tag{C.3}\]
where \(\Delta A = 2.36\,{\rm{x}}\,{\rm{1}}{{\rm{0}}^{ – 5}}\). Some of the figures below will be versus alpha less a starting offset (instead of A). Several markers of interest can be tabulated; \({z_{MAX}} = 14.9213\) at \(A = 0.9992719\), and the turning point at the middle of (blue to red) winter \({A_W} = 1.0413716\) (maximum value of A).
Fig. C1 (below) shows the shape of E in the range 0.999 (winter) to 0.001 (summer) versus alpha less an offset. The bubble discontinuity is at \(A = 0.9992856\) (alpha = 7807.788). Physical time is increasing toward the left of the figure, i.e., the right side is the oldest value.
The history will follow one quarter of a cycle from \(A = 0\) to \(A = {A_W}\)on the red branch. The first segment (Fig. C2) is from the middle of red summer to now (\(0 < A < {A_0}\))—in the future. The duration of this segment is 12.685 billion years.
The next segment (Fig. C3) is from now to the past physical time when \(z = {z_{MAX}}\) (completely in red summer). The duration of this segment is 19.972 billion years (in the past).
The next brief segment goes from maximum z (\(A = 0.9992719\)) to the summer side of the E-discontinuity (\(A = 0.9992855\)). The duration of this brief segment is 17.534 million years.
The last segment (Fig. C4) is from the winter side of the E-discontinuity at \(A = 0.9992856\) to \({A_W} = 1.0413716\) . The duration of this lengthy segment is 150.192 billion years. This quarter lasts 182.867 billion years and a complete cycle will be 731.466 billion years. Winter will amount to 82.2 % of the cycle. Note that z is negative (cosmological blue shift) for the first 367 million years of Fig. C4 (late winter).
Fig. C3 is important since it involves only four universal constants (\(d_1^{ax},\;d_2^{ax},\;d_3^{ax},{\rm{ and}}{L_{ax}}\)) because \(E = 0\). A ladder of distances can be established by observing Cepheid variables and multi-image gravitational lensing of more-distant type 1a supernovae events (a tall order). The ladder is equivalent to Fig. C3 and the four constants can be established from observations. Thus, one can improve Appendix Z and improve the accuracy of NCNG.
Fig. C5 has detail for events of interest when z is earlier than maximum z. Local gravity is turned on at \(z = 12.9532\) and offset alpha = 0.6398. Fusion is turned on at \(z = 13.4303\) and offset alpha = 0.6102.
All the details in these figures will only be accurate in the future if JWST will eventually establish that the maximum value of z is very close to 14.9213, and the fourteen constants of Eq. (C.1) and Eq. (C.2) are not different than the values of this edition. Otherwise, recalculations will be necessary.
Figure C.1 The epoch field function E defined in Eq. (2.17) versus a linear function of the apex field defined above.
Figure C.2 Physical time in the future moving left from now till A is zero (the half point of the red branch).
Figure C.3 The age (distance/c) of a luminous object versus z from now back to the maximum value of z.
Figure C.4 This figure adds a major segment of the age of a cycle, and z gives the temperature of CMB in winter (cold).
Figure C.5 This figure covers the oldest portion of early red summer—a (brief) eventful era.