APPENDIX B: SPRING BUBBLE FORMATION
The formation of bubbles is described in [3] and [4]. The probability of a bubble forming within a space-volume V and an evolution-duration \(\Delta {x^0}\) is \({P_{form}} = {p_B}V\Delta {x^0}\). In red spring, \({p_B}\) is given with sufficient accuracy in [4] as
\[{p_B} = ({L_{ep}}^{ – 4}/4{\pi ^2}){\theta ^2}{e^{ – \theta }}.\tag{B.1}\]
The exponent (\(\theta \)) is a function of \(\lambda (A)\), \({a_{ep}}\), and \(\mathcal{N}_{ep}\) (calculated using the thin wall approximation of [3])
\[
\theta = 2 \pi^2 \, \mathcal{N}_{ep} \, 27 S^4 \varepsilon^{-3}, \tag{B.2}
\]
\[\begin{array}{l}\varepsilon = {V_{ep}}({\phi _{{\rm{false}}}}) – {V_{ep}}({\phi _{{\rm{true}}}}),\\S = \int_{{\phi _{{\rm{false}}}}}^{{\phi _{{\rm{true}}}}} {d\phi } {\{ {V_{ep}}(\phi ) – {V_{ep}}({\phi _{{\rm{true}}}})\} ^{1/2}}.\end{array}\tag{B.3}\]
\[\begin{array}{l}{\phi _{{\rm{true}}}} = \{ – \lambda (A) + {[{a_{ep}}^2 + \lambda {(A)^2}]^{1/2}}\} /2{a_{ep}}^2,\\{\phi _{{\rm{false}}}} = \{ – \lambda (A) – {[{a_{ep}}^2 + \lambda {(A)^2}]^{1/2}}\} /2{a_{ep}}^2.\end{array}\tag{B.4}\]
\({L_{ep}}\) will be a fraction of a meter, whereas the typical expanding bubble radius (R) will be measured in light-years, so the bubble wall will be extremely thin compared to R.
The average energy density of matter in the universe (as defined at the start of section III) is thought to be about 250 Mev per cubic meter. This number comes from GR and \(\Lambda {\rm{CDM}}\), so the value is suspect, but I will use this value throughout this paper. Thus \(\left| {U_{ep}^S} \right| = {\chi _{ep}}{U_{matter}}\), where \({\chi _{ep}} > > 1\) (by hypothesis). Eq. (2.18) then leads to the relation,
\[
\mathcal{N}_{ep} = \chi_{ep} \left(\frac{L_{ep}}{L_m}\right)^4
\left| V_{ep}(\phi_S) \right|^{-1}, \tag{B.5}
\]
where \({L_m} = {\rm{ 1}}{\rm{.70306}}x{10^{ – 4}}\) meters.
To develop an understanding of the bubble formation era, a cubic space-volume of interest is defined in terms of a length (\(V = {L_V}^3\)). The universe is filled with these cubes. \({L_V}\) must be large compared to the radius of the largest bubble that is likely to form within a specific cube during the chaotic period, e.g., \({L_V} = {10^4}{\rm{ ly}}\).
There is a simple constraint that accurately governs (ultra-thin wall) bubble-formation—a new (valid) bubble cannot form within an existing bubble. \({P_{form}}\) is ignorant of this constraint, but it can be used to calculate the probability of an attempt at bubble formation. It is useful to express the probability in terms of the apex field instead of \({x^0}\)
\[d{P_{form}}/dx = – {(dA/d{x^0})_{{A_d}}}^{ – 1}{L_V}^3{(4{\pi ^2}{L_{ep}}^4)^{ – 1}}{\theta ^2}\exp ( – \theta ),\tag{B.6}\]
where \(A = {A_d} – x\) and \(x \ge 0\). The quantities of interest for \(x \to 0\) are as follows:
\[
\begin{array}{l}
\varepsilon = x{\Gamma _1}\;,\\[6pt]
{\Gamma _1} = \frac{100}{96}\left(\frac{\beta_{ep}\Gamma_{ep}}{a_{ep}^3}\right)\left[1 – \left(\frac{\lambda_{ep}}{\beta_{ep}}\right)^2\right],\\[6pt]
S = 0.25502\,a_{ep}^{-2},\\[6pt]
\theta = \Lambda_{\theta}x^{-3},\\[6pt]
\Lambda_{\theta} = 54\pi^2 S^4 \Gamma_1^{-3}\chi_{ep}\left(\frac{L_{ep}}{L_m}\right)^4 \left|V_{ep}(\phi_S)\right|^{-1},\\[6pt]
DA = 0.5\left(\mathcal{F}_S(A_d) + \mathcal{F}_W(A_d)\right),\\[6pt]
\left(\frac{dA}{dx^0}\right)_{A_d} \simeq -DA L_{ax}^{-1}.
\end{array}
\tag{B.7}
\]
The average number of attempts made prior to x is as follows:
\[\begin{array}{l}NA(x) = \Pi \int_0^x {dx’} ({{x’}^{ – 6}})\exp ( – {\Lambda _\theta }{{x’}^{ – 3}}),\\\Pi = D{A^{ – 1}}{L_{ax}}{L_V}^3{(4{\pi ^2}{L_{ep}}^4)^{ – 1}}{\Lambda _\theta }^2.\end{array}\tag{B.8}\]
The leading term of Eq. (B.8) as \(x \to 0\) results,
\[\log (NA) = \log ({\Lambda _\theta }^{ – 1}\Pi /3) – 2\log (x) – (\log (e)){\Lambda _\theta }{x^{ – 3}}.\tag{B.9}\]
If \(NA({x_1}) = 1\) (the first bubble in V), then there will be explosive growth in \(NA\) thereafter if \(\log (e){\Lambda _\theta }{x_1}^{ – 4} > > 1\). Eq. (B.9) has limited usefulness since \(\Pi \) depends on the arbitrary volume of the cube. There is a more useful metric that is independent of cube volume,
\[\Delta (x) = {L_V}^{ – 3}{( – {(dA/d{x^0})_{{A_d}}})^{ – 3}}\int_0^x {dx’} (4\pi /3){(x – x’)^3}(d{P_{form}}/dx’).\tag{B.10}\]
Delta is a ratio–the (approximate) volume of the bubbles created within the cube divided by the volume of the cube. Delta ignores the bubble validity constraint and the fact that bubbles can overlap. One could define the end of the bubble era (\({x_{end}}\)) by choosing\(\Delta ({x_{end}}) = 1\). This estimate will be a lower bound of \({x_{end}}\) for the following reason. The integrand in Eq. (B.10) represents an over-estimate of the true bubble volume since it includes non-valid bubbles and ignores bubble overlap. Thus, the integration reaches 1 at a lesser value of x than the true value for \({x_{end}}\). The explosive growth of bubble attempts insures that \(\log (\Delta )\) can be neglected in what follows. The leading term of Eq. (B.10) as \(x \to 0\) follows:
\[\begin{array}{l}\Delta ({x_{end}}) = \Xi {x_{end}}^{10}\exp ( – {\Lambda _\theta }{x_{end}}^{ – 3}),\\\Xi = (2/81\pi ){({L_{ax}}/{L_{ep}})^4}D{A^{ – 4}}{\Lambda _\theta }^{ – 2}.\end{array}\tag{B.11}\]
Eq. (B.11) is equivalent to
\[\begin{array}{l}{Y^{ – 3}} – 10\log (Y) = F,\\{x_{end}} = {[{\Lambda _\theta }\log (e)]^{1/3}}Y,\\F = \log (\Xi ) + (10/3)\log [{\Lambda _\theta }\log (e)].\end{array}\tag{B.12}\]
It is a simple matter to find Y given F. The factors in Eq. (B.12) are dominated by \({L_{ax}}/{L_m} > > 1\), \({L_m}/{L_{ep}} > > 1\), and \({\chi _{ep}} > > 1\). It is useful to isolate these factors. F can be rewritten in order of size,
\[
F = 4 \log \left(\frac{L_{ax}}{L_m}\right)
+ \frac{4}{3} \left[ \log(\chi_{ep}) – \log\left(\frac{L_m}{L_{ep}}\right) \right]
+ F_0. \tag{B.13}
\]
The value (\({L_{ax}} = 9e9{\rm{ l}}{\rm{.y}}{\rm{.}}\)) will be established in Appendix C. Thus, \(4\log ({L_{ax}}/{L_m}) = 118.80\). The second and third terms govern the energy density of the epoch field, but they are constrained by limits on the size of \({x_{end}}\). If \(\log ({\chi _{ep}}) = 8\), and if \({L_{ep}}\) is the Compton wavelength of a one electron-volt mass particle (plausible values), then the second and third terms equal 7.849. The last term is due to numerical factors and a swarm of factors from Eq. (B.7). A generic set of epoch field constants gives \({F_0} = – 5.599\). Therefore, the last three terms of F contribute 2.25 (less than 2% of the total). The value \(F = 121\) corresponds to \(Y = 0.20615\). Using Eq. (B.12), one finds
\[{x_{end}} = 0.1561\;{\Lambda _\theta }^{1/3}.\tag{B.14}\]
In practice, \({x_{end}}\) will be fixed by the (small) portion of spring that the bubble era is allotted. In that case, Eq. (B.14) will be a constraint on the relation between \({\chi _{ep}}\) and \({L_{ep}}\). The allotment will also fix the scale of bubble size. Furthermore, \({x_{end}}\) will be very small and the explosive bubble-formation criterion will be met. The fall transition has a bubble era, and the same considerations as above are valid. For all transitions, \({x_{end}} > 0\). For red transitions, \({A_{end}} = {A_d} – {x_{end}}\), and for blue, \({A_{end}} = {A_d} + {x_{end}}\).
On average, nearly all the physical time allotted to \({x_{end}}\) is spent waiting for the first bubble to form. Once formed, explosive growth will end the era within several thousand years. The model of Appendix Z will allot 736,000 years for the era.