/ 6 appearing  in:

${\Omega }_{n}^{2}={a}^{2}\left(\eta \right)\cdot {\omega }_{NON-LIN}^{2}\left(k\right)-\left(1-6\cdot \epsilon \right)\cdot \frac{{a}^{″}}{a}={a}^{2}\left(\eta \right)\cdot {\omega }_{NON-LIN}^{2}\left(k\right)={a}^{2}\left(\eta \right)\cdot {F}^{2}\left(k\right)$ (1)

then for small momentum:

${\omega }_{NON-LIN}^{2}\left({\stackrel{˜}{k}}_{0}\right)\approx {\stackrel{˜}{k}}_{0}^{2}$ (2)

if “momentum” ${\stackrel{˜}{k}}_{0}\ll {k}_{P}$ , where we use the same sort of linear approximation used by Mercini  , as specified for Equation (17) of their article  if the Epstein function specified in Equation (1) of the main text has a linear relationship. We write out a full treatment of the dispersion function $F\left(k\right)$  since it permits a clean derivation of the Bogoliubov coefficient which has the deviation function $\Gamma \left({k}_{0},B\right)$ . We begin with  :

${|{\beta }_{k}|}^{2}\equiv {|{\beta }_{n}|}^{2}=\frac{{\mathrm{sinh}}^{2}\left(2\cdot \text{π}\cdot {\stackrel{^}{\Omega }}_{-}\right)+\Gamma \left({k}_{0},B\right)}{{\mathrm{sinh}}^{2}\left(2\cdot \text{π}\cdot {\stackrel{^}{\Omega }}_{+}\right)-{\mathrm{sinh}}^{2}\left(2\cdot \text{π}\cdot {\stackrel{^}{\Omega }}_{-}\right)}$ (3)

where we get an appropriate value for the deviation function $\Gamma \left({k}_{0},B\right)$  based upon having the square of the dispersion function $F\left(k\right)$ obey Equations (1) and (2) above for ${\stackrel{˜}{k}}_{0}\ll {k}_{P}$ . Note, ${k}_{P}$ is a maximum momentum value along the lines Magueijo  suggested for an ${E}_{P}$ Plank energy value.

Part II. Deriving Appropriate $\Gamma \left({k}_{0},B\right)$ Deviation Function Values

We look at how Bastero-Gil  obtained an appropriate $\Gamma \left({k}_{0},B\right)$ value. Basterero-Gil wrote:

$\Gamma \left({k}_{0},B\right)={\mathrm{cosh}}^{2}\left(\frac{\text{π}}{2}\cdot \sqrt{4\cdot B\cdot {\text{e}}^{-{X}_{o}}-1}\right)$ (4)

with

${x}_{0}=\frac{{\stackrel{˜}{k}}_{0}}{{k}_{P}}\ll 1$ (5)

and

${F}^{2}\left(k\right)=\left({k}^{2}-{\stackrel{˜}{k}}_{1}^{2}\right)\cdot {V}_{0}\left(x,{x}_{0}\right)+{k}^{2}\cdot {V}_{1}\left(x-{x}_{0}\right)+{\stackrel{˜}{k}}_{1}^{2}$ (6)

where ${\stackrel{˜}{k}}_{1}<{k}_{P}$ and where ${\stackrel{˜}{k}}_{1}$ is in the Trans-Planckian regime but is much greater than ${k}_{0}$ . We are determining what B should be in Equation (16) of the

main text provided that $F\left(k\right)\approx k$ as $x=\frac{\stackrel{˜}{k}}{{k}_{P}}\to {x}_{0}$ which will lead to specific

restraints we place upon ${V}_{0}\left(x,{x}_{0}\right)$ as well as ${V}_{1}\left(x-{x}_{0}\right)$ above. Following Bastero-Gil  , we write:

${V}_{0}\left(x,{x}_{0}\right)=\frac{C}{1+{\text{e}}^{X}}+\frac{E\cdot {\text{e}}^{X}}{\left(1+{\text{e}}^{X}\right)\cdot \left(1+{\text{e}}^{X-{X}_{O}}\right)}$ (7)

and:

${V}_{1}\left(x-{x}_{0}\right)=-B\cdot \frac{{\text{e}}^{X}}{{\left(1+{\text{e}}^{X-{X}_{O}}\right)}^{2}}$ (8)

When $x=\frac{\stackrel{˜}{k}}{{k}_{P}}\to {x}_{0}\ll 1$ we get  

${F}^{2}\left({k}_{0}\right)\equiv {\omega }_{NON-LIN}^{2}\left({k}_{0}\right)\cong -{k}_{1}^{2}\cdot \left(1-\frac{c}{2}-\frac{E}{4}\right)+{k}_{0}^{2}\cdot \left(\frac{c}{2}+\frac{E}{4}-\frac{B}{4}\right)\cong {k}_{0}^{2}$ (9)

which then implies $0 . Then we obtain:

$\Gamma \left({k}_{0},B\cong {\epsilon }_{+}\right)\cong {\mathrm{cosh}}^{2}\left(\left(\frac{\text{π}}{2}+{\epsilon }_{+}\right)\cdot i\right)\approx {\epsilon }_{+}\ll 1$ (10)

and

${|{\beta }_{k}|}^{2}\equiv {|{\beta }_{n}|}^{2}\cong \frac{{\mathrm{sinh}}^{2}\left(2\cdot \text{π}\cdot {\stackrel{^}{\Omega }}_{-}\right)+{\epsilon }_{+}}{{\mathrm{sinh}}^{2}\left(2\cdot \text{π}\cdot {\stackrel{^}{\Omega }}_{+}\right)-{\mathrm{sinh}}^{2}\left(2\cdot \text{π}\cdot {\stackrel{^}{\Omega }}_{-}\right)}$ (11)

Part III. Finding Appropriate ${\stackrel{^}{\Omega }}_{+}$ and ${\stackrel{^}{\Omega }}_{-}$ Values

We define, following Bastero-Gil 

${\stackrel{^}{\Omega }}_{±}=\frac{1}{2}\cdot \left({\stackrel{^}{\Omega }}_{OUT}±{\stackrel{^}{\Omega }}_{IN}\right)$ (12)

where we have that

${\Omega }^{OUT}=\stackrel{\eta \to \infty }{\to }{\Omega }_{n}\left(\eta \equiv \infty \right)$ (13)

and

${\Omega }^{IN}=\stackrel{\eta \to -\infty }{\to }{\Omega }_{n}\left(\eta \equiv -\infty \right)$ (14)

whereas we have that

${\stackrel{^}{\Omega }}_{\stackrel{˜}{k}}=\frac{{\Omega }_{\stackrel{˜}{k}}}{n}$ (15)

where $\stackrel{˜}{k}$ denotes either out or in. Also:

${\Omega }^{OUT}\cong {\Omega }^{IN}\cong 1$ (16)

${\stackrel{^}{\Omega }}_{+}\cong \left(1-\frac{B}{2}\right)\cdot \frac{1}{n}=\left(1-\frac{B}{2}\right)\cdot \frac{1}{k}\cdot |\frac{\eta }{{\eta }_{C}}|\cong \frac{1}{k}\cdot |\frac{\eta }{{\eta }_{C}}|$ (17)

as well as

${\stackrel{^}{\Omega }}_{-}\cong \frac{B}{2}\cdot \frac{1}{n}\cong 0$ (18)

Appendix Entry 2: How Equation (16) of Text Changes for Varying b Values and Different Dispersion Relationships

Starting with Equation (21) of the main text.

If $\beta =1.05$ and $L=1/2$ , $\left(\frac{k}{{k}_{P}}\right)\to \sqrt{\frac{k}{{k}_{P}}}$ , then $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\cong 0.371$

If $\beta =1.05$ and $L=1$ , $\left(\frac{k}{{k}_{P}}\right)\to \left(\frac{k}{{k}_{P}}\right)$ , then $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\cong 0.263$

If $\beta =1.05$ and $L=2$ , $\left(\frac{k}{{k}_{P}}\right)\to {\left(\frac{k}{{k}_{P}}\right)}^{2}$ , then $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\cong 0.115$

If $\beta =10.5$ and $L=1/2$ , $\left(\frac{k}{{k}_{P}}\right)\to \sqrt{\frac{k}{{k}_{P}}}$ , then $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\cong 1.935×{10}^{-5}$

If $\beta =10.5$ and $L=1$ , $\left(\frac{k}{{k}_{P}}\right)\to \left(\frac{k}{{k}_{P}}\right)$ , then $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\cong 7.347×{10}^{-6}$

If $\beta =10.5$ and $L=2$ , $\left(\frac{k}{{k}_{P}}\right)\to {\left(\frac{k}{{k}_{P}}\right)}^{2}$ , then $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\cong 6.7448×{10}^{-8}$

We need $\beta \cong {10}^{-10}$ with $\frac{{〈{\rho }_{TAIL}〉}_{M}}{{〈{\rho }_{TOTAL}〉}_{M}}\le {10}^{-30}$ to get our results via this Trans-Plankian model to be consistent with physically verifiable solutions to the

cosmic ray problem.

Conflicts of Interest

The authors declare no conflicts of interest.

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