/ 6 appearing [4] in:

Ω n 2 = a 2 ( η ) ω N O N L I N 2 ( k ) ( 1 6 ε ) a a = a 2 ( η ) ω N O N L I N 2 ( k ) = a 2 ( η ) F 2 ( k ) (1)

then for small momentum:

ω N O N L I N 2 ( k ˜ 0 ) k ˜ 0 2 (2)

if “momentum” k ˜ 0 k P , where we use the same sort of linear approximation used by Mercini [2] , as specified for Equation (17) of their article [2] 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 ( k ) [4] since it permits a clean derivation of the Bogoliubov coefficient which has the deviation function Γ ( k 0 , B ) . We begin with [4] :

| β k | 2 | β n | 2 = sinh 2 ( 2 π Ω ^ ) + Γ ( k 0 , B ) sinh 2 ( 2 π Ω ^ + ) sinh 2 ( 2 π Ω ^ ) (3)

where we get an appropriate value for the deviation function Γ ( k 0 , B ) [4] based upon having the square of the dispersion function F ( k ) obey Equations (1) and (2) above for k ˜ 0 k P . Note, k P is a maximum momentum value along the lines Magueijo [3] suggested for an E P Plank energy value.

Part II. Deriving Appropriate Γ ( k 0 , B ) Deviation Function Values

We look at how Bastero-Gil [4] obtained an appropriate Γ ( k 0 , B ) value. Basterero-Gil wrote:

Γ ( k 0 , B ) = cosh 2 ( π 2 4 B e X o 1 ) (4)

with

x 0 = k ˜ 0 k P 1 (5)

and

F 2 ( k ) = ( k 2 k ˜ 1 2 ) V 0 ( x , x 0 ) + k 2 V 1 ( x x 0 ) + k ˜ 1 2 (6)

where k ˜ 1 < k P and where 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 ( k ) k as x = k ˜ k P x 0 which will lead to specific

restraints we place upon V 0 ( x , x 0 ) as well as V 1 ( x x 0 ) above. Following Bastero-Gil [4] , we write:

V 0 ( x , x 0 ) = C 1 + e X + E e X ( 1 + e X ) ( 1 + e X X O ) (7)

and:

V 1 ( x x 0 ) = B e X ( 1 + e X X O ) 2 (8)

When x = k ˜ k P x 0 1 we get [2] [4]

F 2 ( k 0 ) ω N O N L I N 2 ( k 0 ) k 1 2 ( 1 c 2 E 4 ) + k 0 2 ( c 2 + E 4 B 4 ) k 0 2 (9)

which then implies 0 < B ε + 1 . Then we obtain:

Γ ( k 0 , B ε + ) cosh 2 ( ( π 2 + ε + ) i ) ε + 1 (10)

and

| β k | 2 | β n | 2 sinh 2 ( 2 π Ω ^ ) + ε + sinh 2 ( 2 π Ω ^ + ) sinh 2 ( 2 π Ω ^ ) (11)

Part III. Finding Appropriate Ω ^ + and Ω ^ Values

We define, following Bastero-Gil [4]

Ω ^ ± = 1 2 ( Ω ^ O U T ± Ω ^ I N ) (12)

where we have that

Ω O U T = η Ω n ( η ) (13)

and

Ω I N = η Ω n ( η ) (14)

whereas we have that

Ω ^ k ˜ = Ω k ˜ n (15)

where k ˜ denotes either out or in. Also:

Ω O U T Ω I N 1 (16)

which lead to:

Ω ^ + ( 1 B 2 ) 1 n = ( 1 B 2 ) 1 k | η η C | 1 k | η η C | (17)

as well as

Ω ^ B 2 1 n 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 β = 1.05 and L = 1 / 2 , ( k k P ) k k P , then ρ T A I L M ρ T O T A L M 0.371

If β = 1.05 and L = 1 , ( k k P ) ( k k P ) , then ρ T A I L M ρ T O T A L M 0.263

If β = 1.05 and L = 2 , ( k k P ) ( k k P ) 2 , then ρ T A I L M ρ T O T A L M 0.115

If β = 10.5 and L = 1 / 2 , ( k k P ) k k P , then ρ T A I L M ρ T O T A L M 1.935 × 10 5

If β = 10.5 and L = 1 , ( k k P ) ( k k P ) , then ρ T A I L M ρ T O T A L M 7.347 × 10 6

If β = 10.5 and L = 2 , ( k k P ) ( k k P ) 2 , then ρ T A I L M ρ T O T A L M 6.7448 × 10 8

We need β 10 10 with ρ T A I L M ρ T O T A L M 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.

References

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