_{1}

^{*}

We offer evidence that the Trans Plankian hypothesis about Dark energy is incompatible with necessary and sufficient conditions for solving the cosmic ray problem along the lines presented by Magueijo et al. We can obtain conditions for a dispersion relationship congruent with the Trans Planckian hypothesis only if we cease trying to match cosmic ray data which is important in investigating Doubly Special Relativity. This leads us to conclude that the Trans Planckian hypothesis is inconsistent with respect to current astrophysical data when modeled by Doubly Special Relativity and needs to be seriously revised. Or the Doubly Special Relativity Hypothesis needs to be abandoned.

We examine if an alteration of special relativity presented by Magueijo and Smolin [^{2} to obtain a highly non linear dispersion relationship. However, this dispersion relationship does NOT solve the cosmic ray problem for low momentum values [

What Mersini [_{C} and where we have if we can set k ≪ k C

ω K 2 ( k ) ≈ k 2 (1)

which means for low values of momentum we have a linear relationship for dispersion vs. “momentum” in low momentum situations. In addition we also have that

ω K 2 ( k ≫ k C ) ≈ exp ( − k / k C ) → k → ∞ 0 (2)

We also have a specific “tail mode” energy region picked by:

ω K 2 ( k H ) ≡ H 0 2 (3)

to obtain k H . We then have an energy calculation for the “tail” modes:

〈 ρ T A I L 〉 K = 1 2 ⋅ π 2 ⋅ ∫ K H ∞ k d k ∫ ω K ( k ) ⋅ d ω K ⋅ | β k | 2 (4)

which is about 122 orders of magnitude smaller than

〈 ρ T O T A L 〉 K = 1 2 ⋅ π 2 ⋅ ∫ 0 ∞ k d k ∫ ω K ( k ) ⋅ d ω K ⋅ | β k | 2 (5)

allowing us to write

〈 ρ T A I L 〉 K 〈 ρ T O T A L 〉 K ≈ k H 2 M P 4 ⋅ ω K 2 ( k H ) ≈ H 0 2 M P 2 ≈ 10 − 122 (6)

Here, the tail modes (of energy) are chosen as “frozen” during any expansion of the universe. This is for energy modes for frequency regions ω K 2 ( k ) ≤ H 0 2 so that we have resulting ‘tail modes’ of energy obeying Equation (5) above.

We shall next determine what sort of dispersion relationship we can obtain by the revision of special relativity Magueijo [

1) Assume relativity of inertial frames: When gravitational effects can be neglected, all observers in free, inertial motions are equivalent. This means that there is no preferred state of motion.

2) Assume an equivalence principle: Under the effect of gravity, freely falling observers are all equivalent to each other and are equivalent to inertial observers.

3) A new principle is introduced: The observer independence of Planck energy. i.e. that there exists an invariant energy scale which we shall take to be the Planck energy.

4) There exists a correspondence principle: At energy scales much smaller than E P , conventional special and general relativity are true: that is that they hold to first order in the ratio of energy scales to E P . We ask now how can these principles be fashioned into predictions as to energy values, which we shall use to obtain dispersion relationships. Magueijo and Smolin [

E 0 = m 0 ⋅ c 2 1 + m 0 ⋅ c 2 E P (7)

which if m = γ ⋅ m 0 and c set = 1 becomes:

E = m 1 + m E P (8)

We found it useful to work with, instead:

E = m ( 1 + β ⋅ m E P ) 11 ( 1 − m E P ) (9)

with a power of 11 put in the denominator due to string theory dimensions which gives us preferred numerical values we are seeking for the ratio of dark energy over total cosmological energy. If E P A R T I C L E < E P and m = α ⋅ k , then

where we used

Furthermore, if

which if

Note how the cut off value of momentum

tity in dispersion behavior leads to the results seen in

We can contrast this dispersion behavior with:

We set

So we used a tail mode energy expressions as given by

and

so we obtain [

when we are using

stated by Mersini [

We followed Mercinis [

thermal universe, and we proved it in our bogoliubov coefficient calculation. This lead to us picking the “thermality coefficient” [

dition, the ratio of confocal times as given by

We derive this expression in the 1^{st} appendix entry. In addition, we note that Bastero-Gil, in 2008 in the IDM conference, of 2008 in Stockholm, brought up a discussion of the results of [

We evaluate

and set up a numerical parameterization of

with ^{nd} paper.

_{H} could have a wide range of values.

This permitted _{ }in line with de tuning the sensitivity of the ratio results if we use

_{H} value needed to have the frequency

The

The answer is no even after a modification of our dispersion relationship:

With

ation (21) we obtained, for

which has a very different lower bound than the behavior seen in Equation (16). If we pick

We found that the dispersion relationship given in Equation (10) and its limiting behavior shown in Equation (20) gives the lower bound behavior as noted in Equation (16) above for a wide range of possible

above. This was, however, done for a physically unacceptably large

This work is supported in part by National Nature Science Foundation of China grant No. 110752.

Beckwith, A.W. (2017) Checking the Alleged “Incompatibility of the Relic Dark Energy Hypothesis” with Physically Admissible Solutions to the Cosmic Ray Problem of Doubly Special Relativity. Journal of High Energy Physics, Gravitation and Cosmology, 3, 588-599. https://doi.org/10.4236/jhepgc.2017.34045

We derive the Bogoliubov coefficient, which is used in Equation (16) of the main text. We refer to Mersini’s article [

then for small momentum:

if “momentum”

where we get an appropriate value for the deviation function

We look at how Bastero-Gil [

with

and

where

main text provided that

restraints we place upon

and:

When

which then implies

and

We define, following Bastero-Gil [

where we have that

and

whereas we have that

where

which lead to:

as well as

Starting with Equation (21) of the main text.

If

If

If

If

If

If

We need

cosmic ray problem.

Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.

A wide selection of journals (inclusive of 9 subjects, more than 200 journals)

Providing 24-hour high-quality service

User-friendly online submission system

Fair and swift peer-review system

Efficient typesetting and proofreading procedure

Display of the result of downloads and visits, as well as the number of cited articles

Maximum dissemination of your research work

Submit your manuscript at: http://papersubmission.scirp.org/

Or contact jhepgc@scirp.org