We examine if an alteration of special relativity presented by Magueijo and Smolin  , assuming joining the speed of light and Planck energy as a new invariant permits a dispersion relationship which will set dark energy  from the “tail mode” of ultra high momentum contributions (of the universe) markedly lower than the total energy of the universe. We find that the answer is yes after modifying an energy equation of E = MC2 to obtain a highly non linear dispersion relationship. However, this dispersion relationship does NOT solve the cosmic ray problem for low momentum values  . Our derived dispersion relationship matches the Epstein function used by Mercini et al.  only if we cease trying to fit cosmic ray data  which lead to Magueijo  proposing their alteration of special relativity in the first place. We follow Mersini et al.  in their derivation of a Trans Planckian dark energy over total energy ratio. Our results argue that we cannot reconcile the requirements of a solution of the “cosmic ray” problem of special relativity in a manner congruent with Mercinis  ratios of dark matter energy to total energy being calculated via a Bogoliubov coefficient  . The dispersion relationship which we obtained which actually permitted us to calculate the energy of the tail modes of Trans Plankian dark energy  vs. total energy ratio  to have a value less than ten to the minus 30 power mimics the Epstein function  in a manner which contravenes necessary and sufficient conditions  for solving the cosmic ray problem of special relativity. Our calculations imply that a Trans-Planckian dark energy depends upon initial conditions which are too specialized and which do not match up with known astrophysical data obtained as of the 1990s. This is in tandem with Lemoine, Martin, and Uzan  who dispute on the Trans Planckian hypothesis on different grounds.
2. Description of Procedure Used to Obtain Energy Density Ratio
What Mersini  did was to use ultra low dispersion relationship values for ultra high momentum values to obtain “ultra low” energy values which were and remain allegedly “frozen” today  . They found, using the Epstein function for frequency dispersion relationships a range of frequencies , where is the present Hubble rate of expansion. From there, they computed Trans-Planckian dark energy modes which are about 122 to 123 orders of magnitude smaller than the total energy of the universe assumed for their expansion model. Note in this discussion that refers to the dispersion relationship Mercini  derived, while will be a dispersion relationship derived from Magueijo and Smolin’s  modification of special relativity. Mersini  changed a standard linear dispersion relationship to one which has a modified Epstein function with a peak value for frequency given when k = kC and where we have if we can set
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
We also have a specific “tail mode” energy region picked by:
to obtain . We then have an energy calculation for the “tail” modes:
which is about 122 orders of magnitude smaller than
allowing us to write
Here, the tail modes (of energy) are chosen as “frozen” during any expansion of the universe. This is for energy modes for frequency regions so that we have resulting ‘tail modes’ of energy obeying Equation (5) above.
3. Forming a Dispersion Relationship from Magueijo and Smolins Energy Values and Then Subsequently Modifying It
We shall next determine what sort of dispersion relationship we can obtain by the revision of special relativity Magueijo  proposed. Magueijo  states that the energy of an independent particle will not exceed in value, which is the Planck energy. This Planck energy is the inverse of the Planck length defined by cm, where G is the gravitational constant and c is the speed of light. Specifically, Magueijo and Smolin  state that if and only if the rest mass of a particle obtains an infinite value. If we set , we have as an upper bound. This upper bound with respect to particle energy is consistent with respect to four principles elucidated by Magueijo and Smolin  , which are as follows:
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 , conventional special and general relativity are true: that is that they hold to first order in the ratio of energy scales to . 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  obtained a modified relationship between energy and mass:
which if and c set = 1 becomes:
We found it useful to work with, instead:
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 and , then
permits are write of Equation (9) above as (if ):
where we used and which if will lead to the same result as spoken of with the modified Epstein function  , assuming that , so:
Furthermore, if , Equation (10) will give us
which if gives the values seen in Figure 1 below.
Note how the cut off value of momentum is due to as a quan-
tity in dispersion behavior leads to the results seen in Figure 1.
Figure 1. Graph of 1st dispersion relationship against momentum. This gives the desired behavior in line with the Trans Planckian dark energy hypothesis. However, !
We can contrast this dispersion behavior with:
We set and , leading to Figure 2 as given below. Note, if and we recover Equation (9).
So we used a tail mode energy expressions as given by
so we obtain  a “frozen” tail mode energy vs. total energy ratio of
when we are using . Equation (16) has a lower bound as
stated by Mersini  in Equation (6) if we use . Detuning the sensitivity of this ratio to exact for any is extremely important to the viability of our physical theory about how dark matter plays a role in inflationary cosmology.
4. The Bogoliubov Function Used in This Paper
We followed Mercinis  assumption of negligible deviations from a strictly
Figure 2. Graph of 2nd dispersion relationship against momentum which has too broad a width to be useful.
thermal universe, and we proved it in our bogoliubov coefficient calculation. This lead to us picking the “thermality coefficient”  B to be quite small. In ad-
dition, the ratio of confocal times as given by had little impact upon Equation (16). Also, . Therefore,
We derive this expression in the 1st 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  , as useful research results, and then adopted using the results of the document given in  . Then, Bastero-Gill when using the results of  subsequently delineated the size of tail energy density from Dark matter as which is consistent with our findings that our Bogoliubov function as given by Equation (17) may be often approximated by a constant with small effects on calculating the ratio of energy for the tail vs. total energy  given in Equation (6) above.
5. Analytical and Numerical Evaluation of Equation (16)
We evaluate in light of Equation (12) in our Equation (16) integrand. We then obtain:
and set up a numerical parameterization of
with chosen by considerations presented in Mercini’s  2nd paper.
6. Why We Still Were Unable to Match Cosmic Ray Data and Found Our Dispersion Relationship Not Physically Tenable
in Equation (10) was picked so kH could have a wide range of values.
This permitted to be bounded below by a value for in line with de tuning the sensitivity of the ratio results if we use
in the Equation (10) dispersion relationship. We obtain Mercini’s main result  at the expense of not matching cosmic ray data  . We should note that Equation (13) lead to a far broader dispersion curve width as given in Figure 2, which also necessitated a far larger kH value needed to have the frequency as used by Mercini  . This in turn leads to a much bigger value for a lower bound for Equation (16) than what would obtain numerically if we used Equation (10) for dispersion. Detuning the sensitivity of this ratio to be for any is extremely important to the viability of our physical theory about how dark matter plays a role in inflationary cosmology. We find that this result is still not sufficient to match the cosmic ray problem  since Equation (10) gives us:
The whereas we would prefer to find .
7. Can with a Modified Dispersion Relationship?
The answer is no even after a modification of our dispersion relationship:
With , then 3 put in. However, even with a value of put in Equ-
ation (21) we obtained, for and
which has a very different lower bound than the behavior seen in Equation (16). If we pick as suggested by T. Jacobson  to try to “solve” the cosmic ray problem, we then find that Equation (22) approaches unity which thereby throws into question the Trans-Planckian dark energy hypothesis. Indeed, we believe that the entire Trans-Plankian model of Dark energy makes initial conditions, which contravene known astrophysical cosmic ray data  that has been collected in the last decade. Graphically, having even for Equation (21) in Figure 3 creates a dispersion versus momentum graph, which is much greater in width than Figure 1 which has a much larger value. Appendix entry 2 shows us that we still could not match the beta coefficient values  needed to solve the cosmic ray problem of special relativity.
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 values if
Figure 3. Graph of 3rd dispersion relationship against momentum which is still too broad in width, and has .
above. This was, however, done for a physically unacceptably large value  while we wanted, instead in order to solve the cosmic ray problem  . Our additional modifications of dispersion relationships as noted in Appendix 2 still lead to unacceptably large dark energy versus total energy values. We then conclude that the Trans Planckian dark energy hypothesis contravenes known solutions to the cosmic ray problem of special relativity and is thereby in need of substantial revision. And we think that this document should be compared against the predictions given in  , since modification of the Dark Energy hypothesis may impact gravity theories. Their confirmation or rejection will be affecting predictions done by Corda, and require careful analysis, and are integral to a fuller understanding of scalar-tensor gravitational alternatives to General Relativity.
This work is supported in part by National Nature Science Foundation of China grant No. 110752.
Appendix Entry 1: Deriving the Bogoliubov Coefficient for Section III
Part I. Initial Assumptions
We derive the Bogoliubov coefficient, which is used in Equation (16) of the main text. We refer to Mersini’s article  which has a Bogoliubov coefficient which takes into account a deviation function , which is a measure of deviation from thermality  in the spectrum of co moving frequency values over different momentum values. Note that is part of a scale factor and so that “momentum” . Also if we are working with the conformal case of appearing  in:
then for small momentum:
if “momentum” , 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  since it permits a clean derivation of the Bogoliubov coefficient which has the deviation function . We begin with  :
where we get an appropriate value for the deviation function  based upon having the square of the dispersion function obey Equations (1) and (2) above for . Note, is a maximum momentum value along the lines Magueijo  suggested for an Plank energy value.
Part II. Deriving Appropriate Deviation Function Values
We look at how Bastero-Gil  obtained an appropriate value. Basterero-Gil wrote:
where and where is in the Trans-Planckian regime but is much greater than . We are determining what B should be in Equation (16) of the
main text provided that as which will lead to specific
restraints we place upon as well as above. Following Bastero-Gil  , we write:
When we get  
which then implies . Then we obtain:
Part III. Finding Appropriate and Values
We define, following Bastero-Gil 
where we have that
whereas we have that
where denotes either out or in. Also:
which lead to:
as well as
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 and , , then
If and , , then
If and , , then
If and , , then
If and , , then
If and , , then
We need with to get our results via this Trans-Plankian model to be consistent with physically verifiable solutions to the
cosmic ray problem.