1

Dark Matter and Dark Energy from the Higgs Field© Joseph G. Depp, PhD1

May, 2015 (Updated 9/9/2016 for clarity)

(Updated 11/2/2017 to add Appendix C)

Abstract The recent discovery of the Higgs particle[1], the 1998 finding of Sen[2] that the Higgs particle can under certain circumstances be superluminal, and the more recent results from the LHCb experiments[3] have compelled me to reexamine some of the theories and conjectures that had been proposed in the late 1960s and early 1970s[4] [5]. The reexamination is done in the context of current knowledge of the Higgs field. The goal in doing such a reexamination is to obtain a single physical description that provides an explanation for several experimentally observed phenomena...vacuum polarization, dark energy, and dark matter. The proposed hypothesis does not admit the possibility of faster-than-light travel or faster-than-light information transfer. 1. Introduction We begin by examining the Higgs potential for the special case of a traveling soliton. From this examination comes a form of the Higgs Lagrangian density with a velocity-dependent potential. We show that the soliton solution leads to a Hamiltonian energy density that is Lorentz invariant. Next we look at the implications of the velocity-dependent potential in terms of vacuum polarization, dark energy, and dark matter. 2. The Velocity-Dependent Potential We begin with the usual definition of the Lagrangian density for the Higgs field.

L ( )22

0

22

2

1 φφλφµ −−∂= (1)

where 4

202

0

m=φ (2)

We now look for a soliton solution that is moving in the positive x-direction with velocity, v:

−= )(2

tanh2

00 vtxmmφ (3)

and substitute � = �/� and �� = ��.

−= )(2

tanh2 0

00 xxmm βφ (4)

1 PhD in theoretical physics from Carnegie Mellon University, joe_depp@sbcglobal.net © Copyright 2015 Joseph G. Depp. All rights reserved. Cite as: Depp, Joseph G., "Dark Matter and Dark Energy from the Higgs Field", (June, 2015) Library of Congress TXu001971256

2 Inserting φ into the field equations and simplifying we get:

{ }λβ −− 21 sech4

− )(2

00 xx

m β = 0 (5)

The field, φ , is only a solution to the field equations if

( )21 βλ −= (6)

And so we have an expression for the velocity-dependent Higgs Lagrangian density:

L ( )( )22

0

2222

0

12

1

2

1 φφβφφ −−−∂∂−

∂∂=

xx (7)

The Hamiltonian energy density is given by:

H ( )( )22

0

2222

0

12

1

2

1 φφβφφ −−+∂∂+

∂∂=

xx (8)

Inserting the expression forφ , we obtain:

H 8

4

0m

= sech4

− )(2

00 xx

m β (9)

Note that the resulting energy density is Lorentz invariant and that the maximum of the energy density does not depend on β but only on the particle mass. When all the factors of hand c are included, the

maximum, 0H , is given by:

3

02

00 8

=h

cmcmH (10)

For 1<β , we have the usual sombrero potential with the center maximum equal to ( )21 β− 4

0φ as shown

in Figure 1.

Figure 1. Velocity-dependent potential for 1<β

3

For 1>β , the potential has a central potential well with depth ( )12 −β 4

0φ as shown in Figure 2.

Figure 2. Velocity-dependent potential for 1>β

Clearly the potential with 1>β has a minimum at 0=φ and so it constitutes a true vacuum solution as

opposed to the false vacuum solution at 0φφ = when 1<β . It may be conjectured that the light barrier,

which separates the subluminal part of the universe from the superluminal part of the universe, also separates the false vacuum from the true vacuum and so gives stability to the subluminal part of the universe. The velocity-dependent potential can be interpreted as the potential seen by a Higgs particle as it moves through the Higgs field. Even more interesting is that the potential energy of the field smoothly deforms from positive to negative through the value zero at 1=β . When 1=β , the Lagrangian density becomes just the density for a massless boson that moves with the speed of light in a vacuum. Such a massless spin-0 boson is an alternative to the usual candidate of a massless spin-2 boson for the graviton. 3. Superluminal Relativity For more than a hundred years, physicists have accepted the premises contained within the theory of special relativity. One of these premises prohibits the acceleration of a subluminal particle with non-zero rest mass to the speed of light in a vacuum. In this article we are not suggesting that the prohibition can be broken but, rather, we are examining the possibility that certain elementary particles might exceed the speed of light through other means. For nearly as long, physicists have recognized that the mathematics of special relativity seems to contain the possibility of a superluminal universe...a universe in which all the particles travel faster than the speed of light. However, such a particle would have to be very special. It would have to be a boson with zero spin, zero charge, and non-zero rest mass and it would only interact through gravitation. Clearly, such a boson can be found in the description of the Higgs particle. While there might be other such bosons, we will use the term Higgs particle in this article to stand for all such bosons.

4 Historically, theorists have rejected any theory that contained a superluminal massive particle. But, in 1998, string theorists acknowledged that the Higgs particle might, under certain circumstances, exceed the speed of light in a vacuum[ 2]. Let us consider the extension of special relativity to a superluminal universe. For this section, we draw heavily on the work of Hill and Cox[6]. In Figure 3, we have plotted the following equations:

12 <β 12 >β

2

20

1 β−=−

cmE

12

20

−=+

βcm

E

(11)

420

22 cmEP −= −−

420

22 cmEP += ++

Figure 3 Normalized Energy and Momentum versus Particle Beta

5 Clearly one can envision the energy of the particle as a velocity-dependent potential with an infinite barrier at β =1. According to classical physics, no particle with non-zero rest mass can cross the infinite barrier. However, we are considering a Higgs particle that obeys the rules of quantum physics. Therefore, we should consider the possibility that a Higgs particle might quantum-tunnel through the infinite barrier mediated by the Higgs field.

4. Consequences of the Velocity-Dependent Potential 4.1 Vacuum Polarization and Dark Energy

We consider that momentum is conserved in any tunneling that might occur. Assuming that momentum is conserved we have:

420

2420

2 cmEcmE −=+ −+ (12)

Both momentum and energy must be conserved. However, conserving both momentum and energy is not possible without other considerations. Consider a particle in the subluminal universe with energy, −E . When it tunnels to the superluminal universe it will have an excess of energy.

( ) 420

2420

2cmEcmE −=+− −− ε (13)

Solving for the energy surplus, ε, we obtain:

42

02 2 cmEE −−= −−ε (14)

We immediately see that, for tunneling to occur from the subluminal to the superluminal, the energy must be greater than or equal to the square root of two times the rest mass energy. At or above this energy, the tunneled particle will emerge in the superluminal universe as a real particle with energy,

( ) 420

2 2 cmEEE −=−= −−+ ε . This observation provides the first suggested experimental test of the

superluminal conjecture. If there are fewer Higgs particles created in high energy collisions than is expected from the Standard Model, quantum tunneling into the superluminal universe might be the cause. A particle that tunnels from the superluminal universe into the subluminal universe will not have the minimum energy required to emerge as a real particle in the subluminal universe.

( ) 420

2420

2 cmEcmE −+=+ ++ ε (15) Solving for the energy deficit, ε bar, we obtain

++ −+= EcmE 420

2 2ε (16) As can easily be seen, regardless of the energy, +E , with which the superluminal particle starts, it will always have an energy deficit. For reasons explained below we are primarily interested in the case when the energy, +E , is much less than the rest mass energy. Setting +E = 0, we see that such a superluminal

6 particle will, in fact, have an energy deficit of square root of two times the rest mass energy. Such a particle can only briefly exist in the subluminal universe. For the moment, the important feature that we want to highlight is that a Higgs particle tunneling from the superluminal universe to the subluminal universe will appear as a virtual particle. Such a particle can decay into a cascade of virtual bosons and, eventually, virtual fermions only to return to its initial state as a Higgs particle in the superluminal universe. If the density of the Higgs particles in the superluminal universe is high enough then such tunneling will give rise to the well-known phenomena of vacuum polarization in the subluminal universe. We will address the question of the density of Higgs particles in the superluminal universe below. Now let us consider the superluminal Higgs particles. According to general relativity such particles would contribute to the expansion of the universe providing a source for dark energy. (See Appendix B for quantitative details.) In the process, the particles lose energy and eventually go to the ground state while the momentum goes to cmP 0=+ . (When a theory of quantum gravity is found, we expect that there will

be a minimum value of gamma that, while very small, will be non-zero. It will correspond to the ground state of the superluminal particle. See note on p.9.) Now we can also see how the surplus energy is dissipated when a subluminal particle tunnels through to the superluminal universe. The surplus energy, ε, is transferred to the expansion of the universe. A superluminal particle in its ground state can no longer transfer energy or momentum and so can no longer contribute to the expansion of the universe. However, these particles still have a significant role to play. The density of these superluminal Higgs particles may be as high, or higher, than one particle per Planck volume and yet they do not contribute to the overall energy density because they have zero energy. These particles are the primary contributors to the vacuum polarization. Since the discovery[7][8], in 1998, that the expansion of the universe is accelerating, physics has been faced with a paradox, often called the vacuum paradox. The energy density of the universe required to produce the acceleration is many, many orders of magnitude less than that produced by vacuum polarization. However, if superluminal particles in the ground state are producing vacuum polarization then the paradox is resolved. The difference between the density of energetic superluminal particles, associated with the accelerating universe, and the density of superluminal particles in the ground state, associated with vacuum polarization, accounts for the paradox that exists between the calculation of the energy density of the universe based on acceleration and that based on vacuum polarization. 4.2 Dark Matter In the above, we have tacitly assumed that the superluminal universe is superimposed on the subluminal universe. Such a superposition is possible if the only interaction is gravitational. Now consider what happens in the superluminal universe. Classically, the vast majority of particles carry little or no energy but travel at an extremely high speed. Therefore, in the absence of a gravitational interaction with the subluminal universe, the particles would have a uniform density. However, when we add the galactic clumps of matter in the subluminal universe there is a gravitational interaction. The galactic clumps form gravitational wells in the superluminal universe and the gravitational interaction with the superluminal particles form volumes of higher density in the superluminal universe in the neighborhoods corresponding to galaxies in the subluminal universe. When a superluminal particle approaches a gravitating mass its momentum and energy increase. However when the energy of a superluminal particle increases its velocity is reduced. Consequently the particles

7 spend longer in the vicinity of the gravitating mass causing an increase in density around the gravitating mass. Based on these assumptions, a quantitative model of dark matter is derived in Appendix A to this paper. The model is derived from first principles and produces very good agreement with experimental results. The net result is that the Higgs particles in the superluminal universe are the source of the effects that are currently attributed to dark matter. Computer simulations have suggested that cold dark matter (CDM) is required to obtain the clumping of baryonic matter in a time consistent with the evolution of the subluminal universe. However, these models assume that the dark matter is moving at subluminal speeds. We encourage those who have such models to run them again assuming that the dark matter can respond to changes in the baryonic distribution at superluminal speeds. We envision a cyclic process in which baryonic matter clumps a bit, dark matter clumps with it, more baryonic matter clumps, more dark matter clumps, etc. 4.3 Sources of Superluminal Higgs Particles It is important to understand that we are not talking about tachyons. A tachyon is a particle that appears in the quantum theory for subluminal particles and as such is generally accepted as a sign that the theory is incorrect. We are talking about a separate part of the universe wherein all the particles move with a speed greater than the speed of light in a vacuum. The superluminal and subluminal parts of the universe are only coupled gravitationally and through the Higgs field. Where do these particles come from? There are three possible sources that are not mutually exclusive. Obviously we have already discussed one such source, the tunneling of Higgs particles that are created in energetic interactions in the subluminal universe. The second possibility is that Higgs particles became locked into the superluminal universe during inflation shortly after the Big Bang. Inflation, by its very definition, is a time when the universe was expanding at a speed greater than the speed of light in a vacuum. The third source, and perhaps the most controversial source, is the possibility that the Higgs particles are continuing to enter the superluminal universe from an external source. To examine this possibility we need to revisit the conjecture that our universe is the interior of a black hole in a parent universe[9][10][11][12][13]. Matter, that continues to enter the black hole in the parent universe, appears in our universe as Higgs particles that are traveling with a speed greater than the speed of light in a vacuum. This conjecture is supported by the experimental observation that the acceleration in the expanding universe only overcame the gravitational attraction after the universe was about nine billion years old.

8 5. Conclusions We have set forth the conjecture that there is a superluminal universe that allows quantum-tunneling of Higgs particles in both directions and we have suggested several ways to experimentally explore the validity of the conjecture. There are four corollaries that follow: First, the energetic superluminal Higgs particles provide the pressure required to drive the accelerating expansion of the universe as currently attributed to dark energy. Second, the superluminal Higgs particles eventually lose their energy to the expansion of the universe but continue to contribute to vacuum polarization in the subluminal universe and these particles also provide a resolution of the vacuum paradox. Third, the gravitational wells created in the superluminal universe by the clumps of galactic matter in the subluminal universe form volumes of higher density in the superluminal universe resulting in the effects that are currently attributed to dark matter. Fourth, the energetic superluminal Higgs particles are continually resupplied. The hypothesis is falsifiable: First, the LHC is now running with a much higher center-of-mass energy. The model predicts that the production rate for Higgs particles should have a discontinuous reduction for particles with laboratory energies greater than the square root of two times the rest mass of the Higgs particle. Failure to find such a discontinuous reduction in the production rate would invalidate the hypothesis. Second, the model predicts that there is a single scaling constant for dark matter that is independent of the galaxy under consideration (see A.9 and A.10). If the model is applied to a number of different galaxies and it is not possible to find a single value for the scaling constant that adequately satisfies all conditions, then the model would be invalidated.

9 Note on the ground state energy We take a naïve approach to estimating the ground state for a superluminal particle by asking that it’s Compton wavelength be twice (a winding number of two) the circumference of the universe, �, where � is the diameter of the observable universe, currently estimated to be 93 billion light years. In this case we have:

� = ���� = ��� and � = 2� (17)

From which we obtain: ���� = ℏ/( ���) (18) Using the best available values for the parameters above, the numerical value of ���� is 1.779 x 10-45. Whether by design or serendipity, ���� compares favorably with the current best estimate of the gravitational coupling constant:

�� = ����

ℏ� = 1.7518 x 10-45 (19)

where G is the gravitational constant and is the mass of the electron. These two values differ by only 1.6% with an uncertainty of ±1.3% due to experimental uncertainties in � and L.

10 Appendix A: Dark Matter in the Milky Way The hypothesis, set forth in this appendix, is that dark matter is composed of superluminal Higgs particles. Far from a gravitating body, there is a background of superluminal Higgs particles. The background is defined by the dark energy which has an estimated magnitude of 6.9x10-27 kg/m3. The units used in this appendix are the solar mass ("⊙) and the kilo parsec. In these units, the dark energy, $%& , is approximately 1.0x102 "⊙/kpc3 (solar mass per cubic kilo parsec). We assume that dark matter only interacts with subluminal matter and not with itself. The assumption is not necessary but it is consistent with what most experiments to date suggest. If future measurements find that there is a weak interaction within dark matter, such an interaction can be easily accommodated by the model.

(In this appendix we use gamma as the independent variable such that � = ���� = �'��

()�*+ .)

A given particle in the background has an energy, ��, and a corresponding speed, ��. As the particle approaches a gravitating body, the kinetic energy increases but the speed decreases as shown in Fig A1. Conversely, as the particle moves away from the gravitating body, the particle loses energy but gains speed. The net result is that there is an increase in density of particles traveling along a given line in the vicinity of the gravitating body.

Figure A1 Dark Matter Density Increase

When the particle is a distance , from the gravitating body, the energy, �, is given by:

� = �� − �� / 0(1)�1� 234

4� (A.1)

We are calculating the increase in kinetic energy of an in-falling particle from , , where , is the distance from the galatic center to the outer edge of the dark matter halo, the place where the density of dark matter falls to the background density of superluminal particles. The value for , , in the Milky Way, is not well defined in the literature. It varies from 36 kpc to 100 kpc. For this reason, we take , to be a parameter bounded by these values. For many other galaxies, it may be possible to obtain an independent measurement of , from gravitational lensing.

11 "(3), shown in Figs. A2(d) and Fig. A3, is he total visible (baryonic) matter of the gravitating body contained within a radius 3. In principle, "(3) should be included in the integral. However, the following approximation is made:

���� = ����� 6 ��"(,)7 8+4 − +

4�9 (A.1a)

The approximation is exact when , is greater than about 20 kpc since "(,) is a constant beyond about 20 kpc. The approximation introduces only a small error when , is less than 20 kpc. The approximation is made so that we can continue with an analytic discussion rather than having to move immediately to the numerical results. Making such an approximation is in keeping with the spirit of this appendix, which is to show how superluminal particles explain dark matter. There will be ample opportunity later to do more precise calculations for the Milky Way and other galaxies. The next steps are just algebra:

� = �� 6 � 0(4)��� 8+

4 − +4�

9 (A.1b)

� = �� 6 � 4:�4 84�*4

4�9 (A.1c)

Solving for � gives: � = ;'

+*<:�<8<�=<

<� 9 (A.1d)

The resulting speed is:

�� = 1 6 +;'�

?1 − +�

4:4

4�*44�

@� (A.2)

where ,A(,) ≡ 2"(,)7/��.

Figure A2 Visible (Baryonic) Mass in the Milky Way[14]

Recently, Ibata[15] has suggested, based on observations of the Andromeda galaxy, that most large galaxies, including the Milky Way, might have 10%-15% more mass in its outer region which he calls the

12 extended disk. To accommodate this finding we have extended the total visible mass by 15% to 6.95x1010. The total visible mass used in the following calculations is shown in Fig A3.

Figure A3 Total Visible Mass in the Milky Way

Even more recently, data from Chandra X-ray Observatory et. al. suggest that there is a large amount of hot gas that extends for hundreds of thousands of light years. We have made no attempt to include these findings in this model, choosing to wait until there is confirmation. However, we do note that inclusion of these findings in the model is a straightforward modification of the total visible matter as given in Fig. A3. Note that we are justified in expanding (A.2) in terms of small ,A/, since, even for something as large as the Milky Way, ,A/, < 9x10-7 as can be seen in Fig A4.

Figure A4 Rs and Rs/R

To simplify the notation, we make the following definition:

C(,) ≡ 4:(4)4

4�*44�

(A.3)

where 0 E C(,) ≪ 1.

13 Note that the percentage increase in � is very small:

;;'

= ++*G(<)

�≅ 1 6 I(4)

� (A.4)

We note that, to first order in C(,):

)'�

)� = ;'�K+;'�K+*I(4) (A.4a)

The increase in particle density, above the background density, along a given line is given by:

)') − 1 ≃ I(4)

�(+K;'�) (A.5)

Since the particles can come from any direction, we must sum over all directions which yields an expression for the number density of the particles, M%0(,, ��), as a function of , and ��.

M%0(,, ��) = 4M�(��) ?)') − 1@

�= M�(��) ? I(4)

+K;'�@

� (A.6)

M�(��) is the background density of superluminal particles with energy between �� and �� 6 2��, far from the gravitating body. To obtain the contribution of a particle at distance , and with energy � to the dark matter mass density, $%0(,, ��), we must multiply the number density by the particle mass.

$%0(,, ��) = �M%0(,, ��) ≅ �M�(��) ? I(4)+K;'�

@� (A.7)

To obtain the total mass density of dark matter, $%0(,), we must sum over ��.

$%0(,) = �C(,)� / M�(��)O�

+(+K;'�)� d�� (A.8)

We define a density, $�, such that:

$� = � / M�(��)O�

+(+K;'�)� d�� (A.9)

Note that $� is a scaling constant that is independent of the galaxy under consideration. The relevant physical quantity is:

$%0(,) = $�C(,)� (A.10) One of the pieces of experimental evidence, that gave rise to the concept of dark matter, is the observation that most stars in a galaxy rotate around the gravitational center with a speed that is approximately independent of ,. To obtain this dependency, it is necessary that the total mass in the galaxy contained within a radius ,, "QRQST(,), increases approximately linearly with , and that the total dark matter, "%0(,), is the dominant contribution to the total mass.

14 We have that

"%0(,) = 4$� / C(3)�3�234� (A.11)

and "QRQST(,) = "%0(,) 6 "(,) (A.12) where "(,) is obtained from Fig A3. The tangential velocity as a function of , then given by:

U(,) = V"QRQST(,)7/,W+/� (A.13)

Figure A5 Angular Velocities in the Milky Way

To obtain values for the parameters , and $� we use (A.13) and compare the results to experimental data from Fitch, Blitz, and Stark[16] (FBS) as shown in Fig. A5. The symbols denote the experimental data, the solid curve is the best fit obtained by FBS and the dashed curve is the best fit obtained for (A.13) with $� = 4.3x1019 "⊙/kpc3 and , =97 kpc. With these values our model also obtains a tangential velocity of 221 km/s at a distance of 8 kpc and a velocity of 222 km/s at a distance of 8.5 kpc, the approximate distance between the solar system and the galactic center. These values are in good agreement with the experimental value estimated to be 220 km/s at 8.5 kpc. (See the note at the end of this appendix (p.17) for a discussion of the magnitude and interpretation of $�.) The tangential velocity as a function of distance from the galactic center is shown in Fig. A6 out to a distance of , . The maximum tangential velocity is 242 km/s at a distance of 24 kpc.

15

Figure A6 Tangential Velocity

For comparison we also show, in Fig A7, the tangential velocity from Fig A6 overlaid on the velocity data from FBS out to a distance of 20 kpc.

Figure A7 Tangential Velocity Overlaid on FBS Data

The total matter enclosed in a sphere of radius , is shown out to a distance of , in Fig. A8.

Figure A8 Total Matter in the Milky Way

16 The total matter enclosed in a radius of , = 97 kpc is 6.6x1011 "⊙ and, since the total visible mass from Fig. A3 is about 7.0 x 1010 "⊙, the remaining total dark mass is about 5.9x1011 "⊙. The dark matter density, $%0(,), is shown as a function of , out to a distance of , in Fig. A9.

Figure A9 Dark Matter Density

In conclusion, we have generated a hypothesis that there is a superluminal background of Higgs particles that is responsible for the effect attributed to dark matter. We assume that the dark matter effect is predicated on the property of superluminal particles that causes their speed to decrease as the energy increases. With that assumption, we formed a model of dark matter from first principles that explains the physical source of dark matter and yields quantitative results in good agreement with all of the observed features of stellar velocities in the Milky Way.

17 Note on the constant, $� $� is defined in A.9 and has a magnitude of 4.3 x 1019 "⊙/kpc3. Clearly, the physical interpretation of $� is that it is the density of dark matter just outside the event horizon of a Schwarzschild black hole. (See A.3 and A.10.) Consider the mass density, $��, defined by:

$�� = � / M�(��)O� d�� = �X (A.14)

where N is the total number of particles per unit volume. The mass density as seen by a subluminal observer moving relative to the volume associated with $�� is:

$RYA 1Z 1 = � / ���M�(��)O� d�� = ���

�X (A.15)

It seems reasonable to think that, ignoring the factor of +

(+K;'�)� for ��� ≪ 1:

$� = $�� and that $RYA 1Z 1 = $%& (A.16) giving, $� = [\]

;'� (A.17)

Recalling that $%& = 1.0 x 102 "⊙/kpc3 we find that:

��� = [\]

['= 7.3 x 10-18 (A.18)

or ⟨��⟩1�A = 2.7 x 10-9 (A.19)

18 Appendix B: Dark Energy This appendix builds on the work presented in Appendix A. To understand how the effect, known as dark energy, comes from the superluminal Higgs particles, we first consider the standard development of the energy-momentum tensor, as given in most introductory texts on general relativity, for example Adler [17], p262-269. The tensor is composed of two parts,

`ab = "ab 6 cab (B.1) the mass tensor

"ab = $�(�)da(�)db(�) (B.2) and the pressure tensor

cab = e(�)Vda(�)db(�) − fabW/�� (B.3) In principle, the mass density, $�(�), and the pressure density, e(�), are scalar fields. However, when dealing with the universe as a whole, they are usually taken as constants. da(�) is a four-velocity field, as shown below and fab is the spacetime metric.

da(�) = ghi

gA (B.4)

The elements of the mass tensor are given by:

"�� = $��� "�� = $����� "�j = $������j (B.5)

" = $���

klllm 1 �� �j �n

�� ��� ���j ���n

�j ���j �j� �j�n

�n ���n �j�n �n � opppq

We approach creating the energy-momentum tensor for a superluminal Higgs particle in the same way. First, we note that, if the superluminal Higgs particles do not interact or interact only very slightly, then we can assume that the pressure density is zero. Consequently, we can write the energy-momentum tensor for a single particle as:

`(��) = − ����

klllm 1 �� �j �n

�� ��� ���j ���n

�j ���j �j� �j�n

�n ���n �j�n �n� opppq (B.6)

19

Note that the sign change comes from the fact that 8gQgA9

�in the superluminal universe is minus 8gQ

gA9�in

the subluminal universe. We can take the term ��� inside the brackets to obtain:

`(��) = − �

kllllm��� ��(��

� 6 1 )r

) ∙ ∙

∙ (��� 6 1) )r�

)� (��� 6 1) )r)t

)� ∙∙ ∙ ∙ ∙∙ ∙ ∙ ∙o

ppppq (B.7)

The terms with )r

) are just the elements of a unit vector in the direction of the velocity. We now consider

summing a collection of such particles, all with the same value of ��, but random velocity directions.

`(��) = − �M(��)klllm��� 0 0 0

0 (��� 6 1)/3 0 00 0 (��� 6 1)/3 00 0 0 (��� 6 1)/3o

pppq (B.8)

The quantity M(��) is the number of particles per unit volume with value ��. Next, we integrate �� from zero (or ����) to infinity, obtaining:

` = − �X

klllllm��

� 0 0 00 8��

� 6 19 /3 0 00 0 8��

� 6 19 /3 00 0 0 8��

� 6 19 /3opppppq

(B.9)

The standard model of cosmology, based on the FLRW metric, requires as inputs:

`�� = − �X��� = −$%& (B.10)

and v = ̀ ww = �X = $�� (B.11)

See the note at the end of Appendix A (p.17) for a discussion of these values in terms of the superluminal hypothesis.

20 Appendix C: Energy Distribution This appendix builds on work presented in the note on p17. The quantities M�(��), $�, $%& , and N are as defined in Appendix A and as used in A.14-A.19. We would like to investigate the distribution of the superluminal particles, M�(��). Because we are dealing with particles that are assumed to either not undergo mutual interactions or to have minimal mutual interactions, we must search for a method of equilibration. The only realistic possibility is interaction with baryonic matter and the only place where baryonic matter is dense enough and hot enough to provide significant energy transfer is within a star. Therefore, we begin by assuming that the form of the distribution is given by the Maxwell-Jüttner relativistic distribution. Since we are only interested in large values of x, we need only keep the leading term in ��: M�(��) = N���z*{;' (C.1) where

x = �'��

n| (C.2)

We first determine the total number of particles per unit volume, N: X = N� / ��z*{;'2�� = N�/x�O

� (C.3) Next, we calculate the expectation of �� and the expectation of ���:

�� = +} / ��M�(��)2�� = 2/xO

� (C.4)

��� = +

} / ���M�(��)2�� = 6/x�O� (C.5)

However, we have previously calculated (A.15) that ��� is 7.3 x 10-18, from which we obtain

x = 9.1 x 108. From (C.2) we can estimate that the temperature is approximately 2 million K. Red dwarf stars outnumber all other stars by a significant factor and the interior temperature of red dwarf stars is of the order of 10 million Kelvin. Consequently, the estimated temperature, T, compares favorably with our intuitive expectations based on an interaction between the superluminal particles and the baryonic matter deep within a star. Beginning only with the ratio

[\]['

from (A.18), we have shown that the estimated temperature of the

dark energy background, $%& , is consistent with equilibration between the superluminal particles and the core temperature of stars in the universe.

21 References 1. Taylor, Lucas (4 July 2012). "Observation of a New Particle with a Mass of 125 GeV". CMS Public Website. CERN. http://cms.web.cern.ch/news/observation-new-particle-mass-125-gev. Retrieved 4 July 2012.

2. Sen, Ashoke (28 May 1998). "Tachyon Condensation on the Brane Antibrane System" arXiv:hep-th/9805170v3

3. LHCb Website, Summer 2013, http://lhcb-public.web.cern.ch/lhcb-public, (to be published).

4. Feinberg, G. 1967 Possibility of faster-than-light particles. Phys. Rev. 159, 1089–1105. (doi:10.1103/PhysRev.159.1089)

5. Pathria, R. K. (1972). "The Universe as a Black Hole". Nature 240 (5379): 298. Bibcode 1972Natur.240..298P. (doi:10.1038/240298a0)

6. Hill, J. and Cox, B. 2012 Einstein’s special relativity beyond the speed of light Proc. R. Soc. A (2012) 468, 4174–4192. (doi:10.1098/rspa.2012.0340)

7. Adam G. Riess et al. (Supernova Search Team) (1998). "Observational evidence from supernovae for an accelerating universe and a cosmological constant". Astronomical J. 116 (3): 1009–38. arXiv:astro-ph/9805201. Bibcode:1998AJ....116.1009R. doi:10.1086/300499.

8. S. Perlmutter et al. 1999 ApJ 517 565 S. Perlmutter et al. (The Supernova Cosmology Project) (1999). "Measurements of Omega and Lambda from 42 high redshift supernovae". Astrophysical Journal 517 (2): 565–86. arXiv:astro-ph/9812133. Bibcode:1999ApJ...517..565P. doi:10.1086/307221.

9. E. Fahri and A. H. Guth (1987). "An Obstacle to Creating a Universe in the Laboratory". Physics Letters B 183 (2): 149-155. Bibcode 1987PhLB..183..149F. (doi:10.1016/0370-2693(87)90429-1).

10. N. J. Popławski (2010). "Radial motion into an Einstein–Rosen bridge". Physics Letters B 687 (2-3): 110–113. arXiv:0902.1994. Bibcode 2010PhLB..687..110P. (doi:10.1016/j.physletb.2010.03.029).

11. N. J. Popławski (2010). "Cosmology with torsion: An alternative to cosmic inflation". Physics Letters B 694 (3): 181-185. arXiv:1007.0587 (doi:10.1016/j.physletb.2010.09.056).

12. N. J. Popławski (2012). "Nonsingular, big-bounce cosmology from spinor-torsion coupling". Physical Review D 85 (10): 107502. arXiv:1111.4595. (doi:10.1103/PhysRevD.85.107502).

13. A. Retter and S. Heller (2012). "The revival of white holes as Small Bangs". New Astronomy 17 (2): 73-75. arXiv:1105.2776. (doi:10.1016/j.newast.2011.07.003).

14. Remmen, Grant. "A New Assessment of Dark Matter in the Milky Way Galaxy." Journal of Undergraduate Research in Physics 19 (2010): 1.

15. Ibata, R. et al. (2005). "On the accretion origin of a vast extended stellar disk around the Andromeda Galaxy". The Astrophysical Journal 634 (1): 287–313. arXiv:astro-ph/0504164. Bibcode:2005ApJ...634..287I.doi:10.1086/491727

16. Fich, Michel, Leo Blitz, and Antony A. Stark. "The rotation curve of the Milky Way to 2 R (0)." The Astrophysical Journal 342 (1989): 272-284.

17. Adler, Bazin, and Schiffer. “Introduction to General Relativity”, McGraw Hill 1965.