1. The document estimates the maximum intensity of dark matter glow based on Dr. Hewett's Time Symmetric Cosmology theory, which predicts dark matter is a boson resulting from Hawking evaporation of primordial black holes. 2. Using classical approximations, the author estimates the total photon intensity of the Andromeda galaxy's dark matter halo would be 4.67x1024, far greater than its visible light intensity of 3.19x109. 3. However, the author acknowledges the model's assumptions and approximations likely overestimate the intensity by many orders of magnitude, and more rigorous modeling is needed to test if dark matter glow could be detectable.

1. 1
Estimating the Maximum Intensity of Dark Matter Glow
Kevin L. Romans
Texas A&M Kingsville, Texas, 78363, Department of Physics & Geosciences
Submitted December 11, 2014; Revised December 14, 2014
ABSTRACT
In Dr. Lionel Hewett’s theory, Time Symmetric Cosmologyi
(TSC), an attempt to explain the origin of dark matter is made. The
theory predicts that dark matter is a boson that results from the
Hawking Evaporation (/Radiation) of Primordial Black Holes to a
final residual ground state; its predicted mass is 4.5E-34 kg. This
residual Dark Matter Particle (DMP) can form a Bose-Einstein
condensate with another and perforce must decay back down to the
ground state. The most probable mode of decay is the symmetric
release of two photons, again via Hawking Radiation, ~10 nm in
wavelength. Using a series of classical approximations, a simplistic
model was created to estimate the total photon intensity of a given
dark matter halo. Looking at Andromeda a value of 4.67E+24 was
obtained for the dark matter photon intensity while a visible photon
intensity of 3.19E+09 was calculated for comparison. This stark
difference needs to be accounted for with more rigorous modeling
methods.
Key Topics: Time-Symmetric Cosmology, Dark Matter Particles, Photon Intensity

2. 2
I. Introduction
The current standard model of cosmology, the Λ-Cold Dark Matter model ii, does not
present a satisfactory explanation for the origin and properties of Dark Matter (DM). Time-
Symmetric Cosmology (TSC) is an alternative theory to cosmological inflation authored by Dr.
Lionel Hewett of Texas A&M University – Kingsville. Assuming the universe began with a
physical singularity (or creation event) and utilizing the symmetry of time surrounding this
event, TSC is able to correctly predict over twenty cosmological variables such as: the
Cosmological constant, the Density of Vacuum Energy, and the Age of Photon Decoupling. In
developing the model Dr. Hewett was able to explain the origin of DM, the mechanisms behind
its strange properties, and a way to truly see DM beyond mere gravitational interactions.
TSC is composed of two models, Classical and Quantum. The classical model is similar
to the well-known Friedman-Lemaitre-Robertson-Walker model after inflation. The quantum
model expands upon the classical model to predict that the ensemble of first events which
immediately followed the creation event were Primordial Black Holes (PBH). Created out of a
high energy density confined to an infinitesimal spatial extent these PBH are predicted to have
zero velocity relative to their respective timelines and are kinetically cold. They also exhibit
Schwarzschild geometry and evaporate symmetrically via Hawking Radiation as to produce the
expected radiation density in the early universe, the excess of baryonic matter, and observed DM
content of the universe.
These PBH are assumed to radiate down to a ground state leaving a residual black hole.
This tiny particle retains its Schwarzschild characteristics, has all zero quantum numbers except
its mass, and this predicted mass of
#1
is enough to account for the observed DM content; it is a perfect candidate for a Dark Matter
Particle (DMP). Since the DMP has zero spin it is a Boson and two of them may form a Bose-
Einstein Condensate iii. When two DMP condense they will form a particle with twice the ground
state energy and perforce must evaporate via Hawking Radiation. In doing so it must release the
energy of one DMP’s rest energy.
#2
Given this small energy the most likely mode of decay is by the symmetric emission of two
photons each of wavelength,
#3
This suggests that DM should glow at the spectral border between high Ultra-violet and X-ray.
II. The Project Model
Before describing the model let us consider the simplifying assumptions needed. The
target of interest is a spherically symmetric, homogenous DM Halo. In order for the DMP’s to
condense their interaction needs to be perfectly symmetric; they must be in the same quantum
state within uncertainty. For the scope of this article this will simply mean that two particles are
in the same location with the same momentum vector. Finally, the probability of two DMP’s

3. 3
reacting under the above conditions is assumed to be unity. All of these simplifying assumptions
will have the effect of increasing the number and rate of emission events, thus increasing the
intensity of glow.
The goal of the model is to predict the photon intensity of a DM halo as seen from Earth.
In order to get this we first need the photon flux,
( 𝐹𝑙𝑢𝑥) =
(2 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 𝑝𝑒𝑟 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛)
(𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑡𝑖𝑚𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 )
(𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑎 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 𝑖𝑠 𝑎𝑛 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛) #4
A collision here is a classical one in which the cross section of a particle sweeps out a volume
and another identical particle’s geometric center falls into this volume. As a particle sweeps this
volume out in a given time interval it may interact with many particles. However not all of these
collisions will result in an emission event and this collision rate must be scaled by the
appropriate factor.
A. Same State Probability
We begin by deriving the factor that corresponds to the probability that two DMP’s will
be in the same state and therefore emit. This will be tackled in two parts by treating the
probability of them being in the same location and having the same momentum separately; the
total probability factor will simply be a product of the two.
1. Position
To get a handle on the probability that two DMP’s will be in the same location within
some reference volume (V) we need to estimate the uncertainty in size of the particle.
Assumption 1: The size of a 3-D Infinite Square Well (ISW), with ground state energy
equal to the rest mass energy of the DMP, represents the uncertainty in position of the
particle.
#5
We have made the further simplifying assumption that this uncertainty in position is
symmetric with respect to each of the three Cartesian axes (x,y,z). With this we can use the
length of one side of the (ISW) to be the uncertainty in position of the particle along each
dimension. Furthermore, we will use the square of the uncertainty in position along one
dimension (say x) and treat it like the particle’s collision cross-section. Solving for this
uncertainty we get,
#6
Using the Minimum Uncertainty Principle relationship we can get the uncertainty in
momentum of the particle still using the three dimensional symmetry,
#7
The following values have been calculated and are tabulated in Table 1 below,

4. 4
Δx
3.0E -9 m 1.75E -26 kg m/s
Table 1. Calculated uncertainties along one dimension
Assumption 2: The probability that at least two DMP’s share the same location is given by
the ratio of one’s volume to that of the reference volume.
#8
To see the logic behind this consider the following two dimensional example. Imagine
that you are throwing darts into an area A at a constant rate and at random locations. Within this
area there is a smaller area A1 seen in Figure 1. When averaged over time we expect the ratio of
the number of darts that landed in A1 vs A to be equal to the ratio of these areas. This is easily
extended into three dimensions as a ratio of volumes.
If we recognize that in a given reference volume V, filled with N
particles, where each one sees approximately N other particles, then the
probability scales up by a factor of N 2. Also, since the volume is arbitrary
we can divide P1 by another factor of V leading to probability per unit
volume,
#9
where n is the number density of the halo (N/V). This will be calculated by dividing the mass
density of the halo by the mass of the DMP.
#10
2. Momentum
Assumption 3: The DMP’s can be described with Kinetic Molecular Theory where their
source of energy is gravitational.
In this section we shall use the simplifying assumptions that the halo of interest has
DMP’s that are slow moving (relative to light) and have low density (or the distance between the
particles is large). Under these conditions Classical Statistical Mechanics applies. Starting from
Boltzmann Statistics one can derive a momentum vector distribution,
#11
A1
A
Figure 1

5. 5
where p is the magnitude of the momentum , β is , kB is the Boltzmann constant, T is the
absolute temperature, fB is the distribution, and 𝑑3
𝑝 is a volume element in momentum space.
Given some momentum p the distribution times the cube of the uncertainty in momentum
gives us the probability that the DMP’s have momentum within that small interval. However
what we are interested in is how often (or how likely) at each value of p neighboring momenta
are within our calculated uncertainty. This average is taken over all possible values of p,
#12
and evaluating this integral yields,
#13
3. Same State
As stated earlier the probability that at least two DMP’s will be in the same state is the
product of equations 9 and 12. Carrying out the product and simplifying we get the total
probability per unit volume,
#14
B. Relaxation Time
Assumption 4: The relaxation time for a DM halo is equal to the average time between
collisions found in Classical Collision Theory.
Here the relaxation time refers to the average amount of time needed for the system
(particles in the halo) to be randomly shuffled and the probability relations can be reapplied. The
result can be quoted directly from any undergraduate university physics textbook iv,
#15
where the speed v used here is the approximate root-mean-square (rms) speed of the particles in
the halo. Local particles a small radial distance apart will be moving at slightly different speeds.
Although they cannot condense they can still interact gravitationally. Given a small region of
these weakly interacting particles we would expect each one will be jostled in many directions
with varying strengths. Over time this region will come to contain randomly moving particles
(while still maintaining its orbit); this propagation of energy resembles thermal conduction. We
can then invoke the equipartition theorem and the equation for the kinetic energy per particle to
get rid of β,

6. 6
#16
To use equation 14 effectively we restricted our analysis to halos coupled to a parent galaxy with
a known rotation curve. The speed v is the value taken from averaging the speeds along the flat
section of the curve; this average speed will be approximately constant all the way out to the
edge of the parent galaxy. At this point we have everything we need to fill out equation 4.
C. Photon Flux
Plugging in equation 13 for the probability per unit volume, equation 14 in for the
average time between collisions, and use equation 16 to get rid of β we get,
#17
where F is the number of emitted photons per unit time (or photon flux) per unit volume emitted
by a DM halo. We can multiply through by a spherical volume of radius R to get the total photon
flux. Also, if we use the inverse square law of intensity at a distance d, then we can derive an
equation for the number of photons striking a unit area per unit time (or photon intensity),
#18
As a comparison let us consider the visible photon flux that would be emitted by the
parent galaxy. If we know the apparent magnitude of the galaxy, then we can calculate the
visible photon intensity using the Pogson Equation in the form v,
#19
Where Is is the visible photon intensity of the Sun and ms , m are the apparent magnitude of the
Sun and target galaxy respectively.
The calculated values of I and Iv for three galaxies, as well as the pertinent parameters,
are given below in Table 2 and Table 3 respectively on the next page.
Object Mass (kg) M Radius (m) R Distance to Earth (m) d Number Density(m^-3) n Orbital speed (m/s) v I (1/m^2*s)

7. 7
Andromeda 2.98E+42 7.57E+20 2.37E+22 3.64E+12 2.00E+05 4.67E+24
Triangulum 9.94E+40 4.73E+20 2.84E+22 4.98E+11 1.10E+05 6.71E+21
Vergo A 1.19E+43 1.54E+21 5.11E+23 1.73E+12 2.00E+05 9.03E+21
Table 2. Galaxy parameters were sourced from Wikipedia
Object m ms Is (visible 550nm) (1/m^2*s) Iv (visible 550nm) (1/m^2*s)
Andromeda 3.44 -26.7 3.77E+21 3.20E+09
Triangulum 5.72 3.91E+08
Vergo A 9.59 1.11E+07
Table 3. Parameters sourced from Wikipedia
III. Conclusion
As can be seen from comparing the photon intensities in Tables 2 and 3 we notice that the
DM glow is some ten orders of magnitude larger than the parent galaxy’s visible glow. If this
model were true, then someone would have observed this intense light by now even accidentally.
Looking at equation 16 for qualitative reference we can see the high dependence on the
number density n, cross-section 𝛥𝑥2
, and the geometry of the halo. Since they are raised to
second and third powers varying them will significantly alter the photon intensity. A true DM
density profile, such as NFW profile, is much lower that the assumed homogenous blob; it falls
off as the distance from the center of the halo increases. This would also force us to consider a
different halo geometry as this profile tends to form DM webs or strings rather than spherical
clumps. Also, the assumed probability for reaction was taken to be unity, but when comparing
the reaction cross-sections of known particles (this relates to the reaction probability), we can
expect the true reaction cross-section to be much smaller than 𝛥𝑥2
. Finally, although not as
critical, it might be worthwhile to utilize the formalism of Quantum Statistics.
With this ten orders of magnitude leeway, and the fact that more rigorous considerations
should yield a lower DM photon intensity, then the DM should be detectable with modern
telescopes plus the correct monochromatic filter. We now have some justification in searching
out this glow thereby giving TSC its first empirical test.
ACHKNOWLEDGMENTS

8. 8
This research project was made possible by the support of the Texas A&M University –
Kingsville (TAMUK) and its physics faculty. In particular the guidance of Dr. Hewett kept the
author from being hopelessly lost in the dark. The discussions with Mr. Charles Allison, physics
lecturer at TAMUK, concerning astronomy and light were extremely helpful. The constructive
criticism posed by Dr. Daniel Vrinceanu, professor of physics at Texas Southern University, was
vital in discovering fallacies in the logic used. Finally, special thanks to Mr. Jesus Salas, whose
help early on in the project saved the author from pursuing an impossible alternative model.
REFERENCES
i. L. D. Hewett, Time-Symmetric Cosmology, Professor’s handout, (unpublished).
ii. R. J. GaBany, The Formation and Evolution of Galaxies,
(http://www.cosmotography.com/images/galaxy_formation_and_evolution.html).
iii. P. A. Tipler, R. A. Llewellyn, Modern Physics, 6th ed., (W. H. Freeman and Company,
NY, 2012), pp. 351 – 357, pp. 277 – 279, pp. 213 – 214.
iv. H. D. Young, R. A. Freedman, Sears and Zemansky’s University Physics, 12th ed.,
(Pearson Education, CA, 2008), pp. 611 – 625, pp. 629 – 631.
v. D. Scott Birney, Guillermo Gonzalez, David Oesper, Observational Astronomy, 2nd ed.,
(Cambridge University Press, NY, 2010), pp. 85 - 87