A Quantum Approach, Dark Matter/ Galactic Halos -A Quantum Approach

Dr. TJ Gunn Ph.D. Applied Physics, Ph.M., Masters Philosophy

Published: 2 June, 2016

A Quantum Approach, Dark Matter/ Galactic Halos -A Quantum Approach

Traditional quantum theory can be used to construct hypothetical very large-scale gravitational stationary state
structures from traditionally stable atoms and subatomic particles. These so called “gravitational macroeigenstructures”
have potential to explain the composition of extra-galactic dark matter and galactic halos. It is shown
that the eigenstates within these structures can have radiative and stimulated lifetimes that are longer than the age of
the universe, and also that they cannot be easily transformed or “destroyed” by many conventional galactic processes.
Because of the unique nature of stationary states, it is shown that gravitational eigenstructures have the potential to
remain largely undetected, provided certain conditions are met. Speculatively, it is suggested that they could provide a
mechanism for the origin of high-energy cosmic rays, and also that, if these hypothetical structures have been present
from an early time in the history of the universe, then they could have influenced the large-scale structure of the

1. Introduction

The rotation velocity curves of stars in galaxies, the motions of pairs of galaxies and the
behaviour of galaxies in clusters and super-clusters all indicate that the universe contains
considerable quantities of non-luminous or dark matter. The debate over whether this
matter is of a baryonic or non-baryonic form has continued for some time, the general
consensus being that both types are present. Although to date no stable exotic nonbaryonic
particles have been detected, according to the standard model, element
abundances determined from the nucleosynthesis era suggest that dark matter cannot be
2 entirely baryonic (Silk 1995). By applying quantum theory to gravitational potentials on
a large scale, this paper presents and investigates a further alternative: the feasible
existence of large-scale, gravitationally bound eigenstates that are sufficiently
“invisible” to be considered as possible candidates for dark matter. Such a structure will
be referred to as a “gravitational macro-eigenstructure” (GME). The possibility will be
investigated as to whether or not stable and almost totally dark eigenstructures could
exist throughout the universe and contribute towards dark matter, and further, that
GMEs might form the halos within which most or all galaxies are embedded.
Although the effects of quantum theory are generally ignored in all but the earliest times
in the history of the universe, it is clear that the presence of universal eigenstates has the
potential to influence such things as the development of the large-scale structure of the
universe and the formation of galaxies. For example, the elemental ratios determined
during the nucleosynthesis era assume and depend critically on, a uniform distribution of
density of nucleons. Calculation of these ratios would clearly require the presence of
matter in the form of macroscopic eigenstates to be taken into account.
Furthermore, the free particle or localised orbiting particle is a superposition of many
eigenstates and reaction rates will be the average of that superposition. As will be
demonstrated in this paper, the calculated reaction rates for interactions between specific
individual eigenstates, or between individual eigenstates and free particles can therefore
be quite different to those rates for interactions between localised free particles based on
the average of the superposition. It should be noted that this paper does not attempt to explain how or when these
eigenstates might have developed. Nor does it try to trace galactic development based on
the presence of gravitational eigenstates. Instead, it attempts to examine whether it is
theoretically possible to form gravitational eigenstructures, and to ascertain the viability
of gravitational eigenstates as dark matter candidates. Nevertheless, the speculative
conjecture implicit in this paper is that the material universe may have existed partly, or
even predominantly, in the form of (potentially macroscopic) stationary states (either
gravitational or universal standing wave structures) from a very early time, and that
these states may have coexisted with, and expanded concurrently with, the expansion of
space, the formation of the galaxies occurring within eigenstructures to produce the
visible galactic formations that are presently observed.

2. Formation of Macroscopic Eigenstructures

Quantum effects customarily manifest themselves only over atomic dimensions. There
are however many examples of very large wave functions predicted directly from
quantum theory. An s-wave photon emitted from a distant star may be light years across
before it is destroyed or transformed via a transition within the retina of an observer.
Individual visible-wavelength photons from a highly coherent laser can be many metres
long. Particle wave functions exhibit similar expansions that can grow in size with time
at a remarkably fast rate. The state vector probability function for an electron released
from a cubicle box of side 1mm for example will fill an empty 1-litre container in about
1 1/2 seconds, while the corresponding wave function for an interaction initially
constrained to nuclear dimensions, say 5 fm cubed, will reach galactic size within about
1000 years. For a proton this figure is about 500 000 years. (See appendix A.) There is
no reason therefore to suppose that similarly large eigenfunctions could not exist.
The existence of a significant number of filled eigenstates on the macroscopic scale
depends on:

1. the availability of suitably attractive potentials,
2. the existence of potentials that have a suitably large range, so that the eigenstates
formed have appreciable range,
3. eigenvalues that are significantly negative, so that the particle or particles
involved are sufficiently “bound”, and
4. the susceptibility of the eigenstate (or lack of it) to radiative or induced
transitions within the eigenstructure or to transitions induced by external
influences (such as other particles traversing the eigenstate wave function
It is customary to think of quantum mechanics as being associated with electrical or
nuclear potential wells. To have a large effective radius, eigenstates associated with
these potentials require high principal quantum numbers. Of course the difficulty is that,
as the quantum numbers become large enough to produce macroscopic states, the
energies associated with those states become so close to that of a free particle that the
state could not remain stable. A hydrogen atom with n~3000 has an effective radius of
about 10-3 mm and this corresponds to an energy of around one millionth of an eV.
For a gravitational potential, the simple two-particle Schrodinger equation is written as

Eigenstructures with no visible component
It is theoretically possible to have a gravitational eigenstructure that has no visible
component. There are several examples of observations that indicate the presence of
objects that exhibit gravitational forces but are not seen, for example the so-called “great
attractor” and cases of gravitational lensing where the intervening galaxy presumably
responsible for the lensing process cannot be detected (Silk 1995). A halo of eigenstates
surrounding a non-visible massive centre, e.g., a black hole, would be a possibility. A
structure could however be formed entirely of eigenstates. For a small number of
particles, the states are widely spread and have too little binding energy too be stable. As
more particles are added however, the binding energy of the states increases and the
structure becomes more compact. It is first assumed that the eigenstructure consists of a
total mass M (= ∑ ) forming a wave function that is spread uniformly throughout a mi
spherical volume up to a radius r0. It is further assumed that the problem can be
simplified by considering the eigenstates of the individual masses mi rather than the
eigenstate of the structure as a whole. The potential energy V term for a small mass m
iThere are several ways around this
difficulty without resorting to a massive central force. One alternative is to reduce the
occupancy level from 100% to some small fraction, as in the galactic halo model. The
various parameters are intricately connected, but this has the effect of being able to
produce well-bound states that have densities of 105
m-3 and total masses of the same
order as the galactic mass and should be possible because of the long radiative lifetimes.
Likewise, if the mass of the individual particles is reduced substantially (~10-36 kg), the
eigenstructure can remain filled and possess densities and binding energies are in
keeping with observations. An alternative, somewhat more radical hypothesis is to
reduce the size of the structure. This has the effect of producing very high-density
eigenstructures (~ 103
kg m-3 or greater). Whether stationary structures can exist at such
densities is yet another conjecture. The matter forming the individual eigenstates could
be a mixture of any stable (presumably fermionic) particles that maintained net electrical
neutrality. Such eigenstructures might contribute substantially to the extra galactic dark
matter and the closure of the universe. All the arguments applied to the galactic halo in
the previous section (I) are valid here also. If large enough and dense enough structures
existed then the potential exists for the eigenstate particles to be present in the form of
atoms and they may be detectable by the atomic absorption spectra that would be
obtained as the light from more distant sources shone through them.

3. Discussion

The concept of charged or neutral particles in the form of gravitational eigenstructures
differs from that of classical cloud of hydrogen atoms or plasma in several ways that are
worth repeating here:
Firstly although particles in a gas cloud might be considered to have some semi-definite
energy, they are not, even approximately, in eigenstates. One might anticipate that
although particles in a gas cloud will always exhibit more or less dispersed wave
functions, their probability distributions are certainly not stationary. The individual
gravitational eigenstates however have very precise stationary probability distributions
that are not localised any more than the electrons in an atom are localised within it.
Secondly, the ability of an eigenstructure to condense or collapse into a denser object
such as a star or galactic nucleus relies on either the availability of empty lower energy
states with high probability transition coefficients, or a high probability of transfer to
mixed, “non-eigenstates”. This collapse depends on the electrical interactions that hold
condensed matter together. The crucial point is that by choosing the appropriate total
mass, gravitational eigenstructures can be formed where the gravitational interaction is
sufficiently large to form a totally bound and coherent structure, yet the equivalent
particle densities be sufficiently low that no significant collapse involving the traditional
electrical interactions of condensed matter can occur on time scales appropriate to that of
the universe.
Thirdly, because the eigenstate particles are bound to the total structure as a coherent
whole, their behaviour in many interactions (elastic scattering, for example) can be quite
different to that of the equivalent free particle.
Lastly, because of the stationary nature of the wave functions making up the
eigenstructure, charged particles in eigenstate will not radiate (except via state
transitions). Hot ionised gas is normally detectable through X-ray emission or the like,
but even if eigenstate particles exhibit classically equivalent high enough angular
velocities and accelerations to radiate classically, they will remain “X-ray dark”.
It would appear from the investigations presented in this paper that macroscopic
gravitational eigenstates clearly have properties that render them excellent dark matter
candidates. The total mass of the galactic halo can be easily accommodated within the
framework of a suitable eigenstructure formed around the galaxy. Unlike a gas cloud of
ionised or neutral hydrogen, such a structure would be both stable with respect to
gravitational collapse and largely invisible. If sufficient neutral hydrogen is present in
the eigenstructure, it might be detectable through its absorption lines or recombination
radiation following re-ionisation. Similar extra galactic structures with no visible
component consisting of totally or partly filled gravitational eigenstates are also
theoretically possible and could account for some or all of the extra galactic dark matter.
Indeed the observation of the absorption spectra of light from distant quasars reveals that
it has passed through many separate regions of neutral hydrogen travelling at a variety of
speeds. It is suggested that if these hypothetical structures have been present from an
early time in the history of the universe, a considerable amount of matter might have
existed in this form and that the very large structures developed visible components
while the smaller ones have remained as dark structures.
Whilst the gravitational macroscopic eigenstructure hypothesis for dark matter can
explain several issues, there are clearly also many difficulties. If galactic halo
eigenstructures consist of predominantly protons and electrons then where did these
particles come from? Recent observation of scattered light from quasars suggests that, at

1 billion years after the Big Bang, neutral hydrogen formed at the decoupling era was reionised
to the extent of 99.99% although the mechanism for this re-ionisation is not clear
(Haiman and Loeb 1997). Either the eigenstates existed much earlier and retained their
identity or there were processes at this time or earlier to ionise almost all the neutral
hydrogen. A further problem concerns the level of deuterium detected in the universe.
This suggests that there is still more dark matter in the form of more exotic particles. If
however it becomes apparent that the types of structures discussed in this paper do exist
and have been intimately connected with the expansion process since its inception, then
it may be necessary to re-examined the standard model to investigate the effects that
inclusion of stationary states might have on the density distribution and, in particular, on
the processes occurring during the nucleosynthesis era.
Gravitational eigenstructures might be speculatively used to explain two other
significant problems in astronomy, the origin of high-energy cosmic rays and the largescale
structure of the universe.

(i) High Energy Cosmic Rays
A significant problem in astronomy concerns the observation of numbers of very highenergy
cosmic rays. The problem centres on the development of a suitable mechanism
for the production of these since at present, no known physical process can account for
the very high energies observed. The production of such cosmic rays follows quite
naturally however from level decay within eigenstructures that possess a massive central
core. In traditional atomic physics the energy release accompanying electron demotion
within the atom is of the order of at most tens of electron volts and electromagnetic.
Radiation is the only possible type of emission. Conservation laws could also be satisfied
with the emission of electron-positron or proton-antiproton pairs instead of photons. The
energy changes for the inner states near a massive but small central potential provide
potentially vast amounts of surplus energy to be carried away as kinetic energy of the
particles produced and one might expect to see such transitions if the lifetimes of these
inner states are sufficiently short. Although such transitions would probably have taken
place some time ago in the Milky Way, they may be observable in young objects such as
(ii) Large Scale Structure
Another speculative explanation of the large-scale structure of the universe may be made
using macro-eigenstates. When a plate covered with fine sand is caused to resonate by
rubbing its edge with say, a violin bow, standing waves are set up and sand collects in
the nodes of those waves. If during inflation, when light was able to travel many times
the size of the visible universe, standing waves were set up, then the potential exists for
an equivalent three-dimensional array of walls of collecting matter to form. If the
standing waves were electromagnetic or gravitational in origin the matter would tend to
preferentially collect at the nodal surfaces. Alternatively matter eigenstates may have
formed across the entire universe with the consequence that matter would manifest itself
in regions where the probability density was greatest, nevertheless again in sheet-like

4. Conclusion

This paper has not attempted to prove or disprove the existence of macroscopic
gravitational eigenstructures but rather to show, by looking at macroscopic material in
the universe from a quantum perspective, (1) that there is very little in traditional
physical theory that forbids the existence of eigenstructures and (2) that they do provide
a possible (and perhaps reasonable) explanation of the composition of dark matter and
some other cosmological phenomena.
Although the models used have involved extremely gross approximations, the broad
conclusions from these models should remain generally valid when more realistic details
are included, and suggest that gravitational macroscopic eigenstates can be used to
explain some fundamental problems in modern cosmology.
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