The SSCP draws attention to the self-evident, but often under-appreciated, fact that from the smallest observable subatomic particles to the largest cosmological structures, nature is organized in a hierarchical manner. Although the whole hierarchy, in its full detail, is quasi-continuous, the SSCP emphasizes that the cosmological hierarchy is also highly stratified. While the observable portion of the entire hierarchy encompasses nearly 80 orders of magnitude in mass, three narrow mass ranges, each extending for only about 5 orders of magnitude, account for ³ 99% of all mass observed in nature. These dominant mass ranges are referred to as the Atomic, Stellar and Galactic Scales. They constitute the discrete self-similar scaffolding of the observable portion of the quasi-continuous hierarchy. The SSCP proposes that nature’s hierarchy extends far beyond our current observational limits on both large and small scales, and probably is completely unbounded in terms of scale, such that there are no largest or smallest objects (or Scales) in nature.
The SSCP further proposes that the Atomic,
Stellar and Galactic Scales, and all other fundamental Scales of
nature’s
transfinite hierarchy, are rigorously self-similar such that for
each class of particle, composite system or phenomena on any given
Scale, there is a discrete self-similar analogue on all other Scales. Mass
(M), length (R) and time (T) parameters associated with analogues
on neighboring Scales Ψ and Ψ-1
are related by the following set of discrete self-similar transformation
equations.
RΨ = ΛRΨ-1 . (1)
(a) Numbers of Galaxies: Twenty years ago the canonical value for the total number of galaxies in the observable universe was on the order of 100 billion. Today the rough estimate is slightly higher, possibly by up to a factor of four, due to the increased detection of small dim galaxies. In our calculations we will tentatively use a compromise estimate of 2 x 1011 galaxies as the best estimate for the total number of galaxies in the observable universe.
(a) Size of the Observable Universe: In 1986 the canonical estimate for the radius of the observable universe (Ru) was roughly 4,000 Mpc, or 4 Gpc. Today estimates are somewhat higher and there are more candidates for exactly how to determine Ru. One source noted that the “cosmic light horizon is 4.2 Gpc distant” if we accept the value of 13.7 years as the time since the global expansion began. A study by the WMAP collaboration published in 2004 estimated that Ru ≈ 24 Gpc, while a value of 14.4 Gpc was reported in the March 2005 issue of Scientific American. In 2006, a review of cosmology’s progress over the last 100 years noted that our local region of the cosmos had an “observed horizon about 30 Gpc across,” which presumably represents a diameter rather than a radius. For the purposes of this review we will adopt an approximate value of Ru ≈ 15 Gpc, which is ≈ 4.5 x 1028 cm. The volume of the observable universe (Vu) would then be (4 /3)π(Ru)3 ≈ 3.9 x 1086 cm3.
So if we view our metagalaxy as an analogue of a Stellar Scale system, then we can only observe a volume that is slightly larger than the volume of a single atom of that stellar system. The inescapable conclusion is that the vast observable universe constitutes a nearly infinitesimal volume of the Metagalactic Scale object that we inhabit. One needs to think about this, and perhaps do the necessary radius and volume calculations, until the reality of this fact sinks in.
VΨ=0 =
(4/3)π(RΨ=0)3 =
1.92 x 10-20 cm3. (5)
In this volume we find on the order of 2 x 1011 nucleus
analogues with masses that vary from that of the proton at 1 atomic mass
unit (amu), or about 1.67 x 10-24 g, to that of the lead nucleus
with a mass of over 200 amu. To
calculate an approximate density we need to choose an average mass (<MΨ=-1>)
for our nucleus analogues. Since
our knowledge of the distribution of Galactic Scale masses is marginal, we
will calculate 3 density values based on <MΨ=-1> estimates
of 4 amu, 16 amu and 64 amu.
If <MΨ=-1> corresponds
to the mass of a He++ nucleus, then we calculate a density (ΦΨ=0)
appropriate to the proposed galaxies-as-nuclei analogy as follows.
ΦΨ=0 =
(2 x 1011)(<MΨ=-1>) ¸ (4/3)π(RΨ=0)3 (6)
=
(2 x 1011)(6.68 x 10-24 g) ¸ 1.92
x 10-20 cm3
=
6.97 x 107 g/cm3.
If <MΨ=-1> equals
16 amu instead of 4 amu, then ΦΨ=0 would
be increased to 2.79 x 108 g/cm3. If <MΨ=-1> equals
64 amu, then ΦΨ=0 is increased
to 1.11 x 109 g/cm3.
Therefore the density of galactic systems in the observable
portion of our metagalactic system corresponds to a density that is in the
range of roughly 108 g/cm3 to 109 g/cm3 within
the context of our analogy that treats galaxies as nuclei. This density range is not as high as the
densities in the interior of a nucleus (~1014 g/cm3)
or a neutron star, but a density on the order of 108 g/cm3 is
still extremely high. For comparison,
very dense white dwarf stars, which are the precursors of type-I supernovae,
have densities on the order of 106 g/cm3, which is
equal to over 2,000 lbs per teaspoon.
(d) The Local Metagalactic Scale Temperature: Our next goal is to derive an estimate
for the ambient temperature (TΨ=+2)
that applies to the observable portion of our metagalactic system. It
should be fully acknowledged that our estimate can only be a very crude approximation
because of the fairly high levels of uncertainty associated with various
Galactic and Metagalactic Scale parameters. However,
a crude approximation is better than none at all, and even in its rough form
the estimate may tell us something quite important about our local Metagalactic
Scale region.
In the case of a gas of Atomic Scale particles at approximate
thermal equilibrium, there is a relationship between the average kinetic
energy (<Ekin>) of the particles and the ambient temperature
(T):
<Ekin> =
(3/2)kT , (7)
where k is Boltzmann’s constant (1.38 x 10-16 g
cm2/ sec2 oK). The
very-high-energy state and global expansion that characterize the observable
portion of our metagalactic system surely deviate from thermal equilibrium
conditions, but we are only looking for a crude approximation of TΨ=+2. We can express Eq. (7) as
(1/2)m<v>2 = (3/2)kT , (8)
where m is the mass of an individual particle and <v> is
the average velocity of the particles. Eq.
(8) can then be rearranged to give
T
= (1/3)(m/k)<v>2 . (9)
Continuing with our analogy between galaxies and atomic
nuclei, and remembering that the average peculiar velocities for galaxies
is about 700 km/sec, we can solve Eq. (9) for three hypothetical cases wherein
all of the nuclei have masses of 4 amu, 16 amu or 64 amu. If m = 4 amu, then
T
= (1/3)[(6.68 x 10-24 g) ¸ (1.38 x 10-16 g cm2/ sec2 oK)][7
x 107 cm]2 (10)
= 7.9 x 107 oK .
If m = 16 amu, then T = 3.2 x 108 oK,
and if m = 64 amu, then T = 1.3 x 109 oK.
Assuming that we are justified in using our atomic/galactic
analogy, and that the deviation from thermal equilibrium conditions is not
introducing fatal errors, our crude approximation of TΨ=+2 is
roughly 108 oK to 109 oK. This enormous temperature is quite consistent
with expectations based on the hypothesis that the observable portion of
the metagalaxy is self-similar to the interior of a supernova shortly after
detonation. Whereas our numerical
results are subject to various sources of errors, the general conclusion
that TΨ=+2 is very high is
supported by the very large peculiar velocities of galaxies, the very high
density of galactic “particles”, the fully ionized states
of all observable Galactic Scale systems, the frequent and apparently violent
interactions among galaxies, and the global expansion of the observable portion
of the metagalaxy.
(e) Expansion Timescale for Our Local Metagalactic Region: Cosmologists are fairly confident that the global expansion of the observable universe began approximately 13.7 billion years ago. This sounds like a very long period of time, but relative to the Galactic Scale it only amounts to about 30 rotation periods of our Galaxy. Within the context of our galactic/atomic analogy, wherein galaxies are treated as Atomic Scale particles and the local region of the metagalactic object is regarded as the interior of a supernova, the time (tΨ=0) since the detonation occurred can be calculated as follows.
= (13.7 x 109 yr)(3.15 x 107 sec/yr) ¸ 2.7 x 1035
(f) Plasma-Like Distributions of Galaxies and Galactic Clusters: One of the most remarkable metagalactic phenomena, and one that has been well documented throughout the observable universe, is the unique distribution of galaxies. Astronomers initially guessed that the distribution would be fairly random on moderately large scales and approximately homogeneous as one approached the largest observable scales. However, observations gradually showed that the actual distribution of galaxies is quite different from those early assumptions. Astrophysicists now know that the distribution of galaxies and clusters of galaxies is highly filamentous, with Galactic Scale systems preferentially occurring in vast filaments and thin sheets. The high-density filaments and sheets tend to surround “voids” wherein the mass density is quite low. Thus the overall structure in the observable portion of our metagalaxy has been described using terms like: “Swiss cheese”, “honey comb”, “filamentary”, “cosmic webs”, “walls”, “bubble-like voids”, “fractal networks”, etc. This enigmatic distribution has not been convincingly explained by conventional astrophysics, but our supernova analogy offers an appealing interpretation. The same mix of filaments, thin sheets and voids is regularly seen in the most common state of matter in the observable universe: plasmas. When high-energy conditions lead to ionization of neutral matter, the resultant mix of electrons, nuclei and charged ions manifests this highly unique filament/sheet/void morphology due to a complex interplay between the attraction among unlike charges and the repulsion among like charges. Prime examples, which we have all seen before, are the beautiful (and now highly detailed) photographs of the Crab Nebula supernova remnant. In this case, the supernova remnant has expanded and cooled for hundreds of years, but the imprint of the plasma morphology is still readily observable. In the case of the observable universe, the ambient Metagalactic Scale temperature, density and pressure are enormously higher, but the filament/sheet/void character of the distribution of galaxies and galaxy clusters is highly self-similar to the distributions of very-high-energy plasma particles. This interpretation is certainly in keeping with predictions based on our Metagalactic Scale supernova analogy. Put the other way around, if galactic systems did not have this unique plasma-like distribution, but rather had a very homogeneous distribution, then the Metagalactic Scale supernova analogy would have had a serious flaw.
The “tiny” observable portion of our local Metagalactic
Scale environment clearly shows signs that it is in the very-high-energy
domain, and so it needs to be studied from the perspective of high-energy
physics. We can expect the presence
of both stable and unstable “particles”. Recent
studies suggest that the average galaxy has undergone between one
and three merger events since detonation and this shows that the interaction
probabilities are very high. It
would appear that the application of high-energy particle physics to Galactic
and Metagalactic Scale phenomena, when suitably scaled according to the principle
of discrete cosmological self-similarity, has a rich potential for discovery
and unification. Using
the SSCP’s supernova analogy it should be possible to combine cosmology,
supernova astrophysics and subatomic physics to achieve a truly advanced
understanding of cosmological phenomena.
(1) Ubiquity: Galactic Scale electron analogues should be present in large numbers and should be distributed ubiquitously throughout the observable universe.
(3) M e-,Ψ=+1 < 10-3 Mgal: Given that the electron has a mass of about 1836 times less than the proton mass, we can expect that the mass of the Galactic Scale electron analogue will definitely be less than 10-3 times the average mass for a galaxy, but not much less than 10-5 times the average mass for a galaxy. Because of the very high levels of uncertainty in Galactic Scale mass estimates, this criterion can only involve very rough estimates.
(4) Stability: To the degree that it is self-similar to the Atomic Scale electron, the Galactic Scale electron analogue should be a relatively “stable” particle.
(5) Attraction/Repulsion: Just as electrons are attracted to positively charged particles and repelled from negatively charged particles, the Galactic Scale electron analogues should be attracted to Galactic Scale nuclei (e.g., galaxies), but repel each other.
(6) Central Singularity: The Galactic Scale electron analogue should have a central singularity that contains the overwhelming majority of its Galactic Scale mass. From the point of view of a Metagalactic Scale system, it should behave as a “point-like” particle.
These six criteria put strong constraints on any candidate for the Galactic Scale electron analogue, and in fact there is only one possible candidate: globular clusters. Below we will briefly discuss the general agreement between the properties of globular clusters and the six criteria for the Galactic Scale electron analogue.
(1) Globular Clusters Are Ubiquitous: There is a conspicuous presence of large numbers of globular clusters in all well-observed parts of the observable universe.
(2) Half-Light Radii Of About 4 Parsecs: Given the structure of globular clusters: a very dense central core and a spherical halo of stars, there are several ways to measure their radii (Rgc). The most physically meaningful estimates are the core radius (Rc) and the half-light radius (R1/2), while the tidal radius (Rt) is a less diagnostic measure. Studies2 of the globular clusters that are associated with our Milky Way Galaxy have found a mean R1/2 value of 4.4 parsecs and a median R1/2 of about 3.0 parsecs. The core radii for globular clusters in our Local Group of galaxies are typically in the 2 pc to 7 pc range. Therefore, there is a good agreement between Rgc and the predicted Re-,Ψ=+1 value of ≈ 4 parsecs.
(3) Mgc < 10-3 Mgal: The mean mass (Mgc) for globular clusters in our Galaxy has been estimated2 at about 2 x 105 M8. Galaxy mass estimates tend to range from about 107 M8 to 1013 M8. If we assume that <Mgal> ~ 1010 M8, then <Mgc> ~ 2 x10-5<Mgal>, which is within the expected range. According to the SSCP, Galactic Scale mass estimates are severely hampered by three problems: (a) they are based on the use of GΨ=0, instead of the correct Galactic Scale gravitational constant (GΨ=+1 = GΨ=0 ¸ ΛD-1), (b) the estimates do not adequately take into account the singular nature of Galactic Scale mass distributions, and (c) the estimates completely ignore electromagnetic contributions to Galactic Scale mass/energy calculations. Given these problems, arguments based on conventional Galactic Scale mass estimates must be limited to crude approximations of relative mass relationships, such as order of magnitude mass ratios for different types of galactic systems.
(4) Globular Clusters Are Stable: Globular clusters are regarded as among the oldest and most “pristine” of all Galactic Scale systems. Therefore, relative to Galactic Scale considerations, globular clusters have a high degree of stability.
(5) Globular Clusters Are Attracted To Galaxies: Most globular clusters are associated with galaxies, which the SSCP identifies as analogues of positively charged nuclei. It also appears that tightly bound systems of two or more globular clusters are not observed, and may not exist at all. Therefore it appears that globular clusters are attracted to galaxies, but tend to avoid close contact with other globular clusters. These basic tendencies agree with the attraction/repulsion behavior expected for electron analogues.
(6) Central Singularities In Globular Clusters: The number of globular clusters with ultracompact objects detected in their cores has steadily increased over the years, and some astrophysicists hypothesize that a central singularity may be a universal characteristic of globular clusters.
Given the general agreement between the six criteria for Galactic Scale electron analogues and the basic properties of globular clusters, it seems very likely that globular clusters are the self-similar Galactic Scale analogues of electrons. An interesting test of this idea is as follows. The number of globular clusters associated with a galaxy can vary from zero to hundreds, and even to thousands in the case of giant galaxies. If our analogy between Atomic Scale electrons and Galactic Scale globular clusters is valid, then the distribution of electrons in very-high-energy Atomic Scale plasmas will be found to be compatible with the large numbers of globular clusters that are associated with high-mass galaxies.