International Journal of Theoretical Physics,
Vol. 28, No. 6, 669694, 1989
SelfSimilar Cosmological
Model: Introduction and Empirical Tests
Robert L. Oldershaw
Amherst College,
Amherst,
Massachusetts 01002
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ABSTRACT:
After calling attention to the empirical and theoretical motivations for considering the hypothesis of a selfsimilar cosmos, the basic concepts and scaling rules of the SelfSimilar Cosmological Model are presented. The results of a diverse set of 20 falsification tests are then shown to provide strong quantitative support for the uniqueness and broad applicability of the selfsimilar scale transformation equations, which successfully correlate physical parameters of atomic, stellar, and galactic scale systems. Possible implications of these results are discussed.
...the wise man looks into space
and does not regard the small as too little,
nor the great as too big,
for he knows that there is no limit to dimensions.
Laotse
1. INTRODUCTION
1.1. Goals of the Review
The SelfSimilar Cosmological Model (hereafter SSCM) is a heuristic cosmological model that has been developed over the past 10 years in a series of 17 papers by the author (Oldershaw, 19781987b, 1989a,b). The major goal of this review is to introduce the SSCM, its 20 successful falsification tests, and its major predictions to as large and diverse an audience of scientists as possible. Other goals of this review are a reasonably compact summary of the previous work on the SSCM and the identification and clarification of various modifications to the model that have occurred over the past 10 years. The remainder of this section will be concerned with the question of why one should be interested in an unorthodox selfsimilar model of the cosmos. Section 2 will introduce the SSCM in its simplest and most general form and Section 3 will discuss the substantial amount of empirical evidence in favor of cosmological selfsimilarity. The sequel (Oldershaw, 1989c) to this paper will present a more detailed and technical discussion of the SSCM, including its major predictions, implications, and unresolved problems.
1.2. Reasons for Considering a SelfSimilar Cosmological Model
The overwhelming majority of physicists currently think that
the Big Bang cosmology (augmented by Inflation) and the Standard Model of
particle physics (and subsequent unification theories) are unquestionably
the right theories to guide us toward a fully unified understanding of nature;
some even predict that all fundamental questions in physics will be solved
in the near future by pursuing these theoretical paths. On the other hand,
even supporters of these theories admit that their theoretical constructs
are often untestable in a definitive way, that they have had trouble with
most of the few falsification tests that have been identified, and that
they have been unable to anticipate major new observational discoveries.
This situation has been detailed elsewhere (Oldershaw, 1988) and will not
be repeated here in full, but let us briefly consider the current state
of affairs in cosmology. The Big Bang theory has always encountered serious
theoretical problems, such as the flatness problem, the smoothness problem,
and the horizon problem. These problems were "solved" by the
ad hoc addition of an Inflationary episode at about 10^{35} sec
after the Big Bang, but this solution leads to other equally serious problems.
For example, the major prediction of the Inflated Big Bang theory is that
the matter density of the universe equals the critical density (i.e., Omega
= 1), but this prediction has been contradicted by most observationally
based estimates made to date (Rothman and Ellis, 1987). This theory also
leads to potential conflicts between the predicted age of the universe and
the estimated ages of its oldest constituents (Tayler, 1986). Moreover,
the Inflation scenario is totally dependent upon the validity of the GUTs
of particle physics, which are themselves beset by falsifications, arbitrariness,
and testability problems (Pickering, 1984). Even more worrisome is the
fact that the Big Bang theory failed to anticipate major empirical discoveries
of recent years, such as the largescale inhomogeneity in the distribution
of matter, the large deviations from a smooth Hubble flow and, most importantly,
the dark matter constituting more than 90% of the matter of the universe
(Oldershaw, 1988). None of the variations on the Big Bang theme can provide
a convincing explanation for the existence of galaxies, and the Hubble constant
is uncertain by a factor of 2. In short, there is no justification for complacency
with regard to our current state of knowledge in the field of cosmology
(or particle physics).
Secondly, the Big Bang theory makes the extremely suspect assumption that nature's hierarchy ends at about the scales where our observational capabilities end, and that we just happen to find ourselves in the vicinity of the center of that scale range. In the SSCM, on the other hand, there is no spectre of an anthropocentric truncation of nature's hierarchy, since it postulates that the hierarchy extends well beyond current observational limits, and is perhaps completely unbounded.
Finally, modem physics has something of a split personality in that the physics of the microworld is hypothesized to be inherently different from the physics of the macroworld, with a somewhat fuzzy interface between these two realms wherein quantum microphysics rather mysteriously metamorphoses into classical macrophysics. The SSCM hypothesizes that one set of physical laws holds good for all scales of nature's hierarchy. Therefore, the fact that the SSCM is not plagued by these three philosophical problems is another reason for giving it due consideration.
For these five reasons, then, it would appear that the SSCM is worthy of serious attention in spite of its divergence from generally accepted cosmological assumptions.
2. GENERAL DISCUSSION OF THE SELFSIMILAR COSMOLOGICAL MODEL
2.1. Heuristic Status
It should be understood from the outset that the SSCM is still very much in the heuristic stages of development. That is, the SSCM and the properties of nature which led to the formation of its major hypotheses can be described in some detail and the selfsimilar scaling equations that are the heart of the model can be empirically derived and quantitatively tested, but the SSCM cannot as yet answer the basic questions of why nature has a selfsimilar design and why the two dimensionless constants of the scaling equations have the particular values that are found empirically. The heuristic status of the SSCM may be viewed as a shortcoming, but on the other hand it would seem to be unwise to risk stunting the development of the SSCM by encasing the promising heuristic core in a hastily constructed theoretical shell.
Because the SSCM proposes a fundamentally different understanding of nature, and because it would therefore significantly alter ideas in most branches of theoretical physics, it is of considerable importance when studying this model that: (1) all previous theoretical constructs should be regarded as being open to question (a basic tenet of science), and (2) that observational data should take primacy over theoretical assumptions or expectations when conflicts between these two occur. Unfortunately, at present this is not always the case (Pickering, 1984).
2.2. Hierarchical Organization of the Cosmos
It is a selfevident fact that nature has a nested hierarchical organization, though this fact is often taken too much for granted. In our local planetary environment, for example, "elementary particles" combine to form atoms, which are the building blocks of molecules, which compose a vast array of macroscopic objects, which are collected into planets, moons, asteroids, and comets, which are components of the solar system. Taking a more cosmological perspective [in terms of the ubiquity (Oldershaw, 1985) of the building blocks, the breadth of the spatial domain under consideration, and the range of physical scale] it is known that electrons, atomic nuclei, and ions are the primary building blocks of stars, which are thought to be the primary building blocks of galaxies, which are clustered into everlarger aggregations until the limits of our observational abilities are approached.
Considering welldefined classes of relatively stable objects that have mass, several major characteristics of the observable portion of the cosmological hierarchy can be identified. The important question of the form of the dark matter has been discussed previously (Oldershaw, 1986a,d) and will be a primary topic of the sequel (Oldershaw, 1989c) to this paper.
2.3. Discrete SelfSimilarity
Given the highly stratified organization of the observable portion of the cosmological hierarchy, it seemed natural to compare atomic, stellar, and galactic scale systems with regard to similarities and/or dissimilarities. Taking into account the huge differences in spatiotemporal scale which tend to obscure inherent similarities to a degree that is often seriously underestimated, I found that there was a considerable potential for physically meaningful analogies among atomic, stellar, and galactic scale systems. Let us consider several general examples that suggest the possibility of interesting parallels between the physics operating on different scales.
Potential
analogies such as the six listed above, and others discussed in the SSCM references
cited above, emboldened me to consider the speculative hypothesis that atomic,
stellar, and galactic scale systems might be rigorously selfsimilar, i.e.,
that specific systems on a given cosmological scale have specific analogs on
all other cosmological scales, and that the properties of analogs from different
scales are quantitatively related by simple scale transformation equations.
The derivation
of a set of selfsimilar scale transformation equations, which can relate corresponding
length, time, and mass values for analog systems on different scales, was perhaps
the most important step toward quantitative testing of the cosmological selfsimilarity
hypothesis, since these equations would allow one to identify analogs on different
scales, to assess quantitatively their selfsimilarity, and to make definitive
predictions. From Mandelbrot's (1982) basic discussion of selfsimilarity,
a little physics (e.g., velocities should be scale invariant), and a knowledge
of the abovementioned general properties of the cosmological hierarchy, one
can infer that the simplest scaling equations for a highly stratified selfsimilar
hierarchy would be:
R_{ψ} = LR_{ ψ1} 
(1) 
T_{ψ} = LT_{ψ1} 
(2) and 
M_{ψ} = L^{D}M_{ψ1} 
(3) 
where R, T, and M are length, time, and mass values pertaining to analog systems
on neighboring cosmological scales ψ and ψ1,
and where L
and D are dimensionless scaling constants that must, for the present, be determined
empirically. The values of L and
D are found to be approximately 5.2 x 10^{17 }and 3.174, respectively,
and the methods by which these values were arrived at are discussed in Oldershaw
(1986a) and in the sequel (Oldershaw, 1989c) to this paper. In general,
these methods involve identifying a pair of putative analogs for which there
are reasonably accurate mass and radius estimates, and for which the analogy
seems dependable. Ratios
of analogous mass and radius measurements then yield L
and D, since R_{ψ}/R_{ψ1} = L and
M_{ψ}/M_{ψ1} = L^{D}.
The analog pair that was initially used consisted of the solar system, for
which accurate data are available, and a very highly excited Rydberg atom (n » 168),
an atomic scale system whose basic properties are both quantifiable and strongly
analogous to those of the solar system. The fact that L and
D are singlevalued rather than multivalued or continuous reflects the fact
that according to the SSCM, nature's hierarchy is modeled as having discrete
and symmetric stratification.
2.4. Summary of the Basic Model
The SSCM views nature as a highly stratified, nested, and possibly unbounded hierarchy of systems with atomic, stellar, and galactic scale systems comprising a discrete, symmetric framework for the observable portion of the entire quasicontinuous hierarchy. It is further hypothesized that the hierarchy is rigorously selfsimilar such that radii, periods, masses, and in fact any corresponding parameters (Oldershaw, 1986ae, 1987a) associated with analog systems on different scales are correlated by the very simple scale transformations defined in equations (1)(3). Given the currently accepted theoretical models of atomic, stellar, and galactic systems, one might be highly inclined to regard the latter hypothesis as being simply impossible, i.e., of having no chance of applying to the real world. So much more surprising, then, will be the results presented below of actual empirical tests of the hypothesis. Nature, rather than human theoretical constructs, should be the basis upon which we decide the merits or shortcomings of a scientific hypothesis.
3. EMPIRICAL TESTS OF THE SELFSIMILAR COSMOLOGICAL MODEL
3.1. Introductory Notes
Table I (below) presents the results of 20 retrodictive falsification tests of the SSCM. As opposed to definitive predictions (Oldershaw, 1988), which predict unexpected phenomena or the results of empirical experiments before they are known, retrodictive falsification tests determine a theory's ability to "retrodict" previously known data, i.e., they test a theory's consistency with observations. Therefore, retrodictive falsification tests are inherently less stringent than are tests involving definitive predictions. However, to the extent that a theory can pass a large and diverse array of retrodictive falsification tests, our confidence in the theory as a good approximation to natural phenomena is commensurately increased. The final three tests listed in Table I come reasonably close to being classified as true predictions, since they involve relationships that were not thoroughly characterized prior to the tests; several predictions of the SSCM that unquestionably meet the criteria for definitive predictions have been presented before (Oldershaw 1986a,d; 1987a) and will be discussed further in a forthcoming paper (Oldershaw, 1989c).
Table I. Retrodiction Tests of the SSCM
Test # 
Test Parameter 
Reference Parameter 
Reference Value 
Scale Factor 
SSCP
Prediction 
Observed Value 
1 
M dwarf abundance 
H abundance 
90 ± 2% 
 
< » 90% > 
< » 89% > 
2 
K dwarf abundance 
He abundance 
9 ± 2% 
 
< » 9% > 
< » 10% > 
3 
Lower limit R M dwarfs 
Lower limit R for H 
1.6 x 10^{8} cm 
Λ 
8.3 x 10^{9 }cm 
8.7 x 10^{9} cm 
4 
<R> for white dwarfs 
R for He^{+} 
2.1 x 10^{9} cm 
Λ 
1.1 x 10^{9} 
0.9 x 10^{9} 
5 
Lower limit R for white dwarfs 
Lower limit R for atomic ions 
4.2
x 10^{10} to 
Λ 
2.2
x 10^{8
}to 
5.5 x 10^{8} cm 
6 
1.6
x 10^{8
}to 
Λ 
8.3
x 10^{9
}to 
8.7
x 10^{9
}to 

7 
Average M for white dwarfs 
Mass of ^{4}He 
6.7 x 10^{24 }g 
Λ^{}^{D} 
1.14 x 10^{33} g 
1.15 x 10^{33} g 
8 
Lower M for white dwarfs 
^{ }^{3}He/^{4}He M ratio 
0.75 
 

9 
Proton radius 
Schwarschild R of black hole 
Stellar Scale G 
Λ^{D}Λ^{2}/Λ^{3} 
0.81 x 10^{13} cm 
0.8 x 10^{13} cm 
10 
Log K_{S}/K_{A} from J=K_{i}M^{2} 
 
 
Λ^{2}/Λ^{D+1} 
38.51 
38.41 
11 
Log Δ_{S}/Δ_{A} from μ=Δ_{}J 
 
 
19.31 
20.36 

12 
Typical pulsar spin period 
Typical nuclear spin period 
5 x 10^{20} sec 
Λ 
0.03 sec 
0.002 – 
13 
R range for galaxies 
R range for atomic nuclei 
0.8
x 10^{13} to 
Λ^{2} 
2.2
x 10^{22} 
0.9
x 10^{22} to 
14 
Typical galaxy spin period 
Typical nuclear spin period 
5 x 10^{20} sec 
Λ^{2} 
4.3 x 10^{8} years 
4.4 x 10^{8} years 
15 
μ range for neutron stars 
μ range for atomic nuclei  4.5
x 10^{25
}to 
Λ^{1.59 }times Λ^{1.5} 
10^{30.34
}to 
10^{30.3
}to 
16 
Period range for He^{+} 
Period range for white dwarfs 
250850 sec 
Λ^{1} 
4.8
x 10^{16
}to 
5.5
x 10^{16
}to 
17 
Period range for neutron stars 
Period range for atomic nuclei 
1.3
x 10^{22
}to 
Λ
 6.8
x 10^{5
}to 
10.0
x 10^{5
}to 
18 
PeriodRadius
Law 
PeriodRadius
Law 
p^{2} = k_{a}r^{3} 
 

19 
K_{S} values 
k_{a} values 
1.6 x 10^{7}, 2.0 x 10^{8 }sec^{2}/cm^{3} 
Λ^{2}/Λ^{3} 
3.0
x 10^{25}, 
3
x 10^{25}, 
20 
Period
range 
Period
range 
0.2 – 0.8 
Λ^{1} 
3.3
x 10^{14
}to 
» 5
x 10^{14
}to 
Below I will review each test and its results, referencing previous discussions of the test and identifying new data that are applicable. Since all measurements involve uncertainties, the reference, "predicted," and empirical values listed in Table I are estimates and each should be thought of as being preceded by an "approximately equals" symbol. Relevant sources and degrees of uncertainty are discussed in the cited references and in this paper.
An important caveat, already mentioned in Section 2, is that nature does not present us with equivalent samples of atomic, stellar, and galactic scale systems. In terms of numbers of systems and "sampling ratios" (see point 5 of Section 2.2), the values for the atomic, stellar, and galactic scales are about 10^{80}, 10^{23} , and 10^{17}, and 10^{40}, 10^{17}, and 10^{5}, respectively. To put this into bold perspective, what we observe of the galactic scale (Oldershaw, 1986d) is analogous to studying the atomic scale on the basis of observing a mere 10^{11} subatomic particles crammed into a volume roughly comparable to that of a single hydrogen atom. This sample would woefully underrepresent the richness of atomic scale phenomena, and therefore we must bear in mind that the available galactic scale sample is similarly limited. The situation is quite a bit better on the stellar scale, but the caveat against assuming equivalent samples is still very important when making stellaratomic comparisons.
The empirical tests listed in Table I usually have the following format: a reference parameter that has been measured with reasonable accuracy is identified for a class of systems on a given scale, this value is then transformed according equations (1)(3) in order to yield a "predicted" counterpart value for the analogous class of systems on a different scale, and finally the "predicted" value is compared with empirical measurements made on the relevant class of analog systems. Usually atomic scale systems are chosen as the source of reference parameters because our empirical measurements of atomic scale parameters are in general vastly superior to our quantification of stellar or galactic scale parameters.
3.2. Discussion of Individual Tests
The SSCM proposes (Oldershaw, 1986a) that stars with radii greater than about 9 x 10^{9} cm, e.g., main sequence, giant, and supergiant stars, are stellar scale counterparts to atoms in excited, but for the most part neutral, states. Equations (1)(3) predict that the stellar scale hydrogen analog has a mass of about 0.15M_{¤}and the helium analog has a mass that is four times larger, or about 0.6M_{¤}. As anticipated by the SSCM, recent data (Lupton et al., 1987; Low, 1985) show a distinct abundance peak at about 0.62M_{¤} and a much larger peak that falls somewhere between 0.1M_{¤} and 0.2M_{¤}. Since there is a considerable amount of uncertainty involved in estimating stellar masses [note the broadness of the abundance peaks of Lupton et al. (1987)], the stellar scale abundances of H and He analogs will be defined here as the abundances of stars with masses estimated to be in the ranges 0.1M_{¤} to 0.4M_{¤} and 0.45M_{¤} to 0.75M_{¤}, respectively. These mass ranges correspond quite well to the estimated mass ranges of M dwarf and K dwarf stars, and therefore the SSCM predicts that the abundances of M dwarf and K dwarf stars should be about 90% and 9%, respectively. Quantitative determinations of these stellar abundances are exceedingly hard to find in the literature, but Wood (1966) has made a comprehensive attempt, and for his most reliable sample of galaxies the M dwarf abundance ranges from 81% to 95% with an average of 89%, While the K dwarf abundance ranges from 6% to 18% with an average value of 10%. The average values are quite close to the predicted values.
R_{S} = 2G_{ψ}M/c^{2}  (4) 
p^{2} » k_{1}r^{3}  (for l » n)  (5) 
p^{2} » k_{2}r^{3}  (for l << n)  (6) 
where n is the principal quantum number, l is the azimuthal quantum number,
k, is a constant equal to (p_{0})^{2}/(a_{0})^{3},
and k_{2} is a constant equal to (p_{0})^{2}/(2a_{0})^{3}
. The parameter p_{0} is the minimum transition period for hydrogen
and a_{0} is the Bohr radius. From the fact that Rydberg atoms
obey approximate relationships of the form of Kepler's third law, i.e.,
p^{2} » k_{i}r^{3},
it can be predicted that variable stars with radii ³
lR_{¤}, which have been identified
as stellar scale analogs to Rydberg atoms undergoing transitions to lower
energy states (Oldershaw, 1987b), will have periods P and radii R that obey
approximate relationships of the form P^{2} »
K_{i}R^{3}, where the K_{i} represent analogs to
the k_{i}. It has been demonstrated (Oldershaw, 1989b) that a wide
variety of variable stars, including delta Scuti, RR Lyrae, beta Cepheid,
classical Cepheid, and supergiant variables, do indeed obey periodradius
relations of the predicted form. Moreover, it has been shown that the atomic
scale constants k_{1} and k_{2} are quantitatively related
to their stellar scale counterparts K_{1} and K_{2} by the
selfsimilar scaling rules embodied in equations (1)(3). A third PR relationship
with a K_{3} value that is closely related to K_{1} and
K_{2} has been identified for variable stars and has led to the
prediction that an analogous pr relation will be found for a subset of
Rydberg atoms (Oldershaw, 1989b).
(a) Variable
stars represent a heterogeneous mixture of stellar scale counterparts to
atoms and ions.
(b) Values
of n for individual stars can range from 1 to at least 100.
(c) For each
value of n there are n different energy levels due to orbital angular momentum
considerations, i.e., l can vary from 0 to n  l for each value of n.
(d) If spin
considerations are included, then the abovementioned energy levels are
further split into an even larger set of levels.
(e) Since
the energy levels of Rydberg atoms can be significantly shifted by ambient
electric and magnetic fields, the SSCM asserts that an analogous shifting
of energy levels can also occur in the case of their stellar scale counterparts.
Variable stars from different locations within our galaxy, i.e., near the
nucleus, in the outer halo, in the spiral arms, or in globular clusters,
would therefore be expected to have period distributions that are influenced
by differing galactic scale electromagnetic environments.
If careful thought is given to these five considerations, which would
serve to generate a dense "forest" of transition periods for Rydberg
atoms or their analogs, then it is clear that expecting to find textbookstyle
evidence for quantized periods among variable stars, based on a maximum
sample size on the order of 10^{4} periods, is essentially ruled
out at present. Under the existing observational circumstances even less
overt evidence for quantization in atoms or variable stars would still be
very difficult to obtain, since strategy B is precluded by having only a
tiny sample of systems and since strategy A is hampered by our inability
to manipulate the sample or the ambient physical conditions that affect
the sample. However, all is not lost. Granted that blatant examples of
quantization are not to be expected, one might still hope to observe less
overt evidence of quantization in the following manner. Since the sample
size is invariably going to be small, the best strategy is to identify as
homogeneous a subsample of variable stars as possible, with the hope of
minimizing the number of different species, the spread of n and l values,
and the influence of differing ambient physical conditions.
RR Lyrae stars constitute perhaps the best candidate for a class of variable stars that meets the desired criteria. Their masses are found to cluster around 0.6M_{¤} (Stothers, 1981) and therefore the SSCM unambiguously identifies them as primarily helium analogs. The overwelming majority of their radii fall within the range of 4R_{¤} to 7R_{¤} (Stothers, 1981), and from this fact the SSCM identifies (Oldershaw, 1987b) the range of n values for RR Lyrae variables as n = 7 to n = 9. Also, their position on a periodradius graph shows that they represent the l << n case (Oldershaw, 1989b); here we will assume that l £ 2. Therefore, if reasonably large samples of RR Lyrae variables from reasonably homogeneous galactic environments are analyzed in terms of relative frequencies of oscillation periods, then the SSCM anticipates that evidence for discrete, preferred periods will be present, though the statistical significance might be low. The range of the period distribution and the preferred periods for the RR Lyrae stars should be correlated with corresponding He transition periods in a manner consistent with equation (2).
I have investigated the period distributions
for several RR Lyrae subsamples taken from the General Catalogue of Variable
Stars [the Third Edition and its Supplements] (Kukarkin et al, 19691970),
and two useful empirical findings have resulted from these investigations.
First, about 99% of the RR Lyrae stars have periods in the range 0.2  0.8
days. Second, there tend to be recurrent peaks in the subsample period distributions
at periods of
Table II. Distribution of Periods for a Sample of 672 RR Lyrae Variables and Relevant
Scaled Transition Periods for H, He, and Li Atoms
RR Lyrae period distribution  Scaled atomic periods  
DP (days) 
N 
DP (days) 
N 
H 
He 
Li 

0.1500.159 
1 
0.5000.509 
15 
0.383 
0.276 
0.250 

0.1600.169 
0 
0.5100.519 
32ß 
0.558 
0.287 
0.256 

0.1700.179 
2 
0.5200.529 
35 
0.323 
0.362 

0.1800.189 
1 
0.5300.539 
35 
0.326 
0.369 

0.1900.199 
0 
0.5400.549 
30 
0.354 
0.378 

0.2000.209 
1 
0.5500.559 
23 
0.378 
0.398 

0.2100.219 
0 
0.5600.569 
27 
0.389 
0.529 

0.2200.229 
2 
0.5700.579 
22 
0.404 
0.543 

0.2300.239 
4 
0.5800.589 
18 
0.406 
0.579 

0.2400.249 
2 
0.5900.599 
13 
0.424 
0.590 

0.2500.259 
2 
0.6000.609 
14 
0.432 
0.798 

0.2600.269 
5 
0.6100.619 
9 
0.440 
0.866 

0.2700.279 
5 
0.6200.629 
7 
0.464 

0.2800.289 
5 
0.6300.639 
11 
0.474 

0.2900.299 
7 
0.6400.649 
10 
0.478 

0.3000.309 
4 
0.6500.659 
9 
0.513 

0.3100.319 
4 
0.6600.669 
4 
0.518 

0.3200.329 
11ß 
0.6700.679 
0 
0.552 

0.3300.339 
10 
0.6800.689 
2 
0.565 

0.3400.349 
8 
0.6900.699 
6 
0.590 

0.3500.359 
8 
0.7000.709 
3 
0.633 

0.3600.369 
12 
0.7100.719 
3 
0.645 

0.3700.379 
14ß 
0.7200.729 
2 
0.681 

0.3800.389 
6 
0.7300.739 
0 
0.752 

0.3900.399 
5 
0.7400.749 
1 

0.4000.409 
16ß 
0.7500.759 
1 

0.4100.419 
4 
0.7600.769 
0 

0.4200.429 
7 
0.7700.779 
1 

0.4300.439 
16 
0.7800.789 
0 

0.4400.449 
28ß 
0.7900.799 
1 

0.4500.459 
27 
0.8000.809 
2 

0.4600.469 
24 
0.8100.819 
0 

0.4700.479 
43ß 
0.8200.829 
1 

0.4800.489 
25 
0.8300.839 
0 

0.4900.499 
26 
0.8400.849 
0 
Assuming that RR Lyrae variables correspond to the case 7 £ n £ 9, l £ 2, and <n_{2}  n_{1}> = 1, one can compare the stellar scale results with the corresponding transition period data for He, including singlet and triplet systems. For ease of comparison the He periods are scaled up to units of "days," i.e., multiplied by a factor of L in accordance with equation (2). The scaled He periods are determined by the calculation
P = ( h /DE)(L)  (7) 
where P is the transition period scaled up to "days," DE is the energy level separation (Bashkin and Stoner, 1975), L is the scaling constant from equations (1)(3), and h is Planck's constant. The resulting transition period data for He are given in Table II. The range of relevant transition periods for He is 0.276 to 0.752 "days," which is in reasonable agreement with the predicted range of 0.2 to 0.8 "days." And as predicted, the set of 24 transition periods for He contains counterparts to each of the six preferred periods identified in the RR Lyrae sample. In order to test the uniqueness of the correspondence between the He periods and the preferred RR Lyrae periods, the same calculations were undertaken for hydrogen and lithium atoms, and the relevant transition periods for these atoms are also listed in Table II. There is no correlation between the H periods and the preferred RR Lyrae periods. In the case of Li, three of the preferred RR Lyrae periods, including the largest and most discrete peak, have no counterparts among the scaled Li transition periods. Thus, the correspondence between the He and RR Lyrae periods appears to be significant.
This first attempt to investigate the possibility of quantization in the periods of variable stars represents a very approximate test that involves a small sample size and relies on numerous assumptions. Yet the results are reasonably encouraging and they suggest the way to achieve more rigorous quantization tests in the future (Oldershaw, 1989b). Such tests would require much larger samples of variable stars that have been segregated according to galactic location, and they would require highly accurate period, radius, and mass data. It would also be interesting to test whether the relative peak heights of the preferred periods for variable stars match up with the transition probabilities for the corresponding periods of the atomic scale analogs.
3.3. Implications of the Empirical Tests
There are only three plausible explanations for the general agreement between the predictions and the empirical data in the 20 tests discussed above: chance, fudging of various types, or cosmological selfsimilarity. A rough and very conservative calculation of the probability that the agreement could have resulted by chance can be made in the following manner. Assume that the probability of a chance agreement for each test is £ 1/3, i.e., the prediction could be unacceptably high, unacceptably low, or within the error bars of the relevant empirical parameter. Then the maximum probability that the same set of scaling rules could pass 20 such tests by chance is (1/3 )^{20} or one chance in 3,486,784,424 tries, which is to say that chance would be an extremely unscientific explanation for the favorable results presented above.
Fudging, or arbitrarily adjusting a theory so that it comes into agreement with observational data, has always been and still is (Oldershaw, 1988) a standard tool of the theoretician, though one that tends to be used furtively. The question to be considered here is whether fudging could account for the apparent success of the SSCM and its scale transformation equations. If one could make arbitrary choices with regard to the proposed analog pairs, the form of the scaling equations, and/or the values of the constants appearing in the scaling equations, then could the arbitrarily fudged theory pass the 20 falsification tests presented above, or a comparable set, even though nature was not fundamentally selfsimilar? The author's answer to this question, after investigating such matters for over 10 years, is that if selfsimilarity was not a global property of nature, then a fudged theory that could pass these particular falsification tests, or equally fundamental ones, would be hopelessly complicated and arbitrary. Moreover, as one tried to test the theory beyond the data that it was constructed around, it would quickly fail. In contrast, the SSCM has very simple conceptual foundations and scaling equations, the identifications of analog pairs are always based on two or more fundamental properties such as mass, radius, and spin period, and nearly half of the tests (numbers 1012 and 1420) were conceived and conducted after the theoretical foundations of the SSCM, its major analog pair identities, the form of the scaling equations, and the values of L and D had been submitted for publication (Oldershaw, 1986a). In the absence of a convincing demonstration to the contrary, for example, a demonstration that an equally simple and successful alternative to the SSCM can be arbitrarily constructed, the fudging explanation is scientifically untenable. The number, diversity, and fundamental nature of the quantitative falsification tests passed by the SSCM strongly support the contention that nature manifests discrete cosmological selfsimilarity and that equations (1)(3) uniquely relate the physical properties of atomic, stellar, and galactic scale systems.
4. CONCLUSIONS
In this paper the general concepts, and the selfsimilar scale transformation equations, of the SSCM have been discussed, and 20 successful tests have been presented. The simplicity of this model and its ability to quantitatively relate atomic, stellar, and galactic scale phenomena suggest that a new property of nature has been identified: discrete cosmological selfsimilarity. Although the SSCM is still in the early heuristic stage of development, it may be the initial step toward a truly remarkable unification of our considerable, but fragmented, physical knowledge. Major questions yet to be answered concern the exactness of the cosmological selfsimilarity (i.e., is the selfsimilarity accurate only to a factor of about 2, or is it exact?) and the number of scales in the cosmological hierarchy (i.e., finite or infinite?). It has been argued previously that these two questions are interrelated (Oldershaw, 1981b); for example, exact selfsimilarity necessitates an infinite hierarchy. An even more fundamental question is: why should nature be globally selfsimilar and rife with examples of local selfsimilarity?
A forthcoming paper (Oldershaw, 1989c) on the SSCM will discuss the paradigm in more technical detail. It will also review several definitive predictions by which the SSCM can be put to very rigorous tests, and it will discuss major unresolved problems that raise doubts about some aspects of the model. The review will conclude with a discussion of the diverse implications of the SSCM.
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