The Large-Scale Structure of the Physical Universe, Part II: Mathematical Aspects of the Cosmic Bubbles

Reciprocity XX #3, Autumn, 1991

 

Part II: Mathematical Aspects of the Cosmic Bubbles

Cosmic BubbleIn Part I of this Paper (Reciprocity, XX (2), Summer 1991, pp. 5-8), we have highlighted the recent observational findings in the field of astronomy leading to the discovery of large-scale voids in space coupled with the distribution of galaxies as clumps at the peripheries of these voids. We called these voids bubbles. We have demonstrated there how the new facts could be readily explained in a natural way by the Reciprocal System of theory. In the present Part we attempt to develop the mathematical consequences of those concepts delineated in Part I. Since we cannot afford to repeat, Part I must be read in order to be able to follow the present treatment. For ease of referring, section numbers and reference numbers are continued from Part I.

8. Analysis of the Motion in the Bubble

With the knowledge of the origin and nature of the bubbles we can now attempt to evaluate some of their properties. Let:

c Speed of Light 2.99793×1010 cm/s
G Universal constant of gravitation 6.673×10-8 cm3/g.s2
r Radius of the bubble cm
t Time since creation of the bubble s
s Rate of mass inflow into the material sector g/cm.3s
r = s.t Mass density in the bubble at time t, g/cm3
M Total mass of a material aggregate g
M0 Mass of the Sun 1.99×1033 g
d0 Gravitational limit of a consolidated material aggregate cm
k0 A constant 3.5664×1018 cm
P The universal constant of progression 1.044×10-11 cm/s2
v Speed cm/s
a Acceleration = v Dv/Dr, cm/s2

We note from Reference 15 the following:

d0 = k0 (M/M0)½

(1)

P = G.M/d02 = G.M0/k02

(2)

We will first evaluate the expressions for the speed due to progression and the speed due to gravitation in the bubble. In the beginning stages, (see Section 5), the net speed in the entire mass is outward and we have to consider the expressions relevant to motion in equivalent space. Only when gravitation balances (or predominates) progression does the motion come back into the space of the conventional three-dimensional reference frame.

8.1 Speed due to Progression

In the conventional reference system:

ap = vp (Dv/Dr)p = P

(3)

vp = (2.P.r) ½

On the basis of the explanation given in Reference 15 the corresponding speed in equivalent space is given by:

vp,e/v0 = (vp/v0)2/2

where v0, the zero-point speed, is given by:

v0 = (2.G.M/d0) ½ = (2.P.d0)½

(4)

Therefore we get:

vp,e = [alpha] (r/ρ)¼

(5)

where:

a = (P/2.k0) ½ (0.75 M0/p)¼ = 1.7861×10-7 cgs unit

(6)

8.2 Speed due to Gravitation

In the conventional reference system, considering a location at the periphery of the bubble:

ag = vg (Dv/Dr)g = 4/3.p.G.r.r

(7)

vg = (4.p.G.r/3) ½ r

The corresponding speed in equivalent space is given by:

vg,e/v0 = (vg/v0)2/2

Adopting v0 from Eq. <4> we get:

vg,e = ß r¾ r5/4

(8)

where

ß = p (2.G.k0/9) ½ (0.75/M0p)¼ = 2.391×10-3 cgs unit

(9)

8.3 Net Speed

In the conventional reference system, the net speed is (using Equations <3> and <7>)

vn = vp - vg = (2.P.r) ½ - (4.p.G.r/3) ½ r

(10)

and in equivalent space (using Eqs. <5> and <8>)

vn,e = vp,e - vg,e = (a - ß.r.r) (r/r) ¼

(11)

8.4 Zero-Point Radius

We have called the radius of a uniform spherical mass at whose periphery the net speed becomes zero the zero-point radius, rz. Equating Eq. <11> to zero and using Eqs. <6> and <9>, we obtain

r.rz = a/ß = (3.P)/(2.p.G) = 7.47×10-5 g/cm2

(12)

This relationship gives, for any given value of mass density, the corresponding radius where the net speed becomes zero.

8.5 Advent of Criticality

In Section 5 we have set forth that the mass of the expanding bubble reaches a critical state when its actual radius equals the zero-point radius. We have called this radius the critical radius rcr and the corresponding age of the bubble the critical time tcr. Substituting in Eq. <12> r = s.tcr and rz = rcr , and noting that:

rcr = c.tcr

(13)

we get

tcr = (a/(ß.c.s))½ seconds

(14)

Now if the rate of mass inflow, s, could be evaluated, one obtains the time it requires for the bubble to reach criticality and the corresponding size of the bubble. We, therefore, proceed as follows.

8.6 The Universal Constant of Materialization

We may call s the universal constant of materialization, like we call G and P respectively the corresponding universal constants. Noting that r = c.t and r = s.t we rewrite Eq. <11>

vn,e = (a - ß.s.c.t2)(c/s)¼

(15)

At the moment of the quasar exit (that is, the start of the bubble expansion), we take t = 0. Therefore, at this moment, vn,e reduces to

vn,e,0 = a (c/s)¼

(16)

This is an outward speed and can be equated to the speed that is coming in, vi, with the inflowing matter from the cosmic sector, wherein gravitation acts inward in time (equivalent to outward in space). It is not yet attenuated by gravitation in space (as could be seen from ß.s.c.t2 = 0). The inter-sector transition of matter takes place on individual mass unit basis. Normally, the speed effective on unit mass basis is the unit speed c. However, as elaborated in Reference 15, the scalar rotation of atoms that is the origin of gravitation is distributed over 156.444 directions (degrees of freedom) in the time region (the region inside unit space) and 8 directions in the time-space region (the region of motion in three-dimensional space). In the corresponding situation of the cosmic atom, the cosmic gravitation gets distributed over 156.444 directions in the space region (the region inside unit time) and 8 directions in the space-time region (the region of motion in three-dimensional time). Consequently, the incoming speed, vi , is given by

vi = c/(156.444 * 8)

(17)

remembering that the contact between motion in space and motion in time is one-dimensional. Equating Eqs. <16> and <17> we arrive at the important value

s = 9.2679×10-47 g/cm3 s

(18)

9. The Bubble Parameters

We can calculate the critical time by Eq. <14>, the corresponding critical density by rcr = s.tcr , and the total mass of the bubble at criticality:

tcr

1.643×108 years

rcr

1.643×108 light-years

rcr

4.8055×10-31 g/cm3

Mcr

3.7994×1015 Solar masses

We will examine these results one by one to see if they tally with the observations.

9.1 Matter Density

All the above values can be seen to be within the range of corresponding actual observed values. Current estimates of the density (in g/cm3) of matter are as follows18:

Interstellar space 10-24
Space near edge of galaxy 10-28
Intergalactic space 10-31

The calculated critical density is slightly higher than the estimated density in intergalactic space but very near it.

9.2 Globular Clusters

As the net speed at the bubble periphery changes its scalar direction from outward to inward (on reaching criticality), it initiates the collapse of a large number of individual masses of diffuse matter all around the spherical boundary of the bubble. Each of these masses, as it collapses, further splits into a number of aggregates of stellar size, eventually resulting in a Globular Cluster. We will not here enter into detailed discussion of the mechanics of the formation of the Globular Clusters for want of space. The interested reader may refer to Larson.13 At this juncture we would merely want to make an estimation of the collapse time of these Globular Clusters.

Let us consider the condition at the bubble periphery. There the net speed is given by Eq. <10>. Letting r = s.t, r = c.t (strictly r < c.t since gravitation now predominates: but its effect is negligible in the initial stages of the post-critical phase), and x the radius of a proto-Globular Cluster of mass Mg, we have

dx/dt = vn = (2.P.c.t)½ - (4.p.G.s.c2.t3/3) ½

(19)

The equation can now be integrated between the limits x = xg to 0 and t = tcr to tg, where

xg = [Mg/(4.p.rcr/3)]1/3

(20)

The following Table gives the calculated collapse time as a function of the proto-Globular Cluster mass.

Mg (Solar masses) Collapse Time (years)
103 0.41×108
104 0.59×108
105 0.85×108
106 1.22×108

The relationship between the collapse time and Mg obtained by regression is

Collapse Time = 0.138×108 (Mg)0.158

(21)

and indicates that a star of, say, one Solar mass would condense in 0.138×108 years. Thus the individual stars form well before the Globular Cluster as a whole arrives at its final stage of equilibrium.

In passing, we would like to remark that while it is possible for the Globular Cluster to form from a matter density of about 5×10-31 g/cm3 under the gravitational assistance of the bubble as a whole, simple calculation from Eq. <12> shows that, left to itself, it requires a density of nearly 10-26 g/cm3 to accomplish the same result.

9.3 The Bubble Size

The above calculations indicate that it takes nearly 0.4 to 0.6×108 years for the Globular Clusters to form and become observable after the bubble attains criticality. During this period the original bubble continues to expand, though not at the speed of light, at a slightly slower rate. Adding, therefore, a distance of 0.4×108 light-years to the radius at criticality we find that the bubble diameter at this juncture works out to be

2 (1.643×108 + 0.4×108) = 4.1×108 light-years

It must be noted that this result gives the maximum possible size. Beyond this stage the observed size actually decreases because (i) gravitation retards/nullifies the expansion and (ii) continued formation of Globular Clusters and dwarf galaxies shifts the spherical boundary between the visible and the dark matter ever inward, toward the bubble centre. From Eq. <12> we can see that the apparent void radius (equal to the zero-point radius) varies with time as

r = rcr .tcr/t

(22)

Since the number of clusters grows as time passes, their combined gravitational effect draws up the matter at the bubble core and simultaneously they close in on it. A preliminary calculation on the basis of the gravitational limit of the surrounding group of clusters indicates that the last stage of the bubble, before it rapidly dissipates, will occur at a bubble diameter of about 84 million light-years.

The observed bubble sizes reported in the literature range from 60 to 400 million light-years. Broadhurst’s survey,7 though covering only two narrow regions but extending to depths of 2000 Mpc, puts it at 417 million light-years (see Section 1). Thus the results of calculations made on the basis of the Reciprocal System of theory are entirely in agreement with the facts.

9.4 Total Mass

The bubble mass at criticality has been calculated to be 3.8×1015 Solar masses. But as the formation of the Globular Clusters and other galaxies continues in the post-critical stage, the incessant inflow of matter from the cosmic sector adds to the total mass. When the bubble eventually reaches the supercluster stage its mass—that is, the mass of that portion of the original bubble that condenses into groups of clusters and clusters of stars—would be well within the 1016 Solar mass range of the current estimates.

10. Computer Simulations

B.B. Mandelbrot,19 investigating fractal shapes in nature, has studied the distribution of galaxies and clusters of galaxies in three-dimensional space. By postulating the existence of intergalactic voids he tried to evolve models of clustering. His findings are very interesting and pertinent.

He starts with a completely filled space and keeps on removing spherical volumes of matter. Both the size of the spherical hole and the location of its center are chosen randomly. The size of the hole is treated as a Poisson random variable with a distribution

N (>v) } 1/v

(23)

which reads as the number of holes with volume greater than v is inversely proportional to v.

The model is simulated on computer. His results—both the covariance between two points in space and the covariance between two directions in the sky—indicate a very good fit of data. The graphics output shows the views of the material remained after removing the spherical chunks and bear an amazing resemblance to the actual sky maps.

10.1 Unforced Clusters

A rather significant and unforeseen result of Mandelbrot’s model above is that the distribution of the remaining points shows an apparent hierarchical structure. Mandelbrot exclaims: “Each point stands for a whole minicluster…In addition… the miniclusters are themselves clustered. They exhibit such clear-cut hierarchical levels that it is hard to believe that the model involves no explicit hierarchy, only a built-in self-similarity.”20 Or again, “Increasing clustering is not provoked by the concentration of all points around a few of them but by the disappearance of most points, leading to an increasing number of apparent hierarchical levels.”21 Hence he refers to them as “unforced clusters.”

His finding is directly in line with the conclusions which Larson obtains from the Reciprocal System. “…the largest units in which gravitation is effective toward consolidation of its components are the groups of galaxies. These groups begin separating immediately, but until the outward movement produces a clear-cut separation, their identity as distinct individuals is not apparent to observation. Here, then, is the explanation of the large “clusters” and “superclusters” of galaxies. These are not structural units in the same sense as stars or galaxies, or the groups of galaxies that we have been discussing.”22 (Emphasis added.) These are default clusters with apparent hierarchical structure brought into relief by the randomly generated bubbles.

10.2 Difficulties with Mandelbrot’s Model

The above model suffers from two shortcomings, and Mandelbrot has to introduce two ad hoc assumptions to make it successful. These concern the hole size distribution assumed by him (Eq. <23>). Firstly, while the model shows reasonable verisimilitude when limited portions of sky are considered, the overall sky maps are completely wrong in that they include voids as immense as one-tenth of the sky or more. This defect could be traced to the unrealistically large hole sizes allowed by the hyperbolic distribution function N (>v) } 1/v and could be eliminated by imposing an “upper cut-off,” vmax , on the hole size.

Secondly, the unrealistically large number of small-sized holes allowed by this hyperbolic distribution leaves no portion of the sky not covered by the holes. In fact, Mandelbrot imposes the constraint that

P (>v) = 1, for v < 1

(24)

(where P stands for probability) to save the model. It would, therefore, be interesting to see what the Reciprocal System has to offer in this context.

10.3 Distribution of the Hole Size According to the Reciprocal System

According to the Reciprocal System the large-scale universe is in a steady state. That is, both the rate of inflow of matter from the cosmic sector and the rate of final quasar transitions to the cosmic sector are uniform in time (as well as in space) and equal each other. Therefore, for a given volume of space, the number of bubbles created per unit time, which is the number of quasars exiting per unit time, is given by

dN/dt = b

(25)

where b is a constant directly calculable from s and the average mass of a quasar. Assuming an average quasar mass of 109 Solar masses, b works out to be 1.37×10-15 per second per cubic megaparsec of space.

For 0 <= t <= tcr

We have seen that till criticality the radius is given by the relationship r = c.t. Differentiating this we get dt/dr = 1/c , and finally

dN/dr = (dN/dt)(dt/dr) = b/c

(26)

Integrating we have

N1 (>r) = b (rcr - r)/c

(27)

where N1 is the number of bubbles of radii larger than a specified radius r. It may be seen that N1 is the contribution to the bubble population from the pre-critical phase of the bubble evolution.

For t >= tcr

Beyond the critical point, we have seen that the bubble size decreases according to Eq. <22>. We obtain on differentiating it

dt/dr = - rcr . tcr/r2 = - rcr2/c.r2

since rcr = c.tcr by Eq. <13>. Finally

dN/dr = (dN/dt)(dt/dr) = - b.rcr2/c.r2

(28)

On integrating

N2 (>r) = b ((rcr2/r) - rcr)/c

(29)

where, again, N2 is the number of bubbles of radii larger than r. N2 is the contribution to the bubble population from the post-critical phase. We have shown in Section 9.3 that in the post-critical phase there is lower cut-off to the bubble size due to its quick dissipation. Let this lower cut-off radius be r0. On adding N1 and N2 from Eqs. <27> and <29> respectively we get the following total distribution.

For 0 <= r <= r0

N (>r) = b ((rcr2/r0) - r)/c

(30)

For r0 <= r <= rcr

N (>r) = b ((rcr2/r) - r)/c

(31)

We take the one-dimensional analogue of Mandelbrot’s Eq. <23> for the sake of comparison

N (>r) = C'/r

(32)

where C' is a constant. It can readily be seen that the difficulty of unrealistically large number of small-sized holes that occurs in Mandelbrot does not arise here because N (>0) is not infinite but a finite constant (see Eq. <30>). Similarly the difficulty of occurrence of unrealistically large-sized holes does not arise either. This is because there is a maximum possible size, rcr ; and this comes out as a natural consequence of the development of the theory in the case of the Reciprocal System—not as an arbitrary constraint imposed on the model to make it conform to the reality.

11. Summary

The astronomical observations of the recent decade have brought to light the large-scale distribution of galaxies in the universe and the near perfect uniformity of the cosmic microwave background to an extent that has not been possible earlier. An unexpected fact that has come to be established is the ubiquitous occurrence of spherical voids of gigantic proportions throughout space. Current theories are nonplussed.

Larson has shown that galaxies, on reaching an age limit, explosively eject fragments of their cores, imparting to them ultra high speeds. These fragments are quasars. When gravitation is attenuated by distance (time) the net speed of quasars reaches two units, the limit of the material sector. Then gravitation—which always acts inward—ceases to act in space and starts operating in time. This leaves the outward progression of space unchecked and all the constituent matter of the quasar, which hitherto stayed put, is dispersed in all directions in space at the speed of the progression. Thus, centered at the location of the original quasar, a spherical void starts growing.

Since the ejection of quasars and their exit are inevitable stages in the evolution of material aggregates these voids ought to be a universal phenomenon. Preliminary calculations demonstrate that their observed sizes and other parameters are in consonance with the theoretical predictions.

All these latest observational findings that the current theories are at a loss to account for, are logically explained by the Reciprocal System starting from the foundation of its Fundamental Postulates. This Paper, thus, demonstrates once again the cogency and power of the Reciprocal System as a general physical theory.

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