Structure of the Universe

Philip Isaac Friedgut, Ph.D.,
pifriedgut@statcourse.com

1845 Lake St., Huntington Beach CA 92648, USA.

PACS: 04.60

Section I: Origin of the Scalar Field

First
posited by __Guth__^{
}and^{ }__Linde__,
today, it is generally accepted that our universe arose from an exponentially
expanding scalar field of constant energy density. But what is the source of
this energy? Here we propose an extra-dimensional origin.

Before our universe Ù came into existence, there existed an affine space U consisting of at least five spatial and two time dimensions. Our universe of three spatial and one time dimension arose a point at a time via fibers from this pre-existing universe.

The wave equation of a pathway in the product space U x Ù takes the form:

Ψ[X_{E},
X_{I}, τ, t] = A`exp[i(k_{E}.X_{E} – ω_{E}τ)]
+ hexp[i(k_{I}.X_{I} – ω_{I}τ)]

where (X_{E}, X_{I}) is a
point in a D dimensional space-time, D > 6, with spatial components X_{E}
in U while X_{I} represents the familiar 3-dimensional spatial
coordinates of our own universe; the components of (τ, t) are the cosmic time parameter
τ and the Minkowski local-time parameter t.

The fiber bundle results in the exponentially expanding scalar field of constant energy density that is our early universe.

Guth A. H.*, *Phys. Rev. D*.* **23**,
347 (1981).

Linde A. D., Rep. Prog. Phys*.* **42**,
389 (1979).

Section II: Creation of Ordinary Space

As first posited by __Good__, quantum tunneling
after a random interval and at a random point in the scalar field results in
expansion of a fiber terminal into a Planck (space-time) volume of ordinary
space averaging 10^{-34} cm in radius and 10^{-44} seconds in
duration.[1]

The energy of the degenerate pathway is now divided between the Planck volume in our 4-dimensional universe Ù and the source of the fiber in the external universe U, that is, the amplitude of the external portion of the oscillations is reduced.

Ψ[q,τ,
t] = A`exp[i(k_{E}.X_{E} – ω_{E}τ)] +
A``exp[i(k_{I}.X_{I} – ω_{I}τ)]

where A' + A`` = A so that the total energy is conserved.

Good also proposed that such degenerate
pathways are able to subdivide again and again, with the total energy A being
parceled out in any of four ways:

1.
First, between the products of each division,
the process halting only when the resulting Planck-volume of ordinary space is
reduced to some base energy

ahexp[i(k_{I}.X_{I} –
ω_{I}τ)].

2.
Second, through the production of photons of
energy. These photons can travel freely through Ù including the scalar field.

3.
Third, the energy inherent in a pathway may be
partially converted into the kinetic energy and electric fields of electrons
and positrons.

4.
Fourth, the energy inherent in a pathway may
be partially converted into the mass and magnetic fields of quarks.

Good P. I., Physics Essays **23**, 368
(2010).

Bombelli,
J. Lee, D. Meyer, and R. D. Sorkin, Phys. Rev. Lett. 59, 521 (1987).

Requardt,
J. Phys.A 31, 7997 (1998); e-print arXiv:hep-th/9610055v1

Section III: Gravity

As in previous sections, we assume the
universe is built of discrete Planck units of space-time, averaging 10^{-34}
cm on a side and 10^{-44} seconds in duration. We also assume that
almost all such units are in a ground state with energy just greater than zero.

Assume further that the mass of a quark is
confined to a single Planck unit. Thus, this unit would possess energy far in
excess of the ground state. As [V]=[M]=[L^{-1}], the surrounding units
will decrease in volume (of space and time) in an effort to equalize the energy
density.

In order to minimize the overall distortion of
space, masses would be drawn toward one another. As the distance r from a quark
increases, the number of units affected increases as r^{2}, and the
change in volume of each unit decreases as r^{-2}. In particular, two
masses m_{1} and m_{2}, a distance r apart would be drawn
together by a force proportional to m_{1}m_{2}/r^{2}.

Thus, if our assumptions are valid, in static
situations, the gravitational effects of a mass on its surroundings are exactly
the same on the Planck- and macro-scales.

Section IV Formation of Protons and Neutrons

Assume
that the mass of a quark is confined to a single Planck unit, that is, with
99.9% probability it is located somewhere within that unit, much as with 99.9% probability
an electron is confined to a specific orbital.

Subject to these assumptions, as we saw in the
preceding section, the surrounding Planck units will decrease in volume in an
effort to equalize the energy density. As compressing the surrounding space
requires energy, the quarks group themselves so as to minimize the energy
required, that is, so as to minimize their effect on the surrounding space.

We show now that an aggregate of three quarks
will cause less distortion of the surrounding space than three disparate
quarks:

Let E_{o} denote the ground state
energy of a Planck unit, and E the energy equivalent of a quark. Let n_{1}
denote the number of Planck units an isolated quark comes in contact with. Let
r_{1} denote the effective radius of each of the units. Define R
such that a sphere of radius R will contain 99.9% of a Planck unit’s volume.

n_{1}E_{o}/r_{1}^{3}_{
}= E/R^{3} (1)

and

n_{1}r_{1 }= 2πR. (2)

Without loss of generality, define the
units of measurement so that E=1 and R=1.

From (1), we see that r_{1 }=
(n_{1}/E_{o}^{ })^{1/3} and from (2) that n_{1
}= (2π)^{¾}E_{o}^{¼}.

The center of mass of an aggregate of
three quarks may be assumed to be at the center of a sphere of radius 2R or 2
in our revised units and to possess energy 3E or 3.

n_{2}E_{o}/r_{2}^{3}_{
}= 3/8 (3)

and n_{2}r_{2 }= 4π. (4)

So that r_{2 }= (8n_{2}/3E_{o}^{
})^{1/3 }

and n_{2 }= 2^{¾ }(3/8)^{¼}
(2π)^{¾}
E_{o}^{¼}

= 2^{½ }(3/4)^{¼} (2π)^{¾} E_{o}^{¼}

< 3 (2π)^{¾} E_{o}^{¼}

the
number of units in direct contact with three isolated quarks.

Our third and last assumption is that the
quarks possess an oriented magnetic field. Consequently, any aggregation need
be coplanar and can assume one of two possible configurations, one in which the
magnetic fields lie within the same plane so as to cancel one another, and one
in which they form a single field at right angles to the plane occupied by the
quarks.

Section V: Development of a Super-cluster of Galaxies

Let S denote the space occupied by an arbitrary
super-cluster of galaxies. We assume that S may be subdivided into Planck
volumes with Minkowski coordinates of their centers (x,y,z,t). Good (2010)
proposed that S expands as a self-avoiding random walk, so that after N steps,
one would expect to see an oblong form of shape N^{0.6}N^{0.51}N^{0.4}.

We may cover S with the topology T consisting of the open sets with rational coordinates:

Note that T is a Hausdorff
topology and that for any Planck volume V in S we may always find a countable
sequence in T, such that

Let R denote the real line, and Ξ the topology on R derived from the set of open intervals with real endpoints. A function f from S to R will be said to be continuous providing that

which follows from the rationals being dense in the real numbers.

As noted in Section II, subdivision of the degenerate pathways in S is accompanied by the release of energy in the form of photons, high-speed electrons and quarks. As we saw in Section IV, gravitational attraction leads the quarks to group themselves into positively-charged protons to which the negatively-charged electrons are attracted. This leads to the formation of atoms of hydrogen as the electrons are drawn into orbits about the protons.

These atoms are attracted to one another. When the cloud of hydrogen atoms is of sufficiently high-density, the high-temperatures resulting from the numerous photons, leads to the production of helium atoms and the release of even greater quantities of energy in the form of photons.

The mass of these suns attracts all the free atoms in the vicinity. Gravitational attraction is responsible for the formation of sun clusters and of these clusters into galaxies.

The rate of galactic rotation is determined primarily by the still expanding scalar field which surrounds the super-cluster. For by the same arguments we made use of in Section III, the units of ordinary space adjacent to the scalar field will increase in volume (of space and time) in an effort to equalize the energy density.

The negative gravity induced by the scalar field affects the movement of material within the super-cluster. For example, the rotation of matter within galaxies, including our own, does not increase as rapidly with decreasing distance from the galactic hub as Kepler's laws would predict. The gravitational field of the scalar field, retards the rotation of masses to an extent proportional to the speed with which the masses are traveling. The closer a mass is to the gravitational center of a galaxy, the faster it would move if governed by Kepler's laws alone, and the more it will be retarded.

Section VI: Particles and Anti-Particles

Particles and anti-particles are created with equal probability. This does not mean in equal numbers; in fact, quite the contrary is true.

The arc sin law (see, for example, Feller, III.4) tells us that when the numbers of one type of particle exceed the numbers of the other, the situation is likely to remain so indefinitely. Indeed, because the collision of a particle and anti-particle results in their mutual destruction, inevitably only particles or anti-particles will persist.

Let N(t) denote the number of particles and M(t) the number of anti-particles at time t.

Let λΔt denote the probability that a particle is created in the interval Δt. Note that it is independent of t, N(t), and M(t).

Let λΔt denote the probability that an anti-particle is created in the interval Δt. Note that it too is independent of t, N(t), and M(t).

Let μN(t)M(t)Δt denote the probability that a particle/anti-particle pair is annhilated in the interval Δt.

The
probability that more than one event occurs in an interval Δt is of order Δt^{2}.

Let p_{nmt} =
Pr{N(t)=n, M(t)=m}.

Then p_{nmt} =
(1-2λ-μnm)Δtp_{nmt-1 }+ λΔtp_{n-1mt-1}+ λΔtp_{nm-1t-1}+μ(n+1)(m+1)Δtp_{n+1m+1t-1}...(1)

Let A[t] = ΣΣu^{n}v^{m}p_{nmt}

with
the initial conditions ΣΣp_{nmt }= 1
and A[0] = p_{000}.

Note that

From
(1), it is easy to see that as t ↑ If EN(t)→0, then EM(t)→ ∞; If EM(t)→0, then EN(t)→ ∞.

Inevitably, only particles or anti-particles will persist.

We saw in the previous section that particles and anti-particles are confined to the super-cluster in which they arise, thus, it may be that while particles predominate in our own Virgo super-cluster, some other super-cluster may consist solely of anti-particles.

Feller
W. *An Introduction to Probability and Its Applications. *Vol 1. John
Wiley & Sons, New York, 1950.

Section VII: Large-Scale Structure of the Universe

Our
no-longer-Earth-bound telescopes have given us a picture of a universe that is
a large empty expanse broken up, albeit rarely, by archipelagos of light
radiating from super-clusters of galaxies. This is precisely what would be
expected if the rate of expansion of the scalar field exceeded the rate of
formation of volumes of ordinary space from the scalar field. Moreover, this
picture of a largely empty universe suggests that the energy A inherent to each
fiber of the scalar field must correspond to the energy in an entire
super-cluster of galaxies.

In Section II, we posited that the
regions of ordinary space arise within the scalar field independently of one
another in non-overlapping regions of space-time. For if P_{1}, P_{2}
denote the probabilities of one, two events respectively in the same region of
time-space, then P_{2} = P_{1}^{2}/2. And
the probability of k events in the same region is P_{1}^{2}/k!
Clustering of the super-clusters is to be expected and the observed
"Filaments" and "The Great Wall" are natural consequences.

A Poisson distribution of the
super-clusters of galaxies such that Pk = P_{1}^{2}/k! was
first reported by Neyman and Scott in 1952 utilizing the limited data then
available from ground-based telescopes. Recently, Good analyzed the data from
the SDSS survey main and Luminous Red Galaxy (LRG) samples collected by
Liivamagi, Tempel, and Saar. Good made use of the spatial-temporal distribution
of super-clusters of galaxies for z < 0.5 in the catalogue
scl_cat_ls00180_e.dat. A detailed description of the methods he used to create
9600 regions of equal volume may be found in the appendix.

Table I: Spatial Distribution of Superclusters of Galaxies |
||||||

Number/Region |
0 |
1 |
2 |
3 |
4 |
5 |

Observed |
7178 |
2012 |
370 |
38 |
1 |
1 |

Poisson Distribution |
7095 |
2144 |
324 |
32 |
2 |
0 |

Table I
summarizes his findings and compares the observed values the numbers expected
from a theoretical distribution based on the Poisson.

Current
observations by Reiss et al. and Perlmutter et al. tell us that while the
galaxies within a super-cluster are continuously drawn closer together, the
super-clusters themselves are flying apart. Recall that the effects of mass on
ordinary space was described in Section III. Thus the former may be attributed
to the effects of gravity operating within the confines of each super-cluster:
The latter is a consequence of the still-expanding scalar field described in
Section I.

Perlmutter S. et al., Astrophysical J, **517**, 565
(1999).

Riess A. et al., Astronomical J, **116**, 1009
(1998), Astrophysical J, **607**, 665 (2004).

** **

**Appendix**

**Spatial-Temporal
Distribution of Super-clusters of Galaxies for z < 0.5**

Phillip I. Good*

**1. Introduction**

One
would expect to observe a Poisson distribution of the super clusters of
galaxies only if their origins took place independently in non-overlapping
intervals of space and/or time. We would expect their density to decrease with
time, that is, with decreasing redshift z, if they were all created at
approximately the epoch.

A Poisson distribution of the super clusters of
galaxies was first reported by Neyman & Scott (1952) utilizing the limited
data then available from ground-based telescopes. In the next section, we use
the data from the SDSS survey main and Luminous Red Galaxy (LRG) samples
collected by Liivamagi, Tempel & Saar (2010) to confirm that super clusters
do have a Poisson distribution. In subsequent sections, we use the same data to
derive the density of super clusters as a function of space-time.

Two possibilities suggest themselves. First, that
the super clusters of galaxies formed at more or less the same point in time
(give or take a billion years). Alternatively, super clusters arise and continue
to arise independently of one another at different points in both space and
time.

Both models presuppose that the super clusters will
have Poisson distribution in space. But as will be seen in a third section, the
competing models predict quite different spatial-temporal distributions.

2, Observed Number of Super clusters

Suppose
we were to count all the super clusters of galaxies located in an arc of
constant width and height at various distances Ro < R1 < … < Rn from
the observer so that the volumes Vi, i =0,1,…n of the resulting regions were
the same. Note that Vo is a segment of a sphere, while the remaining volumes
are ring segments, chosen so that

A list of super clusters, their luminosity
distances R, and their redshifts z may be found at __http://atmos.physic.ut.ee/~juhan/super/super_lrg/__ in the catalog scl_cat_ls00180_e.dat
for z<0.5.

Ordering the catalog by the redshift, we find that R ≈3021.33z -706.89z2 + 27.54z3. See
Figure 1. In what follows, we employ the smoothed values of R given by this
equation, rather than the less-accurate luminosity distances
to be found in the table.

Figure1

We first set Ro = 460.514 so as to incorporate 110
super clusters of galaxies, yielding Figure 2a.

Figures 2a,b

Denote by Ni, the number of super clusters observed
in the region Vi located at a distance Ri from the Earth. The apparent density
of super clusters is given by .

But as we observe further and further from Earth,
we are going back in time as well as outward in space; at an earlier time, the
universe was smaller. Specifically, the redshift z is a function of the
expansion of the universe, so that a scale factor proportional to the radius of
the universe, a(z), decreases as 1/(1+z); see Schneider (2010, p155). Thus, the
actual density D(zi) of super clusters in a region Vi at redshift zi is . Rescaling
the corrected densities yields Figure 2b.

3. Spatial Distribution of Super clusters of
Galaxies

We
divided the data from catalogue scl_cat_ls00180_e.dat into 9600 equal-sized
regions, based on the 24 distances used to create Figures 2a,b, sixteen equal
subdivisions of the right ascension which ranged from 111 to 261, and 25 equal
subdivisions of the declination which ranged from -3.7 to 69.6. Our findings,
comparing the observed distribution derived from this data set with a
theoretical distribution based on the Poisson, are shown in Table 1:

Table I: Spatial Distribution of Super clusters of Galaxies |
||||||

Number |
0 |
1 |
2 |
3 |
4 |
5 |

Observed |
7178 |
2012 |
370 |
38 |
1 |
1 |

Poisson Distribution |
7095 |
2144 |
324 |
32 |
2 |
0 |

4. Expected Numbers and Density of Super Clusters

The
cold dark lambda model (CDλM) assumes that all super clusters formed at
approximately the same time and the continuing expansion of our universe merely
dilutes their density. Suppose when the super clusters came into existence, the
radius of the universe in scaled units was a(z*) = 1/(1+z*) and that the total
number of super clusters was N*. The density of super clusters at that time was
D*= kN*(1+z*)^{3}.

At a later time, the density will have been diluted by the
expansion of space so that the true density D*(z) will be equal to kN*(1+z)^{3}.
If the CDλM holds, D(z)/D*(z) will be a constant. Figure 3 depicts the observed relationship between D*
and D, casting doubts on the near-simultaneous origin of the super clusters.

Figure 3

Friedgut (2013) proposed that the super clusters arise at random points in time-space with events in non-overlapping regions of time-space occurring independently of one another. The total number that are formed in a fixed period of time is proportional to the size of the universe. The change in density with time is thus more or less constant. The change may be less than, equal to, or greater than the rate of expansion. Were this model true, the observations depicted in Figure 3 would suggest that the rate of creation was less than the rate of expansion.

5. Death and Mergers of Super-Cluster of Galaxies

Before rejecting the CDλM out of hand, consider that other factors may be at work. If we were to observe a similar decline in numbers in populations of businesses or nations, we would attribute it to mergers. We know that galaxies merge, but current observations suggest that the super clusters continue to fly apart rendering mergers impossible, particularly if they all came into existence at about the same time. The Friedgut model also is ruled out for it views super clusters as entirely separate entities, separated by an expanding scalar field, incapable of merging

If we were to observe the population density of a mature population of cells or organisms decreasing with time, we would attribute it to deaths. We know that individual stars are extinguished over time, perhaps so too are entire galaxies, clusters of galaxies and super clusters.

If all the super clusters were created at approximately the same epoch, their population would pass through three phases each lasting billions of years, a phase in which the density increases, a phase in which the numbers remain more or less constant, and a phase in which the sky empties of stars.

Under the Friedgut model, the population of super clusters would pass through two phases, a period before senescence is observed in which the density of super clusters increases and a period in which the numbers of super clusters remain more or less constant.

The catalog data suggest we are currently in the transition between the first and second phases in either case.

If the death of super clusters is the explanation for the decrease in super-cluster density with time, we would expect the median number of galaxies per super-cluster and the median sum of galaxy luminosities to be an increasing function of z. Figure 4 shows that just the opposite is true.

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Figure 4**

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**References**

Friedgut
P.I. Structure of the Universe __http:/statcourse.com/research/Structure.pdf__

Neyman
J., Scott E.L., A theory of the spatial distribution of galaxies, 1952, Ap. J.
116, 144.

Schneider
P., 2010, Extragalactic Astronomy and Cosmology*, *Springer-Verlag,
Berlin.

[1] The view that the universe is built of discrete Planck units of space-time was first suggested by Bombelli et al. and Requardt.