The Big Bang Theory is the prime theory to describe the beginning of the universe. George Gamov first suggested the concept in 1948. The theory explains how the universe emerged from a tremendously dense and hot point about 13.7 billion years ago. This point is called singularity. Precise prediction of background radiation temperature, even before its actual discovery, and recent findings, which suggest that our universe was much hotter in the younger ages, supports the Big Bang Theory. However, physicists have mainly avoided the conundrums of an infinitely compressed zero-sized starting point for Big Bang. The problem arises when we go to diameters smaller than Planck length. We cannot find any meaningful smaller size in our universe. The only way that we can pass beyond this size is, if we exit our materialistic world and think singularity. Our objectivity based scientific laws do not govern in this arena. Any attempt to carry over these laws is doom to assumptions, uncertainty and confusion.

Ordinal Numbers
The question is how the zero point singularity can create the objective space-time universe.  Ordinal numbers can be suggested as a mathematical formalization for The Big Bang theory. Ordinal Numbers are the simplest set of all and is sometimes called empty set or null set.  The empty set is normally shown as,
Ø = { },
Where the empty brackets delineate a set.  Since there is nothing inside the brackets it is an empty set. Let us associate Ø with the number 0. Then we can go further and define a set that its only member is Ø,
Ø = { Ø },
We use our logic to come to the deduction that { Ø } is not the same as  Ø. The set { Ø }  has one member on it, whereas Ø itself has none at all. Here we can associate {  Ø }  with number one. For number two we collect the two previous entities to come to,
Ø = { Ø, { Ø }},
Then we associate 3 with the collection of all the above sets as,
Ø = { Ø, { Ø },{ Ø, { Ø },}}},
The number 4 is associated with the sets containing the members that we have encountered before to come to,
Ø = { Ø, { Ø },{ Ø, { Ø }}, { Ø, { Ø },}}},{ Ø, { Ø },{ Ø, { Ø },}}}},
And so on. ⁵⁶

The concept of ordinal numbers shows how natural numbers can be derived from nothing. On the other hand natural numbers delineate different objective elements in space-time such as units of matter, space and time.  This concept can be used as formalism for the evolving objective universe from a zero-size, non-material singularity at the beginning of time.

Reaching to natural numbers from null sets has other physical manifestations as well. In the Standard Model, subatomic particles such as electron are zero size and mass-less objects. However, they are computable and we use natural numbers to denote them in calculations. In quantum theory waves are reduced to objects which are computable. In my assumption mentioned in the Quantum Mechanics chapter, Super position is in informational domain. The act of state reduction (changing multiple states to one state) is objectification demonstrated by conscious observer. Here again objective reality comes to existence out of a non-material domain. Mind you that singularity in this model is non-material and encompasses the informational domain.

Origin of Matter
Although there are strong evidences to support the main body of the theory, there are different opinions about how the Big Bang event actually happened and progressed.
 In Gasperini and Veneziano's pre-Big Bang scenario, the universe started out as cold and essentially infinite in spatial extent. In addition, they are suggesting that Big Bang is not the initial event in creation of our universe; rather, it is a few steps further.

Guth Inflationary Theory
The inflationary theory is the most popular and accepted version of the Big Bang. In the inflationary model, the initial stage contained a very rapid and huge expansion of the space. This just took a fraction of a second. Then the expansion slowed down for about seven billion years. This period is called deceleration stage where the rate of expansion had been slowed down by the gravity of the matter inside the universe.  Then as space got bigger and thinner, the gravity got weaker. As a result, the expansion of the universe has been accelerating for the past seven billion years since.
Cosmologists consider the cosmological constant the factor behind the accelerating force. This constant is supposed to exert repulsive force to the space-time. It is believed that dark energy with its negative pressure is the source for the cosmological constant. For the expansion to accelerate, we need constant production of dark energy. Where does this energy come from?

Continuity Equation Principle
Continuity Equation Principle (Note 1) requires that the density of matter in a region of space stays the same. In addition, the density of the universe is currently close to the critical density. Which means the gravitational force from the matter/energy density inside the universe can counteract the expansion force and keep the universe almost flat (please see the Flatness Problem chapter). Chances are that the universe density has always been close to critical density.  Recent findings suggest an expanding universe with an accelerating pace.
The dark energy contributes to the density of the region. In fact, seventy percent of the density in the universe is attributed to dark energy. However, since the universe is constantly expanding, we would need constant matter creation for the continuity equation to hold. Where does the matter come from?
What is the origin of the additional matter or dark energy being continuously created? One is led to look at other possible scenarios for matter creation besides the initial Big Bang. We need another source readily accessible by any inch of space to supply the matter needed.  Steven Hawking presents the lead.

“The uncertainty principle of quantum theory means that fields are always fluctuating up and down even in apparently empty space, and have an energy density that is infinite.” ⁶

Please note that infinite energy cannot belong to a finite world. He further continues:     

“The universe may contain what is called Vacuum energy that is present even in apparently empty space…Vacuum energy causes the expansion (of the universe) to accelerate” ⁶

Besides, infinite energy defies the conservation of matter and energy law (first law of thermodynamics). Many quantum mechanical and astrophysical observations put the conservation law under question. Positive cosmological constant may indicate that matter and or energy constantly penetrate inside the space-time. If we assume that dark energy can leak in from Planck's pores throughout our universe, the result would create the exponentially expanding universe.
On the other hand, expansion of the universe requires either the space to stretch or being built from inside. If we believe that space is a continuum, then it has to stretch to provide expansion. However if we take space as a discrete entity, then the building blocks have to be built from within. In this model, space is discrete. Therefore, we have to look for mechanisms that can create space blocks. 

Steady State Theory
Steady-State Theory of Bondi and Gold suggests that, as the universe expands; new particles of matter are continuously created to fill up the gap so that the average density of the matter in the universe remains unchanged. Hoyle suggested “creation field “for creating matter from energy. He proposed presence of negative energy in his creation field to compensate for breaking the conservation of energy law. Dark energy can be the source for space unit creation. Then again, this challenges the conservation law.
If we are so keen to conservation law and would like to keep it anyway, then we have to expand it to contain a source of energy beyond the space-time (singularity in this model). But then if singularity has infinite amount of energy the law looses its meaning. Alternatively, we can leave conservation law for domestic economy and take space-time and singularity as trade partners. Conservation law works very nicely in macrocosm. We can leave the boundaries energy transaction to microcosm and quantum mechanical arena. This is the arena where we encounter violation of the law the most. Roger Penrose in The Large, The Small and Human Mindwrites:
“...All of the other schemes for quantum state reduction, which attempt to solve quantum measurement problem by introducing some new physical phenomena, run into problem with conservation of energy. You find that the normal rules of energy conservation tend to be violated. Maybe this is indeed the case.” ⁵  

Dark Energy
Interpreting the conservation law in this manner will free our imagination, and we can create theories to explain some unexplained findings in astrophysics. The big challenge today is finding a source for dark energy. The presented model offers a solution for the origin of dark energy.
The general acceptance of The Big Bang Theory by cosmologists, does not exclude the possibility of steady mater creation as a contributing factor in formation and expansion of the universe.Of course, E= mc²gives the formula for this energy to create matter. Isn't it true that continuity equationNote 1 should hold everywhere in space?
Expanding universe without matter creation is against the above equation.
If we try to uphold the conservation of energy law, then we have to assume the presence of negative energy in vacuum to compensate for permeation of energy and accelerating expansion of universe. With the above definition, the law is not broken. But if we extend the law to an arena beyond the space-time or question its universality, then we do not have to be worried about hypothesized negative energy. Then we can explore the possibility that zero point energy fluctuation in intergalactic space, can flourish and enter and provide the dark energy for the positive cosmological constant and acceleration of expansion. 

Shape of the Universe(Note 3)
Steady matter creation also suggests a solution for Flatness Problem.Note 2 What does flatness problem mean? If the density of our universe is greater than the critical density, in another word, If the gravitational force created by density of matter in the universe is greater than the expansion force, our universe is ‘closed,’ It means our universe will eventually stop expanding and start contracting. If the density of the universe is equal to the critical density, then we live in a 'flat' universe. Lastly, if the density of the mater in the universe is less than the critical density, then the universe is 'open.' In an open universe, the expansion continues forever. Currently, the best-known value for the critical density is about 1*10^-29 grams per cubic centimeter. Recent measurements indicate that the actual density of our universe is very close to the critical density. Although, matter density of the universe is so close to unstable critical value between perpetual expansion and re-collapse into a big crunch, most recent researches suggest that the rate of expansion is increasing (1998 Perlmutter et al).

Steady Matter Creation
The proposed steady matter creation is needed to compensate for the ever-increasing expansion of the universe. It is needed to keep the actual density of the matter in the universe close to critical density. To accept the steady matter creation we do not have to deny Big Bang model. Matter creation in the Big Bang moment and matter creation in vacuum can follow the same principle. Except that they are working in two different scales, big and small.
New stars are formed through the nebula activity in different parts of the Universe. Old star's dust is the building blocks for new star formation. Is there a chance that newly formed matter is also contributing to star formation or dark matter lenses? Note 3

Notes
1) In electromagnetic theory, the continuity equation is derived from two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density.
2) Check the Flatness Problem chapter.
3) Visible matter is a small portion of the matter present in the universe. In our Galaxy, the Milky Way, just about 10 percent of the matter is visible.  It is suggested that the remaining 90 percent are dark matter which do not radiate so they are not visible.
Visible and dark matter creates only 30 percent of the critical density. The other seventy percent is believed to be in the form of dark energy.

The arguments presented are open for debate. The reader is encouraged to email his/her inputs to correct, modify or develop the contents. Please send your emails to; zpfields@yahoo.ca

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