Geometry of the Universe
The shape of our universe has been matter of debate for a long-time. The main possibilities were the universe being spherical, flat or saddle like. ³¹

        

 

Cosmological Constant
In 1917, Einstein introduced the cosmological constant in order to explain how our universe is not falling into itself because of gravity and stays static. The force presented by cosmological constant supposedly counteracts the gravity forces of the bodies inside the universe. Therefore, it prevents the collapse and stabilizes the universe. According to its value, the constant would allow for an expanding, contracting or static universe. Since at the beginning of twentieth century universe was considered stationary, he chose a particular value for it, which would permit a static universe. The nature of this constant was not known or explained completely at the time.
When in 1929 Hubble's observation showed that universe is expanding and is not static, Einstein himself regarded the cosmological constant unnecessary. It was considered a non- contributory factor and everybody accepted a zero value for it.

Big Bang Theory and Expansion
The introduction of Big Bang Theory introduced the force, which is supposed to be responsible for expansion of universe despite the presence of the gravity. This would be the residual force that is left from the Big Bang explosion. It was anticipated though, that gravity would slow down and ultimately stop the expansion sometime in the future. That would be when the density of matter in the universe exceeds the critical density.
Critical density is that density of overall matter and energy in the universe that if we exceed it, the gravity has the strength to contract the universe. At critical density, the shape of universe will be flat. Above critical density, the shape of it will be convex and spherical. If the density falls bellow critical density, the shape of the universe will be concave and it will expand forever. All of this happens if we take the cosmological constant as zero. Which means if there is no other factor at work to expand the universe?
In January 1998, Alex Pilippenko in Astrophysist Conference in Southern California announced that cosmological constant could no longer be taken as zero. Different cosmologists reached to this point after long-term observation of supernovae IA in longer red shifts (far distances more than 3 billion light years away).

Supernovae research has been used to measure the amount of red shift and expansion rate of the universe

The Wilkinson Microwave Anisotropy Probe (WMAP) the newest probe launched by NASA in 2002 has determined, within the limits of instrument error, that the universe is almost flat.   This is an explanation for the cosmologic constant effect released by NASA.
“The results of the WMAP mission and observations of distant supernova have suggested that the expansion of the universe is actually accelerating which implies the existence of a form of matter with a strong negative pressure, such as the cosmological constant. This strange form of matter is also sometimes referred to as the "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever.” ³¹

Density of the Universe
Since the cosmological constant is slightly positive, we conclude that space is nearly concave. That means that Omega M (world density/critical density) has to be less than 1. Ω M <1
To have a concave universe the sum of Ω M plus the effect of cosmological constant in the expanding universe (Ω Lambda) has to be smaller than one.

Ω M + Ω Lambda <1

Here we need to assume a factor at work, which provides the positive cosmological constant.

Dark Energy
By definition, dark energy is a hypothetical force which exerts negative pressure and is acting against the gravity at large scales.  This energy is not only acts against gravity and prevents Big Crunch. Supposedly, it is also responsible for accelerated expansion of the universe. We can further speculate that this entity is the source for dark matter, as well. In addition, in a discrete space-time model, this energy can also be responsible for creation of space-time blocks.

Dark Energy in this Model
Previously, I have claimed that the source of dark energy is inherent in this model. In Singularity chapter, I have postulated that Zero Point Energy is out of our space-time and is an element of proposed singularity. An out of space ZPF can open our eye to new possibilities and scenarios.
This scenario gives us the good news that big crunch will never happen, because the universe is going to expand forever. Then there comes this feeling of emptiness and the chilling sensation that our descendants will stay lonely in an infinitely vast and cold universe. One may say that, our own galaxy or branch of galaxies, which remain together, has many new places to explore. Why do we have to be so ambitious and greedy to dream about intergalactic traveling? Is there an end to human race greed? Is there an end to evolution's ambition? Are we a dead end branch of evolution? Is there an alternative for cold and disperse universe scenario in the future?

 Zero Point Energy
Ever expanding universe also implies that in a universe with a discrete fabric, space-time units constantly have o be created. In addition, to keep the matter density near critical value, matter has to constantly appear in the newly formed space. For the above to happen, ZPE has to constantly penetrate inside the universe, create space-time blocks, and matter particles. Energy permeation from every miniscule of space forces universe to exponentially expand. Can there be any other explanation for the acceleration? According to Donald Goldsmith, no:
 “Unless we are prepared to reject the tenets of general relativity theory, the only additional term that may appear to produce an acceleration consists of…a cosmological constant of unknown size, but one with the properties we have described – namely, a transparent energy, untappable and untouchable, so far as we can tell, except by its tendency to make the universe expand more rapidly.”²⁹
In 1990, Hideo Kodama of Kyoto University demonstrated the positive energy density equations. It was based on assumption that cosmological constant is positive. In this case, it:
 “predicts a spectrum of quantum fluctuations in space-time. It also has a precise Planck-scale description, which makes use of very elegant mathematics connected to the invariants of graphs and knots.” ⁴⁸
Alfonso Rueda from department of Electrical Engineering California State University suggests that:
“Both mass and the wave nature of matter can be traced back to specific interactions with the electromagnetic zero-point field and possibly the other bosonic vacuum fields. Given this possible reinterpretation of fundamental properties, we suggest that it is premature to take a firm stand against the reality of the zero-point radiation field and its associated energy on the basis of cosmological arguments, especially given the possible relation between quantum vacuum, or zero-point, radiation and dark energy.”⁵⁵

The Conservation Energy Law
Nonzero cosmological constant require infinite source of energy readily available in every miniscule of space. Where does this energy come from? Whatever happened to conservation of energy law?
The first law of thermodynamics (the conservation of energy) states that the total inflow of energy and matter into a closed system must equal the total outflow of energy and matter from the system. It predicts that energy can be converted from one form to another, but it cannot be created or destroyed. If we take the space-time universe as a closed system, we are faced with a big puzzle. How are we going to offer a solution for expanding force, if there is no possibility for creation of energy inside the universe?
Some physicist try to disprove the presence of dark energy by suggesting models like modified gravity as proposed by Damien Easson or even dark geometry as speculated by by Gu-Je-An presented in Cosmo 4 in Toronto, in 2004. Others try to demystify its source and nature. In the proposed model, the so called dark energy is inherently built in into the scenario and because we took the universe as an open system, its source is explained.
Moreover in line with E = mc² this energy upon changing to matter, has also the potential to positively contributes to curvature of space (dark matter). As mentioned above, the nature of dark matter is not known. No body knows if it is similar to ordinary matter or it is antimatter or made of super symmetric particles. If dark matter is ordinary matter it can contribute to star formation.

The Lagoon Nebula consists of hot star dusts which by contraction create new stars

Steady State Theory
The Steady State theory is a model developed in 1984 by Fred Hoyle, Thomas Gold, Hermann Bondi.  In the Steady State model, new matter is continuously created as the universe expands. Therefore, universe can remain in steady and stable condition. After discovery of  microwave bBackground radiationn, astrophysicists gradually distanced themselves from the theory in favor of the Big Bang model for creation of universe.
The Hubble Deep Field photograph taken in 1996 by the Hubble Space Telescope shows the most distant view known so far. It was expected to show the birth of galaxies, but instead shows galaxies looking remarkably like the present ones. This is in favor of steady state theory. The theory claims that universe has been steady during its history. Although the concept has been discredited by numerous evidences, perhaps a modification of that model can still be useful.  Narlikar, Hoyle, Barbidge and some other astrophysicist introduced the Quasi Steady state theory to offer solutions for energy and matter creation through many small Big Bang events, which has been happening throughout the age of universe. Narlikar, Hoyle and Barbidge claim;
“Creation of matter is governed by a conservation law which operates to prevent space–time singularities, which otherwise occur in general relativity…Unless creation of matter is included in physical laws, the laws lack universality.”³⁴
The amount required to preserve the steady state is undetectably small—about a few atoms for every cubic mile each year or roughly a few hundred atoms of hydrogen in the Milky Way Galaxy each year. The required dark energy to maintain the present expansion of universe also is very small.
Trying to deny the acceleration of universe expansion by steady state theorists proves to be a very difficult task. Acceleration has been shown by different methods and recently by WMAP observations. May be the theory have to be modified to accommodate the new observations. This may mean radical and fundamental change. Maybe we need a combination of modified steady state and the Big Bang scenario?

Nancy Kerrigan Problem
So far, supernovae observation revealed that:

Ω M + Ω Lambda = approximate to one
Ω Lambda – Ω M = approximately 0.4

The above figures are obtained by different methods like Supernova Observation, Cosmic Background Radiation studies, Gravitational Lensing and so on. Although the above numbers are not firm yet and still under investigation, they are the cosmologist's best guess.
According to the Big Bang Theory, at the beginning, the matter density was much higher and cosmological constant was much smaller. Gradually matter got less dense and cosmological constant increased. We are at the point where Omega M and Omega Lambda are almost equal. This creates a puzzling issue. Out of all the possible combinations, why should we live in such a period which is favorable to life? This is what Robert Kirshner called Nancy Kerrigan problem referring to famous figure skater after the attack instigated by skating rival, “Why me? Why now?”
Roger Penrose in his resent book The Road to Reality a writes: “The seeming coincidence that Ω lambda and Ω M are of the same general order of size seems like a puzzling coincidence.”⁵⁶
We can either hang on to philosophical answers like weak or strong Anthropic principles, which claim human only can live in this period. In any other periods, the conditions were not suitable for a human to be alive and ask these questions. Or explore the possibility that a stable relation between matter density and cosmological constant effect has existed from the beginning to date and will continue in the future. This can happen if energy has been steadily penetrating inside empty space.
If we take the size of space after initial expansion, as variable and matter density as constant then the energy penetration through more space pores in expanding space will offer an explanation for Ω lambda and Ω M to be at the same general order.  So one can speculate that may be after initial expansion the matter density remained the same by dark energy penetration which compensated the space expansion.

Summary
Here we adopted the zero point fields as the source of dark energy. As mentioned before, the objection against the notion of zero point field is rooted to the fact that such a field is expected to have interaction with electromagnetic radiation inside. This effect is not observed. In this model zero point energy is out of space-time so it bypasses the above problem.
Previously, the question raised that; is there an alternative for cold and disperse universe scenario in the future? If we have constant energy penetration and matter formation in the universe, the future of world may not be as disappointing after all.

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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