**
Chapter 5:
Cosmological Constant**

Why is there a universe with its temporal and spatial order? Why are there atoms, molecules, solar systems, and galaxies? How accurate are the physical laws, which control them? And why does the universe not collapse under its own weight, or fly apart with the speed of light? - How finely is it tuned? What is the cosmological constant? What have some of the world’s leading physicists now found out about this?

**
Michael B. Green.**

Michael B. Green, at the
University of London, is an expert on superstrings. He states in *Scientific
American*, Vol. 255, Sept. 1986 page 56: "Why is the cosmological constant so
close to zero? The constant describes the part of the curvature of the universe
that is not caused by matter; its value has been determined to be zero within
one part in 10^{120}, which is the most accurate measurement in all
science."

**
John D. Barrow**

John D. Barrow is Professor of
Astronomy at the University of Sussex, England. He says in his book *Theories
of Everything *(1991:104, 105): "What is the cosmological constant today? We
know from the effects it would have upon the expansion rate of distant galaxies
that if it does exist then its numerical value must be infinitesimal, less than
10^{-55} per cm². Such a unit is not very illustrative. It is more
illuminating to compare its size with that of the basic unit of the
elementary-particle and gravitational worlds.

Planck length = 4
x 10^{-33} cm.

The cosmological constant must
be less than 10^{-118} when referred to these Planck units of length
rather than centimetres. To have to consider such a degree of smallness is
unprecedented in the entire history of science. Any quantity that is required to
be so close to zero by observation must surely in reality be precisely zero.
That is what many cosmologists believe. But why? What we seem to require is
either a ‘set and forget’ principle which sets the cosmological constant small
initially in a way that ensures that the cosmological constant must be
vanishingly small when the universe has expanded to a large size comparable to
its present dimensions of fifteen billion light years." - Barrow, J. D.
(1991:105).

**
Martin Rees and John Gribbin**

Martin Rees, at the University
of Cambridge, England, is one of the most distinguished astrophysicists. John
Gribbin is a physicist and science-writer. They report: "The universe has
expanded since 10^{-43} second after time ‘zero’, but we cannot find
out, what happened between this point zero and 10^{-43} s. If we are
going back to this moment (= 10^{-43} s), that is, as closely as
possible to what we call the beginning, the universe must have been flat up to
10^{-60}. The flatness is, therefore, the most precisely determined
figure in all physics. Hence, the universe must have been finely tuned with an
extraordinary precision, so that conditions could arise, enabling stars,
galaxies, and life to arise." (1991:33).

Was that pure coincidence?

Martin Rees and John Gribbin:
"Were this really pure coincidence, it would have been such a lucky strike, that
all the other coincidences in the universe would pale beside it. Much more
reasonable it seems to assume, the physical laws somehow required, that the
universe has to be *exactly* flat. After all, the flatness is the *only*
special density. No other value has any cosmic meaning. It seems more reasonable
to
assume, the universe had to be born with
*exactly* the critical
expansion-speed, than to believe, a blind coincidence had caused it to begin
with a deviation of no more than 10^{-60} from the critical value."
(1991:33, 34).

"But if we want to turn to an
exact description of the universe, to Einstein’s mathematical description of
space and time, and if we realize, how crucial the expansion-speed must have
been during the big bang, we find out, that the universe was sitting not only on
the proverbial knife’s edge, but in a much more critical balance. When going
back to the earliest time, where our physical theories are still valid, we find
out, that the important figure, the socalled ‘density-parameter’, is determined
with a precision of 1 to 10^{60}. If this parameter were changed up or
down by only a fraction, represented by a 1, standing sixty places behind the
comma, our universe would be unsuitable for life, as we know it. The fact, that
protons are not living forever, another result of the Grand Unification, also
means, that the universe has no other *preservation-*sizes, than those,
like the electric load, do have a mean value of exactly zero. Together with
inflation, this suggests the thought about a creation from nothing." - Rees, M.
and J. Gribbin (1991:27).

**
Roger Penrose**

Roger Penrose is Professor of
Mathematics at the University of Oxford, England. He writes in his book *The
Emperor’s New Mind* (1989:343): "But in order to start off the universe in a
state of low entropy (= low disorder) - so that there will indeed be a second
law of thermodynamics - the Creator must aim for a much tinier volume of phase
space. ... In order to produce a universe resembling the one in which we live,
the Creator would have to aim for an absurdly tiny volume of the phase space of
possible universes - about 1/10^{123} of the entire volume, for the
situation under consideration. ... This now tells us how precise the Creator’s
aim must have been: namely to an accuracy of one part in 10^{123}.

This is an extraordinary
figure. One could not possibly even write the number down in full, in the
ordinary denary notation: it would be ‘1’ followed by 10^{123}
successive ‘0’s! Even if we were to write the ‘0’ on each separate proton and on
each separate neutron in the entire universe - and we could throw in all the
other particles as well for good measure - we should fall short of writing down
the figure needed. The precision needed to set the universe on its course is
seen to be in no way inferior to all the extraordinary precision that we have
already become accustomed to in the superb dynamical equations (Newton’s,
Maxwell’s, Einstein’s) which govern the behaviour of things from moment to
moment." Penrose, R. (1989:343, 344).

Prof. Roger Penrose states in
his new book *The Large, the Small and the Human Mind*: "What is the
probability that, purely by *chance*, the Universe had an initial
singularity looking even remotely as it does? The probability is less than one
part in 10^{123}. ...What does that say about the precision that must be
involved in setting up the Big Bang? It is really very, very extraordinary. I
have illustrated the probability in a cartoon of the Creator, finding a very
tiny point in that phase space which represents the initial conditions from
which our Universe must have evolved if it is to resemble remotely the one we
live in. To find it, the Creator has to locate that point in phase space to an
accuracy of one part in 10^{123}. If I were to put one zero on each
elementary particle in the Universe, I still could not write the number down in
full. It is a stupendous number." (1997:47, 48).

**
Steven Weinberg**

Steven Weinberg is Professor at
the University of Texas at Austin, Department of Physics and Astronomy. He
states in his book *The First Three Minutes* (1986:139, 140) about the
protons and electrons in the universe:

"The cosmic load per proton can
easily be determined. As far as we know, the mean density of the electric load,
referring to the whole universe, is zero. If the positive load of the earth and
the sun were stronger than the negative load (or the other way around), by only
one to one million million million million million million (10^{36}),
the electric repulsion between them would be stronger, than the attraction,
caused by
gravitation. If the universe is finite and closed, we could
even raise this remark into the rank of a theorem.
The netto-load of the universe must be zero, or else
the electric lines of force would encircle the universe all the time and build
up an infinite electric field. But at the same time, whether the universe is now
open or closed - we can certainly say, that the electrical load of the cosmos
per proton is negligible."

"What about the lepton-number-density of the universe? Due to the fact that the universe has no electric load, we may assume that for every positively loaded proton, there is now exactly one negatively loaded electron. Since the protons make up now about 87 per cent of all nuclear particles in our present universe, one can say, that the number of electrons is about equal to the total number of nuclear particles." (1986:142).

Prof. Steven Weinberg then says
in *Scientific American*, October 1994 page 27: "But one constant does seem
to require an incredible fine-tuning: it is the vacuum energy, or cosmological
constant, mentioned in connection with inflationary cosmologies. Although we
cannot calculate this quantity, we can calculate some contributions to it (such
as the energy of quantum fluctuations in the gravitational field that have
wavelengths no shorter than about 10^{-33} centimeter). These
contributions come out about 120 orders of magnitude larger than the maximum
value allowed by our observations of the present rate of cosmic expansion. If
the various contributions to the vacuum energy did not nearly cancel, then,
depending on the value of the total vacuum energy, the universe either would go
through a complete cycle of expansion and contraction before life could arise or
would expand so rapidly that no galaxies or stars could form.

"Thus, the existence of life of any kind seems to require a cancellation between different contributions to the vacuum energy, accurate to about 120 decimal places. It is possible that this cancellation will be explained in terms of some future theory."

**
Paul Renteln**

Paul Renteln is assistant
professor of physics at the California State University in San Bernadino. He
states in* American Scientist*, Nov.-Dec. 1991 pp. 524, 525 under the
heading, "Quantum Gravity" about the electrons and quarks, and photons and
gluons, and how well they are working together:

In general relativity, "the background is dynamical: the creation and destruction of virtual particles actually warps spacetime, changing the ambient gravitational field. When we consider the gravitational interactions between particles, it is no longer possible to ignore the effects of these virtual fluctuations. Fluctuations in the matter fields (such as electrons and quarks) make negative contributions to the cosmological constant, whereas the carriers of forces (such as photons and gluons) make positive contribution to the constant.

"Our observations of the
universe suggest that the positive and the negative contributions cancel each
other to better than one part in 10^{120}! If the particles did not make
such nearly equal contributions to the constant, our universe would either
collapse upon itself or else expand at a velocity close to the speed of light.
In the absence of a physical principle that explains the high degree to which
the positive and negative contributions are balanced, the small size of the
cosmological constant poses a problem."

**
David Gross**

David Gross was Professor of Physics at Princeton University. He writes about "The Problem of the Cosmological Constant": "Gravitation, however, is a force, which is directly connected with energy. One often mentions, though, that gravitation is connected with mass, but as we have learned from Einstein, mass, according to its nature, is nothing but energy. Since gravitation is directly connected with energy, it ‘knows’, so to say, how much energy a certain object contains, and this holds true also for the universe as a whole: Also the universe contains a certain energy-density." (1989:172).

Also when space is empty?

Prof. David Gross: "Also when
space is empty. One is able to measure empty space, because the universe will
contract itself the stronger, the higher its energy-density is. Thus, one is
able to determine the background-energy-density of the universe, by determining
its global structure. These measurements have been made. One does have here,
though, only an upper limiting value, for the exact value seems to lie very
close to zero. These measurements are actually the most accurate determinations
of a ‘zero-size’, which one has ever been able to do: Its accuracy is 1:10^{120}
in units of the Planck-mass, the natural mass- or energy-scale of gravitation."
(1989:172).

"Let’s assume, for instance,
you would be working on a modern physical theory, which includes gravitation,
and someone asked you, without knowing the observed result: How high according
to your theory would you estimate the background density of the universe? Your
estimate value would be then 10^{120} times greater, than the upper
limit, everyone believes, that its real value is zero. ... Since its
introduction by Einstein, the small value of the Cosmological Constant has
remained a mystery. Again and again it has been found to be zero, zero, and
again zero, though, no one knows, why." - Gross, D. (1989:172).

**
Planck-Time Energy Density**

Wolfram Knapp (1992:65) states:
The critical energy density of the universe is now 10^{-29} g/cm³. And
Michael Turner writes in *Science*, Vol. 262, 5 Nov. 1993 p. 861: The
critical energy density of the universe is 1.88 x 10^{-29} g/cm³.

Professor Venzo de Sabbata,
Università di Bologna, Italy, and C. Sivaram say in *Gravitation and Modern
Cosmology, The Cosmological Constant Problem *(1991:21, 22, 29): "At Planck
time, when the universe was 10^{-43} s old, it had an energy of 10^{19}
GeV, an energy density of 10^{93} g/cm³, and a curvature energy of 10^{66}
cm². The critical energy density is now 10^{-29} g/cm³. And the
curvature energy of the universe is now only 10^{-56} cm²."

Here I noticed: The energy
density of the universe at Planck time, with its 10^{93} g/cm³, and the
critical energy density of the universe, with its 10^{-29} g/cm³ have a
ratio of 1 : 10^{122}. - And the Planck time curvature energy of the
universe, with its 10^{66} cm², and its present curvature energy, of 10^{-56}
cm², do also have a ratio of 1 : 10^{122}. - But why?

Andrei Linde, Dept. of Physics,
at Stanford University, has come from the Lebedev Physical Institute in Moscow,
Russia. He says in his article "Cosmological Constant, Quantum Cosmology and
Anthropic Principle" (1991:102, 115, 116): "The vacuum energy density of the
universe is now 10^{-29} g/cm³. But we should expect, instead, a vacuum
energy density with a Planck density of 10^{94} g/cm³. It is at least
123 orders of magnitude greater than the present vacuum energy density of 10^{-29}
g/cm³. Galaxies are only able to form, and life of our type only becomes
possible, if the vacuum energy density lies between 10^{-29} g/cm³ and
10^{-27}g/cm³. The universe is created in the quantum state with 10^{-29}
g/cm³."

In his book *Particle Physics
and Inflationary Cosmology *(1990:321), Prof. Andrei Linde states about the
energy density: "When the universe was born (not long after the singularity),
its vacuum energy density was -10^{94} g/cm³. And the critical energy
density of the universe is now 2∙10^{-29} g/cm³. The energy density of
the universe, at Planck time, was 10^{94} g/cm³, and its vacuum energy
density was -10^{94} g/cm³. They were balanced."

How can the vacuum energy
density during the "Big Bang" have been only 10^{-29} g/cm³ (so that
galaxies were able to arise), while the energy density of the universe at the
same time (at Planck time) was10^{94} g/cm³? That is a ratio of 1 : 10^{123}!

Alan H. Guth, Massachusetts
Institute of Technology, Cambridge, MA, and Paul Steinhardt, University of
Pennsylvania (1989:57) found out: The cosmological constant, as a fixed mass
density of the vacuum is 1.6 x 10^{-26} kg/m³ (= 1.6 x 10^{-29}
g/cm³). The pressure of the false vacuum is negative, with a magnitude which is
equal to the energy density. The energy density (or pressure) of the universe at
Planck time is 10^{92} J/m³.

What does that mean?

A. H. Guth and P. Steinhardt: "The total energy of any system can be divided into a gravitational part and a nongravitational part. The gravitational part (that is, the energy of the gravitation field itself) is negligible under laboratory conditions, but cosmologically it can be quite important. The nongravitational part is not by itself conserved; in the standard big-bang model it decreases drastically as the early universe expands, and the rate of energy loss is proportional to the pressure of the hot gas. During the era of inflation, on the other hand, the region of interest is filled with a false vacuum that has a large negative pressure. In this case the nongravitational energy increases drastically. Essentially all the nongravitational energy of the universe is created, as the false vacuum undergoes its accelerated expansion. The energy is released when the phase transition takes place, and eventually evolves to become everything that we see, including the stars, the planets, and even ourselves.

"Under these circumstances the gravitational part of the energy is somewhat ill-defined, but crudely speaking one can say that the gravitational energy is negative, and that it precisely cancels the non-gravitational energy. The total energy is then zero and is consistent with the evolution of the universe from nothing." (1989:54).

Important for us is here: The total energy of any system one can divide into a gravitational part and a non-gravitational part. The gravitational energy precisely cancels the nongravitational energy. The one is negative, and the other positive.

**
Alan H. Guth**

Alan H. Guth reports in his new
book *The Inflationary Universe* (1998:22) about the critical energy
density of the universe. "The value of the critical mass density is believed to
lie between 4.5 x 10^{-30} and 1.8 x 10^{-29} grams per cubic
centimeter, depending on the value for the expansion rate (i.e., the Hubble
constant) that one uses in the calculation. By the standards of our everyday
experience, this density is astonishingly low. The critical density corresponds
to somewhere between 2 and 8 hydrogen atoms per cubic yard, a density that is
more than ten million times lower than that of the best vacuum that can be
achieved in the earthbound laboratory."

The energy density of the
universe at the beginning (big bang) was 10^{93} g/cm³ (Guth, A. H.
1998:268). - 10^{93 }g/cm³ : 1.8 x 10^{-29 }g/cm³ = 10^{122}.
This means: The critical energy density of 1.8 x 10^{-29} g/cm³ of the
universe is the 10^{122th} part of 10^{93 }g/cm³ energy density
at Planck-Time, at the beginning.

"Cosmologists use the uppercase Greek letter omega, to denote the ratio of the actual mass density of the universe to the critical density. ... Dicke (one of the two American scientists, who discovered the background radiation), pointed out that the evolution of omega is like a pencil balanced on its point. If the pencil is perfectly balanced, then the laws of classical physics imply that it will stand on its point forever. If the pencil tilts just slightly to the left or right, however, then the tilt will increase rapidly as the pencil falls over. The situation of perfect balance corresponds to a value of omega equal one - a mass density precisely equal to the critical density. If omega is exactly one at any time, then it will remain exactly one forever. However, if omega in the early universe were just slightly less than one, then it would rapidly fall toward zero. Alternatively, if omega in the early universe were just slightly greater than one, then it would rapidly increase without limit." (1998:22, 23).

**
Omega in first 30 seconds**

"Omega will remain one if it begins at exactly one, but a deviation as small as 0.02 will become a large deviation within the time period shown. For omega to remain near one for ten billion years or more, any deviation from one in the early universe must have been extraordinarily small.

"If we ask what the average
mass density of the universe must have been at one second after the big bang, in
order for it to be somewhere between a tenth and twice the critical value today,
the answer is amazing. The mass density at one second must have been equal to
the critical density to an accuracy of better than one part in 10^{15}.
That is, it must have been at least 0.999999999999999 times the critical
density, but no more than 1.000000000000001 times the critical density!

Dicke concluded "that the mass
density at one second must have equaled the critical density to one part in 10^{14}.
If the mass density were less than 0.999999999999999 times the critical value,
he argued, then the density would have dwindled to a negligible value so quickly
that galaxies would never have had time to form. If the mass density at one
second were more than 1.000000000000001 times the critical value, on the other
hand, then the universe would have reached its maximum size and collapsed before
galaxies had a chance to form." - Guth, A. H. (1998:24, 25).

Prof. Alan H. Guth: "Going all
the way back to one second after the big bang, cosmologists estimate a
temperature of ten billion degrees, comparable to the core of a supernova
explosion - the highest temperature known to exist in the universe today. The
mass density was very high, half a million times of water, and the pressure was
an unfathomable 10^{21 }atmospheres. To make contact with the grand
unified theories, ... one would have to thrust the extrapolation all the way
back to 10^{-39} seconds after the big bang, when the temperature was 10^{29°}K.
At that temperature the average energy per particle would be about 10^{16}
GeV (1 GeV = one billion electron volts), the energy at which the new effects
predicted by grand unified theories become significant. The mass density under
these extraordinary conditions would be roughly 10^{84} times higher
than water, the same density as a trillion suns jammed into the volume of a
proton! ... The true history of the universe, going back to "t = 0," remains a
mystery that we are probably still far from unraveling." (1988:86, 87)

"Omega [is] the ratio between
the actual mass density of the universe and the critical energy density. (The
critical density, calculated from the expansion rate, is the density that would
put the universe just on the borderline between eternal expansion and eventual
collapse.) The problem is caused by the instability of the situation in which
omega equals one, which is like a pencil balanced on its point. If omega is
exactly equal to one, it will remain exactly one forever. But if omega differed
from one by a small amount in the early universe, then the deviation would grow
with time, and today omega would be very far from one. Today omega is known to
lie between 0.1 and 2, implying that at one second after the big bang omega must
have been between 0.999999999999999 and 1.000000000000001 (= 10^{15}).
Yet the standard big bang theory offers no explanation of why omega began close
to one.

"With inflation, however, the
flatness problem disappears. The effect of gravity is reversed during the period
of inflation, so all the equations describing the evolution of the universe are
changed. Instead of omega being driven away from one, as it is during the rest
of the history of the universe, during the period of inflation omega is driven
toward one. In fact, it is driven toward one with incredible swiftness. In 100
doubling times. The difference between omega and 1 decreases by a factor of 10^{60}.
With inflation, it is no longer necessary to postulate that the universe began
with a value of omega incredibly close to one. Before inflation, omega could
have been 1,000 or 1,000,000 or 0.001 or 1.000001, or even some number further
from one. As long as the exponential expansion continues for long enough, the
value of omega will be driven to one with exquisite accuracy." (1998:176, 177).

"The cooling robs the particles
of most of their energy, transferring that energy to the gravitational field.
... In contrast to the standard big bang recipe, the inflationary version calls
for only a single ingredient: a region of false vacuum. And the region need not
be very large. If inflation is driven by the physics of grand unified theories,
a patch of false vacuum 10^{-26} centimeters across is all the recipe
demands. ... the mass in this case is only 10^{-32} solar masses. The
sign of an exponent can make a great difference: in more easily recognizable
units, the required mass is about 25 grams, or roughly one ounce! So, in the
inflationary theory the universe evolves from essentially nothing at all, which
is why I frequently refer to it as the ultimate free lunch. ... Does this mean
that the laws of physics truly enable us to create a new universe at will? If we
tried to carry out this recipe, unfortunately, we would immediately encounter an
annoying snag; since a sphere of false vacuum 10^{-26} centimeters
across has a mass of one ounce, its density is a phenomenal 10^{80}
grams per cubic centimeter.

"For comparison, the density of
water is 1 gram per cubic centimeter, and even the density of an atomic nucleus
is only 10^{15} grams per cubic centimeter. To reach the mass density of
the false vacuum, one can imagine starting with water, and then compressing it
to the density of a nucleus. Even with four more increases in density by the
same factor, the density would still be 100,000 times lower than that of the
false vacuum! If the mass of the entire observed universe were compressed to
false-vacuum density, it would fit in a volume smaller than an atom!" (1998:254,
255).

"We really have no way of
knowing whether there might be false vacuum states at mass dimensions much
higher than 10^{80} grams per cubic centimeter. In particular, if there
exists a false vacuum state associated with the unification of gravity with the
other forces, expected to occur at about 10^{19} GeV, then the mass
density would be about 10^{93} grams per cubic centimeter. For this
density, the answer to our probability calculation would be approximately one -
a new universe would be created with just about every attempt!" (1998:268).

"No one knows how to calculate
the energy density of the vacuum, but when particle physicists estimate the
energy associated with this tempest of activity, they come up with a number that
is colossal. Roughly 10^{120} times larger than the largest value
consistent with observations. There are negative contributions to the energy
density as well as positive ones, but no one knows why they should cancel.
Something is happening that we do not understand, suppressing the cosmological
constant by at least 120 orders of magnitude below our expectations.

"Our inability to understand
this suppression is known as the *cosmological constant problem*, and is
generally recognized as one of the outstanding problems in particle theory. If
the cosmological constant is to significantly affect the age of the universe,
this mysterious suppression mechanism must by coincidence stop at almost exactly
120 orders of magnitude. If the cosmological constant were suppressed by125
orders of magnitude, or 150 or 1000 orders of magnitude, then it would be too
small to have any effect. Since there is no known reason why the suppression
should be almost exactly 120 orders of magnitude, many particle physicists find
it hard to believe that the cosmological constant is the right answer to the age
problem." (1998:284, 285).

**
Cosmological constant problem**

"The puzzle of why the
cosmological constant has a value which is either zero, or in any case roughly
120 *orders of magnitude* or more smaller than the value that particle
theorists would expect. Particle theorists interpret the cosmological constant
as a measure of the energy density of the *vacuum*, which they expect to be
large because of the complexity of the vacuum." Guth, A. H. (1998:329).

Result: The energy density of
the universe at the beginning (Planck-Time) was 10^{93} g/cm³ (Guth, A.
H. 1998:268). - 10^{93 }g/cm³ : 1.8 x 10^{-29 }g/cm³ = 10^{122}.
This means: The critical energy density of 1.8 x 10^{-29} g/cm³ of the
universe is the 10^{122th} part of 10^{93 }g/cm³ energy density
at the beginning (Planck-Time). But it does not reach 1 : 10^{125}.

**
Cosmological Constant and its
Critical Energy Density**

Professor Venzo de Sabbata and
C. Sivaram (1991:21, 22, 29) found out: At Planck time, when the universe was 10^{-43}
s old, it had an energy of 10^{19} GeV, an energy density of 10^{93}
g/cm³, and a curvature energy of
10^{66} cm². The critical energy density of
the universe is now 10^{-29} g/cm³. And its curvature energy is now 10^{-56}
cm². - These two figures are connected by 10^{122}:

+10^{66} cm² : 10^{122}
= 10^{-56} cm² x 10^{122} = -10^{66} cm².

This means: +10^{66}
cm² is the positive form of the Planck time curvature energy. By dividing it
through 10^{122} we get 10^{-56} cm², the curvature energy of
our universe of today. Then we multiply 10^{-56} cm² with 10^{122}
and get -10^{66} cm², the Planck time curvature energy in its negative
form. The energy density of the universe at Planck-time is 1.88∙10^{93}
and its critical energy density is 1.88∙10^{-29} g/cm³. Their ratio is 1
: 10^{122}. If the energy density of the universe at Planck-time was
1.88∙10^{94} g/cm³, the ratio is 1 : 10^{123}.

Planck Unit at 10^{-43} s |
Cosmological Constant | Critical Energy Density |

Energy density 1.88∙10^{93} g/cm³ |
1 : 10 |
1.88∙10^{-29} g/cm³ |

Length 4.13∙10^{-33} cm |
1 : 10 |
4.13∙10^{-155} cm |

Time 1.38∙10^{-43} s |
1 : 10 |
1.38∙10^{-165} s |

Mass 5.56∙10^{-5} g |
1 : 10 |
5.56∙10^{-127} g |

Energy 5∙10^{9 }J |
1 : 10 |
5∙10^{-113} J |

Temperature 3.50∙10^{32} K |
1 : 10 |
3.50∙10^{-90} K |

Curvature energy 10^{66} cm² |
1 : 10 |
10^{-56} cm² |

How does all that now fit
together? What does that mean? How can the energy density of the universe at
Planck-time, when it was 10^{-43} second old, be 1.88∙10^{93}
g/cm³ or 1.88∙10^{94} g/cm³, while the critical energy density of the
universe is at the same time 1.88∙10^{-29} g/cm³? And why is their ratio
then 1 : 10^{122} or 1 : 10^{123}? Which one of these two is the
right one? What have scientists found out about this now?

^{ }

**
Cosmological Constant = Critical
Energy Density of Universe**

Lawrence M. Krauss is now chair
of the physics department at Case Western Reserve University. He is also working
at CERN, Geneva, Switzerland. He writes in *Scientific American* January
1999, p. 37, about "Cosmological Antigravity":

"If virtual particles can change the properties of atoms, might they also affect the expansion of the universe? In 1967 Russian astrophysicist Yakob B. Zeldovich showed that the energy of virtual particles should act precisely as the energy associated with a cosmological constant. ... Even if theorists ignore quantum effects smaller than a certain wavelength ... the calculated vacuum energy is roughly 120 orders of magnitude larger than the energy contained in all the matter in the universe.

"What would be the effect of such a humongous cosmological constant? Taking a cue from Orwell’s maxim, you can easily put an observational limit on its value. Hold out your hand and look at your fingers. If the constant were as large as quantum theory naively suggests, the space between your eyes and your hand would expand so rapidly that the light from your hand would never reach your eyes. To see what is in front of your face would be a constant struggle (so to speak), and you would always lose.

"The fact that you can see anything at all means that the energy of empty space cannot be large. And the fact that we can see not only to the ends of our arms but also to the far reaches of the universe puts an even more stringent limit on the cosmological constant: almost 120 orders of magnitude smaller than the estimate mentioned above. The discrepancy between theory and observation is the most perplexing puzzle in physics today. ... The simplest conclusion is that some as yet undiscovered physical law causes the cosmological constant to vanish. But as much as theorists might like the constant to go away, various astronomical observations - of the age of the universe, the density of matter and the nature of the cosmic structure - all independently suggest that it may be here to stay."

"The average density of ordinary matter decreases as the universe expands. The equivalent density represented by the cosmological constant is fixed. So why, despite those opposite behaviors, do the two have nearly the same value today? The consonance is either happenstance, a precondition for human existence (an appeal to the weak anthropic principle) or an indication of a mechanism not currently envisaged." (1999:40).

Prof. L. M. Krauss’ chart shows us here:

·
At and shortly after time zero, the average
density of ordinary matter (of atoms, molecules and so on) was at first 10^{-20}
g/cm³. And it went quickly down then to 10^{-27} g/cm³. But the critical
energy density of the universe has been then already - since it was born - about
2∙10^{-29 }g/cm³.

·
When our universe was 5 billion years old, the
average density of ordinary matter was about 4∙10^{-28} g/cm³. And its
critical energy density (= the fundamental constant) was still fixed at 2∙10^{-29}
g/cm³.

·
Now, the universe is about 12 billion years
old. The density of its ordinary matter is now 3∙10^{-29} g/cm³. But its
critical energy density is still fixed at 2∙10^{-29} g/cm³.

·
When our universe is 20 billion years old, that
is, about 8 billion years from now, the density of ordinary matter will have
sunk still further down below that of the critical energy density of the
universe. It will be then 10^{-30} g/cm³. And the critical energy
density (the cosmological constant), will still be then at 2∙10^{-29}
g/cm³.

Prof. Lawrence M. Krauss
concludes from this: "Some feat of fine-tuning must subtract virtual-particle
energies to 123 decimal places but leave the 124^{th} untouched - a
precision seen nowhere else in nature." (1999:41).

About the "Fate of the Universe" Prof. Lawrence M. Krauss then says: "The cosmological constant changes the usual simple picture of the future of the universe. Traditionally, cosmology has predicted two possible outcomes that depend on the geometry of the universe or, equivalently, on the average density of matter. If the density of a matter-filled universe exceeds a certain critical value, it is ‘closed,’ in which case it will eventually stop expanding, start contracting and ultimately vanish in a fiery apocalypse. If the density is less than the critical value, the universe is ‘open’ and will expand forever. A ‘flat’ universe, for which the density equals the critical value, also will expand forever but at an ever slower rate.

"Yet these scenarios assume that the cosmological constant equals zero. If not, it, rather than matter - may control the ultimate fate of the universe. The reason is that the constant, by definition, represents a fixed density of energy in space. Matter cannot compete: a doubling in radius dilutes its density eightfold. In an expanding universe the energy densities associated with a cosmological constant must win out. If the constant has a positive value, it generates a long-range repulsive force in space, and the universe will continue to expand even if the total energy density in matter and in space exceeds the critical value. (Large negative values of the constant are ruled out because the resulting attractive force would already have brought the universe to an end.) Even this new prediction for eternal expansion assumes that the constant is indeed constant, as general relativity suggests that it should be." - Krauss, M. L. (1999:40).