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 10120, 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 1060. 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 1/10123 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 10123.

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 10123 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 10123. ...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 10123. 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 (1036), 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 10120! 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 1:10120 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 10120 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 1019 GeV, an energy density of 1093 g/cm³, and a curvature energy of 1066 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 1093 g/cm³, and the critical energy density of the universe, with its 10-29 g/cm³ have a ratio of 1 : 10122. - And the Planck time curvature energy of the universe, with its 1066 cm², and its present curvature energy, of 10-56 cm², do also have a ratio of 1 : 10122. - 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 1094 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 10-27g/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 -1094 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 1094 g/cm³, and its vacuum energy density was -1094 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) was1094 g/cm³? That is a ratio of 1 : 10123!

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 1092 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 1093 g/cm³ (Guth, A. H. 1998:268). - 1093 g/cm³ : 1.8 x 10-29 g/cm³ = 10122. This means: The critical energy density of 1.8 x 10-29 g/cm³ of the universe is the 10122th part of 1093 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 1015. 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 1014. 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 1021 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 1029°K. At that temperature the average energy per particle would be about 1016 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 1084 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 (= 1015). 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 1060. 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 1080 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 1015 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 1080 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 1019 GeV, then the mass density would be about 1093 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 10120 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 1093 g/cm³ (Guth, A. H. 1998:268). - 1093 g/cm³ : 1.8 x 10-29 g/cm³ = 10122. This means: The critical energy density of 1.8 x 10-29 g/cm³ of the universe is the 10122th part of 1093 g/cm³ energy density at the beginning (Planck-Time). But it does not reach 1 : 10125.

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 1019 GeV, an energy density of 1093 g/cm³, and a curvature energy of 1066 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 10122:

+1066 cm² : 10122 = 10-56 cm² x 10122 = -1066 cm².

This means: +1066 cm² is the positive form of the Planck time curvature energy. By dividing it through 10122 we get 10-56 cm², the curvature energy of our universe of today. Then we multiply 10-56 cm² with 10122 and get -1066 cm², the Planck time curvature energy in its negative form. The energy density of the universe at Planck-time is 1.88∙1093 and its critical energy density is 1.88∙10-29 g/cm³. Their ratio is 1 : 10122. If the energy density of the universe at Planck-time was 1.88∙1094 g/cm³, the ratio is 1 : 10123.

Planck Unit at 10-43 s Cosmological Constant Critical Energy Density
Energy density 1.88∙1093 g/cm³

1 : 10122

1.88∙10-29 g/cm³
Length 4.13∙10-33 cm

1 : 10122

4.13∙10-155 cm
Time 1.38∙10-43 s

1 : 10122

1.38∙10-165 s
Mass 5.56∙10-5 g

1 : 10122

5.56∙10-127 g
Energy 5∙109 J

1 : 10122

5∙10-113 J
Temperature 3.50∙1032 K

1 : 10122

3.50∙10-90 K
Curvature energy 1066 cm²

1 : 10122

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∙1093 g/cm³ or 1.88∙1094 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 : 10122 or 1 : 10123? 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 124th 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).