This document was converted to html format from Word 7. Since html cannot read simbols like lambda or rho, they will not appear in the text (although they do in the equations). I will do my best to fix this. Hope you'll like the paper. It is intended to be a guide for non-specialists, not a sophisticated scientific analysis on the cosmological constant.

Astronomy 1 Term Paper

Claudiu Simion

**Do We Really Need a Cosmological Constant?
**

**Introduction**

In 1916, Albert Einstein made up his General Theory of Relativity without
thinking of a cosmological constant. The view of that time was that the
Universe had to be static. Yet, when he tried to model such an universe,
he realized he cannot do it unless either he considers a negative pressure
of matter (which is a totally unreasonable hypothesis) or he introduces
a term (which he called *cosmological constant*), acting like a repulsive
gravitational force.

Some years later however, the Russian physicist Friedmann described
a model of an *expanding universe* in which there was no need for
a cosmological constant. The theory was immediately confirmed by Hubble's
discovery of galaxies' red shift. Following from that, Hubble established
the law that bears his name, according to which every two galaxies are
receding from each other with a speed proportional to the distance between
them. That is, mathematically:

V=H D

where H was named *Hubble's constant.*

From this point on, the idea of a cosmological constant was for a time
forgotten, and Einstein himself called its introduction "his greatest
blunder", mostly because it was later demonstrated that a static Universe
would be in an unstable equilibrium and would tend to be anisotropic. In
most cosmological models that followed, the expansion showed in the Hubble's
law simply reflected the energy remained from the *Big Bang*, the
initial explosion that is supposed to have generated the Universe.

It wasn't until relatively recently - 1960's or so, when more accurate
astronomical and cosmological measurements could be made - that the constant
began to reappear in theories, as a need to compensate the inconsistencies
between the mathematical considerations and the experimental observations.
I will discuss these discrepancies later. For now, I'll just say that this
strange parameter, *lambda*- as Einstein called it, became again an
important factor of the equations trying to describe our universe, a repulsive
force to account not against a negative matter pressure, but for *too
small an expansion rate*, as measured from Hubble's law or cosmic microwave
background radiation experiments. I will show, in the next section, how
all these cosmological parameters are linked together, and that it is sufficient
to accurately determine only one of them for the others to be assigned
a precise value. Unfortunately, there are many controversies on the values
of such constants as the Hubble' constant - H, the age of the Universe
t, its density , its curvature radius R, and our friend *lambda*.

Although I entitled my paper with a question, I will probably not be
able to answer it properly, since many physicists and astronomers are still
debating the matter. I will try, however, to point out what are the certainties
- relatively few in number - and the uncertainties - far more, for sure
- that exist at this time in theories describing the large scale evolution
of the Universe. I will emphasize, of course, the arguments for and against
the use of a cosmological constant in such models, and I would like to
make sure that my assistance gets a general view on the subject, in the
way that I could understand it.

**A Few Mathematical Considerations, or What Einstein
Did**

Since this is not a general relativity paper, I will present how Albert
Einstein arrived to the conclusion that a cosmological constant is necessary
for describing a static Universe in the simplest way possible. Imagine
a sphere of radius R which has a mass M included inside its boundaries.
Let m be a mass situated just on the boundary. We can then write:

,
and ,
hence:

where a is the acceleration of mass m, G is the gravitational constant, and p is the pressure of the radiation, which contributes, along with matter density, to the overall density of the Universe.

(Think now at the sphere as our Universe, and at mass m as the farthest galaxy).

At a glance, the Universe cannot be static unless a is zero, so p= -
3 *rho *c^{2}, which is a negative value. This is an unreasonable
hypothesis, so Einstein introduced a repulsive force characterized by the
cosmological constant to adjust this inconvenience and to straighten his
model. I will not reproduce the calculations here, but just imagine that,
instead of writing the energy conservation equation in the form:

E/m = V^{2}/2 - GM/R, you introduce the term (- R^{2}
) in the right side. (1)

How Einstein has calculated it, the cosmological constant has the ultimate expression (in his static model):

.

The curvature radius of the universe can be further determined from
that, as:

As I stated in the introduction, all the fundamental parameters characterizing
the Universe are linked by equations. Ignoring the constants and the computation
details, I will give the to-date accepted relations. Thus, the age of the
universe is connected to the Hubble constant through:

t ~ 1/2H, in a radiation dominated universe, and

t ~ 2/3H, in a matter dominated universe.

The connection between H and the density of the universe (in Einstein
- De Sitter model, but other models do not state anything significantly
different) is:

It is a matter of philosophy to ask which of these parameters is crucial
in understanding the others. They are all intimately linked. From this
point on, we have to rely on what one can actually measure. The density
and the age can only be estimated, unless indirectly determined. The cosmological
constant is not even a certitude. Thus, the one that we eventually have
to deal with is the Hubble constant, which can be calculated observing
the red shift of the far galaxies. But there are plenty of controversies
on its value also, ranging between 50 and 100 km/s/Mpc. One of the accepted
values is 65 __+__ 5 km/s/Mpc. This is also uncertain, since scientists
do not agree on the methods of measuring it, and in some theories it is
not consistent with the age of the Universe as determined from the cosmic
microwave background radiation or globular clusters experiments (see the
New Situations section).

For a better understanding of these issues, let's see what the different
models agree and disagree on.

**Models of Universe**

Concerning the origin of the universe, contemporary views converge to two different hypotheses: the first, and the most popular, is the Big Bang theory, which states that the universe originated from a primordial explosion involving a singularity with infinitely high matter density and infinitely small size. The second, proposed recently, suggests a sinusoidal universe, that starts at a singularity similar to the Big Bang, but with finite density (supposedly the one of the nuclear material), expands for a while to a maximum state, than begins to contract to the singularity from which it was generated. The cycle will then proceed again. The age of such an universe makes no sense unless we calculate it from the most recent explosion. That is why, in the end, the two hypotheses are to converge after one disregards the philosophical implications.

According to the most recent opinions (Hoyle, Burbidge and Narlikar,
March 1997) the accepted cosmological models can be classified in three
large categories, each with their own general characteristics.

- the standard Big Bang cosmologies, with or without inflation.

*Lambda *or no *Lambda* ?

Albert Einstein, one of the most famous scientists of
all times, thinking, probably, whether he should or should not introduce
the cosmological constant.

They usually follow from Einstein's equations of general relativity. Although Einstein-de Sitter model cannot be included in this class, it was it that gave birth to all the models in this category. I will explain why.

When Einstein tried to picture a static universe, he noticed that he
cannot do that unless he introduces the cosmological constant , as a "cosmic
repulsion force" to compensate for the gravitational pressure of matter.
However, this *ab initio* introduction in such a rigorous theory as
the General Relativity did not please anyone (neither Einstein himself).
So mathematicians all over the world began searching for other models that
would cancel this constant.

Before I present what happened next, I would like to point out some of the characteristics of the Einstein - de Sitter model, for a better understanding of why it later proved inadequate.

Its main feature is that it requires the cosmological constant to picture a static universe. From the equations two solutions can be derived, from which just one was worked out by Einstein. According to it, the space has a positive curvature, but the time line is straight, so that no event will reoccur. The 2-dimensional analogy of this is the surface of a cylinder. Some years later, the Dutch mathematician Willem de Sitter discovered the second allowed solution, with an universe in which both space and time are curved. The analogy follows naturally - the surface of a sphere. The assumptions and the calculations of the two scientists were gathered in what is today called the Einstein-de Sitter model.

In 1922 however, the Russian mathematician Alexander Friedmann showed that such a static universe is in an unstable equilibrium. That is why, he argued, any slight change in the general parameters or in the local equilibrium states would generate discrepancies between the behaviors of the different parts of the universe. This result implies anisotropy. However, there was no reason, at that time, to suppose that the universe was anisotropic, and later measurements on cosmic microwave background radiation showed that it is clearly not the case.

Moreover, Friedmann came up with some other allowed solutions, in which, without contradicting General Relativity, the universe needs not to be static. Following from his equations, it can be either expanding or contracting with time. Einstein recognized the importance of this discovery, and called his theory on the cosmological constant the "greatest blunder of my life". Approximately in the same time, the American astronomer Edwin P. Hubble made a very important observation, that would confirm Friedmann's theory: he measured the redshift of several remote galaxies, and decided that they are receding from us with a velocity proportional to the distance to them. This happened in every direction he looked and, unless we suppose we have a privileged position in the universe (which is absurd), the result demonstrates that the universe is, indeed, expanding, and that there is no privileged point such as a center of inflation.

Models included in this category therefore *do not need* a cosmological
constant, because they account for the inflation by supposing that it is
driven mainly by the energy remained from the Big Bang, the explosion that
initiated the universe. In such a model, the natural question is whether
the expansion will ever cease. It can be calculated that , if the density
is over a certain value, (called the *critical density*), then the
inflation will stop, at a certain moment, and a contraction will begin.
If *rho*<*rho*_{c} , the expansion will go for ever.
If *rho*=*rho*_{c}, the universe will asymptotically
tend to a steady size that it will never acquire. *Rho*_{c }was
calculated, and its value is given by the following formula:

(2),

where H is the Hubble constant of the epoch. The main puzzle, again,
is the uncertainty on the value of *H*. Hoyle, Burbidge and Narlikar,
in their paper "On the Hubble constant and the cosmological constant"
list an interesting table showing the different methods used to calculate
H, and the consequent values. They range between 43__+__ 11 km/s/Mpc
and 55__+__ 8 km/s/Mpc. However, they state that there is another current
of opinion, led by G. de Vaucouleurs, which sustains a value of ~ 80-100
km/s/Mpc. As we can see, quite a divergence. Although the values are not
so different, one must realize that a factor of 2 is involved, and therefore
the density will change by a factor of 4. If this means a shift from 0.4*rho*_{c}
to 1.6*rho*_{c}, it will make a lot of difference! By the
way, the ratio *rho*/*rho*_{c} is called the density
parameter, denoted *Omega*.

As a conclusion, this category includes closed (*Omega*>1),
open (*Omega*<1) and flat (*Omega*=1) models. In each case
the age of the universe is less than, greater than or equal to 2/3H. Latest
measurements indicate a tH factor closer to 1 than to 2/3, so the most
plausible hypothesis might be that we live in an universe that will expand
forever. This is consistent with a determined of ~0.2. Of course, these
data are not a certainty….

- the modified Big Bang cosmologies with cosmological constant.

They are more recent than 1) above, and appeared from the discrepancies
between theoretical assumptions and experimental observations. What they
basically do is to reintroduce the cosmological constant , much in the
same way Einstein did, to adjust for the age of the universe. The result
is that you can obtain arbitrarily large ages by modifying . The disadvantage,
of course, is the difficulty of explaining the nature of such a parameter.
The limiting model, however wrong, is Einstein-de Sitter's, that ultimately
implies:

,
and ,

R being the curvature radius of the universe (also see Mathematical Considerations section).

Lately, scientists managed to give some alternative explanations for
the existence of , and they eased the consideration of models included
in this category. I will discuss these interpretations in the last section,
so I will waive them for now, asking the reader to forgive me.

- the quasi-steady state cosmology

The theoretical foundations of this cosmology lie on the *creation
of matter* , by means of a scalar field *m* (Hoyle, Burbidge and
Narlikar), that has ups and downs adding up to an exponential inflation
over which a short-term oscillation is superimposed. This model requires
a cosmological constant, which is, however, negative!

The main advantage of this cosmology is that the age of the universe makes sense only from one minimum of the oscillation to the next. S. Hoyle found that the time span from the last such minimum is approximately 14 billion years. This could be less than the age of the oldest objects. However, in this theory objects like that can easily be accommodated in.

Another feature of this category is that it views both the gravitational
and the cosmological constant as derived from the mass field *m*,
thus they are not anymore fundamental, but a given by the equations:

,

,

where m_{0} is a characteristic constant of the field *m*,
and N is the number nucleons per Planck particle (an ideal object having
the highest conceivable energy as an elementary particle). In the above
equations h/2*pi* was assigned the value 1. The significance of all
these defies the understanding of most non-specialists. What is really
important for this paper is the value obtained for *lambda*, 7.10^{-57}
cm^{-2}, very small, consistent with other recent cosmologies.

Attractive as it is, the above theory follows closely from the steady-state
model, proposed also by Hoyle. Unfortunately, this model proved to be wrong,
because there is strong evidence that the universe was hotter and denser
in the past, thus ruling out the net creation of matter. The quasi-steady
state cosmology tries to make a trade-off between the author's old idea
and the factual observations by saying that the mass curve is a sinusoid,
thus epochs of creation being compensated by periods of loss. Yet, scientists
do not seem to accept his arguments with ease.

Whatever the model of universe, the cosmological constant problem remains
open. Most often the invoked reason for using it is matching the age of
the universe with the actual measurements. However, finding a reasonable
interpretation for such a term is the hardest part of the theories.

**The New Situation**

After Friedmann and Hubble, each one by his own means, proved the viability of an inflationary universe, the cosmological constant was, as I said, forgotten. Yet, no subsequent model (Lemaitre, 1933 and 1934, for example) succeeded in being consistent with the experimental facts. Before any other comments, let's see what were and still are these facts.

Firstly, there is the *microwave background radiation*. By measuring
its characteristics, one can make predictions on the age of the universe,
as this radiation, now with a temperature of 2.7K, is a remnant of the
era immediately posterior to the Big Bang.

Secondly, one can determine the *age of the oldest observable objects*.
How? A way is to measure their redshift, then apply Hubble's law. Another
method is to study the old stars in the globular clusters nearby, supposed
to have ended their main sequence evolution.

Thirdly, one may estimate the total mass of the matter, in order to come up with a value of the density of the universe (I will explain later why that became questionable recently). These are probably the most important tools, up-to-now, even though they are not the only ones.

Nevertheless, they are not infallible. Scientists still argue on the viability of these measurements, and it turns out that insignificant changes in the magnitude of the different quantities involved (such as in the Hubble constant, see previous section) may influence the overall picture of the universe. That is why so many different cosmologies are being described nowadays without being ruled out.

In 1975, Gunn and Tinsley gathered together the observations done at the time and concluded: "New data on the Hubble diagram, combined with constraints on the density of the universe and the ages of galaxies, suggest that the most plausible cosmological models have a positive cosmological constant, are closed, too dense to make deuterium in the Big Bang, and will expand forever." In the following years however, the first evidence for the existence of dark matter was found, and the "constraints on the density of the universe" were not tight anymore.

There might be up to 10 times more dark matter than luminous matter in the universe. The number is, of course, uncertain, but it can be estimated by kinematics calculations -- the mass of the galaxies measured by electromagnetic methods (optical, radio, X-rays, etc) is too small to account for the rotation curves of the gas and stars filling them. We have no other means to determine how much dark matter is up there "in the skies", since it does not emit any radiation that we could detect and analyze. But the fact itself is remarkable, because it pushes the limits on the actual density of the universe up to a point where many speculations can be made.

One other important point: in the last decades, the view on cosmology was driven to new horizons as more and more particle physicists began to work in the field. The quantum physics principles had to be added to the general relativity theory, thus making cosmology even "more complicated".

All these breakthroughs generated a race in describing models of universe.
Some of the old ones were modified, new ones appeared. The most classical
example for the former is the appearance of *lambda *in the left side
of the energy equation (see Mathematical Considerations) -- equivalent
to a negative *lambda *on the right side -- as a characteristic "strength"
of the Big Bang explosion. By adjusting the cosmological constant and the
density parameter (allowed by the discovery of dark matter) scientists
were able to match this type of model with the experiments.

Others have said that, while the value of was *Omega*~ 0.1 up to
the dark matter affair, the new data suggest a magnitude closer to 1. The
simplest and most elegant solution was to assume a flat universe (*Omega*=1)
with *lambda*=0 (more recent calculations tend to prove, however,
that in fact the density parameter is still less than 1).

More sophisticated models were described as well. In Hoyle and Narlikar's
cosmology, ** lambda **is negative, but it does not introduce
a change in the usual results because it depends on time in the same way
as the density . Ernst Schmutzer (1984) made up a "5-dimensional projective
space" cosmology, with a variable gravitational constant. Physicists
like Baum, Hawking, Coleman, Sandage remained loyal to Big Bang models,
and tried to argue for the cosmological term. Why? Because, indeed, even
if the above mentioned invoke "various combinations of non-baryonic
options, cold dark matter, hot dark matter", or quantum effects, none
proved theoretically satisfactory enough and completely consistent with
the observations.

**Problems and Recent Interpretations**

**Figure 1 (Chaisson/McMillan - Astronomy Today) **

Why is it so important to know the magnitude and the sign
of the cosmological constant? The figure shows how this parameter can influence
the dependence of the size of the universe on time. The allowed range,
at least theoretically, is huge, even if we fix the today value of the
Hubble constant. Hypotheses with no Big Bang or picturing a static universe
become as reasonable as the older models, no matter what end they stated
for the universe - Big Crunch or expansion forever.

In quantum physics there has been described an effect that is unimaginable
in real life. It is called the "*tunnel effect*" and states
that, when a particle is closed in a potential hole with finite walls,
there still is a probability that the particle will pass through the walls,
without being provided with external energy. This result is remarkable
and proved to have profound implications in cosmology.

The universe is composed of stars, gas, intergalactic dust and, "recently",
dark matter. Its density, at a maximum value, is on the order of 10^{-28}
kg/m^{3}. This roughly means 1 hydrogen atom in 10,000 liters of
spaces. The volume of such an atom is about 10^{-30} m^{3}.
The rest is *vacuum*, which is supposed to have zero density.

However, quantum physicists managed to derive another strange, but beautiful
result, now regarding the energy of vacuum. Since E=mc^{2} , this
relates immediately with its density. It turns out that there is a small
probability, much in the same way as in the tunnel effect, that vacuum
will have a non-zero density! And since in this kind of physics the macroscopic
state is a superposition of the microscopic states, the vacuum HAS, overall,
a non-zero density.

This is the nicest interpretation that the cosmological constant is
assigned nowadays, although scientists argue whether such quantum analogy
is appropriate in the case. The way it was computed, the vacuum density
is related to *lambda *through:

.

Some other scientists say that, because of the quantum effects, *rho*_{vac
}could be negative, so *lambda *can still have either
sign. The most reasonable (and consistent) hypothesis though is a positive
cosmological constant. Yet, the main argument against this assumption is
that, to build up a model, *lambda *should be too large (i.e. larger
than the to-date determinations, which I will discuss later). However,
Steven Weinberg , Coleman and Hawking, and Venzo de Sabbata show that a
value for the cosmological constant consistent with the behavior of the
early phases universe (when the energy of vacuum was 10^{94} g/cm^{3}!)
could be derived.

Another elegant interpretation for the cosmological constant, somehow
following from the above, is in terms of entropy. As the universe is expanding,
its entropy is increasing, and it can be shown that the maximum value is
10^{120}k_{B}, (k_{B} - the Boltzmann constant).
The relation with *lambda *might be obscure, but I will say that S,
explaining the previous result later.

Now, the problem. How big is the cosmological constant? The calculations
show that it dominates the energy equation ((1), see Mathematical Considerations)
when

,
which means

If we assume a density of about 10^{-28} kg/m^{3}, *lambda*>
5.10^{-38} m^{-2}. This is, however, way too large relative
to what values of the cosmological constant are determined on theoretical
bases.

Indeed, at H=100km/s/Mpc, *rho*_{c
g/cm}^{3 }(see equation 2), and, assuming that the density
of vacuum cannot exceed the critical density by more than 100 times, one
yields a maximum of 10^{-52} m^{-2}, a much smaller value
than above. Moreover, cumulated quantum physics and general relativity
arguments allow us to compute a minimal value of the cosmological term
in the Plank era (the smallest unit of time after the Big Bang that can
be analyzed, 10^{-43} s). I will give the formula not for memorizing
, but because I think is remarkable, depending only on fundamental constants:

,

h is the Plank constant. This value is rather huge: 10^{70}
m^{-2}. So, its contribution at that time was maybe the only significant
one. What is relevant though, is that the ratio *lambda*/ *lambda*_{Pl}
is the smallest dimensionless number in physics: ~10^{-120}! For
some reason, the cosmological constant, if ever existed, had drastically
dropped during the evolution of the universe.

Now the question: if it is non-zero, why is it so small? Or, in other
terms, if it is so small, why do we need it? These are the main points
of view of the day regarding this long-lasting puzzle. The problem is that,
even at a factor of 10^{-120}, the vacuum density exceeds the matter
density by at least 10-100 times! So it can significantly change our view
of the universe.

The connection with entropy is the following: in the Plank era, the
entropy of the universe was computed as 10^{60} k_{B}.
S goes as *lambda*^{-1/2}, and since *lambda*/ *lambda*_{Pl}
is 10^{-120}, the maximum entropy today can be 10^{120}
kB. The significance of this limit is still an enigma for scientists, but
one has to keep in mind that the entropy of a system is intimately linked
to both the temperature and the degree of disorder of that system.

Using quantum approach, physicists have calculated an "allowed" value of the cosmological constant. It turned out that, in particle physics units, ! By the same method, in cosmological units, it has to be 1. The observed value, mentioned in the previous paragraph is, however, 120 orders of magnitude smaller! (because in particle physics units, a maximum value was calculated as 0.5 or so). This is another relevant observation, that might be there to tell us that we do not need Einstein's greatest blunder to model the world we happen to live in.

The puzzle goes on. New data and more accuracy in astronomical measurements are necessary to solve the cosmological constant problem and, subsequently, to find the best model of the evolution of our universe. There is no major, clear trend seen today among the specialists. A final result can hardly be predicted. This is nicely illustrated in an answer I got from a fourth year graduate student in string theory here, at Caltech, when I asked him what is his opinion about this issue. What he said is:

"I am a theoretician. I should say is zero. But nobody knows..."

**References**

- F. Hoyle, G. Burbridge, J. V. Narlikar - On the Hubble constant and the cosmological constant, Monthly Notices of the Royal Astronomical Society, vol. 286 (2), pp.173-182, March 1997
- A. Zichichi, V. de Sabbata, N. Sanchez - Gravitation and Modern Cosmology - The Cosmological Constant Problem, Ettore Majorana International Science Series, 1991
- M. Rowan-Robinson - Cosmology, 2
^{nd}edition, Claredon Press, Oxford, 1991 - P. J. E. Peebles - Principles of Physical Cosmology, Princeton University Press, 1993
- E. W. Kolb, M. S. Turner - The Early Universe, Addison-Wesley Publishing Company,1990
- Chaisson, McMillan - Astronomy Today, 2
^{nd}edition, Prentice Hall, 1996 - P. Coles, F. Lucchin - Cosmology, John Wiley and Sons Ltd., 1995
- P. J. E. Peebles - The Large Scale Structure of the Universe, Princeton University Press, 1980
- S. Weinberg - The First Three Minutes, Basic Books, Inc., 1977
- J. Leslie - Physical Cosmology and Philosophy, Macmillan Publishing Company, 1990
- J. Silk - A Short History of the Universe, Scientific American Library, 1994
- Scientific American - Cosmology +1, 1977
- http://dept.physics.upenn.edu/~myers/ASTR001/L40.html
- http://dept.physics.upenn.edu/~myers/ASTR001/L41.html
- http://astro.caltech.edu/academics/ay1