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 ~ 1/2 H, in a radiation dominated universe, and t ~ 2/3 H, 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, Burridge 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 init io 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 1), open (Omega.