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Do We Really Need A Cosmological Constant?

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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:


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 c2, 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 = V2/2 - GM/R, you introduce the term (- R2 ) 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).




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