# What is a cosmological constant problem

## Einstein's cosmological constant

Ever since Einstein's proposal, theorists have been arguing about its true meaning

A hundred years ago Albert Einstein published his fundamentally new theory of gravity under the title "General Theory of Relativity". The novelty of this theory in comparison with Newton was that the movement of attracting masses could be described as undisturbed inertial movement in curved space.

The usual visualization of Einstein's theory is shown in Fig. 1. The space-time tissue around the earth bends due to gravity, so that e.g. a ball that initially "rolls" straight ahead in this gravitational field is immediately deflected. Instead of postulating an action at a distance between masses, as with Newton, mutual attraction is understood as undisturbed movement in a local space that is now subject to a certain curvature. For example, rays of light that have no rest mass bend following the curvature of the room. This is not so easily understandable in classical theory.

However, Einstein was concerned about the cosmological consequences of the new theory. It is not enough if one can explain how space and time behave in the vicinity of celestial bodies. One must also think of the infinity of space and imagine the boundary conditions. A world in which only the attraction of gravity is at work cannot be static and runs the risk of collapsing. However, Einstein thought that the universe had existed forever and would exist forever.

In 1917 Einstein published the work "Cosmological considerations on the general theory of relativity". There he first discusses Newton's theory and then moves on to his own. We cannot justify the Einstein equation in detail here, but for our purposes it is sufficient to know that it has the form:

*Space-time geometry* = *Distribution of energy and matter*

The physicist John Wheeler described the Einstein equation as follows: "Space tells matter how to move, matter tells space how to bend." So one should not think of space and time as separate from matter and energy.

Based on Einstein's theory, the Russian physicist Alexander Friedmann predicted the expansion of the universe in 1922. The solution of the gravitational equation proposed by Friedmann (apart from a few constants) has the form:

where H_{0} is the so-called "Hubble constant" which indicates the rate of expansion of the universe (actually, H_{0} not constant, but changes slowly). If the curvature of space is positive, the right side of the equation can go back to zero over time and the universe will collapse in the long term. To prevent this (and still without knowing the Friedmann equation) Einstein added on the *left* Side of his equation a constant, the so-called cosmological constant (Fig. 2). This then appears on the *right* Side of the Friedmann equation and gives the rate of expansion of the universe an additional positive boost. The constant behaves like a kind of "antigravity" and allows the universe to expand. Einstein called this constant λ (today the corresponding capital letter Λ is used):

Actually, Λ / 3 appears in the Friedmann equation, but we will abstract from this in the following.

### Geometry and energy side

You can include a constant on the left or right side (with the opposite sign) in an equation. Einstein's original idea, however, was that the cosmological constant was a property of space and should therefore be inserted on the left-hand side, i.e. on the geometry side, of his equation (Fig. 2).

Today, however, the constant is written on the right side, where it is now being interpreted as an additional energy term. This is the mysterious "dark energy" that can even be weighed, but for which physicists have no generally accepted explanation. Unlike a liquid like water, which presses an object inward, the dark energy creates "negative pressure" and makes the space expand even more, as the Friedmann equation requires.

The reason the dark energy is so mysterious is that it has a constant density. As space expands, the average density of the usual energy and matter in the universe is thinned. The number of celestial bodies does not change, but is distributed over a larger volume. Strangely enough, the density of the dark energy remains constant. That is why it has been suggested to interpret the dark energy as the energy of the vacuum, but the corresponding calculations still miss the target by a gigantic order of magnitude.

### Einstein and the creation of matter

In 1931 Einstein was already familiar with the solutions of Friedmann and Lemaitre (a Belgian physicist who, after Friedmann, rediscovered the expanding universe in the equations), but could still not really get used to the idea of an ever thinning universe. That is why he examined in his notebook a possible way out to keep the universe still statistically unchanged despite expansion.

The idea was to exchange dark energy for elementary particles. Since the room is getting bigger and bigger and since dark energy takes up more and more space, one could simply postulate that elementary particles could be generated spontaneously from this energy. The new elementary particles then fill the existing space with new matter and thus prevent it from being diluted by the expansion of the universe. Einstein did not get much further with his reflections in the notebook, since a calculation error caused him to break off the thought exercise. Einstein did not publish anything about it later.

Many years later, the astronomer Fred Hoyle also had Einstein's idea. He was not satisfied with a unique act of creation in the universe and wanted to see the "perfect" cosmological principle in action. The cosmological principle says that no position or direction in space is special, i.e. space is isotropic and homogeneous. But this does not apply to time if there is an act of creation, because then there is a beginning of time behind which we are necessarily located.

But if, from a statistical point of view, the universe has always looked basically the same, then we always need new matter. Fred Hoyle accepted that the space would expand, but, like Einstein in 1931, wanted to constantly fill it up with matter. It didn't have to be very much either, only "three hydrogen atoms per cubic meter in a million years" would be necessary to keep the universe statistically unchanged.

Coincidentally, Fred Hoyle was also the one who gave the "Big Bang" its name. The term was actually meant satirically. Hoyle understood his dynamically stable universe to be more elegant than a world that emerged from a big bang and that transports us to a special time in the life of the universe. The name "Big Bang" was chosen badly, but was so concise that it has become common.

### Weigh the universe

Now astronomers encounter a major problem when reviewing the Friedmann solution. The rate of expansion of the universe can be measured with telescopes (the value of H._{0}) and the energy density is estimated by various means. However, the measured values do not match. There is a lack of matter and energy in the universe to determine the value of H_{0} to explain!

The astronomers' measurements show that our universe is flat on a large scale, i.e. two parallel beams of light keep their distance to infinity. Einstein initially thought that the universe as a whole should have a positive curvature, like space near the sun or the earth. He was only gradually able to make friends with a flat (Euclidean) universe (with zero curvature), which also simplifies the Friedmann equation:

If Einstein's original motivation was to introduce an opposing force against the gravitational collapse of the universe with the cosmological constant, this was no longer necessary when the astronomical observations showed that the universe was expanding. Einstein then rejected the cosmological constant and from then on referred to it as the "greatest donkey of my life" (according to the legend). The measurements showed that the universe will not collapse in the future, even if the rate of expansion may slow down constantly.

If one wants to explain the current rate of expansion of the universe, one cannot ignore the cosmological constant. If the universe is flat, the Hubble constant for an ever expanding universe is given by a "critical energy density":

The existing mass in the universe is not sufficient for this, so that physicists today supplement the critical energy density with dark matter plus a positive cosmological constant. It turns out that baryonic (common) matter represents only about 5% of the critical energy density. Today's Λ-CDM (Λ-Cold Dark Matter) model consists of matter and photons, dark matter and a cosmological constant:

The astronomers came up with this because they "weigh" the universe and enter the mass present everywhere in the calculation. For this purpose, extensive galaxy catalogs are created and their mass is estimated on the basis of their brightness and distance to the solar system, as was done, for example, as part of the "2DF Survey" (Fig. 3). This confirms that the stars' visible mass is insufficient to explain the Hubble expansion rate (Marc Davis, "Cosmology: Weighing the Universe", Nature 2001).

It is not just the stars that are weighed. The masses of matter clouds in interstellar space can also be estimated through the emission of X-rays in order to obtain a complete catalog of the baryonic matter in the universe.

### What does the cosmological constant mean?

Ever since Einstein proposed the cosmological constant, theorists have argued about its true meaning. Some see the constant as a possible correction of the theory of gravity, which could be achieved by other mathematical means. For example, Hugo von Seeliger proposed a possible decrease in gravitational force for Newton's theory of gravity by a factor larger than the square distance, which would only be noticeable in the case of interstellar forces of attraction (Cormac O'Raifeartaigh, Michael O'Keeffe, Werner Nahm, Simon Mitton: "One Hundred Years of the Cosmological Constant: from 'Superfluous Stunt' to Dark Energy").

The constant caused a real headache for Einstein himself. His entire theory of gravity was based, he thought, on "Mach's principle", according to which only the presence of matter and energy would allow space and time to arise. One can, however, postulate a universe in which the cosmological constant contributes the entire critical density, but a universe with space and time completely empty (such a "de Sitter universe" had already been studied by Willem de Sitter in 1917). Such a universe is compatible with Einstein's equation, but does not fulfill the Mach principle. That is why Einstein ultimately dropped Mach's principle.

Today's common explanation for the cosmological constant is that it represents an unknown form of energy associated with the vacuum. In today's physics, the vacuum is not simply a "nothing", it is a stage with a structure in which virtual particles are constantly emerging and disappearing and where fields are housed. The vacuum has a geometric structure and, if so, also an energy that remains constant per cubic meter, even when the space expands.

If this discussion about the cosmological constant was purely academic, since it could be that up to now we have not weighed all matter in the universe correctly (e.g. the black holes in galaxies), it experienced a new urgency than was confirmed in the nineties that space is not only expanding but that it is expanding ever faster today (Fig. 4). In 2011, Brian Schmidt, Saul Perlmutter, and Adam Riess received the Nobel Prize in Physics for this discovery.

This accelerated expansion of space cannot be explained by common or dark matter in the Friedmann equation, since its density is becoming ever thinner. Only the cosmological constant offers a theoretical anchor to justify the accelerated expansion. The cosmological constant creates a negative (expansive) pressure that is less and less balanced by the attraction of gravity.

In today's cosmological models, an energy density parameter Ω is introduced, which represents the sum of the contribution of mass and dark energy, i.e. Ω = Ω_{M.}+ Ω_{λ}. For a Euclidean (flat) universe, Ω = 1 must apply, whereby the contribution of each component can vary. Different cosmological models differ in the weighting of each summand. Today it is assumed that Ω_{λ} contributes up to 70% of the critical energy density (i.e. the energy in the universe).

Although Einstein himself rejected the cosmological constant at some point, various physicists stuck to it because of its explanatory value. The Friedmann equation describes the expansion of the universe, but does not explain it. The cosmological constant could provide a possible justification for this as a "something" that creates the negative pressure mentioned above. That was the opinion of the famous astronomer Sir Arthur Eddington as early as the 1930s.

Today the theoretical interpretation of the cosmological constant or dark energy is one of the most important open problems of modern physics. It could be that a satisfactory explanation can only be found in the context of a novel quantum gravity. So it remains exciting and new research results can certainly be expected in a few years. Twenty years ago many astronomers thought that it was a matter of time before the missing mass in the universe could be found (that is, Ω_{M.}= 1). Today hardly anyone believes in it anymore and understanding the hundred year old cosmological constant is more urgent than ever. (*Raúl Rojas*)

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