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Fernando Quevedo. The string-theory landscape

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String theorists have found a way to describe an accelerating universe that is similar to our own

Einstein introduced the cosmological constant to his equations of general relativity because he believed the universe was static. Faced with evidence that the universe was actually expanding, however, he decided to remove it, later referring to the cosmological constant as the biggest blunder of his life. Recent observations suggest that the expansion of the universe is accelerating, which favours a small, but non-zero, positive cosmological constant with a value of 10-120 in Planck units.

But perhaps Einstein's most serious mistake regarding the cosmological constant, Λ, was to actually believe that he had the right to decide whether or not it should be included in his equations in the first place. If Λ is very small or zero, its value has to be explained. This is probably the greatest puzzle in modern theoretical physics.

De Sitter's solution

Without a cosmological constant, the most symmetric solution of Einstein's equations in the vacuum is the flat, 4D Minkowski space-time of special relativity. In 1917 the Dutch astronomer Willem de Sitter found the analogous solution if the cosmological constant is non-zero. If Λ is positive, the solution is called "de Sitter space" and Λ is the vacuum energy that curves space-time (a negative value of Λ corresponds to what is called anti-de Sitter space).

Einstein immediately rejected the de Sitter solution because it went against his intuition - it implied that space-time can be curved in the absence of matter - although he ended up accepting the idea after some debate.

In de Sitter space the universe expands exponentially, which is the basis of the inflationary model of the universe. Inflation describes an extremely short period of rapid expansion that is thought to have taken place shortly after the Big Bang. The inflationary model solves many of the cosmological problems of the Big Bang model, and has also received strong support from the latest measurements of the cosmic microwave background. In addition to describing inflation, de Sitter space also provides an explanation for the acceleration of universal expansion. In this scenario a very small value of the cosmological constant is the "dark energy" that is driving the expansion of the universe.

Obtaining a de Sitter solution is therefore a major challenge for string theory, which has been the main candidate for a fundamental theory of the universe for almost 20 years. String theory is believed to be a unique theory that unifies all the particles and forces in nature - including gravity - by treating them as infinitesimal 1D strings.

But even if the theory is unique, the number of different universes that appear as solutions to its equations is extremely large. One solution, for example, is a flat Minkowski space-time in 10 dimensions. Another is a universe similar to ours that has four flat space-time dimensions and an additional six dimensions that are curled up at extremely small scales.

These "hidden" dimensions correspond to a class of spaces known as Calabi-Yau spaces. The crux of string theory is that these 6D spaces can fix the physical properties of the observable universe, such as the type and number of elementary particles and the forces that act between them. There are at least 10,000 different Calabi-Yau spaces, each of which is defined by a number of parameters called moduli that determine its size and shape.

Each value of these parameters gives rise to different physics in the 4D observable world, but the trouble is that all values are equally valid solutions. In other words, they all lead to the same vacuum energy (Λ=0) in four dimensions.

Flat land

We can picture this range of solutions as a very boring landscape in which the east-west and north-south directions are the values of the moduli and height is the energy. Since the energy vanishes for each point, this landscape looks like a big, motionless, flat ocean. But in four dimensions the moduli - which come from the full 10D gravitational field - manifest themselves as massless particles called moduli fields. This is a big problem for string theory because nobody has observed such particles. Furthermore, the moduli fields correspond to a multitude of values for observable physical quantities, such as the strength of the electromagnetic interaction, which we know to have essentially fixed values.

An outstanding challenge in string theory is therefore to find ways to "lift" the shape of this landscape and fix the value of the moduli fields by minimizing their energy. If this energy is positive it would indicate a de Sitter solution. Now, Shamit Kachru, Renata Kallosh and Andrei Linde from Stanford University in the US, and Sandip Trivedi from the Tata Institute in India have found evidence for such a solution. The methods they used have all been developed before by string theorists, but their merit was to put together the different techniques in a self-consistent manner. One of the main ingredients of their construction is fluxes of fields that are higher-dimensional generalizations of electromagnetic fields (S Kachru et al. 2003 Phys. Rev. D 68 046005).

Unstable universes

This new solution has several important implications. De Sitter space is only a local minimum of the energy, whereas the global minimum corresponds to a flat 10D space-time. We therefore know that the de Sitter minimum has to be unstable, and that it will ultimately decay to the stable flat 10D minimum via quantum tunnelling (see figure). Fortunately its lifetime is far greater than the age of the universe.


Stringscape - a small part of the string-theory landscape showing the new de Sitter solution as a local minimum of the energy (vertical axis). The global minimum occurs at the infinite size of the extra dimensions on the extreme right of the figure

Once we know that there is one de Sitter solution, it is easy to find many more of them by just changing the values of the fluxes. Sujay Ashok and Michael Douglas of Rutgers University have recently estimated the number of different solutions to be at least 10100, which indicates an extremely rich landscape with many mountains, valleys, oceans and even volcanoes. Each minimum-energy point represents a different universe, and the height of that point is the value of the cosmological constant for that universe. Viewing the solution this way, the probability that one of these universes has a cosmological constant that is as small as is indicated by current experiments is actually non-zero.

This points to an anthropic approach to the cosmological-constant problem: out of the enormous number of solutions of string theory that represent different universes, we happen to live in one that allows our existence. This is an idea that Leonard Susskind of Stanford University has coined "the anthropic landscape of string theory".

However, mentioning anthropic arguments in physics guarantees heated debates, and this is not an exception. At the moment we cannot deny the existence of these numerous solutions. The best we can do is to learn from de Sitter, rather than Einstein, and to follow what the theory has to tell us without prejudice.

November 1, 2003

Об авторе:

Фернандо Квеведо - профессор теоретической физики высоких энергий на кафедре прикладной математики и теоретической физики Кембриджского университета (Великобритания). Известный струнный теоретик, специалист по теории струнных компактификаций и струнной космологии.