# PDF Cosmological Crossroads - An Advanced Course in Mathematical, Physical and String Cosmology

This book has grown out of lectures held at a summer school on cosmology, in response to an ever increasing need for an advanced textbook that addresses the needs of both postgraduate students and nonspecialist researchers from various disciplines ranging from mathematical physics to observational astrophysics. Bridging the gap between standard textbook material in cosmology and the forefront of research, this book also constitutes a modern source of reference for the experienced researcher in classical and quantum cosmology.

## Cosmological Crossroads: An Advanced Course in Mathematical, Physical and - Google книги

Although there are by now many textbooks discussing the subjectatmanylevels,itistruethatnosinglebookhasthecharacteristics wehadinmindwheneditingthisvolume. Wehavetriednottoproducea proceedingsvolumebutmoreamultiauthoredtextbookwhichcouldserveas areferencesourceofcurrentideasincosmology. Webelievethisbookcovers atanintroductorylevelmostoftheissueswhichareconsideredimportant inmoderncosmologicalresearchandcanbereadbyagraduatestudentor researcherwhowishestoacquireareasonableknowledgeofcosmologythat will,wehope,continuetobeofvalueforyearstocome. The 1st Aegean School on Cosmology, and consequently this book, - camepossiblewiththekindsupportofmanypeopleandorganizations.

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## Cosmological Crossroads

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Read more Read less. K; Softcover reprint of hardcover 1st ed. During the transition, the atoms behave in an unusual way. For example, the atoms slow to a halt at a rate that depends on the temperature and on Planck's constant , the fundamental parameter of quantum mechanics, which does not enter into the description of the other phases. This behavior has recently been understood by considering a dual description where properties of the fluid are described in terms of a higher dimensional black hole.

In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing models of real world physics that combine general relativity and particle physics. Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas.

String phenomenology is the part of string theory that attempts to construct realistic or semi-realistic models based on string theory.

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Partly because of theoretical and mathematical difficulties and partly because of the extremely high energies needed to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to any of these models being a correct fundamental description of nature. This has led some in the community to criticize these approaches to unification and question the value of continued research on these problems.

The currently accepted theory describing elementary particles and their interactions is known as the standard model of particle physics. This theory provides a unified description of three of the fundamental forces of nature: electromagnetism and the strong and weak nuclear forces.

Despite its remarkable success in explaining a wide range of physical phenomena, the standard model cannot be a complete description of reality.

### Cosmological crossroads : an advanced course in mathematical, physical and string cosmology

This is because the standard model fails to incorporate the force of gravity and because of problems such as the hierarchy problem and the inability to explain the structure of fermion masses or dark matter. String theory has been used to construct a variety of models of particle physics going beyond the standard model. Typically, such models are based on the idea of compactification.

Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles.

Such compactifications offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct realistic or semi-realistic models of our four-dimensional world based on M-theory. The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution.

Despite its success in explaining many observed features of the universe including galactic redshifts , the relative abundance of light elements such as hydrogen and helium , and the existence of a cosmic microwave background , there are several questions that remain unanswered. For example, the standard Big Bang model does not explain why the universe appears to be same in all directions, why it appears flat on very large distance scales, or why certain hypothesized particles such as magnetic monopoles are not observed in experiments.

Currently, the leading candidate for a theory going beyond the Big Bang is the theory of cosmic inflation. Developed by Alan Guth and others in the s, inflation postulates a period of extremely rapid accelerated expansion of the universe prior to the expansion described by the standard Big Bang theory. The theory of cosmic inflation preserves the successes of the Big Bang while providing a natural explanation for some of the mysterious features of the universe.

In the theory of inflation, the rapid initial expansion of the universe is caused by a hypothetical particle called the inflaton. The exact properties of this particle are not fixed by the theory but should ultimately be derived from a more fundamental theory such as string theory. While these approaches might eventually find support in observational data such as measurements of the cosmic microwave background, the application of string theory to cosmology is still in its early stages. In addition to influencing research in theoretical physics , string theory has stimulated a number of major developments in pure mathematics.

Like many developing ideas in theoretical physics, string theory does not at present have a mathematically rigorous formulation in which all of its concepts can be defined precisely. As a result, physicists who study string theory are often guided by physical intuition to conjecture relationships between the seemingly different mathematical structures that are used to formalize different parts of the theory. These conjectures are later proved by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics.

After Calabi—Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi—Yau manifold.

In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry. Regardless of whether Calabi—Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between different string theories has significant mathematical consequences. The Calabi—Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry , a branch of mathematics concerned with counting the numbers of solutions to geometric questions.

Enumerative geometry studies a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic illustrated on the right is an algebraic variety defined using a certain polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface.

Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi—Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five. This problem was solved by the nineteenth-century German mathematician Hermann Schubert , who found that there are exactly 2, such lines.

## String theory

In , geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is , By the year , most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish. Originally, these results of Candelas were justified on physical grounds.

However, mathematicians generally prefer rigorous proofs that do not require an appeal to physical intuition. Inspired by physicists' work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry. Group theory is the branch of mathematics that studies the concept of symmetry. For example, one can consider a geometric shape such as an equilateral triangle.

There are various operations that one can perform on this triangle without changing its shape. Each of these operations is called a symmetry , and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a group. In this particular example, the group is known as the dihedral group of order 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a finite group.

Mathematicians often strive for a classification or list of all mathematical objects of a given type. It is generally believed that finite groups are too diverse to admit a useful classification. A more modest but still challenging problem is to classify all finite simple groups. These are finite groups which may be used as building blocks for constructing arbitrary finite groups in the same way that prime numbers can be used to construct arbitrary whole numbers by taking products.

This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances. The largest sporadic group, the so-called monster group , has over 10 53 elements, more than a thousand times the number of atoms in the Earth.

A seemingly unrelated construction is the j -function of number theory. This object belongs to a special class of functions called modular functions , whose graphs form a certain kind of repeating pattern. In the late s, mathematicians John McKay and John Thompson noticed that certain numbers arising in the analysis of the monster group namely, the dimensions of its irreducible representations are related to numbers that appear in a formula for the j -function namely, the coefficients of its Fourier series. In , Richard Borcherds constructed a bridge between the theory of modular functions and finite groups and, in the process, explained the observations of McKay and Thompson.

Since the s, the connection between string theory and moonshine has led to further results in mathematics and physics. Harvey proposed a generalization of this moonshine phenomenon called umbral moonshine , [] and their conjecture was proved mathematically by Duncan, Michael Griffin, and Ken Ono. Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein.

Thereafter, German mathematician Theodor Kaluza combined the fifth dimension with general relativity , and only Kaluza is usually credited with the idea. In , the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension—it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor , while much later Brans and Dicke added a scalar component to gravity.

These ideas would be revived within string theory, where they are demanded by consistency conditions. String theory was originally developed during the late s and early s as a never completely successful theory of hadrons , the subatomic particles like the proton and neutron that feel the strong interaction. In the s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu , Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings.

Chew advocated making a theory for the interactions of these trajectories that did not presume that they were composed of any fundamental particles, but would construct their interactions from self-consistency conditions on the S-matrix. The S-matrix approach was started by Werner Heisenberg in the s as a way of constructing a theory that did not rely on the local notions of space and time, which Heisenberg believed break down at the nuclear scale.

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While the scale was off by many orders of magnitude, the approach he advocated was ideally suited for a theory of quantum gravity. Working with experimental data, R. Dolen, D. Horn and C.