* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Here is a very brief outline of the development of string theory, the
Survey
Document related concepts
Relativistic quantum mechanics wikipedia , lookup
Matter wave wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Canonical quantization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Atomic theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Yang–Mills theory wikipedia , lookup
Renormalization group wikipedia , lookup
History of quantum field theory wikipedia , lookup
Renormalization wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Elementary particle wikipedia , lookup
Transcript
1 String Theory: An Introduction and Application to Cosmology Project Report By Paramita Barai In course Phys 8610: High Energy Nuclear and Particle Physics Instructor: Dr. Xiaochun He Department of Physics and Astronomy Georgia State University 10th May, 2004 2 Abstract It has been human curiosity from ancient times to understand the underlying mechanisms of the physical processes going on in the world around us. Through our search we have learnt several exciting facts about some of the very large scales (galaxies) in the universe and also very small atomic and nuclear scales. Powerful physical theories have been formulated like Relativity, Quantum Field Theory etc. which have been extremely successful to describe the nature in their limiting cases. We have learnt that the macroscopic bodies we see around us are made up of tiny particles in a hierarchical structure (e.g. protons, electrons, quarks etc). Space-time and matter have been found to be integrally coupled. However, there are still several questions about the Universe unanswered, several mysteries unsolved. One such aspect is the Grand Unification Theory, the Holy Grail of Physics. This theory is supposed to explain all particles, their interactions and the forces in a single integrated framework. But no such concrete theory has been found till date. STRING THEORY is a theory to explain the nature, which is presently at a preliminary theoretical developmental stage. Originally proposed to describe strong forces, String Theory today is the strongest candidate of a Unified Theory, as it has brought Relativity and Quantum Mechanics into the same footing. The basic concept behind String Theory is that all particles of the universe are composed of tiny (~ Planck length) fundamental strings, whose different modes of vibrations produce the particle zoo and the interactions between them. To do so, String Theory demands the strings to vibrate in 10 spacetime dimensions. String Theory accounts for Gravitation by predicting a massless spin-2 particle (graviton), which no other theory does. Hence in this aspect String Theory holds the promise of being a Grand Unified Theory sometime in future. The problem with String Theory is that, it is not mature enough to predict something that can be tested by present experiments and observations, to decide if the theory is right or wrong. Whatever it predicts is at so high energy that it is much beyond the reach of our technology today. Still, the wonderful theoretical implications of String Theory (if it is true) makes it so attractive. We hope that eventually it can be decided if String Theory is right or not; and if right more and more successful physical predictions can be done with it. 3 Table of Contents Abstract …………………………………………………………………………………………...2 List of figures ……………………………………………………………………………………..5 Chapter 1 String Theory: Basic Concepts 1.1 Introduction: What is String Theory? ………………………………………………………...7 1.2 Brief history and Timeline ……………………………………………………………………8 1.3 Basic characteristics of Strings ……………………………………………………………….9 1.4 From String to Particle ………………………………………………………………………12 Chapter 2 Revolutionary features and Successful Predictions 2.1 More than just strings: Branes ………………………………………………………………16 2.2 How many dimensions ………………………………………………………………………16 2.3 How many string theories? M Theory ………………………………………………………17 2.4 T-duality ……………………………………………………………………………………..18 2.5 Predictions and advantages of String Theory ……………………………………………….20 Chapter 3 Cosmology: A Review .. 3.1 Historical developments ……………………………………………………………………..21 3.2 The Big Bang Model ………………………………………………………………………...22 3.2.1 Qualitative Overview of the model ………………………………………………………..22 3.2.2 The Starting point …………………………………………………………………………24 3.2.3 Phase transitions …………………………………………………………………………...25 3.2.4 Topological defects ………………………………………………………………………..26 Chapter 4 Problems in Big Bang Theory: Inflationary Solution 4.1 Initial Condition Problems …………………………………………………………………..26 4.2 Relic Problems ………………………………………………………………………………30 4.3 Inflation ……………………………………………………………………………………...30 4 4.4 Why Inflation Occurs ………………………………………………………………………..31 4.5 Some Consequences of Inflation: Solution to Flatness and Relic problems ………………..33 4.6 Reheating ……………………………………………………………………………………36 4.7 Inflation and Particle Physics ………………………………………………………………..37 4.8 Perturbations in Inflation ……………………………………………………………………38 4.9 Initial conditions and eternal inflation ………………………………………………………39 Chapter 5 Stringy cosmology: String Theory applied to Cosmology 5.1 The beginning of time ……………………………………………………………………….40 5.2 Extra dimensions and Compactification in Cosmology ……………………………………..42 5.3 The late universe …………………………………………………………………………….46 Chapter 6 Conclusions ……………………………………………………………………………………...48 References ……………………………………………………………………………………….49 5 List of figures 1.1 – Closed String Interaction: The world-sheet of 2 strings joining to form the world-sheet of single string ………………………………………………………………………………….10 1.2 – The Feynman diagram for particle interaction …………………………………………….11 1.3 – The String Interaction diagram when consider higher order terms in perturbation theory..11 1.4 – A mode of string oscillation (assumed to do the calculations simpler) …………………...13 2.1. – The length scale showing the grand unifying feature ……………………………………21 4.1. – Infra-red map of the sky (as observed by COBE and WMAP) showing the smoothness and fluctuations in the CMBR …………………………………………………………………..27 4.2. – 2-Dimensinal analogue of the Possible Curvature of the Universe ………………………28 6 Acknowledgement I would like to thank Dr. He for his cooperation and advice during the course of my project. I learnt to organize my thoughts in a natural rational flow from the several discussions that I had with him, which also helped me to gain more insight about such a complicated topic. His help and guidance about organizing the references and constructing a coherent picture of the topic have been very helpful. I have also gained from his course lectures, to get motivated for this kind of project. Paramita Barai 7 Chapter 1 Introduction 1.1 What is String Theory? The physical idea behind String Theory is utterly simple. The macroscopic objects we see around us have been found to be composed of several small particles. In String Theory we go a step further and say that the fundamental particles are composed of tiny vibrating strings. Instead of many types of elementary point–like particles, we postulate that there is only a single variety of string–like object in nature. The string is not "made up of anything", rather, it is fundamental and other things are made up of it. Similar to musical strings, this basic string can vibrate, and each vibrational mode can be viewed as a point-like elementary particle, just as the modes of a musical string are perceived as distinct notes! In this way all the matter in the universe and their interactions by various forces are unified under string theory as everything is described by vibration of the same string. String theory solves the deep problem of the incompatibility of the two fundamental theories: General Relativity and Quantum Field Theory, by modifying the properties of General Relativity when it is applied to scales on the order of the Planck length. Hence it is the most powerful potential candidate for a Grand Unified Theory (human’s ultimate dream!). But as of now, string theory is only at a very preliminary theoretical stage. There are no experimental or observational proofs that the theory is true. This is because we have not yet developed the technology to approach the high energy range where stringy behavior of particles can come to play. Modern accelerators can only probe down to distance scales around 10–16 cm and hence these loops of strings appear to be point objects. However, the string theoretic hypothesis that all the particles are actually composed tiny loops, changes drastically the way in which the particles interact on the shortest of distance scales. This modification is what allows gravity and quantum mechanics to form a harmonious union. In this chapter I report the historical developments leading to String Theory and then some conceptual foundations of it. Chapter 2 gives some of the revolutionary mind boggling ideas of string theory, which makes it so unique. There I also touch upon the successful 8 predictions of string theory and how it is a potential candidate of the Grand Unified Theory. Then to give a concrete idea on how strings can be applied to solve some of the cosmological problems, I give a brief review of present cosmology, the Big Bang Model in Chapter 3. Then Chapter 4 describes some of the problems of conventional Big Bang Theory, and giving the solution through Inflation, and how does physics of strings come to play there. Chapter 5 gives a very brief introduction to Stringy Cosmology, the kinds of calculations done when apply string theory to cosmology. Finally, Chapter 6 gives the conclusions. 1.2 Brief history and Timeline This section gives a very brief outline of the development of string theory [1]. In 1921 Kaluza-Klein theory was proposed which stated that electromagnetism can be derived from gravity in a unified theory if there are 4 space dimensions instead of 3, and the 4 th dimension is curled into a tiny circle. Kaluza and Klein made this discovery independently. But the idea did not become so popular till 1970 which year marks the official birth of string theory. Three particle theorists; Yoichiro Nambu, Leonard Susskind and Holger Nielsen realized that the dual theories of 1968 to describe particle spectrum could also describe quantum mechanics of oscillating strings. Originally String Theory was formulated to explain the strong force. But it predicted a massless spin–2 particle (the graviton), which did not fit into the standard picture as of that time. Later it was found that Quantum Chromo Dynamics (QCD) could describe strong interactions successfully. String Theory made its revival after some more years. In 1971, Supersymmetry (SUSY) was invented in two contexts at once: in ordinary particle field theory and as a consequence of introducing fermions into string theory. SUSY holds the promise of resolving many problems in particle theory, but when applied to string theory requires equal numbers of fermions and bosons, so it cannot be an exact symmetry of Nature. In 1974 gravitons were predicted to be the carrier of gravitational force. String theory using closed strings fails to describe hadronic physics because the spin–2 excitation has zero mass. However, that makes it an ideal candidate for the missing theory of quantum gravity!! This marks the advent of string theory as a proposed unified theory of all four observed forces in Nature. In 1976 Supergravity was proposed, namely Supersymmetry is added to gravity, making Supergravity. 1980 marks the beginning of the 1st Superstring Revolution. Michael Green and John Schwarz combined string theory with supersymmetry to yield an excitation spectrum that has equal numbers of fermions and bosons, showing that string theory can be made totally supersymmetric. The resulting 9 objects are called Superstrings. Finally in the late 1980’s and 1990’s string theory was accepted by the mainstream physics community as an actual candidate theory uniting quantum mechanics, particle physics and gravity. In 1991–95 several interesting works on stringy black holes in higher dimensions lead to a revolution in understanding how different versions of string theory are related through duality transformations. This brought in a surge of progress towards a deeper nonperturbative picture of string theory. This period also marks the 2nd Superstring Revolution, when Edward Witten and Townsend discovered that the 5 different prevalent string theories were actually manifestations of a single Superstring theory called the M Theory. In 1996 Black Hole entropy was described using Einstein’s relativity and Hawking radiation. There were hints in the past that black holes have thermodynamic properties that need to be understood microscopically. A microscopic origin for black hole thermodynamics was finally achieved in string theory, when Andy Strominger and Cumrun Vafa accounted for black hole entropy using string theory. And the search is going on. Now, in the beginning of this century, physicists all over the world are engaged in studying string theory and predict something meaningful from it. 1.3 Basic characteristics of Strings We are used to thinking of fundamental particles (like electrons) as point–like 0– dimensional objects. A generalization of this is a fundamental string which is an 1–dimensional object [1]. Strings have no thickness but do have a length of the order of the Planck Length, LString LPlanck GN ~ 10 33 cm . 3 c The strings are stretched under tension (as expressed) in order to become excited. TString 1 2 The parameter is called the string parameter and the square root of this number represents the approximate distance scale at which string effects should become observable. In other words, 2 is equal to the square of the string length scale, i.e. ~ LString . The string tension is fantastically high – equivalent to a loading of about 1039 tons. These strings have certain vibrational modes which can be characterized by various quantum numbers such as mass, spin, etc. The basic idea is that each mode carries a set of quantum numbers that correspond to a distinct type of fundamental particle. This is the ultimate 10 unification: all the fundamental particles we know can be described by one object, a string! A string oscillates in space and time, and as it oscillates, being a 1–dimensional entity, it sweeps out a 2–dimensional surface in spacetime that we call a world sheet (compared to the 1– dimensional world line path of a particle). The dynamics of a free relativistic particle is described by requiring the path swept out by it to be extremal in length. By analogy, the dynamics for the string is based on the requirement that the area of the surface swept out is extremal. In addition, there are quantum fields which are defined on the world-sheet, and which enter into the dynamics too [2]. There is essentially only one adjustable constant in the model, the tension of the string, which determines the characteristic mass scale. The characteristic length scale; related to the tension of the string, which is the one free parameter in the theory; is the Planck length. This is so tiny that even on the scale of particle physics the tube is so narrow as to resemble a line – just like the world-line of an elementary particle. Strings interact by splitting and joining. For example the annihilation of two closed strings into a single closed string occurs with an interaction that looks like the following [4]. Fig 1.1. – Closed String Interaction: The world-sheet of 2 strings joining to form the world-sheet of a single string Here the worldsheet of the interaction is a smooth surface. This essentially accounts for another nice property of string theory. It is not plagued by infinities in the way that point particle quantum field theories are. The analogous Feynman diagram (for the above interaction of strings) in a point particle field theory is the following. 11 Fig 1.2. – The Feynman diagram for particle interaction In this case the interaction point occurs at a topological singularity in the diagram (where the 3 world-lines intersect). This leads to a break down of the point particle theory at high energies. The leading contribution to this process of constructing Feynman diagram analogues of string interactions is called a tree level interaction. To compute quantum mechanical amplitudes using perturbation theory we add contributions from higher order quantum processes. Perturbation theory provides good answers as long as the contributions get smaller and smaller as we go to higher and higher orders. Then we only need to compute the first few diagrams to get accurate results. In string theory, higher order diagrams correspond to the number of holes (or handles) in the world sheet, as depicted nicely in the following diagram, which shows several higher order terms. Fig 1.3. – The String Interaction diagram when consider higher order terms in perturbation theory The nice thing about this is that at each order in perturbation theory there is only one diagram. [For a comparison, in point particle field theories the number of diagrams grows exponentially at higher orders.] The problem lies in the fact that to extract answers from diagrams with more than about two handles is very difficult due to the complexity of the mathematics involved in dealing with these surfaces. Perturbation theory is a very useful tool for studying the physics at weak coupling, and most of our current understanding of particle physics and string theory is based on it. However it is far from complete. The answers to many of the deepest questions will only be found once we have a complete non-perturbative description of string theory. 12 Hence it is clear that in the string-theory analogues to Feynman diagrams there are none of the singularities that in field theory lead to infinities. So the Quantum Gravity that is contained in string theory is free from infinities; i.e. string theory contains a consistent quantum theory of gravity, something that had eluded physicists for at least half a century. If one goes to the limit in which the world-sheet tube resembles a line to see how this comes about, it is apparent that Einstein's theory has to be modified, but in a very strictly determined way. The corrections that need to be introduced are quite negligible when one is concerned with the gravitational phenomena so far observed, because quantum effects are far too small to be relevant. String theories are classified according to whether or not the strings are required to be closed loops, and whether or not the particle spectrum includes fermions. In order to include fermions in string theory, there must be a special kind of symmetry called Supersymmetry (SUSY), which means that for every boson (a particle, of integral spin that transmits a force) there is a corresponding fermion (a particle, of half-integral spin that makes up matter). So SUSY relates the particles that transmit forces to the particles that make up matter. Supersymmetric partners to currently known particles have not been observed as of date in particle experiments, but theorists believe this is because supersymmetric particles are too massive to be detected using present-day technology. Particle accelerators could be on the verge of finding evidence for high-energy supersymmetry in the next decade. Evidence for supersymmetry at high energy would be a very compelling evidence that string theory is a good mathematical model for nature at the smallest distance scales. In string theory, all of the properties of elementary particles – charge, mass, spin, etc – come from vibration of the same string. The easiest to see is mass. The more frantic the vibration, the more energy. And since mass and energy are the same thing, higher mass comes from higher vibration. 1.4 From String to Particle String theory proclaims that all the observed particle properties (mass, charge, spin etc) are reflections of the various ways in which a string can vibrate. Just as the strings on a piano or violin have some resonant (natural) frequencies at which they prefer to vibrate – the same holds true for the loops of string theory. Each of the preferred patterns of vibration of a string appears as a particle whose mass, spin and charge are determined by the string's 13 oscillatory pattern. The electron is a string vibrating one way; the up-quark is a string vibrating another way. Force mediator particles like photons, weak gauge bosons, and gluons are yet other resonant patterns of string vibration. There is even a mode describing the graviton which is the particle carrying the force of gravity. Particle properties in string theory are the manifestations of one and the same physical feature: the resonant patterns of vibration of fundamental loops of string. The same idea applies to the forces of nature as well. Hence everything, all matter and all forces, is unified under the microscopic string oscillations – the notes that strings can play [1, 2, 3, 4]. The strings are so tiny (~1016 smaller than the present distances probed by most powerful accelerators) that when we see them over large scale their vibration appears to us as point particles. We are not yet technologically powerful to resolve the stringy nature of particles. But this formalism can be followed mathematically too. In this section, a crude review of the equations of motion of the strings are given starting with the well-known notion of classical strings, aiming to describe particle behavior from there [1]. Consider the wave equation for a classical macroscopic string with a tension T and a mass per unit length . If the string is described in coordinates as in fig. 1.4, where x is the distance along the string and y is the height of the string, and the string oscillates in time t. Fig 1.4. – A mode of string oscillation (assumed to do the calculations simpler) Then the equation of motion is the one-dimensional wave equation 2 2 y x, t T 2 y x, t 2 y x, t v , w x 2 t 2 x 2 where vw is the wave velocity along the string. For the first case, assume this is a nonrelativistic string, i.e. one where the wave velocity much smaller than the speed of light. When solving the equations of motion, we need to know the boundary conditions of the string. If we suppose that the string is fixed at each end and has an unstretched length L. The 14 general solution to this equation can be written as a sum of normal modes, here labeled by the integer n, such that nv w t nv w t nx . yx, t a n cos bn sin sin L L L n 1 The condition for a normal mode is that the wavelength be some integral fraction of twice the string length, or n nv 2L . The frequency of the normal mode is then f n w . The string wave n 2L velocity vw increases as the tension of the string is increased, and so the normal frequency of the string increases as well. Now consider relativistic strings, i.e. where the wave velocity is comparable to the speed of light. According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. In the nonrelativistic string, there exists a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a twodimensional spacetime of its own, where the division between space and time depends upon the observer. The classical equation can be written as, 2 2 X , 2 X , c , 2 2 where and are coordinates on the string world sheet representing space and time along the string, and the parameter c2 is the ratio of the string tension to the string mass per unit length. These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet, S 1 dd 4 h h nm m X n X . In this expression, the spacetime coordinates X of the string are also fields X in a twodimension field theory defined on the surface that a string sweeps out as it travels in space. The partial derivatives are with respect to the coordinates and on the world sheet and hmn is the 2dimensional metric defined on the string world sheet. The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as, 15 1 nc nc X i , x i x i i 2 ni cos i sin L L n0 n n . cos L The string solution above is for an open string with floppy ends i.e. it isn't tied down at either end and so travels freely through spacetime as it oscillates. For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and left-movers, and this difference is important in the supersymmetric heterotic string theory. The analysis till now was for classical string. When we add quantum mechanics by making the string momentum and position obey quantum commutation relations, the oscillator mode coefficients have the commutation relations, m , n m m n . The quantized string oscillator modes wind up giving representations of the Poincaré group (the inhomogeneous Lorentz Group), through which quantum states of mass and spin are classified in a relativistic quantum field theory. So this is how the elementary particle arises in string theory. In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts. This happens because of the minus sign in the spacetime metric, which implies that, m0 , n0 m m n . So there ends up being extra unphysical states in the string spectrum. In 10 spacetime dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore string quantum mechanics is only consistent if the dimension of spacetime is 10. By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in spacetime, with mass (MJ) that depends on the spin (J) according to the formula, J M J , 2 which is the well known relation obtained in particle physics from Regge plots. Boundary conditions are important for string behavior. Strings can be open, with ends that travel at the speed of light, or closed, with their ends joined in a ring. One of the particle states of a closed string has zero mass and 2 units of spin, the same mass and spin as a graviton, the particle that is supposed to be the carrier of the gravitational force. 16 For string theory to be a successful theory of quantum gravity, the average size of a string should be somewhere near the length scale of quantum gravity, called the Planck length, which is about 10–33 cm. Chapter 2 Revolutionary features and Successful Predictions 2.1 More than just strings: Branes Superstring theory is not just a theory of 1-dimensional objects called strings. There are higher dimensional objects in string theory with dimensions from zero to nine, called p-branes (objects with p dimensions). In terms of branes, what we usually call a membrane would be a 2brane, a string would be a 1-brane and a point would be called a 0-brane [2, 3]. Now the question arises how do we visualize a p-brane. In very simple words to explain physically, we can think of a brane as being a slice through the higher dimensional world that string theory says exists. Mathematically speaking, a p-brane is a spacetime object that is a solution to the Einstein’s equations in the low energy limit of superstring theory, with the energy density of the non-gravitational fields confined to some p-dimensional subspace of the 9 space dimensions in the theory. (Point to note, superstring theory lives in 10 spacetime dimensions, which means 1 time dimension plus 9 space dimensions.) For example, in a solution with electric charge, if the energy density in the electromagnetic field was distributed along a line in spacetime, this 1dimensional line would be considered a p-brane with p=1. A special class of p-brane in string theory is called D-brane. Roughly speaking, a Dbrane is a p-brane where the ends of open strings are localized on the brane. A D-brane is like a collective excitation of strings. D-branes are important in understanding black holes in string theory, especially in counting the quantum microstates that lead to black hole entropy, which was a very big accomplishment for string theory. 2.2 How many dimensions? Before string theory won the full attention of the theoretical physics community, the most popular unified theory was an 11–dimensional theory of supergravity, which is supersymmetry 17 combined with gravity. The 11–dimensional spacetime was to be compactified on a small 7– dimensional sphere, leaving 4 spacetime dimensions visible to observers at large distances. This theory didn't work as a unified theory of particle physics, because it doesn't have a sensible quantum limit as a point particle theory. But this 11 dimensional theory did not die. It eventually came back to life in the strong coupling limit of superstring theory in 10 dimensions [5]. Now the question is, how could a superstring theory with 10 spacetime dimensions turn into a supergravity theory with 11 spacetime dimensions? We will see in subsequent sections that, duality relations between superstring theories relate very different theories, equate large distance with small distance, and exchange strong coupling with weak coupling. So there must be some duality relation that can explain how a superstring theory that requires 10 spacetime dimensions for quantum consistency can really be a theory in 11 spacetime dimensions after all. But in our known world around we see only 3–space (x, y, z) and 1–time dimension (t). Then where are these extra 6 (or 7) dimensions which must exist if string theory is to be true. There are two main ways to explain these missing dimensions. One is to propose that these extra dimensions have compactified into a tiny space of size of string scale, which our present technology cannot resolve. The other idea is that we are trapped on the surface of a higher dimensional brane, for which we can’t leave the brane and hence can’t perceive the extra dimensions. 2.3 How many string theories? M Theory At one time, string theorists believed that there were 5 distinct superstring theories: type I, types IIA and IIB, heterotic SO(32) and E8XE8 string theories. Each of these was a valid String Theory differing in the ways of formulation, and each gave valid predictions. The thinking was that out of these five candidate theories, only one was the actual correct Theory of Everything, and that theory was the theory whose low energy limit, with 10 dimensions spacetime compactified down to 4, matched the physics observed in our world today. The other theories would be nothing more than rejected string theories, mathematical constructs not blessed by Nature with existence. But now it is known that the five superstring theories are connected to one another as if they are each a special limiting case of some more fundamental theory. In the mid-nineties it 18 was learned that superstring theories are related by duality transformations known as T-duality and S-duality. These dualities link the quantities of large and small distance, and strong and weak coupling, limits that have always been identified as distinct limits of a physical system in both classical and quantum physics. These duality relationships between string theories have sparked a radical shift in our understanding of string theory, and have led to the reasonable expectation that all five superstring theories -- type I, types IIA and IIB, heterotic SO(32) and E8XE8 -- are special limits of a more fundamental theory. This more fundamental theory, called the M-Theory exists in 11 spacetime dimensions [1, 3]. 2.4 T-duality The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality. This comes about from the compactification of extra space dimensions in a 10–dimensional superstring theory. Let's take one of the 9 directions, Xi in the flat 10–dimensional spacetime, and compactify it into a circle of radius R, so that, X i X i 2R . A particle traveling around this circle will have its momentum quantized in integer multiples of 1/R, and a particle in the nth quantized momentum state will contribute to the total mass squared of the particle as, mn 2 n2 . R2 A string can travel around the circle, too, and the contribution to the string mass squared is the same as above. A closed string can also wrap around the circle, something a particle cannot do. The number of times the string winds around the circle is called the winding number, denoted as w, and w is also quantized in integer units. Tension is energy per unit length, and the wrapped string has energy from being stretched around the circular dimension. The winding contribution Ew to the string energy is equal to the string tension Tstring times the total length of the wrapped string, which is the circumference of the circle multiplied by the number of times w that the string is wrapped around the circle. E w 2wRTstring wR , 19 where as already introduced before, Ls , tells us the length scale Ls of string theory. 2 The total mass squared for each mode of the closed string is n 2 w2 R 2 2 ~ m 2 N N 2 2 R ~ with, N N nw 2 The integers N and Ñ are the number of oscillation modes excited on a closed string in the rightmoving and left-moving directions around the string. The above formula is invariant under the exchange: R R ,n w. In other words, we can exchange compactification radius R with radius '/R if we exchange the winding modes with the quantized momentum modes. This mode exchange is the basis of the duality known as T-duality. If the compactification radius R is much smaller than the string scale Ls, then the compactification radius after the winding and momentum modes are exchanged is much larger than the string scale Ls. So T-duality obscures the difference between compactified dimensions that are much bigger than the string scale, and those that are much smaller than the string scale. T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic SO(32) superstring theory to heterotic E8XE8 superstring theory. A duality relationship between IIA and IIB theory is very unexpected, because type IIA theory has massless fermions of both chiralities, making it a non-chiral theory, whereas type IIB theory is a chiral theory and has massless fermions with only a single chirality. T-duality is something unique to string physics. It's something point particles cannot do, because they don't have winding modes. If string theory is a correct theory of Nature, then this implies that on some deep level, the separation between large vs. small distance scales in physics is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure distance, and how we count the states of the probe. This idea may sound that it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because gravity comes from the metric tensor field that tells us the distances between events in spacetime. There is a similar duality concept called S-duality which relates strong and weak coupling constants with each other [1]. 20 Hence String Theory seems to be very interesting and rich theory where distance scales, coupling strengths and even the number of dimensions in spacetime are not fixed concepts but fluid entities that shift with our point of view. 2.5 Predictions and Advantages of String Theory String Theory has predicted several solutions mechanisms in various fields of physics and hence has turned to be the most powerful potential candidate for the Grand Unified Theory [5, 6, 7, 8]. In low energy limit, string theory predicts the right types of particles and the right types of interactions among them, which corresponds to the precise particle behavior we see in the Standard Model. The string solutions also generally include Yang–Mills gauge theories. String theory explains gravitation, by predicting the massless spin–2 particle, graviton, the quantum of gravitation, approximated at low energies by General Relativity. No other quantum theory has this feature of unifying gravity, which makes string theory ahead of all. String theory predicts Supersymmetry (SUSY) at low energies (the electroweak scale), which says for every fermions existing in the universe there are corresponding bosons. String theory also has several successful predictions in cosmology, to be discussed in the next chapter. String theory holds the potential to combine General Relativity with Quantum theory and give a Grand Unified Theory. This Grand Unification is a kind of Holy Grail of Physics. This aims to formulate a theory that will describe virtually everything of the universe. It will predict the fundamental particles we see around us, and the various forces and interactions between them which combine to give the world around us. Another important feature of the unification is, there should be a very less number of input parameters to the unified theory. Some of the grand unifying feature of String theory are as following. It unifies in one single 4–dimensional quantum theory all known forces and elementary particles. In the process, it reduces the number of parameters that need to be determined by experiments from between 20 and 30 (assuming massive neutrinos) to 1 (the string length or tension), not counting Planck's constant and the velocity of light [9, 10]. Hence, Superstring theory holds out the promise of a unification of all the successes of the standard model with a quantum theory of gravity, with no awkward infinities and with no arbitrary constants to be adjusted once the string tension has been chosen. 21 Fig 2.1 – The length scale showing the grand unifying feature Chapter 3 Cosmology: A Review In most simple words, Cosmology is the study of structure and evolution of the universe; how does the universe look like, how did it begin and evolve to came to the present universe as we see it today [11, 12]. 3.1 Historical developments When Einstein formulated the General Theory of Relativity, he found that it was incompatible with a static universe; as the equations predicted that the universe must either be expanding or shrinking. That time, the notion that the universe is static, was so strong that Einstein altered the equations of relativity, by forcing the Cosmological Constant in General Relativity equation to become zero, in order to allow for a static solution. But then Edwin Hubble found that the universe was indeed expanding by observing far off galaxies receding away from us. So Einstein retracted this alteration, calling it the biggest blunder of his life. From that point onwards the prevailing scientific viewpoint has been that of an expanding universe that at earlier times was much hotter and denser than it is today. Extrapolating this expansion backwards, we find that at a specific time in the past the universe would have been infinitely 22 dense. This time, the beginning of the universe's expansion, has come to be known as the event of BIG BANG. The Big Bang model has been extremely successful at explaining known aspects of the universe and correctly predicting new observations. Nonetheless, there are certain problems with the model. There are several features of our current universe that seem to emerge as strange coincidences in Big Bang theory. There are some predictions of the theory that are in contradiction with observation. These problems have motivated people to look for ways to extend or modify the theory without losing all of the successful predictions it has made. In 1980 a theory was developed by Alan Guth, that solved many of the problems plaguing the big bang model while leaving intact its basic structure. More specifically, this new theory modified our picture of what happened in the 1st fraction of a second of the universe's expansion. This change in our view of that 1st fraction of a second has proven to have profound influences on our view of the universe and the Big Bang itself. This new theory is called INFLATION. 3.2 The Big Bang Model At first I give a broad qualitative discussion of the Big Bang model, followed by some mathematical intricacies. 3.2.1 Qualitative Overview of the Model The most popular and accepted theory about the large-scale structure and history of the universe is the Big Bang model. According to this model the universe is expanding and at very early times was a nearly uniform expanding collection of high energy, high temperature particles. The small random differences in density that existed from one point to another created tiny inhomogeneities called density fluctuations. As the universe expanded and cooled these small inhomogeneities were then amplified by gravity. The matter in regions with slightly higher than average density gravitationally collapsed to form the structures we see today such as clusters and galaxies. Extrapolating the expansion backwards, (i.e. effectively contracting the size of the universe) we get at earlier times a nearly homogeneous fireball having higher temperatures and densities, ultimately reaching infinite density at a moment about 13.7 billion years ago. That moment is called the Big Bang, the ‘moment of creation of our present universe’ in a very crude way. There are no direct evidences of the Big Bang event (as it involves the moment of creation of the Universe, about which we have no idea of). However, we can still extrapolate 23 further backwards and assume the Big Bang model to be an accurate description of the universe all the way back to the Big Bang Singularity (when everything was crunched into a point at infinite temperature and density). But, such a complete extrapolation of the theory is not possible, because of several limitations of our theories of high energy physics. When we think about extrapolating the Big Bang model backwards, we are referring to running the equations of General Relativity backwards to earlier times and higher densities. We know, however, that General Relativity ceases to be valid when we try to describe a region of spacetime whose density exceeds a certain value known as the Planck density, roughly 1093 g/cm3. If we try to consistently apply Quantum Mechanics and General Relativity at such a density we find that quantum fluctuations of spacetime become important, and we have no theory that describes such a situation, as Quantum Mechanics and General Relativity, by themselves, are incompatible with each other. The Planck density is enormous. It corresponds to the mass of 100 billion galaxies being squeezed into a space the size of an atomic nucleus. If we could extrapolate General Relativity all the way back to the Big Bang the universe would have gone from infinite density to the Planck density in roughly 10–43 seconds (the Planck time). So considering something had happened, let 3 minutes after the Big Bang is equivalent to considering that it happened 3 minutes after the time the universe was at Planck density, i.e. the Planck Time. At any time before the Planck time all existing theories break down. This means that all accepted theories like, General Relativity, Quantum Field Theory etc. can’t be defined any more as they give infinite quantities as answers. Since we can't describe physics at densities higher than the Planck density, the best we can do in using the Big Bang model to describe the very early universe is the following. We assume, at some point in the past the density of the universe was above the Planck density. We don't know what physics governs such a case so we can make no predictions based on it. Somehow this super-Planckian state (sometimes called Spacetime Foam) gave rise to at least one region of sub-Planckian density with the right initial conditions to produce the universe we currently see. Here the term "initial conditions" for our universe, means that the state in which the observable part of the universe was in after the density first became sub–Planckian and the universe (or at least this region of it) could thus be described by known theories of physics. So we apply these known theories (Relativity, Quantum Mechanics etc.) from after the Planck Time of the Big Bang Singularity. 24 This Big Bang model after the Planck Time is in perfect accord with the theory of General Relativity, which predicts that a homogeneous universe would expand and cool in exactly that way. Moreover, there have been many observational confirmations of the Big Bang model. Some of these are as following. Distant objects have been found to be moving away from us radially, obtained from measuring redshifts of distant galaxies observing their spectral line shift. The microwave radiation left over from the early universe from the epoch of recombination is seen today in the sky as isotropic blackbody radiation at ~ 2.7 ° K, called the CMBR (Cosmic Microwave Background Radiation). The abundances of light elements formed in the first few minutes after the Big Bang have been measured, and hence an estimate of primordial nucleosysthesis (discussed in next paragraph) have been done. For roughly 3 minutes after the Big Bang the temperature of the universe was so high that protons and neutrons couldn't bind together into nuclei; all the particles had so much energy that the forces that hold nuclei together were too weak to make them stick to each other. Thus for those first 3 minutes the only element in the universe was hydrogen, i.e. single protons not bound to anything else. As the universe expanded and cooled it eventually reached a temperature where the protons and neutrons could bind together, and different elements were formed. The formation of these nuclei from their constituent particles (i.e. protons and neutrons) is known as Nucleosynthesis. String Theory is applicable before the Planck Time. 3.2.2 The Starting point Here I give a review on some of the mathematical principles applied from the time after Planck Time of the Big Bang [15]. This very early universe, had it’s energy density as radiationdominated. The universe on large scales can be accurately described by a perturbed RobertsonWalker metric. On thermodynamic grounds (backed up by evidence from CMB anisotropy) it seems likely that these perturbations are growing rather than shrinking with time, at least in the matter-dominated era; it would require extreme fine-tuning of initial conditions to arrange for diminishing matter perturbations. Thus, the early universe was smoother as well. Let us trace the history of the universe as we reconstruct it given these conditions (in the previous paragraph) plus our current best guesses at the relevant laws of physics. We start at a 25 time infinitesimally after the reduced Planck scale (which corresponds to an energy of, EPlanck 1/ 8G ~ 1018 GeV ). [The “ordinary” Planck scale is simply 1 / G ~ 1019 GeV . It is only an accident of history (Newton's law of gravity predating general relativity, or for that matter Poisson's equation) that it is defined this way".] We imagine an expanding universe with matter and radiation in a thermal state, perfectly homogeneous and isotropic (we will put in perturbations later), and all conserved quantum numbers set to zero (no chemical potentials). Note that asymptotic freedom makes our task much easier; at the high temperatures we are concerned with, QCD (and possible grand unified gauge interactions) are weakly coupled, allowing us to work within the framework of perturbation theory. 3.2.3 Phase transitions The high temperatures and densities characteristic of the early universe typically put matter fields into different phases than they are in at zero temperature and density, and often these phases are ones in which symmetries are restored. Consider a simple theory of a real scalar field with a Z2 symmetry . The potential at zero temperature can be written as V , T 0 2 2 2 4 . Interactions with a thermal background typically give positive 4 contributions to the potential at finite temperature: V , T V ,0 T 2 2 , where = /8 in the theory. 1 At high T, the coefficient of 2 in the effective potential, T 2 2 , will be a 2 positive number, so the minimum-energy state will be one with vanishing expectation value, 0 . The Z2 symmetry is unbroken in such a state. As the temperature declines, eventually the coefficient will be negative and there will be two lowest-energy states, with equal and opposite values of . A zero – temperature vacuum will be built upon one of these values, which are not invariant under the Z2; we therefore say the symmetry is spontaneously broken. The dynamics of the transition from unbroken to broken symmetry is described by a phase transition, which might be either first–order or second–order. A first–order transition is one in which first derivatives of the order parameter (in this case ) are discontinuous; they are 26 generally dramatic, with phases coexisting simultaneously, and proceed by nucleation of bubbles of the new phase. In a second-order transition only second derivatives are discontinuous; they are generally more gradual, without mixing of phases, and proceed by “spinodal decomposition”. 3.2.4 Topological defects Note that, post–transition, the field falls into the vacuum manifold (the set of field values with minimum energy – in our current example it’s simply two points) essentially randomly. It will fall in different directions at different spatial locations x1 and x2 separated by more than one correlation length of the field. In an ordinary FRW universe, the field cannot be correlated on scales larger than approximately H–1, as this is the distance to the particle horizon (as will be discussed later in the section on inflation). If x1 v and x2 v , then somewhere in between x1 and x2, must climb over the energy barrier to pass through zero. Where this happens there will be energy density; this is known as a topological defect (in this case a defect of codimension one, a domain wall). The argument that the existence of horizons implies the production of defects is known as the Kibble mechanism. Chapter 4 Problems in Big Bang Theory: Inflationary Solution Despite the great success of the conventional cosmology, there remain certain problems with extending the Big Bang model back to the Planck Time. These can be categorized as initial condition problems and relic problems. These problems and their solutions are discussed in the following sections. The initial condition problem embodies two interesting conceptual puzzles: flatness and isotropy. The leading solution to these problems is the inflationary universe scenario, which has become a central organizing principle of modern cosmology. 4.1 Initial Condition Problems We want to understand, what state was the universe in when it first dropped below the Planck density; whether it was just a random collection of energy. In order to give rise to a universe at all like the one we see the early universe had to have a number of very precisely tuned features. Since there is no present accepted theory that can predict about what gave rise to the initial conditions for our universe; these features are simply assumed, or introduced into the 27 theory by hand. [This is something standard Big Bang theory can’t predict. But the Grand Unified Theory (GUT) (when formulated) is supposed to give the initial conditions. This is a great triumph of the GUT theory, to predict the initial conditions of the early universe which led to its universe in its present state as we see today]. Two of these features unexplained by Big Bang are homogeneity and flatness. The early universe was nearly homogeneous, i.e. the same or uniform in all places. Our most direct measure of this uniformity comes from observing the Cosmic Microwave Background Radiation (CMBR) that was emitted when the universe was roughly 300,000 years old. The intensity of this radiation is a direct measure of how dense the universe was at that time. Looking at this background radiation coming to us from different directions, and examining its smoothness; shows that the largest density differences from one point to another were about only about one part in 100,000. If the universe really had been such less homogeneous it would not have given rise to the smooth distribution of galaxies we see filling the sky. However if it had been exactly homogeneous, then clumps of matter like galaxies and us would never have emerged at all. The big bang model offers no explanation for why the universe emerged in this nearly—but not perfectly—homogeneous state; so that matter were formed and also at the same time we see the CMBR significantly smooth. Fig. 4.1. – Infra-red map of the sky (as observed by COBE and WMAP) showing the smoothness and fluctuations in the CMBR. 28 The second initial condition has to do with something called curvature. General Relativity says that the universe can be closed, i.e. curved inward like the surface of a ball; or open, meaning curved outward like the surface of a saddle, or flat, meaning it has no curvature. Fig 4.2. – 2-Dimensinal analogue of the Possible Curvature of the Universe These different kinds of curvature cause the universe to evolve in different ways. A closed universe will eventually stop expanding and recollapse, while an open universe will tend to fly apart more and more quickly. A flat universe (i.e. with no curvature) must have a particular density value; mathematically, = 1 (where is the Density Parameter). For the Big Bang model to work the universe at the time of Planck density must have been almost precisely flat; the curvature couldn't have exceeded one part in 1059. If it were slightly more curved than this (closed) it would have recollapsed long ago and if it were slightly less so (open) it would have flown apart so quickly galaxies would never have formed. This apparent coincidence—the universe initially having exactly the curvature required to survive to later times and form galaxies—is known as the flatness problem. 29 Mathematically, the flatness problem comes from considering the Friedmann equation in a universe with matter and radiation but no vacuum energy: H 2 1 M R k2 . 3 3M P a The curvature term –k / a2 is proportional to a-2, while the energy density terms fall off faster with increasing scale factor, M a–3 and R a–4. This raises the question of why the ratio (ka–2)/(/3MP2) isn't much larger than unity, given that a has increased by a factor of perhaps 1028 since the grand unification epoch. Said another way, the point = 1 is a repulsive fixed point – any deviation from this value will grow with time, so why do we observe ~ 1 today? The isotropy problem is also called the horizon problem since it stems from the existence of particle horizons in Friedmann Robertson Walker cosmologies. Horizons exist because only a finite amount of time has passes since the Planck Time, and thus there is only a finite distance that photons can travel within the age of the universe. Consider a photon moving along a radial trajectory in a flat universe.A radial null path obeys; 0 ds 2 dt 2 a 2 dr 2 . So the comoving distance traveled by such a photon between times t1 and t2 is, t2 dt . a t t1 r To get the physical distance as it would be measured by an observer at time t1, we need to multiply r by a(t1). For a universe dominated by an energy density a–n, this becomes, r 1 2 n / 21 , a n 2 a H n/2 where the subscripts refer to some fiducial epoch (the quantity an / 2 H is a constant). The horizon problem is simply the fact that the CMB is isotropic to a high degree of precision, even though widely separated points on the last scattering surface are completely outside each other’s horizons. Choosing a0 = 1, the comoving horizon size today is approximately H0 –1, which is also the approximate comoving distance between us and the surface of last scattering (since, of the comoving distance traversed by a photon between a redshift of infinity and a redshift of zero, the amount between z = ∞ and z = 1100 is much less than the amount between z = 1100 and z = 0). Meanwhile, the comoving horizon size at the time of last scattering was approximately aCMB H0–1 ~ 10–3 H0–1, so distinct patches of the CMBR sky were causally disconnected at recombination. 30 Nevertheless, they are observed to be at the same temperature to high precision. The question then is, how did they know ahead of time to coordinate their evolution in the right way, even though they were never in causal contact? We must somehow modify the causal structure of the conventional Friedmann Robertson Walkar cosmology. 4.2 Relic Problems Another set of problems with the big bang model has to do with the production of exotic particles at high energies. According to our current physical theories we believe that in the hot, dense environment prevalent in the early universe a number of exotic particles would have been produced. The current universe is far too cold to produce the reactions required to make these particles, but if they had been produced in the early universe we would expect some of them to be still detectable today. But we do not see any such exotic particles around us. Although these particles could only have been produced in the first very small fraction of a second after Planck density we would nonetheless expect so many of them to have been produced that they would be quite abundant today. Any particle left over from the early, hot stages of the universe is called a relic particle. The big bang model predicts that we should see such relics, but we don't. 4.3 Inflation The basic idea of INFLATION has to do with the rate at which the universe is expanding. In an expanding universe the distances between galaxies are increasing, and the rate of expansion essentially refers to how long it takes for all of those distances to double. In the standard Big Bang model the universe experiences power law expansion, meaning the doubling time gets longer as the universe expands. For example, in our current power law expansion distances in the universe were roughly half their current value about 10 billion years ago, but they won't be twice their current value until about 30 billion years from now. By contrast, if the doubling time stays constant then the expansion is referred to as exponential. Inflationary theory says that before our current power law expansion there was a brief period of exponential expansion. Exponential growth can be much faster than power-law growth. In the simplest models of inflation the universe would have expanded by a factor of over ten to the ten million in a fraction of a second. There are two obvious questions raised by this idea: What mechanism would cause such an expansion to occur and what would be the consequences if it did? 31 4.4 Why Inflation Occurs In general relativity the rate at which the universe expands depends on the average energy density in the universe. If the density is high the expansion is rapid and the doubling time is small. The relation is of the form: Doubling _ time ~ 1 . Energy _ Density (This relation is for a flat universe. For an open or closed one it is slightly different, but the basic implications do not change). Here the energy density also includes the density of matter because relativity says mass is a form of energy. In general the expansion rate slows down as the universe expands because the average density decreases. If there are 1000 galaxies in some region of space and all distances double then the volume of space occupied by those galaxies will increase eight times. Since the galaxies have the same total mass as before their density will decrease by eight times. If the mass of galaxies were the only form of energy in the universe then every time distances doubled the doubling time would increase by a factor of the square root of eight. In short a universe whose energy consists entirely of mass will experience power law expansion. It turns out, however, that other forms of energy behave differently as the universe expands. For example, the energy density contained in light (which is a form of electromagnetic radiation) decreases faster than the energy density of mass. Every time the universe expands by a factor of two the energy density of light decreases not by eight times, but actually by sixteen. So if there is a lot of light energy in the universe the doubling time increases faster than it would for a universe with only mass energy. Now a problem might occur if we think conventionally with the above facts. This is that the total energy of the universe is apparently not conserved. If a region of space doubled in radius and the energy density in that region did anything other than decrease by a factor of eight then the total energy would change. The resolution of this problem is a somewhat subtle issue in General Relativity and involves a kind of gravitational energy, which we cannot directly observe, associated with the expansion of the universe. I’ll not go to get into this issue in any detail here. It suffices to say that; the total energy including gravitational potential energy is still conserved, however, the amount of energy that we can observe in the universe can change as the universe expands. This gravitational energy is not included in the energy density that 32 determines the expansion rate, and from here on when I refer to the energy density of the universe I will be referring to observable energy, whose density can change in a variety of ways as the universe expands. We have never observed a kind of energy that acts like this, but according to our current theories of physics there is one. This kind of energy is in the form of a field. It is well known that different kinds of fields react differently to the expansion of the universe. For example, the energy density of electromagnetic radiation decreases faster than that of ordinary matter. It turns out that there is one particular kind of field with the property that when its energy density is very large that density decreases very slowly as the universe expands. When its energy density decreases past a certain point it stops behaving this way and starts decreasing at the same rate as ordinary matter. Such a field is called a scalar field. So in order for inflation to have occurred it requires that some scalar field exists and at some point in the history of the universe it had a very large energy density. It's true that we have never to date observed a scalar field, but physicists believe for a variety of theoretical reasons that many of them probably do exist and that we will start to see them in our next generation of particle accelerators (which will be able to probe higher energies). However there is a second requirement of inflation. Having a scalar field with a large energy density is in many ways like having a very strong magnetic field. The idea that the early universe was filled with strong magnetic fields (actually scalar fields) must be justified by proper reasons in a successful theory. The initial conditions for our universe were set by physics that we don't know occurring above the Planck scale. So it might be that somewhere in the universe, after the Planck time, a region emerged where the largest contribution to the energy came from a high-energy scalar field. If that happened then that region (big or small in size) would inflate, almost instantly growing much larger. Very soon this inflationary region would occupy nearly 100% of the total volume. In the standard Big Bang model the entire universe started expanding uniformly at the same rate, at the same moment. In the inflationary scenario the expansion began with an exponential growth in only one small part of the universe while the rest of it either grew in a power law or started shrinking. However this one inflationary region became so big that everything we can see, or will probably ever be able to see, lies within it. Thus the universe appears to us to be uniform, even though on much larger scales it is not. 33 4.5 Some Consequences of Inflation: Solution to Flatness and Relic Problems The main consequences of inflation mostly stem from one simple fact, namely that the expansion experienced during that time was enormous. In fact, during inflation the universe expanded, i.e. all distances increased, by a factor of over 1010 million. This number comes from assuming that the energy density of the scalar field driving inflation started near the Planck density and inflation continued until it had decreased enough to no longer cause quasiexponential expansion. This number is so large that is out of our imagination and concept. By comparison, the total number of elementary particles in the observable universe is much less than 10100. In our universe the differences in energy density from one place to another smoothens due to stretching and expansion of the universe. Suppose that before inflation there was a region one foot wide in which the energy density varied smoothly such that it was twice as great on one side as the other. During inflation the width of this region would expand by perhaps ten to the ten million times, becoming so vast that we could never hope to see from one end of it to the other. Suppose the region of the universe that we observe, which is roughly thirty billion light years across, were somewhere in the middle of this inflated region. The difference in energy density we would see from one side of the observable universe to the other would be far too small for us to measure. Inflation solves the flatness problem. It turns out that in exponential expansion the curvature of the universe decreases. On an intuitive conceptual level, we think of this in terms of rapidly stretching out a piece of rubber. For a closed universe the rubber is like the surface of a sphere and for an open one it is like a saddle shape, but in either case once you stretch it out enough it looks locally flat. This geometric picture is a somewhat simplified account of how inflation solves the flatness problem. The detailed solution requires solving the equations of General Relativity, but when you do you find that after inflation the curvature of the universe will be far too small for us to measure. Inflation also solves the problem of relic particles. Our current theories of particle physics predict that in the hot, dense conditions of the early universe that existed before inflation and even during its early stages various kinds of exotic particles would be produced that we don't 34 observe. There are numerous examples of such particles, like magnetic monopoles and gravitinos, and many of them presumably were created before inflation began. However, any particles that may have existed in this inflationary region before it began inflating are reduced to a density of essentially zero. After inflation the density and temperature were too small to produce these particles. Any monopoles and gravitinos produced before inflation or during its early stages were spread out too thin for us to find, and by the time inflation ends, the density and temperature had dropped too much for them to be produced. In the example of the previous paragraph, if a one foot wide region contained 1030 monopoles before inflation, the odds of a single one ending up within 30 billion light years of us is virtually zero. The same logic applies to particles produced during the early stages of inflation. At some point during inflation the energy density would have dropped enough that such particles could no longer be produced, which explains why we don't see any of them today. All of these results of inflation are theoretically attractive as explanations of the features we observe in the universe at large scales. Probably the most important success of inflation, however, is its explanation of the origin of inhomogeneities in the universe. In the rubber sheet analogy before it was pointed out that, any fluctuations existing before inflation get stretched out so much that we can no longer detect them. However quantum mechanics predicts that some fluctuations will always be produced at small scales. During inflation (specially in the later phases) quantum mechanics causes microscopic fluctuations to be generated and inflation stretches these fluctuations out to large distances. The quantum fluctuations produced early in inflation get stretched out so much that we can't see them. As before it's as if we were standing on a hill so wide it looks flat to us. Fluctuations generated close to the end of inflation, however, produce "hills" whose width is still small enough today for us to see from one end of them to the other. The height of these hills, i.e. the difference in energy density from one place to another, is small because the quantum fluctuations that produced them represented small perturbations in the energy density. When inflationary theory was developed in the 1980s people used this theory of stretched out quantum fluctuations to predict the differences that should be seen in the microwave background radiation from one part of the universe to another. These differences are so small that they weren't detected at all until the 1990s. Fluctuations in CMBR have been detected by the spacecrafts COBE (COsmic microwave Background Explorer) in the 1990’s and by WMAP 35 (Wilkinson Microwave Anisotropy Probe) in the beginning of this century. More than just predicting the existence of these fluctuations, inflation predicts in some detail the shape that they will have. Only in the last few years of the 1990s were measurements taken with enough accuracy to test the detailed predictions of inflation regarding these fluctuations, and to date the data match these predictions perfectly. It is primarily the successful prediction of the form of the microwave background fluctuations that has caused inflation to be generally accepted today by most early universe physicists. Since these small inhomogeneities are what later gave rise to galaxies, stars, and us, all of the structure we see in the universe arose because of quantum fluctuations. Some insight into the mathematical formulation of inflation is given now. Let's consider a period in the early universe when it was dominated by vacuum energy rather than by matter or radiation. (Robertson-Walker metric is assumed, which assumes isotropy from the start). Then the flatness and horizon problems can be simultaneously solved. First, during the vacuumdominated era, / 3MP2 a0 grows rapidly with respect to – k / a2 so the universe becomes flatter with time ( is driven to unity). If this process proceeds for a sufficiently long period, after which the vacuum energy is converted into matter and radiation, the density parameter will be sufficiently close to unity, so that it will not have had a chance to noticeably change into the present era. The horizon problem, meanwhile, can be traced to the fact that the physical distance between any two comoving objects grows as the scale factor, while the physical horizon size in a matter or radiation dominated universe grows more slowly, as rhor ~ a n / 21 H 01 . This can again be solved by an early period of exponential expansion, in which the true horizon size grows to a fantastic amount, so that our horizon today is actually much larger than the naive estimate that it is equal to the Hubble radius H 01 . In fact, a truly exponential expansion is not necessary; both problems can be solved by a universe which is accelerated, a 0 . Typically we require that this accelerated period be sustained for 60 or more e-folds, which is what is needed to solve the horizon problem. It is easy to overshoot, and this much inflation generally makes the present-day universe spatially at to incredible precision. 36 4.6 Reheating Inflation occurs when in some region of space the largest contribution to the energy density comes from a high energy scalar field. In the tradition of naming particles and fields with the suffix on (proton, neutron, photon, etc) the scalar field that caused inflation is called the inflaton. During inflation the universe expands exponentially and the energy density of everything else drops to essentially zero, while the energy density of the inflaton field decreases only very slowly. Inflation ends when this field reaches a low enough energy density that it starts behaving like matter, i.e. when the universe starts experiencing power law expansion. So at the end of inflation essentially all of the energy of the universe is contained in this one, nearly homogeneous field. So when inflation ends, the energy in the inflaton potential is converted into a thermalized gas of matter and radiation, a process known as reheating. The universe after inflation is very close to being homogeneous, flat, and empty of all particles. One of the successes of inflationary theory is that it explains why we don't see relic particles like monopoles and gravitinos today (as discussed in previous section), but somehow we still have to explain how all the particles we do see got here. Since inflation reduces the particle density to virtually zero, we know the particles in the universe today must all have been produced after inflation. Once its energy density becomes small enough to allow power law inflation, the inflaton field becomes highly unstable. (This is another fact about scalar fields that can be derived from field theory). After inflation the energy in the inflaton field would have quickly decayed into other particles and fields until eventually the universe consisted mainly of long-lived forms of energy such as protons, neutrons, electrons, and electromagnetic radiation. We don't know yet how long did it take after inflation for the inflaton to decay into the particles we see today. But what we can say is that, the inflaton field must have decayed by the time of nucleosynthesis, i.e. about three minutes after the end of inflation. Assuming that to be true, the universe at the time of nucleosynthesis would have looked exactly like it does in the Big Bang model (minus relic particles like monopoles). The key difference is that inflation explains why the universe had many of the features it did at that time. This decay of the inflaton field used to be modeled as a perturbative decay of –bosons into other particles; this is a relatively inefficient process, and the temperature of the resulting thermal state cannot be very high. More recently it has been realized that nonlinear effects 37 (parametric resonance) can efficiently transfer energy from coherent oscillations of into other particles, a process referred to as “preheating”. The resulting temperature can be quite a bit higher than had been previously believed. (On the other hand, the inflaton tends to be weakly coupled, which suppresses the reheat temperature.) A proper understanding of the reheating process is of utmost importance, as it controls the production of various relics that we may or may not want in our universe. For example, one of the most beneficial aspects of inflation in the context of grand unification is that it can solve the monopole problem. Essentially, any monopoles will be inflated away, leaving a relic abundance well under the observational limits. It is therefore important that reheating does not reproduce too many monopoles. On the other hand, we do want to reheat to a sufficiently high temperature to allow for some sort of Baryogenesis (formation of baryons from elementary particles) scenario. 4.7 Inflation and Particle Physics It is nevertheless important to try to implement inflation within a believable particle physics model. Some relevant issues are listed below. A great deal of effort has gone into exploring the relationship between inflation and supersymmetry, although simultaneously satisfying the strict requirements of inflation and SUSY turns out to be a difficult task. Hybrid inflation is a kind of model which invokes two scalar fields with a waterfall potential. One field rolls slowly and is weakly coupled, the other is strongly coupled and leads to efficient reheating once the first rolls far enough. Another interesting class of models involve scalar-tensor theories and make intimate use of the conformal transformations relating these theories to conventional Einstein gravity. The need for a flat potential for the inflaton, coupled with the fact that string theory moduli can naturally have flat potentials, makes the idea of modular inflation an attractive one. Specific implementations have been studied, but we probably don't understand enough about moduli at this point to be confident of finding a compelling model. 38 4.8 Perturbations in Inflation A crucial element of inflationary scenarios is the production of density perturbations, which may be the origin of the CMBR temperature anisotropies and the large-scale structure in galaxies that we observe today (qualitatively discussed in the previous sections on Inflation). The idea behind density perturbations generated by inflation is fairly straightforward. Inflation will attenuate any ambient particle density rapidly to zero, leaving behind only the vacuum. But the vacuum state in an accelerating universe has a nonzero temperature, the Gibbons Hawking temperature, analogous to the Hawking temperature of a black hole. For a universe dominated by a potential energy V, the Gibbons Hawking temperature is given by TGH H / 2 ~ V 1 / 2 / M P . Corresponding to this temperature are fluctuations in the inflaton field at each wavenumber k, with magnitude, k TGH . Since the potential is by hypothesis nearly flat, the fluctuations in lead to small fluctuations in the energy density, V . Inflation therefore produces density perturbations on every scale. The amplitude of the perturbations is nearly equal at each wavenumber, but there will be slight deviations due to the gradual change in V as the inflaton rolls. We characterize the fluctuations in terms of their spectrum which describes scalar fluctuations in the metric. These are tied to the energy momentum distribution, and the density fluctuations produced by inflation are adiabatic (or, better, isentropic) – fluctuations in the density of all species are correlated. The fluctuations are also Gaussian, in the sense that the phases of the Fourier modes describing the fluctuations at different scales are uncorrelated. These aspects of inflationary perturbations – a nearly scale-free spectrum of adiabatic density fluctuations with a Gaussian distribution – are all consistent with current observations of the CMBR and large-scale structure, and new data scheduled to be collected over the next decades should greatly improve the precision of these tests. Not only the nearly–massless inflaton gets excited during inflation, but any nearly– massless particle does so. The other important example is the graviton, which corresponds to tensor perturbations in the metric (propagating excitations of the gravitational field). The 39 existence of tensor perturbations is a crucial prediction of inflation, which may in principle be verifiable through observations of the polarization of the CMBR. In practice, however, the induced polarization is very small, and we may never detect the tensor fluctuations even if they are there. Our current knowledge of the amplitude of the perturbations can give us important information about the energy scale of inflation. The tensor perturbations depend on V alone (not its derivatives), so observations of tensor modes yields direct knowledge of the energy scale. If the CMBR anisotropies detected by further space explorers, are due to tensor fluctuations (possible, although unlikely), we can instantly derive, 4 Vinf lation ~ 1016 GeV . (Here, the value of V being constrained is that which was responsible for creating the observed fluctuations; namely, 60 e-folds before the end of inflation.) This is remarkably reminiscent of the grand unification scale, which is very encouraging. Even in the more likely case that the perturbations observed in the CMB are scalar in nature, we can still write, 1/ 4 1/ 4 Vinf 1016 GeV , lation ~ where is the slow-roll parameter. Although we expect to be small, the 1/4 in the exponent means that the dependence on is quite weak; unless this parameter is extraordinarily tiny, it is 1/ 4 15 1016 GeV . The fact that we can have such information about such very likely that Vinf lation ~ 10 tremendous energy scales is a cause for great wonder. 4.9 Initial conditions and eternal inflation The topic of initial conditions for inflation is an especially important as, although inflation is supposed to solve the horizon problem, it is necessary to start the universe simultaneously inflating in a region larger than one horizon volume in order to achieve successful inflation. Presumably we must appeal to some sort of quantum fluctuation to get the universe (or some patch thereof) into such a state. Fortunately, inflation has the wonderful property that it is eternal. That is, once inflation begins, even if some regions cease to inflate there will always be an inflating region with increasing physical volume. This property holds in most models of inflation that we can construct. It relies on the fact that the scalar inflaton field doesn't merely follow its classical equations of motion, but undergoes quantum fluctuations, which can make it temporarily 40 roll up the potential instead of down. The regions in which this happens will have a larger potential energy, and therefore a larger expansion rate, and therefore will grow in volume in comparison to the other regions. One can argue that this process guarantees that inflation never stops once it begins. We can therefore imagine that the universe approaches a steady state (at least statistically), in which it is described by a certain fractal dimension. This means that the universe on ultra–large scales, much larger than the current Hubble radius, may be dramatically inhomogeneous and isotropic, and even raises the possibility that different post-inflationary regions may have fallen into different vacuum states and experience very different physics than we see around us. Certainly, this picture represents a dramatic alteration of the conventional view of a single Robertson Walker cosmology describing the entire universe. Of course, it should be kept in mind that the arguments in favor of eternal inflation rely on features of the interaction between quantum fluctuations and the gravitational field, which are slightly outside the realm of things we claim to fully understand. It would certainly be interesting to study eternal inflation within the context of string theory. Chapter 5 Stringy cosmology: String Theory applied to Cosmology There are many things we do not understand about both cosmology and string theory to make statements about the very early universe in string theory with any confidence. However, it is still worthwhile to speculate about different possibilities, and work towards incorporating these speculations into a more complete picture [15]. 5.1 The beginning of time As the correct place to start is not known, a simple guess can done by considering the action as the low–energy effective action in D dimensions (bosonic, NS-NS part), which is expressed in terms of the Ricci scalar, the dilaton, field strength tensor for the two-form gauge field (which is typically set to zero in papers on cosmology). The existence of the dilaton implies that the theory of gravity described by this action is a scalar-tensor model (reminiscent of BransDicke theory), not pure general relativity. Of course there are good experimental limits on scalar 41 components to the gravitational interaction, but they are only sensitive to low-mass scalars (i.e., long-range forces). With such an action (as described in last paragraph), the cosmological solutions (homogeneous but not necessarily isotropic) have a scale–factor duality symmetry, i.e. for any solution ai t , t , there is also a solution with, a i 1 , 2 ln ai . ai i Thus expanding solutions are dual to contracting solutions. (In fact this is just T-duality, and is a subgroup of a larger O(D – 1, D – 1) symmetry.) The solutions with decreasing curvature are mapped to those with increasing curvature. This feature of the low-energy string action has led to the development of the Pre-BigBang Scenario, in which the universe starts out as flat empty space, begins to contract (with increasing curvature), until reaching a stringy state of maximum curvature, and then expands (as curvature decreases) and commences standard cosmological evolution. There are various questions about the Pre-BB scenario. One is a claim that significant fine tuning is required in the initial phase, in the sense that any small amount of curvature will grow fantastically during the evolutionary process and must be extremely suppressed. Another is the role of the potential for the dilaton. We cannot set this potential to zero on the grounds that the relevant temperatures are much higher than the SUSY–breaking scale TSUSY ~ 103 GeV; supersymmetry is an example of a symmetry which is not restored at high temperatures. Indeed, almost any state breaks supersymmetry. In a thermal background, this breaking is manifested most clearly by the differing occupation numbers for bosons and fermions. More generally, the SUSY algebra Q, Q H Z , with Z a central charge, implies that Q 0 whenever H 0, except in BPS states, which feature a precise cancellation between H and Z. In the real world these are a negligible fraction of all possible states. It is not clear how SUSY breaking affects the Pre-BB idea. Perhaps more profoundly, it seems perfectly likely that the appropriate description of the high curvature stringy phase will be nothing like a smooth classical spacetime. Evidence for this comes from matrix theory, not to mention attempts to canonically quantize general relativity. There are other, non-stringy, approaches to the very beginning of the universe, and it would be interesting to know what light can be shed on them by string theory. One is quantum 42 cosmology, which by some definitions is just the study of the wave function of the universe, although in practice it has the connotation of mini-superspace techniques (drastically truncating the gravitational degrees of freedom and quantizing what is left). There is also the related idea of creation of baby universes from our own. This is in principle a conceivable scheme, as closed universes have zero total energy in general relativity. There is also the hope that string theory will offer some unique resolution to the question of cosmological (and other) singularities; studies to date have had some interesting results, but we don't know enough to understand the Big Bang singularity of the real world. 5.2 Extra Dimensions and Compactification in Cosmology Of all the features of string theory, the one with the most obvious relevance to cosmology is the existence of 6 (or 7?) extra spatial (temporal?) dimensions. The success of our traditional description of the world as a (3+1)-dimensional spacetime implies that the extra dimensions must be somehow inaccessible, and the simplest method for hiding them is Compactification: the idea that the extra dimensions describe a compact space of sufficiently small size that they can only be probed by very high energies. Of course in General Relativity (and even in string theory) spacetime is dynamical, and it would be natural to expect the compact dimensions to evolve. However, the parameters describing the size and shape of the compact dimensions show up in our low-energy world as moduli fields whose values affect the Standard Model parameters. As discussed earlier, we have good limits on any variation of these parameters in spacetime, and typically appeal to SUSY breaking to fix their expectation values. This raises all sorts of questions. Why are three dimensions allowed to be large and expanding while the others are small and essentially frozen? What is the precise origin of the moduli potentials? What was the behavior of the extra dimensions in the early universe? For the most part these are baffling questions, although there have been some provocative suggestions. One is by Brandenberger and Vafa, who attempted to understand the existence of 3 macroscopic spatial dimensions in terms of string dynamics. Consider an n–torus populated by both momentum modes and winding modes of strings. The momentum and winding modes are dual to each other under T-duality (R 1/R), and have opposite effects on the dynamics of the torus: the momentum modes tend to make it expand, and the winding modes tend to make it contract. (It's counterintuitive, but true.) We can therefore have a static universe at the self-dual 43 radius where the two effects are balanced. However, when wound strings intersect they tend to intercommute and therefore unwind. Through this process, the balance holding the torus at the self–dual radius can be upset, and the universe will begin to expand, hopefully evolving into a conventional Friedmann cosmology. But notice that in a sufficiently large number of spatial dimensions, 1-dimensional strings will generically never intersect. (Just as 0-dimensional points will generically intersect in one dimension but not in two or more dimensions.) The largest number in which they tend to intersect is three. So we can imagine a universe that begins as a tiny torus in thermal equilibrium at the self-dual point, until some winding modes happen to annihilate in some three-dimensional subspace which then begins to expand, forming our universe. Of course a scenario such as this loses some of its charm (because of increasing complexity) in a theory which has not only strings but also higher-dimensional branes. An alternate route is to imagine that we are living on a brane. That is to say, that the reason why the extra dimensions are invisible to us is not simply because they are so very small that low-energy excitations cannot probe them, but because we are confined to a threedimensional brane embedded in a higher-dimensional space. We know that we can easily construct field theories confined to branes, for example a U(N) gauge theory by stacking N coincident branes; it is not an incredible stretch to imagine that the entire Standard Model can be constructed in such a way in principle (it hasn't been done yet, because of emergence of surmounting complexities as we go on). Unfortunately, it seems impossible to entirely do away with the necessity of compactification, since there is one force, which we don't know how to confine to a brane, namely gravity (although see below). We therefore imagine a world in which the Standard Model particles are confined to a 3– brane, with gravity propagating in a higher-dimensional bulk which includes compactified extra dimensions. In D spacetime dimensions, Newton's law of gravity can be written ~ mm F D r G D 1D 22 , r ~ where G D is the D–dimensional Newton's constant with appropriate factors of 4 absorbed. If we compactify (D – 4) of the spatial dimensions on a compact manifold of volume V(D–4), the effective 4 dimensional Newton's constant is, 44 ~ G D ~ . G 4 ~ V D 4 ~ We can rewrite this in terms of what we will define as the Planck scale, M P G41/ 2 , and the ~ 1 /( D2) fundamental scale, M G D ,as M P2 ~ M D2V D4 . In conventional compactification, M ~ M P and VD4 ~ M P D4 , so this relation is satisfied in a straightforward way. But we can also satisfy it by lowering the fundamental scale and increasing the compactification volume. Imagine that the compactification manifold has n large dimensions of radius R and (D – 4 – n) dimensions of radius M 1 . Then, M R ~ P M 2 n M 1 . A scenario of this type was proposed by Horava and Witten, who suggested that the gravitational coupling could unify with the gauge couplings of GUT’s by introducing a single large extra dimension with R ~ 1015 GeV 1 . But we can go further. The lowest value we can safely imagine the fundamental scale having is M ~ 10 3 GeV ; otherwise we would have detected quantum gravity at Fermilab or CERN. This value is essentially the desired low-energy supersymmetry breaking scale (i.e. just above the electroweak scale), so it is tempting to explain the apparent hierarchy M / M EW ~ 1015 by trying to move M all the way down to 103 GeV. (Supersymmetry itself can stabilize the hierarchy, but doesn’t actually explain it.). Then we have, R ~ 10 30 n 3 GeV 1 ~ 10 30 n 17 cm. For n = 1, we have a single extra dimension of radius R ~ 1013 cm, about the distance from the Sun to the Earth. This is clearly ruled out, as such a scenario predicts that gravitational forces would fall off as r–3 for distances smaller than 1013 cm. But for n = 2 we have R ~ 10–2 cm, which is just below the limits on deviations from the inverse square law from laboratory experiments. Larger n gives smaller values of R; these are not as exciting from the point of view of having macroscopically big extra dimensions, but may actually be the most sensible from a physics standpoint. 45 So we have a picture of the world as a 3-brane with Standard Model particles restricted to it, and gravity able to propagate into a bulk with extra dimensions which are compactified but perhaps of macroscopic size, with a fundamental scale M ~ 103 GeV and the observed Planck scale simply an artifact of the large extra dimensions. (There is still something of a hierarchy problem, since R must be larger than M to get the Planck scale right.) Such scenarios are subject to all sorts of limits from astrophysics and accelerator experiments, from processes such as gravitons escaping into the bulk. (In these models gravity becomes strongly coupled near 10 3 GeV.) There are also going to be cosmological implications, although it is not precisely clear as yet what these are. Our entire popular notion of the thermal history of the universe for T > 10 3 GeV would obviously need to be discarded. Baryogenesis will presumably be modified. Inflation is a very interesting question, including the issues of inflation in the bulk vs. inflation in the boundary. Of course we don’t know what stabilizes the large extra dimensions, but then again we don't know much about moduli stabilization in conventional scenarios. There is also the interesting possibility of a 3–brane parallel to the one we live on, which only interacts with us gravitationally, and on which the dark matter resides. There are some cosmological problems, though – most clearly, the issue of why the bulk is not highly populated by light particles that one might have expected to be left over from an early high-temperature state; presumably reheating after inflation cannot be to a very high temperature in these models (although we must at the very least have Treheat > 1 MeV to preserve standard nucleosynthesis). Clearly there is a good deal of work left to do in exploring these scenarios. Randall and Sundrum found a loophole in the conventional wisdom that gravity cannot be confined to a brane. They showed that a single extra dimension could be infinitely large, but still yield an effective 4-dimensional gravity theory on the brane, if the bulk geometry were antide Sitter rather than flat. The curvature in the extra dimension can then effectively confine gravity to the vicinity of the brane. In the subsequent times a great deal of effort has gone into understanding cosmological and other ramifications of Randall-Sundrum scenarios. 5.3 The late universe The behavior of gravity and particle physics on extremely short length scales and high energies is a largely uncharted territory, and it is clear that string theory, if correct, will play an 46 important role in understanding this regime. But it is also interesting to contemplate the possibility of new physics at ultra-large length scales and low energies. You might guess that experiments in the zero-energy limit are straightforward to perform, but in fact it requires great effort to isolate yourself from unwanted noise sources in this regime. Cosmology offers a way to probe physics on the largest observable length scales in the universe, and it is natural to take advantage. We have evidence from observations that the expansion of the universe is accelerating. Explaining the observations with a positive vacuum energy V M V4 requires M V ~ 10 3 eV , which is remarkably small in comparison to M SUSY ~ 10 3 GeV 1012 eV , not to mention M Planck ~ 1018 GeV 10 27 eV . It does, of course, induce the irresistible temptation to write MV ~ 2 M SUSY . MP This is a numerological curiosity without a theory that actually predicts it, although it has the look and feel of similar relations familiar from models in which SUSY breaking is communicated from one sector to another by gravitational interactions. Another provocative relation is M V ~ e 1 / 2 M P , where is the fine-structure constant. Again, it falls somewhat short of the standards of a scientific theory, but it does suggest the possibility that the vacuum energy would be precisely zero if it were not for some small nonperturbative effect. More generally, we can classify vacuum energy as coming from one of three categories: true vacua, which are global minima of the energy density; false vacua, which are local but not global minima; and non-vacua, which is a way of expressing the idea that we have not yet reached a local minimum value of the potential energy. For example, we could posit the existence of a scalar field with a very shallow potential. From our analysis in previous sections, the field will be overdamped when V < H, and its potential energy will dominate over its kinetic energy (exactly as in inflation). Such a possibility has been termed quintessence. For a quintessence field to explain the accelerating universe, it must have an effective mass, m V H 0 ~ 10 33 eV , and a typical range of variation over cosmological timescales, 47 ~ M P ~ 1018 GeV . From a particle-physics point of view, these parameters seem somewhat contrived, to say the least. Quintessence models have the benefit of involving dynamical fields rather than a single constant, and it may be possible to take advantage of these dynamics to ameliorate the coincidence problem that ~ M today (despite the radically different time dependences of these two quantities). In addition, there may be more complicated ways to get a time-dependent vacuum energy that are also worth exploring. The moduli fields of string theory could provide potential candidates for quintessence, and the acceleration of the universe more generally provides a rare opportunity for string theory to provide an explanation of an empirical fact. We could also imagine that string theory may have more profound late-time cosmological consequences than simply providing a small vacuum energy or ultralight scalar fields. An interesting move in this direction is to explore the implications of the holographic principle for cosmology. This principle was inspired by our semiclassical expectation that the entropy of a black hole, which in traditional statistical mechanics is a measure of the number of degrees of freedom in the system, scales as the area of the event horizon rather than as the enclosed volume (as we would expect the degrees of freedom to do in a local quantum field theory). In its vaguest (and therefore most likely to be correct) form, the holographic principle proposes that a theory with gravity in n dimensions (or a state in such a theory) is equivalent in some sense to a theory without gravity in n–1 dimensions (or a state in such a theory). Making this statement more precise is an area of active investigation and controversy; see Susskind's lectures for a more complete account. The only context in which the holographic equivalence has been made at all explicit is in the AdS/CFT correspondence, where the non gravitational theory can be thought of as living on the spacelike boundary at conformal infinity of the AdS space on which the gravitational theory lives. Regrettably, we don't live in anti-de Sitter space, which corresponds to a Robertson – Walker metric with a negative cosmological constant and no matter, since our universe seems to feature both matter and a positive cosmological constant. How might the holographic principle apply to more general spacetimes, without the properties of conformal infinity unique to AdS, or for that matter without any special symmetries? A possible answer has been suggested by Bousso, building upon ideas of Fischler and Susskind. The basic idea is to the area A of the boundary of a spatial volume to the amount of entropy S passing through a certain null sheet 48 bounded by that surface. (For details of how to construct an appropriate sheet, see the original references.) Specifically, the conjecture is that S A / 4G . This is more properly an entropy bound, not a claim about holography; however, it seems to be a short step from limiting entropy (and thus the number of degrees of freedom) to claiming the existence of an underlying theory dealing directly with those degrees of freedom. Does this proposal have any consequences for cosmology? It is straightforward to check that the bound is satisfied by standard cosmological solutions7, and a classical version can even be proven to hold under certain assumptions. One optimistic hope is that holography could be responsible for the small observed value of the cosmological constant. Roughly speaking, this hope is based on the idea that there are far fewer degrees of freedom per unit volume in a holographic theory than local quantum field theory would lead us to expect, and perhaps the unwarranted inclusion of these degrees of freedom has been leading to an overestimate of the vacuum energy. It remains to be seen whether a workable implementation of this idea can capture the successes of conventional cosmology. Chapter 6 Conclusions The last several years have been a very exciting time in string theory, as we have learned a great deal about non-perturbative aspects of the theory, most impressively the dualities connecting what were thought to be different theories. They have been equally exciting in cosmology, as a wealth of new data have greatly increased our knowledge about the constituents and evolution of the universe. The two subjects still have a long way to go, however, before their respective domains of established understanding are definitively overlapping. One road towards that goal is to work diligently at those aspects of string theory and cosmology which are best understood, hoping to enlarge these regions until they someday meet. Another strategy is to leap fearlessly into the murky regions in between, hoping that our current fumbling attempts will mature into more solid ideas. Both approaches are, of course, useful and indeed necessary. Cosmology is a wonderful interplay between general relativity and particle physics, the physics of the very large and of the very small. The theory of superstrings has not yet answered some of the open questions (of detail, rather than of principle) in Big Bang cosmology, but I believe it will. In any case it is really awesome that there are such intimate links between the 49 physics at the smallest scales, set by the Planck time and the Planck length, and the physics of the cosmos as a whole with a time scale of the order of fifteen billion years and a length scale of as many light-years. There are after all some sixty orders of magnitude separating the Planck from the cosmic scales. These links have only emerged in this century, and will surely enrich physics in the next. In the meanwhile we can look forward to a lot of good tests of early universe physics in the next couple of decades. High sensitivity probes of the microwave background, searches for waves of gravity surviving from the early universe, and many other experiments are going to give us excellent tests, not only of inflation, but of our understanding of the universe in general. References [1] http://superstringtheory.com/ [The Official String Theory Website] [2] http://www.pbs.org/wgbh/nova/elegant/ [The Elegant Universe] [3] http://www.theory.caltech.edu/people/jhs/strings/ [The Second Superstring Revolution] [4] http://www.sukidog.com/jpierre/strings/ [Superstrings Homapage] [5] http://turing.wins.uva.nl/~rhd/string_theory.html [6] http://www.lassp.cornell.edu/GraduateAdmissions/greene/greene.html [7] http://theory.tifr.res.in/~mukhi/Physics/string.html [8] http://www.damtp.cam.ac.uk/user/gr/public/qg_ss.html [9] http://tena4.vub.ac.be/beyondstringtheory/ [10] http://fy.chalmers.se/tp/Egroup/stringtheory.html [11] http://www.nuclecu.unam.mx/~alberto/physics/string.html [12] http://www.crystalinks.com/superstrings.html [13] http://monopole.ph.qmul.ac.uk/~jmc/EUni/Strings.html [14] http://www.hyper-mind.com/hypermind//universe/content/gsst.htm [15] arXiv:hep-th/0011110 – TASI Lectures: Cosmology for String Theorists