In this chapter we summarize measurements of the spectrum and composition of cosmic rays with energies above 100 TeV and the implications for sources. We noted in Chapter 12 that the knee of the cosmic ray spectrum may coincide with the upper limit of shock acceleration by supernova remnants (SNR). We also noted that, whether the knee reflects the maximum energy of a class of accelerators or a rigidity-dependent change in propagation, the composition should change systematically from light to heavy in the knee region.

Particles with energies greater than 3×1018 eV are generally assumed to be from sources outside of the MilkyWay because they show no sign of the anisotropy that would be expected if they came from sources in the Galactic plane. The hardening of the spectrum at the ankle around this energy is often interpreted as the transition from Galactic to extragalactic cosmic rays [638]. If the knee reflects the upper limit of acceleration by SNR with a maximum energy for protons of ≈ 1015 eV, then the major nuclear groups would follow the Peters cycle in rigidity culminating with Emax(Fe) ≈ 3 × 1016 eV. Interpretation of the ankle as the transition to extragalactic cosmic rays would then require a second kind of Galactic source capable of accelerating protons to ≈ 1017 eV and iron to ≈ 3 × 1018 eV. Hillas [639] suggests this possibility and identifies the contribution of unknown origin as Population B. On the other hand, the population of extragalactic cosmic rays may extend down to lower energy, A 1017 eV, in which case an alternate explanation for the ankle is needed. Berezinsky et al. [308] proposed an extragalactic spectrum dominated by protons to explain the ankle as a pileup effect just below 3 × 1018 eV, the energy above which losses due to e+e− pair production by protons in the CMB become more important than adiabatic expansion (see Figure 10.2).

These conflicting possibilities require a detailed understanding of the spectrum and composition as a function of energy for their resolution.

For 49 out of the 50 centuries of recorded history, the idea of extra dimensions of space was the province of the mystic, the crank and the trickster. For what could be more obvious than the fact that there are three dimensions – length, breadth and height – as every schoolchild knows? Take a brick, for example, which is so reassuringly solid, and seems so uncompromisingly three-dimensional. Yet, for the last century or so, modern atomic science has been telling us that the solid brick is 99.99999999999% empty space. What remains is some little bits of matter, flying around eternally, and held together by electromagnetic pulls and pushes of incredible strength. This may seem bizarre and unreal, but that's how it is.

We have learnt, therefore, that ‘common sense’ – or rather, the evidence of our five senses – is not a trustworthy guide to the world when we try to perceive it at length scales much below the size of an average human. Once we are trained to have an open mind in his way, there is no problem at all in conceiving of more dimensions of space – they just have to be curled up into tiny loops, or spheres or higher dimensional compact objects. If the size of these is below the resolution of the human eye (or any instruments which aid the human eye), these extra dimensions will be invisible, just as atoms and molecules were at the time they were conceived of.

Most scientists encounter curled-up dimensions without realising it, when they study the quantum physics of solids. There, one can set up a wave function of an electron in a solid by invoking a ‘periodic boundary condition’, which is another way of saying that the co-ordinates are curled up. In fact, a three-dimensional solid is actually treated in the textbooks as a 3-torus, which is a complicated topological object that cannot be drawn on paper. If we can get useful and experimentally verifiable results by assuming a block of copper, for example, to be a 3-torus, surely it is not too much of a flight of fancy to assume that the selfsame block also has other dimensions, similarly curled-up, which are too small to be seen?

Introduction

We have seen in the previous chapters that the ADD model of large extra dimensions provides us with a rich phenomenology which is currently being explored in high-energy experiments. However, soon after the ADD model was first studied, it was realised that, in the absence of a mechanism that will stabilise the large radii of compactification, the large disparity between the radii of compactification and the four-dimensional Planck radius shows up as a residual hierarchy. This was historically the main motivation to look for models without large extra dimensions and it was this search that led to the development of the warped extra-dimensional model of Randall and Sundrum [26]. In the meantime, however, a study of Calabi-Yau flux compactifications in Type II B string theory with all moduli stabilised has shown how stable, large-volume compactifications naturally arise in this context. While it seems difficult to generate millimetresized extra dimensions in this scenario, it is easily possible to generate a whole range of large compactification radii and also have physics related to a variety of scales from the Planck scale to the TeV scale [192, 193, 194].

This demonstration of how large extra dimensions can arise in generic string models has taken the sting out of the original motivation for warped extra dimensional models. But there are other reasons, equally compelling, to study warped extra dimensional models. For one, it provides a completely novel way of addressing the hierarchy problem. Moreover, it allows one to think of curvature in the extra dimensions. The ADD model assumes flat extra dimensions. The assumption that is being made is that the presence of the branes does not significantly distort the flat extra-dimensional metric. This is not a good assumption if brane tensions are not negligible. In fact, the possibility of curved extra dimensions brings in a bonus: the cosmological constant in the bulk can be adjusted so as to get an effective vanishing 4-d cosmological constant. The bulk cosmological constant allows one to compensate for the effect of the back-reaction of gravity to the branes and, in effect, from the view-point of an observer on the brane the four-dimensional universe will appear static and flat.

The last waltz

In this book, we have presented a pedagogical review of some of the main developments in the study of brane worlds and extra dimensions. We have focussed on three main ideas: Large Extra Dimensions, Universal Extra Dimensions and Warped Extra Dimensions, and have dealt with these topics in considerable detail. We have also reviewed the theoretical ideas that are a necessary background to delve deep into these topics. However, from the time in the late 1990s when these new ideas of brane worlds and extra dimensions took shape, the subject has arborised in ways which would have been difficult to imagine then. We have concentrated on what in our view constitute the central ideas that would be of interest to students and researchers in high-energy physics but even as we focus on this flow, which we take to be the river we are acutely aware that there are several tributaries that we have not paid attention to. In fact, some of these tributaries have gained enough breadth and depth to be considered rivers in their own right. But we had to limit our task and focus on what we had set out to write about, else this river analogy would have been inadequate and we would have had to contend with an ocean. Nevertheless, it is possibly a good idea to leave the reader with an idea of some of these other topics that have not made it to the centre stage of our presentation. We do precisely this in this final chapter: a random walk, if you like, in the extra-dimensional world. In each case, we present a couple of references which deal with these subjects in much greater detail.

Neutrinos in extra dimensions

The last 15 years have seen the accumulation of a wealth of experimental data on neutrinos and there is now definite evidence for oscillations between the different neutrino generations showing that the three active neutrinos of the Standard Model are massive. More precisely, the experiments give information on the masssplittings, Δm2 between the neutrinos, and these are typically of the order of 10−3 eV2. Typically, one would bring in a right-handed fermion NR which is, as discussed in Chapter 2, a Standard Model singlet.

Historical prelude

As we have already discussed in Chapter 1, it was with the idea of understanding strong interactions that string theory was proposed. The motivation for this, as we discussed, came from the attempt to derive the dynamics of strong interactions from the properties of the S-matrix, such as analyticity, unitarity and crossing symmetry. The amplitude proposed by Veneziano [63], which encrypted these properties of the S-matrix, led to the study of what came to be known as the dual model of hadrons. The realisation that the dynamics of the dual model could be obtained from relativistic strings led to a systematic study of string theory. Nambu [64] and, independently, Goto [65] proposed the form of the classical action for a string and the study of the theory as a model for hadrons was actively pursued. Eventually, this string picture of hadrons had to be abandoned because the spectrum of states of a closed string always contained a massless, spin-2 hadron which had never been observed. The theory also predicted tachyons and there were issues related to the consistency of the theory whose resolution required the theory to be a higher-dimensional theory.

Of course, the graviton perfectly fitted the description of a massless spin-2 particle and it was this observation that made Scherk and Schwarz [24] suggest a reinterpretation of string theory not as a theory of hadrons but as a theory of all interactions, which naturally incorporates gravity. The string scale was taken to be proportional to the Planck scale rather than hadronic scales so that the corrections to classical gravity coming from string theory would be at appropriate short-distance scales. Once it was taken seriously as a candidate for a unified theory of all interactions including gravity then the other problems plaguing the theory could be addressed. The early phase of string theory is reviewed in Refs. [66, 67]. Starting from the mid 1980s to the present, expositions of the developments in string theory have been presented in several excellent books and reviews. For a very incomplete, albeit select, list, see Refs. [68, 69, 70, 71, 72, 73].

The motivation

In the previous chapter, we presented a detailed exposition of the original version of the Randall-Sundrum model, also known as the RS1 model. In this next chapter, we will discuss variations and extensions of the original model. The primary difference is that the gauge fields and the fermions of the Standard Model are no longer confined to the TeV brane and so we first need to discuss how to deal with these fields when they propagate in the extra dimension. This is what we will take up first.

Let us first try to understand the motivation to go beyond the minimal Randall–Sundrum or RS1 model. In the RS1 model, the exponential warp factor involving the fifth-dimensional co-ordinate results in a conformal rescaling of the fields localised on the TeV brane and it thus provides a solution to the gaugehierarchy problem. As explained in the previous chapter, this solution is complete once the stablisation of the model, using the Goldberger-Wise mechanism, is achieved.

Trying to address the hierarchy problem in this fashion, however, forces us to reckon with attendant issues. One issue is that since any mass scale associated with the fields localised on the TeV brane is rescaled towards the infrared by e−kL, it also affects mass scales which need to be large and not get warped down to TeV scales. Such mass scales, for example, are those which suppress higherdimensional operators associated with proton decay, flavour-changing neutral currents (FCNCs) or neutrino masses. In the usual four-dimensional theories, these operators are suppressed by a large mass scale M which is typically in the range of 1012 − 1015 GeV related to the scale of grand unification or left–right symmetry-breaking. But in the RS1 model these get redshifted to roughly a TeV, which then has obvious disastrous consequences.

One way out of this problem could be using discrete symmetries to forbid these operators but this is an unwieldy procedure. A much simpler solution is bulk. The point is that to address the hierarchy problem only the Higgs field needs to be localised on the TeV brane – it is not necessary to have the fermions and gauge bosons so localised.

Space: a historical view

Of all the aspects of the physical world, the mysteries regarding the nature of space and time are the most recondite. Questions related to the nature of space and time have engaged philosophers, mathematicians and physicists alike across histories and cultures. It appears that even in ancient cultures some thinkers concerned themselves with these questions but it was probably with the Greeks that an attempt at a systematic, physical understanding of space and time is discerned. This is not surprising given that the Greek interest in both geometry and mechanics was motivated by physical rather than metaphysical considerations. The famous antinomies of Zeno tell us that even pre-Socratic philosophers had devoted much thought to the understanding of space, time and motion. But somewhere in the period between Zeno and Aristotle, earlier meditations on these issues crystallised and were put down more concretely.

Between the Greek atomists and Aristotle, then, we have been provided two enduring conceptions of space which are broad enough to accomodate the whole gamut of ideas about space that have emerged over the centuries. For Aristotle, space did not have an existence independent of the material world but rather was a relational property of material bodies. Independent of matter, space had no existence: there was no conception of a vacuum. Any statement about space that is made is really a statement about a relation between different material entities. In contrast to Aristotle, and in fact preceding him, there existed another conception of space due to the atomists, prominent among whom were Leucippus and Democritus. The atoms, which were unchanging and eternal, gave rise to the variety and the change perceived in the phenomenal world by coming together in different ways as they moved about in the void. The void or empty space had an existence independent of the atoms that moved in them and was as fundamental as the atoms. These two views, relationalism (of Aristotle) and substantivalism (of the atomists), have been paradigmatic for theories of space that have been developed for over two millenia. While for the substantivalists, geometry has a significance of its own, for the relationalists it is something that can only be inferred from the properties of material bodies.