Why String Theory?

by Joseph Conlon

  While the interactions of moduli are extremely weak, they are not entirely zero. What do these new interactions do? One thing they do is to provide ways for heavy moduli to decay to light particles. Generally, heavy particles always tend to decay into light particles if they can. The Higgs boson decays to bottom quarks. The bottom quark decays to other, lighter quarks. The muon decays to an electron and a muon neutrino. Heavy particles are unstable – they live for a short period of time before falling apart into lighter and more stable daughter particles. The series of decays as heavier particles decay to less heavy ones, which again decay, which again decay, generates a particle decay chain. In these decay chains, weight loss is rapid and irreversible. Once particle decay has occurred, it cannot be undone except by re-colliding particles at high energies using accelerators.

  A decay is one form of interaction, and the speed of decay depends on particle mass and interaction strength. Heavier particles decay quicker, and more strongly interacting particles decay quicker. While a particle decaying via the strong force has a typical lifetime of one million-billion-billionth of a second, one decaying through the weak force lives much longer, and in some special cases the lifetime can reach human timescales. A free neutron lives on average for fifteen minutes before decaying via the weak force to a proton, an electron and an anti-neutrino. This is atypically long – a more normal value for a weak interaction lifetime is provided by the decay of the muon, with a lifetime of around a microsecond.

  The consequence of all this is that if a particle only barely interacts, it can survive for exceedingly long times. Such particles are very hard to create due to their almost non-existent interactions. However, once created, they are also almost impossible to dispose of. As interactions of gravitational strength are the weakest of all, particles that only interact in this way have the longest lifetimes of all. Once they have, with great difficulty, been created, there is nothing one can do other than wait for them to decay.

  Objects that interact weakly live a long time and objects that interact strongly do not: this fact is not confined to particle physics. The safest way to store fragile objets d’art, whether they be jewellery, tapestries or manuscripts, is by hiding them away so that they lie unjostled and untouched by dirty hands and bacteria-laden breathing. Many a librarian responsible for ancient books fears nothing more than the books being read, and grubby fingers poring over them – hence the rush to digitise ancient manuscripts.

  To look for observational consequences of moduli though, we do need a way of making them. The harder it is to make a particle, the more extreme the conditions that are required for an efficient particle factory.

  As we saw in chapter 3, one of the great truths about the early universe is that it was both extremely hot and extremely dense. Our universe at its start was a more rambunctious and callithumpian place than at any time subsequently. It was also more energetic than even the most fiery locations, such as the interior of supernovae, present in the universe today. The gargantuan amounts of energy present in the early universe allowed for the production of particles then that could never be produced at more irenic later epochs.

  In particular, this includes moduli – which despite their exceedingly weak interactions could be made when the universe was very young.4

  Moduli are hard to make, and hard to get rid of. Once moduli have been created, they survive for what is, by the standards of the early universe, an aeon. Aeons are relative, as here even a microsecond can count as a long time. Timescales in the early universe are measured, without blushing, in picoseconds or nanoseconds. As the universe moves through these ages, it grows. The universe at an age of a nanosecond is by volume over a million times larger than it was at the age of a picosecond.

  Moduli live for a long time – what does this mean? This is best understood through ratios. What was the universe like when the moduli were made? What was it like when the moduli decayed? The quantity to compare is the age of the universe at each epoch, and the ratio between these times can be enormous, easily reaching a factor of ten followed by twenty-four zeroes. This ratio in ages is the same as that between the universe today and the universe at only one microsecond old. At the epoch of moduli decay, the universe is vastly older than at the epoch of moduli creation. Such decaying moduli are coelacanths of the early universe, surviving relics of a distant past.

  There is another important aspect of moduli in the early universe. Their Methuselan properties also imply that, by the time moduli eventually decay, they are expected to dominate the energy density of the universe. This last sentence is another way of saying that if you look at what makes up the universe at the time at which the moduli decay, it is expected to consist of moduli, moduli, and moduli – and almost nothing else. As moduli linger while all else decays, the proportion of the energy of the universe that is in the form of moduli is expected to grow and grow, until it is almost one hundred percent by the time that moduli finally decay.5

  In this respect, there is another parallel between the behaviour of moduli and the behaviour of a struck bell. When a bell is struck, the timbre of the sound changes with time. The higher-pitched notes die out more rapidly, while the deep low frequency notes last longer. While the initial strike of the clapper generates a superposition of notes of many frequencies, a couple of seconds later only the most long-lasting low frequency notes can still be heard.

  On this analogy, particles with strong interactions correspond to the high-pitched notes, while the moduli correspond to the deep bass frequencies. As with the deep bass notes of the bell, in the early universe particles that interact strongly soon decay and disappear, while the particles with extremely weak interactions are able to survive for a long time. The universe then goes through a phase where it is dominated by the longest lasting particles – the moduli.

  The summary of the above is that moduli were produced in the extraordinarily early universe, before disappearing through decays to other particles in the merely very early universe. In between – they dominated. Despite this, our first thought is that this does not matter very much. The time at which moduli particles decayed and disappeared was long before there were atoms, let alone stars or galaxies or people. Like the dinosaurs, moduli had an epoch of importance, but have now all disappeared.

  Of course, we actually know rather a lot about dinosaurs. As with the dinosaurs, we can still look for fossil traces of the existence of moduli. The time at which moduli decayed is also late enough that what happened then can be connected onto the physics that we do observe. The equations that governed the universe at this period are equations that we understand. They do not require quantum gravity. They do not require extra dimensions. They do not require spacetime foam. They are the standard and familiar equations of particle physics and general relativity that have been tested repeatedly against experiment, and that we know are correct. If something happened then, we know how to evolve it forward to the present day.

  Let us pause for breath. I have argued above that moduli are important because we expect the universe to go through an epoch where it is filled with moduli and only moduli. The moduli are gone now, but for a time they were everything, and we can hope to find legacies from this era. The history of the universe subsequent to this point is determined by what actually happens when the moduli decay, and it is to this we now turn our attention.

  What happens when a modulus decays? In a decay, the mass-energy of the modulus particle is converted into the energy of the daughter particles produced. The heavier the modulus particle, the more energetic the daughter particles are. Suppose a modulus decayed into two photons, so that the decay results in two photons heading out in opposite directions. As the mass of the modulus particle increases, these photons appear respectively as radio waves, visible light, X-rays or gamma rays.

  For our purposes, there are two distinct possible ways for the decay to occur. The modulus can either decay to particles from the visible universe, or it can decay to particles from the invisible univ
erse. The invisible universe is the dark universe, which is by definition dark: it consists of those parts of the universe that we cannot see, and which do not easily interact with photons.

  What makes up the dark universe? First, it includes dark matter – stuff moving at speeds far lower than that of light, and that is heavy and can be weighed. As discussed in chapter 4, we know that dark matter is present, but we have little firm idea what it actually is. The quest to identify dark matter is one of the largest areas of research in particle physics and cosmology.

  The dark universe also includes dark energy. This represents the energy of space itself. It is present in even the emptiest vacuum, and its magnitude is proportional purely to the volume of space. Dark energy is also hard to understand, but for a different reason. With dark energy, we have a good idea what it is, and observations are entirely consistent with this idea. Our problem is that we have no understanding of why it is there. Quantitative attempts to calculate the amount of dark energy give values that overshoot the data by a factor of at least ten to the power of sixty, and no good resolution of this problem has been found.

  My focus here is on an additional possible element of the dark universe. This is called dark radiation, and it refers to stuff that is both dark and relativistic, moving at or extremely close to the speed of light. The word ‘radiation’ is used in analogy to light. Conventional radiation consists of a massless particle, the photon, moving at the speed of light. The visible universe is full of photons, which we could call ‘visible’ radiation. Dark radiation is the analogue, except with invisible particles.

  This is not actually so exotic. We know that at least one particle in the universe, the photon, travels at the speed of light. Once you accept the existence of dark matter, you accept that there are still unknown aspects to the universe. The history of particle physics, with its repeated discovery of new particles, should render the wise sceptical of any suggestion that all the particles in the universe have now been discovered. The notion that the dark universe contains both relativistic and non-relativistic particles should not be shocking, as it would just mirror the properties of the visible universe.

  If dark radiation exists, how would we ever know? First, relativistic energy density – ‘radiation’ – behaves differently from non-relativistic energy density – ‘matter’. For a fixed total amount of energy, the universe expands more rapidly if that energy is in the form of matter than if it is in the form of radiation. This statement is not at all obvious, but its truth follows from Einstein’s equations of general relativity.

  In an expanding universe, the density of energy in matter also dilutes more slowly than the density of energy in radiation. This can be understood by imagining a fixed number of particles, either relativistic or non-relativistic, within a box whose volume grows as the space inside it is stretched out. As the box grows, the number of particles per unit volume decreases. The volume of the universe is bigger while the total number of particles remains the same. In both cases, the number density of particles per unit volume decreases in exact proportion to the increase in volume.

  The mass, and therefore the energy, of non-relativistic particles remains the same as the universe expands. However, as the volume increases the energies of relativistic particles decrease. In an expanding universe, every part of the universe is running away from every other part. In the same way that an ambulance siren decreases in pitch as an ambulance accelerates away from you, the energies of relativistic particles decrease as the universe expands – and so the overall fraction of energy in radiation decreases.

  As the universe expands and ages, matter becomes more and more important and radiation less and less important. This is the same logic that explains why moduli would have come to dominate the universe. Moduli are long-lived matter, and so they win out over any radiation that was present.

  It follows that the effects of radiation become ever more important as one goes back in time. To look for the effects of dark radiation, we need to look at the young universe. As noted in chapter 3, there is one visible relic from this era. This is the cosmic microwave background, the diffuse glow of microwave light left over from the early universe. If dark radiation were present, the detailed properties of the cosmic microwave background are changed slightly. For a fixed amount of energy pie, if some is given to additional invisible radiation, there is less of the pie left for visible radiation. This leads to small modifications in the rate at which the universe expands, leading to small modifications in the properties of the microwave background.6

  The theory is clear. We know what the implications of dark radiation are. Current experiments are not, however, quite sensitive enough to tell us whether dark radiation is present or not. Hints have come, grown and shrunk. The quality of experimental precision is however improving rapidly, and a decisive verdict is expected over the next decade.

  Let us recall why we introduced dark radiation. We have argued that, if moduli exist, the universe will go through an epoch where its energy is almost entirely in the form of massive moduli particles. The next step in the story of the universe occurs when the moduli decay. As we shall soon see, these decays can lead to dark radiation.

  When moduli decay, their mass-energy is given to daughter particles. As we have seen, there are two basic possibilities. Part of the time, the moduli will decay into the particles of the Standard Model: the electron, the quarks, the W bosons, the Higgs boson and others. When this happens, all the mass-energy of the modulus gets turned into a hot and energetic plasma of quarks, gluons and photons, as if in a superheated oven. Gluons scatter off quarks which emit W bosons which decay into quarks which scatter off quarks … the energy of the original particle is dissipated across hundreds and thousands of particles scattering off one another and exchanging their energy to form a hot thermal bath. This hot thermal bath is the Hot Big Bang. As the universe expands, this hot bath gradually cools down, until it becomes today the afterglow of the Big Bang that is the cosmic microwave background.

  While this is the light side of the decays, our interest is in the dark side. I introduced moduli as extra-dimensional modes of gravity and a necessary consequence of a fundamental theory with more spatial dimensions than three. Gravity is universal. It loves everyone and everything, and no form of energy is exempt from it. The consequence of this is that while moduli will decay into the particles of the Standard Model, they will also, part of the time, decay into the dark sector where interactions with all familiar particles are exceedingly weak.

  The democracy of gravitational interactions implies that moduli interact with the hidden sector just as strongly, or weakly, as they do with the visible sector. Moduli decay to hidden particles at a similar rate to their decays to Standard Model particles, and in decaying to the hidden sector they can produce dark radiation.

  Our focus here is on these invisible decays to the ghosts of the dark sector, the light particles with ephemeral interactions with our visible world. Energy released into these particles does not heat the universe. Instead, such particles continue through the universe as they were formed, travelling at the speed of light but neither scattering nor interacting.

  Are these particles neutrinos? Neutrinos, famously, barely interact with matter. They are the most feebly interacting of all known particles, and they can pass through thousands of kilometres of solid rock as if it were empty space. Indeed, experimental searches for dark radiation are often loosely called searches for additional neutrino species. This reflects the fact that at the time the cosmic microwave background was formed, when the universe was four hundred thousand years old, neutrinos are dark, relativistic particles to which the universe is transparent. Neutrinos behave precisely in the same way that dark radiation does, and so when searching for the effects of additional dark relativistic particles in the cosmic microwave background, it is necessary first to subtract the known contribution from neutrinos.

  Despite all this, in the very beginning the universe was not transparent to
neutrinos. When less than one second old, the universe was so dense that it was opaque to neutrinos. During this period, neutrinos were like any other particle. When a neutrino was produced, it would interact with the thermal bath and deposit its energy back into this hot soup. This is because neutrino interactions, while weak and tenuous, are still vastly stronger than purely gravitational interactions. While a neutron star is dense enough to stop neutrinos, a ‘gravitational neutrino’ would pass straight through such a star in the way neutrinos pass straight through the earth.

  In the next section I will discuss one specific candidate particle for ‘gravitational neutrinos’. The broader general feature, however, is that the universe is transparent to such particles from the moment of their birth. Once they are produced by decaying moduli, they travel freely throughout the universe, passing through whatever they encounter on the way. The fact that the interaction strength is so weak means that as the universe expands such particles continue to travel through the universe, even today, at the speed of light. They are the Flying Dutchmen of the universe; unable to stop and unable to interact, they are condemned to travel as dark radiation permanently and forever.

  Why is this interesting? From a theorist’s perspective, what is striking about dark radiation is that, given relatively few assumptions, its existence is very hard to avoid. This comes from its origin in a moduli-dominated epoch of the universe, when the energy density of the universe lay almost entirely in the form of moduli. These moduli are bridges between the visible sector of Standard Model particles and the gravitational world of ephemeral weakly coupled massless particles. Moduli at the time of their decay are like a pencil balanced on its tip at the edge of a chasm – sometimes the pencil lands safely, and sometimes it disappears into darkness. The mathematics of quantum mechanics guarantees that, given enough moduli, some will always end up decaying to the invisible shadow world of dark radiation.