IT CAN often feel like our days are governed by numbers in the most mundane ways. They tell you that you are going to be late for work, how much interest you are paying on your mortgage and how old you will be when you finally pay it off. But numbers can also be exhilarating. For me, as a cosmologist, the most fantastic numbers have meaning far beyond the scope of our humdrum daily lives. Some are indescribably small, others incomprehensibly large. What unites them is that they can reveal the secrets of how the universe works, and – most excitingly – the deep and meaningful mysteries that continue to confound physicists today. Here are five of the best.

## 10^{-120} : Why is the vacuum energy so small?

To appreciate how small this number is, imagine shrinking a person to the size of a proton, and then again by the same ratio, and again, a total of eight times. The miniature human left would now be 10^{-120} times its original size. This is many orders of magnitude smaller than the Planck length, a fundamental unit of distance at which space, in the classical sense, ceases to make sense.

In nature, exponentially small or large ratios between sizes are rare, and when they happen, physicists are hardwired to seek an explanation. Of course, there are no humans 10^{-120} times the size of a typical person, but this minuscule number does appear when we compare observations to our best and most well-tested theory of the universe: quantum field theory. More specifically, it crops up when we start thinking about vacuum energy – the energy of empty space and the driving force of cosmic expansion. This means understanding this tiny ratio is one of the greatest outstanding problems in fundamental physics.

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Our universe is expanding, the space between distant galaxies growing larger over time, and this process is speeding up. It is generally agreed that this acceleration could be the result of the underlying energy of empty space – energy it gets from virtual particles popping in and out of the vacuum, called vacuum energy. Virtual particles aren’t particles like an electron or a proton that you can see in an experiment. They are a convenient way to imagine ambient quantum fields that push and pull on every particle in the universe, and even on space itself. We know these fields exist because we see their effects in experiments. They should also give the universe a vacuum energy.

Problems start when we calculate how much vacuum energy there is. Based on our current understanding of quantum field theory, the energy density of the vacuum should be enormous. A coffee cup of empty space should have enough energy to blow up every planet in the observable universe more than a trillion trillion trillion trillion times. But this isn’t the case – and for that, we should be grateful, because a universe with this much vacuum energy would be bent into oblivion by gravity. According to the gentle acceleration of the universe’s expansion that we measure with our telescopes, the vacuum energy is actually hundreds of orders of magnitude less than our models predict. In other words, it is 10^{-120} times the value predicted by quantum field theory.

So what gives? A possible answer is that there is something wrong with quantum field theory. Unlikely, since this theory is the most accurately tested in all of physics. Alternatively, we might think there is something missing from Albert Einstein’s theory of gravity – after all, it is only through gravity that we could ever feel the true vacuum energy. The trouble is that Einstein’s theory is also very robust: change it too much and you can unleash all manner of instabilities and mathematical catastrophes.

Some physicists have argued that, out of all the possible universes, one with a tiny value for the vacuum energy is necessary for complex life to have evolved. Such an explanation requires physics to conjure up a multiverse, a vast, traversable landscape of different universes with different particles, properties and laws. At least one universe in the multiverse should look like our own. However, we still don’t have a complete description of a multiverse in any of our fundamental theories.

Can experiments help us find an answer? Up until now, not really. The only experimental probe we have had has been through the accelerated expansion of the universe. But, along with colleagues in Nottingham, UK, as well as in London and L’Aquila, Italy, I am now studying the gravitational properties of vacuum energy directly.

It turns out that the core of a neutron star, the densest type of star in the universe, could be so dense that it triggers a change in the vacuum energy. This could leave an imprint on the size and shape of the neutron star, something we could try to detect with gravitational waves, ripples in space and time that were first detected in 2015. The hope is that through probing some of the most exotic objects in existence, we can finally understand one of the smallest numbers in the universe.

## 0.0000000000000001: Why is the Higgs boson so light?

This next number, relative to the previous one, is colossal. And yet to us, it is still imperceptibly tiny. If you scaled a human down by this amount, they would be a million times smaller than the width of an atom and 10 times smaller than a proton. Just like our first number, this tiny figure expresses a ratio between what we actually see and what our best theories predict. In this case, it is related to the Higgs boson.

When the discovery of this particle was announced on 4 July 2012, it was said to be the crowning glory of the standard model of particle physics, which accurately describes the microscopic dance of all the visible matter in our universe. The Higgs was the final piece of this particle jigsaw, explaining why fundamental particles, including the Higgs boson, have mass. But we knew that something was up with some of the numbers involved. In fact, there was something up with the mass of the Higgs boson itself.

Like all particles, the Higgs is pushed and pulled by the ambient quantum fields I mentioned earlier – the ones that permeate all of space. I think of these fields like a quantum fluid that surrounds the particle, from which it can never escape. This fluid drags on the Higgs, increasing its inertia, or, in other words, increasing its mass. When we calculate how much mass the Higgs should have, we find that it ought to weigh a few micrograms, about the same as a tiny wasp known as a fairyfly. In reality, when it has been measured again and again, the Higgs boson is 1000 trillion times lighter than this. Put another way, it has 0.0000000000000001 times the mass we expected.

The unbearable lightness of the Higgs is troubling for particle physicists. Theorists have come up with a few potential explanations, from extra dimensions to doubling up on the number of particles in some theories, drastically changing the properties of the ambient quantum fluid as we push to higher and higher energies. They have even tried breaking the Higgs up into tiny bits – at microscopic scales, it is no longer the Higgs that interacts with the quantum fields, but its constituent particles, and they could do so very differently.

So far, experiments at CERN, the particle physics facility near Geneva, Switzerland, have shown no evidence to support any of these ideas. However, after a three-year hiatus, the Large Hadron Collider (LHC) has just been turned on for a third run, probing collisions with higher energies and greater precision than before. This means CERN may soon finally get some evidence for one of the hypotheses and discover the reason behind the unexpectedly light nature of the Higgs.

## 10^{80}: Why are there so many particles?

Our third number is huge by any standards. If you were to scale a human up by this amount, they would be 10 billion billion billion billion billion billion times bigger than the observable universe. 10^{80} is actually the number of ordinary particles there are in this universe, so we can expect it to be big. What doesn’t make sense to a physicist like me, however, is why there are any particles at all.

About 5 per cent of the universe is composed of normal matter – the stuff that makes up you and I and all the things we can see, like stars and galaxies. But it also contains antimatter. Antimatter is rare, so rare that it is generally regarded as the most expensive substance in the universe, costing the equivalent of more than $60 trillion per gram to produce in particle colliders. This strange stuff is like matter in a mirror. In fact, if you wanted to change a particle to its antiparticle, you would have to flip it in all directions of space, as well as its electric charge. This hypothetical transition is known as a charge-parity flip, and it takes a negatively charged electron to a positive charged positron. Under the standard model of particle physics, these “flips” are what is known as a symmetry.

According to this model, because of this symmetry, the big bang should have created an equal amount of matter and antimatter. But when matter meets antimatter, the two annihilate. In that case, they would have annihilated each other to leave nothing but a bath of boring radiation. But as the fact of our existence demonstrates, that didn’t happen – and we don’t know why.

One option is that the missing antimatter is out there somewhere, hiding from us. But we should be able to see evidence if that were the case. We might see gamma rays being given off where antimatter meets matter, for example, and we don’t. The reality is we need a mechanism that breaks the symmetries in our physical theories that connect matter to antimatter in exactly equal quantities. This would require an extension of the standard model.

The good news is that ideas for how matter came to dominate the universe can be probed by experiments, ranging from collisions at the LHC to gravitational wave observations. One suggestion is that the Higgs can be broken up into smaller bits. They would allow for new particles that break the symmetry between matter and antimatter, and whose interactions with other fundamental particles could be detected in the third LHC run.

These new particles could also have affected how the Higgs started handing out mass to other particles in the primordial soup of the early universe. The precise details of this could be imprinted on an ancient background of gravitational waves, similar to the cosmic background of microwave radiation we see left over from the big bang. The hope is that this gravitational wave background will be discovered by future space-based detectors, such as the Laser Interferometer Space Antenna (LISA), which could launch in the 2030s, revealing clues as to how and why ordinary matter like us managed to avoid annihilation.

## 10^{10^}^{68}: Does your doppelgänger exist?

Here we have a number so ludicrously large that it is tricky to compare with any other numbers: 10 to the power of 10 to the power of 68 corresponds to 1 followed by 100 million trillion trillion trillion trillion trillion zeroes.

It is a number I have named the doppelgängion, because it relates to the chances of there being another person like you, a doppelgänger, somewhere else in our universe. To be clear, we aren’t talking about doppelgängers in other universes disconnected from our own, as imagined in the many worlds interpretation of quantum mechanics, or any other multiverse theory. Instead, we are asking about our very own universe and the possibility that your doppelgänger is out there – same nose, same eyes, even the same thoughts. It is an idea that goes back to physicist Max Tegmark at the Massachusetts Institute of Technology.

It might seem absurd. But ultimately it comes down to the probability that there is another bit of space, the same size as the one you occupy, in exactly the same quantum state, encoding the likely arrangements of all the particles that make up you.

The thing is, there are only a finite number of quantum states that can describe the volume of space you take up. To compute how many, us cosmologists need to use one of the most remarkable things we have learned about gravity in the past 30 years: it behaves like a hologram, encoding quantum information on the two-dimensional surface of a volume, rather than in the interior. To count the quantum states, you need to imagine the surface that surrounds you, and break it up into the smallest possible building blocks, with dimensions the size of the Planck length. Add up all the different patterns you can generate and you arrive at a doppelgängion.

What this all means is that you are one of a doppelgängion possibilities. So am I, so is a cow and so is a cubic metre of empty space. To appreciate the implications of this, we need to imagine a universe that is far greater than a doppelgängion metres in radius, containing far more than a doppelgängion cubic metres of space. Although the chances of any one cubic metre containing your doppelgänger are very low, if the universe is big enough, you can overwhelm the odds.

Could it be large enough? The observable universe extends to about 47 billion light years, but it doesn’t stop there. Measurements of radiation left over from the big bang – the cosmic microwave background – tell us that it is at least 250 times larger than what we can see. A further clue to the true size of the cosmos could lie in that leftover radiation, which reveals that our region of the universe underwent a burst of accelerated expansion very early on, helping it grow big in a very short space of time. These bursts of inflation could have happened anywhere in the universe and at any moment, turning small pockets of space into something more gargantuan.

This allows us to imagine extremely distant worlds – too distant to ever visit, but that are part of the same universe as ours. With the entire universe constantly inflating in one bit of space or another, the idea of a gigantic universe becomes a little less fanciful. If the universe is large enough, your exact copy will be out there, reading an article about fantastic numbers.

## Infinity: What happens inside black holes?

The biggest number of all. The number that goes on forever. In maths, there are multiple infinities: there is the infinity that counts the whole numbers and the squares, but also higher infinities that count larger sets, like the set of real numbers between 0 and 1. In physics, too, infinities take us to the limits of our knowledge.

If we venture into the deepest innards of a black hole, say, or rewind to the moment the universe was born, we are confronted with what we call a “singularity”. This is a region where the gravitational field becomes infinitely strong. In Albert Einstein’s general theory of relativity, gravity is understood to be the warping of space-time, with planets following curved paths in the distorted space-time around the sun because it is the most efficient way to traverse the geometry that exists there. At a singularity, that warping in space-time blows up to infinity, and Einstein’s law can no longer describe what is going on.

To grasp these singularities, we need a theory capable of describing the strongest and most violent gravitational fields and all the matter that interacts with them. In other words, we need a theory that explains all aspects of the universe.

In my opinion, the most compelling candidate for this is string theory. In it, all the particles of nature are understood to be made up of tiny vibrating one-dimensional entities known as strings. Unlike some of its rivals, we can perform reliable calculations in string theory, and make the infinities disappear. The transition from particles to strings means we no longer need to worry about particles interacting with one another over infinitesimally small distances.

Those tiny distances were responsible for the infinities that broke our equations, so now they can be vanquished. Quite how this plays out for the singularities at the core of a black hole, or at the moment of creation, we don’t yet know. But once we figure out how to do the maths for them, we should be able to understand the creation of our universe.

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