*To see past the blur of the big bang, you need to wrestle with some infuriating infinities*

The subtleties of quantum theory are for the most part of little practical consequence for cosmology. Gravity drives the expansion of the Universe, the formation of galaxies, and the way matter condenses into planets. And gravity as depicted by Einstein's general theory of relativity is classical physics par excellence. Relativity assumes that mass and energy are infinitely divisible, and that the geometry of space and time is smooth and continuous down to the smallest scales. But there's one moment when quantum theory can't be ignored - at the very beginning of the Universe, the big bang itself.

This is because of a problem with the classical explanation for the big bang. Flowing from general relativity is the notion of the big bang as a "singularity" - in other words an instant of infinitesimal duration, occupying an infinitely small volume of space, but containing an infinite density of energy.

Singularities are alarming because they tend to make nonsense of the mathematical equations used to describe them. The good news is that quantum theory seems not to allow a singularity such as the big bang. The bad news is that no one knows what it would put there instead.

Quantum theory tends to rub out classical singularities whenever they show up. Classical theory pictures the elementary particles, for instance, as objects with mass and charge packed into a dimensionless mathematical point. Quantum theory, however, gives them a certain size, because of the way quantum particles can also be seen as waves, with an associated wavelength.

One remarkable consequence of this was Stephen Hawking's breakthrough in 1974 when he realised that black holes are not absolutely black. According to relativity, a black hole is a region of space whose gravity is so strong that nothing, not even light, can escape. It occurred to Hawking, however, that the uncertainty principle makes it harder to say whether a quantum particle was inside or outside a black hole. He showed, loosely speaking, that a black hole could leak particles with a quantum wavelength similar to the hole's radius.

Practically speaking Hawking radiation doesn't make a great deal of difference to most astrophysicists' calculations: a black hole with the mass of the Sun would be about a kilometre across, and it would leak particles with a temperature of less than a millionth of a degree. What Hawking's discovery did for physics was to turn the absolute barrier relativity had erected round a black hole into a fuzzy, porous quantum one.

In the same way, quantum theory must replace the sharply defined singularity of the big bang with a fuzzy blob. Wind the cosmological clock backwards, and there comes a moment when classical theory demands that particles be confined in a space smaller than the uncertainty principle permits. Quantum theory draws a modesty-preserving screen across what would otherwise be a naked singularity. To look behind that screen, and to understand how the Universe might emerge from it, requires a theory that marries quantum principles with classical general relativity. And so far, no one has figured out a way of doing that.

The problem is infinities. In the 1940s, physicists ran into a seemingly intractable barrier while struggling to develop a theory of quantum electrodynamics - the quantum version of Maxwell's classical theory of electromagnetism. Classically speaking, figuring out the force between two charged particles is easy: you put in the size of the two charges, the distance between the particles, plug them into a simple formula and out pops the answer. But the quantum version of this calculation proved irksome, because the space between the two charges isn't quite empty anymore. You simply can't have a true vacuum, because energy and particles can come and go within the fuzziness imposed by the uncertainty principle.

This buzz of activity has some important consequences. The electric field near an electron, for example, affects the particles seething in the quantum vacuum, attracting the positive ones and repelling the negative ones. In effect, the vacuum "shields" the electron's charge. Unfortunately, shielding seems to require an infinite correction to the charge of an electron. Three physicists - Julian Schwinger, Sin-itiro Tomonaga and Richard Feynman - found a way around this problem in 1948 by what was either a stroke of brilliance or a cheap subterfuge. You start with an infinite "bare" charge, make an infinite correction to it so that you end up with an electron charge of finite value, and then proceed as if all was well. A single infinite subtraction, they showed, was enough to fix the problem so that all further calculations were straightforward. This was called "renormalisation".

But it simply won't work for gravity. Try to construct a quantum theory of gravity and the infinities thrown up by the quantum vacuum won't lie down and die. Because energy and mass are equivalent, the energy of gravitational attraction itself generates gravity. Infinities thus pop up at every stage of a quantum gravitational calculation and can't be cancelled out. One subtraction won't do the trick. You have to keep getting rid of new infinities every step of the way and in the end you can't get a sensible answer. How this mismatch between quantum theory and gravity will be resolved no one yet knows. One popular idea these days is superstring theory, which proposes that fundamentally there are no particles. Instead, there are tiny wiggly loops of energy, the equivalent of mathematical lines rather than mathematical points. The particles we observe - quarks, photons, electrons and all the rest - represent different oscillations of the loops of string.

Substituting loops for particles in this way gets rid of any need for renormalisation. It's the point-like nature of the electron that makes quantum electrodynamics so vexing. Replace the point with oscillations of a line, and the infinities don't occur in the first place. What's more, superstring theory contains a loopy oscillation that looks like a "graviton," a hypothetical quantum particle that bears the same relationship to the gravitational field as the photon does to the electromagnetic field.

But fairly major difficulties remain. For starters, the world of superstrings has 10 dimensions, and the only way the theory can explain the four-dimensional world we live in (three space dimensions, plus time), is to wrap up six of the ten dimensions so tightly that we don't see them. The trouble is that the dimensions won't roll up of their own accord. It takes a nudge from a theoretical physicist.

Moreover, though superstring theory is based on particles and interactions, general relativity is fundamentally a theory of geometry and topology. Ultimately, a quantum theory of gravity must tackle this mismatch head-on. Is space-time in fact divided up into little quantum units that connect according to their own laws but produce the seemingly continuous dimensions we are familiar with? What would space and time mean at this elementary, discontinuous level? And where, to return to the original question, did the big bang come from? The frenzied quantum vacuum presumably includes gravitational activity too - perhaps space and time forming and re-forming like foam on a stormy sea, or perhaps with the quantum counterpart of black holes popping in and out of existence too quickly to be caught. In that case, might our whole Universe ultimately be some perfectly ordinary little quantum fluctuation that, for no reason except pure chance, grew a little bigger than the rest and, so to speak, ran away with itself? To all of these questions, no one yet knows the answers.