Topological defects and the magnetic monopole problem

Picture growing a huge crystal from a bucket of molten quartz. You seed it with a small crystal. The seed sets a preferred orientation of the crystal axes; material freezing nearby latches on to it, and the crystal grows very large with all its axes aligned. (The lecturer's example: the huge, very pure optics of the LIGO gravitational-wave detectors, which must be made extremely carefully, or the result is "the world's prettiest paperweight". [He describes them as 100-kilogram crystals; Advanced LIGO's test masses are about 40 kg of fused silica, a glass rather than a grown crystal.])

The Higgs field behaves similarly. At some event 1 the universe cools enough for the Higgs field to break the symmetry, settling — say — into $h_1 = 1$, $h_2 = i$. Regions nearby find a preferred value already set, and it is energetically favourable to fall into line, so a growing region of spacetime takes on that value. But at some event 2, so far from event 1 that the two are out of causal contact, the universe also cools enough. The universe is not in perfect thermal equilibrium, so some places reach the transition a little earlier than others. At event 2 the field might settle into $h_1 = i$, $h_2 = -1$: the same values multiplied by $i$, so the same magnitude and the same energy — perfectly allowed. Its neighbours copy that value. (A student asks whether the symmetry could break differently in different places; that, the lecturer says, is exactly the point.)

Eventually the domains grow into each other. What exactly happens is not known: it depends on details of symmetry breaking and of the grand unified theory, at temperatures around $10^{16}$ GeV, far beyond what experiments reach. But physicists can "parametrise their ignorance": for plausible choices of the theory's parameters, domains with different Higgs values generically produce topological defects.

Across the boundary the field tries to match itself smoothly, rotating from one value to the other through intermediate values. In doing so it may be pushed away from the minimum of $V$, trapping energy at the boundary. The form this takes depends on how the Standard Model is packed into the grand unified theory — there are many ways — and comes in three kinds, each one dimension lower than the last:

1. Domain walls — two-dimensional sheets of mismatched field, possibly gigaparsecs away from us, with anomalous behaviour around them. They would leave signs in the CMB; strong limits now disfavour grand unified theories that produce them. Only some theories predict them.
2. Cosmic strings — one-dimensional lines of trapped field energy, possibly gigaparsecs long. They gravitate strongly, and were once proposed as the seeds of cosmic structure; that is now strongly disfavoured, and if they exist they contribute little to the mass of the universe. Again, only some theories predict them. (To a student puzzled by a "one-dimensional wall": in such theories the field manages to match smoothly almost everywhere, leaving energy trapped only along a line.)
3. Point defects — the field matches smoothly everywhere except at a point, leaving a little knot in the Higgs field. This knot must carry magnetic charge: it is a magnetic monopole. Monopoles are predicted by all grand unified theories.

Only the point defects carry magnetic charge, but all of these defects gravitate, which is how observers try to limit them. Cosmic strings could act as gravitational lenses, and some years before the lecture a pair of identical-looking quasars at just the right separation caused excitement — until better telescopes showed the two objects were unrelated.

At the knot the Higgs field passes through zero. Far away in every direction the field sits at the minimum of $V$, yet its value must change across the region; a theorem — presented in Guth's notes with a figure — then requires a point where $h = 0$. There the potential energy density is huge, so the defect is a massive, particle-like nugget.

Estimate its energy as for electric and magnetic fields: the energy inside a sphere of radius $R$ includes the gradient energy of the Higgs field,

Close to the defect the field is complicated, but far away its gradient falls only as $1/r$, so the energy grows in proportion to $R$ and diverges. The mistake is to leave out the Higgs field's strong coupling to the gauge fields, including electric and magnetic fields. Including it (the calculation is beyond an undergraduate class; Guth's notes give references), the integral converges only if the nugget carries a magnetic charge, whose monopolar magnetic field supplies the cancelling terms. So if the interactions were unified at high energy and then froze out, defects in the Higgs field are inevitably magnetic monopoles.

The full calculation gives the magnetic charge, in Gaussian units:

where $\alpha \approx 1/137$ is the fine-structure constant. Two monopoles attract with a force $68.5^2$ times that between two electrons at the same distance. The mass is

— a single particle as heavy as about $10^{18}$ protons, around a microgram: heavy enough to register on a scale.

How many form? Domains with different Higgs values must start out of causal contact, so their size — the correlation length $\xi$ — is at most about the horizon distance. With roughly one monopole per domain — an estimate usually credited to Tom Kibble —

Setting $\xi$ equal to the horizon distance gives a lower bound on the number of monopoles, and so on their mass density, $\rho_M = m_M n_M$, compared with the critical density. The lecturer, out of time (another professor arrived to claim the room), quotes the result: monopoles alone would give about $10^{20}$ times the critical density, and "the universe is approximately two years old". [The reasoning is left to Guth and the homework. Presumably the second statement means that a universe that dense would have expanded and cooled to today's temperature in only a year or two: for a matter-dominated universe $t = 2/(3H)$, and $\rho = 10^{20}\rho_c$ means $H = 10^{10}H_0$, giving about one year.] Either way, the conventional Big Bang plus grand unification predicts a universe utterly unlike ours. This is the magnetic monopole problem.

• Symmetry breaking starts independently in causally disconnected regions; each picks a random Higgs value and its neighbours follow, forming domains.
• Where domains meet, the field may be trapped away from its minimum, forming topological defects: domain walls (some grand unified theories; strongly limited by observation), cosmic strings (some grand unified theories; disfavoured as the seeds of structure) and point defects (all grand unified theories).
• A point defect has $h = 0$ at its core; its energy is finite only if it carries magnetic charge, $g = e/2\alpha \approx 68.5e$. Its mass is about $10^{18}$ GeV, around a microgram.
• One monopole per horizon volume at the GUT transition gives a monopole density vastly larger than the critical density: the magnetic monopole problem.