USAAAO — 2024 First Round

The problem of magnetic monopoles — that is, the apparent absence of magnetic monopoles in the universe — arises from the fact that some modern physical theories (such as string theory) predict that the number density of magnetic monopoles at the time of their creation was nM(tGUT) ≈ 10⁸² m⁻³. The inflation theory provides a possible solution to this problem, as the exponential expansion of the primordial universe would “dilute” the monopoles. Calculate, approximately, how much the universe expanded during the inflationary period so that today the probability of a single magnetic monopole existing in the observational universe is 1%. Consider that the beginning of inflation coincides with the time of the creation of magnetic monopoles, and that the universe is flat (Euclidean geometry can be used on large scales). You can use that the diameter of the observational universe is 28.5 Gpc, and that between the end of inflation and today, the universe has linearly expanded by a factor of 5 × 10²⁶.