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According to this Wikipedia article on cosmic inflation:

The magnetic monopole problem, sometimes called the exotic-relics problem, says that if the early universe were very hot, a large number of very heavy, stable magnetic monopoles would have been produced.

Inflation solves this "problem" by diluting them as the universe expands (they are created before inflation and diluted during it).

But we have no evidence that such exotic particles even exist! They are predicted by grand unified theories which are in turn scantly supported by evidence.

Also, monopoles require a somewhat awkward modification to the magnetic potential.

So why is the "monopole problem" such a problem in the first place?

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    $\begingroup$ It's partly due to historical factors. When Guth was formulating inflation, grand unified theories were the hot new thing, and the presence of monopoles was essentially the first failed prediction they made. These days, when cosmologists introduce inflation, they rarely mention the monopole problem. $\endgroup$
    – knzhou
    Commented Sep 16, 2022 at 6:30
  • $\begingroup$ @knzhou not sure which cosmologists you are talking about, but the monopole problem (and the relic problem in general) is regularly mentioned in introductory lectures, textbooks, and presentations in my experience. $\endgroup$
    – Kosm
    Commented Sep 16, 2022 at 6:55

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I had this conversation with my cosmology lecturer many years ago. We had just covered the traditional three "problems" that inflation was supposed to solve (the horizon, flatness, and monopole problems). I argued that:

  • The flatness problem isn't a problem because perhaps the Universe is just flat. I didn't see (and still don't) see a strong reason why we should just say that the universe cannot have $\rho = \rho_c$.
  • The monopole problem isn't a problem because maybe monopoles don't exist (same as your question).

Strictly speaking, there is no firm proof against either of these assertions. After all, the universe is always going to have some density, and it could be $\rho_c$. Similarly, it could be that GUTs are wrong and magnetic monopoles don't exist.

However: the horizon problem remains a problem, which is what my lecturer emphasized. If you disbelieve GUTs, then the monopole problem isn't a problem for you either (it's only a problem for people who believe GUTs), but you still need some mechanism to explain the horizon problem.

Phrased alternatively, you could say that the horizon problem is on much firmer footing than the flatness & monopole problems.

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But we have no evidence that such exotic particles even exist!

Exactly. We don't see them, despite the fact that they are predicted by a lot of beyond-the-Standard-Model theories. This means there is a very good chance they might exist, in which case it would be a problem. It is not a critical problem like horizon problem for example, or cosmological constant problem, but it is never-the-less desirable to have possible monopoles diluted. More so, once you include all the other possible defects, like domain walls and cosmic strings.

Also, monopoles require a somewhat awkward modification to the magnetic potential.

See this answer for the distinction between Dirac and 't Hooft-Polyakov monopoles. The latter is/are proper (finite energy, regular) solutions to various field theories with broken symmetries, e.g. GUTs.

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