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Original question: So theory predicts that the monople number density at time of GUT was $10^{82}$ per cubic meter, due to inflation it's diluted and today it's about $10^{-16}$ per cubic Mpc.

This is lower only by 2 orders of magnitude than today's photon number density (~$10^8$ per cubic meter), but still much higher than today's baryon number density. And even back at GUT, the monople number density was higher than baryon number density, and they both will dilute as $1/a^3$.

So how come today we see baryons everywhere, but the monopole is hard to detect because of its "low number density" due to inflation?

Update: I made a mistake in my comparison calculation for today. Also, while trying to extrapolate today's baryon and matter energy density back to GUT for a comparison at that time, I used the ratio between today's temperature and GUT temperature since $T \propto 1/a$, but I failed to realize the temperature after the inflation went back to before the inflation, but $a$ has changed a lot, so by using temperature ratio directly to scale back will not work.

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    $\begingroup$ (a) Do you have a source for the monopole density before and after inflation? Surely those numbers are model dependent? (b) According to wikipedia, the lack of magnetic monopole detections "places an upper limit on the number of monopoles of about one monopole per $10^{29}$ nucleons." So either the model you are referring to is ruled out, or it's some other kind of weakly interacting monopole that wouldn't be detected by the searches described on wikipedia. $\endgroup$
    – Andrew
    Dec 12, 2022 at 1:10
  • $\begingroup$ The source is a textbook, but I just realized I made a mistake in my calculation. Will update the original post. Thanks. $\endgroup$
    – ABC
    Dec 12, 2022 at 17:10
  • $\begingroup$ So... $10^{-16} {\rm Mpc}^{-3} = 3 \times 10^{-84}\ {\rm m}^{-3}$... does that resolve your question? $\endgroup$
    – Andrew
    Dec 12, 2022 at 21:35
  • $\begingroup$ Yeah that's definitely part of the problem. But the other mistake was when extrapolating backwards, which further convinced me of the first mistake. I've updated the post. $\endgroup$
    – ABC
    Dec 12, 2022 at 21:42
  • $\begingroup$ @Andrew, actually I'm confused again. Were there matter particles and photons before inflation? If so they got super-diluted during inflation, but then at the end of inflation the "re-heating" created a bunch of new matter particles and photons so that their density went back to pre-inflation time? Is this correct? Thanks. $\endgroup$
    – ABC
    Dec 12, 2022 at 22:48

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