Astroparticle Physics

12.8 Solution to the Monopole Problem 259This is the largest region where one would expect to find thesame temperature, since any region further away would notbe in causal contact. Now, during inflation, a region of sized expands in proportion to the scale factor R by a factor e N ,where the number of e foldings is from (12.30) N ≈ 100.After inflation ends, the region expands following R ∼ t 1/2up to the time of matter–radiation equality (50 000 a) andthen in proportion to t 2/3 from then until the present, assumingmatter domination. So, the current size of the regionwhich would have been in causal contact before inflation is( ) 1/2 ( ) 2/3d(t 0 ) = d(t i ) e N tmr t0≈ 10 38 m .t f t mr(12.32)This distance can be compared to the size of the currentHubble distance, c/H 0 ≈ 10 26 m. So, the currently visibleuniverse, including the entire surface of last scattering, fitseasily into the much larger region which one can expect tobe at the same temperature. With inflation, it is not true thatopposite directions of the sky were never in causal contact.So, the very high degree of isotropy of the CMB can be understood.region in causal contactHubble distance12.8 Solution to the Monopole Problem“If you can’t find them, dilute them.”AnonymousThe solution to the monopole problem is equally straightforward.One simply has to arrange for the monopoles tobe produced before or during the inflationary period. Thisarises naturally in models where inflation is related to theHiggs fields of a Grand Unified Theory and works, of course,equally well if inflation takes place after the GUT scale. Themonopole density is then reduced by the inflationary expansion,leaving it with so few monopoles that one would notexpect to see any of them.To see this in numbers, let us suppose the monopoles areformed at a critical time t c = 10 −39 s. Suppose, as in theprevious example, the start and end times for inflation aret i = 10 −38 sandt f = 10 −36 s, and let us assume an expansionrate during inflation of H = 1/t i .ThisgivesN ≈ 100e foldings of exponential expansion. The volume containinga given number of monopoles increases in proportion tomonopole densitydilution during inflation