Astroparticle Physics

248 12 InflationΩ at the Planck timeGoing further back in time makes the problem moreacute. Suppose that the universe was radiation dominatedfor times less than t mr going all the way back to the Plancktime, t Pl ≈ 10 −43 s. Then one should take R ∼ t 1/2 ,whichmeans Ṙ ∼ t −1/2 and therefore 1Ω − 1 ∼ kR 2 ∼ kt . (12.6)flatness problem≈ 10 −59 . (12.7)Thatis,tobeabletofindΩ of order unity today, the modelrequires it to be within 10 −59 of unity at the Planck time.As with the horizon problem, the issue is not one ofa prediction that stands in contradiction with observation.There is nothing to prevent Ω from being arbitrarily closeto unity at early times, but within the context of the cosmo-logical model that has been described so far, it could havejust as easily had some other value. And if other values areapriorijust as likely, then it seems ridiculous to believe thatnature would pick Ω ‘by chance’ to begin so close to unity.One feels that there must be some reason why Ω came outthe way it did.fine-tuning of ΩUsing this dependence for times earlier than t mr , one findsfor Ω − 1 at the Planck time relative to the value now:( )Ω(t Pl ) − 1Ω(t 0 ) − 1 = R(tPl ) 2R(t mr )R(t mr )) R(t 0 ) = t ( ) 2/3Pl tmrt mr t 010 −43 ( )s50 000 a 2/3≈50 000 a × 3.2 × 10 7 s/a 1.4 × 10 10 a12.3 The Monopole Problem“From the theoretical point of view onewould think that monopoles should exist,because of the prettiness of the mathematics.”Paul Adrian Maurice Diracphase transitionsin the early universeIn order to understand the final ‘problem’ with the modelthat has been presented up to now, one needs to recallsomething about phase transitions in the early universe. InChap. 9 it was remarked that a sort of phase transition took1 The Boltzmann constant and the curvature parameter are traditionallydenoted with the same letter k. From the context itshould always be clear which parameter is meant.