Paradoxes of the Expansion 157Combining this mass with the number densities in Equations (6.76) and (7.15) the
density parameter of monopoles becomes
훺M=푁M푚M
휌c≃ 2. 8 × 1017. (7.17)
This is in flagrant conflict with the value of the dark energy density parameter
훺휆= 1 −훺m= 0 .72 to which we sall come back later. Thus, yet another paradox. Such
a universe would be closed and its maximal lifetime would be only a fraction of the
age of the present Universe, of the order of
푡max= 휋
2 퐻 0√
훺M
≃40 yr. (7.18)Monopoles have other curious properties as well. Unlike the leptons and quarks,
which appear to be pointlike down to the smallest distances measured (10−^19 m), the
monopoles have an internal structure. All their mass is concentrated within a core
of about 10−^30 m, with the consequence that the temperature in the core is of GUT
scale or more. Outside that core there is a layer populated by the X leptoquark vector
bosons, and outside that at about 10−^17 m there is a shell of W and Z bosons.
The monopoles are so heavy that they should accumulate in the center of stars
where they may collide with protons. Some protons may then occasionally penetrate
in to the GUT shell and collide with a virtual leptoquark, which transforms a d quark
into a lepton according to the reaction
d+Xvirtual→e+. (7.19)Thus monopoles would destroy hadronic matter at a rate much higher than their nat-
ural decay rate. This would catalyze a faster disappearance of baryonic matter and
yield a different timescale for the Universe.
Flatness Problem. Recall that in a spatially flat Einstein–de Sitter universe the cur-
vature parameter푘vanishes and the density parameter is훺=1. This is obvious from
Equation (5.11), where푘and훺are related by
훺− 1 =푘푐^2
푎̇^2
.
The current value of the total density parameter훺 0 is of order unity. This does not
seem remarkable until one considers the extraordinary fine-tuning required: a value
of훺 0 close to, but not exactly, unity today implies that훺 0 (푡)at earlier times must
have been close to unity with incredible precision. During the radiation era the energy
density휀ris proportional to푎−^4. It then follows from Equation (5.4) that
푎̇^2 ∝푎−^2. (7.20)At GUT time, the linear scale was some 10^27 times smaller than today, and since most
of this change occurred during the radiation era
훺− 1 ∝푎^2 ≃ 10 −^54. (7.21)