2.1. Local and global monopoles and domain walls

Monopoles are yet another example of stable topological defects. Their formation stems from the fact that the vacuum expectation value of the symmetry breaking Higgs field has random orientations (<^a> pointing in different directions in group space) on scales greater than the horizon. One expects therefore to have a probability of order unity that a monopole configuration will result after the phase transition (cf. the Kibble mechanism). Thus, about one monopole per Hubble volume should arise and we have for the number density n_monop ~ 1 / H^-3 ~ T_c⁶ / m_P³, where T_c is the critical temperature and m_P is Planck mass, when the transition occurs. We also know the entropy density at this temperature, s ~ T_c³, and so the monopole to entropy ratio is n_monop / s 100 (T_c / m_P)³. In the absence of non-adiabatic processes after monopole creation this constant ratio determines their present abundance. For the typical value T_c ~ 10¹⁴ GeV we have n_monop / s ~ 10^-13. This estimate leads to a present _monop h² 10¹¹, for the superheavy monopoles m_monop 10¹⁶ GeV that are created ⁽⁶⁾. This value contradicts standard cosmology and the presently most attractive way out seems to be to allow for an early period of inflation: the massive entropy production will hence lead to an exponential decrease of the initial n_monop / s ratio, yielding _monop consistent with observations. ⁽⁷⁾ In summary, the broad-brush picture one has in mind is that of a mechanism that could solve the monopole problem by `weeping' these unwanted relics out of our sight, to scales much bigger than the one that will eventually become our present horizon today.

Note that these arguments do not apply for global monopoles as these (in the absence of gauge fields) possess long-range forces that lead to a decrease of their number in comoving coordinates. The large attractive force between global monopoles and antimonopoles leads to a high annihilation probability and hence monopole over-production does not take place. Simulations performed by Bennett & Rhie [1990] showed that global monopole evolution rapidly settles into a scale invariant regime with only a few monopoles per horizon volume at all times.

Given that global monopoles do not represent a danger for cosmology one may proceed in studying their observable consequences. The gravitational fields of global monopoles may lead to matter clustering and CMB anisotropies. Given an average number of monopoles per horizon of ~ 4, Bennett & Rhie [1990] estimate a scale invariant spectrum of fluctuations ( / )_H ~ 30 G ² at horizon crossing ⁽⁸⁾. In a subsequent paper they simulate the large-scale CMB anisotropies and, upon normalization with COBE-DMR, they get roughly G ² ~ 6 × 10^-7 in agreement with a GUT energy scale [Bennett & Rhie, 1993]. However, as we will see in the CMB sections below, current estimates for the angular power spectrum of global defects do not match the most recent observations, their main problem being the lack of power on the degree angular scale once the spectrum is normalized to COBE on large scales.

Let us concentrate now on domain walls, and briefly try to show why they are not welcome in any cosmological context (at least in the simple version we here consider - there is always room for more complicated (and contrived) models). If the symmetry breaking pattern is appropriate at least one domain wall per horizon volume will be formed. The mass per unit surface of these two-dimensional objects is given by ~ ^{1/2 3}, where as usual is the coupling constant in the symmetry breaking potential for the Higgs field. Domain walls are generally horizon-sized and therefore their mass is given by ~ ^{1/2 3} H^-2. This implies a mass energy density roughly given by _DW ~ ³ t^-1 and we may readily see now how the problem arises: the critical density goes as _crit ~ t^-2 which implies _DW(t) ~ ( / m_P)² t. Taking a typical GUT value for we get _DW(t ~ 10^-35sec) ~ 1 already at the time of the phase transition. It is not hard to imagine that today this will be at variance with observations; in fact we get _DW(t ~ 10¹⁸sec) ~ 10⁵². This indicates that models where domain walls are produced are tightly constrained, and the general feeling is that it is best to avoid them altogether [see Kolb & Turner, 1990 for further details; see also Dvali et al., 1998, Pogosian & Vachaspati,2000 ⁽⁹⁾ and Alexander et al., 1999 for an alternative solution].

⁶ These are the actual figures for a gauge SU(5) GUT second-order phase transition. Preskill [1979] has shown that in this case monopole antimonopole annihilation is not effective to reduce their abundance. Guth & Weinberg [1983] did the case for a first-order phase transition and drew qualitatively similar conclusions regarding the excess of monopoles. Back.

⁷ The inflationary expansion reaches an end in the so-called reheating process, when the enormous vacuum energy driving inflation is transferred to coherent oscillations of the inflaton field. These oscillations will in turn be damped by the creation of light particles (e.g., via preheating) whose final fate is to thermalise and reheat the universe. Back.

⁸ The spectrum of density fluctuations on smaller scales has also been computed. They normalize the spectrum at 8 h^-1 Mpc and agreement with observations lead them to assume that galaxies are clustered more strongly than the overall mass density, this implying a `biasing' of a few [see Bennett, Rhie & Weinberg, 1993 for details]. Back.

⁹ Animations of monopoles colliding with domain walls can be found in `LEP' page at http://theory.ic.ac.uk/~LEP/figures.html Back.