Evolution of Quadrupole Moments of Total Energy Density
Corresponding to the energy density plots,
these graphs show the quadrupole moments Q_xx, Q_yy, and Q_zz
of the total energy
density for collapse of the various simulations. The top two graphs
are those models which do not support either strings or monopoles,
while the bottom two graphs have either one or both. The graphs on
the left are simulations which began with spherically symmetric
energy densities, while the graphs on the right are those which
started in an oblate configuration (obtained from the symmetric
configuration by compression along one axis). Each plot shows
either a single straight line or two curves; the straight
line indicates that the collapse was
spherically symmetric, while in the nonsymmetric collapses two of the
quadrupole moments are always precisely equal, since the collapse has
one preferred axis.
Several features of these plots are immediately evident. The models
without strings or monopoles behave "reasonably": the configurations
with no initial quadrupoles stay that way (i.e. they collapse symmetrically),
and the deformed configurations evolve in a regular fashion. The
models with either strings or monopoles are less regular. Three of
the four develop nonzero quadrupoles even when the initial quadrupole
was zero; these are the theories in which the initial configurations are
not truly spherically symmetric, and are therefore unstable to
nonspherical collapse. (This is associated with
defect nucleation.)
The one theory with strings and monopoles in
which the initial configuration is truly spherical, SO(5) breaking
to SO(3)xSO(2)xZ_2, does not develop nonzero
quadrupoles from initially spherical data, but the evolution of the
oblate configuration resembles the other models with defects more than
the defect-free models. Note also that the SO(3)/SO(2) and SO(4)/U(2)
theories behave identically. This makes sense, since the vacuum manifolds
(S^2 and RP^2 = S^2/Z_2, respectively) have locally
identical geometries, and differ only by a discrete identification. Also,
the SO(5)/SO(4) theory is once again practically indistinguishable
from SO(4)/SO(3).
Corresponding integrated energy densities.
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