On the computation of CMBR anisotropies from simulations of topological defects Page: 4 of 18
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equation once for each value of Jkl. For a given type of inhomogeneity, e.g.
adiabatic or isocurvature, the only freedom in the initial condition of a given
mode is the amplitude, and, since the equations are linear, solutions scale
linearly with the initial amplitude.
For topological defects the equations are non-linear so different eigenmodes
are coupled and one does not know the evolution of one mode without taking
into account the evolution of all the other. It is usually simpler to think
about topological defects in real space rather than the space of eigenmodes
(k-space). One reason is that equations describing the evolution of the defects
are spatially local while they are highly non-local in k-space. Another reason
is that causality guarantees that the range interaction does not extend beyond
a certain distance, the causal horizon, while the range of interaction in k-space
is unbounded. Causality allows one to consider a finite patch of the universe
and ignore what goes on outside of that patch, at least for points more than a
horizon distance away from the boundary of the patch.
In order to determine the likelihood that the observed inhomogeneities
in the universe could be explained by topological defects one would need a
description of the statistical ensemble of possible topological defects configu-
rations and their evolution. In general we cannot even find analytic solution
to describe the evolution of specific configurations so the way one tries to de-
termine the statistical properties is through numerical experiment. Namely
one generates a realization of the defect configuration at some early time and
evolves it and the matter surrounding it according to their equations of mo-
tion. One might then try another configuration, and so on, until one has a fair
statistical sample. It is generally believed that the statistical properties of the
defects are ergodic so that one may also obtain a fair sample by considering a
large enough volume of a single realization. One need only consider the obser-
vations predicted for observers situated in different locations in the simulation
volume. Since one needs to build up a fair sample it is important to obtain as
many realizations of the anisotropy as possible.
1 Geometry of Computation
Numerous authors have used simulations of defects to produce realizations of
anisotropy patterns from defects 1,2,5,6,7,8,9,10,11 Some of these have been used
to examine large angle anisotropies 2,5,6,8,9,11 while others have been used to
examine anisotropies on smaller angular scales 1,7,10 The geometry used for
these large angle numerical experiments is shown in fig 1. One must evolve
the defects in a very large simulation volume, as big as the present horizon
and ideally one would like to evolve from time of recombination. Usually one
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Stebbins, A. & Dodelson, S. On the computation of CMBR anisotropies from simulations of topological defects, article, May 1, 1997; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc690215/m1/4/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.