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\section*{Research Proposal}

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Title:  Nonstationary model for a general relativistic cosmic string

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For my doctoral research I began the derivation of an exact general
relativistic model for a cosmic string that takes into account the 
back reaction of the string on the spacetime.  A time
independent model was then considered and the stationary 
Euler-Lagrange-Einstein 
equations were solved simultaneously.  This work led to new insights into the 
characterisitics of the missing angle and the gravitational effects 
generated by the string. The consequences of the model on the new 
physics of the stationary cosmic string and discrepancies from
previous works were also analyzed.

\medskip

I propose to study the dynamical evolution of exact self-gravitating
vortex-like topological defects directly from the Higgs Lagrangian and
Einstein's equations.  By extending the research done for the stationary
case to a time dependent case, one expects to obtain a better physical
picture of a vortex topological defect trapped in our universe.  The 
evolution of the defect is important since it provides a new parameter for
matching conditions in an expanding Friedman-Robertson-Walker universe.
In fact, a specific model for embedding the string in the universe could 
be constructed by adjusting the free parameters which might result in local 
dynamic density perturbations creating anisotropies in the cosmic 
microwave background (see C.~C.~Dyer and Peter S.~S.~Ip, 1988. 
{\sl Mon.~Not.~R.~astr.~Soc.}, {\bf 235}, 895).  Temperature
fluctuations should arise from the gravitational redshift effects 
due to the hills and valleys of the model potential.

\medskip

The set of coupled differential equations solved for the stationary case
now become, with the addition of the time variable, a set of partial
coupled differential equations (see Eric Shaver, 1992.  
{\sl Gen.~Rel.~Grav.}, {\bf 24}, 187).  The knowledge and
mathematical tools developed by studying a simpler case 
can now be applied to attack this more complex problem.  A study of the new
set of equations shows that some characteristics of the cosmic string will
change drastically since the energy-momentum tensor no longer satisfies the
condition that the energy density equals the tension along the string 
and it now has an additional
non-diagonal component representing the vorticity of the
string.  Numerical methods for solving partial differential equations
such as the 
finite-difference method, for our initial value problem, have to 
be investigated carefully to ensure the stability and accuracy 
of the sought-after solutions.

\newpage

The impact of this research on observational cosmology and on the 
theoretical models for the origin of density perturbations in
the universe will be very important. For example, our model will be 
more consistent with the description of topological defects found
in other physical systems, thereby increasing
the accuracy of the predictions related to cosmic string defects 
formed during a grand unified theory phase transition in the early universe. 
Currently, the best available estimates of the cosmic microwave background
anisotropy from strings are those of Bouchet, Bennet, and
Stebbins. However, their predictions rely on the small missing angle 
approximation for strings in Minkowski space 
(see David P.~Bennett, Albert Stebbins and Fran\c{c}ois Bouchet, 1992.
{\sl Ap.~J.~Lett.}, {\bf 399}, L5).  The model I propose to  
develop will be more accurate to compare to the science being done by
the Hubble space telescope in probing the large scale structure
of the universe.  The observations made by the Hubble telescope are
extremely important in order to put constraints on the different 
theoretical models proposed to explain the origin of density 
perturbations in the early universe, later leading on to the
formation of structure that we see now.  The existing models such
as the one involving cosmic strings make
predictions that are distinct but difficult to measure and 
in order to verify if any of these models describe well the presence
of structure in the universe, we need to gather more and better
observations on large distances.  I believe the Hubble space 
telescope will play a major role in those future achievements
and many more.

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