• abstract indices
• cosmic strings
• gravitational waves
• isogonal point
• kinetic theory
• orthonormalization

Phase transitions are milestones in the Universe evolution. When a transition breaks a symmetry, topologically constrained defects can arise in the vacuum field. Some theories predict the formation of string-like defects via Kibble mechanism [3] in the early Universe.
These cosmic strings would be relict of a transition happened before Cosmic Microwave Background decoupling. Thus, detecting them -- or their effects -- could offer precious information about the history of the Universe.
Cosmic strings are expected to behave mostly like relativistic ideal strings, arranged in a network spread across the Universe, with non-trivial gravitational effects.
Advanced gravitational observatories will probably soon be able to detect a stochastic background of gravitational waves. Numerical simulations, together with order of magnitude estimates, regard cosmic string networks as the most promising non-astrophysical source of such signal.
Because of the complex non-linear behaviour of strings, efficient simulations [1] run on supercomputers for several months in order to be accurate. Still, the demand for predictions proceeds faster than computer power.
In recent years, Vitaly Vanchurin and colleagues [4] applied tools from Statistical Physics to strings. Their kinetic approach led to describe a cosmic string network as a non-ordinary fluid evolving in a fixed homogeneous background space.
This thesis work is intended to include gravitational perturbations induced by strings on space-time into the kinetic theory for string fragments.
In order to do that, evolution of strings in a generic space-time, in the context of General Relativity, is studied. The aim is to build tools for a new approach to simulations of cosmic string networks, so that a reduced computational power or, equivalently, an increased scale of the simulation can be achieved.
Differential Geometry is widely used. The formalism is based on abstract index notation.
An orthonormal basis is defined in terms of the coordinate one, in order to better describe string dynamics.
Specializing to the case of an expanding Universe, a foliation into space-like slices is considered, adapting the ADM approach [2] to the context.
Determining the production of gravitational waves and the corresponding dissipative term for a cosmic string fluid constitutes the main intended application of the formalism.


References
[1] B. Allen and E. P. S. Shellard. Cosmic-string evolution: A numerical simulation. Phys. Rev. Lett., 64:119--122, Jan 1990.
[2] Richard Arnowitt, Stanley Deser, and Charles W. Misner. Republication of: The dynamics of General Relativity. General Relativity and Gravitation, 40(9):1997--2027, 2008.
[3] T W B Kibble. Topology of cosmic domains and strings. Journal of Physics A: Mathematical and General, 9(8):1387, 1976.
[4] Daniel Schubring and Vitaly Vanchurin. Transport equation for Nambu-Goto strings. Phys. Rev. D, 89:083530, Apr 2014.