• cosmic strings
• cusps
• effective actions
• gravitational waves

Cosmic strings are one dimensional objects which may have formed during a cosmological phase transition in the early Universe because of a spontaneously broken symmetry in the context of an high-energy model (electroweak theory and GUTs are some examples) in which the vacuum manifold is not simply connected. There are several reasons why such objects are of physical interest: they potentially stand as the most well-motivated example of defects in the context of particle physics, and they have a phenomenological relevance since they can emit a substantial quantity of gravitational waves through a mechanism that is unique to them.

The simplest high energy model that supports stable cosmic strings and that can be studied, without loss of generality, as a reference theoretical framework in which such exotic objects arise is the Abelian Higgs model. The model supports a vortex solution (the Nielsen-Olesen vortex) corresponding to an infinite straight static string. However, all known field theoretic solutions are characterized by an high degree of spacetime symmetry that is improbable to be a realistic feature of cosmological strings. Moreover it is extremely difficult that an exact solution to the Abelian Higgs field equations will ever describe a curved or oscillating string. Thus, in order to study the dynamics of a general string configuration it is necessary to simplify the problem by removing some degrees of freedom. This leads to the construction of a string effective action obtained by modelling the large number of degrees of freedom of the full field theory with only the ones that can be associated with the position of the defect core.
Such effective action can be written as a power expansion in the defect thickness, where the zeroth order takes the form of a Nambu action. Higher order terms are proportional to powers of the ratio between the defect thickness and the curvature radius scale of the world sheet swept by the string during its motion.

In studying the dynamics of cosmic strings, literature has generally focused on the "wire approximation" which consists in considering the defect in the zero thickness limit and therefore in taking only the Nambu term of the full effective action. Nambu strings are described by the sum of two mode functions and different choices for these functions lead to a plethora of different string configurations from infinite strings to closed loops. An interesting property of infinitely thin loop solutions is that some points along the string can reach the speed of light. Such luminal points are generally known as cusps and manifest themselves as sharp features of the string profile. Cusps are of physical interest since the gravitational field of a Nambu string diverges in their proximity and so does the GWs flux emitted by a cusp.

This realistically means that since these loop solutions emit gravitational radiation, the GWs flux of a loop with cusps is dominated by cusps radiation. Of course from a phenomenological point of view we do not expect cusps to be a realistic feature of cosmic strings and for this reason the study of effects beyond the wire approximation is of interest.

The width of a cosmic string is mainly determined by the energy scale at which the phase transition of interest occur. In the case of GUT strings, the thickness is generally comparable to the Compton wavelength of a particle with GUT mass which is approximately 10^-29 cm, while in the case of electroweak strings the thickness is about 10^-17 cm. These quantities are definitely smaller than the typical astrophysical and cosmological scales, thus cosmic strings are usually thought as infinitely thin. So, in general, the zero thickness string described by the Nambu action can be used as a reasonable approximation of a realistic string.

However, when talking about cusps we are referring to local features of a string and locally finite width effects can play a role in the string dynamics, therefore one could wonder if including such effects can be sufficient to prevent cusps formation. This is the main motivation of this thesis.

To do that we decided to study finite width corrections of the so called 2/1 harmonic solution. It belongs to the class of p/q harmonic loop solutions, where (p,q) are two non-zero integers labeling the harmonics of the two mode functions. This solution of the Nambu effective action is characterized by 3 permanent cusps and this gives us a bench test for the cusp behavior when one includes higher order curvature terms (and therefore higher order powers of the string thickness) in the string effective action.

From the GWs emission point of view, this Nambu string is characterized by a divergent power per unit solid angle along the beaming direction of each cusp as well as by a divergent total emitted power. Hence, if finite width corrections regularize these sharp features such divergences would be cured.

In this work we will set the formalism for the study of the problem, and we obtain some initial results. This opens the door to the possibility of a careful exploration of the parameter space of the theory, which will be able to give a detailed picture of the behaviour of the theories in a neighborhood of the Nambu Goto one.