• Next-to-Minimal Supersymmetric Standard Model
• Stochastic Gravitational Waves Background
• Electroweak Baryogenesis
• Electroweak Cosmological Phase Transition

The purpose of this thesis is the study of one of the most promising solutions to the problem of the matter-antimatter asymmetry in the observed universe. The problem can be stated as follows: without very fine tuned initial boundary conditions equal amounts of matter and anti-matter should have been produced during the Big Bang. If this were the case, after annihilation, no matter would have remained.

The solution we are going to explore in this thesis is called Electroweak Baryogenesis and manifests itself within a first-order phase transition (FOPT) induced by Electroweak Spontaneous Symmetry Breaking (EWSSB) in the early universe. During this phase transition bubbles of the (true) vacuum scalar fields start to grow and blend together, inside the electroweak symmetry preserving (false) vacuum the universe previously lived in, until they incorporate the whole space.

Being induced by EWSSB which can be already found in the Standard Model one could argue that the latter is the most natural place where to study this phenomenon but the experimental data collected at LHC about the Higgs physics have already excluded it. What we need in addition to the possibilities offered by the Standard Model is a framework inducing a sufficiently strongly first-order phase transition producing significant departure from thermal equilibrium, baryon number violation and enough CP violation, thus fully satisfying the three Sakharov conditions for successful Electroweak Baryogenesis.

This is not however the only motivation for trying to go beyond the Standard Model. The latter is in fact cursed among others by the hierarchy problem which consists in the largest energy scales of the theory generating too large quantum corrections to the masses of the lower energy particles. These corrections cannot be easily eliminated if not by an incredibly precise and improbable fine-tuning.

By extending the possible space-time symmetries of the lagrangian including a symmetry connecting fermions and bosons in an appropriate way it has been shown that this elimination is not only possible but unavoidable. Such an extension of the old scheme is called Supersymmetry, which is the framework in which we will study Electroweak Baryogenesis.

Supersymmetry is very promising in this sense because, by predicting a superpartner for each presently known particle and many new parameters and possible C and CP violating terms, it could offer an appropriate parameter space for successful baryogenesis.

In this thesis we will begin the exploration of this possibility by studying various aspects of the Electroweak Cosmological Phase Transition (EWCPT) in the Next-to-Minimal Supersymmetric Standard Model (NMSSM), one of the simplest models which already at tree-level can have big enough cubic scalar terms in the lagrangian enhancing the first-order character of the phase transition we are looking for.

The thesis is articulated into three chapters which are organized as follows:

- Chapter 1 deals with the introduction of the EWCPT starting from the building blocks of the Standard Model which is where it was born and which then reveals not to be suitable for the phenomenon we want to explore in this thesis.

The Effective Potential formalism (at zero and finite temperature) which will be used for the study of the vacuum manifold and the basic elements of Supersymmetry which will be necessary to describe the peculiar supersymmetric theory considered will be introduced. We end with an hint on why a supersymmetric theory could perhaps lead to successful baryogenesis;

- Chapter 2 introduces the study of the EWCPT in the NMSSM starting with the constituting elements of this model and building the Effective Potential at finite temperature, including the one-loop quantum and thermal corrections to the zero-temperature classical potential. This gives the shape of the vacuum manifold which is explored first analitically and then more thoroughly with the help of a numerical strategy based on the continuation of stationary points of the potential within the parameter space, which is employed for the first time in this kind of study. Possible patterns of symmetry breaking and corners of parameter space leading to a strongly first-order phase transition are investigated also identifying the relevant marker for the strenght of the FOPT in the ratio between the scalar fields norm and temperature at the critical temperature (when the true and false minima become degenerate);

- Chapter 3 is about the direct study of Electroweak Baryogenesis from the combination of the Euler-Lagrange equations of motion for the scalar fields with the Boltzmann equations which couple them with the cosmological plasma of interacting particles.

We find that the net baryon asymmetry produced during the FOPT is a consequence of C and CP violating scattering at the phase boundary between the broken and unbroken phase, for which a FOPT is required. This biases sphaleron (baryon number violating) transitions which, for a strongly FOPT, are more efficient in the unbroken phase and much less in the broken one thus avoiding washing out the baryon asymmetry ending here due to the expansion of the scalar field bubbles. We see that the baryon asymmetry depends on some parameters describing the phase transition, in particular the bubble wall final velocity acquired as equilibrium between the free expansion of the bubble and the friction with the plasma is reached.

Finally we conclude this last chapter with a brief mention to the main experimental tests for Electroweak Baryogenesis with particular attention to the production of a Stochastic Gravitational Waves Background during the EWCPT which could fall in a detectable range of frequencies for the next generation of interferometers, thus providing another way to test the model.