Spontaneous symmetry breaking in the late Universe and glimpses of early
  Universe phase transitions \`a la baryogenesis
  Spontaneous symmetry breaking is the foundation of electroweak unification
and serves as an integral part of the model building beyond the standard model
of particle physics and it also finds interesting applications in the late
Universe. We review development related to obtaining the late cosmic
acceleration from spontaneous symmetry breaking in the Universe at large
scales. This phenomenon is best understood through Ginzburg-Landau theory of
phase transitions which we briefly describe. Hereafter, we present elements of
spontaneous symmetry breaking in relativistic field theory. We then discuss the
"symmetron" scenario-based upon symmetry breaking in the late Universe which is
realized by using a specific form of conformal coupling. However, the model is
faced with "NO GO" for late time acceleration due to local gravity constraints.
We argue that the problem can be circumvented by using the massless $\lambda
\phi^4$ theory coupled to massive neutrino matter. As for the early Universe,
spontaneous symmetry breaking finds its interesting applications in the study
of electroweak phase transition. To this effect, we first discuss in detail,
the Ginzburg-Landau theory of first order phase transitions and then apply it
to electroweak phase transition including technical discussions on bubble
nucleation and sphaleron transitions. We provide a pedagogical expositions of
dynamics of electroweak phase transition and emphasize the need to go beyond
the standard model of particle physics for addressing the baryogenesis problem.
Review ends with a brief discussion on Affleck-Dine mechanism and spontaneous
baryogenesis. Appendixes include technical details on essential ingredients of
baryogenesis, sphaleron solution, one loop finite temperature effective
potential and dynamics of bubble nucleation.
