This White Paper has been prepared to fulfill SNOWMASS 2021 requirements and it extends the material previously summarized in the four Letters of Interest [1][2][3][4]. The Particle Physics Community Planning Exercise (a.k.a. SNOWMASS ) is organized by the Division of Particles and Fields of the American Physical Society, and this is an effort to bring together the community of theoretical physicists and cosmologists and identify promising opportunities to address the questions described.This White Paper was initiated with the aim of identifying the opportunities in the cosmological field for the next decade, and strengthening the coordination of the community. It is addressed to identifying the most promising directions of investigation, and rather than attempting a long review of the current status of the whole field of research, it focuses on the upcoming theoretical opportunities and challenges described. The White Paper is a collaborative effort led by Eleonora Di Valentino and Luis Anchordoqui, and it is organized in the topics listed below, each of them coordinated by the scientists indicated (alphabetical order):