This book offers a comprehensive and pedagogically oriented introduction to modern cosmology, developed in the spirit of the Lecture Notes in Physics series. The material originates from graduate-level courses and advanced schools, and is designed to bridge the gap between introductory textbooks and current research literature.

In this second edition the historical overview of the development of the discipline of cosmology has been extended, including translations of parts from the original seminal papers. It further includes a discussion of the Hubble tension and a more comprehensive explanation of distances in cosmology. The author also discusses the fundamental problem of the incompatibility of the thermal distributions with an expanding universe and the repercussion of this fact on the cosmological model of the early universe. Chapters 1-4 have been design such that they can cover a basic course in cosmology.

The book begins with an overview of the observational foundations of cosmology, including the discovery of the expanding Universe and the evidence for its main components: radiation, baryonic matter, dark matter, and dark energy. Special attention is given to current open problems, such as the nature of dark components and the emerging cosmological tensions.

A consistent theoretical framework is then established. Starting from Newtonian arguments, the discussion naturally progresses to the relativistic description based on the Friedmann–Lemaître–Robertson–Walker metric. The Friedmann equations are derived and analyzed in detail, together with key concepts such as cosmological distances, horizons, and the ΛCDM model.

The text places strong emphasis on the microphysics of the early Universe. Kinetic theory in an expanding background is developed from first principles, leading to a systematic treatment of thermal equilibrium, particle decoupling, and relic abundances. Applications include Big Bang nucleosynthesis, recombination, and the thermal history of photons and neutrinos.

A substantial portion of the book is devoted to cosmological perturbation theory, presented in a clear and self-contained way. The reader is guided through the formalism of metric perturbations, gauge transformations, and the evolution of scalar, vector, and tensor modes. The Boltzmann equations for photons, neutrinos, and cold dark matter are derived and applied to the study of structure formation and Cosmic Microwave Background anisotropies.

The origin of primordial perturbations is discussed within the framework of inflation, including slow-roll dynamics and observational implications. The statistical description of cosmological fluctuations—random fields, power spectra, and cosmic variance—is developed and connected to modern observational probes.

The final chapters address the evolution of perturbations across cosmic epochs and provide a detailed account of the physics of CMB anisotropies and polarization. Additional material introduces practical tools for data analysis and parameter estimation in cosmology.

Throughout the book, the emphasis is on clarity, logical development, and physical insight, while maintaining a solid mathematical foundation. Numerous intermediate steps, explicit derivations, and selected exercises make the text particularly suitable for self-study as well as for use in graduate courses and specialized schools.

Oliver Fabio Piattella (OFP) obtained his Bachelor’s degree in Astrophysics (2003), Master’s degree (2006), and PhD (2010) in Cosmology and Gravitation from the University of Insubria (Como, Italy), under the joint supervision of Professors Ugo Moschella and Alexander Yu. Kamentchik. During his PhD, he carried out part of his research at the Institute of Gravitation and Cosmology (ICG) of the University of Portsmouth (United Kingdom), collaborating with Professor Marco Bruni.

In 2010, he was a junior postdoctoral fellow of CNPq under the supervision of Professor Júlio C. Fabris; in 2011, he served as a visiting professor and subsequently as an associate professor at the Universidade Federal do Espírito Santo (UFES, Vitória, Brazil). From September 2018 to March 2020, he was a CAPES/Humboldt fellow at the Institute for Theoretical Physics of Heidelberg University (Germany), hosted by Professor Luca Amendola.

In 2016, he contributed to the design and launch of the PPGCosmo doctoral program, where he served as vice-coordinator. From 2022 to 2024, he was a researcher at the Department of Science and High Technology (DiSAT) of the University of Insubria, where he currently holds the position of Associate Professor.

His main research areas are Cosmology, Astrophysics, and Gravitation, with particular interest in dark matter and dark energy problems, modified gravity theories, inflationary models of the Universe, and black hole astrophysics.