The tightest and most robust cosmological results of the next decade will be achieved by bringing together multiple surveys of the Universe. This endeavor has to happen across multiple layers of the data processing and analysis, e.g., enhancements are expected from combining Euclid, Rubin, and Roman (as well as other surveys) not only at the level of joint processing and catalog combination, but also during the post-catalog parts of the analysis such as the cosmological inference process. While every experiment builds their own analysis and inference framework and creates their own set of simulations, cross-survey work that homogenizes these efforts, exchanges information from numerical simulations, and coordinates details in the modeling of astrophysical and observational systematics of the corresponding datasets is crucial.

This book lays out the scientific goals to be addressed by the next-generation ground-based cosmic microwave background experiment, CMB-S4, envisioned to consist of dedicated telescopes at the South Pole, the high Chilean Atacama plateau and possibly a northern hemisphere site, all equipped with new superconducting cameras. CMB-S4 will dramatically advance cosmological studies by crossing critical thresholds in the search for the B-mode polarization signature of primordial gravitational waves, in the determination of the number and masses of the neutrinos, in the search for evidence of new light relics, in constraining the nature of dark energy, and in testing general relativity on large scales.

The expansion of the Universe is understood to have accelerated during two epochs: in its very first moments during a period of Inflation and much more recently, at $z < 1$, when Dark Energy is hypothesized to drive cosmic acceleration. The undiscovered mechanisms behind these two epochs represent some of the most important open problems in fundamental physics. The large cosmological volume at $2 < z < 5$, together with the ability to efficiently target high-$z$ galaxies with known techniques, enables large gains in the study of Inflation and Dark Energy. A future spectroscopic survey can test the Gaussianity of the initial conditions up to a factor of ~50 better than our current bounds, crossing the crucial theoretical threshold of $\sigma(f_{NL}^{\rm local})$ of order unity that separates single field and multi-field models. Simultaneously, it can measure the fraction of Dark Energy at the percent level up to $z = 5$, thus serving as an unprecedented test of the standard model and opening up a tremendous discovery space.

CMB-S4---the next-generation ground-based cosmic microwave background (CMB) experiment---is set to significantly advance the sensitivity of CMB measurements and enhance our understanding of the origin and evolution of the Universe, from the highest energies at the dawn of time through the growth of structure to the present day. Among the science cases pursued with CMB-S4, the quest for detecting primordial gravitational waves is a central driver of the experimental design. This work details the development of a forecasting framework that includes a power-spectrum-based semi-analytic projection tool, targeted explicitly towards optimizing constraints on the tensor-to-scalar ratio, $r$, in the presence of Galactic foregrounds and gravitational lensing of the CMB. This framework is unique in its direct use of information from the achieved performance of current Stage 2--3 CMB experiments to robustly forecast the science reach of upcoming CMB-polarization endeavors. The methodology allows for rapid iteration over experimental configurations and offers a flexible way to optimize the design of future experiments given a desired scientific goal. To form a closed-loop process, we couple this semi-analytic tool with map-based validation studies, which allow for the injection of additional complexity and verification of our forecasts with several independent analysis methods. We document multiple rounds of forecasts for CMB-S4 using this process and the resulting establishment of the current reference design of the primordial gravitational-wave component of the Stage-4 experiment, optimized to achieve our science goals of detecting primordial gravitational waves for $r > 0.003$ at greater than $5\sigma$, or, in the absence of a detection, of reaching an upper limit of $r < 0.001$ at $95\%$ CL.