Astronomers collect light, nothing more. The formalism of radiative transfer is a macroscopic treatment of microscopic interplay between light and matter; it employs macroscopic variables that parameterize microscopic interactions. In this chapter we describe the radiation and photon field and define the fundamental macroscopic quantity – the specific intensity. The geometry of radiative transfer is key as it involves an origin and an observer defined line-of-sight perspective. The observed solid angle is expressed for a cosmologically distant observer, from which flux vectors and the observed flux are derived. The equation of radiative transfer is introduced, including the macroscopic parameters known as the emission and extinction coefficients and the optical depth and mean free path. The solution for pure absorption is given including illustrations of the anatomy of an absorption line in terms of optical depth. The details hidden within a beam-averaged astronomical absorption spectrum are described, followed by a treatment of partial covering, from which the covering factor is derived. Finally, a formal definition of column density is provided.

Studies of the high-ionization metal-line absorbers provide insights into hot diffuse gas that has been processed through stars in galaxies. In the ultraviolet and optical bands, these absorbers have been studied primarily using five-times ionized oxygen (OVI), six-times ionized nitrogen (NV), and seven-times ionized neon (NeVIII). Both OVI and NeVIII arise within the spectral range of the Ly α forest and are thus mostly visible at low redshifts where the Ly α forest line density is much smaller. NV is adjacent to the Ly α line and in principle can be surveyed over the full range of redshift; however, this ion is found in only a narrow range of astrophysical conditions. The population statistics measured include the redshift path density, the equivalent width and column density distributions, the cosmic mass densities, and the kinematics (broadening parameters, velocity splitting distributions, and absorber velocity widths). In this chapter, we discuss multiple observational programs and their reported findings for several of these ions.

The spin-flip of the ground-state electron in neutral hydrogen, known as hyperfine structure, gives rise to the famous 21-cm line in the radio band. The 21-cm line informs us about the “cold” phase of atomic hydrogen in the Universe. In this chapter, we present the basic physics of the 21-cm spin flip and then discuss several surveys of 21-cm absorption and their reported findings. These include blind surveys, galaxy-selected surveys, metal-line selected surveys, and DLA-selected surveys. An exciting measurement known as redshift drift, which would provide a direct measurement of the change in the expansion rate of the Universe, is expected to be highly precise using 21-cm observations. During the Epoch of Reionization, a 21-cm forest, analogous to the Ly α forest, is expected. In fact, this absorption line is expected to trace all the way back to the Dark Ages of the universe and yield empirical insights into the formation of the first stars and black holes.

Handed a spectrum, the work begins. In this chapter, we explain how one takes a spectrum and objectively locates and quantifies the statistically significant absorption features peppered throughout. We describe a continuum normalization method that is objective and provides an error model. Multiple spectra may be co-added to improve signal to noise. For objectively locating absorption features, we present a scanning algorithm weighted by the line spread function and optimized for weak lines. Multiple absorption lines arising in rich absorption systems can be found using autocorrelation methods, and one such method is described. To analyze absorption systems, a systemic absorber redshift is determined, and the wavelength scale of all absorption profiles is converted to and aligned in the absorber’s rest-frame velocity. For high-resolution profiles, methods are presented for measuring equivalent widths and quantifying kinematics directly from pixel flux decrements. These include velocity spreads containing 90% of total optical depth and other flux decrement weighted velocity moments. We conclude with detailed methods for building composite two-point velocity correlation functions.

Every recorded quasar spectrum is a blemished version of an otherwise pure light beam. It is blurred by the atmosphere and suffers interference and scattering when reflected off optical elements. It is imperfectly collimated, impurely dispersed, iteratively refocused, and inefficiently discretized when recorded. It is then converted to analog and re-digitized, which introduces “read” errors to an already noise-ridden Poissonian process of photon counting. To understand spectra, one needs to understand its recording device, the spectrograph. In this chapter, a range of long-slit low-resolution spectrographs and high-resolution echelle spectrographs are described. Grating equations, blaze functions, and cross dispersers are examined in detail. The equations for resolving power and instrumental resolution are derived from first principles, followed by illustrations showing the impact of CCD pixelization and line broadening on recorded absorption lines. We present quantitative models for the recorded counts in observed spectra. Flux calibration is also derived from first principles of telescope characteristics and spectrograph design. Finally, integrated field units are described.

Galaxies do not live alone; they live in groups and clusters; they are surrounded by smaller companions bound in or passing through their dark matter halos. As such, there is some ambiguity when studying the CGM in connection to “isolated” galaxy properties because gas is shared between galaxies and companions. In this chapter, we describe halo occupation distribution (HOD) theory, which characterizes the average distribution of companions associated with a given galaxy. HOD relations guide our quantified definitions of galaxy groups and clusters and provide a formalism within which absorption line studies can be applied to the intragroup medium (IGrM) and the intracluster medium (ICM). The remainder of this chapter covers the characteristic properties of the IGrM and the ICM. The IGrM is primarily discussed in terms of theoretical hydrodynamic simulations of small groups like our own Local Group. Of interest are the dynamic “boundaries” between the individual CGM of the orbiting member galaxies and the common IGrM envelope. Mergers are briefly discussed followed by a detailed characterization of the ICM based on X-ray emission studies.

Empirically demonstrating the association of metal-line absorber lines with galaxies has a long, rich history from the earliest theoretical predictions in the mid 1960s to observational confirmation in the 1990s. From that point onward, quasar absorption line studies became a powerful tool for characterizing the gaseous halos of galaxies. Countless works have provided valuable insights into the chemical, ionization, and kinematic conditions of what is now called the circumgalactic medium. A new concept called the baryon cycle was birthed in which the balance of accretion modes, stellar feedback, gas recycling, and outflow dynamics of galactic gas was found to be closely linked to how baryons respond to dark matter halos of a given mass. Modern theory known as halo abundance matching has helped us empirically connect the average stellar mass to dark matter halos of a given mass. Powerful hydrodynamics simulations tell a story in which the average baryon cycle processes in a galaxy are closely linked to dark matter halo mass. In this chapter, we discuss how synthesizing both the observational data and theoretical insights has yielded a simple composite model of the baryon cycle.

The 1960s through the 1970s was an exciting era of the discovery of quasars. During this time the study of these cosmologically distant luminous sources developed into a powerful tool that changed the course of the science of astronomy. This story runs in parallel with technological advances in both light-gathering capability and computing power. In this chapter, we chart the development of the study of quasars and show how quasar absorption lines provide a tool for studying the properties of diffuse gas across the full dynamic range of astrophysical environment out to the highest redshifts.

The wave model of hydrogenic ions naturally yields transition probabilities. These probabilities are written in terms of three Einstein coefficients, which are determined from “overlap integrals” for spontaneous emission. Under the assumption that a simple dipole describes the moment between the charge densities of the initial and final stationary states of an electron transition, the transition probabilities yield selection rules, emission line intensities, and absorption cross sections. The former governs whether a transition is permitted or forbidden. The amplitudes of the latter two are often written as oscillator strengths. In this chapter, we describe the formalism for determining selection rules and oscillator strengths. We begin with the Schrödinger model and generalize to fine structure transitions for bound-bound transitions. We then address the oscillator strengths of bound-free transitions. Finally, we derive the line spread function describing the natural line width, which depends on the magnitude of the Einstein coefficients and is written in terms of the damping constant. Full expressions for the bound-bound and bound-free absorption cross sections are provided.

It is time to take a deep dive into several of the “key quantitie” introduced in Chapter 3. Above all are the population density functions, which describe the number of absorbers per unit redshift per unit column density (or equivalent width). In this chapter, we present practical equations for obtaining maximum likelihood estimates of the population parameters for commonly adopted distribution functions: the power law, the exponential, and Schechter. Summing absorber counts and/or integrating these parameterized distribution functions in absorber subspaces (i.e., bins of redshift and column density) – or along one axis of the absorber survey space (i.e., across all equivalent widths at fixed redshift, etc.) – allows absorber evolution to be quantified. Examples include the redshift path density, absorber cross sections, the column density and equivalent width distributions, and the mass density of absorbers. We derive these quantities from first principles and then show how they can be computed accounting for the detection completeness, the redshift path sensitivity, and the total redshift sensitivity path of the survey.