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.

Studies of the low-ionization metal-line absorbers provide insights into cool/warm higher-density gas that has been processed through stars in galaxies. These absorbers have been studied primarily using the abundant neutral atoms sodium, oxygen, and carbon (NaI, OI, and CI), as well as the singly ionized ions of carbon, silicon, calcium, and magnesium (CII, SiII, CaII, and MgII). For optical quasar spectroscopy, these ions have limited visibilities over different redshift ranges. The advent of sensitive UV and IR spectrographs expanded the redshift coverage of MgII absorbers from z = 0 to z = 7. However, the redshift visibility of OI, CI, CII, and SiII remain limited because of their far-ultraviolet transitions. 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 the ions.

Astrophysical gases are characterized by their macro variables, which describe the radiation field and the particle field. Radiation can be described by its frequency-dependent energy density or photon number density. Both these quantities can be integrated over frequency, yielding total radiative energy density and/or photon number density. Particles are described by their number and/or mass densities, thermal motions, partial pressures, and the charge density of ions and free electrons. Ion number densities depend on the abundances of elements in the gas and their ionization fractions. In this chapter, we describe the formalisms of the quantities describing the radiation and particle fields in astrophysical gases. We describe the cosmic ultraviolet ionizing background and starburst galaxies ionizing spectra. The principles of particle density and charge density conservation are derived, and the equation of state is presented. This chapter provides the fundamental formalism for studying the micro processes of ionization and the detailed balancing of partially ionized astrophysical gases key to modeling the ionization balance of these gases.

Each quasar sightline provides only a “pencil beam” core sample of the Universe. Quasar absorption lines surveys that employ numerous quasar sightlines aim to measure several key quantities (known as absorber population statistics) that characterize the astrophysical properties of absorbers. Creative methods and experimental approaches have been developed over the past decades. These include rudimentary tomography experiments using close groupings of multiple quasars for measuring transverse properties of absorbers, stacking of spectra, peering “down-the-barrel,” and the use of Gamma-ray bursts. In this chapter, we describe these “key” population statistics and how they are measured in principle. We also describe the innovative ways that multiple sightlines are used experimentally adopting modern techniques.

The abundance of deuterium, an isotope of hydrogen, is sensitive to the ratio of the cosmic baryon density to the photon density. This ratio is fixed (or “frozen in”) well before Big Bang nucleosynthesis begins. Competing with the timescale over which fusion is building up helium-4 nuclei is the timescale for the photodissociation of deuterium and the β-decay rate of free neutrons. Together, these form the highly sensitive “deuterium bottleneck.” In the 1990s, measurements of the cosmic deuterium abundance using quasar absorption line techniques varied by an order of magnitude. After 25 years of effort, the scatter has been reduced to sub-1% precision and the highly sought cosmic D/H ratio has been pinned down. Additional constraints are obtained using the cosmic microwave background, but these can be in tension with quasar absorption line results. In this chapter, we describe efforts to measure the cosmic deuterium abundance and reconcile them with theoretical predictions, which may be limited by the accuracy of the reaction rates used for Big Bang nucleosynthesis calculations.

Over the last quarter century, studies of the circumgalactic medium (CGM) have evolved from small, isolated cottage-industry efforts to a few dozen factory-scale assembly-line collaborations. The advent and continued development of large galaxy surveys, the refinement of photometric redshifts, and the honing of color selection of quasars have all combined to yield more than a million object-searchable catalogs for building large samples of galaxy-quasar pairs on the sky. Though the largest body of work has focused on low- and intermediate-redshifts, where detailed galaxy properties can be measured, wholesale studies of the CGM have now reached redshifts of 4 using Lyman break galaxies (LBGs) and the stacking of the spectra of thousands of Lyman alpha emitters. In this chapter, we provide an overview of CGM studies with a focus on sample building and experimental approaches and techniques. The three main types of survey strategies are discussed. Concepts such as the characterization of CGM absorption properties as a function of impact parameters, covering fractions, and galaxy-absorber morphokinematic and morphospatial analysis are presented.

After a series of observational and theoretical breakthroughs in the 1960s, the Steady State theory was discarded, whereas the Big Bang cosmological paradigm remained viable. This model is described by the Friedmann equations with a Robertson-Walker metric. The metric describes the dynamic spacetime intervals and the Friedmann equations describe the expansion dynamics. The latter are derived from Einstein’s field equations of General Relativity assuming an isotropic and homogeneous medium, conservation of energy density, and an equation of state known as the “continuity equation.” Friedmann’s equations are conveniently written in terms of a time-dependent scale factor, the Hubble constant, and four present-epoch cosmological parameters. Today, we live in an era known as precision cosmology, in which the Hubble constant and cosmological parameters are measured with 1% or better uncertainties. In this chapter, we present an abridged derivation of the Friedmann equations and discuss the cosmological parameters and their temporal evolution in detail. The Robertson-Walker metric is then rewritten in terms of radial and transverse components suitable for convenient practical application.

The atomic physics of excitation, ionization, and recombination is the story of frolicking electrons – like space traveling aliens, these little leptons are busy jumping up and down when bound to their “home planet” atom/ion. Launching freely into space, they adventure out and engage in a series of friendly energy exchanges with fellow particles and photons in an expansive plasma. Landing on and being captured by some other random “planet” atom/ion, the cavorting continues. In this chapter, we follow the dynamic lives of electrons, photons, and ions and present an abridged review of the physics of collisional excitation, ionization, and recombination. We describe photoionization, Auger ionization, direct collisional ionization, excitation auto-ionization, radiative and dielectronic recombination, and charge exchange. We show that detailed balancing and reaction cross sections, rates, and rate coefficients are the heart of chemical-ionization modeling of absorbers. We then present the cosmic photoionization rate of HI, HeII, MgII, CIV, and OVI as a function of redshift. We conclude with a comprehensive treatment of the heating and cooling functions of astrophysical gas.

Surveys answer the big science questions, but they are trickier than one might think. Designing a survey requires careful planning fraught with technical limitations, uncontrolled variables, and implicit sample biasing. Analyzing a large number of individual quasar spectra presents many challenges. In this chapter, we outline the fundamental attributes of a survey, which define its breadth, depth, and completeness over the domain of the survey space. Large surveys require automated algorithms for objectively identifying absorption lines; their success rates for finding true absorption and erroneously identifying false positives must be both objectively and subjectively assessed. We outline a comprehensive strategy, including automated routines, human inspection, and Monte Carlo simulations, for obtaining the best estimate of the number of true absorbers in the spectra. Other key quantities include the redshift path sensitivity and the total redshift sensitivity path of the survey. These can be computed in binned survey subspaces (redshift, etc.) and will be central to estimating absorption population statistics. We conclude with a summary of these complex survey assessment methods.