The Encyclopedia of Cosmology

The following sections are included:

• Dedication
• Preface
• Contents

This part begins with a brief review of basic cosmology and of relevant topics in statistics, and then develops in detail the theory of galaxy formation as understood in modern cosmology; this includes linear perturbation theory, spherical collapse, and baryonic effects. After deriving some essential elements of stellar dynamics and of galactic structure, this part ends with a brief presentation of gravitational lensing.

The theoretical basis of modern cosmology was laid out soon after Einstein’s 1915 discovery of general relativity. Friedmann first worked out models of the expanding Universe, with Lemaitre adding the concepts of redshift and the initial Big Bang. Hubble’s 1929 discovery of cosmic expansion was the first observational milestone. The hot Big Bang theory accumulated more substance as the first theoretical predictions of the cosmic microwave background (CMB) and of Big-Bang nucleosynthesis were made by Alpher, Gamow, and Herman in the 1940’s. A watershed moment in observational cosmology came with the 1965 discovery of the CMB [1], which was the first direct evidence for the hot and dense initial state of the Universe. Meanwhile, astronomical evidence accumulated for the evolution of the Universe over time, as increasingly distant galaxies were discovered. Soon after quasars were identified as high-redshift objects, Gunn & Peterson [2] used their spectra (in 1965) to show that the inter-galactic gas around them was highly ionized; this was the first sign that the gas had undergone cosmic reionization, likely by the stars in early galaxies.

This volume assumes pre-requisite knowledge of basic cosmology and of standard methods of mathematical physics. This chapter presents just a brief review of essential elements of the expansion history of the Universe (more exhaustive expositions are available in standard textbooks such as those cited in the previous chapter).

Since theoretical models of galaxy formation generally make their predictions statistically, in this chapter we briefly introduce a number of important subjects in statistics. These include random fields, correlation functions, and power spectra.

The following sections are included:

• Preview of perturbation theory
  • Mass-radius relations for initial halo mass
  • Linear spherical collapse
  • Examples of linear perturbation growth
• Distribution functions
  • Quantum statistical mechanics
  • Fluid frame and stress tensor
• The collisionless Boltzmann equation and its moments
  • Liouville equation and collisionless Boltzmann equation (CBE)
  • Moments of the CBE
• Eulerian fluid equations in comoving coordinates
  • Coordinate transformation
  • Poisson equation
  • Continuity equation
  • Euler equation
• Linear perturbation theory
  • Longitudinal isentropic (“adiabatic”) modes
  • Transverse modes
  • GR modes
  • No growth
  • The Jeans length
  • Super-horizon perturbations
• Cosmic history of perturbations
• The power spectrum
• Baryons
  • The baryon-photon fluid
  • Silk damping
  • Early history of the Jeans mass
  • Mean molecular weight
  • Optical depth due to reionization
• References

In the previous chapter, we followed the development of density perturbations in the limit of small perturbations, to linear order in the relative fluctuation. Gravity, though, increases the size of the perturbations so that eventually they become nonlinear (i.e. of order unity or greater) in some regions. In this regime, the simple linear approximation is no longer valid, and a full non-linear calculation of gravity is required. In general, such a calculation is highly complex and can only be attempted numerically. However, in the simple case of spherical symmetry, the evolution can be calculated analytically [1], and the results have proven to be extremely useful in developing intuition, and approximate models, for the formation of galactic halos in the real Universe. We focus here on the formation of halos, which are the hosts of galaxies, but note that on small scales, the power spectrum also changes shape due to the non-linear gravitational growth of perturbations, yielding the non-linear power spectrum (e.g. [2]).

In this chapter and the next one, we temporarily put aside cosmology and the large-scale properties of the Universe, and study the evolution of stellar systems, following [1]. Since stellar collisions are extremely rare and do not affect the overall distribution of stars (except in extreme environments such as globular clusters or galactic centers), we can apply the collisionless Boltzmann equation and related results derived in Chap. 4. The results are also valid for other collisionless systems, and some in particular (such as the virial theorem) are quite important also for cold dark matter halos.

In this chapter we look within galaxies and try to decipher their most spectacular observed property, namely, the spiral arms of disk galaxies. We follow [1] in presenting the explanation of spiral structure due to Lin & Shu [2].

Gravitational lensing is a subject that has a wide range of applications in cosmology and astrophysics. Strong lensing (in which there are large lensing distortions and multiply-imaged sources) is used to measure the masses of galactic halos (and cluster cores) and probe their substructure. Weak lensing is used to determine the mass distribution of galaxy clusters and of large-scale structure on cosmological scales. Microlensing is used to find otherwise undetectable stars and planets. There is a book on lensing [1] and many reviews. In this chapter we give a brief introduction to the subject.

In part I we have laid out the basic understanding of galaxies in modern cosmology. We focused on several main topics. One of them is the power spectrum of density fluctuations, which is the key statistical quantity in cosmology, which in particular determines the distribution of galactic halos. Since the early Universe was highly uniform, linear perturbation theory is a natural framework for understanding the evolution of the power spectrum from the initial density fluctuations, perhaps produced during an early period of cosmic inflation, to its present form roughly in the shape of a broken power law. In the late, non-linear regime, collapse and virialization are the principal concepts in the formation of halos. The former is illustrated by the detailed, analytically tractable example of spherical collapse, while the latter can be analyzed using the virial theorem. These various concepts come together in the Press–Schechter model which yields a quantitative understanding of the connection between the initial fluctuations and the final distribution of massive virialized halos.

This part focuses on cosmology and galaxies at high redshift, covering basic theory and recent research results on galaxy formation at early times, the supersonic streaming velocity, and the cosmic milestones of early radiative feedback. These subjects are all brought together under 21-cm cosmology, which is presented in some detail. The topics discussed here are of great interest in current research, and the high levels of activity and intensity are expected to last. Note that while Part I of this volume naturally reflects this author’s personal style and preferences, this is even more so in Part II, which covers many topics involving the author’s own work.

The following sections are included:

• Brief outline
• Detailed introduction
• References

In this chapter we expand on several features of galaxy formation that are particularly important at high redshifts. We first briefly summarize a number of topics related to halos and the baryons within them. We then discuss fluctuations in the number density of galaxies, which are generally important in cosmology, but at high redshift the fluctuations become unusually large and this has significant consequences that reverberate throughout the remaining chapters of this work. Next, we discuss various challenges of numerical simulations and approaches to deal with them. While simulations have become an indispensable tool in cosmology, it is important to bear in mind that they have fundamental limitations, some of them specific to, or worsening at, high redshifts. For example, while simulations at low redshift can be continually tested by and improved based on astronomical observations, this is not currently possible (or is at least far more limited) at high redshift. Finally, we discuss the formation of the very first stars, obviously a subject of great theoretical and numerical interest, and with significant observational traces as well.

An overview of the basic features and early development of 21-cm cosmology was given in Chap. 10. In this section we present the basic physics in greater detail, then focus on some important low-temperature corrections, and discuss the important subject of anisotropy in the 21-cm signal. Finally, we give a brief overview of the observational aspects of 21-cm cosmology, focusing on the power spectrum. More details of 21-cm physics and observational techniques are available in specific reviews of 21-cm cosmology [1–3].

Current observational efforts in 21-cm cosmology (and high-redshift astronomy more generally) are focused on the reionization era (redshift z ∼ 10), with earlier times considered more difficult to observe. However, recent work suggests that at least in the case of 21-cm cosmology, the pre-reionization, z ∼ 20 era of even earlier galaxies may produce very interesting signals that make observational exploration quite promising. One argument for this is based on a recently noticed effect on early galaxy formation that had been previously neglected. We discuss here this supersonic streaming velocity, which has also been reviewed elsewhere [1].

The following sections are included:

• Reionization
• Lyα coupling and Lyman–Werner feedback
• Cosmic heating
• References

Ongoing and planned interferometric observations in 21-cosmology hope to reach a sub-mK sensitivity level [1, 2] (see also Chap. 10). The best current observational upper limit is from PAPER [3]: 22.4 mK at a wavenumber range of k = 0.1 − 0.35 Mpc⁻¹ at z = 8.4, around an order of magnitude away from plausible predictions (or two orders of magnitude in terms of the power spectrum). Global 21-cm experiments (measuring the total sky spectrum) are also being pursued, with the best result thus far (from the EDGES experiment) [4] being a lower limit of Δz > 0.06 for the duration of the reionization epoch. In the next few sections we focus on 21-cm fluctuations, and consider global experiments separately in Sec. 15.5.

The study of the first stars, galaxies, and black holes, and their effect on the 21-cm sky, is entering a critical stage. While this subject has been developing theoretically for several decades, including a great acceleration in recent years, observationally this field is in its infancy. Thus, we are about to experience that pinnacle of excitement that comes with the first clash of a scientific theory with experimental data. In such a collision of theoretical expectations with reality, there are several possible outcomes. The predictions can be perfectly verified, an outcome that will make the successful theorists gleeful and proud, but at the same time will be quite boring. At the other extreme, the predictions can fail completely, making the theorists a laughing-stock, but revealing previously unexpected cosmic events, which makes this possibility the most exciting one. Neither of these extreme possibilities is expected in the case of 21-cm cosmology. The shear magnitude of the uncertainty about high-redshift astrophysics makes the first possibility unlikely, even in the absence of exotic cosmic events such as dark matter decay. On the other hand, complete failure is made unlikely by the fact that the theory is grounded in solid atomic physics as well as models of galaxy formation that are significantly constrained by observations of the current Universe, at one end, and the CMB at the other (initial condition) end. Thus, the most likely outcome is an intermediate one, where the overall framework of theoretical expectations will be confirmed, but with some, hopefully interesting and significant, surprises, such as an unexpected, new class of astrophysical sources (which will be noticed if it dominated one of the types of radiation that drove the 21-cm emission). Regardless of the precise outcome, it is likely that once a clear detection of the 21-cm signal from early cosmic history is achieved, the field will get a big boost, analogous to the development of CMB observations and theory after the first detection of CMB temperature fluctuations by the COBE satellite [1]. This breakthrough moment for 21-cm cosmology will hopefully occur within the next few years, and will be followed up with confirmations and more detailed measurements soon afterwards.