Recombination

The photons of the CMB that we observe come from the epoch at a redshift z ~ 1090 when the cosmic plasma was almost neutral and the timescale for electron–photon collisions became longer than the cosmic expansion timescale. The Universe was then some ~ 380 000 years old. Prior to that time, electrons and photons had been held in thermodynamic equilibrium by the Thomson scattering process, while after that time the baryonic component of the cosmic plasma could evolve as an independent component of the plasma.

This was the time of the decoupling of matter from the radiation field. Shortly after that time most of the CMB photons were able to travel directly to us now without being scattered by free electrons. This gave us a direct view of the early conditions that led to the formation of the cosmic structures we see today, the galaxies, clusters of galaxies and their organisation into what is now known as the ‘cosmic web’.

The Universe did not suddenly become neutral as it emerged from the fully ionised fire-ball phase of the cosmic expansion. The neutralisation, or ‘recombination’, of the cosmic plasma took place over several tens of thousands of years. That is enough to slightly blur the details of the structure that might be observed on the last scattering surface: we are looking into the cosmic photosphere.

Early Thermal History

Astro-particle physics inevitably brings in an even wider domain of physics than the classical cosmology of general relativistic models. Some fine texts have been written on this subject from a variety of points of view and at different levels. The approach used here is to explain the physics of the early Universe with a view to understanding what precision cosmology has to say about the particle and nuclear physics aspects of the earliest moments after the Big Bang.

The first steps in this direction were taken by Gamow and his co-workers during the first decade after the Second World War. Gamow had realised that the wartime research into nuclear physics could answer some important questions about his concept of the early Universe that he had started working on prior to 1939. We can trace back Gamow's interest in a Big Bang Universe to Gamow and Teller (1939a,b) in which he had described a simple idea for the formation of galaxies, and no doubt his interest extended back to his brief association with Friedmann.

Introduction

Almost every civilisation throughout history has had a cosmology of some kind. By this we mean a description of the Universe in which they live based on their state of knowledge. The Vikings, for example, had a complex cosmology in which the world and its inhabitants were controlled by a set of Gods, both good and bad. Nature was ruled not by the laws of physics, but by the forces of nature controlled by the whims of these Gods. However bizarre that may seem to us now, at the time this belief-set dominated human behaviour: its social mores and values.

Today we live in a Universe that is described by physical laws. What is remarkable is that these laws have more often than not been discovered on the basis of laboratory experiments, and subsequently found to work on the vastest scales imaginable. That fact leads us to believe that our explanations of the Universe are a valid description of what is actually happening. We do not need to invoke special laws just to explain the cosmos and our position within it.

The physical laws governing the Universe and its constituents were discovered over a period of several centuries. Some might say this path to realisation started with Copernicus putting the Sun at the centre of everything rather than the Earth. Others might argue that this was merely descriptive and that knowledge of the laws started to emerge following on from the work of Kepler, Galileo and Newton. However one sees it, by the beginning of the 20th century, with Einstein's Theory of Relativity, the scene was set to embark on a journey of observational cosmology which 100 years later would lead to most scientists agreeing that we have a self-consistent theory of the Universe based on known laws of physics. Some, no doubt, would go as far as to say that the current view was incontrovertible.

Just as the early map makers measured and marked out our planet, the cosmographers of the 20th century marked out and mapped the Universe.

Models of the Cosmic Expansion

Several spatially homogeneous solutions of the equations of the General Theory of Relativity were discovered within the first decade following their publication (Einstein, 1916a). At that time relatively little was known about the Universe: it was still uncertain whether or not the nebulae were merely parts of our own Galaxy, although, through the pioneering work of Slipher, it was known that most of them were rushing away from us. Little or nothing was known about the homogeneity or isotropy of the Universe, but the assumption of homogeneity and isotropy would simplify the largely intractable Einstein equations. During the decade following the publication of the Einstein equations several important cosmological solutions were discovered. These are discussed next.

It is interesting in this context to read and compare the texts of Eddington (1923), published before Edwin Hubble's study of the nebulae and his consequent discovery of the cosmic expansion, and the text of Tolman (1934b) which was published shortly thereafter. The search for understanding the Universe in terms of models is beautifully described in Michael Heller's Ultimate Explanations of the Universe (Heller, 2010).

Use of Figures

The publications for which a licence to re-use the material had to be obtained from the publisher, or their agent, are as follows:

1. • Figure 1.6 reproduced from Gamow (1956, Fig. 1. p.1731) with permission (©Elsevier 1956).

2. • Figure 3.2 reproduced from Bortolot et al. (1969, Fig. 1 p.308) with permission, ©1969 American Physical Society.

3. • Figure 3.7 (right) reproduced from Sunyaev and Zel'dovich (1970a, Fig. 1b p.9) with permission from the publishers via CCC 37001813366238.

4. • Figure 3.11 (left) reproduced from Gawiser and Silk (2000, Fig. 4 p.17) with permission from the publishers via CCC 3700450644616.

5. • Figure 4.3 adapted by permission from Macmillan Publishers Ltd: Massey et al. (2007, Nature) ©2007. CCC licence 3730161227530, article æ6.

6. • Figure 27.3 reproduced from Sherwin et al. (2011, Figure 3) with permission from the publishers via CCC 3721150136987. ©2011 by the American Physical Society.

In all cases the full title of the article and other bibliographic data are available from the corresponding entry in the References section. ‘CCC’ refers to permissions gained through the Copyright Clearance Centre and the number following is the granted licence number. These cover situations where the authors’ permission was required but was not available.

NASA copyright policy states that ‘NASA material is not protected by copyright unless noted’. Thus Figure 3.1 is in the public domain.

Unless otherwise noted, images and video on Laser Interferometer Gravitational-wave Observatory (LIGO) public web sites (public sites ending with a ligo.caltech.edu or ligo.mit.edu address) may be used for any purpose without prior permission. See Figure 4.6.

Figure 3.10: The Particle Data Group publishes annual Reports on Particle Physics (RPP which are published in the journal Chinese Physics, C. Since 2014, the figures from RPP are in the public domain (Olive and Particle Data Group, 2014) and author permission is automatically granted. This concerns Figure 27.1 of Scott and Smoot (2014).

The distance to an astronomical source is a fundamental astronomical datum about the source. While astronomers measure sizes of objects in centimeters, meters or kilometers, distances to distant objects are measured in light years or parsecs. Large objects like galaxies and clusters of galaxies are also measured in light years or parsecs.

The light year is simply the distance travelled by light in one year and it is a fairly convenient unit of measure for the distances to stars. However, distances to stars are not measured using the travel time of light rays: nearby stellar distances are measured using parallax. The parallax is the maximum apparent angular shift in position of a star on the sky as the Earth goes from one side of its orbit to the other. This is the same as the angle subtended by the Earth's orbit as viewed from the star.

The parsec is the distance at which the parallax of an object would be 1 arc second. The parallax of the nearest star, 4.2 light years away, is 0.772 arcsec. The relationship between light years and parsecs is 1 parsec = 3.26 light years. Distances to cosmological objects are to be measured in Megaparsecs (Mpc).

Because of the curvature of space the distance as measured by parallax is not the same as the distances measured by brightness, or by diameter.

Magnitudes and all that

It is difficult to avoid discussing the apparent brightness of objects such as stars and galaxies without encountering one of astronomy's major idiosyncrasies – the magnitude scale. Ever since the time of Hipparchus of Niacea (c190BC–120BC), the brightnesses of stars have been measured relative to one another on a logarithmic scale. Around 150BC, Hipparchus had classified the brightest stars he could see as being of ‘magnitude 1’ and the faintest as being of ‘magnitude 6’: a change of 5 magnitudes represented a factor of 100 in brightness. Because the brain perceives increments in brightness logarithmically, each increment of 1 magnitude corresponded to a change in brightness by a factor of 10⅕ = 2.512.

The magnitude scale, as we know it today, was introduced by Pogson (1856). Nowadays we might decide to base a logarithmic brightness scale on powers of two, so that each ‘magnitude’ was twice as bright as its predecessor.

Systems of units are always a problem, despite international agreement to standardise on the SI system. The old cgs system and its multitude of mutually inconsistent variants (Gauss, Lorentz–Heaviside, esu, emu, etc.) is still in use. This is particularly an issue when it comes to using the Maxwell equations. In astrophysics we generally see a mixture of cgs units and subject-specific units, like the magnitude scale for brightness, parsecs for distance, and flux units in the older radio astronomy literature. This is what we might call the Astrophysical System of Units and, like many contemporary authors, I have adhered to that in this text. However, certain issues must be clarified, and in particular the conventions used in the Maxwell equations.

This appendix provides translation between Gaussian cgs units and the SI system for several of the quantities that are used in the text, and provides some update on the values of the fundamental physical constants that have occurred since the decision to fix the speed of light at a given numerical value. The Maxwell equations have a small section to themselves.

SI, MKS and cgs

There has been a slow move in astronomy away from the traditional centimetre-gram-second (CGS) based system of units that evolved in the 19th century towards the metric metre-kilogram-second (MKS) based system of units and the subsequent International System (SI), which is also known as the MKSA system. However, the change has been slow or, as in the case of astronomy, only partial. By the 1950s many fields had adopted their own subject-specific units, chosen largely because they were particularly convenient or simply entrenched in the culture of the discipline. There are perhaps just two reasons for the slowness of this change: teachers today were often taught in cgs or hybrid systems, and used textbooks with which they were familiar. Moving from cgs to SI units takes people out of their comfort zone.

The full details of the system of physical units are maintained and documented by the National Institute of Standards and Technology, NIST, which is maintained by the US

Department of Commerce. The information presented here is for convenience and is largely abstracted from the NIST web site.

Light Propagation in Curved Space-Time

Understanding the physics of light propagation is a vital step in interpreting the observed cosmological redshift. Getting to understand the redshift was not at all straightforward even though it is a consequence of the cosmological models constructed within the context of general relativity. Some of this early story with regard to de Sitter's Universe was retold in Section 15.6.4.

As late as the 1960s and 1970s there was still an active, and often heated, debate as to whether the observed redshifts of galaxies and quasars were of cosmological origin. That was not simply a question of whether or not general relativity could provide the explanation for the phenomenon. The debate was more about whether there was an additional contribution from phenomena intrinsic to the source or from something that might happen along the path of the light rays (as in ‘tired light’). Some of the debate even suggested that the constants of nature might be a function of time, an idea that originated with Dirac (1937, 1938) following up on earlier ideas of Milne and Eddington.

The subject matter in this book covers the theory and observations relating to the earliest epochs of our evolving Universe, from its singular origin to just beyond the epoch of recombination: the first half million years. This is a fundamental period of the Universe's lifetime, and the period from which the key parameters describing it can be determined. As our understanding of the Universe grew, and new observations were found to rule out the simplest models, new physical entities and concepts had to be introduced to ensure consistency between theory and observation. The subsequent period, from recombination to the present epoch, is the era during which stellar and galactic physics takes place, and is not covered in this volume.

Precision Cosmology is a teaching book and a technical reference book which aims to provide a clear and understandable description of the applied mathematics and physics required to establish the high level of precision which can now be achieved for the study of the Universe – Precision Cosmology. It details the background and mathematical tools needed to enable a student embarking on advanced studies and research to understand what others have done and published.

This book is also an experiment in presentation. The printed text is accompanied by extensive online supplements, which provide a high level of didactic mathematical material too extensive to be included in the main text. These supplements are freely available for download and are intended as a source of teaching material for both teachers and students across a range of topics in physics, mathematics and statistics. They are suitable for all levels of university study, from undergraduate to postdoctoral research. The online supplements and other support material are available from

www.BernardJones.net/PrecisionCosmology.html

Figures in the text marked cc are available on this website and are free for use under the terms of the CC-BY-SA Creative Commons License. Exercises and slides for teaching purposes are also to be found on this site.

In this way, much of the book should be accessible to physics and mathematics undergraduates who have had courses in mechanics, electromagnetism and special relativity. The aim is to take the student beyond this basic knowledge, right up to the level of being able to follow and understand research papers, and thus to understand today's cosmology in its proper context as a branch of physics.

Observing the Universe

The increased level of precision with which we can determine the parameters that define our Universe has grown as a result of research by thousands of astronomers and physicists. Although a few lines of research have grown to dominate our thinking in regard to cosmological parameters, we have only arrived at that point due to efforts in a large number of different explorations. The culmination of that effort was perhaps the discovery of the acceleration of the cosmic expansion, and the subsequent efforts to pin down the properties of the expansion using high-precision techniques.

The Radiation Field

The cosmic fireball is a somewhat unusual environment, far from any laboratory experience. The principal difference lies in the intensity of the cosmic radiation field: in the Universe photons outnumber baryons by some eight orders of magnitude.

Some of the material in this chapter is based on the seminal paper of Sunyaev and Zel'dovich (1970). This is one of the most important papers written on the subject of the CMB.

The Equilibrium Distribution of Photons

The radiation field can be most simply defined by telling how many photons there are of a given frequency moving in a particular direction. However, things are complicated somewhat because the radiation field is measured in terms of the flux of energy carried across a unit surface by photons of a given frequency moving into a given solid angle about a direction normal to the surface. For radiation in thermodynamic equilibrium at some temperature, this flux follows the famous Planck law. In this section we discuss the concept of flux and then relate it to the equilibrium distribution function for the photons and so derive the Planck law from basic principles.