×

zbMATH — the first resource for mathematics

Fundamentals of cosmology. (English) Zbl 1017.85001
Berlin: Springer. xi, 302 p. (2001).
This book arose from classes given at the University of Paris and the Ecole Polytechnique (Palaiseau). It is an introduction addressing the issue of producing a treatise on cosmology while providing a necessary background in observational astronomy, astrophysics, elementary particle physics, nuclear physics and general relativity. The last topic is handled in a very simplified way without using the differential geometry machinery. The book stands in spirit in a middle way between theory and a thorough discussion of observation techniques, emphasizing on phenomenology.
Chapter one is an introduction which reproduces in miniature the topics which will be developed later. Here many physical values are quantified and compared so that the reader gets a feeling for the universe. The author defines his science in the following sentence: “The primary aim of cosmology is to determine the thermal, chemical and structural state of the present universe and to explain this state in terms of conditions in the early universe”.
The story starts with the observation of galaxies and clusters of galaxies (the “structure” component of the definition) and Hubble’s view according to which galaxies are receding (from us; this means that one assumes uniformity in order to avoid anthropocentrism) with a velocity proportional to their distance \(R\): \({dR\over dt}=H_0 R+v_p\). This law combined with the observed redshift of photons emitted by the galaxy (Doppler effect?) allows to evaluate its distance. Concerning the chemical composition, the author gives estimates of the density of various components (baryons visible or not, cold dark matter (weakly interacting particles) whose existence is hypothesized because there is not enough of the previous baryons to explain structure formation and its inhomogeneity especially at an early phase, photons the majority coming from the cosmic background radiation and one has around 400 of them in a cubic centimer on average today, the corresponding thermalized neutrinos with around 100 per species in the same volume on average today, a vacuum energy hypothesized to explain the observed acceleration of the expansion of the universe (one associates hypothetical particles from supersymmetrical models). The expansion of the universe is characterized by a scale factor \(a(t)\) which satisfies a nonlinear differential equation, Friedmann’s equation. It is derived here first using the classical Newtonian gravitational potential and Newton’s dynamical principle. The constant of integration is obtained from Hubble’s law today. Various destinies for the universe (expansions, contraction) depend on the value of constants attached to various terms. The vacuum energy would save the universe from imploding. The evolution of temperature is then directly obtained from the redshifts of the cosmic background radiation photons identified with those sent from a receding galaxy and then related to \(a(t)\). The Friedmann equation is refined by adding decelerating relativistic terms (photons, neutrinos) and an expansion independent accelerating one (vacuum energy). This leads to a scenario for structure evolution (radiation dominated first, matter dominated next, cold dark matter dominated subsequently and finally vacuum dominated). The scenario is associated to naturally changing physical phenomena (baryogenesis, nucleosynthesis, atom formation, structure formation). It is pointed out that 1) the values of some cosmological parameters are not always known precisely and that they may vary with time to match data or even be more complicated (random quantities) 2) cold dark matter and vacuum energy are not attested experimentally 3) gravitation is not understood in general (quantum gravity, large scale effects) 4) the physics of the early universe is not understood (asymmetries, excess of matter over anti-matter\(,\dots)\) 5) The physics determining the proportions of radiation, cold dark matter and nonrelativistic matter is not known.
Chapter two is devoted to observational cosmology (stars, supernovae, quasars, galaxies and the measurement of standard parameters and densities)
Chapter three gives a setup for introducing a suitable metric (isotropy, homogeneity) and then derives geodesics with an eye on practical issues like relating photon characteristics and information on the sending object.
Chapter four rederives the Friedmann equation but this time in the context of general relativity.
Chapter five studies various topics associated to the inflation model (the determination of the age of the universe via densities themselves obtained from luminosity and angular distances, the horizon problem and the plausibility of a total density close to the critical one are both related to inflation).
Chapter six deals with the thermal history of the universe and the associated densities of particles. A key aspect is the determination of how the associated thermal equilibrium is broken.
Chapter seven is devoted to issues related to cold dark matter (involving inhomogeneity and anisotropy).
The appendices provide background mathematical and physical information as well as formulae and values of various physical parameters. Each chapter has a set of exercices and some solutions are provided at the end. One finds 163 references and an index. There are some typographical mistakes.
The reviewer carries a doubt since childhood when first confronted with Friedmann’s model: “Is space expanding through us or do we expand with space?”. This is closely related to the two antinomic views according to which a photon is sensitive to the expansion of the universe because of the redshift while galaxies and matter get diluted when the universe expands. Even if there is a difference of scale, these two conceptions should be properly patched together. In the book an answer has not been found but one reads the corresponding passages: p. 23 “The behaviour [the falling of energy density of non-relativistic matter] is simply due to the dilution of the particles” and page 25: “This elegant expression says that the wavelength of a photon scales by the same factor as the universe itself” and there is an associated footnote: “This sometimes leads people to say that the wavelength of a photon grows because space is expanding. While the author does not claim to understand what is meant by this statement, he does not deny that it is conceivable that a meaning can be found. It should however be emphasized that space is apparently not expanding uniformly because the sizes of bound objects like atoms and galaxies do not grow with time”. But uniform expansion is a key assumption on which the whole development is being built!

MSC:
85-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to astronomy and astrophysics
85A40 Cosmology
83F05 Cosmology
PDF BibTeX XML Cite