Cosmology with MATLAB

The following sections are included:

• Preface
• Contents

There are several basic parameters in the SM of cosmology. This section makes a self-contained exposition of the basic facts of big bang (BB) cosmology as they relate to inflation. The SM appears schematically in Figure 1.1. Inflation starts the Universe on a path of rapidly growing size. After inflation the Universe evolves more sedately to the time of CMB creation when photons decouple from matter and then almost freely propagate to us some 14 billion years later. After the “dark ages” stars form and over time matter aggregates, since gravity is always attractive and wins in the long run. Large scale structures (LSS) develop with time. Finally, dark energy (DE) begins to dominate over matter and radiation and the size of the Universe again begins to increase rapidly.

The first section defined the tools needed to study the cosmology of the Big Bang (BB). The time evolution of matter and radiation and the great preponderance of radiation (entropy) means that at early times in the evolution of the Universe, radiation dominated the energy density (RD) while later, the fireball cooled and the Universe became matter dominated (MD). Still later, the unexpected dark energy (ΛD) began to dominate the energy density, as it does today. The time, temperatures and scale factors for these three epochs will be determined later in this section first assuming dominance of a single energy source but later using all sources in numerical calculations.

The issues for BB cosmology identified in the previous section need to be addressed. They can be removed by postulating a period of early and rapid acceleration, driven by an unknown scalar field. The resulting theory will be termed “inflation” in what follows. In this section how inflation solves the BB issues will be explored in some depth.

The inflationary hypothesis solves some outstanding issues in BB cosmology, but at the cost of introducing a new scalar field. However, it also makes additional testable predictions. Specifically it predicts the magnitude of density perturbations and the power and spectrum of the temperature perturbations in the CMB which are nearly independent of the scale of those perturbations. Since H is the only scale during inflation and is approximately constant, an almost scale invariant spectrum of fluctuations is predicted. Because inflation predicts gravity waves, which are as yet undetected, it also makes a major new prediction. An artist's view of the evolution of the fluctuations appears in Figure 4.1, showing both density fluctuations and gravity waves.

After re-heating the Universe evolves in a fashion defined by BB cosmology. The photon-baryon plasma expands and cools until the photons decouple in a manner similar to decoupling of WIMPs and nuclei. The CBM decoupling time discussed in Chapter 2, t_dec is about 4.04 × 10⁵ years and occurs when the temperature was 0.26 eV. The scale, a(t), has since grown by a factor of about 1100. A CMB temperature contour is shown in Figure 5.1. The perturbations in temperature occur on angular scales of a few degrees and are on the 10 μKelvin scale, or parts per 10⁵. The CMB structure has much detailed information encoded in it and it is the earliest period of the Universe directly accessible to observation at present.

The Universe of matter has been assumed to be a uniform fluid so far with small density perturbations. The observed Universe is much more structured as seen in Figure 6.1, a Hubble Space Telescope extreme deep field picture from 2012 [42]. The aim of this chapter is to start to make the connection between the two versions of the structure of the matter in the Universe.

As a good physical theory inflation makes testable predictions; the Universe is flat, density perturbations are almost scale invariant with a spectral index almost one, there are acoustic peaks in the CMB and BAO, perturbations are Gaussian and gravity waves exist with a spectral index near one. Except for the last prediction, the others have been confirmed. The SM of cosmology has been well verified. However, there remains the issue that no known candidates for DM, DE or the scalar field of inflation exist.

The following sections are included:

• Appendix A: Matlab Tools
• Appendix B: Power Law, RD or MD Formulae
• Appendix C: Symbol and Acronym Tables
• Appendix D: MATLAB Script
• References
• Index