In this chapter we present the basics of linear perturbations in cosmology. After a general introduction of cosmological perturbation theory, we work out several cases: (i) single pressureless perfect fluid, (ii) single general perfect fluid, and (iii) two fluids: matter and radiation. We also discuss a number of topics such as the velocity field, the redshift distribution, Boltzmann equations, the matter power spectrum, and the perturbed photon propagation. These provide us with important tools when we confront dark energy models with observations of the cosmic microwave background (CMB) and large-scale structure (LSS).

In this chapter the treatment and notation are fairly standard and the topic is covered in most modern textbooks on cosmology. Readers familiar with cosmological perturbation theory may skip this chapter.

Perturbing General Relativity

In Chapter 2 we have outlined the cosmic expansion history in the homogeneous and isotropic FLRW background. However, our Universe is far richer than this simple picture. A metric that deviates from the FLRW spacetime can be written as the sum of an unperturbed FLRW part plus something else, that we can generally call “perturbed” metric. If the perturbed part is assumed to be small, in a sense to be defined later, then this splitting of the full metric into a background part and a perturbed one leads to extremely useful results. As we all know, physics is to a large extent described by a Taylor expansion to some low order, and cosmology is not an exception.