The Encyclopedia of Cosmology

The following sections are included:

• Preface
• Contents

The cosmology has made enormous progress after the construction of General Relativity (GR) in 1915. The theoretical predictions of GR — such as the existence of black holes and gravitational waves — have been directly/indirectly confirmed by observations. Now, we know that GR is sufficiently dependable to describe the gravitational law in the solar system.

The following sections are included:

• Hubble–Lemaître’s law
• Cosmic microwave background (CMB)
• Newtonian picture of the expansion of the Universe
• References

Einstein developed two theories of relativity in the early 20th century: special relativity and general relativity. Special relativity is a relativistic theory of spacetime between two inertial frames with non-accelerating observers. General Relativity (GR) is a geometric theory of gravitation constructed by extending special relativity to include acceleration in the theory [1]. Einstein developed the theory of GR by assuming an equivalence principle, under which a freely falling frame of reference is physically indistinguishable from an inertial frame. In other words, a freely falling observer in an elevator cannot distinguish whether he is accelerating under the influence of gravity or he is in a local inertial frame without gravity. If the equivalence principle holds, the inertial mass and the gravitational mass are equivalent to each other. According to local gravity experiments in the solar system, it is known that the equivalence principle between inertial and gravitational masses holds in high precision [2].

To discuss the cosmic expansion history, we need to solve the Einstein equations on a time-dependent cosmological background. In doing so, we employ the cosmological principle stating that the Universe is globally homogeneous and isotropic. On small scales there are inhomogeneities responsible for the formation of compact objects, but they can be regarded as small perturbations on the homogeneous and isotropic background. In this section, we derive background equations of motion by solving Einstein equations and then study how the cosmic expansion is affected by matter species in the Universe.

In 1998, the first direct observational evidence of the late-time cosmic acceleration was reported from the observations of supernovae type Ia (SN Ia). Even before this discovery, there were several pieces of evidence of a “missing matter” problem — such as those from the age of the Universe and number counts of faint galaxies. In this chapter, we first review the problems of cosmic age and number counts of galaxies already known in the early 1990s. We then proceed to describe how the cosmic acceleration was discovered from the SN Ia measurements.

To study the growth of structures in the Universe from a primordial era to today, we need to solve perturbed Einstein equations on the FLRW background in the presence of a fluid or a scalar field. We derive the equations of motion of linear cosmological perturbations by using a gauge-invariant formulation of perturbed quantities. In this and the rest of the chapters, we use the natural unit

In Sec. 6.8, we derived the primordial power spectra of scalar and tensor perturbations generated during inflation. They are regarded as initial conditions of perturbations at the onset of the radiation era. The perturbations re-enter the Hubble radius again long after the end of inflation. After this re-entry, the perturbation of photons starts to oscillate with a frequency associated with the sound speed of a coupled photon–baryon fluid. The oscillations of photons are imprinted in CMB temperature anisotropies. After the onset of the matter-dominated epoch, the gravitational instability of primordial perturbations leads to the growth of inhomogeneities to form large-scale structures in the Universe. In this section, we study the evolution of cosmological perturbations from the radiation-dominated epoch to today, paying particular attention to the physics of CMB temperature anisotropies.

In Sec. 5.3, we explained how the late-time cosmic acceleration was discovered from the SN Ia measurements. The discussions given in Sec. 5.3 were based on observational constraints on dark energy at the background level. In this section, we study observational signatures of dark energy extracted from the information of cosmological perturbations. We discuss how the property of dark energy can be constrained from the measurements of CMB, galaxy clusterings, redshift-space distortions, and weak lensing.

The cosmological constant Λ is the simplest model of dark energy. However, if it originates from the vacuum energy, its predicted energy scale is vastly larger than the observed dark energy scale. Since the existence of such large values of Λ spoils the known expansion history of the Universe, the cosmological constant problem was recognized as a serious problem even before the discovery of the late-time cosmic acceleration in 1998 [1]. In this chapter, we first review the fine-tuning problem of Λ by explicitly calculating the vacuum energy and present a number of approaches that aim to tackle this problem.

If the cosmological constant problem is solved in such a way that it vanishes (almost) completely, we need to find an alternative mechanism to explain today’s cosmic acceleration. Broadly speaking, there are two approaches to addressing the dark energy problem. One of them is to introduce a specific form of matter with a negative pressure — which we call modified matter models. Another is known as modified gravity models in which the large-distance modification of gravity drives the cosmic acceleration. In this chapter, we review modified matter models of dark energy, paying particular attention to scalar-field models (quintessence and k-essence) and coupled dark energy with dark matter.

The second approach to the problem of the late-time cosmic acceleration is based on the modification of gravity at large distances. The left hand side of Einstein equations (3.85) is modified in this approach. In terms of the action (3.87), the Lagrangian density R is subject to change. This geometric modification breaks the gauge symmetry of GR, which generally gives rise to additional propagating degrees of freedom. It is possible to construct models in which the new degrees of freedom give rise to the cosmic acceleration. In this chapter, we introduce several representative models of modified gravity with a scalar degree of freedom and study the resulting background cosmological dynamics. In Chap. 12, we consider more general theories with second-order equations of motion and study the evolution of cosmological perturbations and their observational consequences. These theories need to recover the gravitational law close to that of GR in the solar system for the consistency with local gravity experiments. In Chap. 14, we will discuss several mechanisms of how to screen fifth forces mediated by new dynamical degrees of freedom.

In Sec. 11.3, we showed that the construction of second-order theories is important to avoid the Ostrogradski instability. The equations of motion of the Galileon field is of second order on the Minkowski background. If we simply replace partial derivatives of the Minkowski Galileon with covariant derivatives on the curved background, this gives rise to derivatives higher than second order. Adding gravitational counter terms to the action allows one to construct second-order theories in both the field and the metric. The similar prescription is also possible for a general single scalar field ø. Most general scalar–tensor theories with second-order equations of motion are known as Horndeski theories [1]. In this chapter, we will review Horndeski theories and its cosmological applications with the derivations of perturbation equations of motion as well as stability conditions. We will also apply our general results to place observational constraints on concrete modified gravity models of late-time cosmic acceleration from observations associated with the growth of inhomogeneities.

So far, we have studied the cosmological dynamics and observational consequences in scalar-tensor theories with one scalar degree of freedom (DOF). There are models in which the cosmic acceleration is driven by an Abelian vector field A_μ in the presence of derivative interactions [1–4]. The propagation of such an Abelian vector field is mediated by a spin-1 particle. In general, a massive spin-s particle has 2s+1 propagating DOFs. The scalar field is a spin-0 particle, so it has 1 scalar DOF (besides two tensor polarizations in the presence of gravitational interactions).

In modified gravity theories, the breaking of gauge symmetry in GR gives rise to propagating scalar or vector degrees of freedom. Since this mediates the propagation of extra forces coupled to non-relativistic matter, we need some screening mechanism of such fifth forces for the compatibility with local gravity experiments in the solar system. There are two representative mechanisms for suppressing the propagation of fifth forces in local regions of the Universe: (i) chameleon mechanism, and (ii) Vainshtein mechanism.

The effective field theory (EFT) of dark energy is a powerful framework to deal with low-energy scalar degrees of freedom in a systematic and unified way [1–6] (see also Refs. [7, 8] for the EFT of inflation). This approach is based on the expansion of a general four-dimensional action about the flat FLRW background in terms of the perturbations of three-dimensional geometric scalar quantities appearing in the 3 + 1 Arnowitt–Deser–Misner (ADM) decomposition of spacetime [9]. Such geometric scalars consist of a lapse function N as well as traces and squares of an extrinsic curvature K_μv and a three-dimensional intrinsic curvature ⁽³⁾R_µv [10]. The Lagrangian can also depend on a scalar field ϕ and its kinetic energy X, but it is possible to absorb such dependence into the lapse by choosing a unitary gauge in which the field perturbation δϕ vanishes.

The problem of dark energy has been at the heart of astronomical observations and is driving the way cosmologists and particle physicists are trying to understand the nature of the early and late Universe. In fact, it has led to a remarkable number of publications after its first discovery. More than 4600 papers with the word “dark energy” in the title have appeared on the arXiv between 1998 and 2017, with more than 2300 papers containing “cosmological constant”.

The following sections are included:

• Appendix A: Equations of Motion in Horndeski Theories
• Appendix B: Effective Mass Term in Horndeski Theories
• Index