If the cosmological constant problem is solved in a way that Λ completely vanishes, we need to find alternative models of dark energy. As we already mentioned in the Introduction, there a re basically two approaches for the construction of dark energy models. The first approach is based on “modified matter models” in which the energy-momentum tensor Tμν on the r.h.s. of the Einstein equations contains an exotic matter source with a negative pressure. The second approach is based on “modified gravity models” in which the Einstein tensor Gμν on the l.h.s. of the Einstein equations is modified.

It is however important to realize t hat within General Relativity this division is mostly a practical way to classify the variety of dark energy models but does not carry a fundamental meaning. One can always write down Einstein's equations in the standard form Gμν = 8π GTμν by absorbing in Tμν all the gravity modifications that one conventionally puts on the l.h.s.. In other words, one can define a covariantly conserved energy-momentum tensor that equals the Einstein tensor. There is no way, within General Relativity, i.e. by using only gravitational interactions, to distinguish modified matter from modified gravity.

The existence of dark energy is supported by a number of observations. This includes (i) the age of the Universe compared to oldest stars, (ii) supernovae observations, (iii) Cosmic Microwave Background (CMB), (iv) baryon acoustic oscillations (BAO), and (v) large-scale structure (LSS).

Even before 1998 it was known that in a CDM Universe the cosmic age can be smaller than the age of the oldest stars. Dark energy can account for this discrepancy because its presence can make the cosmic age longer. The first strong evidence for the acceleration of the Universe today came however by measuring the luminosity distance of the type Ia supernovae (SN Ia). The CMB observations are also consistent with the presence of dark energy, although the constraint coming from the CMB alone is not so strong. The measurements of BAO have provided another independent test for the existence of dark energy. The power spectrum of matter distributions also favors a Universe with dark energy rather than the CDM Universe. In the following we shall discuss this observational evidence for dark energy. The statistical method used to constrain cosmological parameters will be discussed in Chapter 13. More details on present and future observational aspects to detect dark energy will be presented in Chapter 14.

The age of the Universe

As we alreadymentioned, the inverse of the Hubble constant H0 is a rough measure of the age t0 of the Universe.

The simplest candidate for dark energy is the cosmological constant Λ, which is so called because its energy density is constant in time and space. In fact the ΛCDM model has been systematically proved consistent with a large number of observations. The Lagrangian density for the ΛCDM model is simply given by the linear term in R plus Λ, see Eq. (6.2). Despite its simplicity it is generally difficult to explain why the energy scale of the cosmological constant required for the cosmic acceleration today is very small relative to that predicted by particle physics. As we already mentioned, the vacuum energy density evaluated by summing the zero-point energy of a scalar field is about 10 times larger than the observed dark energy density (for a momentum cut-off around the Planck scale).

The problem of a large value of Λ was present long before the observational discovery of the late-time cosmic acceleration. In fact, even if we had no observational evidence of dark energy we would still need to understand why the cosmological constant vanishes. Models of dark energy alternative to ΛCDM are based on the assumption that Λ is zero or negligible. So the problem of the cosmological constant is to find some mechanism that either makes it vanish or renders it a very small value compatible with the present cosmological density. In the former case the origin of dark energy needs to be explored further, but in the latter case the problems of the cosmological constant and dark energy are solved at the same time.

The observables used to obtain information about the global properties of the Universe are not many: distances, background radiation, source positions and velocities, galaxy shapes as an indicator of lensing shear, galaxy or cluster densities, all of them as functions of redshift. All these observables can in principle be employed to constrain the properties of dark energy. Five methods emerged so far as possibly the best tools for exploring the property of dark energy: SN Ia, CMB, LSS (including BAO), weak lensing and galaxy clusters. In some cases this selection was based on the actual current performance (e.g., the SN Ia method that gave birth to the whole dark energy concept); in others, on good promises for the next decade (e.g., weak lensing).

In Section 5.2 we already addressed the SN Ia method in some detail. In this chapter we discuss the other techniques and their prospects. We also explain the potential of alternativemethods such as age tests, gamma ray bursts, strong lensing, and redshift drift.

Dark energy and the CMB

The physics of CMB is a wonderful playground for cosmologists. The initial conditions are set by inflation; the evolution of perturbations involves the delicate interplay of photons, baryons, neutrinos, dark matter, and dark energy, all coupled either directly or via the gravitational field. Finally, the observation and the analysis of the anisotropies themselves also involve very interesting physics, mathematics, and statistics.

The models of dark energy we will introduce later on are linked to the observations by a precious tool, statistics. Since this world is complicated, we have to average the ups and downs of everyday life to get a sense of the underlying substance. In this chapter we present basic tools of statistics in order to confront dark energy models with observations.

Statistics itself is often divided into descriptive statistics, i.e. how to condense the data in a compact and useful way, and estimation (or inferential, inductive) statistics, i.e. how to derive information on model parameters. We start here with the statistics needed for cosmological perturbation theory, essentially descriptive statistics such as correlation function and power spectrum, and postpone parameter estimation statistics to Chapter 13.

A note on notation is in order here. When there is no real need, we will not use separate notation for an estimator (say, the correlation function) and its expected value. Similarly, when there is no risk of confusion, we will denote vector quantities like position x and wavevector k with unbolded fonts x, k, especially to denote the argument of functions: δ(x) will in general mean the density contrast at a position x. Finally, when an integration, even a multiple one, is extended to the whole space we simply write ∫ d V or ∫ dx. When the domain of integration really matters, then it will be specified.

From the observational data of Supernovae Type Ia (SN Ia) accumulated by the year 1998, Riess et al. [1] in the High-redshift Supernova Search Team and Perlmutter et al. [2] in the Supernova Cosmology Project Team independently reported that the present Universe is accelerating. The source for this late-time cosmic acceleration was dubbed “dark energy.” Despite many years of research (see e.g., the reviews [3, 4, 5, 6, 7]) its origin has not been identified yet. Dark energy is distinguished from ordinary matter species such as baryons and radiation, in the sense that it has a negative pressure. This negative pressure leads to the accelerated expansion of the Universe by counteracting the gravitational force. The SN Ia observations have shown that about 70% of the present energy of the Universe consists of dark energy.

The expression “dark energy” may be somewhat confusing in the sense that a similar expression, “dark matter,” has been used to describe a pressureless matter (a non-relativistic matter) that interacts very weakly with standard matter particles. The existence of dark matter was already pointed out by Zwicky in the 1930s by comparing the dispersion velocities of galaxies in the Coma cluster with the observable starmass. Since darkmatter does not mediate the electromagnetic force, its presence is mainly inferred from gravitational effects on visible matter. Dark matter can cluster by gravitational instability (unlike standard dark energy) so that local structures have been formed in the Universe.

Cosmology is, by and large, the realm of linear gravitational processes. When gravitational instability reaches a regime of non-linearity, astrophysical objects form (galaxies, black hole, stars) and the memory of the global structure of spacetime, and therefore of cosmology, is lost or diluted in new physics and new interactions. There is however an intermediate regime in which gravity is still the only player but effects beyond linearity begin to be observable. This regime lies between the linear perturbation theory we have explored in the previous chapter and the full nonlinear dynamics that can be dealt with only in N-body simulations or by focusing on single objects.

It may be expected that the presence of dark energy will not influence small-scale non-linear processes. This is probably true for standard dark energy, i.e. a smooth component with negligible clustering described by a slowly varying equation of state. However we have learned how rich the possible phenomenology of dark energy is. We cannot exclude that weakly non-linear processes might keep some record on the cosmological conditions in which they developed. We know of at least one such process, the epoch of the beginning of structure formation and, as a consequence, the abundance of collapsed objects.

This chapterwill present the effects of non-linearity in higher-order cosmological perturbation theory that are of interest in dark energy research. Primes here denote differentiation with respect to N = ln a.