### 4. RELATIVISTIC COSMOLOGICAL PERTURBATION THEORY

This section is an expanded version of my fifth lecture at Les Houches. One lecture gave barely enough time to introduce the essential ideas of relativistic perturbation theory: classification of metric perturbations, the linearized Einstein equations, and gauge modes. Understanding the physics of these topics, as well as the relativistic generalizations of my previous lectures, requires a much deeper immersion. Unable to find a pedagogical treatment in the existing literature that matches these needs to my satisfaction, I have developed the subject more fully in these written lecture notes. They are not a complete guide to relativistic perturbation theory but rather a starting point from which the reader may delve into the increasingly rich literature of applications. This section is self-contained and may be read independently of the previous sections, although the reader may find it interesting to contrast the nonrelativistic presentations of sections 1 and 2 with the relativistic treatment given below.

According to the Newtonian perspective of gravity and cosmology, spacetime is flat and absolute, gravity is action at a distance, and particle dynamics is given by Newton's second law F = ma or, equivalently, by Hamilton's principle of least action. The Einsteinian perspective is quite different: spacetime is a curved manifold which evolves causally through the Einstein field equations in response to sources, and particle dynamics is given in absence of nongravitational forces by geodesic motion. In this section I attempt not only to present the essentials of relativistic gravitational dynamics, but also to show how it reduces to and extends Newtonian cosmology in the appropriate limit.

One of the main purposes of these notes is to provide a clear explanation of the scalar, vector, and tensor modes of gravitational perturbations. (We shall follow the customary usage in this subject by referring to different spatial symmetry components as "modes" even when they are not expanded in any basis eigenfunctions. Thus, the "scalar mode" is described, in part, by a field (xµ) that is a scalar under spatial coordinate transformations but is not restricted to being a single Fourier component or other harmonic basis function.) Newtonian gravity corresponds to the former (the scalar mode), while the latter (vector and tensor modes) represent the relativistic effects of gravitomagnetism and gravitational radiation, which have no counterpart in Newtonian gravity although they are similar to electromagnetic phenomena. If the motion of sources is expanded in powers of v/c, the vector and tensor gravitational fields are O(v / c) and O(v / c)2 times the Newtonian field, respectively. On terrestrial scales the vector and tensor modes are extremely weak - they have not been detected in the laboratory, although satellite experiments are planned to search for the former through the Lense-Thirring "gravitomagnetic moment" precession, and large interferometric detectors are being built to measure gravitational radiation - but they could have important consequences for the evolution of large-scale matter and radiation fluctuations, including the production of anisotropy in the microwave background radiation.

The Newtonian limit corresponds to weak gravitational fields (black holes are to be avoided) and slow motions (v2 << c2, for both sources and test particles). For nearly all cosmological applications it is sufficient to consider only weak fields - small perturbations of the spacetime metric around a homogeneous and isotropic background spacetime. At the same time it is usually safe to assume that the gravitational sources are nonrelativistic, although the test particles (e.g., photons) need not be. Because the weak-field, slow source motion limit does not necessarily imply small density fluctuations, we can (and will) investigate nonlinear particle and fluid dynamics even while treating the metric perturbations and source velocities as being small.

In sections 4.2-4.5 we shall develop the machinery for cosmological perturbation theory using the methods developed by Lifshitz, Peebles, Bardeen, Kodama & Sasaki, and others. We discuss the consequences of gauge invariance - the invariance of physical quantities to small changes in the spacetime coordinates - and summarize the standard results in the synchronous gauge of Lifshitz (1946). (1) In section 4.6 we introduce a new gauge that clarifies how general relativity extends Newtonian gravity in the weak-field limit and in section 4.7 we attempt to clarify the physical content of general relativity theory in this limit. In section 4.8 we shall see how simply and clearly the Hamiltonian formulation of particle dynamics follows from general relativity. Finally, in section 4.9 we introduce an alternative fully nonlinear formulation of general relativity due to Ehlers, Ellis and others, and we demonstrate its connection with the Lagrangian fluid dynamics that was discussed in my fourth lecture.

We shall not discuss the relativistic Boltzmann equation nor the classification of isentropic and isocurvature initial conditions. In the nonrelativistic limit, these topics have already been covered in my preceding lectures. Neither shall we discuss the physics of microwave background anisotropy or the evolution of perturbations in specific models. Our aim here is to derive and comprehend the gravitational field equations, not their solution. Although this goal is restricted, we shall see that the physical content is sufficiently rich. After working through these notes the reader may wish to consult one of the many books or articles discussing the detailed evolution for a variety of models (e.g., Lifshitz & Khalatnikov 1963; Peebles & Yu 1970; Weinberg 1972; Peebles 1980; Press & Vishniac 1980; Wilson & Silk 1981; Wilson 1983; Bond & Szalay 1983; Zel'dovich & Novikov 1983; Kodama & Sasaki 1984, 1986; Efstathiou & Bond 1986; Bond & Efstathiou 1987; Ratra 1988; Holtzman 1989; Efstathiou 1990; Mukhanov, Feldman & Brandenberger 1992; Liddle & Lyth 1993; Peebles 1993; Ma & Bertschinger 1994b).

Understanding these notes will not require much experience with general relativity, although some background is helpful. The reader can test the waters by examining the following summary of essential general relativity and differential geometry. While some mathematical formalism is needed to get started, the focus thereafter will remain as much as possible on physics.

1 The same gauge has been proposed recently by Bombelli, Couch & Torrence (1994), who call it "cosmological gauge." However, I prefer the name Poisson gauge because cosmology - i.e., nonzero - is irrelevant for the definition and physical interpretation of this gauge. Although I have seen no earlier discussion of Poisson gauge in the literature, its time slicing corresponds with the minimal shear hypersurface condition of Bardeen (1980). Back.