We present in a manifestly gauge-invariant form the theory of classical linear gravitational perturbations in part I, and a quantum theory of cosmological perturbations in part II. Part I includes applications to several important examples arising in cosmology: a univese dominated by hydrodynamical matter, a universe filled with scalar-field matter, and higher-derivative theories of gravity. The growth rates of perturbations are calculated analytically in most interesting cases. The analysis is applied to study the evolution of fluctuations in inflationary universe models. Part II includes a unified description of the quantum generation and evolution of inhomogeneities about a classial Friedmann background. The method is based on standard canonical quantization of the action for cosmological perturbations which has been reduced to an expression in terms of a single gauge-invariant variable. The spectrum of density perturbations originating in quantum fluctuations is calculated in universe with hydrodynamical matter, in inflationary universe models with scalar-field matter, and in higher-derivative theories of gravity.

The gauge-invariant theory of classical and quantized cosmological perturbations developed in parts I and II is applied in part III to several interesting physical problems. It allows a simple derivation of the relation between temperature anistropes in the cosmic microwave background. radiation and the gauge-invariant potential for metric perturbations. The generation and evolution of gravitational waves is studied. As another example, a simple analysis of entropy perturbations and non-scale-invariant spectra in inflationary universe models is presented. The gauge-invariant theory of cosmological perturbations also allows a consistent and gauge-invariant definition of statistical fluctuations.

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Theory of Cosmological Perturbations

We present geometric derivations of the Smarr formula for static AdS black holes and an expanded first law that includes variations in the cosmological constant. These two results are further related by a scaling argument based on Euler's theorem. The key new ingredient in the constructions is a two-form potential for the static Killing field. Surface integrals of the Killing potential determine the coefficient of the variation of Λ in the first law. This coefficient is proportional to a finite, effective volume for the region outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded from a spatial slice by the black hole horizon. This effective volume also contributes to the Smarr formula. Since Λ is naturally thought of as a pressure, the new term in the first law has the form of effective volume times change in pressure that arises in the variation of the enthalpy in classical thermodynamics. This and related arguments suggest that the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime.

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Enthalpy and the mechanics of AdS black holes

We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequality—an inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. We also review the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.

https://iopscience.iop.org/article/10.1088/1361-6382/aa5c69/pdf

Black hole chemistry: thermodynamics with Lambda

Relativistic Cosmology: The standard model and extensions

We consider the thermodynamics of rotating and charged asymptotically de Sitter (dS) black holes. Using Hamiltonian perturbation-theory techniques, we derive three different first-law relations including variations in the cosmological constant, and associated Smarr formulas that are satisfied by such spacetimes. Each first law introduces a different thermodynamic volume conjugate to the cosmological constant. We examine the relation between these thermodynamic volumes and associated geometric volumes in a number of examples, including Kerr-dS black holes in all dimensions and Kerr-Newman-dS black holes in D=4. We also show that the Chong-Cvetic-Lu-Pope solution of D=5 minimal supergravity—analytically continued to positive cosmological constant—describes black hole solutions of the Einstein-Chern-Simons theory and include such charged asymptotically de Sitter black holes in our analysis. In all these examples we find that the particular thermodynamic volume associated with the region between the black hole and cosmological horizons is equal to the naive geometric volume. Isoperimetric inequalities, which hold in the examples considered, are formulated for the different thermodynamic volumes and conjectured to remain valid for all asymptotically de Sitter black holes. In particular, in all examples considered, we find that for a fixed volume of the observable universe, the entropy is increased by adding black holes. We conjecture that this is true in general.

/pdf/thermodynamic-volumes-and-isoperimetric-inequalities-for-de-15ktzmoh6s.pdf

Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes

Conditions are found for the existence of integral constraints on stress-energy perturbations in general relativity. The integral constraints can be thought of as a general-relativistic generalization of the conservation of energy and momentum of matter perturbations in special relativity. The constraints are stated in terms of a vector field V\ensuremath{\rightarrow}, and the Robertson-Walker spacetimes are shown to have such constraint vectors. Although in general V\ensuremath{\rightarrow} is not a Killing vector, in a vacuum spacetime the constraint vectors are precisely the Killing vectors.

Constraints on Stress Energy Perturbations in General Relativity

The anisotropies in the microwave background radiation in the cosmic-string theory of galaxy formation are calculated using a general-relativistic analysis. On small scales (t-italich-italice-italict-italica-italicapprox. =0.5/sup 0/) we predict /sup 1/2//Tapprox. =4 x 10/sup -6/, on large scales (t-italich-italice-italict-italica-italicapprox. =6/sup 0/) /sup 1/2//Tapprox. =10 /sup -5/. The dominant effect is the Sachs-Wolfe effect from string loops. The above amplitudes are significantly smaller than the best current observational upper bounds.

Microwave Anisotropies From Cosmic Strings

We compute the large-scale anisotropy of the cosmic background radiation due to the gravitational field of clumped matter in open, closed, and flat Friedmann universes. Formulas are derived for the mean-square angular fluctuation of the cosmic background radiation in terms of the two-point correlation function of matter. The results depend importantly on whether or not the matter perturbations locally satisfy the integral constraints which express local conservation of energy and momentum, and we discuss these two possibilities. We examine the behavior as spatial curvature goes to zero.

Jennie H. Traschen

Papers

Constraints on Stress Energy Perturbations in General Relativity

Microwave Anisotropies From Cosmic Strings

Large-scale anisotropy of the cosmic background radiation in Friedmann universes.