A new type of isotropic cosmological models without singularity

Modified Gravity and Cosmology

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

The Search for Supersymmetry: Probing Physics Beyond the Standard Model

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

Quantum Theory of Gravity

The general spherically symmetric, static solution of ${\ensuremath{\nabla}}_{\ensuremath{\nu}}{T}_{\ensuremath{\mu}}^{\ensuremath{\nu}} = 0$ in the exterior Schwarzschild metric is expressed in terms of two integration constants and two arbitrary functions, one of which is the trace of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$. One constant is the magnitude of ${T}_{\mathrm{tr}}$ at infinity, and the other is determined if the physically normalized components of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$ are finite on the future horizon. The trace of the stress tensor of a conformally invariant quantum field theory may be nonzero (anomalous), but must be proportional (here) to the Weyl scalar, $48{M}^{2}{r}^{\ensuremath{-}6}$; we fix the coefficient for the scalar field by indirect arguments to be ${(2880{\ensuremath{\pi}}^{2})}^{\ensuremath{-}1}$. In the two-dimensional analog, the magnitude of the Hawking blackbody effect at infinity is directly proportional to the magnitude of the anomalous trace (a multiple of the curvature scalar); a knowledge of either number completely determines the stress tensor outside a body in the final state of collapse. In four dimensions, one obtains instead a relation constraining the remaining undetermined function, which we choose as ${T}_{\ensuremath{\theta}}^{\ensuremath{\theta}}\ensuremath{-}\frac{{T}_{\ensuremath{\alpha}}^{\ensuremath{\alpha}}}{4}$. This, plus additional physical and mathematical considerations, leads us to a fairly definite, physically convincing qualitative picture of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$. Groundwork is laid for explicit calculations of $〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉$.

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Trace Anomalies and the Hawking Effect

A method known as covariant geodesic point separation is developed to calculate the vacuum expectation value of the stress tensor for a massive scalar field in an arbitrary gravitational field. The vacuum expectation value will diverge because the stress-tensor operator is constructed from products of field operators evaluated at the same space-time point. To remedy this problem, one of the field operators is taken to a nearby point. The resultant vacuum expectation value is finite and may be expressed in terms of the Hadamard elementary function. This function is calculated using a curved-space generalization of Schwinger's proper-time method for calculating the Feynman Green's function. The expression for the Hadamard function is written in terms of the biscalar of geodetic interval which gives a measure of the square of the geodesic distance between the separated points. Next, using a covariant expansion in terms of the tangent to the geodesic, the stress tensor may be expanded in powers of the length of the geodesic. Covariant expressions for each divergent term and for certain terms in the finite portion of the vacuum expectation value of the stress tensor are found. The properties, uses, and limitations of the results are discussed.

Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Quantizing gravity with a cosmological constant

One of the techniques used in quantum field theory in curved space-times to eliminate divergences in the vacuum expectation value of the stress tensor for quantum fields propagating on a classical gravitational background is called covariant geodesic point separation. Beginning with the Schwinger-DeWitt proper-time method we show how to discard divergences in the effective action by renormalization of the coupling constants in a classical gravitational action functional. We then demonstrate how to determine which terms in the vacuum expectation value of the stress tensor vanish when this renormalization is carried out. This is done using the point-separation approach. We give the form of these terms for spin 0, 1/2, and 1 fields, massive or massless, on an arbitrary curved background. The procedure used is covariant and introduces no ambiguities beyond those inherent in any renormalization scheme. We note the appearance of trace anomalies which arise due to the breaking of conformal invariance by the renormalization process and give the form of the anomalies for arbitrary space-time dimension.

S.M. Christensen

Papers

Quantum Theory of Gravity

Trace Anomalies and the Hawking Effect

Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

Quantizing gravity with a cosmological constant

Regularization, renormalization, and covariant geodesic point separation