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

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

A modified gravity, which eliminates the need for dark energy and which seems to be stable, is considered. The terms with positive powers of curvature support the inflationary epoch while the terms with negative powers of curvature serve as effective dark energy, supporting current cosmic acceleration. The equivalent scalar-tensor gravity may be compatible with the simplest solar system experiments.

Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration

Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution

Rapid interstellar travel by means of spacetime wormholes is described in a way that is useful for teaching elementary general relativity. The description touches base with Carl Sagan’s novel C o n t a c t, which, unlike most science fiction novels, treats such travel in a manner that accords with the best 1986 knowledge of the laws of physics. Many objections are given against the use of black holes or Schwarzschild wormholes for rapid interstellar travel. A new class of solutions of the Einstein field equations is presented, which describe wormholes that, in principle, could be traversed by human beings. It is essential in these solutions that the wormhole possess a throat at which there is no horizon; and this property, together with the Einstein field equations, places an extreme constraint on the material that generates the wormhole’s spacetime curvature: In the wormhole’s throat that material must possess a radial tension τ0 with the enormous magnitude τ0∼ (pressure at the center of the most massive of neutron stars)×(20 km)2/(circumference of throat)2. Moreover, this tension must exceed the material’s density of mass‐energy, ρ0 c 2. No known material has this τ0>ρ0 c 2property, and such material would violate all the ‘‘energy conditions’’ that underlie some deeply cherished theorems in general relativity. However, it is not possible today to rule out firmly the existence of such material; and quantum field theory gives tantalizing hints that such material might, in fact, be possible.

Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity

It is argued that, if the laws of physics permit an advanced civilization to create and maintain a wormhole in space for interstellar travel, then that wormhole can be converted into a time machine with which causality might be violatable. Whether wormholes can be created and maintained entails deep, ill-understood issues about cosmic censorship, quantum gravity, and quantum field theory, including the question of whether field theory enforces an averaged version of the weak energy condition.

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Wormholes, time machines, and the weak energy condition.

A pure gravity inflationary model for the Universe is examined which is based on adding an \ensuremath{\epsilon}${R}^{2}$ term to the usual gravitational Lagrangian. The classical evolution is worked out, including eventual particle production and the subsequent join to radiation-dominated Friedmann behavior. We show that this model gives significant inflation essentially independent of initial conditions. The model has only one free parameter which is bounded from above by observational constraints on scalar and tensorial perturbations and from below by both the need for standard baryogenesis and the need for galaxy formation. This requires ${10}^{11}$${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}1/2}$${10}^{13}$ GeV.

The R 2 cosmology: Inflation without a phase transition

Once quantum mechanical effects are included, the hypotheses underlying the positive mass theorem of classical general relativity fail. As an example of the peculiarities attendant upon this observation, a wormhole mouth embedded in a region of high mass density might accrete mass, giving the other mouth a net [ital negative] mass of unusual gravitational properties. The lensing of such a gravitationally negative anomalous compact halo object (GNACHO) will enhance background stars with a time profile that is observable and qualitatively different from that recently observed for massive compact halo objects (MACHO's) of positive mass. While the analysis is discussed in terms of wormholes, the observational test proposed is more generally a search for compact negative mass objects of any origin. We recommend that MACHO search data be analyzed for GNACHO's.

Natural wormholes as gravitational lenses.

A pure gravity cosmology based on the R+eR^2 Lagrangian is known to exhibit inflation for a wide range of initial conditions. In this paper we use the wave function from quantum cosmology to describe this inflation as a chaotic inflationary phase immediately following the quantum creation of the Universe. We evaluate, compare, and discuss the distributions over initial conditions that are fixed by the two boundary-condition proposals of Hartle and Hawking ("no boundary") and Vilenkin ("tunneling from nothing"). We find that among all classical inflationary trajectories that begin on the classical-quantum boundary, those that lead to an inflation of at least 70 e-foldings make up a fraction of ∼exp(-10^12) in the former case and ∼1-exp(-8×10^10) in the latter. Thus, in the simplest interpretation, the observable Universe would be the outcome of a rare event for the first boundary-condition proposal and a typical event for the second.

Michael S. Morris

Papers

Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity

Wormholes, time machines, and the weak energy condition.

The R 2 cosmology: Inflation without a phase transition

Natural wormholes as gravitational lenses.

Initial Conditions for the $R + \epsilon R^2$ Cosmology