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Register a new userSome remarks on the dynamical systems approach to fourth order gravity
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Building on earlier work, we discuss a general framework for exploring the cosmological dynamics of Higher Order Theories of Gravity. We show that once the theory of gravity has been specified, the cosmological equations can be written as a first-order autonomous system and we give several examples which illustrate the utility of our method. We also discuss a number of results which have appeared recently in the literature.
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The Nobel Prize winning confirmation in 1998 of the accelerated expansion of our Universe put into sharp focus the need of a consistent theoretical model to explain the origin of this acceleration. As a result over the past two decades there has been a huge theoretical and observational effort into improving our understanding of the Universe. The cosmological equations describing the dynamics of a homogeneous and isotropic Universe are systems of ordinary differential equations, and one of the most elegant ways these can be investigated is by casting them into the form of dynamical systems. This allows the use of powerful analytical and numerical methods to gain a quantitative understanding of the cosmological dynamics derived by the models under study. In this review we apply these techniques to cosmology. We begin with a brief introduction to dynamical systems, fixed points, linear stability theory, Lyapunov stability, centre manifold theory and more advanced topics relating to the global structure of the solutions. Using this machinery we then analyse a large number of cosmological models and show how the stability conditions allow them to be tightly constrained and even ruled out on purely theoretical grounds. We are also able to identify those models which deserve further in depth investigation through comparison with observational data. This review is a comprehensive and detailed study of dynamical systems applications to cosmological models focusing on the late-time behaviour of our Universe, and in particular on its accelerated expansion. In self contained sections we present a large number of models ranging from canonical and non-canonical scalar fields, interacting models and non-scalar field models through to modified gravity scenarios. Selected models are discussed in detail and interpreted in the context of late-time cosmology.
After the previous work on gravitational frequency shift, light deflection (arXiv:1003.5296) and perihelion advance (arXiv:0812.2332), we calculate carefully the fourth gravity test, i.e. radar echo delay in a central gravity field surrounded by static free quintessence matter, in this paper. Through the Lagrangian method, we find the influence of the quintessence matter on the time delay of null particle is presence by means of an additional integral term. When the quintessence field vanishes, it reduces to the usual Schwarzschild case naturally. Meanwhile, we also use the data of the Viking lander from the Mars and Cassini spacecraft to Saturn to constrain the quintessence field. For the Viking case, the field parameter $alpha$ is under the order of $10^{-9}$. However, $alpha$ is under $10^{-18}$ for the Cassini case.
In this paper we present our point of view on correct physical interpretation of the Bel-Robinson tensor within the framework of the standard General Relativity ({bf GR}), i.e., within the framework of the {bf GR} without supplementary elements like arbitrary vector field, distinguished tetrads field or second metric. We show that this tensor arises as a consequence of the Bianchi identities and, in a natural manner, it is linked to the differences of the canonical gravitational energy-momentum calculated in normal coordinates {bf NC(P)}.
Working within the post-Newtonian (PN) approximation to General Relativity, we use the effective field theory (EFT) framework to study the conservative dynamics of the two-body motion at fourth PN order, at fifth order in the Newton constant. This is one of the missing pieces preventing the computation of the full Lagrangian at fourth PN order using EFT methods. We exploit the analogy between diagrams in the EFT gravitational theory and 2-point functions in massless gauge theory, to address the calculation of 4-loop amplitudes by means of standard multi-loop diagrammatic techniques. For those terms which can be directly compared, our result confirms the findings of previous studies, performed using different methods.
We study a class of almost scale-invariant modified gravity theories, using a particular form of $f(R, G) = alpha R^2 + beta G log G$ where $R$ and $G$ are the Ricci and Gauss-Bonnet scalars, respectively and $alpha$, $beta$ are arbitrary constants. We derive the Einstein-like field equations to first order in cosmological perturbation theory in longitudinal gauge.
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