Scalar-Tensor and Multiscalar-Tensor Gravity and Cosmological Models
We consider scalar-tensor and multiscalar-tensor theories of gravity and their formulations in the Jordan and the Einstein conformal frames. After constructing a generic multi-scalar tensor action, we derive its full equations of motion as well as equatio
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Abstract We consider scalar-tensor and multiscalar-tensor theories of gravity and their formulations in the Jordan and the Einstein conformal frames. After constructing a generic multi-scalar tensor action, we derive its full equations of motion as well as equations for homogeneous isotropic cosmological models in the Jordan frame. We use methods of dynamical systems in the case of two scalar fields to determine the fixed point and conditions for its being an attractor.
1 Introduction For decades mathematical cosmology has been based on Einstein’s theory of general relativity (GR) where the gravitational interaction is described by the metric tensor gμν of a Riemannian spacetime. Present observational data are in good agreement with general relativistic Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology with homogeneous and isotropic flat (k = 0) 3-space, a cosmological constant Λ > 0 and additional cold dark matter (ΛCDM model). However, this model seems to be somewhat phenomenological and fine-tuned: extremely small observational value of Λ, very special initial and/or boundary conditions, etc. From a mathematical point of view it is enticing to consider other possible cosmological models based on theories of gravity appropriately generalizing Einstein’s GR. Such alternative theories may be constructed by supplying extra fields, modifying the standard Einstein-Hilbert action to include an arbitrary function of curvature P. Kuusk (B) · L. Järv · E. Randla Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia e-mail: [email protected] L. Järv e-mail: [email protected] E. Randla e-mail: [email protected] A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, 661 Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_40, © Springer-Verlag Berlin Heidelberg 2014
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invariants, adding extra dimensions and branes, etc. (For a comprehensive review see [1].) The focus of the present paper is scalar-tensor gravity (STG) and its natural extension to multiple scalar fields (MSTG). The former is a paradigmatic example of a versatile modification of GR, while variants of the latter have received more attention only recently and its comprehensive treatment is still lacking. The action functional of a general STG contains up to four arbitrary functions of the scalar field, two of which can be fixed using a group of field redefinitions containing two free functional degrees of freedom [2]. The group transforms STG from one conformal frame and scalar field parametrization to another. A rather natural physical interpretation of STG is obtained in the so-called Jordan conformal frame, where the scalar field Φ is coupled to the scalar curvature R via the F (Φ)R term, but not directly to the matter fields, whereas the scalar field kinetic term involves an arbitrary function Z (Φ); now F (Φ) acts as a variable part of gravitational “constant”. In the so-called Einstein conformal frame the action functional is reminiscent of the Einstein GR with a mini
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