We propose a scenario where inflation is driven by non-minimally coupled massive vector fields. In an isotropic homogeneous universe these fields behave in presicely the same way as a massive minimally coupled scalar field. Therefore our model is very similar to the model of chaotic inflation with scalar field. For vector fields the isotropy of expansion is achived either by considering a triplet of orthogonal vector fields or for the expense of N randomly oriented vector fields. In the last case the substantial anisotropy of the expansion of order 1/ √ N survives until the end of inflation. The lightest vector fields might also force the late time acceleration of the Universe.

Recently, an interesting gravitational model was proposed in order to mimic the effect of Dark Matter. Chamseddine and Mukhanov in the arXiv preprint 1308.5410 have separated the conformal mode of a physical metric in the form of a squared gradient of an auxiliary scalar field. Notably, the variational principle has given a more general equation of motion than that of purely Einsteinian relativity theory, with a possibility of reproducing an effective Dark Matter. In this short paper, we explain the nature of this phenomenon in terms of the class of functions on which the variation takes place. Then we give a more transparent equivalent formulation of the model without an auxiliary metric. Finally, we speculate a bit about possible extensions.

The basics of teleparallel gravity and its extensions are reviewed with particular emphasis on the problem of Lorentz-breaking choice of connection in pure-tetrad versions of the theories. Various possible ways to covariantise such models are discussed. A by-product is a new form of f (T ) field equations.

Cosmological perturbations are considered in f (T ) and in scalar-torsion f (ϕ)T teleparallel models of gravity. Full sets of linear perturbation equations are accurately derived and analysed at the relevant limits. Interesting features of generalisations to other teleparallel models, spatially curved backgrounds, and rotated tetrads are pointed out.1 One may remind that rather than something "extra" or "alternative", the tetrad is necessary to couple matter to gravitation. Only currently, a new preprint claims to consistently supersede the tetrad by a more minimal structure, which may shed new light on the singularity resolution and, surprisingly, the nature of dark matter [4].2 Unfortunately, with a typo (missing prime for the extra perturbation in the space-space field equation) inherited in [22].

In this paper we present a very simple and independent argument for the absence of the Boulware-Deser ghost in the recently proposed potentially ghost-free non-linear massive gravity. The limitation is that, in its simple form, the argument is, in a sense, non-constructive and less explicit than the standard approach. However, the formalism developed here may prove to be useful for discussing the formal aspects of the theory.Comment: 8 pages; minor changes; one of the references and the sign of the potential term are corrected; the reference list is expande