We construct the higher order terms of curvatures in Lagrangians of the scale factor in the D-dimensional Robertson-Walker metric, which are linear in the second derivative of the scale factor with respect to cosmic time. It is shown that they are composed of the Lovelock tensors at the first step; iterative construction yields arbitrarily high order terms. The relation to the former work on higher order gravity is discussed. Despite the absence of scalar degrees of freedom in cosmological models which come from our Lagrangian, it is shown that an inflationary behavior of the scale factor can be found. The application to the thick brane solutions is also studied. 1. Introduction It is well known that higher-derivative gravity has a scalar degree of freedom in general [1–4]. In cosmological models of higher-derivative gravity, the scalar mode is expected to play an important role [5–7]. On the other hand, some cases are also known that higher order terms in curvatures for a gravitational action do not affect cosmological development of a scale factor. For example, it is known that terms which consist of contraction of Weyl tensors in a gravitational Lagrangian do not change evolutional equations for a scale factor in a model with homogeneous and isotropic space. The other special combinations of curvatures are known. In the specific dimension, the Euler form as a Lagrangian does not produce the dynamics of gravity at all, because the action becomes a topological quantity in such a case. The dimensionally continued Euler densities have also been studied [8–17], because of their relation to the effective Lagrangian of string theory, and are found to give no scalar mode since the second derivative of the metric disappears in the action if we perform integration by parts. The absence of scalar modes is interesting for studying black holes in the theory, because the scalar modes lead to singularities, in general, which avoid expected horizons. In recent years, it turned out that there is a special case where a scalar mode disappears in higher-derivative gravity. Originally, this fact was found in research of a three-dimensional theory of massive gravity [18], and an extended version in four dimensions was proposed [19]. The authors of those papers intended to study the renormalizability and unitarity of gravitation theory in a maximally symmetric spacetime. Thus, the absence of a massive scalar mode is at least a necessary condition of such theories referred as critical gravity (in our analysis, we do not care for the other critical values for the
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