Cosmology of non-minimal derivative coupling to gravity in Palatini formalism and its chaotic inflation

Abstract

We consider, in Palatini formalism, a modified gravity of which the scalar field derivative couples to Einstein tensor. In this scenario, Ricci scalar, Ricci tensor and Einstein tensor are functions of connection field. As a result, the connection field gives rise to relation, $h_{\mu \nu }=f{g}_{\mu \nu }$ between effective metric, $h_{\mu \nu }$ and the usual metric $g_{\mu \nu }$ where $f=1-\kappa {\varphi }^{,\alpha }{\varphi }_{,\alpha }/2$. In FLRW universe, NMDC coupling constant is limited in a range of $-2/{\dot{\varphi }}^2<\kappa \le \infty$ preserving Lorentz signature of the effective metric. Slowly-rolling regime provides $\kappa <0$ forbidding graviton from travelling at superluminal speed. Effective gravitational coupling and entropy of blackhole's apparent horizon are derived. In case of negative coupling, acceleration could happen even with $w_{\mathrm{eff}}>-1/3$. Power-law potentials of chaotic inflation are considered. For $V\propto {\varphi }^2$ and $V\propto {\varphi }^4$, it is possible to obtain tensor-to-scalar ratio lower than that of GR so that it satisfies $r<0.12$ as constrained by Planck 2015 \cite{Ade:2015lrj}. The $V\propto {\varphi }^2$ case yields acceptable range of spectrum index and $r$ values. The quartic potential's spectrum index is disfavored by the Planck results. Viable range of $\k$ for $V\propto {\varphi }^2$ case lies in positive region, resulting in less blackhole's entropy, superluminal metric, more amount of inflation, avoidance of super-Planckian field initial value and stronger gravitational constant.

12 pages, no figures, chaotic inflation potential section and appendices added, minor correction on Planck unit

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