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

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, hμν=fgμν between effective metric, hμν and the usual metric gμν where f=1−κϕ,αϕ,α∕2. In FLRW universe, NMDC coupling constant is limited in a range of −2∕ϕ̇2<κ≤∞ preserving Lorentz signature of the effective metric. Slowly-rolling regime provides κ<0 forbidding graviton from traveling 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 weff>−1∕3. Power-law potentials of chaotic inflation are considered. For V∝ϕ2 and V∝ϕ4, 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 (Ade et al., 2016). The V∝ϕ2 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 κ for V∝ϕ2 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.