Cosmology in holographic non-minimal derivative coupling theory:
  constraints from inflation and variation of gravitational constant
  We consider a cosmological model of non-minimal derivative coupling (NMDC) to
gravity with holographic effect from Bekenstein-Hawking entropy using Hubble
horizon IR cutoff. Holographic parameter $c$ is constant in a range, $0 \leq c
< 1$. NMDC effect allows gravitational constant to be time-varying. Definition
of holographic density include time-varying part of the gravitational constant.
NMDC part reduces strength of gravitational constant for $\k > 0$ and opposite
for $\k < 0$. The holographic part enhances gravitational strength. We use
spectral index and tensor-to-scalar ratio to test the model against CMB
constraint. Number of e-folding is chosen to be $N \geq 60$. Potentials, $V =
V_0 \phi^n $ with $n = 2, 4$, and $V = V_0 \exp{(-\beta \phi)}$ are considered.
Combined parametric plots of $\k$ and $\phi$ show that the allowed regions of
the power spectrum index and of the tensor-to-scalar ratio are not overlapping.
NMDC inflation is ruled out and the holographic NMDC inflation is also ruled
out for $0 < c < 1$. NMDC significantly changes major anatomy of the dynamics,
i.e. it gives new late-time attractor trajectories in acceleration regions. The
holographic part clearly affects pattern of trajectories. However, for the
holographic part to affect shape of the acceleration region, the NMDC field
must be in presence. To constrain the model at late time, variation of
gravitational constant is considered. Gravitational-wave standard sirens and
supernovae data give a constraint, $\dot{G}/G|_{t_0} \lesssim 3\times10^{-12}
\, \text{year}^{-1}$ \cite{Zhao:2018gwk} which, for this model, results in $
10^{-12} \, \text{year}^{-1} \, \gtrsim \, {- \kappa}
\dot{\phi}\ddot{\phi}/{M^2_{\p}}\,. $ Positive $\k$ is favored and greater
$c^2$ results in lifting up lower bound of $\k$.
