Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1sim -2/N$, where $N$ is the number of e-foldings. Quadratic chaotic model, Starobinsky model and Higgs inflation or $alpha$-attractors connecting t
hem are typical examples predicting such a relation. We consider the problem in the opposite: given $n_s$ as a function of $N$, what is the inflaton potential $V(phi)$. We find that for $n_s-1=-2/N$, $V(phi)$ is either $tanh^2(gammaphi/2)$ (T-model) or $phi^2$ (chaotic inflation) to the leading order in the slow-roll approximation. $gamma$ is the ratio of $1/V$ at $Nrightarrow infty$ to the slope of $1/V$ at a finite $N$ and is related to $alpha$ in the $alpha$-attractors by $gamma^2=2/3alpha$. The tensor-to-scalar ratio $r$ is $r=8/N(gamma^2 N +1) $. The implications for the reheating temperature are also discussed. We also derive formulas for $n_s-1=-p/N$. We find that if the potential is bounded from above, only $p>1$ is allowed. Although $r$ depends on a parameter, the running of the spectral index is independent of it, which can be used as a consistency check of the assumed relation of $n_s(N)$.
We derive the consistency relations for a chaotic inflation model with a non-minimal coupling to gravity. For a quadratic potential in the limit of a small non-minimal coupling parameter $xi$ and for a quartic potential without assuming small $xi$, w
e give the consistency relations among the spectral index $n_s$, the tensor-to-scalar ratio $r$ and the running of the spectral index $alpha$. We find that unlike $r$, $alpha$ is less sensitive to $xi$. If $r<0.1$, then $xi$ is constrained to $xi<0$ and $alpha$ is predicted to be $alphasimeq -8times 10^{-4}$ for a quartic potential. For a general monomial potential, $alpha$ is constrained in the range $-2.7times 10^{-3}<alpha< -6times 10^{-4}$ as long as $|xi|leq 10^{-3}$ if $r<0.1$.
We study the motion of a charged particle around a weakly magnetized rotating black hole. We classify the fate of a charged particle kicked out from the innermost stable circular orbit. We find that the final fate of the charged particle depends most
ly on the energy of the particle and the radius of the orbit. The energy and the radius in turn depend on the initial velocity, the black hole spin, and the magnitude of the magnetic field. We also find possible evidence for the existence of bound motion in the vicinity of the equatorial plane.
Consistency relations for chaotic inflation with a monomial potential and natural inflation and hilltop inflation are given which involve the scalar spectral index $n_s$, the tensor-to-scalar ratio $r$ and the running of the spectral index $alpha$. T
he measurement of $alpha$ with $O(10^{-3})$ and the improvement in the measurement of $n_s$ could discriminate monomial model from natural/hilltop inflation models. A consistency region for general large field models is also presented.
We derive the slow-roll conditions for a non-minimally coupled scalar field (extended quintessence) during the radiation/matter dominated era extending our previous results for thawing quintessence. We find that the ratio $ddotphi/3Hdotphi$ becomes c
onstant but negative, in sharp contrast to the ratio for the minimally coupled scalar field. We also find that the functional form of the equation of state of the scalar field asymptotically approaches that of the minimally coupled thawing quintessence.
