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 them 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 $xi$, we 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 mostly 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 $alpha$. The 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.
The prospects for direct measurements of inflationary gravitational waves by next generation interferometric detectors inferred from the possible detection of B-mode polarization of the cosmic microwave background are studied. We compute the spectra of the gravitational wave background and the signal-to-noise ratios by two interferometric detectors (DECIGO and BBO) for large-field inflationary models in which the tensor-to-scalar ratio is greater than the order of 0.01. If the reheating temperature $T_{rm RH}$ of chaotic inflation with the quadratic potential is high ($T_{rm RH}>7.9times10^6$ GeV for upgraded DECIGO and $T_{rm RH}> 1.8times 10^{6}$ GeV for BBO), it will be possible to reach the sensitivity of the gravitational background in future experiments at $3sigma$ confidence level. The direct detection is also possible for natural inflation with the potential $V(phi)=Lambda^4 [1-cos(phi/f)]$, provided that $f>4.2 M_{rm pl}$ (upgraded DECIGO) and $f>3.6 M_{rm pl}$ (BBO) with $T_{rm RH}$ higher than $10^8$ GeV. The quartic potential $V(phi)=lambda phi^4/4$ with a non-minimal coupling $xi$ between the inflaton field $phi$ and the Ricci scalar $R$ gives rise to a detectable level of gravitational waves for $|xi|$ smaller than the order of 0.01, irrespective of the reheating temperature.
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 constant 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.
