Kei-ichiro Kubota, Hiroki Matsui, Takahiro Terada
A generalization of inflationary $\alpha$-attractor models was recently proposed by Kallosh and Linde, in which the potential involves logarithmic functions of the inflaton so that the derivative of the potential but not potential itself has a singularity. We find that the models can lead to viable inflationary observables even without the pole in the kinetic term. Also, the generalization with a pole order other than two does not significantly change the functional form of the potential. This allows a systematic analysis of the predictions of this class of models. The models typically predict the scalar spectral index $n_s$ around 0.97 and values of the tensor-to-scalar ratio $r$ observable by LiteBIRD. Taking advantage of the relatively large $n_s$, we discuss the modification of the potential to produce primordial black holes as the whole dark matter and gravitational waves induced by curvature perturbations detectable by LISA and BBO/DECIGO, while keeping $n_s$ in agreement with the Planck/BICEP/Keck data.
Kei-ichiro Kubota, Shun Arai, and Shinji Mukohyama
Gravitational waves travel through the distributions of matter and dark energy during propagation. For this reason, gravitational waves emitted from binary compact objects serve as a useful tool especially to probe the nature of dark energy. The geometrical optics approximation is a conventional way of investigating wave propagation. However, the approximation becomes less accurate as the wavelength approaches the curvature radius of the background, which can occur in generic situations. In this paper, we suggest a formulation for higher-order corrections of the geometrical optics expansion, applied to Horndeski theory which accommodates many dark energy models. At the level of the background, assuming that the derivative of the scalar field is non-vanishing and timelike, we choose the time slices to coincide with the contours of the scalar field. This choice of the background time slices is advantageous as the sound cones of both scalar and tensor gravitational waves are upright with respect to the background time slices whenever the scalar field behaves as a perfect fluid. We then analyze the equations of motion for scalar and tensor components of gravitational waves at the leading and next-to-leading order in the geometrical optics expansion, deriving the evolution equations for their amplitudes under certain conditions. In particular, for Generalized Brans-Dicke theories, we find a simple description of equations for gravitational waves in terms of an effective metric.