We consider inflationary models with the inflaton coupled to the Gauss-Bonnet term assuming a special relation ${\delta }_1=2\lambda {\epsilon }_1$ between the two slow-roll parameters ${\delta }_1$ and ${\epsilon }_1$. For the slow-roll inflation, the assumed relation leads to the reciprocal relation between the Gauss-Bonnet coupling function $\xi (\varphi )$ and the potential $V(\varphi )$, and it leads to the relation $r=16(1-\lambda ){\epsilon }_1$ that reduces the tensor-to-scalar ratio $r$ by a factor of $1-\lambda$. For the constant-roll inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor-to-scalar ratio to the first order of ${\epsilon }_1$ by using the method of Bessel function approximation. The tensor-to-scalar ratio is reduced by a factor of $1-\lambda +\lambda \tilde{\eta }$. Comparing the derived $n_s-r$ with the observations, we obtain the constraints on the model parameters $\tilde{\eta }$ and $\lambda$.

With the enhancement mechanism provided by a noncanonical kinetic term with a peak, the amplitude of primordial curvature perturbations can be enhanced by seven orders of magnitude at small scales while keeping to be consistent with observations at large scales. The peak function and inflationary potential are not restricted in this mechanism. We use the Higgs model and T-model as examples to show how abundant primordial black hole dark matter with different mass and scalar induced secondary gravitational waves with different peak frequency are generated. We also show that the enhanced power spectrum for the primordial curvature perturbations and the energy density of the scalar induced secondary gravitational waves can have either a sharp peak or a broad peak. The primordial black holes with the mass around ${10}^{-14}-{10}^{-12}{M}_{\odot }$ produced with the enhancement mechanism can make up almost all dark matter, and the scalar induced secondary gravitational waves accompanied with the production of primordial black holes can be tested by the pulsar timing arrays and spaced based gravitational wave observatory. Therefore, the mechanism can be tested by primordial black hole dark matter and gravitational wave observations.

The speed of gravitational waves provides us a new tool to test alternative theories of gravity. The constraint on the speed of gravitational waves from GW170817 and GRB170817A is used to test some...

The stochastic signal detected by NANOGrav, PPTA, EPTA, and CPTA can be explained by the scalar-induced gravitational waves. In order to determine the scalar-induced gravitational waves model that best fits the stochastic signal, we employ both single- and double-peak parameterizations for the power spectrum of the primordial curvature perturbations, where the single-peak scenarios include the δ-function, box, lognormal, and broken power law model, and the double-peak scenario is described by the double lognormal form. Using Bayesian inference, we find that there is no significant evidence for or against the single-peak scenario over the double-peak model, with log (Bayes factors) among these models ln ℬ < 1. Therefore, we cannot distinguish the different shapes of the power spectrum of the primordial curvature perturbation with the current sensitivity of pulsar timing arrays.