The possibility that in the mass range around ${10}^{-12}{M}_{\odot }$ most of dark matter is constituted of primordial black holes (PBHs) is a very interesting topic. To produce PBHs with this mass, the primordial scalar power spectrum needs to be enhanced to the order of 0.01 at the scale $k\sim {10}^{12}{\mathrm{Mpc}}^{-1}$. The enhanced power spectrum also produces large secondary gravitational waves at the mHz band. A phenomenological delta function power spectrum is usually used to discuss the production of PBHs and secondary gravitational waves. Based on $k$ and $G$ inflations, we propose a new mechanism to enhance the power spectrum at small scales by introducing a noncanonical kinetic term $[1-2G(\varphi )]X$ with the function $-G(\varphi )$ having a peak. Away from the peak, $G(\varphi )$ is negligible and we recover the usual slow-roll inflation which is constrained by the cosmic microwave background anisotropy observations. Around the peak, the slow-roll inflation transiently turns to ultra slow-roll inflation. The enhancement of the power spectrum can be obtained with generic potentials, and there is no need to fine tune the parameters in $G(\varphi )$ to several significant digits. The energy spectrum ${\mathrm{\Omega }}_{\mathrm{GW}}(f)$ of secondary gravitational waves produced by the model have the characteristic power law behavior ${\mathrm{\Omega }}_{\mathrm{GW}}(f)\sim f^n$ and is testable by pulsar timing array and space based gravitational wave detectors.

We resolve the potential-restriction problem in K/G inflation by introducing nonminimal coupling. In this context, Higgs field successfully drives inflation satisfying cosmic microwave background (CMB) observations while enhancing curvature perturbations at small scales, which in turn accounts for primordial black holes (PBHs) and scalar induced gravitational waves (SIGWs). We then uncover the effect of the noncanonical kinetic coupling function in more detail and study its the observational constraint. Besides, we also give the gauge invariant expression for the integral kernel of SIGWs, which is related to terms propagating with the speed of light. Finally, the non-Gaussian effect on PBH abundance and SIGWs is studied. We find that non-Gaussianity makes PBHs form more easily, but its effect on the energy density of SIGWs is negligible.

The newly proposed island formula for entanglement entropy of Hawking radiation is applied to spherically symmetric 4-dimensional eternal Kaluza–Klein (KK) black holes. The “charge” Q of KK black holes quantifies its deviation from Schwarzschild black holes. The impact of Q on the island is studied at late times. The late-time island, whose boundary is located outside but within a Planckian distance of the horizon, is slightly extended by Q. While the no-island entropy grows linearly, the late-time entanglement entropy is given by island configuration with twice the Bekenstein–Hawking entropy. Thus we reproduce the Page curve for the eternal KK black holes. Compared with Schwarzschild results, the Page time is delayed by a factor $$(1+Q/r_h)$$( 1 + Q / r h ) and the scrambling time is prolonged by a factor $$(1+Q/r_h)^{1/2}$$( 1 + Q / r h ) 1 / 2 . Moreover, the higher-dimensional generalization is presented. Skeptically, there are Planck length scales involved, in which a semi-classical description may break down.

A significant abundance of primordial black hole (PBH) dark matter can be produced by curvature perturbations with power spectrum ${\mathrm{\Delta }}_{\zeta }^2(k_{\mathrm{peak}})\sim \mathcal{O}({10}^{-2})$ at small scales, associated with the generation of observable scalar induced gravitational waves (SIGWs). However, the primordial non-Gaussianity may play a non-negligible role, which is not usually considered. We propose two inflation models that predict double peaks of order $\mathcal{O}({10}^{-2})$ in the power spectrum and study the effects of primordial non-Gaussianity on PBHs and SIGWs. This model is driven by a power-law potential, and has a noncanonical kinetic term whose coupling function admits two peaks. By field-redefinition, it can be recast into a canonical inflation model with two quasi-inflection points in the potential. We find that the PBH abundance will be altered saliently if non-Gaussianity parameter satisfies $|f_{\mathrm{NL}}(k_{\mathrm{peak}},k_{\mathrm{peak}},k_{\mathrm{peak}})|\gtrsim {\mathrm{\Delta }}_{\zeta }^2(k_{\mathrm{peak}})/(23{\delta }_c^3)\sim \mathcal{O}({10}^{-2})$. Whether the PBH abundance is suppressed or enhanced depends on the $f_{\mathrm{NL}}$ being positive or negative, respectively. In our model, non-Gaussianity parameter $f_{\mathrm{NL}}(k_{\mathrm{peak}},k_{\mathrm{peak}},k_{\mathrm{peak}})\sim \mathcal{O}(1)$ takes positive sign, thus PBH abundance is suppressed dramatically. On the contrary, SIGWs are insensitive to primordial non-Gaussianity and hardly affected, so they are still within the sensitivities of space-based GWs observatories and Square Kilometer Array.

The gauge dependence of the scalar induced gravitational waves (SIGWs) generated at the second order imposes a challenge to the discussion of the secondary gravitational waves generated by scalar perturbations. We provide a general formula that is valid in any gauge for the calculation of SIGWs and the relationship for SIGWs calculated in various gauges under the coordinate transformation. The formula relating SIGWs in the Newtonian gauge to other gauges is used to calculate SIGWs in six different gauges. We find that the Newtonian gauge, the uniform curvature gauge, the synchronous gauge, and the uniform expansion gauge yield the same result for the energy density of SIGWs. We also identify and eliminate the pure gauge modes that existed in the synchronous gauge. In the total matter gauge and the comoving orthogonal gauge, the energy density of SIGWs increases as ${\eta }^2$. While in the uniform density gauge, the energy density of SIGWs increases as ${\eta }^6$.