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We investigate the second order energy density perturbation $\delta^{(2)}$ induced by small-scale Gaussian and local-type non-Gaussian primordial curvature perturbations. The relative abundance of primordial black hole is calculated in terms of the probability density function of total energy density perturbation $\delta_r=\delta^{(1)}+\frac{1}{2}\delta^{(2)}$. The effects of second order density perturbation greatly reduce the upper bounds of small-scale power spectra of primordial curvature perturbations by one to two orders of magnitude. For log-normal primordial power spectrum, its amplitude $A_{\zeta}$ is constrained to be about $A_{\zeta}\sim 3\times10^{-3}$. And for local-type non-Gaussianity with $f_{\mathrm{NL}}=10$, the upper bound of $A_{\zeta}$ is about $2.5\times10^{-4}$., Comment: 5 pages, 4 figures

The primordial perturbations will inevitably generate higher order perturbations. We study the second order scalar perturbations generated by the primordial curvature and tensor perturbations in the radiation-dominated era. After presenting all the possible second-order source terms, we obtain the explicit expressions of the kernel functions and the power spectra of the second order scalar perturbations. The contributions from the initial second-order perturbations are considered. We calculate the power spectra of second order scalar perturbations for different tensor-to-scalar ratio $r$.

The process of primordial black hole (PBH) formation is inevitably accompanied by scalar induced gravitational waves (SIGWs). The strong correlation between PBH and SIGW signals may offer a promising approach to detecting PBHs in upcoming gravitational wave experiments, such as the Laser Interferometer Space Antenna (LISA). We investigate third order SIGWs during a radiation-dominated era in the case of the monochromatic primordial power spectrum . For LISA observations, the relationships between the signal-to-noise ratio (SNR) and monochromatic primordial power spectrum are studied systematically, revealing that the effects of third order SIGWs extend the cutoff frequency from to and lead to an approximately 200% increase in the SNR for the frequency band from Hz to Hz, corresponding to PBHs with masses in the range . We find that there is a critical value, , for the amplitude of the monochromatic primordial power spectra, such that when , the energy density of third order SIGWs is larger than that of second order SIGWs.

Recently, from 12 γ-ray Galactic sources, the LHAASO has detected ultrahigh-energy photons up to 1.4 PeV. The γ-ray spectra of the sources J2226+6057, J1908+0621, and J1825-1326 and the suggested origin pulsars near the sources have been published. In our previous work, we studied the hadronic γ-ray spectra of the sources J2226+6057, J1908+0621, and J1825-1326 in terms of the Hertzian dipole model of pulsars. In this paper, we investigate the possibility of the leptonic origin of the γ-ray. We use the Hertzian dipole model to describe the pulsars around the sources. The electrons around the pulsars can be accelerated to PeV by the electromagnetic fields of pulsars. Under the assumption that the initial electrons are uniformly distributed in a spherical shell between and m around the pulsar, we obtain the energy distribution of electrons. The leptonic γ-ray spectra can be calculated through inverse Compton scattering processes. The leptonic γ-ray can roughly conform to the observation of LHAASO.