Accretion-modified Stars in Accretion Disks of Active Galactic Nuclei: the Low-luminosity Cases and an Application to Sgr A$\!^{*}$

In this paper, we investigate the astrophysical processes of stellar-mass black holes (sMBHs) embedded in advection-dominated accretion flows (ADAFs) of supermassive black holes (SMBHs) in low-luminosity active galactic nuclei (AGNs). The sMBH is undergoing Bondi accretion at a rate lower than the SMBH. Outflows from the sMBH-ADAF dynamically interact with their surroundings and form a cavity inside the SMBH-ADAF, thereby quenching the accretion onto the SMBH. Rejuvenation of the Bondi accretion is rapidly done by turbulence. These processes give rise to quasi-periodic episodes of sMBH activities and create flickerings from relativistic jets developed by the Blandford-Znajek mechanism if the sMBH is maximally rotating. Accumulating successive sMBH-outflows trigger viscous instability of the SMBH-ADAF, leading to a flare following a series of flickerings. Recently, the similarity of near-infrared flare's orbits has been found by GRAVITY/VLTI astrometric observations of Sgr A$\!^{*}$: their loci during the last 4-years consist of a ring in agreement with the well-determined SMBH mass. We apply the present model to Sgr A$\!^{*}$, which shows quasi-periodic flickerings. A SMBHH of $\sim 40 M_{\odot}$ is preferred orbiting around the central SMBH of Sgr A$\!^{*}$ from fitting radio to X-ray continuum. Such an extreme mass ratio inspiraling (EMRI) provides an excellent laboratory for LISA, Taiji and Tianqin detection of mHz gravitational waves with strains of $\sim 10^{-17}$, as well as their polarization.