Microlensing of Sub-parsec Massive Binary Black Holes in Lensed QSOs: Light Curves and Size-Wavelength Relation

Sub-parsec binary massive black holes (BBHs) are long anticipated to exist in many QSOs but remain observationally elusive. In this paper, we propose a novel method to probe sub-parsec BBHs through microlensing of lensed QSOs. If a QSO hosts a sub-parsec BBH in its center, it is expected that the BBH is surrounded by a circum-binary disk, each component of the BBH is surrounded by a small accretion disk, and a gap is opened by the secondary component in between the circum-binary disk and the two small disks. Assuming such a BBH structure, we generate mock microlensing light curves for some QSO systems that host BBHs with typical physical parameters. We show that microlensing light curves of a BBH QSO system at the infrared-optical-UV bands can be significantly different from those of corresponding QSO system with a single massive black hole (MBH), mainly because of the existence of the gap and the rotation of the BBH (and its associated small disks) around the center of mass. We estimate the half-light radii of the emission region at different wavelengths from mock light curves and find that the obtained half-light radius vs. wavelength relations of BBH QSO systems can be much flatter than those of single MBH QSO systems at a wavelength range determined by the BBH parameters, such as the total mass, mass ratio, separation, accretion rates, etc. The difference is primarily due to the existence of the gap. Such unique features on the light curves and half-light radius-wavelength relations of BBH QSO systems can be used to select and probe sub-parsec BBHs in a large number of lensed QSOs to be discovered by current and future surveys, including the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), the Large Synoptic Survey telescope (LSST) and Euclid.Comment: 18 pages, 17 figures, accepted for publication in the Astrophysical Journa

Ergodicity for the GI/G/1GI/G/1-type Markov Chain

Ergodicity is a fundamental issue for a stochastic process. In this paper, we refine results on ergodicity for a general type of Markov chain to a specific type or the $GI/G/1$-type Markov chain, which has many interesting and important applications in various areas. It is of interest to obtain conditions in terms of system parameters or the given information about the process, under which the chain has various ergodic properties. Specifically, we provide necessary and sufficient conditions for geometric, strong and polynomial ergodicity, respectively.Comment: 16 page

Photometric properties and luminosity function of nearby massive early-type galaxies

We perform photometric analyses for a bright early-type galaxy (ETG) sample with 2949 galaxies ($M_{\rm r}<-22.5$ mag) in the redshift range of 0.05 to 0.15, drawn from the SDSS DR7 with morphological classification from Galaxy Zoo 1. We measure the Petrosian and isophotal magnitudes, as well as the corresponding half-light radius for each galaxy. We find that for brightest galaxies ($M_{\rm r}<-23$ mag), our Petrosian magnitudes, and isophotal magnitudes to 25 ${\rm mag/arcsec^2}$ and 1\% of the sky brightness are on average 0.16 mag, 0.20 mag, and 0.26 mag brighter than the SDSS Petrosian values, respectively. In the first case the underestimations are caused by overestimations in the sky background by the SDSS PHOTO algorithm, while the latter two are also due to deeper photometry. Similarly, the typical half-light radii ($r_{50}$) measured by the SDSS algorithm are smaller than our measurements. As a result, the bright-end of the $r$-band luminosity function is found to decline more slowly than previous works. Our measured luminosity densities at the bright end are more than one order of magnitude higher than those of Blanton et al. (2003), and the stellar mass densities at $M_{\ast}\sim 5\times10^{11} M_{\odot}$ and $M_{\ast}\sim 10^{12} M_{\odot}$ are a few tenths and a factor of few higher than those of Bernardi et al. (2010). These results may significantly alleviate the tension in the assembly of massive galaxies between observations and predictions of the hierarchical structure formation model.Comment: 43 pages, 14 figures, version accepted for publication in the Astrophysical Journa