On the inspiral of Massive Black Holes in gas-rich galaxy mergers

We present a study on the dynamics of massive BHs in galaxy mergers, obtained from a series of high-resolution N-Body/SPH simulations. The presence of a gaseous component is essential for the rapid formation of an eccentric (Keplerian) BH binary, that resides at the center of a massive (~10^9 Msun) turbulent nuclear disc. Using physically and/or numerically motivated recipes, we follow the accretion history of the BHs during the merger. The mass of the BHs increases as large central inflows of gas occur inside each galaxy, and their mass ratio varies with time. Given the encountered strong degeneracy between numerical resolution and physical assumptions, we suggest here three possible paths followed by the galaxies and the BHs during a merger in order to fulfill the M-sigma relation : Adjustment, Symbiosis, and BH Dominance. In an extremely high resolution run, we resolved the turbulent gas pattern down to parsec scales, and found that BH feedback is expected to be effective near the end of the merger. We then trace the BH binary orbit down to a scale of 0.1 pc modeling the nuclear disc as an equilibrium Mestel disc composed either of gas, gas and stars, or just stars. Under the action of dynamical friction against the rotating gaseous and/or stellar background the orbit circularizes. When this occurs, each BH is endowed with its own small-size (~0.01 pc) accretion disc comprising a few percent of the BH mass. Double AGN activity is expected to occur on an estimated timescale of ~10 Myrs, comparable to the inspiral time. The double nuclear point--like sources that may appear have typical separations of ~10 pc, and are likely to be embedded in the still ongoing starburst.Comment: 10 pages, 5 figures, Proceedings of the Conference "The Multicoloured Landscape of Compact Objects and their Explosive Origins", Cefalu` 200

Birth of massive black hole binaries

If massive black holes (BHs) are ubiquitous in galaxies and galaxies experience multiple mergers during their cosmic assembly, then BH binaries should be common albeit temporary features of most galactic bulges. Observationally, the paucity of active BH pairs points toward binary lifetimes far shorter than the Hubble time, indicating rapid inspiral of the BHs down to the domain where gravitational waves lead to their coalescence. Here, we review a series of studies on the dynamics of massive BHs in gas-rich galaxy mergers that underscore the vital role played by a cool, gaseous component in promoting the rapid formation of the BH binary. The BH binary is found to reside at the center of a massive self-gravitating nuclear disc resulting from the collision of the two gaseous discs present in the mother galaxies. Hardening by gravitational torques against gas in this grand disc is found to continue down to sub-parsec scales. The eccentricity decreases with time to zero and when the binary is circular, accretion sets in around the two BHs. When this occurs, each BH is endowed with it own small-size (< 0.01 pc) accretion disc comprising a few percent of the BH mass. Double AGN activity is expected to occur on an estimated timescale of < 1 Myr. The double nuclear point-like sources that may appear have typical separation of < 10 pc, and are likely to be embedded in the still ongoing starburst. We note that a potential threat of binary stalling, in a gaseous environment, may come from radiation and/or mechanical energy injections by the BHs. Only short-lived or sub-Eddington accretion episodes can guarantee the persistence of a dense cool gas structure around the binary necessary for continuing BH inspiral.Comment: To appear in "2007 STScI Spring Symposium: Black Holes", eds. M. Livio & A. M. Koekemoer, Cambridge University Press, 25 pages, 12 figure

The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity

The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a 1+1 wave equation with a potential $V$, on a field $\Psi_z$. For smooth metric perturbations $\Psi_z$ is singular at $r_s=-6M/(\ell-1)(\ell+2)$, $\ell$ the mode harmonic number, and $V$ has a second order pole at $r_s$. This is irrelevant to the black hole exterior stability problem, where $r>2M>0$, and $r_s <0$, but it introduces a non trivial problem in the naked singular case where $M0$, and the singularity appears in the relevant range of $r$. We solve this problem by developing a new approach to the evolution of the even mode, based on a {\em new gauge invariant function}, $\hat \Psi$ -related to $\Psi_z$ by an intertwiner operator- that is a regular function of the metric perturbation {\em for any value of $M$}. This allows to address the issue of evolution of gravitational perturbations in this non globally hyperbolic background, and to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.Comment: typos corrected, references adde

Unstable fields in Kerr spacetimes

We show that both the interior region $r<M-\sqrt{M^2-a^2}$ of a Kerr black hole and the $a^2>M^2$ Kerr naked singularity admit unstable solutions of the Teukolsky equation for any value of the spin weight. For every harmonic number there is at least one axially symmetric mode that grows exponentially in time and decays properly in the radial directions. These can be used as Debye potentials to generate solutions for the scalar, Weyl spinor, Maxwell and linearized gravity field equations on these backgrounds, satisfying appropriate spatial boundary conditions and growing exponentially in time, as shown in detail for the Maxwell case. It is suggested that the existence of the unstable modes is related to the so called "time machine" region, where the axial Killing vector field is time-like, and the Teukolsky equation, restricted to axially symmetric fields, changes its character from hyperbolic to elliptic

On the orientation and magnitude of the black hole spin in galactic nuclei

Massive black holes in galactic nuclei vary their mass M and spin vector J due to accretion. In this study we relax, for the first time, the assumption that accretion can be either chaotic, i.e. when the accretion episodes are randomly and isotropically oriented, or coherent, i.e. when they occur all in a preferred plane. Instead, we consider different degrees of anisotropy in the fueling, never confining to accretion events on a fixed direction. We follow the black hole growth evolving contemporarily mass, spin modulus a and spin direction. We discover the occurrence of two regimes. An early phase (M <~ 10 million solar masses) in which rapid alignment of the black hole spin direction to the disk angular momentum in each single episode leads to erratic changes in the black hole spin orientation and at the same time to large spins (a ~ 0.8). A second phase starts when the black hole mass increases above >~ 10 million solar masses and the accretion disks carry less mass and angular momentum relatively to the hole. In the absence of a preferential direction the black holes tend to spin-down in this phase. However, when a modest degree of anisotropy in the fueling process (still far from being coherent) is present, the black hole spin can increase up to a ~ 1 for very massive black holes (M >~ 100 million solar masses), and its direction is stable over the many accretion cycles. We discuss the implications that our results have in the realm of the observations of black hole spin and jet orientations.Comment: 14 pages, 7 figures, accepted for publication in Ap

The Effects of Class Size on Student Performance in Reading in the Primary Grades

The purpose of this study was to examine the effect of class size on reading achievement of primary grade students in the Bismarck (North Dakota) Public Schools. Data for the study included using scores from the Gates-MacGinitie Reading Test (Fourth Edition). The results on the Gates-MacGinitie Reading Test were analyzed according to grade, gender, socioeconomic status, race, and class size. Students in the study were identified as having been in either small classes (17 or less) or large classes (22 or more). Three years of data (1999–2002) were analyzed as part of the small class study. A second source of data was gathered from a teacher survey instrument that was distributed to all teachers employed by the Bismarck Public Schools for the 2001–2002 school year who taught in classrooms identified as small or large. This survey measures teachers\u27 perceptions regarding instructional practices, classroom management, and time allotment. Results indicated that there was no significant difference in reading achievement scores of first grade, second grade, or third grade students who were placed in small compared to large classrooms. There was a significant difference in female students\u27 reading achievement when they were placed in small sized classrooms. When the achievement of Native American students was compared to Caucasian students, there was a significant different in reading achievement scores by Caucasian students in small sized classrooms. There was also a significant difference in reading achievement scores by students not on free-reduced meal plans. Consequently, this study does not suggest that small class size is an equalizer for Native American students, males, or students who are economically disadvantaged. The major finding from the teacher survey showed that teachers in large sized classrooms had more current professional development on reading strategies, which may have contributed to the class size achievement showing no significance when in small sized classrooms. By using analysis of variance (ANOVA), the study found that the following items were significantly different when teachers in small sized and large sized classrooms were compared: teaching leans toward students as individuals rather than towards the class in general (.016), time spent on disciplining the class (.008), time working with small groups (.041), time spent with students one on one (.017), time spent working with students on special projects (.004), and time spent developing creative projects for the class (.008)

Nonmodal linear stability of the Schwarzschild black hole

A proof is given that the space L of solutions of the linearized vacuum Einstein equation around a Schwarzschild black hole is parametrized by two scalar fields, which are gauge invariant combinations of perturbed algebraic and differential invariants of the Weyl tensor and encode the information on the odd (−) and even (+) sectors L. These fields measure the distortion of the geometry caused by a generic perturbation and are shown to be pointwise bounded on the outer region r ≥ 2M.publishedVersionFil: Dotti, Gustavo Daniel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Dotti, Gustavo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Física Enrique Gaviola; Argentina.Física de Partículas y Campo

Hyperk\"ahler torsion structures invariant by nilpotent Lie groups

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying non isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.Comment: LateX, 12 page

Maternal employment : enabling factors in context

Maternal employment is still below the overall EU recommended level of 60% in many European countries. Understanding the individual, household and contextual circumstances under which mothers of children of different ages are likely to be employed is crucial to develop strategies capable of increasing maternal employment. This article takes a comparative approach to investigating the characteristics associated with maternal employment in the presence of children aged 0\u20132, 3\u20135, 6\u20139 and 10\u201312 years. We model the probability of being employed full-time, parttime or being a homemaker using EU-SILC data (2004 to 2007) from Germany, Italy, Norway and the United Kingdom \u2013 four countries belonging to different gender and welfare regimes. The results indicate that individual and household characteristics are more relevant in determining mothers\u2019 employment in countries where the state is less supportive towards maternal employment: Italy and to a lesser extent Germany and the UK \u2013 for the period observed

Instability of the negative mass Schwarzschild naked singularity

We study the negative mass Schwarzschild spacetime, which has a naked singularity, and show that it is perturbatively unstable. This is achieved by first introducing a modification of the well known Regge - Wheeler - Zerilli approach to black hole perturbations to allow for the presence of a ``kinematic'' singularity that arises for negative masses, and then exhibiting exact exponentially growing solutions to the linearized Einstein's equations. The perturbations are smooth everywhere and behave nicely around the singularity and at infinity. In particular, the first order variation of the scalar invariants can be made everywhere arbitrarily small as compared to the zeroth order terms. Our approach is also compared to a recent analysis that leads to a different conclusion regarding the stability of the negative mass Schwarzschild spacetime. We also comment on the relevance of our results to the stability of more general negative mass, nakedly singular spacetimes.Comment: 15 page