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We investigate secular evolution in hierarchical quadruple systems as a formation channel of mass-gap black holes (with masses of about $3-5$ $M_{\odot}$) in systems that will eventually lead to binary black hole mergers detectable by ground-based gravitational wave detectors (LIGO/Virgo). We show that in a 3+1 hierarchical system, two episodes of induced mergers would first cause two neutron stars to merge and form a mass-gap black hole, which will subsequently merge with another (more massive) black hole through a second induced merger. We demonstrate that such systems are stable to flybys, and their formation would predict a high mass ratio and eccentric merger of a mass-gap black hole with a more massive black hole companion. Such a formation channel may explain observed gravitational wave events such as the recently-discovered LIGO/Virgo events S190814bv and S190924h., Comment: Accepted for publication at ApJ Letters

The isolated binary evolution model for merging neutron stars (NSs) involves processes such as mass transfer, common-envelope evolution, and natal kicks, all of which are poorly understood. Also, the predicted NS-NS merger rates are typically lower than the rates inferred from the LIGO GW170817 event. Here, we investigate merger rates of NS and black hole (BH)-NS binaries in hierarchical triple-star systems. In such systems, the tertiary can induce Lidov-Kozai (LK) oscillations in the inner binary, accelerating its coalescence, and potentially enhancing compact object merger rates. However, since compact objects originate from massive stars, the prior evolution should also be taken into account. Natal kicks, in particular, could significantly reduce the rates by unbinding the tertiary before it can affect the inner binary through LK evolution. We carry out simulations of massive triples taking into account stellar evolution starting from the main sequence, secular and tidal evolution, and the effects of supernovae. For large NS birth kicks (\sigma_k = 265 km/s), we find that the triple NS-NS merger rate (several hundred Gpc^{-3} yr^{-1}) is lower by a factor of ~2-3 than the binary rate, but for no kicks (\sigma_k = 0 km/s), the triple rate (several thousand Gpc^{-3} yr^{-1}) is comparable to the binary rate. Our results indicate that a significant fraction of NS-NS mergers could originate from triples if a substantial portion of the NS population is born with low kick velocities, as indicated by other work. However, uncertainties and open questions remain because of our simplifying assumption of dynamical decoupling after inner binary interaction has been triggered., Comment: Accepted for publication in ApJ. 16 pages, 8 figures

In statistical data assimilation (SDA) and supervised machine learning (ML), we wish to transfer information from observations to a model of the processes underlying those observations. For SDA, the model consists of a set of differential equations that describe the dynamics of a physical system. For ML, the model is usually constructed using other strategies. In this paper, we develop a systematic formulation based on Monte Carlo sampling to achieve such information transfer. Following the derivation of an appropriate target distribution, we present the formulation based on the standard Metropolis-Hasting (MH) procedure and the Hamiltonian Monte Carlo (HMC) method for performing the high dimensional integrals that appear. To the extensive literature on MH and HMC, we add (1) an annealing method using a hyperparameter that governs the precision of the model to identify and explore the highest probability regions of phase space dominating those integrals, and (2) a strategy for initializing the state space search. The efficacy of the proposed formulation is demonstrated using a nonlinear dynamical model with chaotic solutions widely used in geophysics.

Dense stellar systems such as globular clusters are believed to harbor merging binary black holes (BHs). The evolution of such binaries is driven by interactions with other stars, most notably, binary-single interactions. Traditionally, so-called `strong' interactions are believed to be the driving force in this evolution. However, we recently showed that more distant, i.e., "weak" or "secular" encounters, can have important implications for the properties of merging BH binaries in globular clusters. This motivates more detailed understanding of the effects of secular encounters on a binary. In another previous paper, we analytically calculated expressions for the changes of the eccentricity and angular-momentum vectors taking into account second-order (SO) perturbation theory, and showed that, for highly eccentric binaries, the new expressions give rise to behavior that is not captured by first-order (FO) theory. Here, we extend our previous work to third order (TO) perturbation theory. We also include terms up to and including octupole order. The latter are nonzero for binaries with unequal component masses. In addition, we consider the effects of post-Newtonian terms, and we determine the steady-state distribution due to the cumulative effect of secular encounters by computing the associated angular-momentum diffusion coefficients, and applying the Fokker-Planck equation. Together with our previous work, the results in this paper provide a framework for incorporating the effects of distant encounters on binaries in models of cluster evolution, such as Monte Carlo codes., Comment: Accepted for publication in MNRAS. 17 pages, 8 figures. Code can be found at https://github.com/hamers/flybys

We present the first systematic study on how distant weak interactions impact the dynamical evolution of merging binary black holes (BBHs) in dense stellar clusters. Recent studies indicate that dense clusters are likely to significantly contribute to the rate of merging BBHs observable through gravitational waves (GWs), and that many of these mergers will appear with notable eccentricities measurable in the LISA and LIGO sensitivity bands. This is highly interesting, as eccentricity can be used to distinguish between different astrophysical merger channels. However, all of these recent studies are based on various Monte Carlo (MC) techniques that only include strong interactions for the dynamical evolution of BBHs, whereas any binary generally undergoes orders-of-magnitude more weak interactions than strong. It is well known that weak interactions primarily lead to a change in the binary's eccentricity, which for BBHs implies that weak interactions can change their GW inspiral time and thereby their merger probability. With this motivation, we perform MC simulations of BBHs evolving in dense clusters under the influence of both weak and strong interactions. We find that including weak interactions leads to a notable increase in the number of BBHs that merge inside their cluster, which correspondingly leads to a higher number of eccentric LISA sources. These preliminary results illustrate the importance of including weak interactions for accurately modeling how BBHs merge in clusters, and how to link their emitted GW signals to their astrophysical environment., Comment: 19 pages. 12 figures. comments very welcome

Type Ia supernovae (SNe Ia) could arise from mergers of carbon-oxygen white dwarfs (WDs) triggered by Lidov-Kozai (LK) oscillations in hierarchical triple-star systems. However, predicted merger rates are several orders of magnitude lower than the observed SNe Ia rate. The low predicted rates can be attributed in part to the fact that many potential WD-merger progenitor systems, with high mutual orbital inclination, merge or interact before the WD stage. Recently, evidence was found for the existence of natal kicks imparted on WDs with a typical magnitude of 0.75 km/s. In triples, kicks change the mutual inclination and in general increase the outer orbit eccentricity, bringing the triple into an active LK regime at late stages and avoiding the issue of pre-WD merger or interaction. Stars passing by the triple can result in similar effects. However, both processes can also disrupt the triple. In this paper, we quantitatively investigate the impact of WD kicks and flybys on the rate of WD mergers using detailed simulations. We find that WD kicks and flybys combine to increase the predicted WD merger rates by a factor of ~2.5, resulting in a time-integrated rate of ~1.1e-4 Msun^-1. Despite the significant boost, the predicted rates are still more than one order of magnitude below the observed rate of ~1e-3 Msun^-1. However, many systematic uncertainties still remain in our calculations, in particular the potential contributions from tighter triples, dynamically unstable systems, unbound systems due to WD kicks, and quadruple systems., Comment: Accepted for publication in ApJ. 12 pages, 9 figures

Binary-single interactions play a crucial role in the evolution of dense stellar systems such as globular clusters. In addition, they are believed to drive black hole (BH) binary mergers in these systems. A subset of binary-single interactions are secular encounters, for which the third body approaches the binary on a relatively wide orbit, and such that it is justified to average the equations of motion over the binary's orbital phase. Previous works used first-order perturbation theory to compute the effects of such secular encounters on the binary. However, this approach can break down for highly eccentric binaries, which are important for BH binary mergers and gravitational wave sources. Here, we present an analytic computation using second-order perturbation techniques, valid to the quadrupole-order approximation. In our calculation, we take into account the instantaneous back-reaction of the binary to the third body, and compute corrections to previous first-order results. Using singly-averaged and direct 3-body integrations, we demonstrate the validity of our expressions. In particular, we show that the eccentricity change for highly eccentric binaries can reach a plateau, associated with a large inclination change, and can even reverse sign. These effects are not captured by previous first-order results. We provide a simple script to conveniently evaluate our analytic expressions, including routines for numerical integration and verification., Comment: Accepted for publication in MNRAS. One reference updated. 18 pages, 7 figures. Code can be found at https://github.com/hamers/flybys

Diverse optimization algorithms correctly identify, in finite time, intrinsic constraints that must be active at optimality. Analogous behavior extends beyond optimization to systems involving partly smooth operators, and in particular to variational inequalities over partly smooth sets. As in classical nonlinear programming, such active-set structure underlies the design of accelerated local algorithms of Newton type. We formalize this idea in broad generality as a simple linearization scheme for two intersecting manifolds., Comment: 21 pages, 0 figures

Statistical Data Assimilation (SDA) is the transfer of information from field or laboratory observations to a user selected model of the dynamical system producing those observations. The data is noisy and the model has errors; the information transfer addresses properties of the conditional probability distribution of the states of the model conditioned on the observations. The quantities of interest in SDA are the conditional expected values of functions of the model state, and these require the approximate evaluation of high dimensional integrals. We introduce a conditional probability distribution and use the Laplace method with annealing to identify the maxima of the conditional probability distribution. The annealing method slowly increases the precision term of the model as it enters the Laplace method. In this paper, we extend the idea of precision annealing (PA) to Monte Carlo calculations of conditional expected values using Metropolis-Hastings methods.

Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a well-known conjecture of Crouzeix. Finally, we consider arbitrary complex three-by-three matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semi-algebraic manifold with dimension 12.

Most studies of mass transfer in binary systems assume circular orbits at the onset of Roche lobe overflow. However, there are theoretical and observational indications that mass transfer could occur in eccentric orbits. In particular, eccentricity could be produced via sudden mass loss and velocity kicks during supernova explosions, or Lidov-Kozai (LK) oscillations in hierarchical triple systems, or, more generally, secular evolution in multiple-star systems. However, current analytic models of eccentric mass transfer are faced with the problem that they are only well defined in the limit of very high eccentricities, and break down for less eccentric and circular orbits. This provides a major obstacle to implementing such models in binary and higher-order population synthesis codes, which are useful tools for studying the long-term evolution of a large number of systems. Here, we present a new analytic model to describe the secular orbital evolution of binaries undergoing conservative mass transfer. The main improvement of our model is that the mass transfer rate is a smoothly varying function of orbital phase, rather than a delta function centered at periapsis. Consequently, our model is in principle valid for any eccentricity, thereby overcoming the main limitation of previous works. We implement our model in an easy-to-use and publicly available code that can be used as a basis for implementations of our model into population synthesis codes. We investigate the implications of our model in a number of applications with circular and eccentric binaries, and triples undergoing LK oscillations., Comment: Minor changes to match ApJ publication. 16 pages, 10 figures. Code can be found at https://github.com/hamers/emt

Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically intractable, but we show that computationally-cheap inexact projections may suffice instead. In particular, if one set is defined by sufficiently regular smooth constraints, then projecting onto the approximation obtained by linearizing those constraints around the current iterate suffices. On the other hand, if one set is a smooth manifold represented through local coordinates, then the approximate projection resulting from linearizing the coordinate system around the preceding iterate on the manifold also suffices.

The (yet-to-be confirmed) discovery of a Neptune-sized moon around the ~3.2 Jupiter-mass planet in Kepler 1625 puts interesting constraints on the formation of the system. In particular, the relatively wide orbit of the moon around the planet, at ~40 planetary radii, is hard to reconcile with planet formation theories. We demonstrate that the observed characteristics of the system can be explained from the tidal capture of a secondary planet in the young system. After a quick phase of tidal circularization, the lunar orbit, initially much tighter than 40 planetary radii, subsequently gradually widened due to tidal synchronization of the spin of the planet with the orbit, resulting in a synchronous planet-moon system. Interestingly, in our scenario the captured object was originally a Neptune-like planet, turned into a moon by its capture., Comment: Accepted for publication in ApJL. 7 pages, 5 figures

The distribution of the inner orbital periods of solar-type main-sequence (MS) triple star systems is known to be peaked at a few days, and this has been attributed to tidal evolution combined with eccentricity excitation due to Lidov-Kozai oscillations. Solar-type MS quadruple star systems also show peaks in their inner orbital period distributions at a few days. Here, we investigate the natural question whether tidal evolution combined with secular evolution can explain the observed inner orbital period distributions in quadruple stars. We carry out population synthesis simulations of solar-type MS quadruple star systems in both the 2+2 (two binaries orbiting each other's barycentre) and 3+1 (triple orbited by a fourth star) configurations. We take into account secular gravitational and tidal evolution, and the effects of passing stars. We assume that no short-period systems are formed initially, and truncate the initial orbital period distributions below 10 d accordingly. We find that, due to secular and tidal evolution, the inner orbital period distributions develop tails at short periods. Although qualitatively consistent with the observations, we find that our simulated orbital period distributions only quantitatively agree with the observations for the 3+1 systems. The observed 2+2 systems, on the other hand, show an enhancement of systems around 10 d, which is not reproduced in the simulations. This suggests that the inner orbital periods of 2+2 systems are not predominantly shaped by tidal and secular evolution, but by other processes, most likely occurring during the stellar formation and early evolution., Comment: Minor revision to match MNRAS publication. 15 pages, 13 figures

The idea of partial smoothness in optimization blends certain smooth and nonsmooth properties of feasible regions and objective functions. As a consequence, the standard first-order conditions guarantee that diverse iterative algorithms (and post-optimality analyses) identify active structure or constraints. However, by instead focusing directly on the first-order conditions, the formal concept of partial smoothness simplifies dramatically: in basic differential geometric language, it is just a constant-rank condition. In this view, partial smoothness extends to more general mappings, such as saddlepoint operators underlying primal-dual splitting algorithms.

The Kepler mission has detected a number of transiting circumbinary planets (CBPs). Although currently not detected, exomoons could be orbiting some of these CBPs, and they might be suitable for harboring life. A necessary condition for the existence of such exomoons is their long-term dynamical stability. Here, we investigate the stability of exomoons around the Kepler CBPs using numerical $N$-body integrations. We determine regions of stability and obtain stability maps in the (a_m,i_pm) plane, where a_m is the initial exolunar semimajor axis with respect to the CBP, and i_pm is the initial inclination of the orbit of the exomoon around the planet with respect to the orbit of the planet around the stellar binary. Ignoring any dependence on i_pm, for most Kepler CBPs the stability regions are well described by the location of the 1:1 mean motion commensurability of the binary orbit with the orbit of the moon around the CBP. This is related to a destabilizing effect of the binary compared to the case if the binary were replaced by a single body, and which is borne out by corresponding 3-body integrations. For high inclinations, the evolution is dominated by Lidov-Kozai oscillations, which can bring moons in dynamically stable orbits to close proximity within the CBP, triggering strong interactions such as tidal evolution, tidal disruption, or direct collisions. This suggests that there is a dearth of highly-inclined exomoons around the Kepler CBPs, whereas coplanar exomoons are dynamically allowed., Comment: Accepted for publication by MNRAS. 12 pages, 7 figures

Binaries within the sphere of influence of a massive black hole (MBH) in galactic nuclei are susceptible to the Lidov-Kozai (LK) mechanism, which can drive orbits to high eccentricities and trigger strong interactions within the binary such as the emission of gravitational waves (GWs), and mergers of compact objects. These events are potential sources for GW detectors such as Advanced LIGO and VIRGO. The LK mechanism is only effective if the binary is highly inclined with respect to its orbit around the MBH (within a few degrees of 90 deg), implying low rates. However, close to an MBH, torques from the stellar cluster give rise to the process of vector resonant relaxation (VRR). VRR can bring a low-inclination binary into an `active' LK regime in which high eccentricities and strong interactions are triggered in the binary. Here, we study the coupled LK-VRR dynamics, with implications for LIGO and VIRGO GW sources. We carry out Monte Carlo simulations and find that the merger fraction enhancement due to LK-VRR dynamics is up to a factor of ~10 for the lower end of assumed MBH masses (M_MBH = 1e4 MSun), and decreases sharply with increasing M_MBH. We find that, even in our most optimistic scenario, the baseline BH-BH merger rate is small, and the enhancement by LK-VRR coupling is not large enough to increase the rate to well above the LIGO/VIRGO lower limit, 12 Gpc^{-3} yr^{-1}. For the Galactic Center, the LK-VRR-enhanced rate is ~100 times lower than the LIGO/VIRGO limit, and for M_MBH = 1e4 MSun, the rate barely reaches 12 Gpc^{-3} yr^{-1}., Comment: Accepted for publication in ApJ. 22 pages, 20 figures

This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various enhancements that have been proposed to improve practical performance, as well as of several extensions that have been made in the literature, such as to solve constrained problems. The paper also includes clarification of certain technical aspects of the analysis of gradient sampling algorithms, most notably related to the assumptions one needs to make about the set of points at which the objective is continuously differentiable. Finally, we discuss possible future research directions., Comment: Submitted to: Special Methods for Nonsmooth Optimization (Springer, 2018), edited by A. Bagirov, M. Gaudioso, N. Karmitsa and M. M\"akel\"a

We consider the evolution of stellar hierarchical quadruple systems in the 2+2 (two binaries orbiting each other's barycentre) and 3+1 (triple orbited by a fourth star) configurations. In our simulations, we take into account the effects of secular dynamical evolution, stellar evolution, tidal evolution and encounters with passing stars. We focus on type Ia supernovae (SNe Ia) driven by collisions of carbon-oxygen (CO) white dwarfs (WDs). Such collisions can arise from several channels: (1) collisions due to extremely high eccentricities induced by secular evolution, (2) collisions following a dynamical instability of the system, and (3) collisions driven by semisecular evolution. The systems considered here have initially wide inner orbits, with initial semilatus recti larger than 12 AU, implying no interaction if the orbits were isolated. However, taking into account dynamical evolution, we find that $\approx 0.4$ ($\approx 0.6$) of 2+2 (3+1) systems interact. In particular, Roche Lobe overflow can be triggered possibly in highly eccentric orbits, dynamical instability can ensue due to mass-loss-driven orbital expansion or secular evolution, or a semisecular regime can be entered. We compute the delay-time distributions (DTDs) of collision-induced SNe Ia, and find that they are flatter compared to the observed DTD. Moreover, our combined SNe Ia rates are $(3.7\pm0.7) \times 10^{-6} \,\mathrm{M}_\odot^{-1}$ and $(1.3\pm0.2) \times 10^{-6} \,\mathrm{M}_\odot^{-1}$ for 2+2 and 3+1 systems, respectively, three orders of magnitude lower compared to the observed rate, of order $10^{-3} \, \mathrm{M}_\odot^{-1}$. The low rates can be ascribed to interactions before the stars evolve to CO WDs. However, our results are lower limits given that we considered a subset of quadruple systems., Comment: Accepted for publication in MNRAS. 18 pages, 13 figures

The BFGS quasi-Newton methodology, popular for smooth minimization, has also proved surprisingly effective in nonsmooth optimization. Through a variety of simple examples and computational experiments, we explore how the BFGS matrix update improves the local metric associated with a convex function even in the absence of smoothness and without using a line search. We compare the behavior of the BFGS and Shor r-algorithm updates., Comment: 26 pages

We extend the formalism of a previous paper to include the effects of flybys and instantaneous perturbations such as supernovae on the long-term secular evolution of hierarchical multiple systems with an arbitrary number of bodies and hierarchy, provided that the system is composed of nested binary orbits. To model secular encounters, we expand the Hamiltonian in terms of the ratio of the separation of the perturber with respect to the barycentre of the multiple system, to the separation of the widest orbit. Subsequently, we integrate over the perturber orbit numerically or analytically. We verify our method for secular encounters, and illustrate it with an example. Furthermore, we describe a method to compute instantaneous orbital changes to multiple systems, such as asymmetric supernovae and impulsive encounters. The secular code, with implementation of the extensions described in this paper, is publicly available within AMUSE, and we provide a number of simple example scripts to illustrate its usage for secular and impulsive encounters, and asymmetric supernovae. The extensions presented in this paper are a next step toward efficiently modeling the evolution of complex multiple systems embedded in star clusters., Comment: Accepted for publication in MNRAS. 24 pages, 7 figures