Search

The Zwicky Transient Transient Facility (ZTF) performs a systematic neutrino follow-up program, searching for optical counterparts to high-energy neutrinos with dedicated Target-of-Opportunity (ToO) observations. Since first light in March 2018, ZTF has taken prompt observations for 24 high-quality neutrino alerts from the IceCube Neutrino Observatory, with a median latency of 12.2 hours from initial neutrino detection. From two of these campaigns, we have already reported tidal disruption event (TDE) AT2019dsg and likely TDE AT2019fdr as probable counterparts, suggesting that TDEs contribute >7.8% of the astrophysical neutrino flux. We here present the full results of our program through to December 2021. No additional candidate neutrino sources were identified by our program, allowing us to place the first constraints on the underlying optical luminosity function of astrophysical neutrino sources. Transients with optical absolutes magnitudes brighter that -21 can contribute no more than 87% of the total, while transients brighter than -22 can contribute no more than 58% of the total, neglecting the effect of extinction. These are the the first observational constraints on the neutrino emission of bright populations such as superluminous supernovae. None of the neutrinos were coincident with bright optical AGN flares comparable to that observed for TXS 0506+056/IC170922A, suggesting that most astrophysical neutrinos are not produced during such optical flares. We highlight the outlook for electromagnetic neutrino follow-up programs, including the expected potential for the Rubin Observatory., Submitted to MNRAS, comments welcome!

Close encounters between two initially unbound objects can result in a binary system if enough energy is released as gravitational waves (GWs). We address the scenario in which such encounters occur in merging elliptical galaxies. There is evidence that elliptical galaxies can harbor intermediate-mass black holes. Therefore, these systems are potentially the breeding grounds of sources of gravitational waves corresponding to inspiraling compact objects onto a massive black hole due to the dynamics, the large densities, and the number of compact remnants they contain. We show that this process is efficient for intermediate-mass black holes (IMBHs) with masses ranging from M $\in (10^3,10^5)$ M$_{\odot}$ and results in the formation of intermediate mass-ratio inspirals (IMRIs). We consider a set of IMBHs and smaller black holes with masses $m_2 \in (10,10^3)$ M$_{\odot}$ to estimate the IMRI formation rate. We find rates ranging between 10$^{-8}$ yr$^{-1}$, and 10$^{-5}$ yr$^{-1}$, and the IMRI formation rate per comoving volume in merging galaxies as a function of the redshift. The peak frequencies of the gravitational radiation emitted when these IMRIs are formed are within the detection band of space-borne detectors such as LISA and TianQin; taking into account the observable volume of these detectors, the total amount of IMRI detections per year is significant., 7 pages, 4 figures. Accepted for publication in MNRAS

Peters' formula is an analytical estimate of the time-scale of gravitational wave (GW)-induced coalescence of binary systems. It is used in countless applications, where the convenience of a simple formula outweighs the need for precision. However, many promising sources of the Laser Interferometer Space Antenna (LISA), such as supermassive black hole binaries and extreme mass-ratio inspirals (EMRIs), are expected to enter the LISA band with highly eccentric ($e \gtrsim$ 0.9) and highly relativistic orbits. These are exactly the two limits in which Peters' estimate performs the worst. In this work, we expand upon previous results and give simple analytical fits to quantify how the inspiral time-scale is affected by the relative 1.5 post-Newtonian (PN) hereditary fluxes and spin-orbit couplings. We discuss several cases that demand a more accurate GW time-scale. We show how this can have a major influence on quantities that are relevant for LISA event-rate estimates, such as the EMRI critical semi-major axis. We further discuss two types of environmental perturbations that can play a role in the inspiral phase: the gravitational interaction with a third massive body and the energy loss due to dynamical friction and torques from a surrounding gas medium ubiquitous in galactic nuclei. With the aid of PN corrections to the time-scale in vacuum, we find simple analytical expressions for the regions of phase space in which environmental perturbations are of comparable strength to the effects of any particular PN order, being able to qualitatively reproduce the results of much more sophisticated analyses., Accepted for publication in MNRAS. Comments welcome!

The astrophysical journal / Supplement series 258(2), 22 (2022). doi:10.3847/1538-4365/ac2edf, We describe the public release of the Cluster Monte Carlo (CMC) code, a parallel, star-by-star N-body code for modeling dense star clusters. CMC treats collisional stellar dynamics using Hénon’s method, where the cumulative effect of many two-body encounters is statistically reproduced as a single effective encounter between nearest-neighbor particles on a relaxation timescale. The star-by-star approach allows for the inclusion of additional physics, including strong gravitational three- and four-body encounters, two-body tidal and gravitational-wave captures, mass loss in arbitrary galactic tidal fields, and stellar evolution for both single and binary stars. The public release of CMC is pinned directly to the COSMIC population synthesis code, allowing dynamical star cluster simulations and population synthesis studies to be performed using identical assumptions about the stellar physics and initial conditions. As a demonstration, we present two examples of star cluster modeling: first, we perform the largest (N = 10$^{8}$) star-by-star N-body simulation of a Plummer sphere evolving to core collapse, reproducing the expected self-similar density profile over more than 15 orders of magnitude; second, we generate realistic models for typical globular clusters, and we show that their dynamical evolution can produce significant numbers of black hole mergers with masses greater than those produced from isolated binary evolution (such as GW190521, a recently reported merger with component masses in the pulsational pair-instability mass gap)., Published by Univ. of Chicago Press, Chicago, Ill. [u.a.]

We present a large ${\sim 30" \times 30"}$ spectroscopic survey of the Galactic Center using the SINFONI IFU at the VLT. Combining observations of the last two decades we compile spectra of over $2800$ stars. Using the Bracket-$\gamma$ absorption lines we identify $195$ young stars, extending the list of known young stars by $79$. In order to explore the angular momentum distribution of the young stars, we introduce an isotropic cluster prior. This prior reproduces an isotropic cluster in a mathematically exact way, which we test through numerical simulations. We calculate the posterior angular momentum space as function of projected separation from Sgr~A*. We find that the observed young star distribution is substantially different from an isotropic cluster. We identify the previously reported feature of the clockwise disk and find that its angular momentum changes as function of separation from the black hole, and thus confirm a warp of the clockwise disk ($p \sim 99.2\%$). At large separations, we discover three prominent overdensities of angular momentum. One overdensity has been reported previously, the counter-clockwise disk. The other two are new. Determining the likely members of these structures, we find that as many as $75\%$ of stars can be associated with one of these features. Stars belonging to the warped clockwise-disk show a top heavy K-band luminosity function, while stars belonging to the larger separation features do not. Our observations are in good agreement with the predictions of simulations of in-situ star formation, and argue for common formation of these structures., Comment: Accepted for Publication in APJL

In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalised) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initial data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalize to a class of spaces of non-constant curvature the theorem that spaces of the same constant curvature, dimension and metric tensor sign must be locally isometric. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields., Accepted for publication

Classical and quantum gravity 38(12), 125008 (2021). doi:10.1088/1361-6382/abf441, Gravitational wave astronomy has now left its infancy and has become an important tool for probing the most violent phenomena in our Universe. The LIGO/Virgo-KAGRA collaboration operates ground based detectors which cover the frequency band from 10 Hz to the kHz regime. Meanwhile, the pulsar timing array and the soon to launch LISA mission will cover frequencies below 0.1 Hz, leaving a gap in detectable gravitational wave frequencies. Here we show how a laser interferometer on the moon (LION) gravitational wave detector would be sensitive to frequencies from sub Hz to kHz. We find that the sensitivity curve is such that LION can measure compact binaries with masses between 10 and 100M ��� at cosmological distances, with redshifts as high as z = 100 and beyond, depending on the spin and the mass ratio of the binaries. LION can detect binaries of compact objects with higher-masses, with very large signal-to-noise ratios (SNRs), help us to understand how supermassive black holes got their colossal masses on the cosmological landscape, and it can observe in detail intermediate-mass ratio inspirals at distances as large as at least 100 Gpc. Compact binaries that never reach the LIGO/Virgo sensitivity band can spend significant amounts of time in the LION band, while sources present in the LISA band can be picked up by the detector and observed until their final merger. Since LION covers the deci-Hertz regime with such large SNRs, it truly achieves the dream of multi messenger astronomy., Published by IOP Publ., Bristol

Since 2015 the gravitational-wave observations of LIGO and Virgo have transformed our understanding of compact-object binaries. In the years to come, ground-based gravitational-wave observatories such as LIGO, Virgo, and their successors will increase in sensitivity, discovering thousands of stellar-mass binaries. In the 2030s, the space-based LISA will provide gravitational-wave observations of massive black holes binaries. Between the $\sim 10$-$10^3~\mathrm{Hz}$ band of ground-based observatories and the $\sim10^{-4}$-$10^{-1}~\mathrm{Hz}$ band of LISA lies the uncharted decihertz gravitational-wave band. We propose a Decihertz Observatory to study this frequency range, and to complement observations made by other detectors. Decihertz observatories are well suited to observation of intermediate-mass ($\sim10^2$-$10^4 M_\odot$) black holes; they will be able to detect stellar-mass binaries days to years before they merge, providing early warning of nearby binary neutron star mergers and measurements of the eccentricity of binary black holes, and they will enable new tests of general relativity and the Standard Model of particle physics. Here we summarise how a Decihertz Observatory could provide unique insights into how black holes form and evolve across cosmic time, improve prospects for both multimessenger astronomy and multiband gravitational-wave astronomy, and enable new probes of gravity, particle physics and cosmology., Comment: 13 pages, 1 figure. Published in Experimental Astronomy. Summarising white paper arXiv:1908.11375

The velocity of a gravitational wave (GW) source provides crucial information about its formation and evolution processes. Previous studies considered the Doppler effect on the phase of GWs as a potential signature of a time-dependent velocity of the source. However, the Doppler shift only accounts for the time component of the wave vector, and in principle motion also affects the spatial components. The latter effect, known as “aberration” for light, is analyzed in this paper for GWs and applied to the waveform modeling of an accelerating source. We show that the additional aberrational phase shift could be detectable in two astrophysical scenarios, namely, a recoiling binary black hole (BBH) due to GW radiation and a BBH in a triple system. Our results suggest that adding the aberrational phase shift in the waveform templates could significantly enhance the detectability of moving sources., This work is supported by the National Science Foundation of China Grants No. 11873022 and No. 11991053. A. T. O. is partly supported by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grants No. XDB23040100 and No. XDB23010200. P. A. S. acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, as well as the COST Action GWverse CA16104.

arXiv:2003.08953v1, Tidal disruption events occur rarely in any individual galaxy. Over the last decade, however, time-domain surveys have begun to accumulate statistical samples of these flares. What dynamical processes are responsible for feeding stars to supermassive black holes? At what rate are stars tidally disrupted in realistic galactic nuclei? What may we learn about supermassive black holes and broader astrophysical questions by estimating tidal disruption event rates from observational samples of flares? These are the questions we aim to address in this Chapter, which summarizes current theoretical knowledge about rates of stellar tidal disruption, and compares theoretical predictions to the current state of observations., N.C.S. received financial support from the NASA Astrophysics Theory Research Program (Grant NNX17AK43G; PI B. Metzger). M. K. acknowledges support from NSF Grant No. PHY-1607031 and NASA Award No. 80NSSC18K0639. E.M.R. acknowledges support from NWO TOP grant Module 2, project number 614.001.401. P.A.S. acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, the COST Action GWverse CA16104, and the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303).

The gravitational-wave astronomical revolution began in 2015 with LIGO's observation of the coalescence of two stellar-mass black holes. Over the coming decades, ground-based detectors like LIGO will extend their reach, discovering thousands of stellar-mass binaries. In the 2030s, the space-based LISA will enable gravitational-wave observations of the massive black holes in galactic centres. Between LISA and ground-based observatories lies the unexplored decihertz gravitational-wave frequency band. Here, we propose a Decihertz Observatory to cover this band, and complement observations made by other gravitational-wave observatories. The decihertz band is uniquely suited to observation of intermediate-mass ($\sim 10^2$-$10^4 M_\odot$) black holes, which may form the missing link between stellar-mass and massive black holes, offering a unique opportunity to measure their properties. Decihertz observations will be able to detect stellar-mass binaries days to years before they merge and are observed by ground-based detectors, providing early warning of nearby binary neutron star mergers, and enabling measurements of the eccentricity of binary black holes, providing revealing insights into their formation. Observing decihertz gravitational-waves also opens the possibility of testing fundamental physics in a new laboratory, permitting unique tests of general relativity and the Standard Model of particle physics. Overall, a Decihertz Observatory will answer key questions about how black holes form and evolve across cosmic time, open new avenues for multimessenger astronomy, and advance our understanding of gravitation, particle physics and cosmology., 52 pages, 5 figures, 4 tables. Submitted to Classical & Quantum Gravity. Based upon a white paper for ESA's Voyage 2050 on behalf of the LISA Consortium 2050 Task Force

Physical review letters 127(4), 041102 (2021). doi:10.1103/PhysRevLett.127.041102, Gravitational waves from a source moving relative to us can suffer from special-relativistic effects such as aberration. The required velocities for these to be significant are on the order of 1000 km s$^{���1}$. This value corresponds to the velocity dispersion that one finds in clusters of galaxies. Hence, we expect a large number of gravitational-wave sources to have such effects imprinted in their signals. In particular, the signal from a moving source will have its higher modes excited, i.e., (3,3) and beyond. We derive expressions describing this effect and study its measurability for the specific case of a circular, nonspinning extreme-mass-ratio inspiral. We find that the excitation of higher modes by a peculiar velocity of 1000 km s$^{���1}$ is detectable for such inspirals with signal-to-noise ratios of $���20$. Using a Fisher matrix analysis, we show that the velocity of the source can be measured to a precision of just a few percent for a signal-to-noise ratio of 100. If the motion of the source is ignored, parameter estimates could be biased, e.g., the estimated masses of the components through a Doppler shift. Conversely, by including this effect in waveform models, we could measure the velocity dispersion of clusters of galaxies at distances inaccessible to light., Published by APS, College Park, Md.

We study the formation of intermediate-mass ratio inspirals (IMRIs) triggered by the interactions between two stellar black holes (BHs) and an intermediate-mass BH (IMBH) inhabiting the centre of a dense star cluster. We exploit $N$-body models varying the IMBH mass, the stellar BH mass spectrum, and the star cluster properties. These simulations are coupled with a semi-analytic procedure to characterise the evolution of the remnant IMBH. The IMRIs formation probability attains values $\sim 5-50\%$, with larger values corresponding to larger IMBH masses. IMRIs map out the stellar BH mass spectrum, thus they might be used to unravel BH populations in star clusters harboring an IMBH. After the IMRI phase, an IMBH initially nearly maximal(almost non-rotating) tends to decrease(increase) its spin. If IMBHs grow mostly via repeated IMRIs, we show that only IMBH seeds sufficiently massive ($M_{\rm seed} > 300$ M$_\odot$) can grow up to $M_{\rm imbh} >10^3$ M$_\odot$ in dense globular clusters. Assuming that these seeds form at a redshift $z\sim 2-6$, we find that around $1-5\%$ of them would reach masses $\sim 500-1500$ M$_\odot$ at redshift $z=0$ and would exhibit low-spins, $S_{\rm imbh} < 0.2$. Measuring the mass and spin of IMBHs involved in IMRIs could help unravelling their formation mechanisms. We show that LISA can detect IMBHs in Milky Way globular clusters with a signal-to-noise ratio SNR$=10-100$, or in the Large Magellanic Cloud with an SNR$=8-40$. We provide the IMRIs merger rate for LIGO ($\Gamma_{\rm LIG} = 0.003-1.6$ yr$^{-1}$), LISA ($\Gamma_{\rm LIS} = 0.02-60$ yr$^{-1}$), ET ($\Gamma_{\rm ET} = 1-600$ yr$^{-1}$), and DECIGO ($\Gamma_{\rm DEC} = 6-3000$ yr$^{-1}$). Our simulations show that IMRIs' mass and spin encode crucial insights on the mechanisms that regulate IMBH formation and that the synergy among different detectors would enable us to fully unveil them. (Abridged), Comment: 18 pages, 20 figures, 3 tables. Accepted for publication in A&A

We present a revised version of Peters' (1964) time-scale for the gravitational-wave (GW) induced decay of two point masses. The new formula includes the effects of the first-order post-Newtonian perturbation and additionally provides a simple fit to account for the Newtonian self-consistent evolution of the eccentricity. The revised time-scale is found by multiplying Peters' estimate by two factors, $R(e_0)= 8^{1-\sqrt{1-e_0}}$ and $Q_{\rm f}(p_0) = \exp \left(2.5 (r_{\rm S}/p_0) \right)$, where $e_0$ and $p_0$ are the initial eccentricity and periapsis, respectively, and $r_{\rm S}$ the Schwarzschild radius of the system. Their use can correct errors of a factor of 1-10 that arise from using the original Peters' formula. We apply the revised time-scales to a set of typical sources for existing ground-based laser interferometers and for the future Laser Interferometer Space Antenna (LISA), at the onset of their GW driven decay. We argue that our more accurate model for the orbital evolution will affect current event- and detection-rate estimates for mergers of compact object binaries, with stronger deviations for eccentric LISA sources, such as extreme and intermediate mass-ratio inspirals. We propose the correction factors $R$ and $Q_{\rm f}$ as a simple prescription to quantify decay time-scales more accurately in future population synthesis models. We also suggest that the corrected time-scale may be used as a computationally efficient alternative to numerical integration in other applications that include the modelling of radiation reaction for eccentric sources., Accepted for publication in MNRAS

When two black holes merge in a dense star cluster, they form a new black hole with a well-defined mass and spin. If that "second-generation" black hole remains in the cluster, it will continue to participate in dynamical encounters, form binaries, and potentially merge again. Using a grid of 96 dynamical models of dense star clusters and a cosmological model of cluster formation, we explore the production of binary black hole mergers where at least one component of the binary was forged in a previous merger. We create four hypothetical universes where every black hole born in the collapse of a massive star has a dimensionless Kerr spin parameter, $\chi_{\rm birth}$, of 0.0, 0.1, 0.2, or 0.5. We show that if all stellar-born black holes are non-spinning ($\chi_{\rm birth} = 0.0$), then more than 10% of merging binary black holes from clusters have components formed from previous mergers, accounting for more than 20% of the mergers from globular clusters detectable by LIGO/Virgo. Furthermore, nearly 7% of detectable mergers would have a component with a mass $\gtrsim 55M_{\odot}$, placing them clearly in the mass "gap" region where black holes cannot form from isolated collapsing stars due to the pulsational-pair instability mechanism. On the other hand, if black holes are born spinning, then the contribution from these second-generation mergers decreases, making up as little as 1% of all detections from globular clusters when $\chi_{\rm birth} = 0.5$. We make quantitative predictions for the detected masses, mass ratios, and spin properties of first- and second-generation mergers from dense star clusters, and show how these distributions are highly sensitive to the birth spins of black holes., Comment: 15 pages, 8 figures, matches version accepted by PRD

The detection of the gravitational waves emitted in the capture process of a compact object by a massive black hole (MBH) is known as an extreme-mass ratio inspiral (EMRI), it represents a unique probe of gravity in the strong regime, and it is one of the main targets of the Laser Interferometer Space Antenna (LISA). The possibility of observing a compact-object EMRI at the Galactic Center (GC) when LISA is taking data is very low. However, the capture of a brown dwarf, an X-MRI, is more frequent because these objects are much more abundant and can plunge without being tidally disrupted. An X-MRI covers some $\ensuremath{\sim}{10}^{8}$ cycles before merger, and hence stays on band for millions of years. About $2\ifmmode\times\else\texttimes\fi{}{10}^{6}\text{ }\text{ }\mathrm{yrs}$ before merger they have a signal-to-noise ratio (SNR) at the GC of 10. Later, $1{0}^{4}\text{ }\text{ }\mathrm{yrs}$ before merger, the SNR is of several thousands, and $1{0}^{3}\text{ }\text{ }\mathrm{yrs}$ before the merger a few $1{0}^{4}$. Based on these values, this kind of EMRIs is also detectable at neighbor MBHs, albeit with fainter SNRs. We calculate the event rate of X-MRIs at the GC taking into account the asymmetry of pro- and retrograde orbits on the location of the last stable orbit (LSO). We estimate that at any given moment, and using a conservative approach, there are of the order of $\ensuremath{\gtrsim}20$ sources in band. From these, $\ensuremath{\gtrsim}5$ are circular and are located at higher frequencies, and about $\ensuremath{\gtrsim}15$ are highly eccentric and are at lower frequencies. Because of their proximity, X-MRIs represent a unique probe of gravity in the strong regime. The mass ratio for a X-MRI at the GC is $q\ensuremath{\sim}{10}^{8}$, i.e., 3 orders of magnitude larger than stellar-mass black hole EMRIs. Since backreaction depends on $q$, the orbit more closely follows a standard geodesic, which means that approximations work better in the calculation of the orbit. X-MRIs can be sufficiently loud so as to track the systematic growth of their SNR, which can be high enough to bury that of MBH binaries.

Current observations of the Galactic Centre (GC) seem to display a core-like distribution of bright stars from ~5 arcsec inwards. On the other hand, we observe young, massive stars at the GC, with roughly 20-50 per cent of them in a disc, mostly in the region where the bright giants appear to be lacking. In a previous publication we put the idea forward that the missing stars are deeply connected to the presence of this disc. The progenitor of the stellar disc is very likely to have been a gaseous disc that at some point fragmented and triggered star formation. This caused the appearance of overdensity regions in the disc that had high enough densities to ensure stripping large giants of their atmospheres and thus rendering them very faint. In this paper, we use a stellar evolution code to derive the properties that a red giant would display in a colour-magnitude diagram, as well as a non-linearity factor required for a correct estimate of the mass loss. We find that in a very short time-scale, the red giants leave their standard evolutionary track. The non-linearity factor has values that not only depend on the properties of the clumps, but also on the physical conditions of the giant stars, as we predicted analytically. According to our results, envelope stripping works, moving stars on a short time-scale from the giant branch to the white dwarf stage, thus rendering them invisible to observations.© 2019 The Author(s)., PA-S acknowledges support from the Ramon y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, as well as the COST Action GWverse CA16104. This work was supported by the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303 and 11873022). PA-S is indebted with Marta Masini and Luisa Seoane for their encouragement and countless hours of remote support during his visit to the KIAA in Beijing. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 614922. RS acknowledges financial support from the State Agency for Research of the Spanish MCIU through the 'Center of Excellence Severo Ochoa' award for the Instituto de Astrof ' isica de Andalucia (SEV-2017-0709). RS also acknowledges financial support from the national grant PGC2018-095049-B-C21 (MCIU/AEI/FEDER, UE).

The future space-borne detectors will provide the possibility to detect gravitational waves emitted from extreme mass ratio inspirals of stellar-mass compact objects into supermassive black holes. It is natural to expect that the spin of the compact object and cosmological constant will affect the orbit of the inspiral process and hence lead to the considerable phase shift of the corresponding gravitational waves. In this paper, we investigate the motion of a spinning test particle in the spinning black hole background with a cosmological constant and give the order of motion deviation induced by the particle's spin and the cosmological constant by considering the corresponding innermost stable circular orbit. By taking the neutron star or kerr black hole as the small body, the deviations of the innermost stable circular orbit parameters induced by the particle's spin and cosmological constant are given. Our results show that the deviation induced by particle's spin is much larger than that induced by cosmological constant when the test particle locates not very far away from the black hole, the accumulation of phase shift during the inspiral from the cosmological constant can be ignored when compared to the one induced by the particle's spin. However when the test particle locates very far away from the black hole, the impact from the cosmological constant will increase dramatically. Therefore the accumulation of phase shift for the whole process of inspiral induced by the cosmological constant and the particle's spin should be handled with caution., 12 pages, 2 figures, 7 tables

The detection of a gravitational capture of a stellar-mass compact object by a massive black hole (MBH) will allow us to test gravity in the strong regime. These sources form via two-body relaxation, by exchanging energy and angular momentum, and inspiral in a slow, progressive way down to the final merger. The range of frequencies is localised in the range of millihertz in the case of MBH of masses $\sim 10^6\,M_{\odot}$, i.e. that of space-borne gravitational-wave observatories such as LISA. In this article I show that, depending on their orbital parameters, intermediate-mass ratios (IMRIs) of MBH of masses between a hundred and a few thousand have frequencies that make them detectable (i) with ground-based observatories, or (ii) with both LISA and ground-based ones such as advanced LIGO/Virgo and third generation ones, with ET as an example. The binaries have a signal-to-noise ratio large enough to ensure detection. More extreme values in their orbital parameters correspond to systems detectable only with ground-based detectors and enter the LIGO/Virgo band in particular in many different harmonics for masses up to some $2000,\,M_{\odot}$. I show that environmental effects are negligible, so that the source should not have this kind of complication. The accumulated phase-shift is measurable with LISA and ET, and for some cases also with LIGO, so that it is possible to recover information about the eccentricity and formation scenario. For IMRIs with a total mass $\lessapprox 2000\,M_{\odot}$ and initial eccentricities up to $0.999$, LISA can give a warning to ground-based detectors with enough time in advance and seconds of precision. The possibility of detecting IMRIs from the ground alone or combined with space-borne observatories opens new possibilities for gravitational wave astronomy., Comment: Accepted for publication PRD. Added brief comment about how the joint detection can help to split various degeneracies

Sagittarius A* (SgrA*) lying in the Galactic Centre $8$ kpc from Earth, hosts the closest supermassive black hole known to us. It is now inactive, but there is evidence indicating that about six million years ago it underwent a powerful outburst where the luminosity could have approached the Eddington limit. Motivated by the fact that in extragalaxies the supermassive black holes with similar masses and near-Eddington luminosities are usually strong X-ray emitters, we calculate here the X-ray luminosity of SgrA*. For that, we assume that the outburst was due to accretion of gas or the tidal disruption of a star. We show that these cases could precipitate on Earth a hard X-ray (i.e. $h\nu>2~{\rm keV}$) flux comparable to that from the current quiescent sun. The flux in harder energy band $20~{\rm keV}, Comment: Minor changes, accepted for publication

The models currently used in the detection of gravitational waves (GWs) either do not consider a relative motion between the center-of-mass of the source and the observer, or usually only consider its effect on the frequencies of GWs. However, it is known for light waves that a relative motion not only changes the frequencies but also the brightness of the source, the latter of which is called the `beaming effect'. Here we investigate such an effect for GWs and find that the observed amplitude of a GW signal, unlike the behavior of light, is not a monotonic function of the relative velocity and responds differently to the two GW polarizations. We attribute the difference to a rotation of the wave-vector, as well as a reorientation of the GW polarizations. We find that even for velocities as small as $0.25\%$ of the speed of light, ignoring the aforementioned beaming effect could induce a systematic error that is larger than the designated calibration accuracy of LIGO. This error could lead to an incorrect estimation of the distance and orbital inclination of a GW source, or result in a spurious signal that appears to be incompatible with general relativity., 10 pages, 4 figures

The dynamical processing of black holes in the dense cores of globular clusters (GCs), makes them efficient factories for producing binary black holes (BBHs). Here we explore the population of BBHs that form dynamically in GCs and may be observable at mHz frequencies or higher with LISA. We use our Monte Carlo stellar dynamics code, which includes gravitational radiation reaction effects for all BH encounters. By creating a representative local universe of GCs, we show that up to dozens of these systems may be resolvable by LISA with signal-to-noise ratios of at least 5. Approximately one third of these binaries will have measurable eccentricities ($e > 10^{-3}$) in the LISA band and a small number ($\lesssim 5$) may evolve from the LISA band to the LIGO band during the LISA mission., Accepted for publication in Physical Review D

In currently favoured hierarchical cosmologies, the formation of massive black hole binaries (MBHBs) following galaxy mergers is unavoidable. Still, due the complex physics governing the (hydro)dynamics of the post-merger dense environment of stars and gas in galactic nuclei, the final fate of those MBHBs is still unclear. In gas-rich environments, it is plausible that turbulence and gravitational instabilities feed gas to the nucleus in the form of a series of cold incoherent clumps, thus providing a way to exchange energy and angular momentum between the MBHB and its surroundings. Within this context, we present a suite of smoothed-particle-hydrodynamical models to study the evolution of a sequence of near-radial turbulent gas clouds as they infall towards equal-mass, circular MBHBs. We focus on the dynamical response of the binary orbit to different levels of anisotropy of the incoherent accretion events. Compared to a model extrapolated from a set of individual cloud-MBHB interactions, we find that accretion increases considerably and the binary evolution is faster. This occurs because the continuous infall of clouds drags inwards circumbinary gas left behind by previous accretion events, thus promoting a more effective exchange of angular momentum between the MBHB and the gas. These results suggest that sub-parsec MBHBs efficiently evolve towards coalescence during the interaction with a sequence of individual gas pockets., Comment: 18 pages, 17 figures. Accepted for publication by MNRAS. Find companion paper at arXiv:1801.06179 Animations available at http://multipleclouds.xyz/movies/

Context. The distribution of stars around a massive black hole (MBH) has been addressed in stellar dynamics for the last four decades by a number of authors. Because of its proximity, the centre of the Milky Way is the only observational test case where the stellar distribution can be accurately tested. Past observational work indicated that the brightest giants in the Galactic centre (GC) may show a density deficit around the central black hole, not a cusp-like distribution, while we theoretically expect the presence of a stellar cusp. Aims. We here present a solution to this long-standing problem. Methods. We performed direct-summation N-body simulations of star clusters around massive black holes and compared the results of our simulations with new observational data of the GC's nuclear cluster. Results. We find that after a Hubble time, the distribution of bright stars as well as the diffuse light follow power-law distributions in projection with slopes of Γ ≈ 0.3 in our simulations. This is in excellent agreement with what is seen in star counts and in the distribution of the diffuse stellar light extracted from adaptive-optics (AO) assisted near-infrared observations of the GC. Conclusions. Our simulations also confirm that there exists a missing giant star population within a projected radius of a few arcsec around Sgr A∗. Such a depletion of giant stars in the innermost 0.1 pc could be explained by a previously present gaseous disc and collisions, which means that a stellar cusp would also be present at the innermost radii, but in the form of degenerate compact cores.© ESO, 2017., The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. [614922]. PAS acknowledges support from the Ramon y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain. This work has been partially supported by the CAS President's International Fellowship Initiative.

Context. The existence of dynamically relaxed stellar density cusps in dense clusters around massive black holes is a long-standing prediction of stellar dynamics, but it has so far escaped unambiguous observational confirmation. Aims. In this paper we aim to revisit the problem of inferring the innermost structure of the Milky Way's nuclear star cluster via star counts, to clarify whether it displays a core or a cusp around the central black hole. Methods. We used judiciously selected adaptive optics assisted high angular resolution images obtained with the NACO instrument at the ESO VLT. Through image stacking and improved point spread function fitting we pushed the completeness limit about one magnitude deeper than in previous, comparable work. Crowding and extinction corrections were derived and applied to the surface density estimates. Known young, and therefore dynamically not relaxed stars, are excluded from the analysis. Contrary to previous work, we analyse the stellar density in well-defined magnitude ranges in order to be able to constrain stellar masses and ages. Results. We focus on giant stars, with observed magnitudes K = 12.5-16, and on stars with observed magnitudes K ≈ 18, which may have similar mean ages and masses than the former. The giants display a core-like surface density profile within a projected radius R ≤ 0.3 pc of the central black hole, in agreement with previous studies, but their 3D density distribution is not inconsistent with a shallow cusp if we take into account the extent of the entire cluster, beyond the radius of influence of the central black hole. The surface density of the fainter stars can be described well by a single power-law at R < 2 pc. The cusp-like profile of the faint stars persists even if we take into account the possible contamination of stars in this brightness range by young pre-main sequence stars. The data are inconsistent with a core-profile for the faint stars. Finally, we show that a 3D Nuker law provides a good description of the cluster structure. Conclusions. We conclude that the observed density of the faintest stars detectable with reasonable completeness at the Galactic centre, is consistent with the existence of a stellar cusp around the Milky Way's central black hole, Sagittarius A∗. This cusp is well developed inside the influence radius of Sagittarius A∗and can be described by a single three-dimensional power-law with an exponent γ = 1.43 ± 0.02 ± 0.1. This corroborates existing conclusions from Nbody simulations performed in a companion paper. An important caveat is that the faint stars analysed here may be contaminated significantly by dynamically unrelaxed stars that formed about 100 Myr ago. The apparent lack of giants at projected distances of R ≲ 0.3 pc (R ≲ 8″) of the massive black hole may indicate that some mechanism may have altered their distribution or intrinsic luminosity. We roughly estimate the number of possibly missing giants to about 100.© ESO 2017., The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. [614922].PAS acknowledges support from the Ramon y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain. This work has been partially supported by the CAS President's International Fellowship Initiative.

Massive black hole binaries (MBHBs) represent an unavoidable outcome of hierarchical galaxy formation, but their dynamical evolution at sub-parsec scales is poorly understood, due to a combination of uncertainties in theoretical models and lack of firm observational evidence. In gas rich environments, it has been shown that a putative extended, steady circumbinary gaseous disc plays an important role in the MBHB evolution, facilitating its coalescence. How gas on galactic scales is transported to the nuclear region to form and maintain such a stable structure is, however, unclear. If, following a galaxy merger, turbulent gas is condenses in cold clumps and filaments that are randomly scattered, gas is naturally transported on parsec scales and interacts with the MBHB in discrete incoherent pockets. The aim of this work is to investigate the gaseous structures arising from this interaction. We employ a suite of smoothed-particle-hydrodynamic simulations to study the formation and evolution of gaseous structures around a MBHB constantly perturbed by the incoherent infall of molecular clouds. We investigate the influence of the infall rate and angular momentum distribution of the clouds on the geometry and stability of the arising structures. We find that the continuous supply of incoherent clouds is a double-edge sword, resulting in the intermittent formation and disruption of circumbinary structures. Anisotropic cloud distributions featuring an excess of co-rotating events generate more prominent co-rotating circumbinary discs. Similar structures are seen when mostly counter-rotating clouds are fed to the binary, even though they are more compact and less stable. In general, our simulations do not show the formation of extended smooth and stable circumbinary discs, typically assumed in analytical and numerical investigations of the the long term evolution of MBHBs. (Abridged), 22 Pages, 17 Figures. To be submitted to MNRAS

There is compelling evidence that most -if not all- galaxies harbour a super-massive black hole (SMBH) at their nucleus, hence binaries of these massive objects are an inevitable product of the hierarchical evolution of structures in the universe, and represent an important but thus-far elusive phase of galaxy evolution. Gas accretion via a circumbinary disc is thought to be important for the dynamical evolution of SMBH binaries, as well as in producing luminous emission that can be used to infer their properties. One plausible source of the gaseous fuel is clumps of gas formed due to turbulence and gravitational instabilities in the interstellar medium, that later fall toward and interact with the binary. In this context, we model numerically the evolution of turbulent clouds in near-radial infall onto equal-mass SMBH binaries, using a modified version of the SPH code GADGET-3. We present a total of 12 simulations that explore different possible pericentre distances and relative inclinations, and show that the formation of circumbinary discs and discs around each SMBH ('mini-discs') depend on those parameters. We also study the dynamics of the formed discs, and the variability of the feeding rate onto the SMBHs in the different configurations., Accepted for publication in MNRAS, 17 pages, 13 figures. Animations available at http://www.astro.puc.cl/~fgarrido/animations

The formation of compact stellar-mass binaries is a difficult, but interesting problem in astrophysics. There are two main formation channels: In the field via binary star evolution, or in dense stellar systems via dynamical interactions. The Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected black hole binaries (BHBs) via their gravitational radiation. These detections provide us with information about the physical parameters of the system. It has been claimed that when the Laser Interferometer Space Antenna (LISA) is operating, the joint observation of these binaries with LIGO will allow us to derive the channels that lead to their formation. However, we show that for BHBs in dense stellar systems dynamical interactions could lead to high eccentricities such that a fraction of the relativistic mergers are not audible to LISA. A non-detection by LISA puts a lower limit of about $0.005$ on the eccentricity of a BHB entering the LIGO band. On the other hand, a deci-Hertz observatory, like DECIGO or Tian Qin, would significantly enhance the chances of a joint detection, and shed light on the formation channels of these binaries., Submitted

The space-based Laser Interferometer Space Antenna (LISA) will be able to observe the gravitational-wave signals from systems comprised of a massive black hole and a stellar-mass compact object. These systems are known as extreme-mass-ratio inspirals (EMRIs) and are expected to complete $\sim 10^4$-$10^5$ cycles in band, thus allowing exquisite measurements of their parameters. In this work, we attempt to quantify the astrophysical uncertainties affecting the predictions for the number of EMRIs detectable by LISA, and find that competing astrophysical assumptions produce a variance of about three orders of magnitude in the expected intrinsic EMRI rate. However, we find that irrespective of the astrophysical model, at least a few EMRIs per year should be detectable by the LISA mission, with up to a few thousands per year under the most optimistic astrophysical assumptions. We also investigate the precision with which LISA will be able to extract the parameters of these sources. We find that typical fractional statistical errors with which the intrinsic parameters (redshifted masses, massive black hole spin and orbital eccentricity) can be recovered are $\sim 10^{-6}$-$10^{-4}$. Luminosity distance (which is required to infer true masses) is inferred to about $10\%$ precision and sky position is localized to a few square degrees, while tests of the multipolar structure of the Kerr metric can be performed to percent-level precision or better., Comment: 13 figures, 22 pages; updated to match published version

Context. This is the second of three papers that search for the predicted stellar cusp around the Milky Way's central black hole, Sagittarius A∗, with new data and methods. Aims. We aim to infer the distribution of the faintest stellar population currently accessible through observations around Sagittarius A∗. Methods. We used adaptive optics assisted high angular resolution images obtained with the NACO instrument at the ESO VLT. Through optimised PSF fitting we removed the light from all detected stars above a given magnitude limit. Subsequently we analysed the remaining, diffuse light density. Systematic uncertainties were constrained by the use of data from different observing epochs and obtained with different filters. We show that it is necessary to correct for the diffuse emission from the mini-spiral, which would otherwise lead to a systematically biased light density profile. We used a Paschen α map obtained with the Hubble Space Telescope for this purpose. Results. The azimuthally averaged diffuse surface light density profile within a projected distance of R ≲ 0.5 pc from Sagittarius A∗ can be described consistently by a single power law with an exponent of Γ = 0.26 ± 0.02 ± 0.05, similar to what has been found for the surface number density of faint stars in Paper I. Conclusions. The analysed diffuse light arises from sub-giant and main-sequence stars with K ≈ 19-22 with masses of 0.8-1.5 M. These stars can be old enough to be dynamically relaxed. The observed power-law profile and its slope are consistent with the existence of a relaxed stellar cusp around the Milky Way's central black hole. We find that a Nuker law provides an adequate description of the nuclear cluster's intrinsic shape (assuming spherical symmetry). The 3D power-law slope near Sgr A∗ is γ = 1.13 ± 0.03 ± 0.05. The stellar density decreases more steeply beyond a break radius of about 3 pc, which corresponds roughly to the radius of influence of the massive black hole. At a distance of 0.01 pc from the black hole, we estimate a stellar mass density of 2.6 ± 0.3 × 10 M pc and a total enclosed stellar mass of 180 ± 30 M. These estimates assume a constant mass-to-light ratio and do not take stellar remnants into account. The fact that a flat projected surface density is observed for old giants at projected distances R ≲ 0.3 pc implies that some mechanism may have altered their appearance or distribution.© ESO, 2017., The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No [614922]. PAS acknowledges support from the Ramon y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain. This work has been partially supported by the CAS President's International Fellowship Initiative. F.N.L. acknowledges financial support from a predoctoral contract of the Spanish Ministerio de Educacion, Cultura y Deporte, code FPU14/01700.

A wide variety of outstanding problems in astrophysics involve the motion of a large number of particles ($N\gtrsim 10^{6}$) under the force of gravity. These include the global evolution of globular clusters, tidal disruptions of stars by a massive black hole, the formation of protoplanets and the detection of sources of gravitational radiation. The direct-summation of $N$ gravitational forces is a complex problem with no analytical solution and can only be tackled with approximations and numerical methods. To this end, the Hermite scheme is a widely used integration method. With different numerical techniques and special-purpose hardware, it can be used to speed up the calculations. But these methods tend to be computationally slow and cumbersome to work with. Here we present a new GPU, direct-summation $N-$body integrator written from scratch and based on this scheme. This code has high modularity, allowing users to readily introduce new physics, it exploits available high-performance computing resources and will be maintained by public, regular updates. The code can be used in parallel on multiple CPUs and GPUs, with a considerable speed-up benefit. The single GPU version runs about 200 times faster compared to the single CPU version. A test run using 4 GPUs in parallel shows a speed up factor of about 3 as compared to the single GPU version. The conception and design of this first release is aimed at users with access to traditional parallel CPU clusters or computational nodes with one or a few GPU cards., Comment: minor modifications, version of the code includes now post-Newtonian corrections

One of the key astrophysical sources for the Laser Interferometer Space Antenna (LISA) are the inspirals of stellar-origin compact objects into massive black holes in the centres of galaxies. These extreme-mass-ratio inspirals (EMRIs) have great potential for astrophysics, cosmology and fundamental physics. In this paper we describe the likely numbers and properties of EMRI events that LISA will observe. We present the first results computed for the 2.5 Gm interferometer that was the new baseline mission submitted in January 2017 in response to the ESA L3 mission call. In addition, we attempt to quantify the astrophysical uncertainties in EMRI event rate estimates by considering a range of different models for the astrophysical population. We present both likely event rates and estimates for the precision with which the parameters of the observed sources could be measured. We finish by discussing the implications of these results for science using EMRIs., Comment: 12 pages, 3 figures, 5 tables. These proceedings are a summary of the results presented in arXiv:1703.09722

The formation and evolution of protoplanetary discs remains a challenge from both a theoretical and numerical standpoint. In this work we first perform a series of tests of our new hybrid algorithm presented in Glaschke, Amaro-Seoane and Spurzem 2011 (henceforth Paper I) that combines the advantages of high accuracy of direct-summation N-body methods with a statistical description for the planetesimal disc based on Fokker-Planck techniques. We then address the formation of planets, with a focus on the formation of protoplanets out of planetesimals. We find that the evolution of the system is driven by encounters as well as direct collisions and requires a careful modelling of the evolution of the velocity dispersion and the size distribution over a large range of sizes. The simulations show no termination of the protoplanetary accretion due to gap formation, since the distribution of the planetesimals is only subjected to small fluctuations. We also show that these features are weakly correlated with the positions of the protoplanets. The exploration of different impact strengths indicates that fragmentation mainly controls the overall mass loss, which is less pronounced during the early runaway growth. We prove that the fragmentation in combination with the effective removal of collisional fragments by gas drag sets an universal upper limit of the protoplanetary mass as a function of the distance to the host star, which we refer to as the mill condition., Submitted

Triple supermassive black holes (SMBH) can form during the hierarchical mergers of massive galaxies with an existing binary. Perturbations by a third black hole may accelerate the merging process of an inner binary, for example through the Kozai mechanism. We analyze the evolution of simulated hierarchical triple SMBHs in galactic centers, and find resonances in the evolution of the semi-major axis, the eccentricity and the inclination, for both the inner and the outer orbits of the triple system, which are not only Kozai like. Through resonant oscillations, SMBH can trigger a significant increase of the inner SMBH binary eccentricity shortening the merger timescale expected from gravitational wave (GW) emission. As hierarchical triple SMBHs may be frequent in massive galaxies, the influence of orbital resonances is of great importance to our understanding of black hole coalescence and gravitational wave detection. Although Kozai mechanism is believed to play an important role in this process, detailed studies on the pattern of these resonances is necessary.

Kozai-Lidov (KL) oscillations can accelerate compact object mergers via gravitational wave (GW) radiation by driving the inner binaries of hierarchical triples to high eccentricities. We perform direct three-body integrations of high mass ratio compact object triple systems using Fewbody including post-Newtonian terms. We find that the inner binary undergoes rapid eccentricity oscillations (REOs) on the timescale of the outer orbital period which drive it to higher eccentricities than secular theory would otherwise predict, resulting in substantially reduced merger times. For a uniform distribution of tertiary eccentricity ($e_2$), ~40% of systems merge within ~1-2 eccentric KL timescales whereas secular theory predicts that only ~20% of such systems merge that rapidly. This discrepancy becomes especially pronounced at low $e_2$, with secular theory overpredicting the merger time by many orders of magnitude. We show that a non-negligible fraction of systems have eccentricity > 0.8 when they merge, in contrast to predictions from secular theory. Our results are applicable to high mass ratio triple systems containing black holes or neutron stars. In objects in which tidal effects are important, such as white dwarfs, stars, and planets, REOs can reduce the tidal circularization timescale by an order of magnitude and bring the components of the inner binary into closer orbits than would be possible in the secular approximation., Accepted to MNRAS. 15 pages, 11 figures

We explore the hardening of a massive black hole binary embedded in a circum-binary gas disc when the binary and the gas are coplanar and the gas is counter-rotating. The secondary black hole, revolving in the direction opposite to the gas, experiences a drag from gas-dynamical friction and from direct accretion of part of it. Using two-dimensional (2D) hydrodynamical grid simulations we investigate the effect of changing the accretion prescriptions on the dynamics of the secondary black hole which in turn affect the binary hardening and eccentricity evolution. We find that realistic accretion prescriptions lead to results that differ from those inferred assuming accretion of all the gas within the Roche Lobe of the secondary black hole. Different accretion prescriptions result in different disc's surface densities which alter the black hole's dynamics back. Full 3D SPH realizations of a number of representative cases, run over a shorter interval of time, validate the general trends observed in the less computationally demanding 2D simulations. Initially circular black hole binaries increase only slightly their eccentricity which then oscillates around small values (, abridged. Accepted for publication A&A

Gravitational waves are a prediction of general relativity, and with ground-based detectors now running in their advanced configuration, we will soon be able to measure them directly for the first time. Binaries of stellar-mass black holes are among the most interesting sources for these detectors. Unfortunately, the many different parameters associated with the problem make it difficult to promptly produce a large set of waveforms for the search in the data stream. To reduce the number of templates to develop, one must restrict some of the physical parameters to a certain range of values predicted by either (electromagnetic) observations or theoretical modeling. In this work we show that "hyperstellar" black holes (HSBs) with masses $30 \lesssim M_{\rm BH}/M_{\odot} \lesssim 100$, i.e black holes significantly larger than the nominal $10\,M_{\odot}$, will have an associated low value for the spin, i.e. $a, Accepted for publication MNRAS

Two coalescing black holes (BHs) represent a conspicuous source of gravitational waves (GWs). The merger involves 17 parameters in the general case of Kerr BHs, so that a successful identification and parameter extraction of the information encoded in the waves will provide us with a detailed description of the physics of BHs. A search based on matched-filtering for characterization and parameter extraction requires the development of some $10^{15}$ waveforms. If a third additional BH perturbed the system, the waveforms would not be applicable, and we would need to increase the number of templates required for a valid detection. In this letter, we calculate the probability that more than two BHs interact in the regime of strong relativity in a dense stellar cluster. We determine the physical properties necessary in a stellar system for three black holes to have a close encounter in this regime and also for an existing binary of two BHs to have a strong interaction with a third hole. In both cases the event rate is negligible. While dense stellar systems such as galactic nuclei, globular clusters and nuclear stellar clusters are the breeding grounds for the sources of gravitational waves that ground-based detectors like Advanced LIGO and Advanced VIRGO will be exploring, the analysis of the waveforms in full general relativity needs only to evaluate the two-body problem. This reduces the number of templates of waveforms to create by orders of magnitude.

Dense stellar systems such as globular clusters, galactic nuclei and nuclear star clusters are ideal loci to study stellar dynamics due to the very high densities reached, usually a million times higher than in the solar neighborhood; they are unique laboratories to study processes related to relaxation. There are a number of different techniques to model the global evolution of such a system. In statistical models we assume that relaxation is the result of a large number of two-body gravitational encounters with a net local effect. We present two moment models that are based on the collisional Boltzmann equation. By taking moments of the Boltzmann equation one obtains an infinite set of differential moment equations where the equation for the moment of order $n$ contains moments of order $n+1$. In our models we assume spherical symmetry but we do not require dynamical equilibrium. We truncate the infinite set of moment equations at order $n=4$ for the first model and at order $n=5$ for the second model. The collisional terms on the right-hand side of the moment equations account for two-body relaxation and are computed by means of the Rosenbluth potentials. We complete the set of moment equations with closure relations which constrain the degree of anisotropy of our model by expressing moments of order $n+1$ by moments of order $n$. The accuracy of this approach relies on the number of moments included from the infinite series. Since both models include fourth order moments we can study mechanisms in more detail that increase or decrease the number of high velocity stars. The resulting model allows us to derive a velocity distribution function, with unprecedented accuracy, compared to previous moment models.