The next generation of ground-based gravitational-wave detectors will observe coalescences of black holes and neutron stars throughout the cosmos, thousands of them with exceptional fidelity. The Science Book is the result of a 3-year effort to study the science capabilities of networks of next generation detectors. Such networks would make it possible to address unsolved problems in numerous areas of physics and astronomy, from Cosmology to Beyond the Standard Model of particle physics, and how they could provide insights into workings of strongly gravitating systems, astrophysics of compact objects and the nature of dense matter. It is inevitable that observatories of such depth and finesse will make new discoveries inaccessible to other windows of observation. In addition to laying out the rich science potential of the next generation of detectors, this report provides specific science targets in five different areas in physics and astronomy and the sensitivity requirements to accomplish those science goals. This report is the second in a six part series of reports by the GWIC 3G Subcommittee: i) Expanding the Reach of Gravitational Wave Observatories to the Edge of the Universe, ii) The Next Generation Global Gravitational Wave Observatory: The Science Book (this report), iii) 3G R D: R D for the Next Generation of Ground-based Gravitational Wave Detectors, iv) Gravitational Wave Data Analysis: Computing Challenges in the 3G Era, v) Future Ground-based Gravitational-wave Observatories: Synergies with Other Scientific Communities, and vi) An Exploration of Possible Governance Models for the Future Global Gravitational-Wave Observatory Network.

The extent of mixed regions around convective zones is one of the biggest uncertainties in stellar evolution. One-dimensional overshooting descriptions introduce a free parameter (fov) that is, in general, not well constrained from observations. Especially in small central convective regions, the value is highly uncertain due to its tight connection to the pressure scale height. Long-term multi-dimensional hydrodynamic simulations can be used to study the size of the overshooting region as well as the involved mixing processes. Here we show how one can calibrate an overshooting parameter by performing two-dimensional Maestro simulations of zero-age-main-sequence stars ranging from 1.3 to 3.5 M⊙. The simulations cover the convective cores of the stars and a large fraction of the surrounding radiative envelope. We follow the convective flow for at least 20 convective turnover times, while the longest simulation covers 430 turnover time scales. This allows us to study how the mixing as well as the convective boundary itself evolve with time, and how the resulting entrainment can be interpreted in terms of overshooting parameters. We find that increasing the overshooting parameter fov beyond a certain value in the initial model of our simulations changes the mixing behaviour completely. This result can be used to put limits on the overshooting parameter. We find 0.010 < fov < 0.017 to be in good agreement with our simulations of a 3.5 M⊙ mass star. We also identify a diffusive mixing component due to internal gravity waves that is active throughout the convectively stable layer, but it is most likely overestimated in our simulations. Furthermore, applying our calibration method to simulations of less massive stars suggests a need for a mass-dependent overshooting description where the mixing in terms of the pressure scale height is reduced for small convective cores.

Observations with next-generation ground-based detectors further enhanced with multi-messenger (electromagnetic and neutrino) detections will allow us to probe new extreme astrophysics. Target sources included: core-collapse supernovae, continuous emission from isolated or accreting neutron stars, and bursts from magnetars and other pulsars.

We present a new method for solving the 3D gravitational potential of a density field on the Yin–Yang grid. Our algorithm is based on a multipole decomposition and is completely symmetric with respect to the two Yin–Yang grid patches. It is particularly efficient on distributed-memory machines with a large number of compute tasks, because the amount of data being explicitly communicated is minimized. All operations are performed on the original grid without the need for interpolating data onto an auxiliary spherical mesh.

The birth properties of neutron stars yield important information on the still debated physical processes that trigger the explosion and on intrinsic neutron-star physics. These properties include the high space velocities of young neutron stars with average values of several 100 km/s, whose underlying "kick" mechanism is not finally clarified. There are two competing possibilities that could accelerate NSs during their birth: anisotropic ejection of either stellar debris or neutrinos. We here present new evidence from X-ray measurements that chemical elements between silicon and calcium in six young gaseous supernova remnants are preferentially expelled opposite to the direction of neutron star motion. There is no correlation between the kick velocities and magnetic field strengths of these neutron stars. Our results support a hydrodynamic origin of neutron-star kicks connected to asymmetric explosive mass ejection, and they conflict with neutron-star acceleration scenarios that invoke anisotropic neutrino emission caused by particle and nuclear physics in combination with very strong neutron-star magnetic fields.

We discuss torsional oscillations of highly magnetized neutron stars (magnetars) using two-dimensional, magneto-elastic-hydrodynamical simulations. Our model is able to explain both the low-and high-frequency quasi-periodic oscillations (QPOs) observed in magnetars. The analysis of these oscillations provides constraints on the breakout magnetic-field strength, on the fundamental QPO frequency, and on the frequency of a particularly excited overtone. By performing a new set of simulations, we are able to derive for the first time empirical relations for a self consistent model including a superfluid core which describe these constraints quantitatively. We use these relations to generically constrain properties of high-density matter in neutron stars, employing Bayesian analysis. In spite of current uncertainties and computational approximations, our model-dependent Bayesian posterior estimates for SGR 1806-20 yield a magnetic-field strength (B) over bar similar to 2.1(-1.0)(+1.3) x 10(15) G and a crust thickness of Delta r = 1.6(-0.6)(+0.7) km, which are both in remarkable agreement with observational and theoretical expectations, respectively (1 sigma error bars are indicated). Our posteriors also favour the presence of a superfluid phase in the core, a relatively low stellar compactness, M/R < 0.19, indicating a relatively stiff equation of state and/or low-mass neutron star, and high shear speeds at the base of the crust, c(s) > 1.4 x 10(8) cm s(-1). Although the procedure laid out here still has large uncertainties, these constraints could become tighter when additional observations become available.

The spatial and velocity distributions of nuclear species synthesized in the innermost regions of core-collapse supernovae can yield important clues about explosion asymmetries and the operation of the still disputed explosion mechanism. Recent observations of radioactive 44 Ti with high-energy satellite telescopes ( Nuclear Spectroscopic Telescope Array [ NuSTAR ], INTEGRAL ) have measured gamma-ray line details, which provide direct evidence of large-scale explosion asymmetries in SN 1987A and in Cassiopeia A (Cas A) even by mapping of the spatial brightness distribution ( NuSTAR ). Here we discuss a 3D simulation of a neutrino-driven explosion, using a parameterized neutrino engine, whose 44 Ti distribution is mostly concentrated in one hemisphere pointing opposite to the neutron star (NS) kick velocity. Both exhibit intriguing resemblance to the observed morphology of the Cas A remnant, although neither the progenitor nor the explosion was fine-tuned for a perfect match. Our results demonstrate that the asymmetries observed in this remnant can, in principle, be accounted for by a neutrino-driven explosion, and that the high 44 Ti abundance in Cas A may be explained without invoking rapid rotation or a jet-driven explosion, because neutrino-driven explosions generically eject large amounts of high-entropy matter. The recoil acceleration of the NS is connected to mass ejection asymmetries and is opposite to the direction of the stronger explosion, fully compatible with the gravitational tugboat mechanism. Our results also imply that Cas A and SN 1987A could possess similarly “one-sided” Ti and Fe asymmetries, with the difference that Cas A is viewed from a direction with large inclination angle to the NS motion, whereas the NS in SN 1987A should have a dominant velocity component pointing toward us.

We propose a simple ansatz for estimating the value of the numerical resistivity and the numerical viscosity of any Eulerian MHD code. We test this ansatz with the help of simulations of the propagation of (magneto)sonic waves, Alfven waves, and the tearing mode instability using the MHD code Aenus. By comparing the simu- lation results with analytical solutions of the resistive-viscous MHD equations and an empirical ansatz for the growth rate of tearing modes we measure the numerical viscosity and resistivity of Aenus. The comparison shows that the fast-magnetosonic speed and wavelength are the characteristic velocity and length, respectively, of the aforementioned (relatively simple) systems. We also determine the dependance of the numerical viscosity and resistivity on the time integration method, the spatial reconstruction scheme and (to a lesser extent) the Riemann solver employed in the simulations. From the measured results we infer the numerical resolution (as a function of the spatial reconstruction method) required to properly resolve the growth and saturation level of the magnetic field amplified by the magnetorotational instability in the post-collapsed core of massive stars. Our results show that it is to the best advantage to resort to ultra-high order methods (e.g., 9th-order Monotonicity Preserving method) to tackle this problem properly, in particular in three dimensional simulations.

Extremely strong magnetic fields of the order of 10(15) G are required to explain the properties of magnetars, the most magnetic neutron stars. Such a strong magnetic field is expected to play an important role for the dynamics of core-collapse supernovae, and in the presence of rapid rotation may power superluminous supernovae and hypernovae associated to long gamma-ray bursts. The origin of these strong magnetic fields remains, however, obscure and most likely requires an amplification over many orders of magnitude in the protoneutron star. One of the most promising agents is the magnetorotational instability (MRI), which can in principle amplify exponentially fast a weak initial magnetic field to a dynamically relevant strength. We describe our current understanding of the MRI in protoneutron stars and show recent results on its dependence on physical conditions specific to protoneutron stars such as neutrino radiation, strong buoyancy effects and large magnetic Prandtl number.

During the common envelope (CE) phase, a giant star in a binary system overflows its Roche lobe and unstable mass transfer leads to a spiral-in of the companion, resulting in a close binary system or in a merger of the stellar cores. Dynamo processes during the CE phase have been proposed as a mechanism to generate magnetic fields that are important for forming magnetic white dwarfs (MWDs) and for shaping planetary nebulae. Here, we present the first magnetohydrodynamics simulations of the dynamical spiral-in during a CE phase. We find that magnetic fields are strongly amplified in the accretion stream around the 1 M-circle dot companion as it spirals into the envelope of a 2 M-circle dot RG. This leads to field strengths of 10-100 kG throughout the envelope after 120 d. The magnetic field amplification is consistent with being driven by the magnetorotational instability. The field strengths reached in our simulation make the magnetic field interesting for diagnostic purposes, but they are dynamically irrelevant. They are also too small to explain the formation of the highest fields found in MWDs, but may be relevant for luminous red novae, and detecting magnetic fields in these events would support the scenario as proposed here.

Stochastic differential equations describe well many physical, biological, and sociological systems, despite the simplification often made in their derivation. Here the usage of simple stochastic differential equations to characterize and classify complex dynamical systems is proposed within a Bayesian framework. To this end, we develop a dynamic system classifier (DSC). The DSC first abstracts training data of a system in terms of time-dependent coefficients of the descriptive stochastic differential equation. Thereby the DSC identifies unique correlation structures within the training data. For definiteness we restrict the presentation of the DSC to oscillation processes with a time-dependent frequency ω(t) and damping factor γ(t). Although real systems might be more complex, this simple oscillator captures many characteristic features. The ω and γ time lines represent the abstract system characterization and permit the construction of efficient signal classifiers. Numerical experiments show that such classifiers perform well even in the low signal-to-noise regime.

We investigate a method to construct parametrized progenitor models for core-collapse supernova simulations. Different from all modern core-collapse supernova studies, which rely on progenitor models from stellar evolution calculations, we follow the methodology of Baron & Cooperstein to construct initial models. Choosing parametrized spatial distributions of entropy and electron fraction as a function of mass coordinate and solving the equation of hydrostatic equilibrium, we obtain the initial density structures of our progenitor models. First, we calculate structures with parameters fitting broadly the evolutionary model s11.2 of Woosley et al. (2002). We then demonstrate the reliability of our method by performing general relativistic hydrodynamic simulations in spherical symmetry with the isotropic diffusion source approximation to solve the neutrino transport. Our comprehensive parameter study shows that initial models with a small central entropy (a parts per thousand(2)0.4 k(B) nucleon(-1)) can explode even in spherically symmetric simulations. Models with a large entropy (a parts per thousand(3)6 k(B) nucleon(-1)) in the Si/O layer have a rather large explosion energy (similar to 4 x 10(50) erg) at the end of the simulations, which is still rapidly increasing.

The magnetorotational instability (MRI) is a promising mechanism to amplify the magnetic field in fast rotating protoneutron stars (PNS). The diffusion of neutrinos trapped in the PNS induces a transport of momentum, which can be modelled as a viscosity on length-scales longer than the neutrino mean free path. This neutrino-viscosity can slow down the growth of MRI modes to such an extent that a minimum initial magnetic field strength of greater than or similar to 10(12) G is needed for the MRI to grow on a sufficiently short time-scale to potentially affect the explosion. It is uncertain whether the magnetic field of fast rotating progenitor cores is strong enough to yield such an initial magnetic field in PNS. At MRI wavelengths shorter than the neutrino mean free path, on the other hand, neutrino radiation does not act as a viscosity but rather induces a drag on the velocity with a damping rate independent of the wavelength. We perform a linear analysis of the MRI in this regime, and apply our analytical results to the PNS structure from a one-dimensional numerical simulation. We show that in the outer layers of the PNS, the MRI can grow from weak magnetic fields at wavelengths shorter than the neutrino mean free path, while deeper in the PNS MRI growth takes place in the viscous regime and requires a minimum magnetic field strength.

The magnetorotational instability (MRI) is considered to be a promising mechanism to amplify the magnetic field in fast-rotating protoneutron stars. In contrast to accretion discs, radial buoyancy driven by entropy and lepton fraction gradients is expected to have a dynamical role as important as rotation and shear. We investigate the poorly known impact of buoyancy on the non-linear phase of the MRI, by means of three-dimensional numerical simulations of a local model in the equatorial plane of a protoneutron star. The use of the Boussinesq approximation allows us to utilize a shearing box model with clean shearing periodic boundary conditions, while taking into account the buoyancy driven by radial entropy and composition gradients. We find significantly stronger turbulence and magnetic fields in buoyantly unstable flows. On the other hand, buoyancy has only a limited impact on the strength of turbulence and magnetic field amplification for buoyantly stable flows in the presence of a realistic thermal diffusion. The properties of the turbulence are, however, significantly affected in the latter case. In particular, the toroidal components of the magnetic field and of the velocity become even more dominant with respect to the poloidal ones. Furthermore, we observed in the regime of stable buoyancy the formation of long-lived coherent structures such as channel flows and zonal flows. Overall, our results support the ability of the MRI to amplify the magnetic field significantly even in stably stratified regions of protoneutron stars.

We have recently presented a new approach for numerical relativity simulations in spherical polar coordinates, both for vacuum and for relativistic hydrodynamics. Our approach is based on a referencemetric formulation of the Baumgarte-Shapiro-Shibata-Nakamura equations, a factoring of all tensor components, as well as a partially implicit Runge-Kutta method, and does not rely on a regularization of the equations, nor does it make any assumptions about the symmetry across the origin. In order to demonstrate this feature we present here several off-centered simulations, including simulations of single black holes and neutron stars whose center is placed away from the origin of the coordinate system, as well as the asymmetric head-on collision of two black holes. We also revisit our implementation of relativistic hydrodynamics and demonstrate that a reference-metric formulation of hydrodynamics together with a factoring of all tensor components avoids problems related to the coordinate singularities at the origin and on the axes. As a particularly demanding test we present results for a shock wave propagating through the origin of the spherical polar coordinate system.

We couple internal torsional, magneto-elastic oscillations of highly magnetized neutron stars (magnetars) to their magnetospheres. The corresponding axisymmetric perturbations of the external magnetic field configuration evolve as a sequence of linear, force-free equilibria that are completely determined by the background magnetic field configuration and by the perturbations of the magnetic field at the surface. The perturbations are obtained from simulations of magneto-elastic oscillations in the interior of the magnetar. While such oscillations can excite travelling Alfven waves in the exterior of the star only in a very limited region close to the poles, they still modulate the near magnetosphere by inducing a time-dependent twist between the foot-points of closed magnetic field lines that exit the star at a polar angle greater than or similar to 0.19 rad. Moreover, we find that for a dipole-like background magnetic field configuration the magnetic field modulations in the magnetosphere, driven by internal oscillations, can only be symmetric with respect to the equator. This is in agreement with our previous findings, where we interpreted the observed quasi-periodic oscillations in the X-ray tail of magnetar bursts as driven by the family of internal magneto-elastic oscillations with symmetric magnetic field perturbations.

In this paper we report on what we believe is the first successful implementation of relativistic hydrodynamics, coupled to dynamical spacetimes, in spherical polar coordinates without symmetry assumptions. We employ a high-resolution shock-capturing scheme, which requires that the equations be cast in flux-conservative form. One example of such a form is the :Valencia" formulation, which has been adopted in numerous applications, in particular in Cartesian coordinates. Here we generalize this formulation to allow for a reference-metric approach, which provides a natural framework for calculations in curvilinear coordinates. In spherical polar coordinates, for example, it allows for an analytical treatment of the singular r and sin(\theta) terms that appear in the equations. We experiment with different versions of our generalized Valencia formulation in numerical implementations of relativistic hydrodynamics for both fixed and dynamical spacetimes. We consider a number of different tests -- non-rotating and rotating relativistic stars, as well as gravitational collapse to a black hole -- to demonstrate that our formulation provides a promising approach to performing fully relativistic astrophysics simulations in spherical polar coordinates.

Our numerical simulations show that axisymmetric, torsional, magneto-elastic oscillations of magnetars with a superfluid core can explain the whole range of observed quasi-periodic oscillations (QPOs) in the giant flares of soft gamma-ray repeaters. There exist constant phase, magneto-elastic QPOs at both low (f<150 Hz) and high frequencies (f>500 Hz), in full agreement with observations. The range of magnetic field strengths required to match the observed QPO frequencies agrees with that from spin-down estimates. These results strongly suggest that neutrons in magnetar cores are superfluid.

In the absence of symmetry assumptions most numerical relativity simulations adopt Cartesian coordinates. While Cartesian coordinates have some desirable properties, spherical polar coordinates appear better suited for certain applications, including gravitational collapse and supernova simulations. Development of numerical relativity codes in spherical polar coordinates has been hampered by the need to handle the coordinate singularities at the origin and on the axis, for example by careful regularization of the appropriate variables. Assuming spherical symmetry and adopting a covariant version of the Baumgarte-Shapiro-Shibata-Nakamura equations, Montero and Cordero-Carri\'on recently demonstrated that such a regularization is not necessary when a partially implicit Runge-Kutta method is used for the time evolution of the gravitational fields. Here we report on an implementation of the Baumgarte-Shapiro-Shibata-Nakamura equations in spherical polar coordinates without any symmetry assumptions. Using a partially implicit Runge-Kutta method we obtain stable simulations in three spatial dimensions without the need to regularize the origin or the axis. We perform and discuss a number of tests to assess the stability, accuracy and convergence of the code, namely weak gravitational waves, ``hydro-without-hydro'' evolutions of spherical and rotating relativistic stars in equilibrium, and single black holes.

We study numerical evolutions of nonlinear gravitational waves in moving-puncture coordinates. We adopt two different types of initial data---Brill and Teukolsky waves---and evolve them with two independent codes producing consistent results. We find that Brill data fail to produce long-term evolutions for common choices of coordinates and parameters, unless the initial amplitude is small, while Teukolsky wave initial data lead to stable evolutions, at least for amplitudes sufficiently far from criticality. The critical amplitude separates initial data whose evolutions leave behind flat space from those that lead to a black hole. For the latter we follow the interaction of the wave, the formation of a horizon, and the settling down into a time-independent trumpet geometry. We explore the differences between Brill and Teukolsky data and show that for less common choices of the parameters---in particular negative amplitudes---Brill data can be evolved with moving-puncture coordinates and behave similarly to Teukolsky waves.