This paper presents solutions to Einstein's equation, and the numerical methods used to construct them, that describe simple cosmological models on manifolds with compact nonorientable spatial slices. These solutions have been constructed on a selection of manifolds having positive, negative, and vanishing spatial scalar curvatures. One example is shown to be indistinguishable locally from a homogeneous Friedman cosmological model, and others are constructed with significant inhomogeneities. Together, these examples are used to explore the strengths and limitations of the numerical methods used in this study, and to test the code used to implement them.

This study explores the nonlinear interactions between gravitational waves in an expanding universe. Random ensembles of gravitational waves are evolved numerically in cosmological models whose spatial volumes increase by more than an order of magnitude during these simulations. These evolutions show the spectra of the gravitational waves evolving as energy cascades away from the wavelengths at the peak of the initial spectra toward both longer and shorter wavelengths. Evolutions performed at different gravitational wave amplitudes are shown to produce cascades that scale with amplitude in the way predicted by the four-wave scattering models of gravitational-wave turbulence.

The relativistic inverse stellar structure problem determines the equation of state of the stellar matter given a knowledge of suitable macroscopic observable properties (e.g. their masses and radii) of the stars composed of that material. This study determines how accurately this equation of state can be determined using noisy mass and radius observations. The relationship between the size of the observational errors and the accuracy of the inferred equation of state is evaluated, and the optimal number of adjustable equation of state parameters needed to achieve the highest accuracy is determined.

Parametric representations of the high-density nuclear equation of state are used in constructing models for interpreting the astrophysical observations of neutron stars. This study explores how accurately equations of state with strong first-order phase transitions can be represented using spectral or piecewise analytic methods that assume no a priori knowledge of the location or the strength of the phase transition. The model equations of state used in this study have phase transitions strong enough to induce a gravitational instability that terminates the sequence of stable neutron stars. These equations of state also admit a second sequence of stable stars with core matter that has undergone this strong first-order phase transition (possibly driven by quark deconfinement). These results indicate that spectral representations generally achieve somewhat higher accuracy than piecewise analytic representations having the same number of parameters. Both types of representation show power-law convergence at approximately the same rate.

Causal parametric representations of neutron-star equations of state are constructed here using Chebyshev polynomial based spectral expansions. The accuracies of these representations are evaluated for a collection of model equations of state from a variety of nuclear-theory models and also a collection of equations of state with first- or second-order phase transitions of various sizes. These tests show that the Chebyshev based representations are convergent (even for equations of state with phase transitions) as the number of spectral basis functions is increased. This study finds that the Chebyshev based representations are generally more accurate than a previously studied power-law based spectral representation, and that pressure-based representations are generally more accurate than those based on enthalpy.

Neutron stars provide a unique opportunity to study strongly interacting matter under extreme density conditions. The intricacies of matter inside neutron stars and their equation of state are not directly visible, but determine bulk properties, such as mass and radius, which affect the star's thermal X-ray emissions. However, the telescope spectra of these emissions are also affected by the stellar distance, hydrogen column, and effective surface temperature, which are not always well-constrained. Uncertainties on these nuisance parameters must be accounted for when making a robust estimation of the equation of state. In this study, we develop a novel methodology that, for the first time, can infer the full posterior distribution of both the equation of state and nuisance parameters directly from telescope observations. This method relies on the use of neural likelihood estimation, in which normalizing flows use samples of simulated telescope data to learn the likelihood of the neutron star spectra as a function of these parameters, coupled with Hamiltonian Monte Carlo methods to efficiently sample from the corresponding posterior distribution. Our approach surpasses the accuracy of previous methods, improves the interpretability of the results by providing access to the full posterior distribution, and naturally scales to a growing number of neutron star observations expected in the coming years.

The effectiveness of the hyperbolic relaxation method for solving the Einstein constraint equations numerically is studied here on a variety of compact orientable three -manifolds. Convergent numerical solutions are found using this method on manifolds admitting negative Ricci scalar curvature metrics, i.e., those from the H3 and the H2 x S1 geometrization classes. The method fails to produce solutions, however, on all the manifolds examined here admitting non -negative Ricci scalar curvatures, i.e., those from the S3, S2 x S1, and the E3 classes. This study also finds that the accuracy of the convergent solutions produced by hyperbolic relaxation can be increased significantly by performing fairly low-cost standard elliptic solves using the hyperbolic relaxation solutions as initial guesses.

This paper explores the use of low-dimensional parametric representations of neutron-star equations of state that include discontinuities caused by phase transitions. The accuracies of optimal piecewise-analytic and spectral representations are evaluated for equations of state having first- or second-order phase transitions with a wide range of discontinuity sizes. These results suggest that the piecewise-analytic representations of these nonsmooth equations of state are convergent, while the spectral representations are not. Nevertheless, the lower-order (2 <= Nparms <= 7) spectral representations are found to be more accurate than the piecewise-analytic representations with the same number of parameters.

A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as non-overlapping coordinate charts on these manifolds. It uses solutions to the two- and three-dimensional biharmonic equations in a sequence of steps that increase the differentiability of the reference metrics across the interfaces between cubic regions. This method is algorithmic and has been implemented in a computer code that automatically generates these reference metrics. Examples of three-manifolds constructed in this way are presented here, including representatives from five of the eight Thurston geometrization classes, plus the well-known Hantzsche-Wendt, the Poincare dodecahedral space, and the Seifert-Weber space.

We report on a new search for continuous gravitational waves from NS 1987A, the neutron star born in SN 1987A, using open data from Advanced LIGO and Virgo's third observing run (O3). The search covered frequencies from 35-1050 Hz, more than five times the band of the only previous gravitational wave search to constrain NS 1987A [B. J. Owen et al., ApJL 935, L7 (2022)]. It used an improved code and coherently integrated from 5.10 days to 14.85 days depending on frequency. No astrophysical signals were detected. By expanding the frequency range and using O3 data, this search improved on strain upper limits from the previous search and was sensitive at the highest frequencies to ellipticities of 1.6e-5 and r-mode amplitudes of 4.4e-4, both an order of magnitude improvement over the previous search and both well within the range of theoretical predictions.

Neutron stars provide a unique laboratory for studying matter at extreme pressures and densities. While there is no direct way to explore their interior structure, X-rays emitted from these stars can indirectly provide clues to the equation of state (EOS) of the superdense nuclear matter through the inference of the star's mass and radius. However, inference of EOS directly from a star's X-ray spectra is extremely challenging and is complicated by systematic uncertainties. The current state of the art is to use simulation-based likelihoods in a piece-wise method which relies on certain theoretical assumptions and simplifications about the uncertainties. It first infers the star's mass and radius to reduce the dimensionality of the problem, and from those quantities infer the EOS. We demonstrate a series of enhancements to the state of the art, in terms of realistic uncertainty quantification and a path towards circumventing the need for theoretical assumptions to infer physical properties with machine learning. We also demonstrate novel inference of the EOS directly from the high-dimensional spectra of observed stars, avoiding the intermediate mass-radius step. Our network is conditioned on the sources of uncertainty of each star, allowing for natural and complete propagation of uncertainties to the EOS.

The interiors of neutron stars reach densities and temperatures beyond the limits of terrestrial experiments, providing vital laboratories for probing nuclear physics. While the star's interior is not directly observable, its pressure and density determine the star's macroscopic structure which affects the spectra observed in telescopes. The relationship between the observations and the internal state is complex and partially intractable, presenting difficulties for inference. Previous work has focused on the regression from stellar spectra of parameters describing the internal state. We demonstrate a calculation of the full likelihood of the internal state parameters given observations, accomplished by replacing intractable elements with machine learning models trained on samples of simulated stars. Our machine-learning-derived likelihood allows us to perform maximum a posteriori estimation of the parameters of interest, as well as full scans. We demonstrate the technique by inferring stellar mass and radius from an individual stellar spectrum, as well as equation of state parameters from a set of spectra. Our results are more precise than pure regression models, reducing the width of the parameter residuals by 11.8% in the most realistic scenario. The neural networks will be released as a tool for fast simulation of neutron star properties and observed spectra.

The interior of a neutron star is a unique astrophysical laboratory for studying matter at extreme densities and pressures beyond what is replicable in terrestrial experiments. While there is no direct way to simulate the interior of these stars, one promising avenue to learning more about the equation of state (EOS) of such matter is through X-rays emitted from the star's surface. The current state-of-the-art method for inference of EOS from a star's X-ray spectra uses piece-wise, simulation-based likelihoods that rely on theoretical assumptions complicated by systematic uncertainties. To reduce the dimensionality of the problem, this method infers macroscopic properties of the star (mass and radius) from emitted X-ray spectra, and from those quantities infers the EOS. This work approaches the same problem using machine learning techniques, demonstrating a series of enhancements to the current state-of-the-art by realistic uncertainty quantification and reducing the need for theoretical assumptions. We also demonstrate novel inference of the EOS directly from high-dimensional simulated X-ray spectra from neutron stars that negate the need for a piece-wise approach. This inference allows for a natural propagation of uncertainties from the X-ray spectra by conditioning the discussed networks on realistic sources of uncertainty for each star.

We report on a search for continuous gravitational waves (GWs) from NS 1987A, the neutron star born in SN 1987A. The search covered a frequency band of 75–275 Hz, included a wide range of spin-down parameters for the first time, and coherently integrated 12.8 days of LIGO data below 125 Hz and 8.7 days of LIGO data above 125 Hz from the second Advanced LIGO–Virgo observing run. We found no astrophysical signal. We set upper limits on GW emission as tight as an intrinsic strain of 2 × 10 −25 at 90% confidence. The large spin-down parameter space makes this search the first astrophysically consistent one for continuous GWs from NS 1987A. Our upper limits are the first consistent ones to beat an analog of the spin-down limit based on the age of the neutron star and hence are the first GW observations to put new constraints on NS 1987A.

Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature.

Spectral representations have been shown to provide an efficient way to represent the poorly understood high-density portion of the neutron-star equation of state. This paper shows how the efficiency and accuracy of those representations can be improved by a very simple change.

Received 21 September 2021DOI:https://doi.org/10.1103/PhysRevD.104.109902© 2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasGravitational wavesGravitation, Cosmology & Astrophysics

We describe directed searches for continuous gravitational waves from twelve well localized non-pulsing candidate neutron stars in young supernova remnants using data from Advanced LIGO's second observing run. We assumed that each neutron star is isolated and searched a band of frequencies from 15 to 150\,Hz, consistent with frequencies expected from known young pulsars. After coherently integrating spans of data ranging from 12.0 to 55.9 days using the F-statistic and applying data-based vetoes, we found no evidence of astrophysical signals. We set upper limits on intrinsic gravitational wave amplitude in some cases stronger than 10^{-25}, generally about a factor of two better than upper limits on the same objects from Advanced LIGO's first observing run.

Observations suggest that our universe is spatially flat on the largest observable scales. Exactly six different compact orientable three-dimensional manifolds admit flat metrics. These six manifolds are therefore the most natural choices for building cosmological models based on the present observations. This paper briefly describes these six manifolds and the harmonic basis functions previously developed for representing arbitrary scalar fields on them. The principal focus of this paper is the development of new harmonics for representing arbitrary vector and second-rank tensor fields on these manifolds. These new harmonics are designed to be useful tools for analyzing the dynamics of electromagnetic and gravitational fields on these spaces.

The purpose of this note is to clarify the conditions under which the first-order generalize-harmonic representation of the vacuum Einstein evolution system is linearly degenerate.