3+1 geodesic equation and images in numerical spacetimes

Vincent, F.; Gourgoulhon, Éric; Novak, Jérôme

doi:10.1088/0264-9381/29/24/245005

Cited by 32 publications

(36 citation statements)

References 28 publications

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“…Although numerical integration of the interior perturbation equations is fast, we are ultimately interested in calculating properties of a radiation field incident on a telescope so as to evaluate a likelihood function (Section 2.2.1). A matched stable analytic exterior solution permits faster geodesic computation than do numerical exterior solutions because computation of the affine connection requires numerical post-processing of metric data (as in, e.g., Vincent et al 2011Vincent et al , 2012. Moreover, an approximate slow-rotation symmetry can be exploited (see Nättilä & Pihajoki 2017, and references therein), which is useful because (CPU-bound) null geodesic computation is expensive for non-Kerr spacetimes due to the coupled system of ordinary non-linear second-order differential geodesic equations (see, e.g., Dexter & Agol 2009;Chan et al 2013).…”

Section: Exterior Spacetime Solutions For Modelling Observed Radiationmentioning

confidence: 99%

On parametrized cold dense matter equation-of-state inference

Riley

Raaijmakers

Watts

2018

Monthly Notices of the Royal Astronomical Society

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Constraining the equation of state of cold dense matter in compact stars is a major science goal for observing programmes being conducted using X-ray, radio, and gravitational wave telescopes. We discuss Bayesian hierarchical inference of parametrised dense matter equations of state. In particular we generalise and examine two inference paradigms from the literature: (i) direct posterior equation of state parameter estimation, conditioned on observations of a set of rotating compact stars; and (ii) indirect parameter estimation, via transformation of an intermediary joint posterior distribution of exterior spacetime parameters (such as gravitational masses and coordinate equatorial radii). We conclude that the former paradigm is not only tractable for large-scale analyses, but is principled and flexible from a Bayesian perspective whilst the latter paradigm is not. The thematic problem of Bayesian prior definition emerges as the crux of the difference between these paradigms. The second paradigm should in general only be considered as an ill-defined approach to the problem of utilising archival posterior constraints on exterior spacetime parameters; we advocate for an alternative approach whereby such information is repurposed as an approximative likelihood function. We also discuss why conditioning on a piecewise-polytropic equation of state model -currently standard in the field of dense matter study -can easily violate conditions required for transformation of a probability density distribution between spaces of exterior (spacetime) and interior (source matter) parameters.3 Note that, as stated above, one important caveat regards the assumption of BNS components sharing the same cold EOS -during inspiral -as stars which do not exist in compact relativistic binaries. In principle, if there is evidence that model complexity needs to be increased, these two classes of star could be assumed to only share a subset of (continuous, phenomenological) EOS parameters, whilst another subset of EOS parameters are used to emulate temperature dependence. 4 We use bold font to denote both a set of abstract parameters, and a point in the associated parameter space. For instance, θ ∈ T where T ⊂ R n , where the coordinates of point θ represent parameter values. 13 Or, interchangeably, a point in a space of parameters defined as random variables. 14 That is, all elements of X are ordered countably infinite tuples, but each coordinate is restricted to a compact subdomain of R, and a tuple of coordinates is a point in R ∞ .

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Section: Exterior Spacetime Solutions For Modelling Observed Radiationmentioning

confidence: 99%

On parametrized cold dense matter equation-of-state inference

Riley

Raaijmakers

Watts

2018

Monthly Notices of the Royal Astronomical Society

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“…Moreover, numerical relativistic simulations are currently becoming one of the most important tools of theoretical cosmology [1, 2, 7-9, 27, 46, 47, 50, 74, 75]. While the null geodesic tracking is a fairly standard problem in numerical relativity [10,69], extracting the position drift effects is not. It is, of course, possible to do it using ray tracing together with the shooting method: we begin with one null geodesic connecting the source and the observer and then search by trial and error for another one, connecting observer's and emitter's worldline at a slightly later moment.…”

Section: Introductionmentioning

confidence: 99%

Geometric optics in general relativity using bilocal operators

2019

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We consider the standard problem of observational astronomy, i.e. the observations of light emission from a distant region of spacetime in general relativity. The goal is to describe the changes between the measurements of the light performed by a sample of observers slightly displaced with respect to each other and moving with different 4-velocities and 4-accelerations. In our approach, all results of observations can be expressed as functions of the kinematic variables, describing the motions of the observers and the emitting bodies with respect to their local inertial frames, and four linear bilocal geodesic operators, describing the influence of the spacetime geometry on light propagation. The operators are functionals of the curvature tensor along the line of sight. The results are based on the assumption that the regions of emissions and observations are sufficiently small so that the spacetime curvature effects are negligible within each of them, although they are significant for the light propagation between them. The new formulation provides a uniform approach to optical phenomena in curved spacetimes and, as an application, we discuss the problem of a fully relativistic definition of the parallax and position drifts (or proper motions). We then use the results to construct combinations of observables which are completely insensitive to the motion of both the observer and the emitter. These combinations by construction probe the spacetime geometry between the observation and emission regions and in our formalism we may express them as functionals of the Riemann tensor along the line of sight. For short distances one of these combinations depends only on the matter content along the line of sight. This opens up the possibility to measure the matter content of a spacetime in a tomographylike manner irrespective of the motions of the emitter and the observer.Even if A enters the differential equation linearly, the general solution contains a nonlinear dependence.

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“…The open-source ray tracing code Gyoto (see Vincent et al 2011Vincent et al , 2012, and http://gyoto.obspm.fr) is used to compute null geodesics backwards in time, from a distant observer towards the neutron star. Photons are traced in the neutron star's metric as computed by the Lorene/nrotstar code.…”

Section: Ray Tracingmentioning

confidence: 99%

Accurate Ray-tracing of Realistic Neutron Star Atmospheres for Constraining Their Parameters

Vincent

Bejger

Różáńska

et al. 2018

ApJ

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Thermal dominated X-ray spectra of neutron stars in quiescent transient X-ray binaries and neutron stars that undergo thermonuclear bursts are sensitive to mass and radius. The mass-radius relation of neutron stars depends on the equation of state that governs their interior. Constraining this relation accurately is thus of fundamental importance to understand the nature of dense matter. In this context we introduce a pipeline to calculate realistic model spectra of rotating neutron stars with hydrogen and helium atmospheres. An arbitrarily fast rotating neutron star with a given equation of state generates the spacetime in which the atmosphere emits radiation. We use the Lorene/nrotstar code to compute the spacetime numerically and the Atm24 code to solve the radiative transfer equations self-consistently. Emerging specific intensity spectra are then ray-traced through the neutron star's spacetime from the atmosphere to a distant observer with the Gyoto code. Here, we present and test our fully relativistic numerical pipeline. To discuss and illustrate the importance of realistic atmosphere models we compare our model spectra to simpler models like the commonly used isotropic color-corrected blackbody emission. We highlight the importance of considering realistic model-atmosphere spectra together with relativistic ray tracing to obtain accurate predictions. We also insist on the crucial impact of the star's rotation on the observables. Finally, we close a controversy that has been appearing in the literature in the recent years regarding the validity of the Atm24 code.

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3+1 geodesic equation and images in numerical spacetimes

Cited by 32 publications

References 28 publications

On parametrized cold dense matter equation-of-state inference

On parametrized cold dense matter equation-of-state inference

Geometric optics in general relativity using bilocal operators

Accurate Ray-tracing of Realistic Neutron Star Atmospheres for Constraining Their Parameters

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