On parametrized cold dense matter equation-of-state inference

Abstract

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 ∞ .

Constraining the Equation of State of High-density Cold Matter Using Nuclear and Astronomical Measurements

“…Here, we focus on the statistical approaches themselves. Our method is generally consistent with other methods that are fully Bayesian, e.g., among recent papers Lackey & Wade (2015), Agathos et al (2015), Alvarez-Castillo et al (2016), and Riley et al (2018). The non-parameteric approach of Landry & Essick (2019) is also worth consideration.…”

“…For example, if two M (R) curves obtained from different equations of state cross, then the inversion is clearly singular at the crossing point. Another difficulty with this approach has been emphasized by Riley et al (2018) and Raaijmakers et al (2018), in the context of EOS models that have separately parameterized segments at different densities, such as models that use a sequence of polytropes. They point out that some neutron stars might not have a central density large enough to reach the highest density in the EOS model.…”

Constraining the Dense Matter Equation of State with Joint Analysis of NICER and LIGO/Virgo Measurements

“…In the future however when a population of neutron star observables is available this can quickly become computationally intractable-depending on the sampling algorithm applied-as the parameter vector θ grows linearly with the number of observed stars. One can perform the parameter estimation sequentially in this case, where the prior p(θ | M) is updated after each iteration and sampled from in the next (see also Figure 2.1 in Riley et al 2018). 7…”

Equation of state sensitivities when inferring neutron star and dense matter properties

“…It is a less computationally intensive alternative to full direct inference of EOS parameters from the data. As outlined in Riley et al (2018), this assumption only holds when the prior on mass and radius, which is implicitly defined in the proportionality, is sufficiently non-informative 4 . A second assumption is that the datasets of different observed neutron stars are independent, which allows us to separate the likelihoods and rewrite Eq.…”