Direct astrophysical tests of chiral effective field theory at supranuclear densities

Recent observations of neutron stars with gravitational waves and x-ray timing provide unprecedented access to the equation of state (EoS) of cold dense matter at densities difficult to realize in terrestrial experiments. At the same time, predictions for the EoS equipped with reliable uncertainty estimates from chiral effective field theory ($\chi \text{EFT}$) allow us to bound our theoretical ignorance. In this work, we analyze astrophysical data by using a nonparametric representation of the neutron-star EoS conditioned on $\chi \text{EFT}$ to directly constrain the underlying physical properties of the compact objects without introducing modeling systematics. We discuss how the data alone constrain the EoS at high densities when we condition on $\chi \text{EFT}$ at low densities. We also demonstrate how to exploit astrophysical data to directly test the predictions of $\chi \text{EFT}$ for the EoS up to twice nuclear saturation density, in order to estimate the density at which these predictions might break down. We find that the existence of massive pulsars, gravitational waves from $\mathrm{GW}170817$, and NICER observations of $\text{PSR}\text{J}0030+0451$ favor $\chi \text{EFT}$ predictions for the EoS up to nuclear saturation density over a more agnostic analysis by as much as a factor of seven for the quantum Monte Carlo (QMC) calculations used in this work. While $\chi \text{EFT}$ predictions using QMC are fully consistent with gravitational-wave data up to twice nuclear saturation density, NICER observations suggest that the EoS stiffens relative to these predictions at or slightly above nuclear saturation density. Additionally, for these QMC calculations, we marginalize over the uncertainty in the density at which $\chi \text{EFT}$ begins to break down, constraining the radius of a $1.4M_{\odot }$ neutron star to $R_{1.4}=11.{40}_{-1.04}^{+1.38}$ ($12.{54}_{-0.63}^{+0.71}$) km and the pressure at twice nuclear saturation density to $p\left(2n_{\mathrm{sat}}\right)=14.2_{-8.4}^{+18.1}$ ($28.7_{-15.0}^{+15.3}$) $\mathrm{MeV}/{\mathrm{fm}}^3$ with massive pulsar and gravitational-wave (and NICER) data.

Figure 1

Prior (shaded) and posterior (lines) distributions for the pressure (in $\mathrm{MeV}/{\mathrm{fm}}^3$) at $0.5n_{\mathrm{sat}}$, $1n_{\mathrm{sat}}$, $2n_{\mathrm{sat}}$, and $3n_{\mathrm{sat}}$ when conditioning on data from only PSRs and GWs (left) or PSRs, GWs, and NICER observations (right). Each cluster of vertically offset curves represents the distributions for the pressure at a specific density, with higher densities centered at higher pressures. We show both the prior conditioned on QMC $\chi \text{EFT}$ calculations with local chiral interactions up to $2n_{\mathrm{sat}}$ (${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$; shaded gray region) and the prior conditioned on $\mathrm{MBPT}$ $\chi \text{EFT}$ calculations with nonlocal interactions up to $n_{\mathrm{sat}}$ (${\mathrm{MBPT}}_{{\text{N}}^3\text{LO,nl}}^{\left(2013\right)}$; shaded blue region), both with nuclear-theory agnostic extensions to higher densities. We also show posteriors obtained from a completely nuclear-theory agnostic analysis of the astrophysical data [27] (red and magenta lines) as well as those obtained after initially conditioning on the ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ up to $2n_{\mathrm{sat}}$ (black lines) and ${\mathrm{MBPT}}_{{\text{N}}^3\text{LO,nl}}^{\left(2013\right)}$ predictions up to $n_{\mathrm{sat}}$ (blue lines). Our results quantify the extent to which the $\chi \text{EFT}$ calculations agree with the completely agnostic posterior, and therefore the astrophysical data.

Figure 2

Bayes factors for astrophysical data conditioned on theoretical predictions up to different $p_{\max }$ compared with a completely agnostic analysis [27] with only massive PSRs and GW data (left) and massive PSRs, GWs, and NICER observations (right). We plot the evidence ratios against the median density from the prior theoretical uncertainty at each $p_{\max }$. Both panels show the results assuming ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ calculations (black), ${\mathrm{MBPT}}_{{\text{N}}^3\text{LO,nl}}^{\left(2013\right)}$ calculations (blue), and artificial theories that are softer (yellow) or stiffer (green) than the ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ results at densities above $n_{\mathrm{sat}}$. Shaded regions denote $1\sigma$ Monte Carlo sampling uncertainties, and we highlight $B_{\mathrm{agnostic}}^{\mathrm{theory}}<1$ as the region where the prior with no information from nuclear theory is preferred over priors conditioned on nuclear theory results.

Figure 3

90% symmetric credible regions for the (top) pressure at each density and (bottom) radius at each mass. We show (left) prior processes conditioned on only ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ predictions up to approximately (from lighter shading to darker shading) $0.5n_{\mathrm{sat}},n_{\mathrm{sat}},\mathrm{and}2n_{sat}$, (middle) posteriors conditioned on massive PSRs and GW data, as well as (right) posteriors conditioned on massive PSRs, GWs, and NICER observations. In all panels, analogous results obtained with a completely agnostic analysis not conditioned on $\chi \text{EFT}$ [27] are shown in as colored lines.

Figure 4

Fractions of the prior that are still viable after conditioning on astrophysical observations when trusting ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ up to different $p_{\max }$. More informative observations reduce our uncertainty relative to the prior, leaving only a fraction of the prior still viable (see Appendix pp2).

Figure 5

Posterior 90% symmetric credible regions for the radius at each mass a priori and conditioned on different data sets with the ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ predictions. Shaded regions show the posteriors conditioned on the theory predictions up to $p_{\max }=0.45$, 1.93, and $10.9\mathrm{MeV}/{\mathrm{fm}}^3$ (approximately $0.5n_{\mathrm{sat}},n_{\mathrm{sat}}$, and $2n_{\mathrm{sat}}$ in ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$) and colored lines show the analogous 90% symmetric credible regions from an agnostic analysis not conditioned on field theory predictions [27]. We show the constraints obtained a priori (far left), with only massive PSRs (middle left), with massive PSRs and GWs (middle), massive PSRs and NICER (middle right), and with massive PSRs, GWs, and NICER measurements (far right).

Figure 6

Posterior 90% symmetric credible regions for the pressure at each density a priori and conditioned on different data sets with the ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ predictions (top), the soft FFT (middle), and the stiff FFT (bottom). Shaded regions show the posteriors conditioned on the theory predictions up to $p_{\max }=0.45$, 1.93, and $10.9\mathrm{MeV}/{\mathrm{fm}}^3$ (approximately $0.5n_{\mathrm{sat}},n_{\mathrm{sat}}$, and $2n_{\mathrm{sat}}$ in ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$) and colored lines show the analogous 90% symmetric credible regions from an agnostic analysis not conditioned on field theory predictions [27]. From left to right, we show the constraints obtained a priori, with only massive PSRs, with massive PSRs and GWs, with massive PSRs and NICER measurements, and with all astrophysical data. Compare with Fig. 3.

Figure 7

Correlations between the pressures (in $\mathrm{MeV}/{\mathrm{fm}}^3$) at different densities. Prior predictions from both the ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ (gray) and ${\mathrm{MBPT}}_{{\text{N}}^3\text{LO,nl}}^{\left(2013\right)}$ (blue) $\chi \text{EFT}$ calculations are shown in shaded regions, and we note that the ${\mathrm{MBPT}}_{{\text{N}}^3\text{LO,nl}}^{\left(2013\right)}$ prior transitions to an agnostic process at lower densities than the example ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ prior shown here. We also show the soft (yellow) and stiff (green) FFT priors as shaded regions. Posteriors conditioned on just PSR data (cyan) as well as those conditioned on PSRs and GWs (red) and PSRs, GWs with either ${\mathrm{QMC}}_{{\text{N}}^2\text{LO,l}}^{\left(2018\right)}$ (black) or ${\mathrm{MBPT}}_{{\text{N}}^3\text{LO,nl}}^{\left(2013\right)}$ (blue) predictions are also shown as solid lines. This figure focuses on posteriors conditioned on PSR and GW observations, while Fig. 8 shows the effects of NICER observations.

Figure 8

Correlations between the pressures (in $\mathrm{MeV}/{\mathrm{fm}}^3$) at different densities. This figure shows the same set of priors as Fig. 7, but shows posteriors conditioned on PSRs, GWs, and NICER instead of those conditioned on only PSRs and GWs.