Reconstructing the neutron-star equation of state from astrophysical measurements

The properties of matter at ultrahigh densities, low temperatures, and with a significant asymmetry between protons and neutrons can be studied exclusively through astrophysical observations of neutron stars. We show that measurements of the masses and radii of neutron stars can lead to tight constraints on the pressure of matter at three fiducial densities, from 1.85 to 7.4 times the density of nuclear saturation, in a manner that is largely model independent and that captures the key characteristics of the equation of state. We demonstrate that observations with 10% uncertainties of at least three neutron stars can lead to measurements of the pressure at these fiducial densities with an accuracy of 0.11 dex or $\simeq 30\%$. Observations of three neutron stars with 5% uncertainties are sufficient to distinguish at a better than $3\sigma$ confidence level between currently proposed equations of state. In the electromagnetic spectrum, such accurate measurements will become possible for weakly magnetic neutron stars during thermonuclear flashes and in quiescence with future missions such as the International X-ray Observatory.

Figure 1

Mass-radius relations for neutron stars, computed using the complete (solid lines) as well as the parametric (dashed lines) forms of three equations of state.

Figure 2

The dependence of the calculated mass-radius relations of neutron stars on the parametric equation of state. The first three panels show the change in the predicted relation when the values of the parameters $P_1$, $P_2$, and $P_3$ are varied by 25% in each direction away from the best-fit values for the equation of state FPS. The last panel shows the change in he predicted relation when the value of the parameter ${\rho }_0$ is reduced to $1/2$ and $1/4$ of its best-fit value for the same equation of state.

Figure 3

Simulated values of the masses and radii of three neutron stars that obey the FPS equation of state (solid dots). The hatch-filled regions and the ellipses indicate 5% and 10% uncertainties in each measurement, respectively.

Figure 4

The probability density over the pressure $P_2$ at 3.7 times the nuclear saturation density, evaluated when the other two parameters of the equation of state are set to the best-fit values for equation of state FPS. This is the probability that the parametric equation of state reproduces the set of simulated data shown in Fig. 3 that have 10% measurement uncertainties. The vertical dashed line indicates the true pressure at density ${\rho }_2$ of the FPS equation of state.

Figure 5

Contours of 68.2% ($1\sigma$) and 95.4% ($2\sigma$) confidence levels for the values of pairs of parameters of the equation of state necessary to describe the simulated data shown in Fig. 3. In each panel, the probability density is marginalized over the parameter not shown. The parameters $P_1$, $P_2$, and $P_3$ are the pressures at 1.85, 3.70, and 7.40 times the nuclear saturation density, respectively. In all panels, the corresponding true pressures for a sample of proposed equations of state are shown for comparison. The left panels correspond to a 5% uncertainty in the measurements, whereas the right panels correspond to a 10% uncertainty. A 5% measurement error is sufficient to distinguish to a better than $3\sigma$ level not only between equations of state that have different underlying physics but also between ones with similar mass-radius predictions.

Figure 6

The tightening of the constraints on the equation-of-state parameter $P_1$ compared to the upper-left panel in Fig. 5, when a $1.3M_{\odot }$ neutron star is included in the simulated data.

Figure 7

The constraints on the polytropic indices ${\Gamma }_1$ and ${\Gamma }_2$, which appear in the alternative parametrization of the equation of state, obtained for the same simulated data as in Fig. 6. This parametrization leads to significantly correlated uncertainties even for the wide range of neutron-star masses considered here and for 5% errors in the measurements.