Rapidly rotating neutron stars: Universal relations and EOS inference

We provide accurate universal relations that allow to estimate the moment of inertia $I$ and the ratio of kinetic to gravitational binding energy $T/W$ of uniformly rotating neutron stars from the knowledge of mass, radius, and moment of inertia of an associated nonrotating neutron star. Based on these, several other fluid quantities can be estimated as well. Astrophysical neutron stars rotate to varying degrees, and, although rotational effects may be neglected in some cases, not modeling them will inevitably introduce bias when performing parameter estimation. This is especially important for future, high-precision measurements coming from electromagnetic and gravitational wave observations. The proposed universal relations facilitate computationally cheap equation of state inference codes that permit the inclusion of observations of rotating neutron stars. To demonstrate this, we deploy them into a recent Bayesian framework for equation of state parameter estimation that is now valid for arbitrary, uniform rotation. Our inference results are robust up to around percent-level precision for the generated neutron star observations, consisting of the mass, equatorial radius, rotation rate, as well as co- and counterrotating $f$-mode frequencies, that enter the framework as data.

Figure 1

A corner plot of the dataset on which we construct the universal relations. Each black dot in the $M$ vs $R$ diagram represents one of 83011 NS models in our dataset. Their density distribution is indicated by the red contour lines. The top and right diagram show the corresponding histograms of radius and mass, respectively. The darker lines correspond to the $M\text{−}R$ curves of the EOSs which are visible because our dataset contains some additional near-spherical models; see the main text for more details.

Figure 2

A scatter plot of the rescaled moment of inertia $I_n$ against the fractional angular rotation rate ${\mathrm{\Omega }}_n$ for the 1550 sequences of stars in our dataset. As the graph shows a very large number of data points, we display an inset, in which we enlarge a small region of the graph; it becomes visible that the majority of the data points gather within 2% (indicated by gray lines, not shown in the main graph) of the polynomial fit (black solid line) and there are some with a larger deviation from the fit. Upon inspection, it turns out that the latter belong to neutron stars with rather small masses $M\lesssim 1.2M_{\odot }$. The relative error may appear large in the inset; however, the fit is made for the rescaled quantity $I_n$ which suffers from truncation errors. The relative error for the moment of inertia $I$ is bounded by 1.9%; see the discussion in the main text for details.

Figure 3

A stacked histogram of the relative deviations of our 83011 data points from the universal relation Eq. (2). The majority of data points (roughly 80%) deviate less than 0.4% from the fit, and the maximum relative error is 1.9%. The relative errors are subdivided corresponding to the fractional rotation rate ${\mathrm{\Omega }}_n$ of the neutron star model; this shows that the largest deviations from the correct value appear for rotation rates ${\mathrm{\Omega }}_n\approx 0.7–0.9$.

Figure 4

The scaled moment of inertia of neutron stars at the Kepler limit against the radius-scaled moment of inertia of the nonrotating neutron star with the same central energy density as the rotating stars. The bottom panel shows the relative deviation of the data points from the fit (black solid line); the maximum relative error is 2.2%.

Figure 5

A scatter plot of the rescaled $(T/W{)}_n$ against the fractional angular rotation rate ${\mathrm{\Omega }}_n$ for the 1550 sequences in our dataset. An inset shows an enlargement of a small region of the graph; almost all data points are inside a very narrow band (with less than 1% relative error which is indicated by gray lines; not shown in the main graph) around the polynomial fit (black solid line). The few individually visible points outside the dense band have a relative error of less than 2%.

Figure 6

The scaled value of $T/W$ of neutron stars at the Kepler limit against the mass-scaled moment of inertia of the nonrotating neutron star with the same central energy density as the rotating stars. The bottom panel shows the relative errors of the data points; the maximum relative deviation from the best fit is 3.2%.

Figure 7

A stacked histogram of the relative errors of the universal relation for $T/W$; they remain below 4.3% for all models in our dataset. The relative errors are subdivided corresponding to the fractional rotation rate ${\mathrm{\Omega }}_n$ of the neutron star model; this shows that the deviation from the correct value is largest for models with a small rotation rate, since the absolute value of $T/W$ is small for these.

Figure 8

A stacked histogram of the relative errors as defined in Eq. (8) when estimating the counterrotating $f$-mode frequency using Eq. (7). Since relative errors for low frequencies are an unreliable measure for accuracy, we split the set of relative errors into three groups depending on the absolute value of the $f$-mode frequency (as extracted from the time evolution). When the magnitude of the $f$-mode is larger than 500 Hz (shown in green), the relative errors peak at 3% and are larger than 10% only in a few cases. For smaller magnitudes of the $f$-mode frequency (shown in orange and red), the relative error naturally increases. However, there are still many instances where even low-frequency $f$-modes are estimated with only a small relative error.

Figure 9

A histogram of the relative errors when estimating the corotating branch of the $f$-mode using Eq. (9); we use the newly proposed universal relation for the moment of inertia and the universal relation for the mass from Ref. [37] to estimate the required bulk quantities of the rotating stars. We calculate the relative error as given in Eq. (8). For the majority of neutron star models, our estimate is accurate to 2%–3%; in general, the relative error stays below 10%.

Figure 10

Results of the MCMC analysis using observations of slowly rotating neutron stars listed in Table 3 as described in the main text. The two corner plots compare the results using the old framework with slow-rotation approximations (left panel) and the new framework based on universal relations that allow rapid rotation (right panel). The blue lines indicate the injected EOS parameters, and black dashed intervals define the {0.16, 0.5, 0.84} quantils. The results of both frameworks are almost identical, which reconfirms that the slow-rotation approximation is justified for observations of neutron stars that rotate sufficiently slowly (cf. Ref. [42]).

Figure 11

The same as Fig. 10 but for observations of moderately rotating neutron stars as shown in Table 4. At rotation rates of 350–550 Hz or axis ratios of $\mathfrak{r}\approx 0.96–0.97$, the stars’ oblateness is no longer negligible and we observe a strong bias in the posteriors, in particular, for ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_3$, when using the old framework. The new framework reconstructs the PP parameters well.

Figure 12

The same as Fig. 10 but for observations of rapidly rotating neutron stars as shown in Table 5. With axis ratios of $\mathfrak{r}\approx 0.73–0.90$, the stars are far from sphericity and, as a result, the old framework fails completely. The new framework built on the universal relations for rapid rotation, again, yields very useful results.

Figure 13

Here, we show MCMC results using different EOS database resolutions for slowly rotating neutron star observables from Table 3. The slow-rotation approximation (SR, blue) and universal relation (UR, orange) methods have been applied with EOS grid resolutions of 30, 40, and 50 (different line styles). Black vertical lines indicate the correct EOS parameters.

Figure 14

Here, we show MCMC results using different EOS database resolutions for the moderately rotating ones from Table 4. The chosen resolutions and definition of lines are the same as those in Fig. 13.