Constraining the neutron star equation of state with gravitational wave signals from coalescing binary neutron stars

Recently exploratory studies were performed on the possibility of constraining the neutron star equation of state (EOS) using signals from coalescing binary neutron stars, or neutron star–black hole systems, as they will be seen in upcoming advanced gravitational wave detectors such as Advanced LIGO and Advanced Virgo. In particular, it was estimated to what extent the combined information from multiple detections would enable one to distinguish between different equations of state through hypothesis ranking or parameter estimation. Under the assumption of zero neutron star spins both in signals and in template waveforms and considering tidal effects to 1 post-Newtonian (1PN) order, it was found that $\mathcal{O}(20)$ sources would suffice to distinguish between a stiff, moderate, and soft equation of state. Here we revisit these results, this time including neutron star tidal effects to the highest order currently known, termination of gravitational waveforms at the contact frequency, neutron star spins, and the resulting quadrupole-monopole interaction. We also take the masses of neutron stars in simulated sources to be distributed according to a relatively strongly peaked Gaussian, as hinted at by observations, but without assuming that the data analyst will necessarily have accurate knowledge of this distribution for use as a mass prior. We find that especially the effect of the latter is dramatic, necessitating many more detections to distinguish between different EOSs and causing systematic biases in parameter estimation, on top of biases due to imperfect understanding of the signal model pointed out in earlier work. This would get mitigated if reliable prior information about the mass distribution could be folded into the analyses.

Figure 1

The tidal deformability parameter $\lambda (m)$ as a function of neutron star mass for three different EOSs: a soft one (SQM3), a moderate one (H4), and a stiff one (MS1). Adapted from [18]. Curves are fitted quartic polynomials, whose residuals are shown in the lower subplot. Only masses within the unshaded region $[1,2]M_{\odot }$ will be considered in our analyses.

Figure 2

The quadrupole parameter $a(m)$ as a function of neutron star mass for the three different EOSs in Fig. 1. The horizontal dashed line indicates the value for black holes, which is $a=1$ [74]. Only masses within the unshaded region $[1,2]M_{\odot }$ will be considered in our analyses.

Figure 3

The frequencies $f_{\mathrm{LSO}}$ and $f_{\text{contact}}$ as functions of $m_1$, $m_2$ for the EOSs shown in Fig. 1.

Figure 4

Phase contributions of the QM effect and tidal effects up to different PN orders as functions of GW frequency for a $(1.35,1.35)M_{\odot }$ binary with a stiff EOS (MS1). The QM contribution from each NS scales quadratically with its spin and is shown here for ${\chi }_1={\chi }_2=0.1$. The dashed vertical lines indicate the contact and LSO frequencies.

Figure 5

Hypothesis ranking in the case where the EOS of the simulated sources is MS1. As in [25], spins are set to zero both in the injections and the templates, and masses are distributed uniformly on the interval $[1,2]M_{\odot }$, so that any qualitative differences with previous results come from considering tidal effects to higher PN order, and terminating waveforms at contact or LSO, whichever comes sooner. (Left panel) The cumulative distributions of the log odds ratios ${\ln }^{(20)}{O}_{\mathrm{MS}1}^{\mathrm{EOS}}$ for catalogs of 20 sources each, where EOS is in turn H4, SQM3, and PP. Note how the EOSs are ranked according to how dissimilar they are to the correct one: PP differs the most and is indeed the most deprecated, followed by SQM3 and H4. (Right panel) The fraction of catalogs for which PP, SQM3, and H4 are correctly ranked lower than MS1, as a function of the number of events per catalog. What is shown are the medians and 95% confidence intervals obtained from combining individual sources into catalogs in 1000 different ways.

Figure 6

The same as in Fig. 5, but this time with a relatively strongly peaked Gaussian distribution for the injected component masses (while still using a flat mass prior in the analyses). To approach the discernibility of EOS seen in Fig. 5, we now need $\mathcal{O}(100)$ sources per catalog. Even then, H4, the EOS that is closest to the injected MS1, cannot be distinguished from it.

Figure 7

The same as in Fig. 6, except that simulated sources have (anti-)aligned spins sampled from a Gaussian distribution centered at zero and with ${\sigma }_{\chi }=0.02$. However, the prior on spins used in the recovery is uniform on the interval $[-0.1,0.1]$. (Left panels) Examples of cumulative distributions of log odds ratios for the injected EOSs versus the other ones considered, for catalogs of 100 sources each. From top to bottom the injected EOS is MS1, H4, and SQM3, respectively. (Right panels) The fraction of catalogs for which the correct EOS is ranked higher than each of the others in turn, as a function of the catalog size. Here too, medians and 95% confidence intervals are shown, obtained from combining sources into catalogs in 1000 different ways.

Figure 8

Evolution of the medians and 95% confidence intervals in the measurement of $c_0=\lambda (m_0)$, the tidal deformability at the reference mass $m_0=1.35M_{\odot }$, for the cases where the injected EOS is MS1, H4, or SQM3. Both in the injections and the templates, spins are set to zero, and the injected mass distribution is uniform on the interval $[1,2]M_{\odot }$.

Figure 9

The same as in Fig. 8, but this time the signals have component masses drawn from a strongly peaked Gaussian distribution; on the other hand, the prior distribution for the masses used in the analysis of the data is still taken to be uniform on $[1,2]M_{\odot }$. Note how large systematic errors appear. The effect of the mass prior is further investigated in the Appendix.

Figure 10

The same as in Fig. 9, but now the signals not only have Gaussian distributed masses, but nonzero spins as well. Systematic errors remain, and statistical errors have increased due to the larger parameter space that needs to be probed.

Figure 11

The same as in Fig. 9, but this time using a Gaussian prior for the component masses that exactly matches the injected mass distribution. The significant biases that were seen before have largely disappeared.

Figure 12

The same as in Fig. 11, but now with spins switched on. Again we use a Gaussian prior for the component masses that matches the injected distribution. Here too, the biases have been mitigated.