Measuring the neutron star tidal deformability with equation-of-state-independent relations and gravitational waves

Gravitational wave measurements of binary neutron star coalescences offer information about the properties of the extreme matter that comprises the stars. Despite our expectation that all neutron stars in the Universe obey the same equation of state, i.e. the properties of the matter that forms them are universal, current tidal inference analyses treat the two bodies as independent. We present a method to measure the effect of tidal interactions in the gravitational wave signal—and hence constrain the equation of state—that assumes that the two binary components obey the same equation of state. Our method makes use of a relation between the tidal deformabilities of the two stars given the ratio of their masses, a relation that has been shown to only have a weak dependence on the equation of state. We use this to link the tidal deformabilities of the two stars in a realistic parameter inference study while simultaneously marginalizing over the error in the relation. This approach incorporates more physical information into our analysis, thus leading to a better measurement of tidal effects in gravitational wave signals. Through simulated signals we estimate that uncertainties in the measured tidal parameters are reduced by a factor of at least 2—and in some cases up to 10—depending on the equation of state and mass ratio of the system.

Figure 1

Ratio of the areas of the 90% credible intervals of the ${\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_2$ two-dimensional posterior probability density when the EoS-independent relation is imposed in the analysis and when the individual tidal deformabilities are presumed to be independent for different EoSs, mass ratios, and an SNR of 30 (black) and 15 (red). The method proposed here leads to a reduction in the measurement error for ${\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_2$ by a factor between 2 and 10.

Figure 2

The 90% contours of the marginalized posterior probability density for ${\mathrm{\Lambda }}_i$ and $m_i$ for each NS when using the EoS-independent relation (green contours) and when the components’ tidal deformabilties are independent (orange contours) for a signal created with the H4 EoS, an SNR of 30 and a mass ratio of 1. Overplotted is the dimensionless tidal deformability as a function of the mass for the 3 EoS models studied here. All EoSs are consistent with the posteriors derived under independent tidal deformabilties at the 90% level, but only H4 (the correct EoS) is consistent with the data if we use the EoS-independent relation.

Figure 3

EoS-independent relation between the antisymmetric and the symmetric combination of the tidal deformabilities (top panel) for different mass ratios, and the mass ratio for different values of the symmetric tidal deformability (bottom panel). The scatter around the EoS-independent relation represents its $\lesssim 10\%$ relative error.

Figure 4

Prior probability density distributions for the various dimensionless tidal deformability parameters. The top panel shows the prior densities when employing the EoS-independent relation. The ${\mathrm{\Lambda }}_s$ prior is chosen to be uniform between 0 and 5000, while the prior of the other tidal parameters is informed by the ${\mathrm{\Lambda }}_a={\mathrm{\Lambda }}_a({\mathrm{\Lambda }}_s,q;\overrightarrow{b})$ relation. The prior on the mass ratio is the result of flat priors on the individual masses. The bottom panel shows the prior densities for the analyses without the EoS-independent relation included. Here the ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ priors are chosen to be uniform between 0 and 10 000, which in turn defines the remaining tidal parameters.

Figure 5

Two-dimensional and one-dimensional marginalized posterior density function for the dimensionless tidal deformabilities of the two stars for signals of SNR 30, the H4 (top row), MS1 (middle row), and WFF1 (bottom row) EoSs and $q=1$ (first column), $q=0.85$ (second column), $q=0.65$ (third column), and $q=0.5$ (fourth column). The black dot and black dashed lines indicate the injected values in each plot. Each signal is analyzed with (green lines) and without (orange lines) the EoS-independent relation between ${\mathrm{\Lambda }}_a$ and ${\mathrm{\Lambda }}_s$ and the contours shown represent the 50% and 90% credible regions. In all cases, the EoS-independent relation leads to better measurement of the tidal parameters.

Figure 6

Similar to Fig. 5 but for simulated signals with SNR 15.

Figure 7

Marginalized posterior for $\tilde{\mathrm{\Lambda }}$ for various EoSs and SNR 15 (top panel) and 30 (bottom panel), and a mass ratio of 1 when assuming that the individual tidal deformabilities are independent (dashed lines), and when we employ the EoS-independent relation (solid lines). For low SNR there are apparent differences between the $\tilde{\mathrm{\Lambda }}$ posteriors obtained under there two approaches. At higher SNR both posteriors are very similar, as expected from the fact that $\tilde{\mathrm{\Lambda }}$ is a well-measured parameter.

Figure 8

Marginalized posterior density for the mass ratio for H4, SNR 15 (top) and SNR 30 (bottom) when assuming that the individual tidal deformabilities are independent (dashed lines), and when we employ the EoS-independent relation (solid lines). The vertical black lines denote the injected values for the mass ratios (the $q=1$ line is not discernible).