Finite tidal effects in GW170817: Observational evidence or model assumptions?

After the detection of gravitational waves caused by the coalescence of compact objects in the mass range of neutron stars, GW170817, several studies have searched for an imprint of tidal effects in the signal, employing different model assumptions. One important distinction is whether or not to assume that both objects are neutron stars and obey the same equation of state. Some studies assumed independent properties. Others assume a universal equation of state, and in addition that the tidal deformability follows certain phenomenological relations. An important question is whether the gravitational-wave data alone constitute observational evidence for finite tidal effects. All studies provide Bayesian credible intervals, often without sufficiently discussing the impact of prior assumptions, especially in the case of lower limits on the neutron-star tidal deformability or radius. In this article, we scrutinize the implicit and explicit prior assumptions made in those studies. Our findings strongly indicate that existing lower credible bounds are mainly a consequence of prior assumptions combined with information gained about the system’s masses. Importantly, those lower bounds are typically not informed by the observation of tidal effects in the gravitational-wave signal. Thus, regarding them as observational evidence might be misleading without a more detailed discussion. Further, we point out technical problems and conceptual inconsistencies in existing studies. We also assess the limitations due to systematic waveform model uncertainties in a novel way, demonstrating that differences between existing models are not guaranteed to be small enough for an unbiased estimation of lower bounds on the tidal deformability. Finally, we propose strategies for gravitational-wave data analysis designed to avoid some of the problems we uncovered.

Figure 1

Top: Interpolated tidal deformability $\overline{\mathrm{\Lambda }}$ of a BNS system as a function of $\overline{M}$ (bottom axis) and $q$ (top axis). The curves correspond to various EOS, where only a representative set is labeled and the others are shown in green. Causality-violating EOS used in [10] are shown in purple. The sequences terminate when $M_1(q)$ reaches the maximum mass (marked by symbols). The shaded areas are regions unreachable when using the universal relations from [10], either implying negative tidal deformabilites (red) or by construction (black). Bottom: Comparison of $\overline{\mathrm{\Lambda }}(\overline{M})$ between the BNS case and mixed NS-BH case, for three EOS. The solid curves refer to the BNS case, the dashed (dotted) curves to a mixed NS/BH binary where the heavier (lighter) object is the BH.

Figure 2

Universal relations from [10, 19], evaluated for fixed chirp mass. The solid curves show the parameters $-\overline{S}$, $\overline{\mathrm{\Lambda }}$ resulting for fixed mass ratios, varying ${\mathrm{\Lambda }}_s$ up to 5000 (the upper limit used in [19]). Note at very small ${\mathrm{\Lambda }}_s$, there is an additional branch with $\overline{S}>0$ (not shown) due to the poles of the universal relations. The dashed line represents $-\overline{S}$ from [11], which does not depend on $q$ or $\overline{\mathrm{\Lambda }}$. The labeled symbols mark the values computed for a representative set of nuclear physics EOS, colored according to mass ratio like the universal relation curves. Grey plus symbols show additional EOS, and grey crosses mark causality-violating EOS used in [10, 19].

Figure 3

Universal relations from [10, 19], evaluated for fixed chirp mass. The curves show the slope parameter $\overline{S}$ versus $\overline{M}$ for fixed values of ${\mathrm{\Lambda }}_s$ and varying mass ratio. The curves diverge where ${\mathrm{\Lambda }}_1$ becomes negative. The symbols mark the values obtained for the same EOS as in Fig. 2.

Figure 4

Universal relation used in [10] for NS compactness as a function of tidal deformability (green curve). The dashed red curve shows a similar universal relation used in [11]. The labeled symbols and plus markers show values obtained from various EOS (the crosses correspond to causality-violating EOS) for the masses of the lighter (blue) and heavier (yellow) NS in GW170817 when assuming a mass ratio $q=0.7$.

Figure 5

Prior distribution of $\overline{\mathrm{\Lambda }}$ obtained using the universal relation ${\mathrm{\Lambda }}_a({\mathrm{\Lambda }}_s,q)$ with a flat prior for ${\mathrm{\Lambda }}_s$ in the range (0,5000), fixed values for the mass ratio, and the given chirp mass. The logarithmic x-axis allows one to see the gaps in the distribution. Note however that the curves show the probability density of $\overline{\mathrm{\Lambda }}$, not $\log (\overline{\mathrm{\Lambda }})$.

Figure 6

Like Fig. 5, but showing the prior distribution for the slope $\overline{S}$. In addition, the vertical line represent the unique slope given by the universal relations used in [11].

Figure 7

Like Fig. 5, but showing the prior distribution for the tidal deformability ${\mathrm{\Lambda }}_2$ of the lighter NS. Although the x-axis is logarithmic, the curves show the probability density of ${\mathrm{\Lambda }}_2$, not $\log ({\mathrm{\Lambda }}_2)$

Figure 8

The solid blue and red curves show the fifth percentile of the NS radii derived from the prior used in [10], restricted to $\tilde{\mathrm{\Lambda }}<800$. Those are not boundaries of a two-dimensional distribution; rather the percentiles are computed for each mass ratio $q$ and plotted versus $M_1(q)$ (blue) and $M_2(q)$ (red). The dashed blue and red curves show the 95th percentiles of the full prior. The black and red areas correspond to values excluded by the universal relations in [10]. The lower credible bounds inferred in [10] for the radii of the two NS using the universal relations are shown as two vertical green lines. For comparison, we also show the mass-radius diagrams for a representative set of nuclear physics EOS models. The curves are cut at the mass of the lighter NS in a binary where the heavier NS mass is maximal (with our example chirp mass).

Figure 9

Like Fig. 8, but for the universal relations from [11]. The green shaded area marks the statistical bounds reported for the “common radius” in [11], not including the systematic errors, and based on a prior that excludes tidal deformabilities below the causality limit. The brown shaded area shows the causality-violating region obtained by combining the bound $\tilde{\mathrm{\Lambda }}\le 51$ derived in [29] and the universal relations from [11]. For comparison, we also show the radii derived using $C_u$ (the version from [11]) for fixed values of $\mathrm{\Lambda }$.

Figure 10

Bottom: Prior distribution of effective tidal deformability $\tilde{\mathrm{\Lambda }}$ derived from the EOS-agnostic flat ${\mathrm{\Lambda }}_1$, ${\mathrm{\Lambda }}_2$ prior. The dashed curve shows the full prior marginalized over all mass ratios; the other curves show the prior restricted to various mass ratios. Top: modified prior distribution for $\tilde{\mathrm{\Lambda }}$ obtained by dividing the original prior by the prior after marginalizing out the mass ratio. The different curves show the modified prior restricted to the same mass ratios as the bottom panel. We note the prior density at $\tilde{\mathrm{\Lambda }}=0$ is small, but not 0.

Figure 11

Prior distribution of the slope parameter $\overline{S}$ resulting from a flat prior in ${\mathrm{\Lambda }}_1$, ${\mathrm{\Lambda }}_2$ space, restricted to $0\le {\mathrm{\Lambda }}_i\le 5000$, as used in [7]. We show priors for various fixed mass ratios. Although we only show a restricted range, the prior support extends to $\pm \infty$. The shaded band marks the range spanned by our standard EOS set for mass ranges $q>0.7$. The dashed line marks the value $\overline{S}$ corresponding to the universal relations used in [11].

Figure 12

Systematic error of $\overline{\mathrm{\Lambda }}$, $\overline{M}$ due to BNS waveform modeling, with respect to selected reference points (black circles). The error is estimated with two measures. The contours show $D=1$ (black) and $D=2$ (grey) according to Eq. (18) based on comparing IMRPhenomD_NRTidal and TaylorF2(6PN) waveform models. The markers labeled Best 1 and Best 2 show the location of the best match for each of the models to the other one evaluated at the reference point. For comparison, we also show the curves obtained for different EOS in $\overline{\mathrm{\Lambda }}$, $\overline{M}$ space, and the HPD bound $\tilde{\mathrm{\Lambda }}>70$ from [7].