Speed of sound constraints from tidal deformability of neutron stars

The upper bound of the speed of sound in dense nuclear matter is one of the most interesting but still unsolved problems in nuclear physics. Theoretical studies in connection with recent observational data of isolated neutron stars as well as binary neutron stars systems offer an excellent opportunity to shed light on this problem. In the present work, we suggest a method to directly relate the measured tidal deformability (polarizability) of binary neutron stars system (before merger) to the maximum neutron star mass scenario and possible upper bound on the speed of sound. This method is based on the simple but efficient idea that while the upper limit of the effective tidal deformability favors soft equations of state, the recent high measured values of neutron star mass favor stiff ones. In the present work, first, using a simple well established model we parametrize the stiffness of the equation of state with the help of the speed of sound. Second, in comparison with the recent observations by LIGO/VIRGO collaboration of two events, GW170817 and GW190425, we suggest possible robust constraints. Moreover, we evaluate and postulate, in the framework of the present method, what kind of future measurements could help us to improve the stringent of the constraints on the neutron star equation of state.

Figure 1

Mass vs. radius for an isolated neutron star, for the two cases of speed of sound. The green (red) lines correspond to the upper (lower) bound. The purple diagonal shaded region corresponds to NICER's observation (data taken from Ref. [50]), while the blue upper (orange lower) shaded region corresponds to the higher (smaller) component of GW170817 event (data retrieved from Ref. [18]). The solid (dashed) contour lines describe the $90\%$ ($50\%$) confidence interval.

Figure 2

Pressure vs. rest mass density. The green (red) lines correspond to the upper (lower) limit for the speed of sound. Between the same kind of linestyle, the upper (lower) limit for the speed of sound coresponds to higher (lower) curves for each branch. The shaded region corresponds to the GW170817 estimation (data from Ref. [18]). The solid black line corresponds to the APR1-MDI EoS [13, 28]. The vertical black lines indicate the transition density $n_{\mathrm{tr}}$.

Figure 3

The tidal deformability $\tilde{\mathrm{\Lambda }}$ as a function of the binary mass ratio $q$ for the event (a) GW170817 and (b) GW190425. The corresponding upper observation limits for $\tilde{\mathrm{\Lambda }}$ are also indicated, with the grey shaded region marking the excluded area. The red (green) lines correspond to the ${(v_s/c)}^2=1/3[{(v_s/c)}^2=1]$ limit. Between the same kind of linestyle, the upper (lower) limit for the speed of sound corresponds to higher (lower) curves for each case. On the left panel, the black dashed line (with arrows) indicates the upper limit for $\tilde{\mathrm{\Lambda }}$ from the reanalysis of the signal (see Ref. [19]). On the right panel, the black dashed line (with arrows) corresponds to the upper bound for $\tilde{\mathrm{\Lambda }}$ [20].

Figure 4

${\mathrm{\Lambda }}_1\text{−}{\mathrm{\Lambda }}_2$ diagram for the event (a) GW170817 and (b) GW190425. On the left panel, the shaded region corresponds to the posterior sample of the event [19]. The solid (dashed) contour line indicates the $90\%$ ($50\%$) credible region. The grey dashed line corresponds to the ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2=\tilde{\mathrm{\Lambda }}$ case. The EoSs with $n_{\mathrm{tr}}=1,1.5n_0$ are beyond the credible regions. On the right panel, the shaded regions correspond to three different waveform models [20]. The PhenomPNRT-LS model (orange) is the one we use for the upper limit on $\tilde{\mathrm{\Lambda }}$. We notice that these posteriors disfavor low values of $\tilde{\mathrm{\Lambda }}$, therefore their regions cannot provide appropriate constraints [20].

Figure 5

Dependence of the effective tidal deformability $\tilde{\mathrm{\Lambda }}$ on the transition density $n_{\mathrm{tr}}$ (in units of saturation density $n_0$) at the maximum mass configuration for the two speed of sound bounds $v_s=c/\sqrt{3}$ and $v_s=c$ and for the event (a) GW170817 (top panel) and (b) GW190425 (bottom panel). The corresponding upper observation limits for $\tilde{\mathrm{\Lambda }}$ [19, 20] as well as the compatible lower transition density values are also indicated for both events. The red (green) arrow marks the accepted region of transition density for the $v_s=c/\sqrt{3}$ ($v_s=c$) case. The red lower (green upper) curved shaded region corresponds to the $v_s=c/\sqrt{3}$ ($v_s=c$) limit. The purple intermediate shaded region indicates the predictions for middle values of speed of sound bound between the two limits of our study.

Figure 6

The effective tidal deformability $\tilde{\mathrm{\Lambda }}$ as a function of the maximum mass for the two speed of sound bounds $v_s=c/\sqrt{3}$ and $v_s=c$ and for the event (a) GW170817 (top panel) and (b) GW190425 (bottom panel). The corresponding upper observation limits for $\tilde{\mathrm{\Lambda }}$ (black dashed lines with arrows, see Refs. [19, 20]), the compatible maximum mass shaded regions, for $v_s=c/\sqrt{3}$ (left red) case, $v_s=c$ case (right green) and the middle cases (purple), as well the current observed maximum neutron star mass $M=2.{14}_{-0.09}^{+0.10}{M}_{\odot }$ (blue shaded vertical region, see Ref. [23]) are also indicated. The red left (green right) arrow marks the accepted region of maximum mass $M_{\max }$ for $v_s=c/\sqrt{3}$ ($v_s=c$) case. The green (red) curved dashed line describes the fitted Eq. (16).

Figure 7

The tidal deformability $\tilde{\mathrm{\Lambda }}$ as a function of $R_{1.4}$ for both events and boundary cases of speed of sound. The dashed (dash-dotted) horizontal black line corresponds to the upper limit on $\tilde{\mathrm{\Lambda }}$ for GW190425 (GW170817) event, taken from Refs. [19, 20]. The grey shaded regions show the excluded areas. The arrows indicate the allowed values of $R_{1.4}$ for each case. The purple dotted curve demonstrates the proposed expression by Refs. [70, 71].