Tidal deformabilities and neutron star mergers

Finite size effects in a neutron star merger are manifested, at leading order, through the tidal deformabilities of the stars. If strong first-order phase transitions do not exist within neutron stars, both neutron stars are described by the same equation of state, and their tidal deformabilities are highly correlated through their masses even if the equation of state is unknown. If, however, a strong phase transition exists between the central densities of the two stars, so that the more massive star has a phase transition and the least massive star does not, this correlation will be weakened. In all cases, a minimum deformability for each neutron star mass is imposed by causality, and a less conservative limit is imposed by the unitary gas constraint, both of which we compute. In order to make the best use of gravitational wave data from mergers, it is important to include the correlations relating the deformabilities and the masses as well as lower limits to the deformabilities as a function of mass. Focusing on the case without strong phase transitions, and for mergers where the chirp mass $\mathcal{M}\le 1.4M_{\odot }$, which is the case for all observed double neutron star systems where a total mass has been accurately measured, we show that the ratio of the dimensionless tidal deformabilities satisfy ${\mathrm{\Lambda }}_1/{\mathrm{\Lambda }}_2\sim q^6$, where $q=M_2/M_1$ is the binary mass ratio; $\mathrm{\Lambda }$ and $M$ are the dimensionless deformability and mass of each star, respectively. Moreover, they are bounded by $q^{n_{-}}\ge {\mathrm{\Lambda }}_1/{\mathrm{\Lambda }}_2\ge q^{n_{0+}+q{n}_{1+}}$, where $n_{-}<n_{0+}+q{n}_{1+}$; the parameters depend only on $\mathcal{M}$, which is accurately determined from the gravitational-wave signal. We also provide analytic expressions for the wider bounds that exist in the case of a strong phase transition. We argue that bounded ranges for ${\mathrm{\Lambda }}_1/{\mathrm{\Lambda }}_2$, tuned to $\mathcal{M}$, together with lower bounds to $\mathrm{\Lambda }(M)$, will be more useful in gravitational waveform modeling than other suggested approaches.

Figure 1

Left: Binary mass ratio $q$ as a function of chirp mass $\mathcal{M}$ for known double neutron star (DNS) systems [19]. $\mathcal{M}$ for GW170817 is indicated by the vertical dashed line. Right: Spin parameters for pulsars in known DNS systems. For both figures, curves represent possible values for systems in which the total mass, but not $q$, is accurately known; the minimum value of $q$ is determined by $M_2>1.1M_{\odot }$. Red curves and points indicate systems for which the merger timescale ${\tau }_{\mathrm{GW}}$ is longer than the age of the Universe.

Figure 2

Left panel: Permitted values of masses and radii for different assumptions about the minimum neutron star maximum mass $M_{\max }$ A minimum value of $p_1=8.4\mathrm{MeV}{\mathrm{fm}}^{-3}$ was assumed. Right panel: Permitted values of pressure and energy density for different assumptions about $M_{\max }$.

Figure 3

The dimensionless tidal deformability for individual stars as a function of mass for various equations of state are marked by dots, which are color-coded by their radii. Those configurations lying between the lower solid or colored dashed lines and the upper-most solid line originate from equations of state which satisfy the indicated $M_{\max }$ constraint. $p_{1,\min }=8.4\mathrm{MeV}{\mathrm{fm}}^{-3}$ and $p_{1,\max }=30\mathrm{MeV}{\mathrm{fm}}^{-3}$ were assumed.

Figure 4

Bounds of $\mathrm{\Lambda }{\beta }^6$ as a function of mass for piecewise polytropes as constrained by $M_{\max }$ and $p_{1,\min }$. $p_{1,\max }=30\mathrm{MeV}{\mathrm{fm}}^{-3}$ is assumed. Solid curves are lower bounds for the indicated $M_{\max }$. Upper bounds for $p_{1,\min }=3.74(8.4)\mathrm{MeV}{\mathrm{fm}}^{-3}$ are shown by dashed (dot-dashed) curves. Note that for $1.1M_{\odot }<M<1.6M_{\odot }$ and $M_{\max }>2M_{\odot }$ that $\mathrm{\Lambda }{\beta }^6$ is constant to about $\pm 12\%$ (dotted lines).

Figure 5

Similar to Fig. 3, except that the dimensionless binary tidal deformability as a function of chirp mass is displayed, with stellar pairs indicated with dots colored according to the value of $R_{1.4}$ for each assumed equation of state. $M_{\max }$ only affects the lower bound. $p_{1,\min }=8.4\mathrm{MeV}{\mathrm{fm}}^{-3}$ is assumed.

Figure 6

Similar to Fig. 4, except that bounds of the quantity $\tilde{\mathrm{\Lambda }}[G\mathcal{M}/(R_{1.4}{c}^2){]}^6$ are displayed. Solid (dashed) lines show bounds for $p_{1.\min }=8.4(3.74)\mathrm{MeV}{\mathrm{fm}}^{-3}$. The chirp mass of GW170817 is shown by the vertical dotted line.

Figure 7

Symbols show the upper and lower bounds on ${\mathrm{\Lambda }}_2{q}^6/{\mathrm{\Lambda }}_1$ as a function of $q$ for hadronic stars as determined from piecewise polytropes assuming $\mathcal{M}=1.188M_{\odot }$ for GW170817. The two lower bounds correspond to lower limits $p_{1,\min }=3.74\mathrm{MeV}{\mathrm{fm}}^{-3}$ (crosses) and $8.4\mathrm{MeV}{\mathrm{fm}}^{-3}$ (asterisks). The approximate bounds given by Eq. (20) are shown as black curves.

Figure 8

The same as Fig. 7 but for general chirp mass ranges (color) for hadronic stars. For clarity, lower bounds using $p_{1,\min }=3.74\mathrm{MeV}{\mathrm{fm}}^{-3}$ are not shown.

Figure 9

The same as Fig. 7 but showing the deformability-mass correlation predicted by Refs. [15, 24] over all chirp masses. The upper and lower bounds from Eq. (20) for the GW170817 chirp mass of $1.188M_{\odot }$ are indicated as dashed lines. The left panel shows the mean value of the quantity ${\mathrm{\Lambda }}_2{q}^6/{\mathrm{\Lambda }}_1$ as a function of $q$ and ${\mathrm{\Lambda }}_s={\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2$ (indicated by color). The right panel show mean values as asterisks and their estimated $\pm 1\sigma$ uncertainty ranges.

Figure 10

The mass-radius curves for self-bound configurations parametrized by the sound speed squared, $s$. Quantities are normalized relative to their values for the maximum mass solution.

Figure 11

The dimensionless deformability as a function of $M/M_{\max }=y(w_0)/y_{s,\max }$ for self-bound stars parametrized by a constant sound speed $c_s^2/c^2=s$. Dotted curves show cubic polynomial fits using Eq. (24).

Figure 12

The binary deformability as a function of $\mathcal{M}/M_{\max }$ for the maximally compact self-bound stars with $s=1$. Binary pairs are shown by points color coded according to their mass ratio $q$. The solid curve is a quintic polynomial approximation for the lower boundary using Eq. (25).

Figure 13

Symbols show the upper and lower bounds on ${\mathrm{\Lambda }}_2{q}^6/{\mathrm{\Lambda }}_1$ as a function of $q$ for hybrid stars as determined from piecewise polytropes assuming $\mathcal{M}=1.188M_{\odot }$, appropriate to GW170817. The lower bound corresponds to $p_{1,\min }=8.4\mathrm{MeV}{\mathrm{fm}}^{-3}$. The upper bound corresponds to $M_{\max }\ge 2M_{\odot }$ and $p_{1,\max }=30\mathrm{MeV}{\mathrm{fm}}^{-3}$. The approximate bounds given by Eq. (29) are shown as black curves.

Figure 14

The same as Fig. 13 but for general chirp mass ranges (color) for binaries with one hybrid star. For clarity, lower bounds using $p_{1,\min }=3.74\mathrm{MeV}{\mathrm{fm}}^{-3}$ are not shown.

Figure 15

The variation of the upper and lower bounds to $\mathrm{\Lambda }(M)$ for hadronic stars as the boundary densities $n_1$ and $n_2$ are changed. $M_{\max }=2.0M_{\odot }$ is assumed.