Quasiequilibrium sequences of binary neutron stars undergoing dynamical scalarization

We calculate quasiequilibrium sequences of equal-mass, irrotational binary neutron stars in a scalar-tensor theory of gravity that admits dynamical scalarization. We model neutron stars with realistic equations of state (notably through piecewise polytropic equations of state). Using these quasiequilibrium sequences we compute the binary’s scalar charge and binding energy versus orbital angular frequency. We find that the absolute value of the binding energy is smaller than in general relativity, differing at most by $\sim 14\%$ at high frequencies for the cases considered. We use the newly computed binding energy and the balance equation to estimate the number of gravitational-wave (GW) cycles during the adiabatic, quasicircular inspiral stage up to the end of the sequence, which is the last stable orbit or the mass-shedding point, depending on which comes first. We find that, depending on the scalar-tensor parameters, the number of GW cycles can be substantially smaller than in general relativity. In particular, we obtain that when dynamical scalarization sets in around a GW frequency of $\sim 130\mathrm{Hz}$, the sole inclusion of the scalar-tensor binding energy causes a reduction of GW cycles from $\sim 120\mathrm{Hz}$ up to the end of the sequence ($\sim 1200\mathrm{Hz}$) of $\sim 11\%$ with respect to the general-relativity case. (The number of GW cycles from $\sim 120\mathrm{Hz}$ to the end of the sequence in general relativity is $\sim 270$.) We estimate that when the scalar-tensor energy flux is also included the reduction in GW cycles becomes of $\sim 24\%$. Quite interestingly, dynamical scalarization can produce a difference in the number of GW cycles with respect to the general-relativity point-particle case that is much larger than the effect due to tidal interactions, which is on the order of only a few GW cycles. These results further clarify and confirm recent studies that have evolved binary neutron stars either in full numerical relativity or in post-Newtonian theory, and point out the importance of developing accurate scalar-tensor-theory waveforms for systems composed of strongly self-gravitating objects, such as binary neutron stars.

Figure 1

Scalar charge (left $y$ axis) or scalar mass (right $y$ axis) as a function of the orbital angular frequency normalized to the tensor mass at infinite separation (lower $x$ axis) or as a function of the frequency of GWs defined by $f_{\mathrm{GW}}\equiv \mathrm{\Omega }/\pi$ from a binary neutron star with $m=2.7M_{\odot }$ (upper $x$ axis). Upper panel (a) shows results for APR4 EOS. Black solid curve with open circles, red with open squares, green with open diamonds, and blue with open triangles are, respectively, for the cases $B=9.0$, 8.7, 8.4, and 8.0. Lower panel (b) shows results for H4 EOS. Black solid curve with open circles, red with open squares, and green with open diamonds are, respectively, for the cases $B=9.5$, 9.0, and 8.5. (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 2

Scalar charge as a function of the orbital angular frequency normalized to the tensor mass at infinite separation (lower $x$ axis) or as a function of the frequency of GWs (upper $x$ axis). Both curves are computed for the case of APR4 EOS $B=8.7$, but the value of ${\phi }_0$ is set to $1\times {10}^{-5}$ (red dashed with open squares) and $1\times {10}^{-6}$ (black solid with open circles).

Figure 3

Binding energy as a function of the orbital angular frequency normalized to the tensor mass at infinite separation (lower $x$ axis) or as a function of the frequency of GWs from a binary neutron star with $m=2.7M_{\odot }$ (upper $x$ axis). Upper panel (a) shows results for APR4 EOS. Black short-dashed, red long-dashed, green dot-short-dashed, and blue dot-long-dashed curves are, respectively, the cases: $B=9.0$, 8.7, 8.4, and 8.0. Lower panel (b) shows results for H4 EOS. Black short-dashed, red long-dashed, and green dot-dashed curves are, respectively, the cases: $B=9.5$, 9.0, and 8.5. In both panels, the purple solid curve is drawn by using a quasiequilibrium sequence in general relativity (GR). Cyan dashed and dark-green dot-dashed curves refer to the 3PN and 4PN binding energy in general relativity, respectively [51] (see also Refs. [52, 53, 54]). (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 4

Same as Fig. 3 but for the total angular momentum. Cyan dashed and dark-green dot-dashed curves refer to the 3PN and 4PN angular momentum in general relativity, respectively. Those curves are calculated using Refs. [51, 55]. (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 5

Same as Fig. 3 but for the relative change in the central baryonic rest-mass density of a neutron star. (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 6

Same as Fig. 3 but for the evolution of the orbital angular frequency (left $y$ axis) or the frequency of GWs (right $y$ axis) from a binary neutron star with $m=2.7M_{\odot }$. The gravitational energy flux is the choice (i). (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 7

Same as Fig. 6 but for the choice (ii) for the gravitational energy flux. In the general-relativity case, the 3.5PN energy flux computed in general relativity is used. These are the same data as used in Fig. 6 (shown as “GR”). (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 8

Same as Fig. 3 but for the evolution of the number of GW cycles. The gravitational energy flux is the choice (i). (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 9

Same as Fig. 8 but for the choice (ii) for the gravitational energy flux. In the general-relativity case, the 3.5PN GR energy flux is used. These are the same data used in Fig. 6 (shown as “GR”). (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 10

Scalar charge (left $y$ axis) or scalar mass (right $y$ axis) as a function of the orbital angular frequency normalized to the tensor mass at infinite separation (lower $x$ axis) or as a function of the frequency of GWs defined by $f_{\mathrm{GW}}\equiv \mathrm{\Omega }/\pi$ from an unequal-mass binary neutron star with $m=3.39M_{\odot }$ (upper $x$ axis). The equation of state is a polytropic one with $\mathrm{\Gamma }=2$. Note that the vertical axis is in a linear scale.

Figure 11

Same as Fig. 10 but for an equal-mass binary neutron star with $m=3.03M_{\odot }$. Note that the vertical axis is in a logarithmic scale.

Figure 12

Scalar mass of APR4 $B=8.7$ as a function of the cube root of the total number of collocation points, $\sqrt[3]{N_r\times N_{\theta }\times N_{\widehat{\phi }}}$. The upper panel is the scalar mass at orbital angular frequency of $G_{\mathrm{N}}m\mathrm{\Omega }=0.00754$, and the lower one is for $G_{\mathrm{N}}m\mathrm{\Omega }=0.0206$. $\mathrm{A}\mathrm{P}\mathrm{R}4\mathrm{B}=8.7:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 13

We plot the function $(1+{\alpha }_{\phi }^2{)}^{10/3}$ that appears in Eq. (c1) versus $x\equiv (G_{\mathrm{N}}m\mathrm{\Omega }{)}^{2/3}$ (lower $x$ axis) and the orbital angular frequency normalized to the tensor mass at infinite separation (upper $x$ axis). Left panel (a) shows results for APR4 EOS. Black short-dashed, red long-dashed, green dot-short-dashed, and blue dot-long-dashed curves are, respectively, the cases: $B=9.0$, 8.7, 8.4, and 8.0. Right panel (b) shows results for H4 EOS. Black short-dashed, red long-dashed, and green dot-dashed curves are, respectively, the cases: $B=9.5$, 9.0, and 8.5. (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.

Figure 14

Gravitational-wave energy flux as a function of the orbital angular frequency normalized to the tensor mass at infinite separation. Left panel (a) shows results for APR4 EOS. Black short-dashed, red long-dashed, green dot-short-dashed, and blue dot-long-dashed curves are, respectively, the cases: $B=9.0$, 8.7, 8.4, and 8.0. Right panel (b) shows results for H4 EOS. Black short-dashed, red long-dashed, and green dot-dashed curves are, respectively, the cases: $B=9.5$, 9.0, and 8.5. Those results are calculated by using Eq. (c1). In both panels, the cyan dashed curve refers to the 3.5PN energy flux in general relativity. (a) $\mathrm{A}\mathrm{P}\mathrm{R}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. (b) $\mathrm{H}4:1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}-1.35{\mathrm{M}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$.