New public code for initial data of unequal-mass, spinning compact-object binaries

The construction of constraint-satisfying initial data is an essential element for the numerical exploration of the dynamics of compact-object binaries. While several codes have been developed over the years to compute generic quasiequilibrium configurations of binaries comprising either two black holes, or two neutron stars, or a black hole and a neutron star, these codes are often not publicly available or they provide only a limited capability in terms of mass ratios and spins of the components in the binary. We here present a new open-source collection of spectral elliptic solvers that are capable of exploring the major parameter space of binary black holes (BBHs), binary neutron stars (BNSs), and mixed binaries of black holes and neutron stars (BHNSs). Particularly important is the ability of the spectral-solver library to handle neutron stars that are either irrotational or with an intrinsic spin angular momentum that is parallel to the orbital one. By supporting both analytic and tabulated equations of state at zero or finite temperature, the new infrastructure is particularly geared toward allowing for the construction of BHNS and BNS binaries. For the latter, we show that the new solvers are able to reach the most extreme corners in the physically plausible space of parameters, including extreme mass ratios and spin asymmetries, thus representing the most extreme BNS computed to date. Through a systematic series of examples, we demonstrate that the solvers are able to construct quasiequilibrium and eccentricity-reduced initial data for BBHs, BNSs, and BHNSs, achieving spectral convergence in all cases. Furthermore, using such initial data, we have carried out evolutions of these systems from the inspiral to after the merger, obtaining evolutions with eccentricities $\lesssim {10}^{-4}-{10}^{-3}$, and accurate gravitational waveforms.

Figure 1

The typical bispherical domain decomposition used in this work. Depicted with different shadings are the various coordinate domains where: regions A are the excised regions of each BH; regions B have a spherical outer radius with an adapted inner radius shared A; regions C, D, and E are the bispherical domains, and region F is the compactified region. Note that this decomposition is rotationally symmetric with respect to the $x$-axis.

Figure 2

Spectral convergence of the asymptotic quantities described in Sec. 2c and of the orbital frequency $\mathrm{\Omega }$ for an equal-mass BBH system. Shown are the absolute values of the variations of the quantity $X$ at a given effective resolution $\overline{N}$ given by Eq. (67) with respect to the corresponding quantity at the largest effective resolution ${\overline{N}}_{\max }=52$. Clearly the variations decrease exponentially for all quantities considered.

Figure 3

Behavior of the binding energy [cf. Eq. (24)] as a function of the dimensionless orbital frequency for sequences of irrotational BBH, BNS, and BHNS binaries (colored filled circles), when compared with the combined with the corresponding 4PN prediction given by Eq. (b5) (colored solid lines). For binaries having the same components, we have considered both equal-mass binaries ($q=1$) and unequal-mass binaries ($q=0.5$ for BBHs and $q=0.6$ for BNSs). Finally, in the case of the BHNS binaries, we have computed a rather extreme equal-mass configurations.

Figure 4

Same as in Fig. 3 but for equal-mass, BNS configurations that are either irrotational ($\chi =0$) or spinning ($\chi =\pm 0.3$), with spins aligned to the orbital angular momentum. Also in this case, the colored solid lines refer to the 4PN predictions (b5), which provide an accurate estimate even in the presence of high spins in the range of the given orbital frequencies.

Figure 5

Left: orbital trajectories of a representative BBH configuration reproducing the properties of the GW150914 event; shown with the light-orange track is the orbit of the primary black hole, while the light-blue track refers to the secondary. Middle: evolution of the coordinate separation $r(t)$ of the GW150914 ID when considering only the quasiequilibrium assumption (black solid line), the 3.5 PN estimates for $\dot{a}$ and $\mathrm{\Omega }$ (blue solid line), or after the fourth iteration (ECC4) of the eccentricity-reduction procedure (red solid line). The inset shows the time derivative of the coordinate separation, $\dot{r}(t)$, for the same datasets. Right: gravitational-wave strain of the $\ell =m=2$ multipole of the $+$ polarization for the ECC4 dataset.

Figure 6

Evolution of the differences in the gravitational-wave phase computed from the $\ell =m=2$ multipole of the $+$ polarization produced by BBH configurations representative of the GW150914 event. Different lines contrast the difference when considering either different effective ID resolutions, i.e., $\overline{N}=24$, 38, 42), or different evolution resolutions, i.e., low resolution ($\mathrm{\Delta }{x}_{\mathrm{LR}}$), medium resolution ($\mathrm{\Delta }{x}_{\mathrm{MR}}$), and high resolution ($\mathrm{\Delta }{x}_{\mathrm{HR}}$). Note that the contribution of the ID to the final error budget is always subdominant at the evolution resolutions employed here.

Figure 7

Dimensionless spin $\chi$ as function of the stellar spin frequency parameter $\omega$ for a sequence of BNS configurations using a single polytrope with $K=123.6$ and $\mathrm{\Gamma }=2$. The numerical data (open symbols) is compared with the interpolating functions reported in Refs. [54, 70, 71], indicating the very good agreement.

Figure 8

Representative example of the iterative eccentricity reduction for a rapidly spinning BNS system modelled with the TNTYST and with $M_{\infty }=2.7$, $q=0.6875$, ${\chi }_1=0$ and ${\chi }_2=0.6$. Shown is the evolution of the proper separation between the two stars when using only the quasiequilibrium ID (black line; QE), or when utilizing the 3.5PN estimates for $\mathrm{\Omega }$ and $\dot{a}$ (blue line; 3.5 PN), or when employing the ID from the final step of the eccentricity-reduction procedure (red line; ECC4). Note that the QE condition leads to enormous eccentricities for such a highly spinning binary.

Figure 9

Same as in Fig. 6, but for an equal-mass irrotational BNS system modelled using the SLy EOS. Also in this case, the differences are computed either for different effective ID resolutions ($\overline{N}=29$, 38, 47) or for different evolution resolutions (LR, MR, HR). Also in this case, the resolution evolution provides the largest contribution to the error budget at least for the resolutions considered here, although increasing the ID resolution can reduce the phase difference for HR evolutions.

Figure 10

Two-dimensional cuts through the $(x,y)$ (top row) and $(x,z)$ planes (bottom row) of two extreme BNS systems modeled with the TNTYST EOS and having a very small mass ratio ($q=0.455$ corresponding to $M_1=2.2M_{\odot }$, $M_2=1.0M_{\odot }$). The left column refers to an irrotational binary (${\chi }_1=0$, ${\chi }_2=0$), while the right one to a very large spin asymmetry (${\chi }_1=0.6$, ${\chi }_2=0$); the latter is the most extreme BNS configuration considered here. The panels concentrate on the more massive component, but the insets offer views of the whole binaries, where the secondary is marked in red.

Figure 11

Same as in Fig. 5 when referring to extreme BNS configurations modeled with the TNTYST EOS (see also Fig. 10). The top row reports the orbital trajectories, the evolution of the proper separation after different eccentricity reductions, and the gravitational-wave strain for a irrotational BNS with mass ratio $q=0.455$. The bottom row reports the same quantities for a BNS with the same mass ratio but extreme spin asymmetry, ${\chi }_1=0$, ${\chi }_2=0.6$. Note that the large angular momentum of the spinning binary leads to more orbits and to a metastable merged object rather than to a black hole.

Figure 12

Same as in Figs. 5 and 11 but for a BHNS configuration. Note that the right panel reports the $\ell =m=2$ multipole the ${\psi }_4$ Weyl scalar [(both the real part (red solid line) and its norm (black solid line)] in order to highlight the very short ringdown that would not be visible in the gravitational-wave strain.