Initial-data contribution to the error budget of gravitational waves from neutron-star binaries

As numerical calculations of inspiraling neutron-star binaries reach values of accuracy that are comparable with those of black-hole binaries, a fine budgeting of the various sources of error becomes increasingly important. Among such sources, the initial data are normally not accounted for, the rationale being that the error on the initial spacelike hypersurface is always far smaller than the error gained during the evolution. We here consider critically this assumption and perform a comparative analysis of the gravitational waveforms relative to essentially the same physical binary configuration when computed with two different initial-data codes, and then evolved with the same evolution code. More specifically, we consider the evolution of irrotational neutron-star binaries computed either with the pseudospectral code lorene, or with the newly developed finite-difference code cocal; both sets of initial data are subsequently evolved with the high-order-evolution code whiskythc. In this way we find that although global quantities of the system, like the mass and angular momentum, have differences of the order of $\lesssim 0.02\%$, local quantities, like rest-mass density, extrinsic curvature or angular velocity, show pointwise differences that are much larger, of the order of $\lesssim 1\%$. These local differences are then responsible for a dephasing in the gravitational waves at the merger time (after approximately three orbits) of $\sim 1.4$ radians. Our results highlight the importance of using initial data that are pointwisely the same when comparative studies are done and physical parameters are estimated.

Figure 1

Structure of a two-dimensional cross section of the cocal grids (colored spherical coordinates) overlaid on a Cartesian coordinate system used for the evolution of the initial data. Here we assume that the $z=0$ plane of the evolutionary grid coincides with the corresponding cocal plane. Evolution gridpoints $P(x_p,y_p,z_p)$ inside the sphere of radius $R_i$ are interpolated from the coordinate patches COCP-1 or COCP-2 depending on whether $x_p\le 0$ or $x_p>0$. Points outside that sphere are interpolated from the asymptotic region patch ARCP. Note that the figure is not to scale; in particular, the size of the sphere of radius $R_i$ is much larger than the size of the inner boundary $S_a$ of ARCP. The outer boundary of ARCP is not shown here and extends to very large values when compared to the compact-object sizes. Typical values are $r_b=100$, $d_s=2.5$, $d=1.25$, $R_s=98.75$, $R_i=69.125$, and $r_a=5.0$. The point where the neutron star’s surface intersects the positive $x$ axis of the COCP takes values $r_s\le 1$ and is in general different for the two stars. The cusps that appear in the figure are pictorial and do not represent any model used in this work.

Figure 2

Left column: From top to bottom, initial data quantities relative to the metric function $g_{xx}={\psi }^4$, the lapse function $\alpha$, the $y$ component of the shift, the $xy$ component of the extrinsic curvature, and the rest-mass density $\rho$, as computed by cocal (red lines) and lorene (blue lines). The $x$ axis is the positive $x$ axis of the Cartesian grid, with $x=0$ corresponding to the center of mass of the binary. Right column: Relative difference between cocal and lorene as computed from Eq. (23).

Figure 3

The same as Fig. 2 for the $y$ component of the three-velocity relative to the Eulerian observers and the corresponding Lorentz factor.

Figure 4

Hamiltonian (top) and momentum-constraint violations (bottom) for the $y$ component of the shift (${\beta }^y$) along the $x$ axis for the irrotational binary system at the initial time. The origin $x=0$ corresponds to the center of mass of the binary, with the surface of the star to be located at $x\approx 6M_{\odot }$ and at $x\approx 24M_{\odot }$.

Figure 5

Logarithmic violations of the constraint equations shown at three different times in the different columns: initial time ($t=0$), just after the beginning of the evolution ($t=30.72M_{\odot }$), and one orbit later ($t=660.48M_{\odot }$). From top to bottom, the first row shows the violations of the Hamiltonian constraint on the $(x,y)$ plane from cocal, while the second row shows the corresponding violations from lorene. The third and fourth rows show the violations of the $y$ component of the momentum constraint from cocal and lorene, respectively. Note that all panels show data on the three finest levels of refinement, with two borders clearly visible. The bounding box in the $(x,y)$ plane encompassing each of the panels spans roughly the range $[0,50]\times [-25,25]M_{\odot }$. The oval shape indicates the neutron-star surface at every moment.

Figure 6

Constraint violation $L_2$ norms for cocal and lorene as a function of time. The first row shows the Hamiltonian constraint, while the second row represents the $y$ component of the momentum constraint. Each color refers to a given resolution for the evolution grid: low (red), medium (green), and high (blue).

Figure 7

First row: Real part of $({\mathrm{\Psi }}_4{)}_{22}$ extracted at $\overline{r}=450M_{\odot }$ as a function of the retarded time for both cocal Hs3.5d (left panel) and lorene (right panel) initial data and for the three evolution resolutions (L, M, H). On each plot, the dashed line denotes the evolution with the highest resolution of the other code initial-data set so that the dephasing between the two data sets becomes apparent. Second row: Dephasing between different resolutions and the rescaled dephasing between the high and medium resolutions, assuming a convergence order $p=2.75$. The left panel is for cocal, while the right one is for lorene. Third row: Relative phase difference for the $\ell =m=2$ mode of ${\mathrm{\Psi }}_4$ with respect to the Richardson-extrapolated value (computed assuming a convergence order of $p=2.75$).

Figure 8

Convergence order $p$ as a function of time as computed by Eq. (28) for cocal (top panel) and lorene (bottom panel) initial data. The average values for cocal (lorene), i.e., $p=2.75\pm 0.16$ $(2.81\pm 0.18)$, are computed as arithmetic averages over the time interval $[650,1500]M_{\odot }$, where outlier data points $p<1$ and $p>4$ are excluded from the average and represent the uncertainty range.

Figure 9

Difference between the Richardson-extrapolated phases for cocal and lorene initial data using the three resolutions L, M, and H. Shown with dashed vertical lines are the times relative to one, two, and three orbital periods, while the green vertical line at $t-r_{\star }=1614M_{\odot }$ marks the merger—i.e., the time when $|{\mathrm{\Psi }}_4|$ reaches its first maximum.

Figure 10

The same as in Fig. 2, but for a corotating binary. The dashed green line refers to the irrotational solution in Fig. 2, which has a very similar mass (cf. Tables 3 and 5).

Figure 11

Hamiltonian (top) and momentum violations (bottom) for the $y$ component of the shift (${\beta }^y$) along the $x$ axis for corotating (solid lines) neutron star binaries. Dashed lines are the corresponding irrotational cocal and lorene violations as they appear in Fig. 4. The origin $x=0$ corresponds to the center of mass of the binary, with the surface of the stars located at $x\approx 6M_{\odot }$ and at $x\approx 24M_{\odot }$. Grid parameters used in cocal are those of Hs3.0d.