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The spatially conformally flat approximation (CFA) is a viable method to deduce initial conditions for the subsequent evolution of binary neutron stars employing the full Einstein equations. Here we analyze the viability of the CFA for the general relativistic hydrodynamic initial conditions of binary neutron stars. We illustrate the stability of the conformally flat condition on the hydrodynamics by numerically evolving ~100 quasicircular orbits. We illustrate the use of this approximation for orbiting neutron stars in the quasicircular orbit approximation to demonstrate the equation of state dependence of these initial conditions and how they might affect the emergent gravitational wave frequency as the stars approach the innermost stable circular orbit.

The epoch of gravitational wave astronomy has now begun with the first detection [

To date there have been numerous attempts to calculate theoretical templates for gravitational waves from compact binaries based upon numerical and/or analytic approaches (see, e.g., [

When binary neutron stars are well separated, the post-Newtonian (PN) approximation is sufficiently accurate [

Indeed, the templates generated by PN approximations, unless carried out to fifth and sixth order [

Numeric and analytic simulations [

The spatially conformally flat approximation to GR was first developed in detail in [

Here, we summarize the original CFA approach and associated general relativistic hydrodynamics formalism developed in [

This paper is organized as follows. In Section

The solution of the field equations and hydrodynamic equations of motion were first solved in three spatial dimensions and explained in detail in the 1990s in [

One starts with the slicing of space-time into the usual one-parameter family of hypersurfaces separated by differential displacements in a time-like coordinate as defined in the (

In Cartesian

One consequence of this conformally flat approximation to the three-metric is that the emission of gravitational radiation is not explicitly evolved. Nevertheless, one can extract the gravitational radiation signal and the back reaction via a multipole expansion [

As a third gauge condition, one can choose separate coordinate transformations for the shift vector and the hydrodynamic grid velocity to separately minimize the field and matter motion with respect to the coordinates. This set of gauge conditions is key to the present application. It allows one to stably evolve up to hundreds and even thousands of binary orbits without the numerical error associated with the frequent advocating of fluid through the grid.

Given a distribution of mass and momentum on some manifold, then one first solves the constraint equations of general relativity at each time for a fixed distribution of matter. One then evolves the hydrodynamic equations to the next time step. Thus, at each time slice a solution to the relativistic field equations and information on the hydrodynamic evolution is obtained.

The solutions for the field variables

In a similar manner [

Finally, the momentum constraints yields [

We note that, in early applications of this approach, the source for the shift vector contained a spurious term due to an incorrect transformation between contravariant and covariant forms of the momentum density as was pointed out in [

To solve for the fluid motion of the system in curved-space time it is convenient to use an Eulerian fluid description [

By introducing the usual set of Lorentz contracted state variables it is possible to write the relativistic hydrodynamic equations in a form which is reminiscent of their Newtonian counterparts [

In terms of these state variables, the hydrodynamic equations in the CFA are as follows: the equation for the conservation of baryon number takes the form

In the quasicircular orbit approximation (neglecting angular momentum in the radiation field), this system has a Killing vector corresponding to rotation in the orbital plane. Hence, for these calculations the angular momentum is well defined and given by an integral over the space-time components of the stress-energy tensor [

To find the orbital frequency detected by a distant observer corresponding to a fixed angular momentum we employ a slightly modified derivation of the orbital frequency compared to that of [

In the ADM conformally flat (

A key additional ingredient, however, is the implementation of a grid three velocity

For the orbit calculations illustrated here we model corotating stars, that is, no spin in the corotating frame. This minimizes matter motion on the grid. However, we note that there is need at the present time of initial conditions for arbitrarily spinning neutron stars and the method described here is equally capable of supplying those initial conditions.

As a practical approach the simulation [

To evolve stars at large separation distance it is best [

Schematic representation of the field and hydrodynamics grid used in the simulations. The inner blue grid represents the higher resolution matter grid and the outer white grid represents the field grid.

As noted in [

The time steps

The first condition is known as the Courant condition, that is, a search over all zones

The Newtonian sound speed is given by the variation of pressure with density. In relativity the wave speed is given instead by the adiabatic derivative of the pressure with respect to the relativistic inertial density. In terms of relativistic variables the local sound speed in zone

The second condition is a search for the zone with minimum time for material to flow across a zone

The third condition is introduced to maintain stability of the artificial viscosity algorithm. The viscous equations are analogous to a diffusion equation in four velocity with a diffusion coefficient

The time step

Figure

Comparison of the orbital angular velocity

Figure

Plot of the error in the central density versus the number of zones across the star. It is clear that there is only a 1% error with ≈15 zones across the star. Increasing the number of zones across the star so that there are >35 zones across the star produces less than a 0.1% error.

This figure illustrates that here is only a 1% error in central density with ≈15 zones across the star, while increasing the number of zones across the star to >35 produces less than a 0.1%. In the illustrations below we maintain

As an illustration of the orbit stability Figure ^{2} and the Courant parameter set to

Plot of the orbital angular velocity, ^{−1}) versus cycle number. When

Figure

Figure

Contours of the lapse function (left) and central density (right) at cycle numbers 0 (a), 5,200 (b), and 25,800 (c) corresponding to roughly 0, 5, and 19 orbits.

Contours of the central density for the binary system at the approximate number of orbits as labelled.

We note, however, that this orbit is on the edge of the ISCO. As such it could be unstable to inspiral even after many orbits. Figures

Plot of the orbital angular velocity, ^{2}) goes over ~10 obits and then becomes unstable to inspiral and merger after ~10^{4} cycles. The stable two runs (^{−2} and ^{−2}) were run for 100,000 cycles and ≈100 orbits.

Plot of the central density, ^{2}.

Figure ^{−2}) shown on Figure ^{−2} are stable. Indeed, for these cases, after about the first 3 orbits the orbits continue with almost no discernible change in orbit frequency or central density.

As mentioned previously, the numerical relativistic neutron binary simulations of [

One hope in the forthcoming detection of gravitational waves is that a sensitivity exists to the neutron star equation of state. For illustration we utilize several representative equations of state often employed in the literature. These span a range from relatively soft to stiff nuclear matter. These are used to illustrate the EoS dependence of the initial conditions. One EoS often employed is that of a polytrope, that is, ^{−3}. These parameters, with ^{−3}, produce an isolated star having a radius = 17.12 km and baryon mass =

Table presenting central density, baryon mass, and gravitational mass for the five adopted equations of state.

EoS | ^{15} g cm^{−3}) | | |
---|---|---|---|

GLN | 1.56 | 1.54 | 1.40 |

MW | 1.39 | 1.54 | 1.40 |

LS 220 | 0.698 | 1.54 | ~1.40 |

LS 375 | 0.492 | 1.54 | ~1.40 |

| 0.474 | 1.50 | 1.40 |

In Figure

EoS index

Table

Orbital parameters for each EoS.

EoS | ^{2}) | ^{−1}) | | | | ^{−3}) |
---|---|---|---|---|---|---|

GLN | | 666.5 | 71.62 | 57.67 | 1.390 | |

| 592.34 | 77.82 | 62.81 | 1.391 | | |

| 475.05 | 88.06 | 73.53 | 1.394 | | |

| 391.75 | 100.34 | 84.31 | 1.396 | | |

| ||||||

MW | | 780.92 | 65.22 | 51.52 | 1.391 | |

| 671.85 | 71.18 | 57.24 | 1.393 | | |

| 602.80 | 76.94 | 61.86 | 1.394 | | |

| 482.30 | 86.91 | 72.36 | 1.396 | | |

| 300.46 | 116.13 | 100.8 | 1.399 | | |

| 235.72 | 136.93 | 119.74 | 1.401 | | |

| ||||||

LS 220 | | 523.59 | 90.77 | 77.34 | 1.403 | |

| 472.08 | 97.53 | 83.08 | 1.404 | | |

| 389.96 | 109.78 | 94.84 | 1.405 | | |

| 327.04 | 122.51 | 107.10 | 1.407 | | |

| ||||||

LS 375 | | 490.09 | 97.09 | 83.92 | 1.404 | |

| 442.40 | 103.95 | 90.04 | 1.405 | | |

| 366.67 | 116.65 | 102.50 | 1.406 | | |

| 307.80 | 130.72 | 115.60 | 1.407 | | |

| ||||||

Polytrope | | 804.70 | 63.30 | 51.20 | 1.395 | |

| 826.03 | 67.85 | 55.18 | 1.396 | | |

| 762.37 | 74.64 | 61.72 | 1.397 | | |

| 624.33 | 85.87 | 72.71 | 1.399 | | |

| 532.83 | 94.04 | 80.45 | 1.400 | | |

| 477.19 | 101.34 | 86.95 | 1.400 | |

As expected, the central densities are much higher for the relatively soft MW and GLN equations of state. Also, the orbit angular frequencies are considerably lower for the extended mass distributions of the stiff equations of state than for the more compact soft equations of state. These extended mass distributions induce a sensitivity of the emergent gravitational wave frequencies and amplitude due to the strong dependence of the gravitational wave frequency to the quadrupole moment of the mass distribution.

The physical processes occurring during the last orbits of a neutron star binary are currently a subject of intense interest. As the stars approach their final orbits it is expected that the coupling of the orbital motion to the hydrodynamic evolution of the stars in the strong relativistic fields could provide insight into various physical properties of the coalescing system [

Figure

Computed gravitational wave frequency,

In the (post)

Although this is a gauge-dependent comparison, for illustration we show in Figure

Indeed, a striking feature of Figure

Also, for a soft EoS the orbit becomes unstable to inspiral at a larger separation. At least part of the difference between the soft and stiff EoSs can be attributed to the effects of the finite size of the stars which are more compact for the soft equations of state [

We note that, for comparable angular momenta, our results are consistent with the EoS sensitivity study of [^{2} (in our units) with a corresponding gravitational wave frequency of 530 Hz at a proper separation of 46 km. This EoS is comparable to the polytropic, MW, and GLN EoSs shown on Figure ^{2} and an ADM mass of ^{2} for the same baryon mass of

The main parameter characterizing the last stable orbit in the post-Newtonian calculation is the ratio of coordinate separation to total mass (in isolation)

The relativistic hydrodynamic equilibrium in the CFA remains as a viable approach to calculate the initial conditions for calculations of binary neutron stars. In this paper we have illustrated that one must construct initial conditions that have run for at least several orbits before equilibrium is guaranteed. We have demonstrated that beyond the first several orbits the equations are stable over many orbits (~100). We also have shown that such multiple orbit simulations are valuable as a means to estimate the location of the ISCO prior to a full dynamical calculation. Moreover, we have examined the sensitivity of the initial condition orbit parameters and initial gravitational wave frequency to the equation of state. We have illustrated how the initial condition orbital properties (e.g., central densities, orbital velocities, and binding energies) and location of the ISCO are significantly affected by the stiffness of the EoS.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work at the University of Notre Dame (Grant J. Mathews) was supported by the US Department of Energy under Nuclear Theory Grant DE-FG02-95-ER40934 and by the University of Notre Dame Center for Research Computing. One of the authors (N. Q. Lan) was also supported in part by the National Science Foundation through the Joint Institute for Nuclear Astrophysics (JINA) at UND and in part by the Vietnam Ministry of Education (MOE). N. Q. Lan would also like to thank the Yukawa Institute for Theoretical Physics for their hospitality during a visit where part of this work was done.