4.2 Quasi-equilibrium formalisms

It has long been known that the GW emission from eccentric binaries is very efficient at radiating away angular momentum relative to the radiated energy [226]; as a result, the orbital eccentricity decreases as a binary inspirals, so that orbits should be very nearly circular long before they enter the detection range of ground-based interferometers. The only exception could be from a dynamical capture process that would create a binary with a significant eccentricity and very small orbital separation. Such eccentric binaries have been predicted to form in the nuclear cluster of our galaxy (see, e.g., [213Jump To The Next Citation Point]) and in core-collapsed globular clusters [127, 167Jump To The Next Citation Point]. However, at present, no formalism exists to construct initial data for such systems, besides superposing the individual components with sufficiently large initial separations to minimize constraint violations.

Using this circularity of primordial binaries as a starting point, one may use the constraint equations of GR, along with an assumption of quasi-circularity, to derive sets of elliptic equations describing compact binary configurations. For both QE and dynamical calculations, most groups typically make use of the Arnowitt–Deser–Misner (ADM) 3+1 splitting of the metric [9], which foliates the metric into a set of three-dimensional hypersurfaces by introducing a time coordinate. The resulting form of the metric, which is completely general, is written

gμν ≡ (− α2 + βiβi)dt2 + 2 βidt dxi + γijdxidxj, (7 )
where α is known as the lapse function, βi the shift vector, and γij the spatial three-metric intrinsic to the hypersurface. We are following the standard relativistic notation here where Greek indices correspond to four-dimensional quantities and Latin indices to spatial three-dimensional quantities. Thus, the shift vector is a 3-vector, raised and lowered with the spatial 3-metric γij rather than the spacetime 4-metric gμν. To simplify matters, one typically defines a conformal factor ψ that factors out the determinant of the 3-metric, such that
1āˆ•12 ψ ≡ [det(γij)] , (8 )
introducing the conformal 3-metric &tidle;γij ≡ ψ− 4γij with unit determinant. While the 3-metric is a fundamental component of the geometric structure of the spacetime, the lapse function and shift vector are gauge quantities that simply reflect our choice of coordinates. Thus, while one often determines the lapse and shift in order to construct a appropriately “stationary” solution in the relevant coordinates between neighboring time slices, their values are often replaced to initialize dynamical runs with more convenient choices and thus different assumptions about coordinate evolution in time.

The field equations of general relativity take the deceptively simple form

1 G μν ≡ R μν − -gμνR = 8πT μν, (9 ) 2
where Gμν is the Einstein tensor, R μν and R the Ricci curvature tensor and the curvature scalar, and T μν the stress-energy tensor that accounts for the presence of matter, electromagnetic fields, and other physical effects that contribute to the mass-energy of the spacetime. Since GR is a second-order formulation, valid initial data must include not only the metric but also its first time derivative. It generally proves most convenient to introduce the time derivative of the metric after subtracting away the Lie derivative with respect to the shift, yielding a quantity known as the extrinsic curvature, Kij:
(∂t − ā„’ β)γij ≡ − 2αKij. (10 )
Both the 3-metric and extrinsic curvature are symmetric tensors with six free parameters.

For systems containing NSs, one must consider the effects of nuclear matter through its presence in the stress-energy tensor μν T. It is common to assume that the matter has the EOS describing a perfect, isotropic fluid, for which the stress energy tensor is given by

T μν ≡ (ρ0 + ρ0šœ€ + P )uμuν + P gμν, (11 )
where ρ0, šœ€, P and uμ are the fluid’s rest-mass density, specific internal energy, pressure, and 4-velocity, respectively. Many calculations further assume that the NS EOS is described by an adiabatic polytrope, for which
Γ P = (Γ − 1)ρ0šœ€ = kρ0, (12 )
where Γ is the adiabatic index of the gas and k a constant, though a number of models designed to incorporate nuclear physics and/or strange matter condensates have also been constructed and studied (see Sections 4.4 and 6 below).

The problem in constructing initial data is not so much producing solutions that are self-consistent within GR, but rather to specify a sufficient number of assumptions to fully constrain a solution. Indeed, there are only four constraints imposed by the equations of GR, known as the Hamiltonian and momentum constraints. The Hamiltonian constraint is found by projecting Einstein’s equations twice along the direction defined by a normal observer, and describes the way stress-energy leads to curvature in the metric (see, e.g., [28Jump To The Next Citation Point] for a thorough review):

2 ij R + K − KijK = 16π ρ, (13 )
where R is the scalar curvature of the 4-metric, K = Ki i is the trace of the extrinsic curvature, and
2 00 0 2 ρ ≡ n ⋅ T ⋅ n = α T = ρh (αu ) − P (14 )
is the total energy density seen by a normal observer. The third term indicates that the total energy density is found by projecting the stress-energy tensor in the direction of the unit-length timelike normal vector n, whose components are given by
n μ = (− α, 0,0,0). (15 )
In the final expression h ≡ 1 + šœ€ + Pāˆ•ρ0 is the specific enthalpy of the fluid, and the combination Γ n ≡ αu0 represents the Lorentz factor of the matter seen by an inertial observer. The notation here makes use of the standard summation convention, in which repeated indices are summed over.

Projecting Einstein’s equations in the space and time directions leads to the vectorial momentum constraint

i DiK j − DiK = 8πji, (16 )
where Di represents a three-dimensional covariant derivative and ji ≡ ρ0h Γ nui is the total momentum seen by a normal observer.

4.2.1 The Conformal Thin Sandwich formalism

In order to specify all the free variables that remain once the Hamiltonian and momentum constraints are satisfied, two different techniques have been widely employed throughout the numerical relativity community. One, known as the Conformal Transverse-Traceless (CTT) decomposition, underlies the Bowen–York [52] solution for black holes with known spin and/or linear momentum that is widely used in the “moving puncture” approach. To date, however, the CTT formalism has not been used to generate NS-NS initial data, and we refer readers to [284Jump To The Next Citation Point, 63] for descriptions of the CTT formalism applied to BH-NS and BH-BH initial data, respectively.

To date, most groups have used the Conformal Thin Sandwich (CTS) formalism to generate QE NS-NS data (see [28] for a review, [13, 137Jump To The Next Citation Point] for the initial steps in the formulation, and [326, 327Jump To The Next Citation Point, 333, 69] for derivations of the form in which it is typically used today). One first specifies that the conformal 3-metric is spatially flat, i.e., &tidle;γij = δij, where δij is the Kronecker delta function. Under this assumption, the only remaining parameter defining the spatial metric is the conformal factor ψ, which serves the role of a gravitational potential. Indeed, in the limit of weak sources, it is linearly related to the standard Newtonian potential. Next, one specifies that there exists a helical Killing vector, so that, as the configuration advances forward in time, all quantities remain unchanged when properly rotated at constant angular velocity in the azimuthal direction. This is sufficient to fix all but the trace of the extrinsic curvature, with the other components forced to satisfy the relation

[ ] Kij = − --1-- ∇i βj + ∇j βi − 2γij∇kβk . (17 ) 2 αψ4 3

The trace of the extrinsic curvature K remains a free parameter in this approach. While one may choose arbitrary prescriptions to fix it, most implementations choose a maximal slicing of the spatial hypersurfaces by setting K = ∂tK = 0. Under these assumptions the Hamiltonian and momentum constraints, along with the trace of the Einstein equations, yield five elliptic equations for the lapse, shift vector, and conformal factor:

( ) 2 5 1- ij ∇ ψ = − ψ 8K Kij − 2πρ , (18 ) ∇2 (α ψ) = α ψ5(7-ψ4KijKij + 2πψ4(ρ + 2S ), (19 ) 8 2 i 1- j i 4 ij − 6 4 i ∇ β + 3 ∇ ∇j β = 2α ψ K ∇j(α ψ ) + 16α ψ j , (20 )
[ ] S = (g μν + n μnν)Tμν = Sj = 3P + (ρ + P ) 1 − Γ −n 2 (21 ) j
is the trace of the stress-energy tensor projected twice in the spatial direction. While these five equations are linked and the right-hand sides are nonlinear, they are amenable to solution using iterative methods. Boundary conditions are set by assuming asymptotic flatness: at large radii, the metric takes on the Minkowski form so α → 1, ψ → 1, and i βrot → Ω × āƒ—r. We note that a purely corotating shift term yields zero when we apply the left-hand side of Eq. 20View Equation, so we may subtract it away and solve the equation with a boundary condition of zero instead.

The breakdown in Eqs. 18View Equation, 19View Equation, and 20View Equation is not unique. The Meudon group [125Jump To The Next Citation Point, 124Jump To The Next Citation Point], to pick one example, has often chosen to define ν ≡ ln α and β ≡ ln(αψ2 ), and replace Eqs. 18View Equation and 19View Equation with the equivalent pair

2 4 ij i 4 ∇ ν = ψ K Kij − ∇iν ∇ β + 4π ψ (ρ + S ), 2 3 4 ij 1 i i 4 ∇ β = -ψ K Kij − -(∇iν ∇ ν + ∇i β∇ β) + 4πψ S. 4 2

This approach is sufficient to define the field component of the configuration, but one still needs to solve for the matter quantities as well. One starts by assuming that there is a known prescription for reconstructing the density, internal energy, and pressure from the enthalpy h. Next, one has to assume some model for the spin of the NS. While corotation is often a simpler choice, since the velocity field of the matter is zero in the corotating frame, the more physically reasonable condition is irrotational flow. Indeed a realistic NS viscosity is never sufficiently large to tidally lock the NS to its companion during inspiral [45Jump To The Next Citation Point, 146Jump To The Next Citation Point]. If we define the co-momentum vector wi = hui, irrotational flow implies the vanishing of its curl:

∇ × w = ∇ μw ν − ∇ νwμ = ∂ μwν − ∂νw μ = 0, (22 )
which allows us to define a velocity potential Ψ such that w ≡ ∇ Ψ. Using these quantities, one may write down the integrated Euler equation
hα Γ nāˆ•(1 − γijUiU j0) = const., (23 )
where the 3-velocity i U of the fluid with respect to an Eulerian observer is given by
i i 0 U i ≡ u-+-β--u-, (24 ) Γ n
and the orbital 3-velocity Ui 0 with respect to the same observer by Ui = βiāˆ•α 0. For details on the ways in which one may construct an elliptic equation for the velocity potential, we refer to the derivation in [125Jump To The Next Citation Point].

To date, all QE sequences and dynamical runs in the literature have assumed that NSs are either irrotational or synchronized, but it is possible to construct the equations for arbitrary NS spins so long as they are aligned [29, 309]. While suggestions are also given there on how to construct QE sequences with intermediate spins using the new formalism, none have yet appeared in the literature. Similarly, a formalism to add magnetic fields self-consistently to QE sequences has been constructed [314], as current dynamical simulations typically begin from data assuming either zero magnetic fields or those that only contribute via magnetic pressure.

4.2.2 Other formalisms

The primary drawback of the CTS system is the lack of generality in assuming the spatial metric to be conformally flat, which introduces several problems. The Kerr metric, for example, is known not to be conformally flat, and conformally flat attempts to model Kerr BHs inevitably include spurious GW content. The same problem affects binary initial data: in order to achieve a configuration that is instantaneously time-symmetric, one actually introduces spurious gravitational radiation into the system, which can affect both the measured parameters of the initial system as well as any resulting evolution.

Other numerical formalisms to specify initial data configurations in GR have been derived using different assumptions. Usui and collaborators derived an elliptic set of equations by allowing the azimuthal component of the 3-metric to independently vary from the radial and longitudinal components [319Jump To The Next Citation Point, 318], finding good agreement with the other methods discussed above. A number of techniques have been developed to construct helically symmetric spacetimes in which one actually solves Einstein’s equations to evaluate the non-conformally-flat component of the metric, which are typically referred to as “waveless” or “WL” formalisms [260, 50, 291]. In terms of the fundamental variables, rather than specifying the components of the conformal spatial metric by ansatz, one specifies instead the time derivative of the extrinsic curvature using a physically motivated prescription. These methods are designed to match the proper asymptotic behavior of the metric at large distances, and may be combined with techniques designed to enforce helical symmetry of the metric and gauge in the near zone (the near zone helical symmetry, or “NHS” formalism) to produce a global solution [315Jump To The Next Citation Point, 334, 316Jump To The Next Citation Point]. QE sequences generated using this formalism [315] have shown that the resulting conformal metric is indeed non-flat, with deviations of approximately 1% for the metric components, and similar differences in the system’s binding energy when compared to equivalent CTS results. They suggest [316Jump To The Next Citation Point] that underestimates in the quadrupole deformations of NS prior to merger may result in total phase accumulation errors of a full cycle, especially for more compact NS models.

QE formalisms reflect the assumption that binaries will be very nearly circular, since GW emission acting over very long timescales damps orbital eccentricity to negligible values for primordial NS-NS binaries between their formation and final merger. Binaries formed by tidal capture and other dynamical processes, which may be created with much smaller initial separations, are more likely to maintain significant eccentricities all the way to merger (see, e.g., [213] for a discussion of such processes for BH-BH binaries) and it has been suggested based on simple analytical models that such mergers, likely occurring in or near dense star clusters, may account for a significant fraction of the observed SGRB sample [167]. However, more detailed modeling is required to work out accurate estimates of merger rates given the complex interplay between dynamics and binary star evolution that determines the evolution of dense star clusters, and given the large uncertainties in the distributions of star cluster properties in galaxies throughout the universe. No initial data have ever been constructed in full GR for merging NS-NS binaries with eccentric orbits since the systems are then highly time-dependent, while the calculations performed to evolve them generally use a superposition of two stationary NS configurations [122Jump To The Next Citation Point].

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