GRAVITATIONAL RADIATION FROM MAGNETICALLY FUNNELED SUPERNOVA FALLBACK ONTO A MAGNETAR

Accretion during fallback has been studied thoroughly in the context of collapsars, "mild" core-collapse events in which part of the stellar mantle initially explodes then stalls and implodes. Numerical simulations of collapsars that treat the hydrodynamics, shock physics, and neutrino transport in detail find accretion rates in the range 10⁻⁴–10⁻² M_☉ s⁻¹ lasting for 10³–10⁴ s (MacFadyen et al. 2001; Zhang et al. 2008). Fallback passes through early and late stages, with $\dot{M}_{\rm early} \approx 10^{-3} \eta t^{1/2} \,M_{\odot } \, {\rm s^{-1}}$ and $\dot{M}_{\rm late} \approx 50\, t^{-5/3}\, M_{\odot } \, {\rm s^{-1}}$, respectively, where t is the time after core bounce (in seconds), and η depends sensitively on the explosion energy, E_s, with 0.1 ⩽ η ⩽ 10 for 0.3 ⩽ E_s/10⁵¹ erg ⩽ 1.2. The transition occurs at t ∼ 10² s, after the oxygen shell falls in. We adopt the convenient parameterization $\dot{M}_{\rm a} = (\dot{M}_{\rm early}^{-1} + \dot{M}_{\rm late}^{-1})^{-1}$ for the total accretion rate, $\dot{M}_{\rm a}$, introduced by Piro & Ott (2011). For η in the above range, $\dot{M}_{\rm a}$ is mostly high enough to overcome the outward ram pressure of the magnetar's neutrino- and Poynting-flux-driven wind, even when the system is spherically symmetric (Piro & Ott 2011; Piro & Thrane 2012).

The angular velocity of the magnetar, Ω(t), evolves in response to three torques, N_dip, N_a, and N_gw, defined below, according to $I\dot{\Omega } = N_{\rm dip} + N_{\rm a} + N_{\rm gw}$, where I = 0.35 MR² is the moment of inertia in terms of the stellar mass M and radius R (Lattimer & Prakash 2001). Figure 6 of Lattimer & Prakash (2001) gives 0.2 ⩽ I/MR² ⩽ 0.5 for a range of plausible equations of state; we adopt a central value here. The magnetic dipole torque N_dip obeys the standard vacuum formula, N_dip = −μ²Ω³/(6 c³), as long as the light cylinder (radius c/Ω) lies inside the hydromagnetic lever arm, given by the nominal Alfvén radius $r_{\rm m} = \mu ^{4/7} (G\!M)^{-1/7} \dot{M}_{\rm a}^{-2/7}$. For millisecond magnetars, this is a fair approximation, setting aside extended-dipole corrections (Melatos 1997). The accretion torque N_a takes either sign, depending on the ratio of its mechanical and magnetic components, and is controlled by the fastness parameter ω = (r_m/r_c)^3/2, where r_c = (GM/Ω²)^1/3 is the corotation radius (Ghosh & Lamb 1979). Here we follow Eksi et al. (2005) and Piro & Ott (2011) and set $N_{\rm a} = (1-\omega) (G\!Mr_{\rm m})^{1/2} \dot{M}_{\rm a}$. A more careful treatment of the disk–magnetosphere boundary and hence N_a, to include episodic accretion in a corotation-limited disk (Rappaport et al. 2004; D'Angelo & Spruit 2011) and radiation pressure (Andersson et al. 2005),² lies outside the scope of this paper; our aim here is to explore how magnetic funneling affects magnetar fallback irrespective of the specific mode of disk accretion. The gravitational wave torque N_gw is treated phenomenologically, following Piro & Ott (2011), by setting N_gw = −N_aθ(β − β_c), where θ(...) is the Heaviside step function, β(M, R, Ω) is the ratio of the star's kinetic and gravitational potential energies, and β_c is the threshold for radiation-reaction-driven (β_c = 0.14), viscosity-driven (0.14), and bar-mode (0.27) instabilities. Qualitatively, gravitational wave friction acts to cancel the other torques, once a suitable instability is triggered, with N_gw ≈ −N_a − N_dip and |N_dip| ≪ |N_a| typically. In Section 2.2, we add to N_gw the secular contribution from the quadrupole associated with a polar magnetic mountain.

The mass of the star, M(t), evolves according to $\dot{M} = \dot{M}_{\rm a} \theta (1-\omega)$, where $\dot{M}_{\rm a}$ is the parameterized fallback rate above, and the propeller effect shuts off accretion for ω > 1. This oversimplifies things: even under propeller conditions, some of the infalling plasma penetrates to the stellar surface due to instabilities at the disk–magnetosphere boundary (Rappaport et al. 2004; Romanova et al. 2005; D'Angelo & Spruit 2011). We bundle the uncertainties surrounding propeller leakage specifically and disk dynamics generally into the parameter η, an approach justified approximately elsewhere through one-zone, Shakura–Sunyaev disk calculations (Piro & Ott 2011; Piro & Thrane 2012). To stay consistent with previous work, the initial mass is set to M(0) = 1.4 M_☉, the distinction between gravitational and baryonic mass is ignored, and R is held constant as M increases, a fair approximation for most equations of state until just before black hole formation (Lattimer & Prakash 2001).

The strong magnetic field of a magnetar funnels the fallback accretion flow preferentially onto the magnetic poles to form a mountain. Funneling is imperfect, due to Rayleigh–Taylor mixing on the boundary of the polar flux tube and filamentation caused by obliquity, but simulations confirm that it occurs for a wide range of parameters and geometries (Romanova et al. 2003; Kulkarni & Romanova 2008). As matter splashes down onto the poles, it slides sideways under its own weight, dragging along magnetic field lines by flux freezing. Thus the magnetic field is compressed at the equator and pushes back to confine the mountain, while the radial component of the polar field and hence μ are reduced (Melatos & Phinney 2001; Payne & Melatos 2004).

To calculate accurately the magnetic field structure B(x) and density profile ρ(x) of a polar mountain is a subtle task. Order-of-magnitude pressure balance arguments underestimate the mass quadrupole ∼10⁴ fold by implicitly imposing outflow boundary conditions at the edge of the polar cap and hence underestimating the compression of, and tension in, the equatorial field, e.g., Brown & Bildsten (1998, and references therein). To avoid this, one must solve the force balance equation (quasistatic equilibrium; Alfvén crossing timescale $\ll M_{\rm a} / \dot{M}_{\rm a}$) simultaneously with a mass–flux constraint equation enforcing flux freezing and equatorial magnetic compression. Writing B = (rsin θ)⁻¹∇ψ(r, θ) × ∇ϕ in spherical polar coordinates (r, θ, ϕ), where ψ(r, θ) is a scalar flux function, we obtain

$$\begin{equation} \hspace{-30pt}\Delta ^2\psi = - \frac{d\!F(\psi)}{d\psi } \left\lbrace 1 - \frac{(\Gamma -1) (\phi -\phi _0)}{\Gamma K^{1/\Gamma } [F(\psi)]^{(\Gamma -1)/\Gamma }} \right\rbrace ^{1/(\Gamma -1)} \end{equation} \tag{ 1 }$$

and

$$\begin{equation} F(\psi) = \frac{K}{(2\pi)^\Gamma } \left(\frac{dM}{d\psi } \right)^\Gamma \left[ \int _C ds\, r\sin \theta | \nabla \psi |^{-1} \left\lbrace \dots \right\rbrace ^{1/(\Gamma -1)} \right]^{-\Gamma } \end{equation} \tag{ 2 }$$

for the force balance equation and mass–flux constraint respectively for a polytropic equation of state $P=K\rho ^\Gamma$, where P is the pressure, ϕ(r) is the gravitational potential in the Cowling approximation, ϕ₀ is the gravitational potential at the stellar surface before accretion begins, Δ² is the Grad–Shafranov operator, the integral in Equation (2) is computed along the field line ψ = constant, and the braces {...} in Equation (2) contain the same expression as the braces in Equation (1). The reader is referred to the literature for a detailed derivation of Equations (1) and (2) and a key to the notation (Payne & Melatos 2004; Priymak et al. 2011; Mukherjee et al. 2013a). A vital point concerns the barometric function F(ψ) and mass–flux ratio dM/dψ, i.e., the mass (initial plus accreted) per unit flux enclosed within (ψ, ψ + dψ). Some authors guess F(ψ) in Equation (1) and drop Equation (2) for convenience (Brown & Bildsten 1998; Melatos & Phinney 2001; Mukherjee et al. 2013a), but in general such a guess produces the wrong amount of equatorial magnetic compression. In reality F(ψ) is determined uniquely by accretion through the relevant initial value problem and dM/dψ. In equilibrium, this path-specific information is stored in Equation (2), which maps one-to-one the initial (pre-accretion) and final states to preserve flux freezing at all intermediate steps, as required by the mass continuity and magnetic induction equations of ideal magnetohydrodynamics (MHD; Priymak et al. 2011).

Figure 1 displays the equilibrium structure of a typical magnetic mountain ∼10η^−1/2 s after fallback begins, with initial magnetic dipole moment μ_i = 5 × 10³² G cm³, accreted mass M_a = 1.2 × 10⁻³M_☉, and line-tying boundary conditions (see Sections 2.1 and 2.2 in Priymak et al. 2011 for details). From the solid contours, one observes that the magnetic field is compressed into an equatorial band, with B_r ≪ B_θ near the pole and B_θ ≪ B_r on either side of the equator. The magnetic tension, directed along the field-line radius of curvature, confines the accreted matter into a mound, whose isodensity surfaces are drawn as dashed contours. The equilibrium is calculated for M_a = 1.6 M_c, where M_c = 3 × 10⁻³(μ_i/10³³ G cm³)² M_☉ is the characteristic burial mass for a polytropic equation of state with nondegenerate neutrons (model D in Priymak et al. 2011). Other polytropes with 1 ⩽ Γ ⩽ 5/3 give qualitatively similar results (see Figure 8 in Priymak et al. 2011). By constructing a quasistatic sequence of equilibria with increasing M_a, one finds that the magnetic dipole and mass quadrupole moments scale roughly as μ = μ_i(1 + M_a/M_c)⁻¹ and = (M_a/M_☉)(1 + M_a/M_c)⁻¹ respectively (Shibazaki et al. 1989; Payne & Melatos 2004; Zhang & Kojima 2006; Priymak et al. 2011); polar magnetic burial screens μ, even as the mountain builds up, and |B| rises at the equator. In Figure 1, for example, we have μ = 0.6 μ_i and = 5 × 10⁻⁴ for M_a = 1.6 M_c, within 50% of the predictions from the rule-of-thumb formulae. The formula for , which implies ⩽ M_c/M_☉, is deliberately conservative, to avoid overpredicting the resulting gravitational wave signal. In reality, rises gradually above M_c/M_☉ for M_a ≳ M_c in time-dependent MHD simulations that grow the mountain from scratch (see Figure 5 in Vigelius & Melatos 2009b), accompanied by transient, localized, loss-of-equilibrium events (cf. instabilities), where the magnetic field pinches off at isolated points to form topologically disconnected loops without disrupting the main body of the mountain (Klimchuk & Sturrock 1989; Payne & Melatos 2004; Vigelius & Melatos 2008; Mukherjee et al. 2013a).

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It is natural to wonder whether the stressed magnetic configuration in Figure 1 is stable on the Alfvén timescale τ_A. The answer, counterintuitively, seems to be yes. Independent MHD simulations by two groups using the solvers ZEUS (Payne & Melatos 2007; Vigelius & Melatos 2008) and PLUTO (Mukherjee et al. 2013a, 2013b) confirm stability in two and three dimensions. A self-consistent solution of Equations (1) and (2), when imported into ZEUS or PLUTO, initially experiences the undular submode of the Parker instability, but it is not disrupted; after releasing a "magnetic blister," the mountain settles down to a new equilibrium with reduced by 50%–70% and mass loss ≲ 1% (Vigelius & Melatos 2008). Magnetic line tying at the inner boundary stabilizes the undular submode and switches off the interchange submode and growing modes in the continuous spectrum completely (Vigelius & Melatos 2008, 2009a). Mukherjee et al. (2013a) found unstable, pressure-driven, filamentary, toroidal modes at the periphery of filled and hollow mountains, but (1) these modes do not disrupt the mountain overall, consistent with Vigelius & Melatos (2008); (2) they are suppressed when fixed-gradient boundary conditions at the edge of the polar cap are replaced by north–south symmetry at the equator, to allow properly for stabilization by the equatorial magnetic belt (cf. Litwin et al. 2001); and (3) they arise from equilibria satisfying Equation (1) but not Equation (2) (see above). Grid-refinement tests buttress these findings (Vigelius & Melatos 2008; Mukherjee et al. 2013a). For example, the linear growth rate of the undular submode is observed to vary ∝ (grid  scale)^−1/2 and is grid-limited, as theory predicts, but the final, saturation values of μ and are found to be independent of the grid scale (Vigelius & Melatos 2008).

A magnetic mountain does not relax resistively or by sinking on protomagnetar timescales (≲ 10⁴ s; Vigelius & Melatos 2009c; Wette et al. 2010), nor is it disrupted by resistive instabilities. The buried polar magnetic field and hence μ resurrect on the shorter of the Ohmic diffusion and Hall timescales τ_d and τ_H, with min(τ_d, τ_H) ≳ 1 yr even with enhanced accretion-driven heating (Vigelius & Melatos 2009c; Viganò et al. 2013). Likewise, subduction and meridional redistribution occur long after the magnetar spins down and stops emitting gravitational waves at a detectable level, if no black hole forms (Choudhuri & Konar 2002). The values of μ and calculated from Equations (1) and (2) are within a factor of two of ZEUS mountains grown from scratch on a soft surface, as long as the boundary at r = R_in in Figure 1 is set deep enough that the mountain is light compared to the substrate, its base does not move much laterally, and magnetic line tying stays a good approximation (Wette et al. 2010). In MHD simulations where the electrical resistivity is boosted artificially, the mountain is stable on the tearing mode timescale, (τ_dτ_A)^1/2 ∼ 10⁴ s (Vigelius & Melatos 2008).

Figure 2 displays contours of the peak gravitational wave strain h_max = max[h₀(t)] as a function of μ_i and the initial spin period P_i. The maximum is computed for 0 ⩽ t ⩽ 10⁴ s and typically occurs at t ≈ t_pk = 85η^−6/13 s, i.e., the accretion timescale, which does not depend on μ_i or P_i. To clarify the physics, we examine separately the consequences of magnetic funneling and hydrodynamic instabilities in the top and bottom panels of Figure 2. To help the reader interpret Figure 2, we also plot representative examples of the star's rotational evolution and dominant torque components (N_a, N_gw ≫ N_dip) in Figure 3 for four scenarios (top to bottom rows), defined by whether or not magnetic funneling occurs, hydrodynamic instabilities are switched on, and μ_i is low (solid curves) or high (dashed curves). The rotational evolution differs between the scenarios, as we now discuss.

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In the top panel, we switch off artificially the instabilities, so that they contribute neither to h₀ nor to N_gw, and focus on the radiation from the mountain, with characteristic strain h₀(t) = 2 GΩ(t)²(t)I(t)/(c⁴D), where D is the distance to the source. The solid contours describe what happens, when the mountain quadrupole grows with t, yet μ stays constant—an unlikely scenario, given the physics of magnetic burial in Section 2.2, but still instructive as a stepping stone to the full problem. For μ_i ≳ 0.7 × 10³³ G cm³, the star quickly attains magnetocentrifugal equilibrium [ω ≈ 1, Ω ≈ 7 × 10³(μ/10³³ G cm³)^−6/7 rad s⁻¹] at t ≈ t_pk, getting there either by spinning up by accretion (P_i ≳ 1 ms) or spinning down by the propeller effect (P_i ≲ 1 ms). It then spins down gradually over ∼10³ s mainly under the action of N_gw, with N_a assisting the deceleration for t ≳ 5 × 10² s, and N_dip contributing ∼10% of the total torque (see Figure 3, top row, right panel). The instability threshold is never crossed; β peaks at ≈0.05, so it is consistent to switch off the instabilities in this regime. The peak wave strain increases gently with μ_i according to $\epsilon \propto M_{\rm c} \propto \mu _{\rm i}^2$ (M_a ≫ M_c), $\Omega \propto \mu _{\rm i}^{-6/7}$ (magnetocentrifugal equilibrium), and hence $h_{\rm max} \propto \epsilon \Omega ^2 \propto \mu _{\rm i}^{2/7}$. The h_max scaling steepens at low P_i, where the solid contours turn upward, because the propeller physics asserts itself earlier.

For μ_i ≲ 0.7 × 10³³ G cm³ in the above scenario (Figure 3, top row, dashed curves), the star spins up toward its centrifugal limit,³ entering the propeller phase relatively late at t ≳ 5 × 10² s ≫ t_pk. It then spins down, but more gradually than for μ_i ≳ 0.7 × 10³³ G cm³ because, by the time the star enters the propeller phase, $N_{\rm a} \propto \dot{M}_{\rm a}$ is well below its peak (N_dip ∼ N_a here). The gravitational wave signal is also weaker, as the solid contours indicate, and as expected from $h_{\rm max} \propto \mu _{\rm i}^{2/7}$. However, the predicted h_max is unrealistic, because β promptly exceeds β_c at t ≈ t_pk in the low-μ_i regime, i.e., the switched-off instabilities would switch on rapidly in reality to emit gravitational radiation and spin down the star through N_gw. The latter behavior is discussed further below with reference to the bottom panel of Figure 2.

Now suppose that μ diminishes through magnetic burial, while the mountain grows, and the instabilities remain artificially switched off. The results are described by the dashed contours in the top panel of Figure 2; see also the second row of Figure 3. The contours in Figure 2 are nearly horizontal. Burial suppresses the propeller effect observed at low P_i and high μ_i in the solid contours, because at t ≳ t_pk we have M_a ≫ M_c and hence μ ≪ μ_i everywhere in the plotted region. Moreover, with μ ≪ μ_i everywhere, N_a behaves similarly for low and high μ_i; compare the dashed and solid black curves in the second row of Figure 3. The accretion torque rapidly spins up the star toward torque balance (N_a + N_gw ≈ 0), while maintaining ω < 1 for all t except t ≲ 10 s ≪ t_pk, when M_a is still less than M_c. The wave strain h_max∝Ω² is ≈4 times greater than for the solid contours, because Ω peaks at the stall frequency (Bildsten 1998), which turns out to be about double the magnetocentrifugal equilibrium frequency (see above). The scaling for the stall frequency, $\Omega \propto \epsilon ^{-2/5} \propto \mu _{\rm i}^{-4/5}$, implies $h_{\rm max} \propto \epsilon \Omega ^2 \propto \mu _{\rm i}^{2/5}$, which is confirmed by inspecting the dashed contours. The trend with P_i is weak, as the propeller is inactive. The stall frequency is reached at t ≈ t_pk, after which $\dot{M}_{\rm a}$ and hence N_a drop away rapidly, and h₀(t) decays on the characteristic gravitational wave spin-down timescale $I\Omega /N_{\rm gw} \propto \mu _{\rm i}^{-4/5}$, evaluated at the stall point, with IΩ/N_gw ∼ 10³ s typically (noting |N_dip| ≪ |N_gw|). The decay timescale influences the detectability of the signal, as discussed in Section 3.2. For μ_i ≳ 1 × 10³³ G cm³, β peaks below β_c, so switching off artificially the instabilities does not affect the conclusions. At lower μ_i, the conclusions are affected, as discussed below.

Finally, let us switch on the hydrodynamic instabilities. The results including (dashed contours) and excluding (solid contours) μ reduction by magnetic burial are presented in the bottom panel of Figure 2 and the third and fourth rows of Figure 3. The instability-sourced contribution to the wave strain is given by h₀(t) = [5GN_a(t)]^1/2/[8c³D²Ω(t)]]^1/2 from torque balance (N_a = N_gw; see Section 2.1 and Piro & Ott 2011).⁴ We set the threshold at β_c = 0.14; the results are qualitatively similar for β_c = 0.27. Consider first the solid contours. They are distorted at the left edge of the plot (P_i ≲ 2 ms), because the propeller effect spins down the star initially, before Ω rises to a local maximum ≳ 3 × 10³ rad s⁻¹ at t ≈ t_pk. Along the lower edge of the plot, for μ_i ≲ 0.3 × 10³³ G cm³ (where the contours display kinks), accretion spins up the star toward its centrifugal limit, triggering hydrodynamic instabilities; β stays above β_c for t ≳ t_pk, overshooting as far as β ≈ 0.2, and the star spins down electromagnetically on the timescale ∼10³(μ_i/10³³ G cm³)⁻² s, with N_gw balancing $N_{\rm a}\propto \dot{M}_{\rm a}$, as the instabilities operate. For μ_i ≳ 0.3 × 10³³ G cm³, no instabilities are triggered, β < β_c decreases monotonically with t, and the rotational evolution in the first and third rows of Figure 3 is similar.

On the other hand, the dashed contours in Figure 2 describe what happens when μ is reduced by burial. Across the whole plot, we find ω < 1 for all t, i.e., there is no propeller effect. Likewise N_dip is insignificant. For μ_i ≳ 1 × 10³³ G cm³, no instabilities are triggered. The star spins up to the stall frequency, satisfying N_a ≈ N_gw (Bildsten 1998), then spins down in response to the mountain component of N_gw (with |N_gw| ≈ 1.5|N_a|) on the timescale ${\sim} I\Omega /N_{\rm gw} \propto \mu _{\rm i}^{-4/5}$. The latter evolution resembles the dashed contours in the top panel of Figure 2 and the dashed curves in the second row of Figure 3, as the mountain quadrupole is significant. For μ_i ≲ 1 × 10³³ G cm³, instabilities are triggered, N_a drops sharply due to the instability back-reaction (see solid curve at t ≈ 70 s in Figure 3, fourth row, right panel), and Ω ∼ 10⁴ rad s⁻¹ decays slowly under the action of N_gw and N_dip, which is much reduced by burial.

An accurate estimate of the signal-to-noise ratio (S/N) for a protomagnetar undergoing fallback requires detailed calculations of the waveform and Monte Carlo simulations of the search pipeline, both of which lie outside the scope of this paper. However, it is useful to convert the wave strain predictions in Figure 2 into a rough detectability measure, in order to clarify the relative importance of the peak strain, h_max, and signal lifetime, T_obs. To give a flavor of what is possible, we consider two extremes: (1) a matched filter search, which assumes optimistically that one can track the phase of the signal coherently for its duration, and (2) an excess cross-power search targeting a well-localized electromagnetic counterpart, which does not assume any phase model at all.

The key factors governing detectability are summarized in Table 1. Four scenarios emerge from Section 3.1, classified according to whether the maximum Ω (achieved at t ≈ t_pk) is set by magnetocentrifugal equilibrium or gravitational radiation stalling (C or G, first column), and whether gravitational wave emission is dominated by a magnetic mountain or hydrodynamic instabilities (M or I, second column).⁵ Scalings are presented for h_max and T_obs (third and fourth columns) in terms of the normalized initial magnetic dipole moment, $\tilde{\mu } = \mu _{\rm i} / (10^{33}\,{\rm G\,cm^3})$, and the fallback parameter, η. The source lifetime generally satisfies T_obs ≫ t_pk, i.e., h₀(t) rises faster than it decays. It is set by the propeller effect (T_obs ≈ IΩ/N_a, with ω ≈ 1.05) or the back-reaction from mountain gravitational radiation (T_obs ≈ IΩ/N_gw) in the scenarios C or G, respectively. The matched filter S/N is calculated from ${\rm (S/N)}_{\rm mf}^2 \approx 32 h_{\rm max}^2 T_{\rm obs} / [375 S_h(f=\Omega /\pi)]$ (e.g., Equation (13) in Vigelius & Melatos 2009b), where S_h(f) is the one-sided detector noise power spectral density at the observing frequency f. We thereby assume that most of the power is emitted at twice the spin frequency (Jaranowski et al. 1998)—almost certainly an oversimplification for instabilities and possibly also for a mountain, if it wobbles in response to vigorous accretion (see footnote 4 and Payne & Melatos 2006a). In all table entries the source distance is normalized to 1 Mpc, and (S/N)_mf is quoted in terms of $\tilde{S}_h = S_h(f=0.2\,{\rm kHz}) / (10^{-47} \, {\rm Hz^{-1}})$, with $\tilde{S}_h = 1.4$, 0.90, and 6.7 × 10⁻³ for zero-detuning high-power Advanced LIGO, neutron-star-inspiral-optimized Advanced LIGO, and the conventional Einstein Telescope, respectively (Bennett et al. 2010, and references therein). Interferometer configurations optimized for f ≲ 40 Hz (e.g., black-hole-inspiral-optimized Advanced LIGO, xylophone Einstein Telescope) are not considered in this paper, where we are interested in signals with f ≳ 0.2 kHz and hence S_h(f)∝f². The final column quantifies roughly the reduction in detection distance expected when replacing a matched filter with an excess cross-power search, based on the results in Table 1 in Piro & Thrane (2012).

Table 1. Gravitational Wave Detection: Accretion and Emission Scenarios

Spin Limita	Signalb	1023 DMpc hmaxc	Tobs (102 s)d	$\tilde{S}_h^{1/2} \,D_{\rm Mpc} \,{\rm (S/N)}_{\rm mf}$e	Dmf/Dcpf
C	M	$2.0 \tilde{\mu }^{2/7} \eta ^{60/91}$	$8.8 \tilde{\mu }^{-8/7} \eta ^{-30/91}$	$3.7 \tilde{\mu }^{4/7} \eta ^{15/91}$	∼14
C	I	$6.3 \tilde{\mu }^{3/7} \eta ^{20/91}$	As above	$12 \tilde{\mu }^{5/7} \eta ^{-25/91}$	As above
G	M	$6.7 \tilde{\mu }^{2/5} \eta ^{4/13}$	$1.2 \tilde{\mu }^{-4/5} \eta ^{-8/3}$	$3.8 \tilde{\mu }^{4/5} \eta ^{-46/39}$	∼6
G	I	$7.3 \tilde{\mu }^{2/5} \eta ^{4/13}$	As above	$4.2 \tilde{\mu }^{4/5} \eta ^{-46/39}$	As above

Notes. In all entries, $\tilde{\mu } = \mu _{\rm i} /(10^{33} \, {\rm G\,cm^3})$ denotes the normalized initial magnetic dipole moment, and η is the fallback accretion parameter defined in Section 2.1. ^aAccretion mechanism that determines the maximum value of Ω at t ≈ t_pk: magnetocentrifugal equilibrium (C) or gravitational radiation stalling (G). ^bGravitational wave emission mechanism: magnetic mountain (M) or hydrodynamic instabilities (I). ^cPeak wave strain h_max at t ≈ t_pk; D_Mpc denotes the source distance measured in Mpc. ^dSignal duration T_obs = IΩ/N_a (C) or IΩ/N_gw (G) in units of 10² s; see Section 3.2. ^eSignal-to-noise ratio (S/N)_mf for a matched filter search; $\tilde{S}_h = S_h(0.2\,{\rm kHz}) / (10^{-47}\,{\rm Hz^{-1}})$ denotes the normalized detector noise power spectral density at 0.2 kHz. ^fMatched filter (D_mf) and excess cross-power (D_cp) detection distances; ratio calibrated against Table 1 in Piro & Thrane (2012).

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Table 1 demonstrates that the prospects for detecting a nearby protomagnetar remain respectable, as stated by previous authors (Piro & Ott 2011; Piro & Thrane 2012), when the physics of magnetic funneling and burial is included. For example, a Local Group object with μ_i = 5 × 10³³ G cm³ and D = 4 Mpc reaches (S/N)_mf = 42 in a matched filter search with the Einstein Telescope just from its magnetic mountain emission, even before adding instabilities.⁶ The S/N drops to 3.6 for Advanced LIGO, still marginally detectable, while the effective detection distance is ∼6 times smaller for a more realistic excess cross-power search (see below). Table 1 contains two main trends: (1) gravitational radiation stalling leads to higher h_max and shorter T_obs than magnetocentrifugal equilibrium, so both evolutionary pathways produce similar S/Ns; and (2) the mountain and instability emission are comparable, with the latter typically being stronger and favored by higher μ_i and lower η.

The excess cross-power algorithm (Thrane et al. 2011; Prestegard et al. 2012), a generalized form of the stochastic radiometer statistic (Ballmer 2006), does not assume a priori how the phase evolves. It is therefore a fairer guide to search performance. It is implemented in two steps: one computes a spectrogram S/N(t; f) of the S/N, which is proportional to the cross-correlation of the strains at two interferometers (or a larger network more generally); then one scans the spectrogram for a contiguous track of positive-valued pixels using a clustering algorithm (after excising environmental noise artifacts), in order to get a total S/N for the track. A detection threshold of S/N ≈ 23 can be achieved realistically with Advanced LIGO, assuming a false alarm rate of 0.1%, a false dismissal rate of 50%, and a 1 ks × 1.7 kHz on-source region divided into 0.5 s × 1 Hz pixels. Tripling the on-source time increases the threshold by ≈10%.

Detailed Monte Carlo simulations of the excess cross-power S/N lie outside the scope of this paper; there is considerable uncertainty surrounding the waveforms, when hydrodynamic instabilities operate, and even the magnetic mountain is unlikely to be a static quadrupole, when $\dot{M}_{\rm a}$ is so high during fallback. Instead, we calibrate against the results presented by Piro & Thrane (2012). Four of the cases presented in Table 1 in the latter reference, with η = 1 and maximum stellar mass 2.5–2.9 M_☉, are representative of the results in Section 3.1 and Figure 2: the emission frequency ranges from 0.79 kHz to 2.3 kHz, and the duration of the burst ranges from 0.21 ks to 0.35 ks, consistent with the Ω(t) evolution underlying Figure 2. In the four cases, the ratio of detection distances for the matched filter and cross-power searches ranges from 6.2 to 14. Other examples computed by Piro & Thrane (2012) are consistent with this range; at one extreme, for η = 0.3, the duration increases to 0.64 ks and the detection distance ratio is 19. As a rough guide, we adopt these results here (final column, Table 1), by applying a 14 fold reduction to the detection distance for magnetocentrifugal equilibrium, where T_obs is longer, and a 6 fold reduction for gravitational stalling, where T_obs is shorter. When the phase evolution is unknown, and a template search is prohibitive computationally, excess cross-power performs better relative to a matched filter the longer the signal lasts, as the t–f pixel tracker asserts its advantage.