DIRECTED SEARCHES FOR BROADBAND EXTENDED GRAVITATIONAL WAVE EMISSION IN NEARBY ENERGETIC CORE-COLLAPSE SUPERNOVAE

Cosmological gamma-ray bursts (GRBs) and core-collapse supernovae (CC-SNe) are the most extreme transients in the sky. The latter are quite frequent (e.g., Cappellaro et al. 2015), about once per fifty years in the Milky Way (Diehl et al. 2006) and over once per decade in nearby galaxies such as M51 (D ≃ 8 Mpc) and M82 (D ≃ 4 Mpc). CC-SNe of Type Ib/c have a branching ratio of about 1% into normal long GRBs (LGRBs) (Della Valle et al. 2003; van Putten 2004; Della Valle 2006; Guetta & Della Valle 2007). CC-SNe are generally factories of neutron stars and black holes. A relativistic inner engine of this kind gives a unique outlook on potentially powerful emissions in gravitational waves, which may be probed by upcoming gravitational wave observations by LIGO–Virgo (Abramovici et al. 1992; Acernese et al. 2006, 2007) and KAGRA (Somiya 2012; KAGRA 2014).

Modern robotic optical surveys of the local transient universe (LTU) such as P60 (Drout et al. 2011) and LOSS (Li et al. 2011) provide an increasingly large number of nearby CC-SNe with a yield of tens of events per year within a distance of about 100 Mpc. Higher yields are expected from Pan-STARRS (Scolnic et al. 2011) and the Zwicky Transient Factory (S. R. Kulkarni 2014, private communication; Bell et al. 2015). At present, SN 2010br (z = 0.0023) (Chomiuk & Soderberg 2010; Nevski et al. 2010) exemplifies a nearby SN Ib/c discovered by traditional means in the constellation Ursa Major.

SNe Ib/c stand out as energetic events, and are aspherical and radio-loud (Mazzali et al. 2005; Taubenberger et al. 2009; Modjaz et al. 2014) featuring mildly relativistic ejecta (e.g., Soderberg et al. 2008, 2010; Della Valle 2010). Formed in the core collapse of relatively massive progenitor stars stripped of their hydrogen and helium envelope, they may originate in compact stellar binaries with intra-day periods (e.g., Paczyński 1998; Della Valle 2010; Heo et al. 2016 and references therein) or they have Wolf–Rayet progenitors (Woosley 1993; Woosley & Bloom 2006; Smartt 2009). These considerations suggest that SNe Ib/c are engine-driven. If they are driven by newly formed black holes, those with progenitors in compact binaries should be rich in angular momentum owing to conservation of angular momentum in collapse (van Putten 2004), which, in fact, may be near-extremal in producing LGRBs (van Putten 2015b). Explosions driven by angular momentum-rich central engines provide a natural candidate to account for the observed powerful aspherical explosions (Bisnovatyi-Kogan 1970), especially so with outflows taken to their relativistic limits (MacFadyen & Woosley 1999).

The SN II event SN 1987A provided first-principle evidence for the formation of high-density matter from its >10 MeV neutrino burst. Its current aspherical remnant shows that it must have been rich in angular momentum. This leaves us but one step away from emission in gravitational waves by non-axisymmetric mass motion, from fall-back matter of the progenitor envelope. This outlook is especially relevant in the presence of feedback by central engines onto matter falling in, notably so from rotating black holes (van Putten 1999; van Putten & Levinson 2003). Powerful feedback derived from an angular momentum-rich energy reservoir may drive secular instabilities and sustain long-duration emission (van Putten 2012), much beyond what would be expected outside canonical core-collapse scenarios.

Non-axisymmetric fall-back matter can hereby produce ascending chirps up to several hundred Hz (Piro & Pfahl 2007; Levinson et al. 2015), while non-axisymmetric waves from the innermost stable circular orbit (ISCO) can produce descending chirps down to several hundred Hz in the process of feedback by the black hole (van Putten 2008a). As the latter takes place on the scale of the Schwarzschild radius of the system, the total energy radiated in gravitational waves can reach a fraction of order unity of the total spin energy of the central engine, i.e., a few tenths of a solar mass (van Putten 2001). This combined outlook points to broadband extended gravitational wave emission (BEGE) from events producing black holes (Figure 1), which may further include gravitational wave emission from turbulent mass motion. Probing these events for the nature of their inner engine, therefore, requires a broadband detection algorithm sensitive to both ascending and descending chirps.

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For data on gravitational wave strain from the ISCO around rapidly rotating black holes (van Putten 2008a), signal injection experiments demonstrate a sensitivity distance D ≃ 100 Mpc at advanced LIGO sensitivity for energetic Type Ib/c events with an energy output of a few tenths of a solar mass in waves from the ISCO (van Putten 2001; van Putten et al. 2011b). This sensitivity distance gives an appreciable volume of the local universe with the aforementioned event rate of Type Ib/c events. A similar sensitivity distance may hold for accretion flows with sufficiently rapid cooling (Levinson et al. 2015, and references therein). In frequency, emission from the ISCO accurately carries detailed information on the size of the black hole as defined by the Kerr metric. Of particular interest, therefore, is identification of the associated evolution of the black hole spin in a secular change in gravitational wave frequency from matter at and about the ISCO (van Putten et al. 2011a). CC-SNe during upcoming advanced LIGO–Virgo and KAGRA observations will provide us with the means to test this hypothesis, provided they are captured sufficiently nearby and a preferably well-sampled optical supernova light curve is detected.

Here, we consider SN 2010br as an illustrative example covered by the sixth science run of LIGO (Vallisneri et al. 2015) around the time of its discovery on 2010 April 10. Its light curve is sparsely sampled, however, and it shows an absolute magnitude of M_R ≃ −12.3 (m_R ≃ 17.7 using a distance modulus of 30). It is therefore intrinsically faint or the discovery was late after maximal luminosity. Despite these caveats, SN 2010br provides a rare opportunity, given the overall event rate of SNe Ib/c of about 100 per year within a distance of 100 Mpc. SN 2010br therefore poses an interesting and rare but challenging example motivating the present development of a dedicated pipeline to search for BEGE from nearby core-collapse events, in preparation for future systematic approaches based on triggers provided by robotic optical transient surveys of the local universe (Heo et al. 2016). It exemplifies more broadly directed searches for unmodeled burst and transient sources (Aasi et al. 2015). For concreteness, we selected a period of two to four weeks before its discovery, which conceivably covers the true time of onset t₀.

We set out to probe for BEGE in SN 2010br by application of time-sliced matched filtering (TSMF), previously developed for analysis of noisy time series of TAMA 300 and BeppoSAX (van Putten et al. 2014). For high-density matter at and about the ISCO around a stellar-mass black hole as a source of gravitational waves, we focus on high frequencies and apply a bandpass filter of 350–2000 Hz (Figure 2). The quality of noise at 350–2000 Hz is markedly different from its low-frequency counterpart below 350 Hz. At high frequency, shot noise arising from finite photon counts is nearly white. It is also of much smaller amplitude than the seismic-dominated noise below 350 Hz. Measured by standard deviation (STD) of strain data from the LIGO detector at Louisiana (L1) and Hanford (H1) as shown in Figure 2, high-frequency noise is about three orders of magnitude smaller than low-frequency noise.

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In focusing on the central engine, we seek to identify gravitational wave emission that may permit complete calorimetry on the explosion process in Type Ib/c events and their associated LGRBs, which should be intimately related to the evolution of angular momentum in aforementioned accretion flows and rotating black holes. Our focus is hereby distinct from the (many) other channels of gravitational wave emissions associated with the formation and spin-down of neutrons stars, and the formation of black holes following core collapse of the former. Their gravitational wave signatures tend to be different and of relatively short duration (e.g., Rees et al. 1974; Lipunov 1983; Fryer et al. 2002; Duez et al. 2004; Lipunova et al. 2009; Ott 2009; Fryer & Kimberly 2011; Cerdá-Durán et al. 2013), and fall outside the scope and intent of the present search algorithm.

Recently, several search algorithms for chirplike behavior have been developed based on pattern recognition in Fourier-based spectrograms (Prestegard & Thrane 2009; Sutton et al. 2010; Thrane & Coughlin 2013, 2014; Abbott et al. 2016; Coughlin et al. 2015; Gossan et al. 2015). However, chirps inherently show spreading of energy in frequency space (see Figure 3), i.e., spreading over pixels in such spectrograms. (Fourier-based analysis is optimal for horizontal segments in ft-diagrams.) While allowing large data sets to be analyzed relatively fast, these methods are suboptimal for chirps, relative to the theoretical maximum permitted by matched filtering.

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Our approach is different. We consider a two-dimensional analysis, comprising a range of frequencies and their time rate of change. We endeavor to recover maximal sensitivity to conform to the theory of matched filtering in the limit of large template banks of size $O{(\tau \mathrm{max}f[0,T])}^{2},$ where τ refers to an intermediate timescale of chirp template duration in TSMF. In van Putten et al. (2014), we demonstrate that such large template banks are sufficient to extract complex broadband turbulence from noisy long-duration time series. The proposed approach, however, relies heavily on modern supercomputing for realistic searches.

In Sections 2-3, we first revisit existing evidence for LGRBs from rotating black holes and prospects for long-duration chirps in gravitational waves. We next describe our new butterfly filter in the time–frequency domain using TSMF (Sections 4-5). TSMF realizes near-maximal sensitivity in the application to noisy time series such as gamma-ray light curves of the BeppoSAX catalog (van Putten et al. 2014) and gravitational strain data (van Putten 2015b). An illustration of its extreme sensitivity to broadband signals is the identification of a broadband Kolmogorov spectrum up to 1 kHz (in the laboratory frame) in bright GRBs with a mean photon count of 1.26 photons per 0.5 ms bin. The chirp templates used are superpositions of ascending and descending chirps by superposition, thus removing any bias to the sign of the slope df/dt of a chirp. We here port this method to the noisy time series of gravitational wave strain data of LIGO S6. In Section 6, detection criteria are defined specific for BEGE, applied to two weeks of S6 data prior to SN 2010br. In Section 7, we conclude with an outlook on systematic probes of nearby events provided by robotic optical surveys.

Originating in SNe Ib/c, normal LGRBs derive from either black holes or neutron stars, the latter in the form of hot proto-neutron stars with possibly superstrong magnetized fields (Thompson 1994; Usov 1994; Metzger et al. 2011; Piro & Ott 2011; Piro & Thrane 2012). At a confidence level of greater than 4σ, SGRBs originate in mergers (van Putten et al. 2014). SGRBs should hereby be associated with black holes and especially so for the long-lasting soft extended emission (EE) in the Swift class of SGRBEEs; the final outcome of a merger of a neutron star with another neutron star or companion black hole is always the same: a stellar-mass black hole with practically the mass as the binary progenitor. The black hole from the former should be relatively rapidly rotating (e.g., Baiotti et al. 2008; van Putten 2013), whereas the spin of the latter is not expected to be very different from its original spin prior to the merger and may be diverse (van Putten 1999). As mentioned, the phenomenology of SNe Ib/c points to an explosion mechanism powered by a compact inner engine rich in angular momentum.

On this basis, evidence for LGRBs from rotating black holes derives from their association with SNe Ib/c and SGRB(EE)s in the following ways (Table 1).

• 1.  
  Universality. X-ray afterglows to LGRBs from rotating black holes point to X-ray afterglows and also to SGRBs (van Putten & Ostriker 2001), which is confirmed by weak X-ray afterglows in the Swift event GRB 050509B (Gehrels 2005) and the HETE-II event GRB 050709 (Fox et al. 2005; Hjörth et al. 2005; Villasenor et al. 2005). As a common inner engine, they also explain the observed extension of the Amati relation ${E}_{p,i}\propto {E}_{\mathrm{iso}}^{\alpha }$ for LGRBs between the energy ${E}_{p,i}$ at the maximum of the νF_ν spectrum in the rest frame of the source and the isotropic-equivalent energies E_iso (Amati et al. 2002; Amati 2006), where α ≃ 0.5, to the soft EE in the Swift class of SGRBEEs (van Putten et al. 2014). The BATSE duration T₉₀ ≃ 2 s (respectively T₉₀ ≃ 20 s) of SGRBs (LGRBs) can be explained by hyper- and suspended accretion states onto slowly (rapidly) rotating black holes (van Putten & Ostriker 2001).
• 2.  
  Long durations. The proposed feedback of rapidly rotating black holes onto matter at the ISCO (van Putten 1999; van Putten & Levinson 2003) gives canonical timescales of tens of seconds for the lifetime T_spin of initially rapidly rotating black holes, i.e., T_spin is consistent with the observed T₉₀ durations in the BATSE catalog. A model-predicted correlation ${E}_{\gamma }\propto {E}_{p,i}{T}_{90}^{\beta }$ for the true energy in gamma rays E_γ (corrected for beaming) is found in the data with T₉₀ ≃ T_spin from Swift and HETE-II (van Putten 2008b) and BATSE (Shahmoradi & Nemiroff 2015), where β ≃ 0.5. The time evolution of feedback has been tested against normalized light curves extracted from the 1493 LGRBs in the BATSE catalog, showing fits to model light curves from black holes losing angular momentum against high-density matter at the ISCO. The fits are especially tight for very long-duration events (T₉₀ > 20 s), and more so than for feedback on matter further out or for model light curves from spin-down of rapidly rotating neutron stars (van Putten & Gupta 2009; van Putten 2012). The same mechanism can account for the anomalously long durations of EE in SGRBEEs, which defy any dynamical timescale in mergers, upon associating SGRBEEs with mergers involving rapidly rotating black holes. The latter naturally derive from binary mergers of neutron stars with another neutron star or a rapidly rotating black hole companion.
• 3.  
  Ample energy reservoir. Calorimetry on the kinetic energies of supernovae associated with LGRBs reveals some hyperenergetic events. In the scenario of Bisnovatyi-Kogan (1970), a few require rotational energies E_rot in their inner engines well in excess of the maximal spin energy ${E}_{c}\lt {10}^{53}$ erg of a rapidly rotating (proto-)neutron star (van Putten et al. 2011b). Rapidly rotating neutron stars or their magnetar variety are hereby ruled out as universal inner engines to LGRBs, and most certainly so as universal inner engines to both LGRBs and SGRB(EE)s. Instead, the spin energy ${E}_{\mathrm{spin}}\;=\;6\times {10}^{54}\;{\rm{erg}}(M/10{M}_{\odot })$ ${[\mathrm{sin}(\lambda /4)/\mathrm{sin}(\pi /8)]}^{2}$ $\left(-\frac{\pi }{2}\leqslant \lambda \lt \frac{\pi }{2}\right)$ of rapidly rotating black holes can accommodate the most extreme events, even at moderate efficiencies.
• 4.  
  Intermittency in the prompt GRB light curves shows a positive correlation with brightness (Reichert et al. 2001). (A few of the smoothest less-luminous events show "fast rise exponential decay" (FRED) gamma-ray light curves (Reichert et al. 2001; van Putten & Gupta 2009).) On the shortest timescales, intermittency has been identified with the inner engine, rather than processes downstream of the ultrarelativistic outflows powering the observed gamma rays (Piran & Sari 1997; Sari & Piran 1997). Extreme luminosities naturally derive from intermittent accretion onto the putative black hole (van Putten 2015a). In the presence of, e.g., violent instabilities in the inner accretion disk or torus, the mass accumulated about the ISCO will be variable, resulting in on- and off-states described by a duty cycle τ/T given by the ratio of the durations τ of the on-state and a recurrence time T. The resulting mean luminosity of output in intermittently launched magnetic winds scales with the inverse of the duty cycle: $\langle {L}_{w}^{i}\rangle \propto \;T/\tau$.
• 5.  
  No signature of (proto-)pulsars. A broadband Kolmogorov spectrum has recently been extracted from the 2 kHz BeppoSAX light curves by high-resolution matched filtering using 8.64 million chirp templates. The results reveal a smooth extension up to a few kHz in the comoving frame of relatively bright LGRBs (van Putten et al. 2014). There is no "bump" around a few kHz, as might be expected from (proto-)pulsars.

Table 1.  Observational Evidence for Long Gamma-Ray Bursts from Rotating Black Holes

Instrument	Observation/Analysis	Result
Swift	LGRBs with no SN, SGRBEE	Mergers' extended emission (1)
	Amati spectral-energy correlation	Universal to LGRB and SGRBEE (1)
	Discovery X-ray afterglow, SGRBs	SGRB 050509B (2)
HETE-II	Discovery X-ray afterglow, SGRBs	SGRB 050709 (3)
BeppoSAX	Discovery X-ray afterglow, LGRBs	Common to LGRB and SGRB(EE) (1)
	Broadband Kolmogorov spectrum	No signature of (proto-)pulsars (4)
BATSE	${T}_{90}^{\mathrm{SGRB}}\simeq 2\;{\rm{s}}$, ${T}_{90}^{\mathrm{LGRB}}\simeq 20\;{\rm{s}}$	Short–hard, long–soft GRBs (5)
	ms time variability	Compact relativistic engines (6)
	Normalized light curves, LGRBs	BH spin-down against ISCO (7)
Optical	LGRB association to SNe Ib/c	Branching ratio < 1% (8)
	Calorimetry, SN kinetic energies	Erot > Ec in some GRB-SNe (9)

Note. (1) Revisited in van Putten (2014), (2) Gehrels (2005), (3) Villasenor et al. (2005), Fox et al. (2005), Hjörth et al. (2005), (4) van Putten et al. (2014), (5) Kouveliotou et al. (1993), (6) Sari & Piran (1997), Piran & Sari (1997), Kobayashi et al. (1997) (see also Kobayashi et al. 1997; van Putten 2000; Nakar & Piran 2002), (7) van Putten & Gupta (2009), van Putten (2012), Nathanail et al. (2015), see further van Putten (2008b), Shahmoradi & Nemiroff (2015), (8) van Putten (2004), Guetta & Della Valle (2007), (9) in the model of Bisnovatyi-Kogan (1970), the rotational energy E_res exceeds the maximal spin energy E_c of a (proto-)neutron star in some hyperenergetic events (van Putten et al. 2011a).

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In light of the above, we here consider accretion flows onto initially near-extremal black holes as the putative inner engine to LGRBs (van Putten 2015b) and, by extension, a fraction of the progenitor SNe Ib/c. Various aspects point to gravitational wave emission from non-axisymmetric mass motion.

Currently established experimental results are summarized in Table 2. Decade-long radio observations tracking the evolution of the Hulse–Taylor binary neutron star system shows orbital decay by gravitational wave emission in accord with the linearized equations of general relativity to within 1%. These gravitational wave emissions include the quadrupole and various higher harmonics arising from the strongly elliptical orbit (Peters & Mathews 1963). This observational result establishes gravitational wave emission produced by multipole mass emissions in rotating systems with an ample energy reservoir in angular momentum. It demonstrates gravitational wave emission from a Newtonian tidal field that is rotating to leading order. Mathematically, the latter acts as a source term to the linearized hyperbolic part of the Einstein equations (e.g., van Putten & Eardley 1996). As such, this mechanism is completely general and its applications should extend to rotating tidal fields in any other self-gravitating system, such as non-axisymmetric accretion flows in core-collapse events. The particular outcome will depend on the prospects for generating non-axisymmetric instabilities and the source of angular momentum driving the gravitational wave emission, i.e., orbital angular momentum, leading to contraction as in the Hulse–Taylor binary system, or spin angular momentum, leading to relaxation toward an approximately Schwarzschild spacetime of slowly rotating central engines.

3.1. Type I: Ascending chirps

Non-axisymmetric accretion flows have been widely considered to explain the observed quasi-periodic oscillations (QPO) in X-ray binaries or flaring in SgrA*, e.g., by magnetic stresses (Tagger et al. 1990; Tagger & Pellat 1999; Tagger 2001; Tagger & Melia 2006; Tagger & Varnière 2006; Lovelace & Romanova 2014), and may have counterparts in core-collapse events with potential relevance to gravitational wave emission (Gammie 2001; Kobayashi & Meszaros 2003; Mejia et al. 2005; Rice et al. 2005; Piro & Pfahl 2007; Hadley & Fernandez 2014; Lovelace & Romanova 2014), including close to the ISCO stimulated by enhanced pressure, by heating or by magnetic fields due to feedback by a rotating black hole (van Putten & Levinson 2003; Bromberg et al. 2006). Accretion onto the black hole may further excite quasi-normal mode ringing of the event horizon (e.g., Araya-Góchez 2004). For stellar-mass black holes produced in CC-SNe, however, their frequencies tend to be above the sensitivity bandwidth of ground-based detectors LIGO–Virgo and KAGRA.

In the extended accretion disk, cooling is perhaps most important in driving the formation of non-axisymmetric waves and structures. Self-gravity may hereby lead to fragmentation that, upon infall, will produce ascending chirps (Piro & Pfahl 2007). For a recent detailed discussion on these ascending chirps, see, e.g., Gossan et al. (2015). Alternatively, non-axisymmetric wave patterns of sufficient amplitude may become sufficiently luminous in gravitational waves, such that their radiation in angular momentum dominates over viscous angular momentum loss. If so, these wave patterns likewise produce ascending chirps (Levinson et al. 2015). These alternatives serve to illustrate various generic and potentially natural conditions for ascending chirps to derive from non-axisymmetric accretion flows, converting orbital angular momentum into gravitational waves.

In Levinson et al. (2015), we discuss a detailed framework for gravitational wave emission from wave patterns in accretion disks (Figure 3). A primary control parameter is the efficiency in angular momentum loss by quadrupole gravitational radiation over viscosity-mediated transport, parameterized by

$$\begin{eqnarray}&&P=\displaystyle \frac{\xi }{\alpha },\end{eqnarray} \tag{ 1 }$$

where ξ is a dimensionless amplitude of a quadrupole mass inhomogeneity and α refers to the α-disk model, parameterizing the kinematic viscosity $\nu \;=\;\alpha {c}_{s}H$ in terms of the isothermal sound speed c_s and the vertical scale height $H(r)=\eta r$ of the disk. In what follows, we shall write $\alpha \;=\;0.1{\alpha }_{-1}$, η = 0.1η₋₁, $P=10{P}_{1}$ in the inner region about the black hole of mass $M=10{M}_{1}\;{M}_{\odot }$ at a distance $D={D}_{2}\;\times 100\;{\rm{Mpc}}$. The resulting spectrum of the dimensionless characteristic strain amplitude h_char(f) is parameterized by a break set by the critical radius r_b (if greater than the ISCO radius r_ISCO), within which gravitational radiation losses dominate over outward viscous transport in angular momentum. Figure 1 shows the characteristic break frequency f_b in the turn over of the green curves, satisfying

$$\begin{eqnarray}&&{f}_{b}=430\;{\eta }_{-1}^{9/2}{M}_{1}^{-1}{P}_{-1}^{-3}{\left(\displaystyle \frac{\dot{M}}{{M}_{\odot }{{\rm{s}}}^{-1}}\right)}^{-3/2}\;{\rm{Hz}}.\end{eqnarray} \tag{ 2 }$$

The condition ${r}_{b}\gt {r}_{\mathrm{ISCO}}$ implies ${f}_{b}\lt 2{f}_{\mathrm{ISCO}}$, where for a 10 M_⊙ black hole $430\ {\rm{Hz}}\lt 2{f}_{\mathrm{ISCO}}\lt 3000\ {\rm{Hz}}$, depending on the spin parameter a/M of the black hole, with $2{f}_{\mathrm{ISCO}}\;=\;1600\;{\rm{Hz}}$ for a/M = 0.95 as an example. From the orientation-averaged characteristic strain amplitude (Flanagan & Hughes 1998; Cutler & Thorne 2002)

$$\begin{eqnarray}&&{h}_{\mathrm{char}}(f)=\displaystyle \frac{\sqrt{2}}{\pi D}\sqrt{\left|\displaystyle \frac{{\rm{\Delta }}E}{{\rm{\Delta }}f}\right|},\end{eqnarray} \tag{ 3 }$$

where ${\rm{\Delta }}E/{\rm{\Delta }}f$ is the one-sided spectral-energy density at gravitational wave frequency f, we have

$$\begin{eqnarray}{h}_{\mathrm{char}}(f)=\kappa \left\{\begin{array}{ll}{\left(\displaystyle \frac{f}{{f}_{b}}\right)}^{1/6} & (f\lt {f}_{b})\\ {\left(\displaystyle \frac{f}{{f}_{b}}\right)}^{-1/6}\sqrt{2-{\left(\displaystyle \frac{{f}_{b}}{f}\right)}^{2/3}} & ({f}_{\mathrm{isco}}\gt f\gt {f}_{b})\end{array}\right.\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&\kappa \;=\;\sqrt{8}\times {10}^{-22}{D}_{2}^{-1}{M}_{1}^{1/3}{\left(\displaystyle \frac{\dot{M}\tau }{0.1{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{f}_{b}}{1000\mathrm{Hz}}\right)}^{-1/6}\end{eqnarray} \tag{ 5 }$$

in terms of the mass accretion rate $\dot{M}$ and a lifetime τ of the disk pattern. Equation (4) applies generally, also to ${f}_{b}\gt {f}_{\mathrm{ISCO}}$ for which ${h}_{\mathrm{char}}\;=\;\kappa {(f/{f}_{b})}^{1/6}$ in the range of f less than or equal to the quadrupole gravitational wave frequency at the ISCO.

3.2. Type II: Descending Chirps

The gravitational wave luminosity of a mass moment I_lm in a torus about the ISCO, expressed in terms of the quantum numbers l and m of spherical harmonics, satisfies (Thorne 1980; Bromberg et al. 2006)

$$\begin{eqnarray}&&{L}_{{GW}}\propto {{\rm{\Omega }}}_{T}^{2\;l+2}{I}_{{lm}}^{2},\end{eqnarray} \tag{ 6 }$$

where m = l is the most luminous (Bromberg et al. 2006). To leading order ${{\rm{\Omega }}}_{T}^{2}\propto {r}^{-3}$, thus (6) satisfies

$$\begin{eqnarray}&&{L}_{{GW}}\propto \displaystyle \frac{1}{{r}^{m+3}}.\end{eqnarray} \tag{ 7 }$$

Gravitational wave luminosity hereby tends to reach maximum at the ISCO. We here attribute I_lm to instabilities caused by enhanced thermal and magnetic pressures induced by feedback from the black hole (van Putten 2002; Bromberg et al. 2006), provided it spins rapidly $({{\rm{\Omega }}}_{H}\gt {{\rm{\Omega }}}_{T})$. To be more precise, the feedback is envisioned to exceed the luminosity in magnetic winds with ${{\rm{\Omega }}}_{H}/{{\rm{\Omega }}}_{\mathrm{ISCO}}\gt 1$ (up to 1.4396 according to the Kerr metric). For instance, if the solution of Shakura & Sunyaev (1973) serves as a leading-order approximation of the inner disk, the total energy output E_H onto the ISCO exceeds energy losses ${E}_{w}^{*}$ in magnetic winds whenever $a/M\geqslant 0.4433$ (van Putten 2012). Equivalently, the initial rotational energy merely exceeds the maximal rotational energy by about 9%, which is a rather mild condition. The excess ${E}_{H}-{E}_{w}^{*}\gt 0$ is available to gravitational waves and MeV neutrinos. The gravitational wave spectra from these instabilities tend to be dominated by the lowest order multipole mass moments. Consequently, a dominant output in gravitational waves is expected in quadrupole emission from the ISCO. Consider the quadrupole emission formula (Peters & Mathews 1963): in geometrical units,

$$\begin{eqnarray}&&{L}_{{GW}}\;=\;\displaystyle \frac{32}{5}{(\mu {\rm{\Omega }})}^{\displaystyle \frac{10}{3}}\end{eqnarray} \tag{ 8 }$$

scaled to ${L}_{0}={c}^{5}/G=3.64\times {10}^{59}$ erg s⁻¹, where Ω denotes the orbital angular velocity and μ the chirp mass. Expanding into small mass perturbations $\delta m$ and ignoring gray-body factors that may arise from proximity to the black hole, for a dimensionless inhomogeneity $\xi \;=\;\delta m/{M}_{T}$ and a torus of mass $\sigma \;=\;{M}_{T}/M$ at r_ISCO around a black hole of mass M, (8) reduces to

$$\begin{eqnarray}&&{L}_{{GW}}\;=\;2\times {10}^{51}\;{\left(\displaystyle \frac{\xi }{0.1}\right)}^{2}{\left(\displaystyle \frac{\sigma }{0.01}\right)}^{2}{\left(\displaystyle \frac{4M}{{r}_{\mathrm{ISCO}}}\right)}^{5}\;{\text{erg s}}^{-1}.\end{eqnarray} \tag{ 9 }$$

The observed instantaneous dimensionless strain at a source distance D satisfies

$$\begin{eqnarray}&&h=0.7\times {10}^{-23}\;{M}_{1}{D}_{2}^{-1}\;\left(\displaystyle \frac{\xi }{0.1}\right)\left(\displaystyle \frac{\sigma }{0.01}\right){\left(\displaystyle \frac{f}{600{\rm{Hz}}}\right)}^{\frac{2}{3}},\end{eqnarray} \tag{ 10 }$$

where $f=2{f}_{\mathrm{orb}}$ in the Newtonian approximation $2\pi {f}_{\mathrm{orb}}\;=\;{M}^{-1}{(M/a)}^{3/2}$ and $M={M}_{1}\;\times 10{M}_{\odot }$.

Mediated by relativistic frame-dragging, rotating black holes may sustain (9) for extended durations by sustained feedback onto matter at the ISCO via an inner torus magnetosphere (van Putten 1999). Non-relativistic frame-dragging has recently been established experimentally by LAGEOS-II and Gravity Probe B. Feedback will be particularly prominent in the presence of intermittencies (van Putten 2015a), e.g., arising from various instabilities in accretion flows and the feedback mechanism itself through an inner torus magnetosphere. In deriving (9) from angular momentum in spin of the black hole, the latter will gradually slow down with an accompanying expansion of the ISCO. The result is a descending chirp.

Exact solutions to such descending chirps derive from solutions to feedback in the Kerr metric (van Putten 2008a, 2012), defined by the initial-value problem of conservation of total energy and angular momentum:

$$\begin{eqnarray}&&\dot{M}\;=\;{{\rm{\Omega }}}_{T}\dot{J},\;\;\dot{J}\;=\;-\kappa {e}_{k}({{\rm{\Omega }}}_{H}-{{\rm{\Omega }}}_{T}),\end{eqnarray} \tag{ 11 }$$

where Ω_H and Ω_T denote the angular momentum of, respectively, the black hole and a torus in suspended accretion at the ISCO. Normalized light curves of LGRBs in BATSE point to initially near-extremal black holes (van Putten 2015b). Important to a rapidly rotating black hole is its lowest energy state that preserves the maximal horizon flux by an equilibrium value of Carter's magnetic moment (Carter 1968). Thus, $\kappa \;\simeq \;{uM}$ describes the strength of the feedback, parameterized by the ratio $u\;\simeq \;1/15$ of the total energy ${E}_{B,p}$ in the poloidal magnetic field relative to the kinetic energy ${E}_{k}$ of a torus at the ISCO (van Putten & Levinson 2003). The energy per unit mass satisfies ${e}_{k}\simeq \frac{1}{2}{v}^{2}e$, where $v={Mz}{{\rm{\Omega }}}_{\mathrm{ISCO}}$ and e denotes the specific energy at the ISCO in the Kerr metric (Bardeen et al. 1972).

In re-radiation of input received by (11), matter at the ISCO effectively acts as a catalytic converter of the rotational energy of the black hole, wherein the amplitude ξ in (9)–(10) is determined self-consistently (van Putten 2012). Numerical integration of (11) hereby defines descending chirps based on the evolution of the orbital frequency f_ISCO(t) at the ISCO as a function of time t. Specifically, multipole mass moments at the ISCO radiate at multiples ${{mf}}_{\mathrm{ISCO}}$(m = 2, 3, ...), e.g., from ISCO waves (van Putten 2002).

According to (11), solutions are determined by two fixed points, namely, when the angular velocity of the black hole and that of matter at the ISCO are the same: initially at t = 0 and at late times as t. The first features an initial strengthening in the feedback process, the latter an essentially exponential decay in frequency with time. By the second fixed point, the gravitational wave frequency f(t) as a function of time and total duration T is well approximated by an overall exponential decay in frequency,

$$\begin{eqnarray}&&{f}_{\mathrm{ISCO}}(t)={f}_{1}+({f}_{0}-{f}_{1}){e}^{-{at}/T}\end{eqnarray} \tag{ 12 }$$

where a is a dimensionless scale. The late-time asymptotic frequency of a black hole of initial mass M satisfies a limited range (van Putten et al. 2011b):

$$\begin{eqnarray}&&595\;{\rm{Hz}}\left(\displaystyle \frac{10{M}_{\odot }}{M}\right)\leqslant {f}_{{GW}}\leqslant 704\;{\rm{Hz}}\left(\displaystyle \frac{10{M}_{\odot }}{M}\right)\end{eqnarray} \tag{ 13 }$$

for ${f}_{{GW}}\;=\;2{f}_{\mathrm{ISCO}}$ due to a finite dependence of the total energy output on the initial black hole spin. (The lowest frequency corresponds to an initially extremal black hole with maximal energy output.) Asymptotic analysis shows a dominant emission in gravitational waves over any accompanying output in jets, MeV neutrinos, and magnetic winds (van Putten & Levinson 2003; van Putten 2015b). The total energy output typically reaches a fraction of order unity of the initial rotational energy of the black hole.

Figure 3 illustrates the combined outlook of ascending and descending chirps.

The characteristics of BEGE include (a) trajectories f(t) in time–frequency space that are non-constant in frequency, i.e., with finite slopes df/dt, and (b) a probable loss of phase coherence over a large number of wave periods, due to its origin in (magneto)hydrodynamic mass motion. This suggests searches for BEGE by applying a bandpass filter to df/dt, here in the form of finite bandwidths of chirp templates of intermediate duration $\tau \;=\;1\;{\rm{s}}$, for correlation with strain data by application of aforementioned TSMF. Our chirp templates are extracted by time-slicing a long-duration model chirp.

By using superpositions of ascending and descending chirps, our complex chirp templates have no bias to the sign of slope in the slope df/dt of a chirp (van Putten et al. 2014). When using banks of millions of templates, TSMF densely covers a range in frequency and a range in the time rate of change of frequency. This power of TSMF is demonstrated by identifying complex and broadband Kolmogorov spectra in light curves of LGRB up to 1 kHz (in the observer's frame of reference), extracted from BeppoSAX light curves with, on average, merely 1.26 photons in each bin of 500 μs.

In the present application to LIGO time series, we partition LIGO data into frames of one minute, comprising 64 s of $n={2}^{18}$ samples. At a downsampled frequency of 4096 Hz, the LIGO Open Science Center data give a sensitivity bandwidth of 0–2000 Hz.

With $\tau \;=\;1\;{\rm{s}}$, each choice of model parameters f₀ and T gives 64 chirp templates by time-slicing, having a mean frequency f_i and bandwidth ${B}_{i}={\rm{\Delta }}{f}_{i}$, where $i=1,2,\ldots \;64$. The distribution of chirps represents a cover of the time–frequency plane (t, f) with "butterflies" $(f,{df}/{dt})$, defined by a frequency f at its vertex and a range of slopes df/dt defined by the template bank (Figure 4). This approach enables the capture of a wide variety of trajectories in the (t, f)-plane comprising ascending and descending chirps. A recent demonstration of the power of this approach is the identification of broadband turbulence in the noisy GRB time series of LGRBs from the BeppoSAX catalog (van Putten et al. 2014). Fourier transforms correspond to the degenerate case of butterflies with zero opening angle. In what follows, SNR refers to signal-to-noise ratio, and in a slight abuse of notation SN(t) and SN(f) refer to SNR as a function of time and frequency, respectively.

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TSMF calculates correlations SN(t) by convolving chirp templates with strain from normalized Pearson coefficients. For chirp templates of intermediate duration $\tau \;=\;1$ s applied to LIGO frames of 2¹⁸ samples, these correlations satisfy a truncated distribution that is close to Gaussian with unit variance, where the truncation is set by the correlation length n. These truncations satisfy a skewed Gaussian with a probabilty density function (PDF, Figure 5)

$$\begin{eqnarray}&&p(x)=\frac{n}{\sigma }\sqrt{\displaystyle \frac{2\;}{\pi }}{\rm{erf}}{(y)}^{n-1}{e}^{-{y}^{2}},\end{eqnarray} \tag{ 14 }$$

with $y=x/\sqrt{2}\sigma$ for a standard deviation σ in the data, derived from the probability $P(\lt x)={\int }_{0}^{x}p(s){ds}$ that all n correlations are less than x.

For a given LIGO frame and model parameters, TSMF identifies, out of 64 one-second slices (4096 samples each), the chirp templates with maximal correlation SN(f), where f denotes the frequency of the associated chirp (Figure 4). Thus, SN(f) is essentially uniform in frequency (white), and its PDF is a skewed Gaussian (Figure 5). It should be emphasized that SN(f) is the pseudo-spectrum, in contrast to a real spectrum defined by the average $\langle \mathrm{SN}(f)$ $/\sqrt{{\rm{\Delta }}f}\rangle$ over chirps of frequency f with bandwidth ${\rm{\Delta }}f$ (van Putten et al. 2014).

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To efficiently search for BEGE, we propose a two-step process in which a coarse-grained search identifies epochs of interest for fine-grained searches using a large number of chirp templates. For the signals of interest, the latter approaches the theoretical limit of sensitivity of matched filtering using up to 10 million chirp templates (van Putten et al. 2014; van Putten 2015b). A relatively coarse-grained search using fewer templates will enable an initial scan of epochs of long duration, e.g., weeks or more. Here, we defined coarse- and fine-grained searches by the number of scaled parameters in the long-duration model chirps (12). Preferred parameters are, for instance, overall durations T and scaling of the frequencies f₀ and f₁ in (12). Upon scaling of T only, Figure 6 demonstrates the sensitivity using 64,000 chirp templates. It is suboptimal by a factor of about 0.6 relative to aforementioned fine-grained searches.

To identify events of interest at SN > 6, the coarse-grained search shown in Figure 6 is suitable to scan a large epoch such as weeks prior to a nearby CC-SN.

Matched filtering gives a theoretical upper bound for the effective dimensionless strain, satisfying (van Putten 2001)

$$\begin{eqnarray}&&{h}_{\mathrm{eff}}\simeq \left(\displaystyle \frac{{M}_{1}}{{D}_{2}}\right){\left(\displaystyle \frac{{E}_{-1}^{{GW}}}{{M}_{1}}\right)}^{\displaystyle \frac{1}{2}}\simeq {10}^{-21},\end{eqnarray} \tag{ 15 }$$

where $D=100{D}_{2}$ Mpc, $M=10{M}_{1}\;{M}_{\odot }$, and ${E}^{{GW}}\;=\;0.1{E}_{-1}^{{GW}}{M}_{\odot }$ denotes the energy output in gravitational waves. The theoretical upper bound is based on a perfect match to a model template over the full duration of the burst in the face of Gaussian noise. In TSMF, matched filtering is partitioned over intermediate time intervals τ of phase coherence, since no (predictable) phase coherence is expected over the full duration T of the burst for the present (magneto)hydrodynamic source under consideration. Results for single chirp templates of duration τ hereby define a partition over $N={T}_{90}/\tau$ time slices, each satisfying (van Putten et al. 2011b; van Putten 2015b)

$$\begin{eqnarray}&&{h}_{\mathrm{eff}}^{\prime }\simeq \displaystyle \frac{1}{\sqrt{N}}{h}_{\mathrm{eff}}\sim {10}^{-22}\end{eqnarray} \tag{ 16 }$$

when T matches T₉₀ of the burst, here identified with the durations of tens of seconds of LGRBs.

To combine results (16) obtained from different slices, consider the partition of a time interval [0, T] in intervals ${I}_{i}=[{t}_{i-1},{t}_{i}]({t}_{i}=(i-1){\rm{\Delta }}t$, $i=1,2,\cdots \;,\;N$), and slicing of a long-duration model chirp f(t) into N chirp templates on the I_i. When f(t) is strictly monotonic, these chirp templates have non-overlapping frequency intervals ${F}_{i}=[{f}_{0i},{f}_{1i}]$, ${f}_{0i}\;=\;\mathrm{min}f[I],{f}_{1i}\;=\;\mathrm{max}f({I}_{i})$. The joint probabilities over different slices are hereby joint probabilities over different frequency bins. Since LIGO shot noise is essentially white (Figure 5), matched filtering on the I_i will be statistically independent. Joint probabilities over various I_i hereby reduce to ordinary probability products. The equivalent σ_tot to N confidence levels σ_i in candidate detectors on the I_i is thus derived from the associated probabilities p_i as

$$\begin{eqnarray}{\sigma }_{\mathrm{tot}} & = & \sqrt{2}\;{{\rm{erfc}}}^{-1}\left(\displaystyle \prod _{i=1}^{N}{p}_{i}\right),\;\;{p}_{i}={\rm{erfc}}\left(\displaystyle \frac{{\sigma }_{i}}{\sqrt{2}}\right),\\ \langle {\sigma }_{i}\rangle & = & \displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\;{\sigma }_{i},\end{eqnarray} \tag{ 17 }$$

where the latter expresses the mean of the confidence levels σ_i for each time slice. Numerical evaluation shows to good approximation (Figure 7) that

$$\begin{eqnarray}&&{\sigma }_{\mathrm{tot}}\simeq \sqrt{N}\langle {\sigma }_{i}\rangle \end{eqnarray} \tag{ 18 }$$

for the total confidence level. (The right-hand side is a lower bound on σ_tot.) This result recovers the familiar scaling with the square root of the number of wave periods in matched filtering applied over the original time interval [0, T]. In searches for strictly monotonic chirps, we consider the output of TSMF in terms of a pseudo-spectrum shown in Figure 5, expressed by confidence levels σ_i over the adjacent frequency intervals F_i. Consequently, TMSF recovers the ideal sensitivity limit (15) at full resolution in a bank of about ${(\tau \mathrm{max}f[0,T])}^{2}$templates.

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BEGE originates from a secular evolution of the putative inner engine of a CC-SN event. It hereby describes broadband emissions that cover a certain frequency range. A sufficiently strong signal should appear in both L1 and H1 with a correlation in the SN(f) over a certain frequency bandwidth. BEGE can be searched for by first selecting individual events from L1 and H1 in the tail of their distributions of SN(f), then searching for coincidences. Figure 8 shows a scatter plot of SN(f) > 6 in each detector, which comprises a few tenths of 1% of the data. Coincidences on the timescale of minutes reduce this to about nine events out of a total of about 20,000 minutes covering two weeks of data. However, there are no multiple events covering a certain bandwidth within any one-minute epoch that might suggest a genuine BEGE signal. Nor is there any correlation in these nine SN(f) values from L1 and H1, even though L1 and H1 are very similar. For this reason, these coincidences appear to be spurious events of instrumental origin (including signal injection tests).

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Modern robotic optical surveys promise to provide a wealth of nearby CC-SNe events, which provide attractive opportunities for directed searches of their potential emission in gravitational waves associated with newborn neutron stars and black holes. In particular, SNe Ib/c stand out for being more likely to produce black holes than neutron stars by their association with LGRBs, calorimetry on some of the hyperenergetic events, and the seamless unification of LGRBs and SGRB(EE)s by the inner engines of black holes. Their event rate is about two orders of magnitude greater than that for neutron stars, making any gravitational wave emission from fall-back matter onto rotating black holes an attractive opportunity for LIGO–Virgo and KAGRA. At a distance of about 10 Mpc, SN 2010br provides a challenging example of an exceptionally nearby event.

Gravitational radiation from astrophysical sources generally represents conversion of angular momentum. Ascending and descending chirps result from conversion of orbital and spin angular momentum, respectively. In fall-back matter onto the ISCO of a rotating black hole, they may meet and partially overlap at about a few hundred Hz, i.e., in the shot-noise-dominated sensitivity bandwidth of LIGO–Virgo and KAGRA, and may be long-lasting in representing viscous processes in accretion and feedback from the central black hole.

To search for BEGE, we here presented a dedicated pipeline to search in nearby events. As (magneto)hydrodynamic sources of gravitational waves, phase coherence will be limited to intermediate timescales, representing tens or at most hundreds of wave periods associated with orbiting in an accretion disk. To search for long-duration chirps of either sign, we apply a butterfly filter in the time–frequency domain in the form of a bandpass for the slope df/dt by application of TSMF, using a large bank of chirp templates of intermediate duration τ each with finite bandwidth. Here, τ is generally chosen to be of the order of 0.1–1 s.

For concreteness, we here considered the nearby event SN 2010br. It may have been intrinsically weak or it was discovered rather late after its true time of onset. We applied our search method to two weeks of S6 data starting one month prior to the Type Ib/c SN 2010br. With a chirp resolution in 1024 steps, TSMF comprised 64,000 chirp templates of various frequencies and bandwidths. Our two-week analysis of both L1 and H1 takes about one month of computation on a 64 core mini-supercomputer by embarrassingly parallel computing (e.g., Foster 1995). Our task was distributed over various personal computers using a cloud operating system originally developed for remote sensing (van Putten & van Putten 2007); any other software for distributed computing on multicore systems (e.g., Mighell 2012; Singh et al. 2013) will serve the same purpose—to realize essentially optimal sensitivity by exploiting a diversity in modern supercomputing configurations.

Critical to a potentially successful probe is a well-determined true time of onset t₀ obtained from a well-sampled optical light curve, preferably with an uncertainty of less than one week. With the poorly sampled light curve of SN 2010br, however, we face considerable uncertainty in t₀. This motivated the development of a two-step search, starting with a coarse-grained TSMF to identify events of interest, here defined by S/N > 6. These events are downselected by correlating the results from L1 and H1, here down to 9 and 23 events for template banks of 32,000 and 64,000 chirps, respectively.

These particular events can be followed up by in-depth searches, i.e., (i) a two-dimensional scaling in duration and frequency of the model templates (12) and (ii) coincidence analysis, by further considering time delays between L1 and H1 detections at specific chirp templates that should reflect the geographic distance between the two detector sites. The first is exemplified in van Putten (2015b). The latter is readily included by appending to the TSMF output the offset $\delta t$, resolved down to the sampling time of 1/4096 s in SN(t) in the slices that produce the maximal SN(f).

In the present case, however, the (economized, coarse-grained) coincidence events in Figure 8 do not carry any signature of a common excitation of L1 and H1 by a broadband gravitational wave signal. Rather than further analysis (using a larger bank of chirp templates) of these particular events, it appears that a search epoch longer than the present two-week period prior to SN 2010br may be opportune.

The method presented here is proposed as a new pipeline for systematic probes for BEGE from nearby energetic core-collapse events provided by robotic optical surveys in the upcoming era of advanced LIGO–Virgo and KAGRA. A detection of BEGE promises identification of their inner engine by complete calorimetry on their energetic output.

The author thanks the referees for constructive comments. This research is based in part on a Basic Research Grant (2015) from the Korean National Research Foundation and made use of LIGO S6 data from the LIGO Open Science Center (losc.ligo.org), provided by the LIGO Laboratory and LIGO Scientific Collaboration. LIGO is funded by the U.S. National Science Foundation. Some of this work was supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, MEXT Grant-in-Aid for Scientific Research on Innovative Areas 24103005, JSPS Core-to-Core Program, A. Advanced Research Networks, and the joint research program of the Institute for Cosmic Ray Research, University of Tokyo. Parallel computations were performed using a Cloud OS VPGEONET.

Additional supporting information is available online (van Putten 2016), include the following:

ISCOWAVES: Fortran program on bifurcation-diagram ISCO waves.

ISCOCHIRP: MatLab program on descending ISCO chirps for injection experiments and TSMF.