FIRST SEARCH FOR GRAVITATIONAL WAVES FROM THE YOUNGEST KNOWN NEUTRON STAR

The Cas A CCO was first discovered as an X-ray point source in first-light images taken by the Chandra X-ray Observatory (Tananbaum 1999). It was subsequently identified in other satellite data dating back to the year 1979; the X-ray flux appears to have been constant since then (Pavlov et al. 2000). Optical and near-infrared searches have not found the CCO, and have all but ruled out the presence of an accretion disk from fallback, and a binary companion (Ryan et al. 2001; Kaplan et al. 2001; Fesen et al. 2006a; Wang et al. 2007). The absence of the latter is puzzling, since the light-echo spectrum of the supernova, of type IIb, implies that the progenitor was stripped of hydrogen by a companion (Young et al. 2006; Krause et al. 2008). One possibility is that the progenitor was the product of binary companions merging during a common envelope phase (Krause et al. 2008).

The CCO appears to be a neutron star with a low surface magnetic field (an anti-magnetar). Blackbody fits to the X-ray spectrum by Pavlov et al. (2000) and Chakrabarty et al. (2001) implied a temperature too high, and an emitting area too small, to be consistent with emission from the whole surface of a cooling neutron star, or from the inner region of an accretion disk around a black hole. Pavlov et al. (2000) proposed a model of a strongly magnetized neutron star (a magnetar) with hot polar caps, which would, however, lead to X-ray pulsations which have not been observed (see below). Light echoes from the explosion have been interpreted as signs of a flare, reminiscent of a soft gamma repeater, occurring in the year 1953 (Krause et al. 2005). Such a flare, if single, could also have been due to a one-time phase transition (Gotthelf & Halpern 2008). More recently, however, this interpretation of the echoes, and thus the evidence for the flare, has been discounted (Kim et al. 2008; Dwek & Arendt 2008). Recently, Ho & Heinke (2009) combined previous X-ray spectra (Hwang et al. 2004; Pavlov & Luna 2009), carefully adjusted for instrumental effects, and fitted them to various light-element atmosphere models. Their best fit was for nearly isotropic emission from the entire surface of a neutron star with a carbon atmosphere, low magnetic field, and mass and radius within normal ranges.

The spin period of the neutron star is unknown. McLaughlin et al. (2001) searched for radio pulses (including possible binary orbital periods as short as a few hours) and found none at periods as short as 1–10 ms, depending on dispersion measure, making the CCO much more radio quiet than any known radio pulsar under 10⁴ years old. Searches for X-ray pulsations at periods as short as 2 ms (Chakrabarty et al. 2001; Murray et al. 2002; Mereghetti et al. 2002; Pavlov & Luna 2009) have produced at best a marginal candidate for a period at 12 ms, which has not been confirmed. No pulsar wind nebula has been detected (Hwang et al. 2004; Pavlov & Luna 2009).

If the CCO in Cas A is an anti-magnetar, it may be spinning fast enough to emit periodic gravitational waves above 100 Hz, where LIGO is most sensitive.

To date, seven supernova remnant CCOs are known: three have observed spin periods and a fourth has an observed periodicity that may be due to binary orbital motion (De Luca 2008; Gotthelf & Halpern 2009). The fastest-spinning CCOs have rotation periods of ∼100 ms, which are much slower than the longest rotation periods covered by this search (20 ms for gravitational waves from a non-axisymmetric distortion and 13 ms for r-modes; see Section 3). Only one of the CCOs has a measurable spin-down (Halpern & Gotthelf 2010) and the others have tight upper limits. This indicates that the CCOs have magnetic fields much less than those of typical radio pulsars, and rotation periods that have not changed significantly since birth. If Cas A spins as slowly and constantly as these objects, it is not detectable by LIGO. It is difficult, however, to definitively extrapolate the general properties of CCOs from a sample of three or four.

Young neutron stars, such as Cas A, may be the most likely to retain non-axisymmetries from the violent circumstances of their births. For example, the formation of the crust during an epoch of perturbations such as r-modes (Lindblom et al. 2000; Wu et al. 2001) could lead to an irregular shape. Gravitational radiation may also be generated by the continuing non-axisymmetry of the r-modes themselves (Owen et al. 1998), which may last for up to thousands of years (Arras et al. 2003) depending on the composition and viscosity of the star.

Indirect upper limits on gravitational-wave emission from Cas A can be estimated using a method similar to the spin-down limit for known pulsars (Wette et al. 2008; Owen 2010). If the star was born spinning at least ∼20% more rapidly than it is now, and the spin-down evolution has been dominated by the emission of gravitational waves, the unknown spin frequency and frequency derivative can be eliminated in favor of the known age to place a rough upper limit on the gravitational-wave emission. For Cas A, the indirect limit on the intrinsic gravitational-wave strain h₀ (see Section 2.2) is

$$\begin{equation} h_0 \lesssim 1.2 \times 10^{-24} D_{3.4}^{-1} \tau _{300}^{-1/2} I_{45}^{1/2} \,, \end{equation} \tag{ 1 }$$

where D_3.4 is the distance to Cas A in units of 3.4 kpc, τ₃₀₀ is its age in units of 300 yr, and I₄₅ is its principal moment of inertia in units of 10⁴⁵gcm². The indirect limit on h₀ is independent of frequency.

The choice of a fiducial age of 300 yr for Cas A, at the young end of the range estimated by Fesen et al. (2006b), is conservative in that it gives a larger search parameter space (see Section 2.3). It also raises the indirect limit by ∼10%, a small effect compared to the uncertainties in the distance (of order 10%), and the principal moment of inertia, which may be up to three times higher than its fiducial value (see Abbott et al. 2007c). The uncertainties in the direct upper limits presented in Section 3 are on the order of 10%–15%, due to uncertainties in the calibration of the LIGO detectors (of order 10%; see Section 2.1), and systematic uncertainties in the search pipeline (of order 5%; see Section 2.2). Equation (1) assumes gravitational waves from a mass quadrupole; the equivalent limit for r-modes is higher by tens of percent (Owen 2010).

The indirect limit on h₀ may be converted into an indirect limit on the equatorial ellipticity

$$\begin{equation} \epsilon \lesssim 3.9 \times 10^{-4} \tau _{300}^{-1/2} I_{45}^{-1/2} f_{100}^{-2} \,, \end{equation} \tag{ 2 }$$

where f₁₀₀ is the gravitational-wave frequency in units of 100 Hz. Due to the uncertainties in neutron star parameters mentioned above, this limit on is overall uncertain by roughly a factor two. The indirect limit on h₀ also implies an indirect limit on gravitational waves from r-modes (Owen 2010), in terms of their amplitude

$$\begin{equation} \alpha \lesssim 0.14 \tau _{300}^{-1/2} f_{100}^{-3} \,, \end{equation} \tag{ 3 }$$

where the uncertainty due to the moment of inertia and other properties of the star, while more complicated, is roughly a factor of 2–3. If α varies over the observation time, this limit applies to the rms value of α over time. See Owen (2010) for precise definitions of α and , translations to other quantities used in the literature (including h₀), and more discussion of uncertainties.

Wette et al. (2008) showed that a search for Cas A of 12 days of LIGO S5 data in the band 100–300 Hz is feasible and can expect to beat the indirect limits. The Cas A indirect limits on h₀ and are comparable to the spin-down limits for the Crab pulsar, which have been beaten by searches of LIGO S5 data (Abbott et al. 2008b, 2010). The indirect limits on α are lower and therefore more interesting than for the Crab if Cas A is emitting at 100–300 Hz. The indirect limits are also comparable to the best upper limits achieved by all-sky searches for periodic gravitational waves (Abbott et al. 2009b, 2009c). The all-sky searches covered longer spin-down timescales and thus were not sensitive to Cas A if it is emitting gravitational waves near the indirect limit.

LIGO is a network of three interferometric detectors: a 4 km arm-length detector in Livingston, Louisiana (L1) and two detectors of 4 km (H1) and 2 km (H2) arm lengths co-located in Hanford, Washington.

The S5 science run (Abbott et al. 2009d) is LIGO's fifth and most recently completed science run. It commenced at 2005 November 4, 16:00 UTC at Hanford, and at 2005 November 14, 16:00 UTC at Livingston; it ended at 2007 October 1, 00:00 UTC. The S5 run collected over one year of science data coincident among all three detectors, with an overall triple-coincidence duty cycle of 54%. Interruptions caused by environmental disturbances, as well as scheduled breaks for maintenance and commissioning of equipment, accounted for the downtime. During S5, the detectors were operating at very near their design sensitivities. The strain noise of the two 4 km detectors was on average less than 3 × 10⁻²³Hz^−1/2 at their most sensitive frequencies (around 140 Hz) and less than 5 × 10⁻²³Hz^−1/2 over 100–300 Hz, and generally improved (as did the duty cycle) over the course of the run.

This search uses science data from only the L1 and H1 detectors. Data from the H2 detector are less sensitive, but carry the same computational cost to search. A small percentage of the acquired science data are excluded by data quality controls, which identify times when the data are known to be unsuitable for analysis. This includes, e.g., data taken when the output photodiodes of an interferometer were saturated, when the calibration of the data was ill-defined, when high winds were measured at the Hanford observatory, and 30 s before an interferometer lost lock. The remaining science data are calibrated (Abbott et al. 2010, 2009d; Abadie et al. 2010) to produce a (discontinuous) time series of gravitational-wave strain, h(t). The time series is then broken into 30 minute segments. Because not every continuous section of h(t) is an integer multiple of 30 minutes in length, some science data are discarded in this process. Finally, each 30 minute segment is high-pass filtered above 40 Hz and Fourier transformed⁶⁵ to form Short Fourier Transforms (SFTs) of h(t). The SFTs are the input data to the search pipeline.

The maximum uncertainties in the calibration of h(t) are 10.4% in amplitude and 45 in phase for H1, and 14.4% in amplitude and 42 in phase for L1; all uncertainties are constant in time to within 1% (Abadie et al. 2010). The uncertainties in the calibration amplitude contribute to the overall systematic uncertainty in the upper limits presented in Section 3. The analysis method used in this search (see Section 2.2) is sensitive only to the relative phase uncertainty between detectors. Even for a worst-case relative phase uncertainty of ∼10°, the resulting difference in the signal frequency between detectors would still be much less than the mismatch allowed for by the search template bank (see Section 2.4). Thus, the phase uncertainties do not affect the results of this search.

The search is restricted to a data set spanning a maximum of 12 days (Wette et al. 2008). The computational cost of a coherent search for Cas A scales with the 7th power of the timespan between the first and last timestamp of the analyzed SFTs, while the sensitivity scales only with the square root of the observation time. Increasing the timespan from 12 days would therefore rapidly increase the computational cost, for a negligible improvement in sensitivity. To select the 12 day data set, we compute a figure of merit for each possible data set spanning 12 days, chosen from the available S5 SFTs. The figure of merit, which is proportional to the estimated power signal-to-noise ratio, is given by ∑_k,f[S_h(f)]⁻¹, where S_h is the strain noise power spectral density at frequency f in the SFT numbered k, and the summation is over all SFTs in the data set and over the frequency band 100–300 Hz. We select the data set with the maximum value of the figure of merit.

At the time this search was conducted, SFTs were available from the beginning of S5 until 2007 April 19 UTC. In this period, ∼9% of the H1 science data and ∼13% of the L1 science data are excluded, either due to data quality vetoes or due to the segmentation of h(t) during SFT generation. The 12 day data set selected for the search begins at 2007 March 20, 20:56:37 UTC and ends at 2007 April 1, 20:50:04 UTC. It contains a total of 934 SFTs (445 from L1 and 489 from H1) and an average of 9.7 days of data from each detector.

The data are searched using the $\mathcal {F}$-statistic, a coherent matched filtering technique used to search for periodic gravitational waves using multiple detectors (Jaranowski et al. 1998; Cutler & Schutz 2005). Matched filtering requires an accurate model, or template, of the signal. The template models the response of a detector to the two polarizations ("+" and "×") of the gravitational-wave signal emitted by a rotating neutron star. In addition to the sky position, the barycentered gravitational-wave frequency, and its derivatives, the signal template has four parameters related to amplitude and polarization: the intrinsic strain h₀, initial phase constant ϕ₀, inclination angle ι of the star's rotation axis to the line of sight, and polarization angle ψ. The $\mathcal {F}$-statistic is the logarithm of the likelihood ratio analytically maximized over these unknown parameters. The value of the $\mathcal {F}$-statistic is usually quoted as $2\mathcal {F}$.

In transforming a signal from the reference frame of the star to that of the detector, the intrinsic frequency of the source is modulated by Doppler effects due to the sidereal and orbital motion of Earth with respect to the source. The transformation requires the right ascension α and declination δ of the star, which for Cas A are known to high precision: α = 23^h23^m27943 ± 005, and δ = 58°48'4251 ± 04 (Fesen et al. 2006a). The sky resolution of the $\mathcal {F}$-statistic is, for a data set spanning 12 days, much coarser than the measured uncertainties in α and δ (Whitbeck 2006), and so no search over sky position is required. The search is conducted over the remaining unknown function in the signal template: the instantaneous frequency of the source as observed at the solar system barycenter, f(t). This is modeled as

$$\begin{equation} f(t) \approx f + \dot{f}(t-t_0) + \frac{1}{2} \ddot{f}(t-t_0)^2 \,, \end{equation} \tag{ 4 }$$

where the initial frequency f, first spin-down $\dot{f}$, and second spin-down $\ddot{f}$ (all evaluated at the start time of the data set, t = t₀) constitute the search parameters. In previous searches for periodic gravitational waves, it has not been necessary to include a second spin-down, but one is required for this search due to the young age of Cas A (Wette et al. 2008).

The signal template does not allow for the possibility that Cas A glitched during the 12 days spanned by the data. On the other hand, even the most frequently glitching pulsar does so only a few times per year, and glitch population statistics do not clearly indicate that the youth of Cas A is sure to mean more frequent glitches (Yuan et al. 2010). The worst case would be a glitch at the midpoint of the 12 days span, since in other cases the template would pick up the longer of the pre- and post-glitch coherent stretches. Amplitude signal-to-noise accumulates as the square root of observation time, so the worst loss would be a factor of ∼2.

The $\mathcal {F}$-statistic implicitly assumes a uniform prior on the polarization angle ψ, and therefore the standard data analysis and upper limit procedures remain valid for gravitational-waves signals from r-modes, even though the signal template used in this search assumes a mass quadrupole (Owen 2010).

The Cas A search uses the ComputeFStatistic_v2 implementation of the $\mathcal {F}$-statistic, which is available as part of the LALSuite software package.⁶⁶ Values of the $\mathcal {F}$-statistic returned by ComputeFStatistic_v2 have an uncertainty of up to 5%, due to practical computation issues and optimizations (see Prix 2010).

The range of the gravitational-wave frequency, 100 Hz ⩽ f ⩽ 300 Hz, is chosen based on the estimate in Wette et al. (2008) of the frequency band over which a search of LIGO S5 data could beat the indirect limit on h₀ at reasonable computational cost.

The ranges of the spin-down parameters $\dot{f}$ and $\ddot{f}$ are chosen to be (Wette et al. 2008)

$$\begin{equation} {-}\frac{f}{\langle \min n - 1 \rangle \tau } \le \dot{f}\le -\frac{f}{\langle \max n - 1 \rangle \tau } \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \frac{(\min n) \dot{f}^2}{f} \le \ddot{f}\le \frac{(\max n) \dot{f}^2}{f}, \end{equation} \tag{ 6 }$$

respectively, where the braking index n is defined below. The age of Cas A, τ, is chosen to be 300 yr (as discussed in Section 1.2). Note that the range of $\dot{f}$ depends on f, and the range of $\ddot{f}$ depends on both f and $\dot{f}$. The resulting shape of the three-dimensional parameter space of f, $\dot{f}$, and $\ddot{f}$ is depicted in Figure 1.

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The choices of $\dot{f}$ and $\ddot{f}$ ranges are motivated by the desire to cover a wide range of astrophysically motivated possibilities, expressed in terms of the braking index. In Equation (6), we use the definition of the instantaneous braking index $n = f \ddot{f} \dot{f}^{-2}$; in Equation (5), the angled brackets denote the average value of n over the lifetime of Cas A. If the dominant emission mechanism driving the spin-down of Cas A has changed over its lifetime, these two quantities will be different. Therefore, in the spirit of trying to cover the broadest imaginable parameter space, we do not constrain the instantaneous and averaged values of n to be the same. Instead, we search over values of n (both instantaneous and averaged) between 2 and 7. This range covers the possibilities that Cas A is spinning down primarily due to magnetic dipole radiation (n = 3) or to gravitational waves generated by a mass quadrupole (n = 5) or by constant-α r-modes (n = 7). This range also covers the braking indices of nearly all known pulsars, which are typically between 2 and 3 (Livingstone et al. 2006). The exception is the Vela pulsar, for which n ≈ 1.4 (Lyne et al. 1996). Extending n to lower values would dramatically increase the computational cost of the search.

The search consists of computing the $\mathcal {F}$-statistic over a finite bank of templates, whose parameters are given by points within the search parameter space. The search uses a template bank generation algorithm (Wette 2009) which locates the parameter space points at the vertices of a body-centered cubic lattice. This minimizes the number of points per unit volume required to cover the parameter space (see, e.g., Conway & Sloane 1988; Prix 2007b). The points are spaced using the $\mathcal {F}$-statistic parameter space metric (Whitbeck 2006; Prix 2007a) to ensure that the maximum expected fractional loss in signal-to-noise ratio (known as the mismatch) will never exceed 20%. Due to strong correlations between the frequency and spin-down parameters and the irregular shape of the parameter space, it was necessary to place additional templates outside of the parameter space to fully cover the parameter space boundaries, particularly for $\ddot{f}$. As a result, the number of templates searched, N ≈ 7 × 10¹², is an order of magnitude larger than that estimated in Wette et al. (2008). This resulted in a greater computational cost but does not greatly affect the sensitivity of the search, which is only weakly dependent on the number of templates (see Section 2.6).

The search is divided into ∼21,500 independent computational jobs by partitioning the range of f into small bands, each with approximately equal numbers of templates, resulting in band widths of 1.8 mHz (at f = 100 Hz) to 20 mHz (at f = 300 Hz). The search completed in ∼3.5 days on ∼5000 cores of the ATLAS computer cluster at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Hanover, Germany. From each search job, only the 0.01% of templates with the largest values of $2\mathcal {F}$ were recorded.

Data from the LIGO detectors are known to contain stationary or nearly stationary spectral lines originating from instrumental and environmental noise. The $\mathcal {F}$-statistic depends upon a robust estimator of the power spectral density of the noise. The implementation used in this search (see Section 2.2) uses a spectral running median with a window size of 50 SFT bins, or 27.8 mHz (Abbott et al. 2007a). Lines that are narrower than the median window size will remain in the data and may result in spuriously large values of the $\mathcal {F}$-statistic.

To identify frequency bands where the $\mathcal {F}$-statistic may have been contaminated in this manner, we search for prominent narrow lines in power spectra of the searched H1 and L1 data. This procedure identified 11 frequency bands, which are listed in Table 1, along with the largest value of the $\mathcal {F}$-statistic found in each band. Templates whose instantaneous frequency f(t) at any time falls within any of these bands are excluded from the remainder of the search pipeline. Approximately 2 × 10⁶ templates (a fraction ∼3 × 10⁻⁶ of the total number of templates) are excluded in this manner. Four of the bands contain a largest value of $2\mathcal {F}$ in the range 51–59, which would not be regarded as statistically significant gravitational-wave candidates (see Section 2.6).

Table 1. Frequency Bands Containing Spuriously Large Values of the $\mathcal {F}$

Frequency Band	$2\mathcal {F}_{{\rm max}}$	Origin
108.860 ± 0.018	90	Pulsar hardware injection no. 3
119.877 ± 0.019	72	Sideband of 60 Hz harmonic
128.000 ± 0.017	56	16 Hz harmonic
139.225 ± 0.058	70	L1-only line
139.510 ± 0.017	72	L1-only line
144.751 ± 0.056	110	L1-only line
179.812 ± 0.018	51	Sideband of 60 Hz harmonic
185.630 ± 0.055	59	L1-only line
193.005 ± 0.046	66	L1-only line
193.391 ± 0.018	73	Pulsar hardware injection no. 8
209.265 ± 0.017	54	L1-only line

Notes. Frequency bands (Column 1) identified during post-processing as containing spuriously large values of the $\mathcal {F}$-statistic (the maximum of which are given in Column 2), and a brief description of their origin (Column 3). See the text for details.

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At certain times during S5, 10 simulated periodic gravitational-wave signals were injected into the LIGO detectors at the hardware level by mechanically oscillating the detector mirrors (Abbott et al. 2008a). Four of the injections had frequencies within the search band. The Cas A search data set was not selected with regard to times when the hardware injections were active. As a result, ≲1 day of the searched data (∼4.2% of the H1 data, and ∼9.2% of the L1 data) contain the injections, and the effective intrinsic strains of the injections are reduced by a factor of ∼10. After accounting for this reduction, two of the injections have strains of h₀ ≲ 10⁻²⁵ and are undetectable by this search. The remaining two injections (designated nos. 3 and 8) have reduced strains of h₀ ∼ 1.63 × 10⁻²⁴ and 1.59 × 10⁻²⁴, respectively. These injections are found by the Cas A search at frequencies consistent with the parameters of the injected signals (Wette 2009). Neither injection is at the sky position of Cas A, but are nevertheless detected due to their strength, and the poor sky localization of short-duration (i.e., less than 1 day) signals arising from global correlations in the signal parameter space (Prix & Itoh 2005; Pletsch 2008)

The remaining nine non-injection bands are ruled out as gravitational-wave candidates for the following reasons. Two contain harmonics of the 60 Hz power mains frequency, and one contains a harmonic of the 16 Hz data acquisition buffering frequency (Abbott et al. 2004a). The remaining six bands contain narrow instrumental lines which occur only in the L1 detector. We would expect a convincing gravitational-wave candidate to be seen in both detectors. Four of these six lines (at ∼139.2 Hz, 144.7 Hz, 185.6 Hz, and 193.0 Hz) are definitively identified with environmental noise, by correlating the gravitational wave data with data from environmental monitors (e.g., magnetometers, accelerometers, and microphones). The sources of the remaining two lines, at ∼139.5 Hz and at ∼209.2 Hz, are not conclusively identified. The value of the $\mathcal {F}$-statistic associated with the ∼209.2 Hz line, $2\mathcal {F}= 54$, is smaller than the largest value of $2\mathcal {F}$ expected from the search in the absence of a gravitational-wave signal (see the next section) and is therefore not a candidate for a gravitational-wave signal from Cas A. The value of the $\mathcal {F}$-statistic associated with the ∼139.5 Hz line, $2\mathcal {F}= 72$, is also not statistically significant; as may be deduced from Figure 2, there is a ∼5% probability that the search would return a higher largest value of $2\mathcal {F}$ without a gravitational-wave signal being present in the data.

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The largest value of the $\mathcal {F}$-statistic returned by the search after post-processing is denoted $2\mathcal {F}^{\star }$. This is our most promising candidate for a gravitational-wave signal from Cas A. The probability density of $2\mathcal {F}^{\star }$, under the assumption that no gravitational-wave signal from Cas A is present in the searched data, is given by

$$\begin{equation} p(2\mathcal {F}^{\star }) = N p\big(\chi ^2_4; 2\mathcal {F}^{\star }\big) \left[ \int _{0}^{2\mathcal {F}^{\star }} \hspace{-1.5em} d(2\mathcal {F}) \, p\big(\chi ^2_4; 2\mathcal {F}\big) \right]^{N-1} \,, \end{equation} \tag{ 7 }$$

where N is the number of searched templates and $p(\chi ^2_4; 2\mathcal {F})$ denotes the probability density of a central χ² distribution with four degrees of freedom (i.e., the distribution of $2\mathcal {F}$ in the absence of any signal and assuming Gaussian noise). If the value of $2\mathcal {F}^{\star }$ returned by the Cas A search (see Section 3) is within the range of probable values expected from $p(2\mathcal {F}^{\star })$, it is not statistically significant. If, on the other hand, the value of $2\mathcal {F}^{\star }$ is extremely unlikely to be drawn from $p(2\mathcal {F}^{\star })$, the candidate signal is worthy of further investigation.

Equation (7) assumes that each of the N values of $2\mathcal {F}$ is statistically independent. This is not strictly true, however, as we expect the $2\mathcal {F}$ values of neighboring templates to be correlated due to the close template spacing. Therefore, in Equation (7), N should be substituted with the number of statistically independent templates, N_i < N. An empirical estimate of the number of statistically independent templates found that N_i ≈ 0.88N for this search (Wette 2009). A reduction in N shifts the distribution of $p(2\mathcal {F}^{\star })$ toward lower values of $2\mathcal {F}$, and thus increases the statistical significance of a candidate signal. For a reduction of N to 0.88N, however, the shift is negligible, and can be ignored; indeed, Figure 2 plots Equation (7) with N_i = N. If, taking an extreme example, N_i = 0.1N, the position of the maximum of $p(2\mathcal {F}^{\star })$ would be shifted from $2\mathcal {F}\sim 66$ to ∼62 (see Figure 2), but the significance of the $2\mathcal {F}^{\star }$ found by this search (see Section 3) would not be greatly increased. Therefore, the uncertainty in the number of statistically independent templates does not alter the significance of $2\mathcal {F}^{\star }$, nor the conclusions reached in Section 3. The determination the number of statistically independent templates from first principles is an interesting area for further investigation.

If, after examining $2\mathcal {F}^{\star }$ (see Section 2.6), we conclude that no gravitational signal has been detected, we proceed to set 95% confidence upper limits on the intrinsic strain h₀, the ellipticity , and the r-mode amplitude α. The upper limits are determined using Monte Carlo injections, following the procedure described in Abbott et al. (2007a). The search frequency band is first partitioned into 400 sub-bands of width 0.5 Hz; an upper limit is set separately for each sub-band. The choice of 0.5 Hz is small enough that the time-averaged noise floor of the detectors is approximately constant over each band and is large enough to keep the computational cost reasonable. We denote the largest values of $2\mathcal {F}$ found by the full Cas A search in each sub-band by $2\mathcal {F}^{\star }_{{\rm s\hbox{-}b}}$; they serve as the false alarm thresholds in each band. For each sub-band, a population of 5000–6000 periodic gravitational-wave signals, with h₀ fixed and all other parameters randomly chosen, are each in turn injected into the search data set and then searched for using the same analysis method used in the full Cas A search. We record the largest value of $2\mathcal {F}$ found by the searches for each of the injected signals. The fraction of these $2\mathcal {F}$ values that are greater than $2\mathcal {F}^{\star }_{{\rm s\hbox{-}b}}$ gives the confidence corresponding to the fixed value of h₀.

To expedite the injection procedure, we use an analytic model of the distribution of the population of injected signals to provide an educated guess at the value of h₀ required for 95% confidence (Wette 2009). We then perform the injection procedure once, at that h₀, to check that the recovered fraction really is 95%. We find that the analytic model slightly overestimates the required h₀, and so the 95% confidence upper limits presented in Section 3 are conservative.