# Empirically extending the range of validity of parameter-space metrics for all-sky searches for gravitational-wave pulsars

## Abstract

All-sky searches for gravitational-wave pulsars are generally limited in sensitivity by the finite availability of computing resources. Semicoherent searches are a common method of maximizing search sensitivity given a fixed computing budget. The work of Wette and Prix [Phys. Rev. D 88, 123005 (2013)] and Wette [Phys. Rev. D 92, 082003 (2015)] developed a semicoherent search method which uses metrics to construct the banks of pulsar signal templates needed to search the parameter space of interest. In this work we extend the range of validity of the parameter-space metrics using an empirically-derived relationship between the resolution (or mismatch) of the template banks and the mismatch of the overall search. This work has important consequences for the optimization of metric-based semicoherent searches at fixed computing cost.

###### pacs:

04.80.Nn, 95.55.Ym, 95.75.Pq, 97.60.Jd## I Introduction

The pursuit of the first direct detection of gravitational waves ended with the observation of the merger of two binary black holes Abbott *et al.* (2016).
Other classes of gravitational-wave sources (see e.g. Pitkin *et al.* (2011); Riles (2013); LIGO Scientific Collaboration and Virgo Collaboration (2014) for reviews) may also be detected by the LIGO Abbott *et al.* (2009); Aasi *et al.* (2015a), Virgo Acernese *et al.* (2015), and KAGRA Somiya (2012) observatories, as construction and commissioning of these detectors continues over the coming years.

Rapidly-rotating neutron stars which may be radiating continuous, quasisinusoidal gravitational waves – *gravitational-wave pulsars*, for short – are one potential source.
Data from the LIGO and Virgo observatories has been searched for gravitational waves from known electromagnetic pulsars (e.g. Aasi *et al.*, 2014a, 2015b) and the low-mass X-ray binary Scorpius X-1 (e.g. Aasi *et al.*, 2015c), gravitational-wave pulsars in supernova remnants Abadie *et al.* (2010); Aasi *et al.* (2015d) and at the Galactic center Aasi *et al.* (2013a), and all-sky searches for gravitational-wave pulsars, both isolated (e.g. Aasi *et al.*, 2013b, 2014b, 2014c, 2016) and in binary systems Aasi *et al.* (2014d).

The detection of gravitational-wave pulsars presents a number of challenges.
The gravitational wave amplitude scales with the potential nonaxial deformability of a neutron star; while *maximum* deformations have been studied e.g. in Johnson-McDaniel and Owen (2013), the scale of realistic deformations that might exist in the population of Galactic neutron stars, and hence the number of detectable gravitational-wave pulsars, remain largely unknown. Furthermore, a search for gravitational-wave pulsars using the most sensitive search method – coherent matched filtering against a known signal template – is computationally feasible only in a few circumstances, e.g. searches targeting electromagnetic pulsars whose sky position and frequency evolution are accurately known.

These challenges have motivated the development of a variety of data-analysis techniques, in a quest to gain maximum sensitivity within computational constraints.
These techniques include optimized signal template banks (e.g. Messenger *et al.*, 2009; Wette, 2014; Pisarski and Jaranowski, 2015), semicoherent search methods which trade sensitivity for reduced computing cost (e.g. Brady and Creighton, 2000; Krishnan *et al.*, 2004; Dergachev, 2010; Pletsch, 2010; Messenger, 2011; Wette, 2015), and follow-up procedures for potentially interesting candidate signals (e.g. Shaltev and Prix, 2013; Behnke *et al.*, 2015).

This paper continues a series of papers Wette and Prix (2013); Wette (2014, 2015) which have developed a semicoherent search method for isolated gravitational-wave pulsars.
In common with other semicoherent methods, the input gravitational-wave data are partitioned in time into a number of *segments*, each of which is searched by coherent matched filtering against a *coherent* template bank for each segment.
Detection statistics from each segment are then summed together using a distinct *semicoherent* template bank for the overall search.
The method utilizes the idea of a parameter-space *metric* Balasubramanian *et al.* (1996); Owen (1996); Prix (2007a), which determines both the resolution of the coherent template banks of each segment, and that of the semicoherent template bank of the overall search.

The resolutions of the template banks are typically quantified by the maximum *mismatch*: the fraction of signal-to-noise ratio lost when a signal in the input data does not precisely match any one of the search templates.
The parameter-space metric models the mismatch as a distance measure between the parameters of the signal and that of a template.
Typically, the resolution of the semicoherent bank is much finer than that of the coherent banks; in particular the method developed in Wette (2015) predicts that the semicoherent bank requires a much larger number of templates than previously estimated Brady and Creighton (2000); Pletsch (2010).

Indeed, the large number of templates in the semicoherent bank leads to the following problem.
The parameters describing the search setup – the number of segments, the time span of each segment, and the maximum mismatches allowed in the coherent and semicoherent template banks – may be optimized under fixed constraints on computing cost and available input data using the framework of Prix and Shaltev (2012).
Preliminary studies have found that, under computing cost and data constraints similar to previous searches (e.g. Aasi *et al.*, 2013b) performed on the distributed computing project Einstein@Home ^{1}

The parameter-space metric is, however, only an approximate model based on a Taylor expansion of the mismatch, and hence has a limited range of validity.
Previous work suggests that the metric accurately predicts mismatch values less than (see e.g. Fig. 10 of Prix (2007a) and Fig. 7 of Wette and Prix (2013)), and becomes increasingly inaccurate at higher values.
In order to satisfy the computing cost constraint in the search optimization described above, however, the maximum mismatch allowed in the semicoherent bank must typically be *much greater* than , i.e. beyond the range of validity of the metric.

It would appear therefore that, under reasonable computing cost constraints, the parameter-space metric *alone* cannot be used to reliably predict mismatch; independent investigations into the performance of Einstein@Home all-sky searches have reached a similar conclusion Papa and Walsh (2015).
Note too that the sensitivity of a search is generally degraded as the maximum allowed mismatch is increased; it is therefore unclear whether, at the large maximum semicoherent mismatch required to satisfy computing cost constraints, the sensitivity of a metric-based search as proposed by Wette and Prix (2013); Wette (2014, 2015) would be competitive with other semicoherent methods.

This situation motivates the work described in this paper: a study of the relationship between the mismatches of the coherent and semicoherent template banks predicted by the metric, and the actual mismatch of the overall search as measured by searching for software-generated signals in simulated data. After reviewing background information in Section II, and the methodology of the simulations used to measure actual mismatch in Section III, the results of the study are presented in Sections IV and V. The conclusions drawn from the study are presented in Section VI.

## Ii Background

This section reviews the theory of semicoherent gravitational-wave pulsar searches, and the associated parameter-space metrics. Further details can be found in Wette and Prix (2013); Wette (2015) and references therein.

The signal template for a gravitational-wave pulsar Jaranowski *et al.* (1998) is a function of time at the detector, the four parameters which determine the amplitude modulation of the signal, and the vector of parameters which determine its phase evolution.
The latter parameters, in the case of all-sky searches for isolated pulsars, are the sky position of the pulsar, its initial frequency at some reference time , and its spindown parameters of spindown order .

The -statistic Jaranowski *et al.* (1998); Cutler and Schutz (2005) performs matched filtering of the input gravitational-wave data against the template , and analytically maximized over the parameters .
If the are unknown (as is the case for all-sky searches), a search is performed by computing the detection statistic over a bank of templates whose parameters are drawn from the search parameter space of interest.
(Here the subscript indexes a single data segment, whose time span is denoted .)
In the vicinity of a signal with parameters , the value of follows a noncentral distribution with 4 degrees of freedom and noncentrality parameter ; in Gaussian noise it follows a central distribution with 4 degrees of freedom.

The mismatch determines what fraction of the signal with parameters is not recovered when computing the -statistic using a template with parameters . It is defined in terms of the noncentrality parameter by (e.g. Prix and Shaltev, 2012; Wette, 2015):

(1) |

where is the noncentrality parameter when the template is perfectly matched to the signal. A degree of mismatch is unavoidable, as a signal will never exactly match any template in the bank. Consequentially, template banks are constructed (e.g. Owen, 1996; Prix, 2007b; Wette, 2014) so as to minimize the potential mismatch to some maximum .

Mismatch can be modeled Balasubramanian *et al.* (1996); Owen (1996); Prix (2007a) as the distance between signal and template parameters with respect to the parameter-space metric :

(2) |

The metric arises from a second-order Taylor expansion of Eq. (1) with respect to small parameter offsets :

(3) |

Note that whereas by construction the actual mismatch of Eq. (1) may never exceed 1.0, the mismatch predicted by Eq. (2) may become arbitrarily large as increases.

The full metric of Eq. (3) is often approximated by a simpler expression, the phase metric , which discards the amplitude modulation parameterized by the . The components of the matrix are

(4) |

where is the signal phase, are the derivatives with respect to the th parameter in , and the operator denotes time-averaging over the time span .

It follows from Eq. (4) that, for any parameters in in which the signal phase is linear, the corresponding components of will be independent of ; the mismatch with respect to a signal will therefore not depend on the parameters of that signal.
This is particularly convenient for template placement since, for such a metric, template banks can be constructed using regular lattices that minimize the number of templates, and hence the number of matched filtering operations (e.g. Jaranowski and Królak, 2005; Prix, 2007b; Wette, 2014).
The signal phase is linear in the frequency and spindown parameters Jaranowski *et al.* (1998), but not in commonly-used parameterizations of the sky, e.g. by right ascension and declination .

The work of Wette and Prix (2013) developed an approximation to which is independent of all parameters : the supersky metric . The metric is derived by embedding in a higher-dimensional space which includes an additional sky position parameter; in this space is linear in all parameters. Then, the vector in the now 3-dimensional sky parameter space is identified, along which the mismatch is the least sensitive to the parameter offsets ; this vector is the eigenvector associated with the smallest eigenvalue of the sky–sky components of the embedded phase metric. Finally, the embedded phase metric is projected back onto a subspace perpendicular to this vector, which removes one sky position parameter and results in the metric . Numerical simulations Wette and Prix (2013) found that generally predicts mismatches measured by searching for software-generated signals in simulated data with a relative error %, up to maximum mismatches of .

The metric applies only to a fully-coherent analysis of a single data segment. In Wette (2015), the supersky metric is generalized to a semicoherent analysis, in which the coherent analyses of data segments are combined together. For each semicoherent template in the bank , appropriate -statistic values are chosen from each segment and summed to give the -statistic as a function of :

(5) |

Typically the are chosen by nearest-neighbor interpolation: in each segment, the chosen are the parameters with the smallest mismatch to with respect to the metric .

The semicoherent supersky metric is used to construct the semicoherent template bank , just as the coherent metrics are used to construct the coherent template banks in each segment. The metric is derived following a similar procedure to that of ; the chief difference is that its starting point is the phase metric summed over segments Brady and Creighton (2000). Numerical simulations using a range of search setups – parameterized by the number of segments , the time span of each segment, the total time spanned by all segments, and the maximum mismatches and of the coherent and semicoherent template banks respectively – found to also be a useful predictor of actual mismatch, with relative errors typically %, up to maximum mismatches of Wette (2015).

## Iii Numerical simulations at large metric mismatches

In Wette and Prix (2013); Wette (2015) the supersky metric was validated by performing numerical simulations which compare the mismatch predictions of the metric to mismatches measured by searching simulated data for software-generated signals. Those simulations limited the maximum mismatches of both the coherent and semicoherent template banks to Wette and Prix (2013) and Wette (2015). In this paper we reperform the same simulations with a much wider range of maximum mismatches . The simulation procedure, which is otherwise very similar to that used in Wette and Prix (2013); Wette (2015), is briefly described in this section.

A total of 48 pairs of maximum semicoherent and coherent mismatches are used; these are listed in Table 1. The simulations used a variety of search setups, parameterized by , where is the segment duty cycle, i.e. the fraction of the total time span which falls within a segment. The chosen parameters are days, days, and ; these are a subset of the parameters used in Wette (2015).

Signal parameters are generated with: uniform sky positions; spindowns such that , where is the width of the metric ellipse bounding box of , and is given by Eq. (11) of Wette (2014); and frequencies Hz. The nearest semicoherent template to the signal is determined assuming a semicoherent template bank constructed from an lattice Conway and Sloane (1988) using the metric with maximum mismatch chosen from one of the pairs in Table 1. Similarly, in each segment the nearest coherent template to the semicoherent template is determined assuming a coherent template bank constructed from an lattice using the metric with maximum mismatch chosen from the same pair in Table 1.

Metric mismatches between the signal and various templates are then calculated ^{2}

(6) |

The semicoherent metric mismatch between and the nearest in the semicoherent template bank is

(7) |

with . Finally, the average coherent metric mismatch between and the nearest in the coherent template banks of each segment is

(8) |

with .

The metric mismatches given above are then measured using the -statistic ^{3}*et al.* (2009) are generated at times , where UTC 2015-01-01 00:00:00.
The data comprise no noise (but see the discussion in Section VI) and a simulated signal with random amplitude parameters and chosen phase parameters .
The -statistic is computed in each segment at , the nearest to , and the nearest to , and the total -statistic mismatch is computed using

(9) |

where the denominator equals the noncentrality parameter . The total -statistic mismatch is also computed assuming no nearest-neighbor interpolation, i.e. that in every segment:

(10) |

This procedure is repeated times for all 3456 combinations of , , , , and . A total of coherent -statistic values were computed.

## Iv Mean -statistic mismatch

Coefficients | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

lattice | lattice | |||||||

In this section, the results of the simulations described in Section III are used to investigate the relationship between predicted metric mismatch and actual -statistic mismatch, in the limit of large metric mismatches.

Figure d plots histograms of the mismatches measured using the -statistic [via Eqs. (9)] as a function of the search setup parameters , , , and . As increases, the means of the histograms also increase, as expected, but at a slower rate, and do not saturate at 100% mismatch even when . For example, for day, days, and (left-most subplot in Fig. a), the mean of the histogram for which is 0.17; for , the mean is 0.29; and for , the mean is 0.53. This suggests that the number of semicoherent templates could be reduced significantly (by increasing ) while limiting losses in signal-to-noise ratio (as measured by ) to a reasonable level.

As an example, note that the number of semicoherent templates scales with where is the number of parameter-space dimensions (see e.g. Eq. (29) of Wette (2014)), and that the search sensitivity scales with Wette (2012). By increasing from 0.5 to 10.9, one would save a factor of in the number of semicoherent templates. The corresponding loss in sensitivity from increasing the mean measured mismatch from 0.17 to 0.53 is . An optimization procedure could potentially recover this loss, however, by reinvesting the computational power saved from the reduction in the number of semicoherent templates, e.g. by increasing ; see the discussion in Section VI.

From Fig. d we see that the mean also increases with (compare the left, middle, and right columns), and with (compare e.g. Figs. a and b), but is largely independent of (compare e.g. Figs. a and c).

Figure d plots the mean -statistic mismatch as a function of the maximum metric mismatches , for days and the 4 values of days. In keeping with Fig. d, one sees that increases with , but at a slower rate which is roughly a constant per decade in . For example, with day (Fig. a) and (i.e. ), increases from 0.035 to 0.21 as increases from 0.1 to 0.9, i.e. at a rate of per decade in ; then increases to 0.52 at , at a slower rate of per decade in . Even at , has yet to reach 100% mismatch.

The rate of increase of is slightly lower at day (Fig. a) than for day (Figs. b– d); consequentially, the mean -statistic as approaches 100 is lower at day ( at ) than for day (). As is increased, the rate of increase of with decreases still further, and becomes essentially zero for . The behavior of is only weakly dependent on , and for this reason is fixed to 240 days in Fig. d.

We find the following empirical fit to as a function of , , , and :

(11) |

where

(12) | ||||

(13) |

and the fitted coefficients through are listed in Table 2.
Note that Equation (11) uses as parameters the *mean* semicoherent and coherent metric mismatches and , instead of the *maxima* and respectively.
For template banks generated using lattice template placement Wette (2014), the ratios and are given by the number of parameter-space dimensions and the type of lattice employed; values for up to dimensions and for and lattices are listed in Table 3.
An empirical fit to the standard deviations of the -statistic mismatch, i.e. the widths of the histograms plotted in Fig. d, is given in Appendix A.

Over the 576 values of parameterized by used for fitting, the root-mean-square relative error to was minimized to %. Each value of was weighted by the standard deviation of the means of as a function of and , the two simulation parameters (see Section III) not included in the fit; these standard deviations are typically .

The empirical fit is plotted ^{4}

## V -statistic mismatch as function of search parameters

Equation (11), derived in the previous section, gives us a tool for predicting, with reasonable confidence, the mean -statistic mismatch , as a function of the mean metric mismatches , out to large . One might also use this tool to improve the mismatch predicted of the metric between a signal and its nearest semicoherent and coherent templates , by replacing [Eq. (6)] with

(14) |

where is computed from via Eq. (7), and is computed from the via Eq. (8).
Here, the empirical fit provides the absolute *scaling* of the -statistic mismatch, while the metric provides *directional* information, i.e. how the -statistic mismatch changes in the direction of a vector relative to some other vector .
Of course, given that Eq. (11) is fitted to the -statistic mismatch averaged over signal and template parameters , one would expect Eq.(14) to not necessarily to be an accurate predictor of for a *particular* .

In this section we examine the accuracy to which Eq. (14) models the -statistic mismatch as a function of the search parameters of the semicoherent supersky metric: the sky position , spindown and frequency . The supersky metric coordinates are detailed in Wette and Prix (2013); briefly, are components of the sky position vector in a preferred reference frame (which approaches the equatorial and ecliptic reference frames in the limit of short and long observation times respectively), and the are equal to plus a sky-position-dependent offset. The numerical simulations described in Section V recorded the signal and template parameters for each computed ; from these parameters the mismatch predicted by the metric plus the empirical fit may be computed via Eq. (14).

Figure b compares and , at fixed days, day, and , as a function of the parameter offsets between and , namely ,
, , and .
Figure a plots mismatch as functions of distinct pairs of parameter offsets, all other parameter offsets being approximately zero ^{5}

As expected, mismatches are zero (darkest color) when signal and template are perfectly matched, and increase monotonically (to lighter colors) in response to any offsets between signal and template parameters. The behavior of the -statistic mismatch as a function of offsets is generally well-modeled by , as shown by the similarity of the corresponding subplots in Fig. a. This indicates that is a reasonable model for even out to large . It also implies that, while the derivation of the parameter-space metric (see Section II) loses the correct absolute scaling of the -statistic mismatch at large , it does retain the correct directional information.

The one exception to the above, in Fig. a, is the mismatch behavior with respect to offsets in the sky position parameter . Comparing the first row of subplots (above the diagonal) in Fig. a with the first column (below the diagonal), we see that increases more quickly as a function of than does . This effect is more readily apparent in Fig. b, where we plot and as functions of individual parameter offsets – namely , , , and – and where the other 3 offsets are allowed to vary over their simulated ranges. We see that, as a function of (left-most subplot in Fig. b), (gray shaded area) increases more rapidly that (black solid line); at , whereas .

The reason for this discrepancy is likely due to numerical issues in computing the supersky metric, which requires the eigenvalues of the sky–sky block of a precursor metric Wette and Prix (2013). At day, however, the precursor metric is highly ill-conditioned Prix (2007a); Wette and Prix (2013), which may lead to inaccurate computation of the eigenvalues. It is likely that, in this instance, the – component of the supersky metric, which is proportional to the largest eigenvalue, has been inaccurately computed. To illustrate this, we re-plot re-scaled by , and see that the re-scaled now follows closely. This indicates a systematic error in the supersky metric as a function of , which we surmise is most likely due to inaccurate computation of the largest eigenvalue.

Figures b and b compare and in a similar manner to Fig. b, but at larger and . As increases to 3 days (Fig. b) and 5 days (Fig. b), the discrepancy between and as a function of largely disappears; compare the left-most plots in Figs. b, b, and b. This is expected, since the precursor metric becomes better-conditioned as increases, and therefore the computation of the eigenvalues becomes more reliable.

Some discrepancies between and as a function of and are evident in Figs. a and a. Compare the second row of subplots (above the diagonal), which display banded and/or cross-shaped features, with the second column of subplots (below the diagonal), where those features are absent. This is likely due to assumptions made in deriving the supersky metric Wette and Prix (2013). Briefly, the metric tries to model the orbital motion of the Earth by a second-order Taylor expansion, which can then be absorbed into the frequency and spindown parameters; a small component of the residual orbital motion, i.e. that component which cannot be modeled by a Taylor expansion, is then discarded. This introduces an error into the supersky metric which is generally small, but is also proportional to parameter offsets; hence at large , and hence and , the effect of this error is magnified. Nevertheless, as can be seen from Figs. b and b, generally tracks the average as a function of individual offsets.

Finally, note the small feature in the left-most subplot in Fig. b at , which does not appear at ; one would expect to be insensitive to the sign of . This is likely due to a minor issue in the implementation of the transformation from supersky to physical coordinates, where for very small template banks [i.e. small and high ] neighboring templates in can end up in opposite hemispheres when mapped to . This is not an issue for template banks of realistic densities.

## Vi Discussion

The study described in this paper was motivated by the realization that, under realistic computing cost constraints, a semicoherent search based on the parameter-space metric of Wette (2015) could not be performed with maximum semicoherent mismatches within the range of validity of the metric, i.e. .
This situation was not realized by previous work of the semicoherent metric Brady and Creighton (2000); Pletsch (2010) since those works do not accurately predict the number of semicoherent templates once days Wette (2015).
Other semicoherent search methods (e.g. Krishnan *et al.*, 2004) do not use an explicit metric to describe the parameter space.

The key finding of this paper is that the mean -statistic mismatch increases only slowly with ; see Fig. d. As discussed in Section I, it appears likely that an all-sky semicoherent search based on the metrics of Wette and Prix (2013); Wette (2015) would have to operate at a high , in order to satisfy reasonable computing cost constraints. The results presented here give us some confidence that, despite a high , the mean -statistic mismatch of such a search, and hence its sensitivity, would remain competitive. A thorough examination of the sensitivity of such a search is planned for future work.

Note, however, that the relationship between and need not be known *a priori* in order to implement a semicoherent search based on Wette and Prix (2013); Wette (2014, 2015).
First, a semicoherent template bank can be constructed for any , however large; it is only when estimating the sensitivity of such a search that one must correctly translate between and .
Second, the nearest coherent template in each segment, used to compute the summed -statistic of Eq. (5), are determined by the coherent metrics in each segment; preliminary search optimization has found that, while may be large, is likely to remain small, i.e. within the range of validity of the coherent metric.

The search setup optimization procedure of Prix and Shaltev (2012) assumes that the relationship between and is strictly proportional; see Eq. (23) of that paper. A possible extension to the method of Prix and Shaltev (2012) could be to allow instead an arbitrary functional relationship, e.g. that given by Eq. (11). This could potentially lead to improved optimal search setups, since the optimization would be aware that increasing do not increase , and hence degrade sensitivity, as much as previously assumed.

When deriving the parameter-space metric, it is standard to assume (e.g. Prix, 2007a; Wette and Prix, 2013) an ideal case where the noncentrality parameter (which is proportional to the signal-to-noise-ratio Wette (2015)) is large, i.e. the data being searched contains little or no noise.
The addition of noise will certainly reduce the recovered signal-to-noise-ratio (and consequentially the noncentrality parameter), whether at a mismatched template [] or at perfect match [].
The mismatch is, however, a *ratio* of noncentrality parameters [Eq. (1)], so a constant reduction in noncentrality parameter does not affect the mismatch.
It is probable that noise would affect and , since slightly different data would be used in the computation of each.
We expect that such an effect would average out over a large number of templates, however, and therefore hypothesize that the parameter-space metric would still provide a useful prediction of the *mean* measured mismatch, even in the realistic case of noisy data.

###### Acknowledgements.

I thank Reinhard Prix and Maria Alessandra Papa for valuable discussions. Numerical simulations were performed on the ATLAS computer cluster of the Max-Planck-Institut für Gravitationsphysik. This paper has document number LIGO-P1600162.## Appendix A Standard deviation of -statistic mismatch

Coefficients | ||||||
---|---|---|---|---|---|---|

We also find an empirical fit to the standard deviations of the -statistic mismatch. The fit is a function of , , and the standard deviations and of the coherent and semicoherent metric mismatches; ratios of these quantities to the maxima and respectively are listed in Table 3. The empirical fit is given by

(15) |

where the fitted coefficients , and through are listed in Table 4. Over the 576 values of used for fitting, the root-mean-square relative error to was minimized to %.

### Footnotes

- http://www.einsteinathome.org/
- Note that this paper uses a slightly different notation than Prix and Shaltev (2012); Wette (2015) to distinguish between the various metric mismatches. The total metric mismatch, including contributions from both semicoherent and coherent template banks, is denoted in this paper, as opposed to