Flat parameter-space metric for all-sky searches for gravitational-wave pulsars

All-sky, broadband, coherent searches for gravitational-wave pulsars are computationally limited. It is therefore important to make efficient use of available computational resources, notably by minimizing the number of templates used to cover the signal parameter space of sky position and frequency evolution. For searches over the sky, however, the required template density (determined by the parameter-space metric) is different at each sky position, which makes it difficult in practice to achieve an efficient covering. Previous work on this problem has found various choices of sky and frequency coordinates that render the parameter-space metric approximately constant but that are limited to coherent integration times of either less than a few days or greater than several months. These limitations restrict the sensitivity achievable by hierarchical all-sky searches and hinder the development of follow-up pipelines for interesting gravitational-wave pulsar candidates. We present a new flat parameter-space metric approximation and associated sky and frequency coordinates, which do not suffer from these limitations. Furthermore, the new metric is numerically well conditioned, which facilitates its practical use.

Figure 1

Illustration of the behavior of the relative error $\epsilon ({\mu }_{\mathrm{solid}},{\mu }_{\mathrm{dashed}})$ between mismatches computed by a reference metric ${\mu }_{\mathrm{solid}}$ and a transformed metric ${\mu }_{\mathrm{dashed}}$. Top: the reference and transformed metrics are plotted as solid and dashed ellipses. The transformed metric has been (left to right), inflated, deflated, and rotated relative to the reference metric. The circles and pluses represent 50 random points, sampled uniformly with respect to the reference metric; points where $\epsilon ({\mu }_{\mathrm{solid}},{\mu }_{\mathrm{dashed}})\le 0$ are plotted as circles, and points where $\epsilon ({\mu }_{\mathrm{solid}},{\mu }_{\mathrm{dashed}})>0$ are plotted as pluses. Bottom: histograms of $\epsilon ({\mu }_{\mathrm{solid}},{\mu }_{\mathrm{dashed}})$ of the 50 plotted points.

Figure 2

Relative errors as a function of $T$ between mismatches calculated from the $\mathcal{F}$-statistic, ${\mu }_0$, and predicted by: (a) the phase metric using linear phase model I, $\epsilon ({\mu }_0,{\mu }_{\mathrm{lpI}})$, for ${\mu }_0\le 0.2$; (b) the phase metric using linear phase model II, $\epsilon ({\mu }_0,{\mu }_{\mathrm{lpII}})$ for ${\mu }_0\le 0.2$; (c) the global correlation metric, $\epsilon ({\mu }_0,{\mu }_{\mathrm{gc}})$ for ${\mu }_0\le 0.2$ and $T<30$ days; (d), (e) the super-sky metric, $\epsilon ({\mu }_0,{\mu }_{\mathrm{ss}})$ for ${\mu }_0\le 0.2$ and $0.2\le {\mu }_0\le 0.6$ respectively; and (f) the reduced supersky metric, $\epsilon ({\mu }_0,{\mu }_{\mathrm{rss}})$ for ${\mu }_0\le 0.2$. Plotted are the median (solid line), the 25th–75th percentile range (error bars), and the 2.5th (dotted line) and 97.5th (dashed line) percentiles. Only first spindown is used.

Figure 3

Median magnitude of the relative error $|\epsilon ({\mu }_0,{\mu }_{\mathrm{gc}})|$, as a function of $\alpha$ and $\delta$, for ${\mu }_0\le 0.2$. The ecliptic equator is overplotted in black. Left: median $|\epsilon ({\mu }_0,{\mu }_{\mathrm{gc}})|$ over time spans $1\le T\le 29\mathrm{days}$ and over all simulation reference times $\Delta t_0$ and maximum frequencies $f_{\max }$; see the Appendix. Right: median $|\epsilon ({\mu }_0,{\mu }_{\mathrm{gc}})|$ at fixed values of $T$, $\Delta t_0$, and $f_{\max }$; axis ranges and color values are the same as for the left-hand-side plot. Only the first spindown is used.

Figure 4

Metric ellipsoids of the supersky metric ${\mathbf{g}}_{\mathrm{ss}}$, at five example points, for $T=4\mathrm{days}$ and ${\mu }_{\max }=30$. Their intersections with the sky sphere $|\overrightarrow{n}|=1$ reproduce the physical sky metric.

Figure 5

Median magnitude of the relative error $|\epsilon ({\mu }_0,{\mu }_{\mathrm{ss}})|$, as a function of $\alpha$ and $\delta$, for ${\mu }_0\le 0.2$. The ecliptic equator is over-plotted in black. Left: median $|\epsilon ({\mu }_0,{\mu }_{\mathrm{ss}})|$ over time-spans $1\le T\le 29\mathrm{days}$, and all simulation reference times $\Delta t_0$ and maximum frequencies $f_{\max }$; see the Appendix. Right: median $|\epsilon ({\mu }_0,{\mu }_{\mathrm{ss}})|$ at fixed values of $T$, $\Delta t_0$, and $f_{\max }$; axis ranges and color values are the same as for the left-hand-side plot. Only first spindown is used.

Figure 6

Probability density of simulated coordinate offsets $|\Delta \alpha |$ and $|\Delta \delta |$ (top row) and the minimum of $\Delta \widehat{\Omega }$ [Eq. (24)] at fixed $|\Delta \alpha |$ and $|\Delta \delta |$ (bottom row), over all simulation reference times and maximum frequencies and at $T=1$ day (left column) and $T=7\mathrm{days}$ (right column).

Figure 7

Mean (solid line) and standard deviation (error bars) of ${\mu }_0$ as a function of ${\mu }_{\mathrm{ss}}$, over all simulation reference times and maximum frequencies, and for (a) $T=1\mathrm{day}$ and (b) $T=7\mathrm{days}$. The dotted lines plot ${\mu }_0={\mu }_{\mathrm{ss}}$; the dashed lines plot ${\mu }_0={\mu }_{\mathrm{ss}}-0.38{\mu }_{\mathrm{ss}}^2$.

Figure 8

True mismatch region $\mu \le {\mu }_{\max }$ of the sky metric (solid line) and the local metric ellipse in $(\alpha ,\delta )$ given by Eq. (25) (dotted line) at four example points (black dots) for $T=2\mathrm{days}$ and ${\mu }_{\max }=0.3$.

Figure 9

Relative errors (a) $\epsilon ({\mu }_0,{\mu }_{\alpha \delta })$ and (b) $\epsilon ({\mu }_{\mathrm{ss}},{\mu }_{\alpha \delta })$ between mismatches predicted by the sky metric in $\alpha –\delta$ coordinates, ${\mu }_{\alpha \delta }$, predicted by the supersky metric, ${\mu }_{\mathrm{ss}}$, and calculated from the $\mathcal{F}$ statistic, ${\mu }_0$, as a function of $T$ for ${\mu }_0\le 0.2$ and ${\mu }_{\mathrm{ss}}\le 0.2$, respectively. Plotted are the median (solid line), the 25–75th percentile range (error bars), and the 2.5th (dotted line) and 97.5th (dashed line) percentiles. Only the first spindown is used.

Figure 10

Metric ellipsoids of the supersky metric, in aligned coordinates $(n_a,n_b,n_c)$ [see Sec. 5c], for $T=4\mathrm{days}$ and ${\mu }_{\max }=5$ at five example points. By removing the $c$ dimension, we produce a projected metric on the two-dimensional $a–b$ plane.

Figure 11

Condition numbers of the supersky metric, in Systeme International units, ${\mathbf{g}}_{\mathrm{ss}}$ (points, dotted line), and after diagonal rescaling, ${\overline{\mathbf{g}}}_{\mathrm{ss}}$ (points, solid line); the diagonal-rescaled metric ${\overline{\mathbf{g}}}_{\mathrm{ss}}^{\prime }$, derived in Sec. 5b (triangles); and the diagonal-rescaled metrics ${\overline{\mathbf{g}}}_{\mathrm{ss}}^{††}$ (squares) and ${\overline{\mathbf{g}}}_{\mathrm{ss}}^{†}$ (circles), derived in Sec. 5c. Plotted as functions of $T$ are the mean (lines) and the minimum-to-maximum range (error bars) of condition numbers over all simulation reference times $\Delta t_0$; see the Appendix. Only the first spindown is used.

Figure 12

Relative error in computing the smallest eigenvalue of the supersky metric in Systeme International units, including the first spindown (points) and a second spindown (circles). Plotted as functions of $T$ are the mean (lines) and the minimum-to-maximum range (error bars) of errors over all simulation reference times $\Delta t_0$; see the Appendix.

Figure 13

Metric ellipsoids of the transformed supersky metric ${\mathbf{g}}_{\mathrm{ss}}^{\prime }$, in equatorial coordinates $\overrightarrow{n}$, for $T=4\mathrm{days}$ and ${\mu }_{\max }=5$, at five example points and their intersections with the sky sphere $|\overrightarrow{n}|=1$.

Figure 14

Relative errors $\epsilon ({\mu }_{\mathrm{ss}},{\mu }_{\mathrm{rss}})$ between mismatches predicted by the supersky metric, ${\mu }_{\mathrm{ss}}$, and reduced supersky metric, ${\mu }_{\mathrm{rss}}$, as a function of $T$, for ${\mu }_{\mathrm{ss}}\le 0.2$. Plotted are the median (solid line), the 25–75th percentile range (error bars), and the 2.5th (dotted line) and 97.5th (dashed line) percentiles. Only the first spindown is used.

Figure 15

Median magnitude of the relative error $|\epsilon ({\mu }_{\mathrm{ss}},{\mu }_{\mathrm{rss}})|$, as a function of $\alpha$ and $\delta$, for ${\mu }_{\mathrm{ss}}\le 0.2$. The ecliptic equator is overplotted in black. Left: median $|\epsilon ({\mu }_{\mathrm{ss}},{\mu }_{\mathrm{rss}})|$ over time spans $1\le T\le 121\mathrm{days}$ and over all simulation reference times $\Delta t_0$ and maximum frequencies $f_{\max }$; see the Appendix. Right: median $|\epsilon ({\mu }_{\mathrm{ss}},{\mu }_{\mathrm{rss}})|$ at fixed values of $T$, $\Delta t_0$, and $f_{\max }$; axis ranges and color values are the same as for the left-hand-side plot. Only the first spindown is used.

Figure 16

(a) Ratio of diagonalized supersky eigenvalues $R$ and (b) orientation of the $a–b$ plane $\beta$, as functions of $T$ and $\Delta t_0$. Only the first spindown is used.

Figure 17

Relative errors (a) $\epsilon ({\mu }_0,{\mu }_{\mathrm{rss}})$ and (b) $\epsilon ({\mu }_{\mathrm{ss}},{\mu }_{\mathrm{rss}})$ between mismatches calculated from the $\mathcal{F}$ statistic, ${\mu }_0$, and predicted by the supersky metric, ${\mu }_{\mathrm{ss}}$, and reduced supersky metric, ${\mu }_{\mathrm{rss}}$, as a function of $T$, for ${\mu }_0\le 0.2$ and ${\mu }_{\mathrm{ss}}\le 0.2$, respectively. Plotted are the median (solid line), the 25–75th percentile range (error bars), and the 2.5th (dotted line) and 97.5th (dashed line) percentiles. Both the first and second spindowns are used.

Figure 18

(a) Ratio of diagonalized supersky eigenvalues $R$ and (b) orientation of $a–b$ plane $\beta$ as functions of $T$ and $\Delta t_0$. Both the first and second spindowns are used.

Figure 19

Median absolute relative errors, as functions of $T$, between mismatches calculated from the $\mathcal{F}$ statistic and predicted by the supersky metric, $|\epsilon ({\mu }_0,{\mu }_{\mathrm{ss}})|$ (points); the reduced supersky metric, $|\epsilon ({\mu }_0,{\mu }_{\mathrm{rss}})|$ (circles); the linear phase model I metric, $|\epsilon ({\mu }_0,{\mu }_{\mathrm{lpI}})|$ (crosses); and the global correlation metric, $|\epsilon ({\mu }_0,{\mu }_{\mathrm{gc}})|$ (squares), for ${\mu }_0\le 0.2$. Only the first spindown is used.

Figure 20

Mean condition numbers, as functions of $T$, of the diagonally rescaled supersky (points), reduced supersky (circles), and linear phase model I (crosses) metrics, over all simulation reference times $\Delta t_0$; see the Appendix. Only the first spindown is used.