Directed searches for continuous gravitational waves from binary systems: Parameter-space metrics and optimal Scorpius X-1 sensitivity

We derive simple analytic expressions for the (coherent and semicoherent) phase metrics of continuous-wave sources in low-eccentricity binary systems for the two regimes of long and short segments compared to the orbital period. The resulting expressions correct and extend previous results found in the literature. We present results of extensive Monte Carlo studies comparing metric mismatch predictions against the measured loss of detection statistics for binary parameter offsets. The agreement is generally found to be within $\sim 10\%–30\%$. For an application of the metric template expressions, we estimate the optimal achievable sensitivity of an Einstein@Home directed search for Scorpius X-1, under the assumption of sufficiently small spin wandering. We find that such a search, using data from the upcoming advanced detectors, would be able to beat the torque-balance level [R. V. Wagoner, Astrophys. J. 278, 345 (1984); L. Bildsten, Astrophys. J. 501, L89 (1998).] up to a frequency of $\sim 500–600\mathrm{Hz}$, if orbital eccentricity is well constrained, and up to a frequency of $\sim 160–200\mathrm{Hz}$ for more conservative assumptions about the uncertainty on orbital eccentricity.

Figure 1

Coherent long-segment regime: results for metric tests on ${\tilde{g}}^{\mathrm{LS}}$ of Sec. 4b1, for an orbital period scale of $P_0=19$ h and varying segment length $\mathrm{\Delta }T$ from 3 to 39 d in steps of 4 d. (a) Relative error $ϵ({\tilde{\mu }}_0,\tilde{\mu })$ vs eccentricity $e$. The dashed vertical line denotes the boundary between a “low-$e$” range $e\in [10^{-5},0.1]$ (left-column plots) and a “high-$e$” range $e\in [0.1,0.9]$ (right-column plots). (b, c): Relative error $ϵ$ vs observation time $\mathrm{\Delta }T/P$. The solid lines in panels (a, b, c) denote the median value, the error bars correspond to the 25th–75th percentiles, and the circles and the stars denote the 2.5th and 97.5th percentiles, respectively. (d, e): Mismatch histogram of measured mismatches ${\tilde{\mu }}_0/{\tilde{\mu }}_{\max }$ and theoretical distribution in a ${\mathbb{Z}}_6$ lattice. (f, g): Measured mismatch ${\tilde{\mu }}_0$ vs predicted $\tilde{\mu }$.

Figure 2

Semicoherent long-segment regime: results for metric tests on ${\widehat{g}}^{\mathrm{LS}}$ of Sec. 4b2 for the orbital period scale of $P_0=2$ h and fixed segment length of $\mathrm{\Delta }T=1$ d. (a) Relative error $ϵ({\widehat{\mu }}_0,\widehat{\mu })$ vs eccentricity $e$. The dashed vertical line denotes the boundary between a low-$e$ range $e\in [10^{-5},0.1]$ (left-column plots) and a high-$e$ range $e\in [0.1,0.9]$ (right-column plots). (b, c) Relative error $ϵ$ vs observation time $T_{\mathrm{obs}}$. The solid lines in panels (a, b, c) denote the median value, the error bars correspond to the 25th–75th percentiles, and the circles and the stars denote the 2.5th and 97.5th percentiles, respectively. (d, e) Mismatch histogram of measured mismatches ${\widehat{\mu }}_0/{\widehat{\mu }}_{\max }$ and theoretical distribution in a ${\mathbb{Z}}_6$ lattice. (f, g) Measured mismatch ${\widehat{\mu }}_0$ vs predicted $\widehat{\mu }$.

Figure 3

Effective template dimension $k^{\prime }$ of Eq. (83) vs segment length $\mathrm{\Delta }T/P$ for the software-injection studies shown in Fig. 4. Solid line: mean value, error bars: maximum and minimum of $k^{\prime }$.

Figure 4

Coherent short-segment regime: results for metric tests on ${\tilde{g}}^{\mathrm{SS}}$ of Sec. 4c1 using $u$ coordinates, for an orbital period scale of $P_0=80\text{d}$ and coherent observation times in the range $\mathrm{\Delta }T\in [2,25]d$. (a) Relative error $ϵ({\tilde{\mu }}_0,\tilde{\mu })$ vs eccentricity $e$. The dashed vertical line denotes the boundary between a low-$e$ range $e\in [10^{-5},10^{-3}]$ (left-column plots) and a high-$e$ range $e\in [10^{-3},10^{-2}]$ (right-column plots). (b, c) Relative error $ϵ$ vs observation time $\mathrm{\Delta }T/P$. The solid lines in panels (a, b, c) denote the median value, the error bars correspond to the 25th–75th percentiles, and the circles and the stars denote the 2.5th and 97.5th percentiles, respectively. (d, e) Mismatch histogram of measured mismatches ${\tilde{\mu }}_0/{\tilde{\mu }}_{\max }$ and theoretical distribution in a ${\mathbb{Z}}_6$ lattice. (f, g) Measured mismatch ${\tilde{\mu }}_0$ vs predicted $\tilde{\mu }$. Note that in all plots except (b, c) the range of relative segment length was restricted to $\mathrm{\Delta }T/P\le 0.2$, where the short-segment limit approximation is seen to be valid.

Figure 5

Semicoherent short-segment regime: results for metric tests on ${\widehat{g}}^{\mathrm{SS}}$ of Sec. 4c2 for an orbital period scale of $P_0=10$ d and a fixed segment length of $\mathrm{\Delta }T=1$ d. (a) Relative error $ϵ({\widehat{\mu }}_0,\widehat{\mu })$ vs eccentricity $e$. The dashed vertical line denotes the boundary between a low-$e$ range $e\in [10^{-5},0.1]$ (left-column plots) and a high-$e$ range $e\in [0.1,0.9]$ (right-column plots). (b, c) Relative error $ϵ$ vs observation time $T_{\mathrm{obs}}$. The solid lines in panels (a, b, c) denote the median value, the error bars correspond to the 25th–75th percentiles, and the circles and the stars denote the 2.5th and 97.5th percentiles, respectively. (d, e) Mismatch histogram of measured mismatches ${\widehat{\mu }}_0/{\widehat{\mu }}_{\max }$ and theoretical distribution in a ${\mathbb{Z}}_6$ lattice. (f, g) Measured mismatch ${\widehat{\mu }}_0$ vs predicted $\widehat{\mu }$.

Figure 6

Sensitivity depth ${\mathcal{D}}^{\text{Sco}\mathrm{X}1}$ of Eq. (84) for Scorpius X-1 torque-balance level Eq. (85) vs CW frequency $\overline{f}$. Different line styles correspond to different progressive aLIGO sensitivities (early, mid, late, and final) [53, 54, 55]. For each detector sensitivity, three different emission models are shown, namely, top: mountain ($\overline{f}=2\nu$), mid: r mode ($\overline{f}\approx 4\nu /3$), bottom: precession ($\overline{f}\approx \nu$), respectively.

Figure 7

Effective number of (coherent or semicoherent) templates per dimension ${\mathcal{N}}_{\lambda }$ of Eq. (70) (in the long-segment regime) as a function of CW frequency $\overline{f}$ for the Scorpius X-1 parameter space given in Table 1, using $\pm 3$-sigma ranges. The template numbers are evaluated for ${\mu }_{\max }=0.1$ and mean values of $a_{\mathrm{p}}$ and $\mathrm{\Omega }$. Note that assuming a $2\pi$ uncertainty in $\omega$ entails ${\mathcal{N}}_{\{e,\omega \}}={\mathcal{N}}_e^2\pi$. The template count ${\mathcal{N}}_{\mathrm{\Omega }}$ is the only one depending on the observation time $T$ and is shown for $T=1\mathrm{d},10\mathrm{d},90\mathrm{d}$.

Figure 8

Maximal allowed SFT length $T_{\mathrm{sft}}$ of Eq. (c2) for a Scorpius X-1 demodulation $\tilde{\mathcal{F}}$-statistic search as a function of frequency $\overline{f}$, for three different tolerated maximal mismatches ${\mu }_{\mathrm{sft}}$.