Prospects for Detecting Gravitational Waves from Eccentric Subsolar Mass Compact Binaries

Gravitational-wave astronomy has gradually evolved into a routine method for observing compact binaries comprising black holes and/or neutron stars. Up to now, over 50 gravitational-wave events have been identified by Advanced LIGO (Aasi et al. 2015) and Virgo (Acernese et al. 2015), all from compact binary coalescences (Nitz et al. 2019; Abbott et al. 2020). These detections have had significant astrophysical implications, including inferring stellar population properties (The LIGO Scientific Collaboration 2020a), testing the validity of general relativity (The LIGO Scientific Collaboration et al. 2020b), and determining the value of the Hubble constant (Abbott et al. 2017).

In addition to routine detection, gravitational-wave data analysis is also searching for exotic hypothesized objects, such as primordial black holes (Abbott et al. 2018a, 2019b; Nitz & Wang 2021). Primordial black holes are hypothesized to form by direct collapse in dense regions of the early universe (Zel'dovich & Novikov 1967; Hawking 1971; Carr & Hawking 1974) and are consistent with the properties required for a candidate of cold dark matter (Carr & Kuhnel 2020). A variety of astrophysical observations including from gravitational lensing have searched for primordial black holes and placed upper limits on the abundance with their null results, which shows that primordial black holes with a single mass are not likely to account for all dark matter (see, e.g., Carr & Kuhnel 2020 and Green & Kavanagh 2021 for recent reviews). Nevertheless, as gravitational-wave astronomy provides new opportunities to observe stellar-mass black holes, we investigate the event rate and prospects for searching for gravitational waves emitted from binary primordial black holes in the LIGO/Virgo sensitive band (∼10–1000 Hz). In particular, we focus on subsolar mass binary mergers with nonzero eccentricity.

There are two viable ways for primordial black holes to form binaries. In the early universe, after their initial formation, a nearby pair of primordial black holes can form a binary if the gravitational attraction is strong enough to decouple them from the cosmic expansion (Nakamura et al. 1997; Sasaki et al. 2016; Boehm et al. 2021). To be detected by gravitational-wave detectors, the binaries would have had sufficient time to circularize their orbits, thus the gravitational-wave signals are not expected to have eccentricity by the time they enter the LIGO and Virgo sensitive band. On the other hand, primordial black hole binaries may also form by dynamical capture due to gravitational-wave braking in the late universe in dark matter halos. A significant feature for this formation channel is the retention of nonzero eccentricity in the LIGO/Virgo band if the binaries form at close separation. This latter scenario is the focus of this work.

Bird et al. (2016) has studied the event rate of GW150914-like, i.e., a 30M_⊙ binary black hole merger. Their results show that the event rate is consistent with the empirical rate estimated from GW150914 (Abbott et al. 2016a, 2016b). However, it is challenging to confirm that GW150914 is indeed primordial in origin and not the result of stellar evolution, leaving the question still open. Cholis et al. (2016) has investigated the eccentricity distribution for 30M_⊙ binaries in the late-universe scenario. Meanwhile, given conventional stellar evolution models, subsolar mass (≤1M_⊙) compact binary coalescence can only be due to primordial black holes, instead of stellar products, because they are lighter than the minimum mass of neutron stars (Timmes et al. 1996; Suwa et al. 2018). Therefore, in this work we only consider the mass range [0.01, 1] M_⊙, which is a smoking gun for primordial black holes, extending the work of Bird et al. (2016) and Cholis et al. (2016) to subsolar mass region. Primordial black holes in this mass range are predicted by a variety of formation theories, e.g., early-universe quantum chromodynamics phase transition (Jedamzik 1997; Byrnes et al. 2018; Carr et al. 2021).

In Section 2, assuming a delta function distribution of mass, we first estimate the event rate and eccentricity distribution of subsolar mass primordial black hole binaries formed in dark matter halos. The merger rate of subsolar mass binaries is shown to be higher than that of 30M_⊙ binary primordial black holes, as considered in Bird et al. (2016), by one to two orders of magnitude. With design sensitivity, the Advanced LIGO and Virgo observatories would be expected to detect one 1M_⊙–1M_⊙ binary with eccentricity e^{10 Hz} ≥ 0.1 at a gravitational-wave frequency of 10 Hz with ∼10 years of observation, if the fraction of the primordial black hole in dark matter is 100%. Third generation detectors would detect primordial black hole binaries with a single mass varying from 0.01M_⊙ to 1M_⊙ and eccentricity distribution derived from this late-universe formation channel. Nondetection in the future would put constraints on the event rate modeling. In Section 3, we investigate the capability of the current match-filtering search pipeline using a quasi-circular waveform template bank to identify signals with eccentricity by simulated gravitational-wave data. Results show that 41% and 6% of the signals would be missed compared to the idealized maximum for 0.1M_⊙–0.1M_⊙ and 1M_⊙–1M_⊙ binaries, respectively, arising from the mismatch of gravitational-wave signals from eccentricity. Section 4 presents the discussions and conclusions.

In this section we consider the event rate and eccentricity distribution of compact binaries formed in the late universe through two-body encounters for a delta function mass distribution which we allow to vary within [0.01, 1]M_⊙. More complex mechanisms may also induce eccentric binaries, such as three-body interaction via Kozai–Lidov effects (Kozai 1962; Lidov 1962), but they are not considered in this work and investigations will be left for the future.

We briefly review the two-body dynamics for primordial black hole binary formation following Maggiore (2008), Peters & Mathews (1963), Turner (1977), O'Leary et al. (2009), Bird et al. (2016), and Cholis et al. (2016). Specifically, we derive the overall event rate for primordial black hole binary coalescence with a component mass M_PBH = 0.01/0.1/1 M_⊙. To aid in comparison to other works in this field, we state our eccentricity distributions at a fiducial dominant-mode gravitational-wave frequency of 10 Hz, which is denoted by e^{10 Hz}.

Two black holes can form a binary in an encounter due to gravitational-wave emission. The binary formation criterion is that the released gravitational-wave energy δ E_GW exceeds the two-body kinetic energy:

$$\begin{eqnarray}&&\delta {E}_{\mathrm{GW}}\geqslant \displaystyle \frac{1}{2}\mu {v}_{\mathrm{rel}}^{2},\end{eqnarray} \tag{ 1 }$$

where μ = m₁ m₂/(m₁ + m₂) is the reduced mass, m₁ and m₂ are the component masses of the binary, and v_rel is the relative velocity of two black holes. In this work we only consider equal-mass primordial black hole binaries, therefore we set m₁ = m₂ = M_PBH.

At a Newtonian order, the released gravitational-wave energy is (Peters & Mathews 1963; Turner 1977)

$$\begin{eqnarray}&&\delta {E}_{\mathrm{GW}}=\displaystyle \frac{8}{15}\displaystyle \frac{{G}^{7/2}}{{c}^{5}}\displaystyle \frac{{\left({m}_{1}+{m}_{2}\right)}^{1/2}{m}_{1}^{2}{m}_{2}^{2}}{{r}_{p}^{7/2}}f(e),\end{eqnarray} \tag{ 2 }$$

where G is the Newton gravitational constant, c is the speed of light, r_p is the closest distance at encounter, which is the pericenter distance for an elliptical orbit, and f(e) is a function of eccentricity e. By inserting $f(e)=425\pi /(32\sqrt{2})$ when e = 1 into Equations (1) and (2), one obtains the cross section for binary formation

$$\begin{eqnarray}&&\sigma =2\pi {\left(\displaystyle \frac{85\pi }{6\sqrt{2}}\right)}^{2/7}\displaystyle \frac{{G}^{2}{\left({m}_{1}+{m}_{2}\right)}^{10/7}{\left({m}_{1}{m}_{2}\right)}^{2/7}}{{c}^{10/7}{v}_{\mathrm{rel}}^{18/7}},\end{eqnarray} \tag{ 3 }$$

where $\sigma =\pi {b}_{\max }^{2}$ by definition, and b_max is the maximum impact parameter. Meanwhile, the impact parameter b is related to r_p by

$$\begin{eqnarray}&&{r}_{p}=\displaystyle \frac{{b}^{2}{v}_{\mathrm{rel}}^{2}}{2G({m}_{1}+{m}_{2})}\end{eqnarray} \tag{ 4 }$$

when $G({m}_{1}+{m}_{2})\gg {{bv}}_{\mathrm{rel}}^{2}$, which is satisfied in this study.

Since we aim to investigate the scenario where primordial black holes are dark matter, we assume the number density of primordial black holes follows that of dark matter. Specifically, for a dark matter halo with virial mass M_vir, we use the Navarro–Frenk–White (NFW) density profile (Navarro et al. 1996) to model the mass density

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\rho }_{c}}{\tfrac{r}{{r}_{c}}{\left(1+\tfrac{r}{{r}_{c}}\right)}^{2}},\end{eqnarray} \tag{ 5 }$$

where r is the radial distance to the halo center, and ρ_c and r_c are the characteristic mass density and characteristic radius of the NFW profile, respectively.

The binary primordial black hole formation rate is given by Bird et al. (2016) as

$$\begin{eqnarray}&&{ \mathcal R }({M}_{\mathrm{vir}})=4\pi {\int }_{0}^{{R}_{\mathrm{vir}}}\displaystyle \frac{{r}^{2}}{2}{\left(\displaystyle \frac{{f}_{\mathrm{PBH}}{\rho }_{\mathrm{NFW}}(r)}{{M}_{\mathrm{PBH}}}\right)}^{2}\langle \sigma {v}_{\mathrm{rel}}\rangle {dr},\end{eqnarray} \tag{ 6 }$$

where R_vir is the halo virial radius, f_PBH is the mass fraction of primordial black holes in dark matter, and the angle bracket is averaged with respect to the relative velocity distribution, which is modeled to be a Maxwell–Boltzmann distribution

$$\begin{eqnarray}&&P({v}_{\mathrm{rel}})={N}_{0}\left[\exp \left(-\displaystyle \frac{{v}_{\mathrm{rel}}^{2}}{{v}_{\max }^{2}}\right)-\exp \left(-\displaystyle \frac{{v}_{\mathrm{vir}}^{2}}{{v}_{\max }^{2}}\right)\right]\end{eqnarray} \tag{ 7 }$$

in which v_vir is the circular velocity at virial radius, v_max is the maximum circular velocity, and N₀ is the normalization factor. As demonstrated by Bird et al. (2016), the merger happens shortly after the formation of a binary, therefore the formation rate (Equation (6)) is identical to the event rate for coalescence of binary primordial black holes. Equation (6) can be integrated analytically, the result is

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal R }({M}_{\mathrm{vir}}) & = & {\left(\displaystyle \frac{85\pi }{6\sqrt{2}}\right)}^{2/7}\displaystyle \frac{2\pi {G}^{2}{f}_{\mathrm{PBH}}^{2}{M}_{\mathrm{vir}}^{2}D({v}_{\max })}{3{{cr}}_{s}^{3}{g}^{2}(C)}\\ & \times & \left[1-\displaystyle \frac{1}{{\left(1+C\right)}^{3}}\right],\end{array}\end{eqnarray} \tag{ 8 }$$

where C = R_vir/r_c is the concentration parameter for a dark matter halo, $g(C)=\mathrm{ln}(1+C)-C/(1+C)$ and

$$\begin{eqnarray}&&D({v}_{\mathrm{rel}})={\int }_{0}^{{v}_{\mathrm{vir}}}P({v}_{\mathrm{rel}};{v}_{\max }){\left(\displaystyle \frac{2v}{c}\right)}^{3/7}{dv}.\end{eqnarray} \tag{ 9 }$$

Finally, the overall event rate for binary primordial black hole coalescence is obtained by taking the sum of the contributions from all dark matter halos

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{PBH}}=\int { \mathcal R }({M}_{\mathrm{vir}})\displaystyle \frac{{dN}}{{{dM}}_{\mathrm{vir}}}{{dM}}_{\mathrm{vir}}.\end{eqnarray} \tag{ 10 }$$

We have made use of the python code colossus (Diemer 2018) to generate the dark matter halo mass function ${dN}/{{dM}}_{\mathrm{vir}}$ using the parameterized formula fitting to an N-body simulation proposed by Tinker et al. (2008), and the concentration–dark matter halo mass relation proposed by Prada et al. (2012). We have tested systematics by using Raeymaekers (2007) for N-body simulation fits for the dark matter halo mass function, and Dutton & Macciò (2014) for the concentration–halo mass relation. The results show a different choice of dark matter halo population modeling will not change the final event rate by orders of magnitude.

Assuming f_PBH = 100%, the result of Equation (10) is shown in Figure 1. The main contribution to the integral in Equation (10) is from low-mass dark matter halos, therefore the result depends sensitively on the lower end for halo mass. To illustrate the dependence on lower mass cutoff ${M}_{\mathrm{vir}}^{\mathrm{low}}$, we plot ${{ \mathcal R }}_{\mathrm{PBH}}$ as a function of ${M}_{\mathrm{vir}}^{\mathrm{low}}$ in Figure 1. Also note that the integrand of Equation (10) does not depend on M_PBH due to compensation between the primordial black hole number density and the cross section. To determine the lower limit of the dark matter halo mass for M_PBH = 0.01/0.1/1 M_⊙, we follow the criterion by Equation (11) of Bird et al. (2016) requiring that the dark matter halo evaporation timescale due to dynamical relaxation exceeds the dark energy domination timescale (∼3 × 10⁹ yr). This choice results in the lower integral limit of Equation (10) being 0.3/3/21 M_⊙ for M_PBH = 0.01/0.1/1 M_⊙. Choosing a lighter dark matter halo cutoff would contribute more binary primordial black hole merger events if the halos would not have evaporated at present and the NFW profile is still preserved. High resolution dark matter N-body simulations are needed to precisely resolve the smallest structure of subsolar mass primordial black hole halos but this is beyond the scope of this paper.

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With the above choices of lower integral cutoff, as shown in Figure 1, the event rate for M_PBH = 0.01/0.1/1M_⊙ is 528/140/45 Gpc⁻³ yr⁻¹. As a comparison, for M_PBH = 30M_⊙, the event rate is ${ \mathcal O }(1)$ Gpc⁻³ yr⁻¹, as computed in Bird et al. (2016). Therefore, the merger rate of subsolar primordial black holes increases by one to two orders of magnitude compared to that of 30M_⊙.

To compare with observations, the sensitive volume, which is defined to be the orientation-averaged volume assuming a match-filtering signal-to-noise ratio (S/N) > 8 (Abbott et al. 2018b), is ${ \mathcal O }({10}^{-7})/{ \mathcal O }({10}^{-4})/{ \mathcal O }({10}^{-2})$ Gpc³ for M_PBH = 0.01/0.1/1M_⊙ for Advanced LIGO at its design sensitivity. Given the above estimated event rate, under our assumption of delta function mass distribution, the second generation detectors are capable of detecting M_PBH = 1M_⊙ sources with a ${ \mathcal O }(1)$ year run, and would not be likely to observe 0.1M_⊙ and 0.01M_⊙ sources. Third generation detectors such as the Cosmic Explorer (Reitze et al. 2019) and Einstein Telescope (Punturo et al. 2010) are expected improve the sensitive volume by three orders of magnitude compared to the second generation, thus they would be able to detect a delta function mass of M_PBH = 0.1M_⊙ source every ${ \mathcal O }(0.1)$ yr, and a 0.01M_⊙ source with a ${ \mathcal O }(10)$ yr observation time. Conversely, a nondetection in the future would put constraints on the abundance of primordial black holes formed in this late-universe channel.

We also compare the binary coalescence rate to that derived from primordial black hole binary formation in the early universe. Using the derivation by Sasaki et al. (2016), the merger rate is ${ \mathcal O }({10}^{8})/{ \mathcal O }({10}^{7})/{ \mathcal O }({10}^{6})$ Gpc⁻³ yr⁻¹ for M_PBH = 0.01/0.1/1 M_⊙ assuming f_PBH = 100%. Therefore, an early-universe formation scenario dominates the event rate for subsolar mass compact binary coalescence, but this formation channel is not expected to yield eccentric binaries because they would have been circularized in the local universe. There are also model uncertainties under active investigation about what fraction of primordial black hole binaries would be disrupted after formation in the early universe (Ali-Haïmoud et al. 2017; Raidal et al. 2019; Jedamzik 2020; Vaskonen & Veermäe 2020). If a significant fraction are disrupted, the merger rate for the early-universe formation channel would be suppressed. Recently, Boehm et al. (2021) also pointed out that the merger rate of the early-universe formation channel is overestimated by reanalysis of the primordial black hole binary formation criterion.

Compact binaries formed through dynamical interaction may retain eccentricity within the ground-based gravitational-wave detector frequency band. In the following we determine the eccentricity distribution for binary primordial black hole merger events following O'Leary et al. (2009) and Cholis et al. (2016).

Right after formation, the initial semimajor distance of a binary is given by

$$\begin{eqnarray}&&{a}_{0}=\displaystyle \frac{{{Gm}}_{1}{m}_{2}}{2| {E}_{f}| },\end{eqnarray} \tag{ 11 }$$

where ${E}_{f}=1/2\mu {v}_{\mathrm{rel}}^{2}-\delta {E}_{\mathrm{GW}}$. Combining Equations (4) and (11) and inserting the relation r_p,0 = a₀(1 − e₀), one obtains an expression for the initial eccentricity at binary formation of

$$\begin{eqnarray}&&{e}_{0}({m}_{1},{m}_{2},{v}_{\mathrm{rel}},b)=\sqrt{1-\displaystyle \frac{2| {E}_{f}| {b}^{2}{v}_{\mathrm{rel}}^{2}}{({m}_{1}+{m}_{2}){m}_{1}{m}_{2}}}.\end{eqnarray} \tag{ 12 }$$

After formation of the binary, the semimajor distance a and the eccentricity e gradually decay due to gravitational-wave emission; the time evolution equation is given by (Peters 1964)

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{da}}{{dt}} & = & -\displaystyle \frac{64}{5}\displaystyle \frac{{G}^{3}{m}_{1}{m}_{2}({m}_{1}+{m}_{2})}{{c}^{5}{a}^{3}{\left(1-{e}^{2}\right)}^{7/2}}\left(1+\displaystyle \frac{73}{24}{e}^{2}+\displaystyle \frac{37}{96}{e}^{4}\right),\\ \displaystyle \frac{{de}}{{dt}} & = & -\displaystyle \frac{304}{15}\displaystyle \frac{{G}^{3}{m}_{1}{m}_{2}({m}_{1}+{m}_{2})e}{{c}^{5}{a}^{4}{\left(1-{e}^{2}\right)}^{5/2}}\left(1+\displaystyle \frac{121}{304}{e}^{2}\right).\end{array}\end{eqnarray} \tag{ 13 }$$

Combining Equation (13) yields the expression for da/de which can be integrated analytically, the result is

$$\begin{eqnarray}&&a(e)={a}_{0}\displaystyle \frac{\kappa (e)}{\kappa ({e}_{0})},\end{eqnarray} \tag{ 14 }$$

where κ(e) is a function of e

$$\begin{eqnarray}&&\kappa (e)=\displaystyle \frac{{e}^{12/19}}{1-{e}^{2}}{\left(1+\displaystyle \frac{121}{304}{e}^{2}\right)}^{870/2299}.\end{eqnarray} \tag{ 15 }$$

Substituting Equations (11) and (12) and $a=\sqrt[3]{G({m}_{1}+{m}_{2})/{\pi }^{2}{f}_{\mathrm{GW}}^{2}}$ from Kepler's third law where the frequency f_GW = 10 Hz, the nonlinear Equation (14) for e^{10 Hz} can be solved by numerically finding the root.

As e^{10 Hz} depends on e₀, which in turn depends on the initial relative velocity v_rel and the impact parameter b for two black holes, we compute the relative velocity distribution by taking the derivative of ${{ \mathcal R }}_{\mathrm{PBH}}$ with respect to v_rel. The results are presented in Figure 2. It can be seen that the initial relative velocities for M_PBH = 0.01/0.1/1 M_⊙ peak at 5/10/25 m s⁻¹, respectively, which approximately correspond to the typical velocity of the lightest dark matter halo in the integral Equation (10), because the merger events from the low-mass dark matter halo dominate the event rate.

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We use a Monte Carlo method to simulate a population of binary primordial black holes with masses 0.01/0.1/1M_⊙, respectively, each containing ∼10⁶ sources, to obtain the distribution of e^{10 Hz}. The square of the impact parameter, b², is chosen to be uniformly distributed in $[{r}_{\mathrm{ISCO}}^{2},{b}_{\max }^{2}]$ (Cholis et al. 2016), where r_ISCO is the innermost stable circular orbit for a Schwarzschild black hole (6 times the Schwarzschild radius), b_max is given by Equation (3). The initial relative velocity is drawn from the distribution in Figure 2.

The results for the distribution of e^{10 Hz} are shown in Figure 3 by the solid lines with the corresponding probability density distribution shown on the left vertical axis. To assist understanding, we also plot the cumulative probability distribution in Figure 3 as shown by the dashed lines converging to 100% at the right vertical axis. As seen from the figure, lighter primordial black holes tend to retain larger eccentricity. For M_PBH = 0.01/0.1/1 M_⊙, up to 89%/40%/12% of the total inspiral events have e^{10 Hz} ≥ 0.01, and up to 29%/9%/3% have e^{10 Hz} ≥ 0.1. To connect these estimates with observations, assuming f_PBH = 100%, we find that second generation detectors would be expected to detect an eccentric binary coalescence with M_PBH = 1M_⊙ within a few years observation time, while the third generations are expected to probe the eccentric events down to 0.01M_⊙ with 10³ times the sensitive volume than Advanced LIGO and Virgo. In addition, third generation detectors are designed to increase sensitivity at frequencies of ∼2–5 Hz, where the eccentricity will be more significant. Third generation detectors show promise for detecting or constraining our understanding of eccentric binaries.

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We further investigate the impact of M_PBH and v_rel on the eccentricity distribution. In Figure 4, the mass M_PBH is fixed to 0.1 M_⊙, and the initial relative velocity v_rel is chosen to be 20/200/2000 m s⁻¹. The e^{10 Hz} results show that a higher relative velocity leads to the formation of more eccentric binaries. The cutoff for each distribution at the lowest eccentricity in Figure 4 represents the binaries that would have been circularized the most at the gravitational-wave frequency 10 Hz and are from those with the widest separation at initial formation and thus the lowest initial eccentricity, as can be seen from Equation (12).

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The eccentricity distribution for fixed v_rel but different masses are in shown in Figure 5. The relative velocity is chosen to be 2 km s⁻¹, which is the typical velocity for 10⁶ M_⊙ dark matter halos. The black hole mass is chosen to be 0.01/0.1/1/30 M_⊙. As considered by Cholis et al. (2016) and shown in Figure 5, a majority of 30 M_⊙ binary black holes have eccentricity near 0 at 10 Hz. However, the subsolar mass binaries all have nonzero peaks for eccentricity. In particular, for 0.01 M_⊙ primordial black holes in the 10⁶ M_⊙ dark matter halos, all binaries have high eccentricity not lower than 0.7 at formation. Similarly, the lowest eccentricity cutoff for each distribution represents the most circularized binaries in the simulated sources.

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The above numerical results show that subsolar mass compact binaries with high initial relative velocity tend to be more eccentric compared to the more massive binary black holes with low relative velocity. Therefore, the impact of eccentricity on targeted gravitational-wave searches for subsolar mass compact binary coalescences deserves a more detailed study. We present our investigation in the next section.

There has been one search targeting gravitational waves from eccentric compact binary coalescence using model-based matched filtering (Nitz et al. 2020), where the authors searched for signals from eccentric binary neutron star mergers. The LIGO and Virgo Scientific Collaborations have also performed a nonmodeled generic search for eccentric binary black hole mergers (Abbott et al. 2019a). Most targeted searches for compact binary coalescence use quasi-circular orbit waveform template banks, including the direct searches for subsolar mass black holes (Abbott et al. 2018a, 2019b; Nitz & Wang 2021),

As demonstrated in Section 2, subsolar mass primordial black holes can retain nonzero eccentricity in the LIGO and Virgo frequency band if they form in the late universe through dynamical capture. We aim to quantitatively study the potential loss in detection sensitivity when using only quasi-circular waveform template banks to search for signals from eccentric binary primordial black holes. To quantify the S/N of gravitational-wave detection, we define the following noise-weighted inner product

$$\begin{eqnarray}&&({h}_{1},{h}_{2})=4{\mathfrak{R}}\int \displaystyle \frac{{h}_{1}(f){h}_{2}^{* }(f)}{{S}_{n}(f)},\end{eqnarray} \tag{ 16 }$$

where h₁ and h₂ are signals or gravitational-wave templates in the Fourier domain and S_n (f) is the one-sided noise power spectral density. In this section we only consider the Advanced LIGO design sensitivity in the frequency range of [20, 1024] Hz. The matched-filter S/N between the detector output s and the template h is defined to be

$$\begin{eqnarray}&&\rho =\displaystyle \frac{(s,h)}{\sqrt{(h,h)}}.\end{eqnarray} \tag{ 17 }$$

To measure the similarity between two waveform templates h₁ and h₂, the overlap function is defined as the normalized inner product

$$\begin{eqnarray}&&{ \mathcal O }({h}_{1},{h}_{2})=\displaystyle \frac{({h}_{1},{h}_{2})}{\sqrt{({h}_{1},{h}_{1})({h}_{2},{h}_{2})}}.\end{eqnarray} \tag{ 18 }$$

Two templates may be different up to a constant time and phase in the Fourier domain, thus the match function between two waveforms is given by maximizing over the offset of coalescence time t_c and an overall phase ϕ_c

$$\begin{eqnarray}&&{ \mathcal M }({h}_{1},{h}_{2})=\mathop{\max }\limits_{{t}_{c},{\phi }_{c}}\left({ \mathcal O }({h}_{1},{h}_{2}{e}^{i(2\pi {{ft}}_{c}-{\phi }_{c})})\right).\end{eqnarray} \tag{ 19 }$$

There have been several waveform approximants modeling eccentric compact binary coalescence, such as Moore et al. (2016), Huerta et al. (2018), Liu et al. (2020), and Chiaramello & Nagar (2020). In this work, we utilize the model proposed by Moore & Yunes (2019, 2020) and Moore et al. (2018), referred to as TaylorF2e, ³ for speedy waveform generation. TaylorF2e is an inspiral waveform developed in the Fourier domain and is accurate to the third post-Newtonian order. The eccentricity is valid up to at least 0.8 in the low-mass case ∼1M_⊙ (Moore & Yunes 2020), which is sufficient for our purpose to investigate the eccentricity of subsolar mass compact binaries.

In order to extract gravitational-wave signals in the data, a targeted search (Usman et al. 2016; Messick et al. 2017) builds a precalculated template bank to match filter the data. To measure the detection ability of a template bank built with quasi-circular waveforms to search for eccentric binaries, we first generate a template bank by a geometric method based on hexagonal lattice placement (Cokelaer 2007; Brown et al. 2012) using the quasi-circular waveform approximant TaylorF2 (Buonanno et al. 2009). For this study, we assume the component black holes will have negligible spin. The template bank is designed to recover sources with component masses in the range [0.1, 1] M_⊙. The bank is discrete in the parameter space, and we require the maximum mismatch due to discreteness is no higher than 3% by construction. The above setting results in a bank with 6,995,517 templates.

Next we generate simulated gravitational-wave data with a source population using the TaylorF2e approximant. The mock data are then compared with every template in the bank using Equation (19) to obtain the maximum match "fitting factor" over the entire template bank. A fitting factor of x would lead to a 1 − x³ loss of the detection rate for a search with a fixed amount of noise, as the gravitational-wave search S/N would be decreased to x times the theoretically optimal value.

In Figure 6, we generate eccentric subsolar mass binary black hole gravitational-wave signals and compute the fitting factor with respect to the noneccentric template bank. Each point in the figure represents a mock signal injection.

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In the left panel of Figure 6, the binaries are chosen to have equal mass. M_PBH and e^{10 Hz} are uniformly distributed in [0.1, 1] M_⊙ and [0, 0.2], respectively. The fitting factor for e^{10 Hz} and the chirp mass ${M}_{\mathrm{chirp}}={\left({m}_{1}{m}_{2}\right)}^{3/5}{\left({m}_{1}+{m}_{2}\right)}^{1/5}$ (∼0.87M_PBH for equal mass) is presented. The figure also illustrates the 97% and 80% contour lines obtained from fitting the plot. The figure shows the trend that the fitting factor gets worse for higher eccentricity, and also for lighter binaries because of their longer duration. For a component mass m = 0.1/1M_⊙, the fitting factor drops below 97% with e^{10 Hz} ≥ 0.003/0.20.

In the right panel of Figure 6, we consider the impact on fitting factor from different mass ratios by fixing M_chirp = 0.4 M_⊙. The mass ratio is uniformly distributed in [1, 4] and the eccentricity is uniformly in [0, 0.2]. As shown, the 97% and 80% fitting factor lines only depend weakly on the mass ratio and decrease for higher mass ratio.

We also consider more realistic distributions for e^{10 Hz} as derived in Figure 3 of Section 2. We generate two groups of mock data each with ∼2000 signals using TaylorF2e. The component mass is 0.1M_⊙ and 1M_⊙, respectively, and only equal-mass binaries are considered. The inclination angle is chosen to be uniformly distributed in [−π/2, π/2], the polarization angle is uniformly in [0, 2π], and the source sky location is isotropic. The eccentricity distribution is drawn from Figure 3 for 0.1M_⊙ and 1M_⊙ sources accordingly.

The cumulative probability with respect to the fitting factor is shown in Figure 7. The vertical dashed line in Figure 7 denotes 97%, which all the signals with zero eccentricity should exceed by construction of the template bank. However, due to the existence of eccentricity, for M_PBH = 0.1M_⊙, up to 68% of the signals have a fitting factor lower than 97%, and 41% have a fitting factor lower than 80%. For M_PBH = 1 M_⊙, 15% of the signals have a fitting factor lower than 97%, and 6% of the signals have a fitting factor lower than 80%. Also note that, when the fitting factor is low, the mismatch between signals and templates may be significant enough to cause issues with signal-based vetoes (Allen 2005) which would further reduce the detection rate of eccentric sources. Therefore our results based on the fitting factor are a conservative estimation for the loss of detection rate.

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Given that the fitting factor can only measure the fractional recovered S/N for a single injected signal, we use the effective fitting factor FF_eff (Buonanno et al. 2003; Harry et al. 2014, 2016) to account for the overall fraction for detection loss for a whole group of simulated signals. The effective fitting factor is defined as a mean average of fitting factors weighted by the signal amplitude

$$\begin{eqnarray}&&{\mathrm{FF}}_{\mathrm{eff}}={\left(\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}{\mathrm{FF}}_{i}^{3}{\sigma }_{i}^{3}}{{\displaystyle \sum }_{i=1}^{N}{\sigma }_{i}^{3}}\right)}^{1/3},\end{eqnarray} \tag{ 20 }$$

where the subscript i denotes the ith mock injection, N is the total number of signals, FF_i is the fitting factor, and ${\sigma }_{i}=\sqrt{({h}_{i}| {h}_{i})}$ is the optimal S/N for the ith signal. An effective fitting factor of x would lead to a 1 − x³ loss of detection rate.

For our two groups of injection with M_PBH = 0.1 M_⊙ and 1M_⊙, FF_eff = 83% and 95.8%, respectively, which corresponds to an overall loss of 42% and 12% of signals compared to the idealized maximum. For comparison, if only selecting the injected signals with e^{10 Hz} = 0, the effective fitting factor is 99.7% for M_PBH = 0.1M_⊙ and 97.8% for M_PBH = 1M_⊙, which corresponds to a 0.9% and 6.4% loss, which arises from the discreteness of template bank. After subtracting the signals missed due to template bank discreteness, for M_PBH = 0.1M_⊙ and M_PBH = 1 M_⊙ sources within the late-universe dynamical encounter model, we conclude that the eccentricity effects can lead to a loss of 41% and 6% of the signals, respectively, by a targeted search with a quasi-circular waveform template bank using Advanced LIGO at its designed sensitivity. For primordial black holes with an extended mass in the range [0.1, 1]M_⊙, we expect the the corresponding results for loss of detection should lie in between these.

In this work, we investigate the prospects for detecting gravitational-wave signals from eccentric subsolar mass binary black hole inspirals, which would be a signature for primordial black hole binaries formed by dynamical capture in the local universe. The merger rate and eccentricity distribution for binaries assuming a delta function mass distribution varying in [0.01, 1]M_⊙ are derived. For 1M_⊙–1 M_⊙ binaries, the detection rate is ${ \mathcal O }(1)$ yr⁻¹ for Advanced LIGO and Virgo with design sensitivity, if primordial black holes account for a majority of dark matter, and 12%(3%) of the sources have e^{10 Hz} ≥ 0.01(0.1), thus eccentric signals may be detected with a few years of observation based on the assumed model. For 0.1–0.1M_⊙ and 0.01–0.01 M_⊙ binaries the detection rate would be too low for the second generation ground-based detectors, but it is promising for detection by the third generation detectors. Conversely, nondetection in the future would put constraints on our understanding of this formation channel.

We also investigate what fraction of the eccentric binaries can be missed by using quasi-circular waveform template bank. Results show that current targeted searches can miss 41% and 6% of the events with component masses of 0.1M_⊙ and 1M_⊙, respectively, with Advanced LIGO designed sensitivity, due to the eccentricity of primordial black hole binaries arising from dynamical encounter in the local universe.

For the formation scenario of primordial black hole binaries, we only consider the two-body direct capture through gravitational-wave braking. Other mechanisms such as three-body interaction via Kozai–Lidov effects (Kozai 1962; Lidov 1962) may further improve the event rate, and would be interesting to consider in a future work.

Third generation gravitational-wave detectors such as the Cosmic Explorer (Reitze et al. 2019) and Einstein Telescope (Punturo et al. 2010) are being planned. The detection volume is expected to increase by three orders of magnitude compared to the second generation. Moreover, the sensitive band can be pushed down to ∼2 Hz, at which the eccentricity for primordial black hole binaries is more significant than later at 10 Hz as considered in this work. Third generation detectors are even more promising for hunting for eccentric gravitational-wave signals.

The code and data associated with this work are available at https://github.com/gwastro/prospects-subsolarmass-ecc.

We acknowledge the Max Planck Gesellschaft for support and the Atlas cluster computing team at AEI Hannover.