LISA Capture Sources: Approximate Waveforms, Signal-to-Noise Ratios, and Parameter Estimation Accuracy

TL;DR

The paper develops a fast, approximate post-Newtonian family of waveforms for LISA capture sources—stellar-mass COs inspiraling into $M\sim10^6 M_⊙$ MBHs—incorporating eccentricity, pericenter precession, and Lense–Thirring precession, and couples them to a low-frequency LISA response. Using these templates, it analyzes harmonic content, SNR buildup, and Fisher-matrix–based parameter estimation, highlighting how higher harmonics and multi-year observations improve mass, spin, sky localization, and distance inferences. The study finds that for a typical $10 M_⊙$ CO at $D o1$ Gpc and SNR~30, MBH and CO masses can be determined to ~$10^{-4}$ fractional precision, $S/M^2$ to ~ $10^{-4}$, and the sky location to ~ $10^{-3}$ sr, with extrinsic parameters remaining moderately well constrained; for galactic-center captures, spin can be measured to parts in $10^{-3}$ and distances to ~ a few percent. These results are illustrative, modular, and intended to guide future, more accurate waveform models and data-analysis strategies ahead of LISA launch.

Abstract

Captures of stellar-mass compact objects (COs) by massive ($\sim 10^6 M_\odot$) black holes (MBHs) are potentially an important source for LISA, the proposed space-based gravitational-wave (GW) detector. The orbits of the inspiraling COs are highly complicated; they can remain rather eccentric up until the final plunge, and display extreme versions of relativistic perihelion precession and Lense-Thirring precession of the orbital plane. The strongest capture signals will be ~10 times weaker than LISA's instrumental noise, but in principle (with sufficient computing power) they can be disentangled from the noise by matched filtering. The associated template waveforms are not yet in hand, but theorists will very likely be able to provide them before LISA launches. Here we introduce a family of approximate (post-Newtonian) capture waveforms, given in (nearly) analytic form, for use in advancing LISA studies until more accurate versions are available. Our model waveforms include most of the key qualitative features of true waveforms, and cover the full space of capture-event parameters (including orbital eccentricity and the MBH's spin). Here we use our approximate waveforms to (i) estimate the relative contributions of different harmonics (of the orbital frequency) to the total signal-to-noise ratio, and (ii) estimate the accuracy with which LISA will be able to extract the physical parameters of the capture event from the measured waveform. For a typical source (a $10 M_\odot$ CO captured by a $10^6 M_\odot$ MBH at a signal-to-noise ratio of 30), we find that LISA can determine the MBH and CO masses to within a fractional error of $\sim 10^{-4}$, measure $S/M^2$ (where $S$ and $M$ are the MBH's mass and spin) to within $\sim 10^{-4}$, and determine the sky location of the source to within $\sim 10^{-3}$ stradians.

Figures (17)

• Figure 1: The MBH-CO system: setup and notation. $M$ and $\mu$ are the masses of the MBH and the CO, respectively. The axes labeled $x{-}y{-}z$ represent a Cartesian system based on ecliptic coordinates (the Earth's motion around the Sun is in the x--y plane). The spin $\vec{S}$ of the MBH is parametrized by its magnitude $S$ and the two angular coordinates $\theta_K,\phi_K$, defined (in the standard manner) based on the system $x{-}y{-}z$. $\vec{L}(t)$ represents the (time-varying) orbital angular momentum; its direction is parametrized by the (constant) angle $\lambda$ between $\vec{L}$ and $\vec{S}$, and by an azimuthal angle $\alpha(t)$ (not shown in the figure). The angle $\tilde{\gamma}(t)$ is the (intrinsic) direction of pericenter, as measured with respect to $\vec{L}\times\vec{S}$. Finally, $\Phi(t)$ denotes the mean anomaly of the orbit, i.e., the average orbital phase with respect to the direction of pericenter.
• Figure 2: Evolution of orbits in our model, for a system composed of a $1 M_{\odot}$ CO inspiralling into a $10^6 M_{\odot}$ (non-spinning) MBH. The dashed line represents the last stable orbit (LSO). Each of the solid lines shows the $\nu-e$ trajectory of a system with given initial data (the orbit evolves in time "from bottom to top"). The four dots plotted along each trajectory indicate, from bottom to top, the state of the system 10, 5, 2, and 1 years before the LSO.
• Figure 3: Same as in Fig. \ref{['fig:evo1']}, for a $10 M_{\odot}$ CO inspiralling into a $10^6 M_{\odot}$ MBH.
• Figure 4: GW signal from a $10M_{\odot}$ CO spiralling into a (non-spinning) $10^6M_{\odot}$ MBH at $D=1\,$Gpc: case where eccentricity at the last stable orbit (LSO) is $e_{\rm LSO}=0.3$. The curve labeled '$S_{\rm inst}$' shows LISA's sky-averaged instrumental noise level, $h^{\rm inst}_n(f)$. The dashed line is an estimate of LISA's overall noise level, $h_n(f)$, including the effect of stochastic-background "confusion" due to WD binaries (both galactic and extra-galactic). The convex curves show the amplitudes $h_{c,n}$ of the first 10 $n$-harmonics of the GW signal, over the last 10 years of evolution prior to the final plunge. Along each of these curves we marked 3 dots, indicating (from left to right) the GW amplitude 5, 2, and 1 years before the plunge. The orbital eccentricity 10, 5, 2, and 1 years before plunge is 0.77, 0.67, 0.54, and 0.46, respectively. The orbital frequency 10, 5, 2, and 1 years before plunge is 0.23, 0.41, 0.70, and 0.94 mHz, respectively. The frequency at the LSO is 1.65 mHz.
• Figure 5: Same as in Fig. (\ref{['fig:SN1']}), but for inspiral of a $1M_{\odot}$ CO into a $10^6M_{\odot}$ MBH. The orbital eccentricity 10, 5, 2, and 1 years before plunge is 0.46, 0.40, 0.35, and 0.32, respectively. The orbital frequency $\nu$ 10, 5, 2, and 1 years before plunge is 0.94, 1.16, 1.39, and 1.51 mHz, respectively. The frequency at the LSO is 1.65 mHz.