Science with the space-based interferometer LISA. V: Extreme mass-ratio inspirals

TL;DR

This work assesses the potential of the space-based LISA detector to observe extreme mass-ratio inspirals (EMRIs) formed by stellar-mass compact objects around massive black holes with $M\sim 10^4$–$10^7\,M_\odot$. It builds a family of astrophysical EMRI population models, incorporating MBH mass functions, spin distributions, cusp erosion/regrowth after MBH mergers, EMRI rates, and CO properties, and pairs them with two computationally efficient kludge waveform families (AKS and AKK) to forecast detections and parameter-estimation precision using a Fisher-matrix approach. The study finds that the intrinsic EMRI rate can vary by up to about $10^3$ due to astrophysical uncertainties, yet LISA should detect at least a few EMRIs per year and up to a few thousand under optimistic assumptions; typical detected systems inhabit MBHs of $M\sim 10^5$–$10^6\,M_\odot$ at redshifts $z\lesssim 2$–$3$, with intrinsic parameter precisions of $\sim 10^{-6}$–$10^{-4}$ for masses and spins, distance accuracy around $10\%$, sky localization to a few square degrees, and percent-level constraints on possible deviations from the Kerr quadrupole moment. These measurements will illuminate MBH demographics, the stellar environments of galactic nuclei, and fundamental tests of general relativity, including no-hair theorems. The results demonstrate that robust EMRI science with LISA requires careful accounting of astrophysical uncertainties and waveform-model systematics, but promises transformative insights into gravity and cosmology.

Abstract

The space-based Laser Interferometer Space Antenna (LISA) will be able to observe the gravitational-wave signals from systems comprised of a massive black hole and a stellar-mass compact object. These systems are known as extreme-mass-ratio inspirals (EMRIs) and are expected to complete $\sim 10^4$-$10^5$ cycles in band, thus allowing exquisite measurements of their parameters. In this work, we attempt to quantify the astrophysical uncertainties affecting the predictions for the number of EMRIs detectable by LISA, and find that competing astrophysical assumptions produce a variance of about three orders of magnitude in the expected intrinsic EMRI rate. However, we find that irrespective of the astrophysical model, at least a few EMRIs per year should be detectable by the LISA mission, with up to a few thousands per year under the most optimistic astrophysical assumptions. We also investigate the precision with which LISA will be able to extract the parameters of these sources. We find that typical fractional statistical errors with which the intrinsic parameters (redshifted masses, massive black hole spin and orbital eccentricity) can be recovered are $\sim 10^{-6}$-$10^{-4}$. Luminosity distance (which is required to infer true masses) is inferred to about $10\%$ precision and sky position is localized to a few square degrees, while tests of the multipolar structure of the Kerr metric can be performed to percent-level precision or better.

Figures (13)

• Figure 1: MBH density mass function $\mathrm{d} n/\mathrm{d}\, \log_{10}M$ for the self-consistent model popIII at redshift $0$ (solid), $1$ (long dashed), $2$ (short dashed) and $3$ (dotted). The approximation provided by Eq. (\ref{['mfapprox']}) is shown as a thin straight black line. Also shown in brown is the redshift-independent pessimistic mass function as given by Eq. (\ref{['mfpess']}). The shaded area represent constraints from Shankar et al. 2009ApJ...690...20S (light orange) and Shankar 2013CQGra..30x4001S (green).
• Figure 2: Left panel: Cusp regrowth time $t_\mathrm{cusp}$ as a function of the total MBH binary mass. Solid, long-dashed and short-dashed curves are for $q=1,0.1,0.01$ respectively. Red curves assume $V_\mathrm{k}=0$ whereas blue curves assume $V_\mathrm{k}/V_\mathrm{esc}=0.6$. Right panel: Mass deficit normalized to $M$ as a function of binary mass ratio for $M=10^5M_\odot$(short dashed), $M=10^6M_\odot$ (long dashed), and $M=10^7M_\odot$ (solid). Blue and green dots are mass deficits computed by Khan et al. 2012ApJ...749..147K.
• Figure 3: Cusp regrowth effect for the popIII model. Left panel: The average differential number of mergers per unit redshift (i.e. Eq. (\ref{['eq:nmergsingle']}) integrated over $q$) $\mathrm{d} N_m/\mathrm{d} z$ experienced by each individual MBH of mass $\log_{10} M=4.5,5,5.5,6,6.5$ from darker-thicker to lighter-thinner. Center panel: The solid curves are the values of $N_\mathrm{m}(M,z)$ given by Eq. (\ref{['nmeanmerg']}), and the dashed curves are the corresponding probabilities of retaining a cusp given by Eq. (\ref{['probcusp']}). Right panel: The differential number of MBHs $\mathrm{d} N/\mathrm{d} z$ across the Universe in the three different mass bins that are potential EMRI hosts, either ignoring cusp disruption (solid lines) or taking it into account (dashed lines).
• Figure 4: Top panel: The adjusted EMRI rate computed according to Eq. (\ref{['correctedrate']}). The three (central) thick lines assume $N_\mathrm{p}=10$ and correspond to the pessimistic (KormendyHo13, short-dashed orange), fiducial (Gultekin09, solid turquoise) and optimistic (GrahamScott13, long-dashed violet) $M$--$\sigma$ relations. The two thin turquoise lines show the rates for the fiducial model, but assuming $N_\mathrm{p}=0$ (lower curve) and $N_\mathrm{p}=100$ (upper curve). Lower panel: The average time $t_\mathrm{EMRI}$ that a MBH of a given mass is surrounded by a stellar cusp, and is therefore a potential EMRI source, as implicitly defined by Eq. (\ref{['temri']}). The curves are for the same three different $M$--$\sigma$ relations in the top panel.
• Figure 5: The redshift at which the sky-averaged SNR of a prograde, circular, equatorial EMRI into a MBH with spin $a=0.99$ reaches the threshold $\varrho = 20$. The horizon is shown as a function of intrinsic MBH mass and for the two different choices of the compact object mass used in these studies, $m=10M_\odot$ and $m=30M_\odot$. The horizon is computed using accurate Teukolsky fluxes and using a Newtonian inspiral truncated either at the Schwarzschild ISCO, labelled "AKS", or at the Kerr ISCO, labelled "AKK". Individual sources may be detected to even larger distances if their orientation is near optimal.