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

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.

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. (5) is shown as a thin straight black line. Also shown in brown is the redshift-independent pessimistic mass function as given by Eq. (6). The shaded area represent constraints from Shankar et al. [72] (light orange) and Shankar [73] (green).

Figure 2

Left panel: Cusp regrowth time $t_{\text{cusp}}$ 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 $M=10^5{M}_{\odot }$ (short dashed), $M=10^6{M}_{\odot }$ (long dashed), and $M=10^7{M}_{\odot }$ (solid). Blue and green dots are mass deficits computed by Khan et al. [84].

Figure 3

Cusp regrowth effect for the popIII model. Left panel: The average differential number of mergers per unit redshift (i.e. Eq. (18) 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. (20), and the dashed curves are the corresponding probabilities of retaining a cusp given by Eq. (21). 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. (31). 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. (28). 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 LSO, labeled “AKS,” or at the Kerr LSO, labeled “AKK.” Individual sources may be detected to even larger distances if their orientation is near optimal.

Figure 6

SNR distribution for detectable events with AKS and AKK waveforms for all considered models.

Figure 7

Top panel: Event rates (detected and intrinsic) as function of the astrophysical model. Lower panel: Fraction of detection with $\rho >20$ with respect to the total number of EMRIs featuring a central MBH with $10^4{M}_{\odot }<M<10^7{M}_{\odot }$, considering all events at $z<4.5$ ($z<6.5$ for M4).

Figure 8

MBH mass distribution for detectable events with AKS and AKK waveforms.

Figure 9

Redshift distribution for detectable events with AKS and AKK waveforms. A maximum source redshift of 4.5 is assumed for all models except M4, where the maximum redshift is 6.5.

Figure 10

Fraction of the intrinsic EMRI population detectable by LISA as a function of source-frame total mass and redshift, for models M1 and M4 and with AKS and AKK waveforms.

Figure 11

Distribution over observed EMRIs of the expected statistical errors ($1\sigma$ uncertainties as computed using the Fisher matrix) in the measurement of intrinsic parameters: central MBH redshifted mass (top left), spin (top right), CO mass (bottom left) and eccentricity at plunge (bottom right). The dashed lines mark the first, second and third quartile of the distributions.

Figure 12

Distribution of the statistical errors in the measurement of EMRI extrinsic parameters: luminosity distance (left panel; $1\sigma$ uncertainty as computed using the Fisher matrix) and sky localization [right panel; the area of an ellipse with probability $1-\exp (-1)$ of containing the source]. The dashed lines mark the first, second and third quartile of the distributions. In the plot for the sky position, a horizontal solid red line marks an error of $10{\mathrm{deg}}^2$.

Figure 13

Distribution of the statistical error ($1\sigma$ uncertainties as computed using the Fisher matrix) in measurement of the deviation of the MBH’s quadrupole moment away from the Kerr value. The dashed lines mark the first, second and third quartile of each distribution.