Detecting intermediate-mass ratio inspirals from the ground and space

The detection of the gravitational capture of a stellar-mass compact object by a massive black hole (MBH) will allow us to test gravity in the strong regime. The repeated, accumulated bursts of gravitational radiation from these sources can be envisaged as a geodesic mapping of space-time around the MBH. These sources form via two-body relaxation, by exchanging energy and angular momentum, and inspiral in a slow, progressive way down to the final merger. The frequencies fall in the millihertz range for MBHs with masses $\sim {10}^6{M}_{\odot }$, i.e., that of space-borne gravitational-wave observatories such as LISA. In this article we show that, depending on their orbital parameters, intermediate-mass ratio inspirals (IMRIs) of MBHs with masses between a hundred and a few thousand $M_{\odot }$ have frequencies that make them detectable (i) with ground-based observatories, or (ii) with both LISA and ground-based observatories (such as advanced LIGO/Virgo) and third-generation observatories [such as the Einstein Telescope (ET)]. The binaries have a signal-to-noise ratio large enough to ensure detection. More extreme values of the orbital parameters correspond to systems that are only detectable with ground-based detectors and in particular enter the LIGO/Virgo band in many different harmonics for masses up to $2000M_{\odot }$. We show that environmental effects are negligible, so the source should not have this kind of complication. The accumulated phase shift is measurable with LISA and ET, and for some cases also with LIGO, so that it is possible to recover information about the eccentricity and formation scenario. For IMRIs with a total mass $⪅2000M_{\odot }$ and initial eccentricities up to 0.999, LISA can give a advanced warning to ground-based detectors with seconds of precision. The possibility of detecting IMRIs from the ground alone or combined with space-borne observatories opens new possibilities for gravitational-wave astronomy.

Figure 1

Characteristic amplitude of the first ten harmonics emitted during the evolution of an IMRI. The solid V-shaped curve on the left corresponds to LISA’s intrinsic noise, and the two U-shaped curves on the right correspond to ET (lower $h_c$ values) and Advanced LIGO. The mass of the IMBH is fixed to $M_{\mathrm{MBH}}=100M_{\odot }$ and the mass of the compact object is $m_{\mathrm{CO}}=10M_{\odot }$. The source is assumed to be located at a distance of $D=500\mathrm{Mpc}$. Each panel corresponds to a binary with different initial values for the semimajor axis or eccentricity. The figure shows a few instants of time on the second harmonic in the evolution of the binary before the final merger. The total amount of time for the binary to merge from the initial values of the semimajor axis and eccentricity is given in each panel, $T_{\mathrm{mrg}}$. The square symbol corresponds to 1 year before it. The rest of the harmonics also display the same instants of time using the same symbol but without a text label. The value of the eccentricity at that particular moment and the pericenter value $R_{\mathrm{p}}$ as a function of the Schwarzschild radius $R_{\mathrm{S}}$ are also shown. Additionally, the full waveform of the system in the LIGO sensitivity curve as approximated by the IMRPhenomD algorithm presented in Refs. [53, 54] is depicted in the right panel by the dashed, orange curve.

Figure 2

Same as in Fig. 1 but for $M_{\mathrm{MBH}}=300M_{\odot }$ and $m_{\mathrm{CO}}=30M_{\odot }$ and with different labels in the evolution. Notice the displacement of the frequency peaks, which wander from the LISA band to the LIGO/Virgo one (top left panel to the bottom right one).

Figure 3

Same as in Fig. 1 but for $M_{\mathrm{MBH}}=500M_{\odot }$ and $m_{\mathrm{CO}}=26$, which is based on the relativistic stellar-dynamical simulation of Ref. [39]. The left panel corresponds to the kind of eccentricity in that work and the right one corresponds to a more extreme one, with different labels in the evolution.

Figure 4

Same as in Fig. 1 but for $M_{\mathrm{MBH}}=1000M_{\odot }$ and $m_{\mathrm{CO}}=30$ and with different labels in the evolution.

Figure 5

Same as in Fig. 1 but for $M_{\mathrm{MBH}}=2000M_{\odot }$ and $m_{\mathrm{CO}}=5$ and with different labels in the evolution.

Figure 6

Same as in Fig. 1 but for $M_{\mathrm{MBH}}=3000M_{\odot }$ and $m_{\mathrm{CO}}=30$ and with different labels in the evolution.

Figure 7

(Left panel) As in Fig. 1 but for $M_{\mathrm{MBH}}=100M_{\odot }$ and $m_{\mathrm{CO}}=30$ and with different labels in the evolution. (Right panel) Same as in the left panel but taking into account stellar dynamics (see text). We adopt an ambient stellar density of $2\times {10}^5{M}_{\odot }{\mathrm{pc}}^{-3}$, $K_0=0.1$, and a one-dimensional velocity dispersion of $\sigma =15\mathrm{km}/\mathrm{s}$.

Figure 8

Plus polarization $h_{+}$ for the two systems of Fig. 1 from the eccentricity of entrance into the LIGO bandwidth, i.e., when the frequency is of 10 Hz as approximated by the Fourier-transformed time-domain Taylor T4 algorithm of Ref. [61], which includes the effects of mild orbital eccentricity ($\lesssim 0.2$). The orange dot-dashed curve and the green dashed curve of higher eccentricity correspond to the left and right panels of Fig. 1, respectively. The solid grey curve shows the LIGO zero-detuned high-power design sensitivity.

Figure 9

(Left panel) Accumulated SNR in ET as a function of the time to plunge, $T_{\mathrm{mrg}}$, in seconds, for the IMRI in the bottom-right panel of Fig. 2. we show the individual contributions of the first 100 harmonics and the total. (Right panel) Same for the right panel of Fig. 3.

Figure 10

Same as in Fig. 9 but for the same IMRI system in the left panel of Fig. 3, and in years. we show the individual contributions of the first ten harmonics, but the total SNR takes into account the contribution of the first 100, which are not displayed.

Figure 11

Same as in Fig. 10 but taking into account only the SNR accumulated 10 (left panel) and 5 (right panel) years before the merger. See discussion in text.