Intermediate-mass-ratio inspirals in the Einstein Telescope. II. Parameter estimation errors

We explore the precision with which the Einstein Telescope will be able to measure the parameters of intermediate-mass-ratio inspirals, i.e., the inspirals of stellar mass compact objects into intermediate-mass black holes (IMBHs). We calculate the parameter estimation errors using the Fisher Matrix formalism and present results of Monte Carlo simulations of these errors over choices for the extrinsic parameters of the source. These results are obtained using two different models for the gravitational waveform which were introduced in paper I of this series. These two waveform models include the inspiral, merger, and ringdown phases in a consistent way. One of the models, based on the transition scheme of Ori and Thorne [A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000).], is valid for IMBHs of arbitrary spin; whereas, the second model, based on the effective-one-body approach, has been developed to cross-check our results in the nonspinning limit. In paper I of this series, we demonstrated the excellent agreement in both phase and amplitude between these two models for nonspinning black holes, and that their predictions for signal-to-noise ratios are consistent to within 10%. We now use these waveform models to estimate parameter estimation errors for binary systems with masses $1.4M_{\odot }+100M_{\odot }$, $10M_{\odot }+100M_{\odot }$, $1.4M_{\odot }+500M_{\odot }$, and $10M_{\odot }+500M_{\odot }$ and various choices for the spin of the central IMBH. Assuming a detector network of three Einstein Telescopes, the analysis shows that for a $10M_{\odot }$ compact object inspiralling into a $100M_{\odot }$ IMBH with spin $q=0.3$, detected with a signal-to-noise ratio of 30, we should be able to determine the compact object and IMBH masses, and the IMBH spin magnitude to fractional accuracies of $\sim {10}^{-3}$, $\sim {10}^{-3.5}$, and $\sim {10}^{-3}$, respectively. We also expect to determine the location of the source in the sky and the luminosity distance to within $\sim 0.003$ steradians and $\sim 10\%$, respectively. We also compute results for several different possible configurations of the detector network to assess how the precision of parameter determination depends on the network configuration.

Figure 1

Sensitivity curve for the Einstein Telescope. The Advanced LIGO noise curve is also shown for reference.

Figure 2

We show inspiral, merger, and ringdown gravitational waveforms for compact objects of mass $10M_{\odot }$ (top left panel) and $1.4M_{\odot }$ (top right panel), inspiraling into a $500M_{\odot }$ BH with spin parameter $q=0.3$. The bottom panel shows the gravitational waveform for a binary system consisting of a compact object of mass $10M_{\odot }$ and a nonspinning ($q=0$) IMBH of mass $500M_{\odot }$, computed using both the EOB and transition models. In each case, the various extrinsic parameters were chosen randomly.

Figure 3

Distribution of parameter measurement error estimates computed in the Monte Carlo simulations. We show results for the system with $m=10M_{\odot }$, $M=100M_{\odot }$, $q=0.9$. The panels show the error distributions in the following order, top row, from left to right, $\Delta (\ln m)$, $\Delta (\ln M)$; middle row, $\Delta q$, $\Delta t_0$; bottom row, $\Delta {\theta }_S$, $\Delta {\varphi }_S$.

Figure 4

Left panel: expected fractional measurement errors, $\Delta$, in the mass of the CO and the mass of the IMBH for a fixed value of the IMBH spin parameter $q=0.3$. There is a pair of candlesticks for each of the four binary systems. For each binary, the candlestick representing the CO’s mass error estimate is to the left of that for the IMBH’s mass error estimate. The error bars indicate the lower and upper quartiles found in the Monte Carlo simulation. Right panel: expected measurement errors in the IMBH’s spin parameter for two different values for the spin parameter $q$. For each system, the candlestick corresponding to a central IMBH with $q=0.3$ is to the right of that associated with the error estimate for a system with $q=0.9$. The continuous, long dashed, dashed, and dotted lines represent systems B1: $m=10M_{\odot }$, $M=100M_{\odot }$, B2: $m=1.4M_{\odot }$, $M=100M_{\odot }$, B3: $m=10M_{\odot }$, $M=500M_{\odot }$, and B4: $m=1.4M_{\odot }$, $M=500M_{\odot }$, respectively.

Figure 5

Arranged from left to right in the panel, we show the expected measurement errors in the CO’s mass, IMBH mass and IMBH spin parameter, as a function of the network configuration for binary systems with masses $10M_{\odot }+100M_{\odot }$ and IMBH spin $q=0.3$. In each group the candlesticks are arranged in order C1–C5 from left to right. As before, the CO and IMBH mass errors are quoted as fractional errors, i.e., these are the errors in $\ln (m)$, $\ln (M)$.