Verifying the no-hair property of massive compact objects with intermediate-mass-ratio inspirals in advanced gravitational-wave detectors

The detection of gravitational waves from the inspiral of a neutron star or stellar-mass black hole into an intermediate-mass black hole (IMBH) promises an entirely new look at strong-field gravitational physics. Gravitational waves from these intermediate-mass-ratio inspirals (IMRIs), systems with mass ratios from $\sim 10:1$ to $\sim 100:1$, may be detectable at rates of up to a few tens per year by Advanced LIGO/Virgo and will encode a signature of the central body’s spacetime. Direct observation of the spacetime will allow us to use the “no-hair” theorem of general relativity to determine if the IMBH is a Kerr black hole (or some more exotic object, e.g., a boson star). Using modified post-Newtonian (pN) waveforms, we explore the prospects for constraining the central body’s mass-quadrupole moment in the advanced-detector era. We use the Fisher information matrix to estimate the accuracy with which the parameters of the central body can be measured. We find that for favorable mass and spin combinations, the quadrupole moment of a non-Kerr central body can be measured to within a $\sim 15\%$ fractional error or better using 3.5 pN order waveforms; on the other hand, we find the accuracy decreases to $\sim 100\%$ fractional error using 2 pN waveforms, except for a narrow band of values of the best-fit non-Kerr quadrupole moment.

Figure 1

Standard deviations of $Q_{\mathrm{anom}}$ as a function of central body mass (with ${\chi }_1=0.9$) for an NS-IMBH system (companion mass of $1.4M_{\odot }$). The minimum standard deviation is waveform dependent (at $17M_{\odot }$ for 2 pN and at $37M_{\odot }$ for 3.5 pN). In our dimensionless units, the quadrupole moment for these systems is $|Q|=0.81$.

Figure 2

$1\mathrm{\text{−}}\sigma$ uncertainty estimates for $Q_{\mathrm{anom}}$ as a function of central body mass and the companion object mass. The lines indicate contours of constant $1\mathrm{\text{−}}\sigma$ error in $Q_{\mathrm{anom}}$. The structure of the plots is an interplay between the proximity of the ISCO frequency to the detector’s peak sensitivity (as determined by the total mass) and the preference for more extreme mass ratios to better probe the strong-field of the central body. Both plots are for spin ${\chi }_1=0.9$; plots for other spin values are qualitatively similar, but decreasing the spin leads to an overall deterioration of accuracy and some shifting of the contour lines, due in part to the change in ISCO frequency.

Figure 3

The $1\mathrm{\text{−}}\sigma$ uncertainty in $Q_{\mathrm{anom}}$ as a function of the best-fit value of $Q_{\mathrm{anom}}$, for several choices of the central body mass and spin magnitude. The solid lines are spin ${\chi }_1=0.9$ systems, and the dashed lines are ${\chi }_1=0.5$. In both cases, the standard deviations dip to a local minimum, $Q_{\mathrm{anom}}^{(\min )}$, which varies with pN order. Using the 3.5 pN waveforms, the fractional error plateaus are sufficiently low that it should be possible to distinguish between a Kerr black hole and a more exotic object (e.g., a boson star). The plots cut off the higher-mass systems in the 2 pN case and ignore the medium spin $100M_{\odot }$ system, as these fall outside our numerical cutoff for stable inversion of the Fisher matrix.

Figure 4

Two-dimensional error ellipses from a $\eta \mathrm{\text{−}}{Q}_{\mathrm{anom}}$ Fisher matrix (with other parameters held fixed), evaluated near the value of $Q_{\mathrm{anom}}=Q_{\mathrm{anom}}^{(\min )}$ (in solid black). The colored ellipses are small deviations from the minimum ($\pm 0.1$ in dashed blue and $\pm 0.5$ in dotted red).

Figure 5

The fraction of orbital cycles contributed by the 2 pN verses the 0 pN terms as a function of the anomalous quadrupole moment. The two shaded regions are for two typical IMRI systems with spins ranging from 0 (solid black) to 0.9 (dashed red). Note that the initial crossover between the nonspinning and highly spinning cases arises from the sign change in ${\alpha }_4$ at low $Q_{\mathrm{anom}}$. The validity of the approximation that ignores higher-order $Q$-dependent terms comes into doubt when the number of fractional cycles is high.