Using LISA EMRI sources to test off-Kerr deviations in the geometry of massive black holes

TL;DR

The paper investigates whether LISA-detectable EMRIs can test the Kerr nature of massive black holes by treating the spacetime quadrupole moment Q as an independent parameter and constraining its deviation from the Kerr value. It extends analytic kludge EMRI waveforms to include an off-Kerr Q and uses a Fisher-information approach to forecast the precision Delta(Q/M^3) for various MBH masses at SNR ~100. Results show Delta(Q/M^3) spanning roughly 10^-4 to 10^-2, depending on MBH mass, with weaker dependence on orbital eccentricity and spin, indicating a strong potential for Kerr-ness tests with EMRIs. These findings underscore the practical ability of LISA to probe spacetime geometry near MBHs and motivate further refinement of waveform models and multipole analyses.

Abstract

Inspirals of stellar-mass compact objects into $\sim 10^6 M_{\odot}$ black holes are especially interesting sources of gravitational waves for LISA. We investigate whether the emitted waveforms can be used to strongly constrain the geometry of the central massive object, and in essence check that it corresponds to a Kerr black hole (BH). For a Kerr BH, all multipole moments of the spacetime have a simple, unique relation to $M$ and $S$, the BH's mass and spin; in particular, the spacetime's mass quadrupole moment is given by $Q=- S^2/M$. Here we treat $Q$ as an additional parameter, independent of $M$ and $S$, and ask how well observation can constrain its difference from the Kerr value. This was already estimated by Ryan, but for simplified (circular, equatorial) orbits, and neglecting signal modulations due to the motion of the LISA satellites. Here we consider generic orbits and include these modulations. We use a family of approximate (post-Newtonian) waveforms, which represent the full parameter space of Inspiral sources, and exhibit the main qualitative features of true, general relativistic waveforms. We extend this parameter space to include (in an approximate manner) an arbitrary value of $Q$, and construct the Fisher information matrix for the extended parameter space. By inverting the Fisher matrix we estimate how accurately $Q$ could be extracted from LISA observations. For 1 year of coherent data from the inspiral of a $10 M_{\odot}$ BH into rotating BHs of masses $10^{5.5} M_{\odot}$, $10^6 M_{\odot}$, or $10^{6.5} M_{\odot}$, we find $Δ(Q/M^3) \sim 10^{-4}$, $10^{-3}$, or $10^{-2}$, respectively (assuming total signal-to-noise ratio of 100, typical of the brightest detectable EMRIs). These results depend only weakly on the eccentricity of the orbit or the BH's spin.

Figures (4)

• Figure 1: Distribution of $\Delta\tilde{Q}$ for a selection of $(\theta_S,\phi_S,\theta_K,\phi_K,\lambda)$ values, and fixed values of $(\mu,M,e_{\rm LSO})=(10 M_{\odot},10^6 M_{\odot},0.15)$. The mean, standard deviation (STD), and median of the distribution are calculated twice, for comparison, based on data from two different matrix-inversion methods: singular-value decomposition (SVD) and Gauss-Jordan elimination (see discussion in the text). We find that the choice $(\theta_S,\phi_S,\theta_K,\phi_K,\lambda)= \left(\frac{2\pi}{3},\frac{5\pi}{3},\frac{\pi}{2},0,\frac{\pi}{3}\right)$ yields $\Delta\tilde{Q}$ very close to the median value. The median values for the masses $M=10^{5.5}$ and $10^{6.5} M_{\odot}$ were obtained in a similar manner, and are given in Eq. (\ref{['median']}). The standard deviations in $\log_{10}\Delta\tilde{Q}$ were found to be $\sim 0.13$ and $\sim 0.17$ for $M=10^{5.5}$ and $10^{6.5} M_{\odot}$, respectively.
• Figure 2: Results for $\Delta\tilde{Q}$---the measurement accuracy in $\tilde{Q}\equiv Q/M^3$---for a selection of values of $M$, $e_{\rm LSO}$, and $\tilde{S}\equiv S/M^2$. Left and right panels display $\Delta\tilde{Q}$ for $10 M_{\odot}$ and $0.6 M_{\odot}$ COs, respectively. Both plots are for one year of data (the last year of inspiral), and are normalized to SNR$= 100$. [In the left panel we have discarded a few of the data points, corresponding to low MBH mass with high LSO eccentricity: For these points the initial eccentricity exceeds $0.7$, and our PN evolution model cannot be trusted (see discussion in the text).]
• Figure 3: Comparison of the values of $\Delta\tilde{Q}$ obtained using two different matrix inversion methods. The 108 sample points are the same ones displayed in Fig. \ref{['fig:median']}; i.e., they correspond to 108 different values for $(\theta_S,\phi_S,\theta_K,\phi_K,\lambda)$, and fixed values of $(\mu,M,e_{\rm LSO}, \tilde{Q})$. Circles are results using singular-value decomposition, while pluses correspond to inversion of the same matrices using Gauss-Jordan elimination. Generally the two results lie close together, with deviations smaller than $\sim 20 \%$ in all cases. This is sufficient agreement for our purpose.
• Figure 4: $\Delta\tilde{Q}$ for BH inspirals, for a lower-order version of our "kludge" evolution equations (given explicitly in the text). Comparison of this Figure with the left panel of Fig. \ref{['fig:Deltaq1']} provides a check on the robustness of our results.