Mapping spacetimes with LISA: inspiral of a test-body in a `quasi-Kerr' field

TL;DR

This work proposes a practical spacetime-mapping framework for LISA EMRIs by introducing a quasi-Kerr metric that perturbs Kerr via a small quadrupole deviation, derived from the Hartle-Thorne exterior solution. It analyzes geodesic motion using a perturbative approach around Kerr, identifying near-Kerr integrals of motion and deriving equatorial orbital frequencies that differ from Kerr in the strong-field regime. By constructing approximate kludge waveforms, it demonstrates that even modest quadrupole deviations can cause substantial waveform dephasing and low overlaps with Kerr templates, implying potential SNR losses and a confusion problem where Kerr and non-Kerr signals can mimic each other under certain parameter mappings. The results underscore the need for non-Kerr waveform models and future work on non-equatorial orbits and quasi-Teukolsky-type perturbations to enable robust spacetime-mapping with LISA.

Abstract

The future LISA detector will constitute the prime instrument for high-precision gravitational wave observations.LISA is expected to provide information for the properties of spacetime in the vicinity of massive black holes which reside in galactic nuclei.Such black holes can capture stellar-mass compact objects, which afterwards slowly inspiral,radiating gravitational waves.The body's orbital motion and the associated waveform carry information about the spacetime metric of the massive black hole,and it is possible to extract this information and experimentally identify (or not!) a Kerr black hole.In this paper we lay the foundations for a practical `spacetime-mapping' framework. Our work is based on the assumption that the massive body is not necessarily a Kerr black hole, and that the vacuum exterior spacetime is stationary axisymmetric,described by a metric which deviates slightly from the Kerr metric. We first provide a simple recipe for building such a `quasi-Kerr' metric by adding to the Kerr metric the deviation in the value of the quadrupole moment. We then study geodesic motion in this metric,focusing on equatorial orbits. We proceed by computing `kludge' waveforms which we compare with their Kerr counterparts. We find that a modest deviation from the Kerr metric is sufficient for producing a significant mismatch between the waveforms, provided we fix the orbital parameters. This result suggests that an attempt to use Kerr waveform templates for studying EMRIs around a non-Kerr object might result in serious loss of signal-to-noise ratio and total number of detected events. The waveform comparisons also unveil a `confusion' problem, that is the possibility of matching a true non-Kerr waveform with a Kerr template of different orbital parameters.

Figures (6)

• Figure 1: Left panel: periastron shift difference $\Delta\phi_{\rm K} - \Delta\phi_{\rm qK}$ as a function of the deviation $\epsilon$. Right panel: number of cycles ${\cal N}$ required to accumulate $\pi/2$ difference in periastron shifts (defined by eqn.(\ref{['Ncycl']})) as function of $\epsilon$. For both panels, we have considered two orbits: $a=0.5M, e=0.5$ and $p=10M,\, p=15M$.
• Figure 2: Difference in the radial period $T_r$ Kerr and quasi-Kerr orbits with $a=0.5M, e=0.5$ and $p=10M,\, p=15M$.
• Figure 3: Number of cycles to accumulate $\pi/2$ difference between Kerr and quasi-Kerr periastron shifts as a function of $\epsilon$ and eccentricity $e$. We have fixed the other two parameters: $p=10M$, $a=0.5M$.
• Figure 4: Comparing quasi-Kerr (solid curve) and Kerr (dashed curve) approximate hybrid waveforms for the orbit $p= 10 M, e=0.5, a=0.5 M$ and for $\epsilon = 0.15$. In addition, we include a Kerr waveform taking into account backreaction on the orbit (assuming the same initial orbit), represented by the dashed curve. All waveforms are shown at a time window close to the radiation reaction timescale $T_{\rm RR}$.
• Figure 5: Right panel: radiation reaction timescale $T_{\rm RR}$ (see main text for definition), as a function of mass ratio $\mu/M$. Left panel: overlaps (expressed in $\%$) between quasi-Kerr and Kerr waveforms for the same orbit $p=10 M, e=0.5, a = 0.5 M$ and for $\epsilon = 0.07, 0.15$, truncated at $t=T_{\rm RR}(\mu/M)$. The solid line is the spline interpolation between the data points.