Gravitational waves from extreme-mass-ratio inspirals using general parametrized metrics

Future space-borne interferometers will be able to detect gravitational waves at ${10}^{-3}$ to ${10}^{-1}\mathrm{Hz}$. In this band extreme-mass-ratio inspirals (EMRIs) can be promising gravitational-wave sources. In this paper, we investigate the possibility of testing the hypothesis that the small body is moving in Kerr spacetime against the alternative that the small body is moving in a parametrized non-Kerr metric by matching gravitational waveforms. EMRI snapshots from either equatorial geodesics or inclined geodesics suffer from the “confusion problem.” Our results show that, within the time scale before significant (radiation-driven) orbital evolution takes place, small and moderate deviations from the Kerr spacetime [$|{\delta }_i|<1$ in the notation of Ni, Jiang, and Bambi J. Cosmol. Astropart. Phys. 09 (2016) 014] can be discerned only when the Kerr spin parameter is extreme. In geodesic cases, most waveforms related to a non-Kerr metric can be mimicked by the waveform templates produced from a Kerr black hole. However, when radiation reaction is taken into consideration, the signals that are originally degenerate with each other will gradually separate and finally break the confusion. Depending on the mass ratio and other parameters of the system, the time needed to break the degeneracy can vary from several hours to several months.

Figure 1

Event horizon in the $xz$ plane with the influence of ${\delta }_1$, where $x\equiv \sqrt{r^2+a^2}\sin \theta \cos \varphi$ and $z\equiv r\cos \theta$. The dashed lines of different colors indicate the inner horizon $r_{-}$, and the solid lines indicate the outer horizon $r_{+}$.

Figure 2

“Plus” and “cross” components of gravitational waves emitted by equatorial orbits in non-Kerr and Kerr spacetimes, where we vary $(M,a)$ or $(e,p)$ to equate the orbital frequencies. The non-Kerr waveform is $({\delta }_1,a,M,e,p)=(0,2,0.5,2\times {10}^5,0.5,6.0)$. The yellow and red solid lines are the “plus” and “cross” components of the non-Kerr waveforms. The black and violet dotted lines are the “plus” and “cross” components of Kerr waveforms with the same orbital frequencies as non-Kerr orbits when varying $(M,a)$. The blue and green dashed lines are the “plus” and “cross” components of Kerr waveforms with the same orbital frequencies as non-Kerr orbits when varying $(e,p)$.

Figure 3

Distribution of the overlap between waveforms defined by $({\delta }_1,a,M,e,p)=(0.2,0.5,2\times {10}^5,0.5,6)$ and $({\delta }_1,a,M,e,p)=(0,0.5,2\times {10}^5,e_{\text{Kerr}},p_{\text{Kerr}})$ in the ($e_{\text{Kerr}}$, $p_{\text{Kerr}}$) plane. The blue cross marks the same ${\omega }_r$ and ${\omega }_{\varphi }$ at $(e_{\text{Kerr}},p_{\text{Kerr}})=(0.409248,6.481170)$. The grid size is $50\times 50$.

Figure 4

Distribution of the overlap between waveforms with $({\delta }_1,a,M,e,p)=(0.2,0.5,2\times {10}^5,e_{\mathrm{KRZ}},p_{\mathrm{KRZ}})$ and Kerr waveforms with identical ${\omega }_r$ and ${\omega }_{\varphi }$ by varying $e_{\text{Kerr}}$, $p_{\text{Kerr}}$ (left panel) and varying $M$, $a$ (right panel) in the $(e_{\mathrm{KRZ}},p_{\mathrm{KRZ}})$ plane. The grid size is $100\times 100$. The blank region contains unstable orbits.

Figure 5

Relation between the relative varying spin/mass values when equating the orbital frequencies, i.e., the orbits of $({\delta }_1,a_0,M_0,e,p)$ and $(0,a_0+\delta a,M_0+\delta M,e,p)$ have the same orbital frequency in the $\frac{\delta a}{a_0}-{\delta }_1$ (left) and $\frac{\delta M}{M_0}-{\delta }_1$ (right) planes.

Figure 6

The same as in Fig. 5, but for ${\delta }_2$ (top panels) and ${\delta }_4$ (bottom panels).

Figure 7

The same as in Fig. 5, but only for the $\delta a/a_0-{\delta }_2$ relations for different orbits.

Figure 8

Ranges of parameters that lead to degeneracy. The ranges are projected onto ${\delta }_i$-$a_{\mathrm{KRZ}}$ parameter plane with other parameters set to $e=0.5$, $p=6.0$ and $M=2\times 10^5$. Each point on the plane indicates an EMRI system with $e=0.5$, $p=6.0$, $M=2\times 10^5$ and ${\delta }_i$, $a_{\mathrm{KRZ}}$ set by the coordinate in the plot. Colors are indexed as follows. Blue: the system is degenerate with another one in Kerr spacetime by frequency matching. Red: the corresponding system in Kerr spacetime by frequency matching has a spin larger than 1 ($a_{\text{Kerr}}>1$). Yellow: the system does not exist. Left panel: for ${\delta }_1$ while ${\delta }_2$ is set to 0. Right panel: for ${\delta }_2$ while ${\delta }_1$ is set to 0.

Figure 9

Orbits and GW waveforms of a KRZ orbit and a Kerr orbit with the same orbital frequencies. Top left panel: Time series of motion in the $r$ direction for the last 5000 s. Top right panel: Time series of motion in the $\theta$ direction for the last 5000 s. Middle left panel: Trajectories for the last 5000 s. Middle right panel: Projection onto the $x\text{−}y$ plane of the trajectories for the last 5000 s. Bottom panel: “Plus” component of the EMRI waveform for the last 20 000 s. The orbital parameters and BH spin/mass are $(e,p,\iota ,\mathrm{spin},M)=(0.432,6.819,0.780,0.5,{10}^6)$. The deformation parameter of the KRZ orbit is ${\delta }_1=0.2$.

Figure 10

The same as in Fig. 8 but for inclined orbits, and where the waveform at each point is determined by setting the initial $E$, $L_z$ to be the same as in the corresponding Kerr orbit. See text for details.

Figure 11

Top panel: EMRI signals from geodesic orbits with $({\delta }_1,a,M,e,p)=(0.2,0.5,200000,0.5,6)$ (blue solid line) and $({\delta }_1,a,M,e,p)=(0,0.617491,201020,0.5,6)$ (red dotted line). Bottom panel: Same as in the top panel but for evolving orbits with a mass ratio of ${10}^{-4}$.

Figure 12

Evolution of the fitting factor between EMRI signals with $({\delta }_1,a,M,e,p)=(0.2,0.5,200000,0.5,6)$ and $({\delta }_1,a,M,e,p)=(0,0.617491,201020,0.5,6)$ for different mass ratios ($\nu$). Blue lines: $\nu ={10}^{-4}$. Red lines: $\nu =5\times {10}^{-5}$. Green lines: $\nu ={10}^{-5}$.

Figure 13

Top panel: EMRI signals from geodesic orbits with $({\delta }_1,a,M,e,p,\iota )\approx (0.2,0.5,1000000,0.432,6.819,0.780)$ (blue solid line) and $({\delta }_1,a,M,e,p,\iota )\approx (0,0.553,937173,0.432,7.172,0.780)$ (red dotted line). Bottom panel: Same as in the top panel but for evolving orbits with a mass ratio of ${10}^{-4}$.

Figure 14

Left panel: Evolution of the fitting factor between EMRI signals with $({\delta }_1,a,M,e,p,\iota )\approx (0.2,0.5,1000000,0.432,6.819,0.780)$ and $({\delta }_1,a,M,e,p,\iota )\approx (0,0.553,937173,0.432,7.172,0.780)$. Blue solid line: $\nu ={10}^{-4}$. Red dotted line: $\nu =5\times {10}^{-5}$. Green dashed line: $\nu ={10}^{-5}$. Right panel: Same as in the left panel, but for lower mass ratios. Blue solid line: $\nu ={10}^{-6}$. Red dotted line: $\nu ={10}^{-7}$. Green dashed line: $\nu ={10}^{-8}$.

Figure 15

Evolution of the fitting factor between EMRI signals with $({\delta }_1,a,M,e,p,\iota )\approx (0.2,0.5,1000000,0.502,12.020,0.779)$ and $({\delta }_1,a,M,e,p,\iota )\approx (0,0.538,945615,0.502,12.511,0.779)$ for different mass ratios $\nu$. Blue solid line: $\nu =5\times {10}^{-3}$. Red dotted line: $\nu ={10}^{-3}$. Green dashed line: $\nu ={10}^{-4}$. Yellow dash-dotted line: $\nu ={10}^{-5}$.

Figure 16

The relation between the “dephasing time” (the time required for the fitting factor of two signals to drop below 0.97) and mass ratio $\nu$. Blue triangles: Match between signals defined by $({\delta }_1,a,M,e,p,\iota )\approx (0.2,0.5,1000000,0.502,12.020,0.779)$ and $({\delta }_1,a,M,e,p,\iota )\approx (0,0.538,945615,0.502,12.511,0.779)$. Red dots: Match between signals defined by $({\delta }_1,a,M,e,p,\iota )\approx (0.2,0.5,1000000,0.502,12.020,0.779)$ and $({\delta }_1,a,M,e,p,\iota )\approx (0,0.538,945615,0.502,12.511,0.779)$. Dashed lines are fitted scaling laws.