Gravitational waves from extreme mass ratio inspirals in nonpure Kerr spacetimes

To investigate the imprint on the gravitational-wave emission from extreme mass ratio inspirals (EMRIs) in nonpure Kerr spacetimes, we have studied the kludge waveforms generated in highly accurate, numerically generated spacetimes containing a black hole and a self-gravitating, homogeneous torus with comparable mass and spin. In order to maximize their impact on the produced waveforms, we have considered tori that are compact, massive, and close to the central black hole, investigating under what conditions the LISA experiment could detect their presence. Our results show that for a large portion of the space of parameters the waveforms produced by EMRIs in these black hole-torus systems are indistinguishable from pure Kerr waveforms. Hence, a “confusion problem” will be present for observations carried out over a time scale below or comparable to the dephasing time.

Figure 1

Schematic classification of the two regions of the spacetime. For equatorial orbits in region I (i.e. internal orbits) the mass and angular momentum of the Kerr black hole coincide with the mass and angular momentum of the black hole. For equatorial orbits in region II (i.e. external orbits) the mass and angular momentum of the Kerr black hole coincide with the total mass and angular momentum of the black hole-torus system.

Figure 2

Kludge waveforms around the dephasing time for a small body with mass $m=1M_{\odot }$ moving in the spacetime $B$ of Table . The solid line shows the waveform produced by a geodesic with given latus rectum and eccentricity in spacetime $B$, while the dot-dashed line refers to a geodesic with the same latus rectum and eccentricity (in (Q)BL coordinates) in a Kerr spacetime with $M_{\mathrm{Kerr}}=M_{\mathrm{tot}}$ and $J_{\mathrm{Kerr}}=J_{\mathrm{tot}}$. The dotted line and the circles are instead the waveforms produced by an orbit with the same $r$- and $\varphi$-frequencies as obtained by adjusting $(\delta p,\delta e)$ or $(\delta M,\delta J)$, respectively.

Figure 3

Overlap between waveforms produced in spacetime $A$ by external orbits and waveforms produced in a Kerr spacetime with mass $M_{\mathrm{Kerr}}=M_{\mathrm{tot}}$ and spin $J_{\mathrm{Kerr}}=J_{\mathrm{tot}}$. The orbits all have the same $r$- and $\varphi$-frequencies as obtained by suitably changing the latus rectum and the eccentricity, with positive values of $p_{\mathrm{QBL}}$ referring to prograde orbits, and negative ones to retrograde orbits. The different lines mark the margins of the different relevant regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane, with the dashed line representing the outer “edge of the torus,” the solid line representing the innermost stable bound orbits for a Kerr spacetime with mass $M_{\mathrm{Kerr}}=M_{\mathrm{tot}}$ and spin $J_{\mathrm{Kerr}}=J_{\mathrm{tot}}$, and the dot-dashed line limiting the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where bound stable orbits have been studied. A high overlap in large regions of the space of parameters indicates that a confusion problem is indeed possible in this spacetime for observational time scales below or comparable to the dephasing time, although this confusion disappears for orbits with small eccentricities and close to the innermost bound stable orbits.

Figure 4

The same as in Fig. 3 but for spacetime $B$. Note that in this case the confusion problem is less severe and indeed not present for orbits near the outer edge of the torus (i.e., with $p_{\mathrm{QBL}}/M_{\mathrm{tot}}\lesssim 30$) and with eccentricities $e_{\mathrm{QBL}}\lesssim 0.2$.

Figure 5

The same as in Fig. 4 but for internal orbits, with the solid line marking those orbits whose periastron lies on the event horizon, the dashed line representing the inner “edge of the torus,” and finally the crossed-solid line marking the innermost bound stable orbits in a Kerr spacetime with mass and spin $M_{\mathrm{Kerr}}=M_{\mathrm{BH}}$ and $J_{\mathrm{Kerr}}=J_{\mathrm{BH}}$. Again, the dot-dashed line limits the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where bound stable orbits have been studied, but in contrast to the case of external orbits, these regions correspond to practically all the bound stable orbits not crossing the torus. Note that in this case the confusion problem is absent, with $\mathcal{O}\lesssim 0.2$.

Figure 6

Overlap between waveforms produced in spacetime $A$ by external orbits and waveforms produced in a Kerr spacetime with mass $M_{\mathrm{Kerr}}+\delta M=M_{\mathrm{tot}}+\delta M$ and spin $J_{\mathrm{Kerr}}+\delta J=J_{\mathrm{tot}}+\delta J$ by orbits with the same latus rectum and eccentricity [in (Q)BL coordinates] and the same $r$- and $\varphi$-frequencies. Here too, the dashed line represents the outer “edge of the torus,” the solid line the innermost stable bound orbits for a Kerr spacetime with mass $M_{\mathrm{Kerr}}=M_{\mathrm{tot}}$ and spin $J_{\mathrm{Kerr}}=J_{\mathrm{tot}}$, and the dot-dashed line limits the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where bound stable orbits have been studied. An overlap $\mathcal{O}>0.95$ is present in all of the relevant regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane.

Figure 7

The same as in Fig. 6 but for spacetime $B$. Note that also in this case the overlap is very high ($\mathcal{O}>0.99$) in almost all of the relevant regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane, with the exception of a very small set of orbits very close to the torus, for which Eqs. (38, 39) have no solutions (these orbits correspond to the blank regions inside the dot-dashed line).

Figure 8

The same as in Fig. 7 but for internal orbits, with the solid line marking those orbits whose periastron lies on the event horizon, the dashed line representing the inner “edge of the torus,” and finally the crossed-solid line marking the innermost bound stable orbits in a Kerr spacetime with mass and spin $M_{\mathrm{Kerr}}=M_{\mathrm{BH}}$ and $J_{\mathrm{Kerr}}=J_{\mathrm{BH}}$. Again, the dot-dashed line limits the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where bound stable orbits have been studied, but in contrast to the case of external orbits these regions correspond to practically all the bound stable orbits not crossing the torus. Note that in this case the confusion problem is present in most of the relevant regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane, becoming slightly less severe only for the largest allowed eccentricities and for a very small set of orbits, very close to the torus, for which Eqs. (38, 39) have no solutions (these orbits correspond to the blank regions inside the dot-dashed line).

Figure 9

Relative mass correction $\delta M/M_{\mathrm{Kerr}}=\delta M/M_{\mathrm{tot}}$ in the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where the overlap plotted in Fig. 7 is above 0.95. Note that far from the system $\delta M/M_{\mathrm{tot}}$ approaches zero, as one would expect.

Figure 10

The same as in Fig. 9 but for internal orbits. In this case the deviations are computed as $\delta M/M_{\mathrm{Kerr}}=\delta M/M_{\mathrm{BH}}$ in the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where the overlap plotted in Fig. 8 is above 0.95.

Figure 11

Variations of the spin $a_{\mathrm{Kerr}}+\delta a\equiv a_{\mathrm{tot}}+\delta a=(J_{\mathrm{tot}}+\delta J)/(M_{\mathrm{tot}}+\delta M{)}^2$ in the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where the overlap plotted in Fig. 7 is above 0.95. We recall that for external orbits in spacetime $B$, we have $a_{\mathrm{Kerr}}=a_{\mathrm{tot}}=0.224$ (cf. Table ).

Figure 12

The same as in Fig. 11 but for internal orbits. In this case the corrections are computed as $a_{\mathrm{Kerr}}+\delta a=a_{\mathrm{BH}}+\delta a\equiv (J_{\mathrm{BH}}+\delta J)/(M_{\mathrm{BH}}+\delta M{)}^2$ in the regions of the $(p_{\mathrm{QBL}},e_{\mathrm{QBL}})$ plane where the overlap plotted in Fig. 8 is above 0.95. We recall that for internal orbits in spacetime $B$, we have $a_{\mathrm{Kerr}}=a_{\mathrm{BH}}=-1.74\times {10}^{-3}$ (cf. Table ).