Observable signature of a background deviating from the Kerr metric

By detecting gravitational wave signals from extreme mass ratio inspiraling sources (EMRIs) we will be given the opportunity to check our theoretical expectations regarding the nature of supermassive bodies that inhabit the central regions of galaxies. We have explored some qualitatively new features that a perturbed Kerr metric induces in its geodesic orbits. Since a generic perturbed Kerr metric does not possess all the special symmetries of a Kerr metric, the geodesic equations in the former case are described by a slightly nonintegrable Hamiltonian system. According to the Poincaré-Birkhoff theorem, this causes the appearance of the so-called Birkhoff chains of islands on the corresponding surfaces of section in between the anticipated KAM curves of the integrable Kerr case, whenever the intrinsic frequencies of the system are at resonance. The chains of islands are characterized by finite width, i.e. there is a finite range of initial conditions that correspond to a particular resonance and consequently to a constant rational ratio of intrinsic frequencies. Thus while the EMRI changes adiabatically by radiating energy and angular momentum, by monitoring the frequencies of a signal we can look for a transient pattern, in the form of a plateau, in the evolution of their ratio. We have shown that such a plateau is anticipated to be apparent in a quite large fraction of possible orbital characteristics if the central gravitating source is not a Kerr black hole. Moreover, the plateau in the ratio of frequencies is expected to be more prominent at specific rational values that correspond to the strongest resonances. This gives a possible observational detection of such non-Kerr exotic objects.

Figure 1

(a) The shape of the 2-dimensional well $V_{\mathrm{eff}}(\rho ,z)$ near $V_{\mathrm{eff}}(\rho ,z)=0$ along the line $z=0$ for the MN metric with parameters $\chi =0.9$, $q=0.95$ and orbital parameters $E=0.95$, $L_z=3M$. (b) The CZVs ($V_{\mathrm{eff}}(\rho ,z)=0$) on the $(\rho ,z)$ plane for the same set of parameters with (a).

Figure 2

A schematic representation of a surface of section of a nonintegrable system of 2 degrees of freedom. On the surface there are depicted two KAM curves; one outside and one inside the chain of islands. The particular chain of islands in between consists of three islands ($m=3$, $n=1$), each one arising around each of the three stable points (shown as filled circles). Each island consists of a whole set of KAM curves nested within each other. Between the successive islands there are three unstable points (shown as open circles) from which the asymptotic curves (gray curves) emanate surrounding the islands by a thin chaotic layer (here magnified). The arrows indicate the flow around the left-most island. The ${\mathbf{u}}_0$ indicates the central fixed point around which the KAM curves and the Birkhoff islands are formed.

Figure 3

(a) The surface of section of the outer region on the $(\rho ,\dot{\rho })$ plane for the parameter set $E=0.95$, $L_z=3M$, $\chi =0.9$, $q=0.95$. ${\mathbf{u}}_0$ indicates the fixed point at the center of the main island. (b) The rotation number vs $\rho$ along the line $\dot{\rho }=0$ of the surface of section presented in (a). Embedded in (b) is a detail of the rotation curve around the $2/3$-resonance.

Figure 4

The rotation curve along the line $\dot{\rho }=0.05$ which is crossing the $1/2$-resonance on the surface of section of Fig. 3a.

Figure 5

The width $w$ of the period-3 left-most island of stability along the line $\dot{\rho }=0$ (the thickest region) for different sets of the $E$, $L_z$, $q$, and $\chi$ parameters. (a) The $w$ vs the angular momentum $L_z$ when $E=0.95$, $q=0.95$, $\chi =0.9$. (b) The $w$ vs the energy $E$ when $L_z=3M$, $q=0.95$, $\chi =0.9$. (c) The $w$ vs the quadrupole deviation $q$ when $L_z=3M$, $E=0.95$, $\chi =0.9$. (d) The $w$ vs the spin parameter $\chi$ when $L_z=3M$, $E=0.95$, $q=0.95$. Apparent nonsmooth behavior of the plots is an artifact of the accuracy used to measure the width.

Figure 6

The radial positions ${\rho }_c$ of 6 different locations of interest along the line $\dot{\rho }=0$ on the surface of section $z=0$ for the same sets of parameters $E$, $L_z$, $q$, $\chi$ as in Fig. 5. The diamonds represent the central point ${\mathbf{u}}_0$ of the main island, the black squares represent the center of the multiplicity-3 left-most island, the open triangles and open circles represent the outer and inner boundary of the outer region, respectively, while the black triangles and black circles represent the inner and outer boundary of the inner region, respectively. When the open and the black circles merge the outer and the inner regions get connected through a neck.

Figure 7

(a) The surface of section of the inner region on the $\rho$, $\dot{\rho }$ plane for the parameter set $E=0.95$, $L_z=3M$, $\chi =0.9$, $q=0.95$. (b) A detail of (a) showing islands of stability surrounding a period-3 orbit.

Figure 8

In the left column various bound orbits of the inner region (Fig. 7) are shown as projections on the $(\rho ,z)$ plane. The projected orbits are: (a) a chaotic orbit, (c) a regular orbit of the main island, (e) a regular orbit of the $2/3$-resonance. The right column shows the corresponding evolution of the value of $V_{\mathrm{eff}}$ along the orbit as a function of the instantaneous $\rho$ coordinate. The embedded figure in (b) shows a detail near the upper edge of $V_{\mathrm{eff}}$ ($V_{\mathrm{eff}}\simeq 0$).

Figure 9

The CZV for parameter values $E=0.95$, $L_z=2.995M$, $\chi =0.9$, $q=0.95$, where the outer and the inner region are connected through a short neck.

Figure 10

(a) A detail of the surface of section in the neighborhood of the neck which joins the inner and the outer regions. (b) The rotation curve along the $\dot{\rho }=0$ line of the surface of section presented in (a). Embedded in (b) is the irregular variation of the rotation number just outside the left side of the $2/7$-plateau. This irregular behavior is due to the chaotic layer surrounding the corresponding island.

Figure 11

The time $\Delta t_r$ needed by nongeodesic orbits to cross the chain of islands belonging to the $2/3$-resonance as a function of their initial conditions $\rho (0)$ (the initial value of the $\rho$-coordinate) along the line $\dot{\rho }=0$, $z=0$. The parameters used are $\mu /M=8\times {10}^{-5}$, $q=0.95$, $\chi =0.9$, $E(0)=0.95$, $L_z(0)=3M$.

Figure 12

The black thick lines mark the succession of every third point on the $z=0$ surface of section of a nongeodesic orbit for two different types of entrapment in the $2/3$-resonance. (a) Entrapment of a nongeodesic orbit that makes 3 loops while inside the resonant island and (b) entrapment of a nongeodesic orbit that makes almost one loop. The arrows show the flow of the orbits on the $\rho$, $\dot{\rho }$ plane. In (a) and (b) the blue (light gray) big dot indicates the point at which the corresponding nongeodesic orbit crosses the border (blue or light gray thin line) and enters the left-most island of the $2/3$-resonance, while the red (dark gray) big dot indicates the point at which the nongeodesic orbit crosses the border (red or dark gray thin line) and exits the $2/3$-island of stability. Note the drift of the island during the evolution of the orbit. For both cases $\mu /M=8\times {10}^{-5}$, $q=0.95$, $\chi =0.9$, $E(0)=0.95$, $L_z(0)=3M$. The nongeodesic orbits of (a) and (b) correspond to the first and the third point from the left on the diagram of Fig. 11 respectively.

Figure 13

The evolution of the ratio ${\Omega }_R/{\Omega }_{\Theta }$ as function of the coordinate time $t$ for (a) the nongeodesic orbit shown in Fig. 12a, 12b the nongeodesic orbit shown in Fig. 12b. The vertical dashed lines demarcate the time intervals that the nongeodesic orbit spends in the interior of the $2/3$-resonance.

Figure 14

The evolution of the two polar frequencies ${\Omega }_R$ and ${\Omega }_{\Theta }$ as functions of the coordinate time $t$ for the nongeodesic orbit shown in Fig. 12a before, during and after the crossing of the $2/3$-resonance.

Figure 15

(a) The Fourier spectrum of the $\rho$-coordinate for a geodesic orbit. Embedded in (a) is the evolution of the $\rho$-coordinate along the proper time that produces this Fourier spectrum. The side frequency is apparent as a small secondary peak on the left of the fundamental frequency peak. The data-series are quite long here; namely $\Delta {\tau }_{\mathrm{tot}}=5\times {10}^{4}M$. However small the secondary peak may be, it causes significant problems when one tries to determine the fundamental frequencies of the adiabatically varying orbit by analyzing a data-series 10 times shorter in order to monitor the adiabatic evolution of the frequencies. In that case the fundamental-frequency peak is 10 times broader and the secondary peak slightly alters its shape. (b) The Fourier spectrum of the $R$-coordinate for the same geodesic orbit as in (a). Embedded in (b) is the evolution of the $R$-coordinate along the proper time that produces this Fourier spectrum. The $R$-coordinate has practically no modulation and the corresponding Fourier spectrum has no secondary peaks.