Gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

Newly formed black holes are expected to emit characteristic radiation in the form of quasinormal modes, called ringdown waves, with discrete frequencies. LISA should be able to detect the ringdown waves emitted by oscillating supermassive black holes throughout the observable Universe. We develop a multimode formalism, applicable to any interferometric detectors, for detecting ringdown signals, for estimating black-hole parameters from those signals, and for testing the no-hair theorem of general relativity. Focusing on LISA, we use current models of its sensitivity to compute the expected signal-to-noise ratio for ringdown events, the relative parameter estimation accuracy, and the resolvability of different modes. We also discuss the extent to which uncertainties on physical parameters, such as the black-hole spin and the energy emitted in each mode, will affect our ability to do black-hole spectroscopy.

Figure 1

Value of ${ϵ}_{\mathrm{rd}}$ required to detect the fundamental mode with $l=m=2$ (detection being defined by a SNR of 10) at $D_L=3\mathrm{Gpc}$. For illustrative purposes here we pick the fundamental mode with $l=m=2$, but the dependence on $(n,l,m)$ is very weak. The three curves correspond to $j=0$ (solid), $j=0.8$ (dashed) and $j=0.98$ (dot-dashed), where $j=J^2/M=a/M$ is the dimensionless angular momentum parameter of the hole. The “pessimistic” prediction from numerical simulations of head-on collisions is ${ϵ}_{\mathrm{rd}}={10}^{-3}$ (as marked by the dashed horizontal line), so we should be able to see all equal-mass mergers with a final black-hole mass larger than about $\sim {10}^5{M}_{\odot }$ (the vertical line is just a guide to the eye). The dip in the curves is a consequence of white-dwarf confusion noise in the LISA noise curve.

Figure 2

Errors (multiplied by the signal-to-noise ratio $\rho$) in measurements of different parameters for the fundamental $l=m=2$ mode as functions of the angular momentum parameter $j$. Solid (black) lines give $\rho {\sigma }_j$, dashed (red) lines $\rho {\sigma }_M/M$, dot-dashed (green) lines $\rho {\sigma }_A/A$, dot-dot-dashed (blue) lines $\rho {\sigma }_{\varphi }$, where ${\sigma }_k$ denotes the estimated rms error for variable $k$, $M$ denotes the mass of the black hole, and $A$ and $\varphi$ denote the amplitude and phase of the wave.

Figure 3

Critical SNR ${\rho }_{\mathrm{crit}}$ required to resolve the fundamental mode ($n=0$) from the first overtone ($n^{\prime }=1$) with the same angular dependence ($l=l^{\prime }$, $m=m^{\prime }$). We assume the amplitude of the overtone is one tenth that of the fundamental mode. Solid lines refer to $m=l,\dots ,1$ (bottom to top), the dotted line to $m=0$, and dashed lines to $m=-1,\dots ,-l$ (bottom to top).

Figure 4

Critical SNR ${\rho }_{\mathrm{both}}$ required to resolve both the frequency and the damping time of the fundamental mode ($n=0$) from the first overtone ($n^{\prime }=1$) with the same angular dependence ($l=l^{\prime }$, $m=m^{\prime }$). We assume the amplitude of the overtone is one tenth that of the fundamental mode. Solid lines refer to $m=l,\dots ,1$ (top to bottom), the dotted line to $m=0$, and dashed lines to $m=-1,\dots ,-l$ (top to bottom, unless indicated). In the color versions, we used black for the modes with $l=|m|$, red for those with $0<|m|<l$ and blue for $m=0$. For $l=m$ the critical SNR grows monotonically as $j\to 1$.

Figure 5

Frequency $f_{lmn}$ (left) and quality factor $Q_{lmn}$ (right) for the fundamental modes with $l=2,3,4$ and different values of $m$. Solid lines refer to $m=l,\dots ,1$ (from top to bottom), the dotted line to $m=0$, and dashed lines refer to $m=-1,\dots ,-l$ (from top to bottom). Quality factors for the higher overtones are lower than the ones we display here.

Figure 6

Fundamental $l=2$ QNM frequencies of the Kerr black hole in the range $j\in [0,0.99]$. Solid (blue) lines correspond to $m=2$, dot-dashed (black) lines to $m=1$, dotted (red) lines to $m=0$, dashed (green) lines to $m=-1$, and short dashed (violet) lines to $m=-2$.

Figure 7

Comparison of the SNR for inspiral and ringdown waveforms. In the left panel we use the Barack-Cutler noise-curve; in the right panel we use the same noise curve, but we do not include white-dwarf confusion noise. The thick (black) line marked by “inspiral” is the (angle-averaged) SNR for the inspiral of two equal-mass black holes at $D_L=3\mathrm{Gpc}$. The other sets of lines (red and blue in color versions) show the SNR for the $l=m=2$ mode using the $\delta$-function approximation, assuming a ringdown efficiency ${ϵ}_{\mathrm{rd}}=3\%$ and 0.1%, respectively. Solid, dashed and dot-dashed lines correspond to $j=0$ (Schwarzschild), $j=0.8$ and $j=0.98$ respectively.

Figure 8

Dependence of the SNR on redshift, for both inspiral (continuous lines) and ringdown (dashed and dot-dashed lines). We choose a ringdown efficiency ${ϵ}_{\mathrm{rd}}=1\%$ and consider the cases $j=0$, $j=0.8$ and $j=0.98$. From top to bottom the lines in each panel correspond to $z=0.54$ (black in color versions; $D_L=3\mathrm{Gpc}$), $z=1$ (red), $z=2$ (green), $z=5$ (blue) and $z=10$ (purple). The dashed lines are obtained from the full integral, the dot-dashed lines use the $\delta$-function approximation.

Figure 9

$Q_{lmn}/Q_{lmn}^{\prime }$ for the fundamental mode with $l=2$ (left) and $l=4$. Solid lines refer to $m>0$, dotted lines to $m=0$, dashed lines to $m<0$, and different shades (colors) denote different values of $m$, as indicated. Notice that this factor increases with $j$ when $m>0$ and decreases with $j$ when $m=-2$. The factor blows up as $j\to 0$ for $m=0$, and as $j\to j_{\mathrm{crit}}$ for certain values of $m<0$. This explains most qualitative features of the error plots.

Figure 10

Comparison between different approaches to compute the error (multiplied by the SNR) for the fundamental mode for $(l,m)=(2,2)$ and $(2,-1)$. Solid lines use the numerical implementation of the full expressions obtained by Mathematica, and a numerical calculation of the derivatives of $Q_{lmn}$ and $f_{lmn}$. Dashed and dotted lines use expressions for the Fisher matrix to leading order in $1/Q_{lmn}$; derivatives of $Q_{lmn}$ and $f_{lmn}$ are evaluated numerically for the former, and using the fitting functions given in Appendix  for the latter. Black (red) lines refer to calculations of the error in $j$ ($M$) using the FH “doubling prescription”. The plot shows that all these different calculations are in excellent agreement with each other. However, sometimes using the fits can produce only order-of-magnitude estimates of the errors: this happens for counter-rotating modes, where an accurate calculation of the derivatives of $Q_{lmn}$ and $f_{lmn}$ is more important (dotted lines in the right panel deviate significantly from the true answer obtained by taking numerical derivatives of the QNM tables). Finally (and only in the right panel) we plot, in blue, results obtained using Eqs. (4.20a), (4.20b) in Finn’s paper [6], a numerical implementation of the full expressions obtained by Mathematica, and a numerical calculation of the derivatives of $Q_{lmn}$ and $f_{lmn}$. Finn’s formula would lie on top of the other lines in the left plot (for $l=m=2$), but it gives a slightly different prediction for the errors on the counter-rotating mode with $l=2$, $m=-1$.

Figure 11

Correlation between the mass and angular momentum parameters for the fundamental mode and different values of $(l,m)$. For counter-rotating modes $j$ and $M$ are actually anticorrelated ($r_{jM}<0$), and we plot the modulus of the correlation. The solid line uses the numerical implementation of the full expressions obtained by Mathematica, and a numerical calculation of the derivatives of $Q_{lmn}$ and $f_{lmn}$. The dashed line uses an expansion to leading order in $1/Q_{lmn}$ of the correlation matrix, and a numerical calculation of the derivatives of $Q_{lmn}$ and $f_{lmn}$. The dotted line uses again a Taylor expansion, but this time we evaluate the derivatives of $Q_{lmn}$ and $f_{lmn}$ using the fitting functions given in Appendix . The minimum in the correlation between mass and angular momentum (for $l=m=2$) disappears when we use the numerical derivatives. The behavior of $r_{jM}$, as computed numerically, is qualitatively consistent with the entry marked by $f_{Ma}$ in Echeverria’s Table  [5], so the fake minimum really seems to be due to the inaccuracy of the fit.

Figure 12

Errors in measurements of different parameters for the fundamental mode with different values of $(l,m)$, as functions of the angular momentum parameter $j$. Solid (black) lines give $\rho {\sigma }_j$, dashed (red) lines $\rho {\sigma }_M/M$, dot-dashed (green) lines $\rho {\sigma }_A/A$, dot-dot-dashed (blue) lines $\rho {\sigma }_{{\varphi }_{lmn}}$.

Figure 13

Errors in angular momentum ${\sigma }_j$ (black) and mass ${\sigma }_M/M$ (red) for: $l=m=2$ (top left), $l=2$, $m=0$ (top center), $l=2$, $m=-2$ (top right); $l=4$, $m=2$ (bottom left), $l=4$, $m=0$ (bottom center), $l=4$, $m=-2$ (bottom right), for a source at $D_L=3\mathrm{Gpc}$ with ${ϵ}_{\mathrm{rd}}=3\%$. Solid lines refer to $j=0$, dashed lines to $j=0.8$, dot-dashed lines to $j=0.98$. Errors scale with ${\rho }^{-1}$, hence they are proportional to ${ϵ}_{\mathrm{rd}}^{-1/2}$.

Figure 14

Scaled errors ($\rho {\sigma }_M$, $\rho {\sigma }_j$) for two-mode measurements of mass and angular momentum, with $l=m=m^{\prime }=2$. Solid lines refer to $l^{\prime }=3$, dashed lines to $l^{\prime }=4$.

Figure 15

Scaled errors ($\rho {\sigma }_M$, $\rho {\sigma }_j$) for two-mode measurements of mass and angular momentum, with $l=m=l^{\prime }=2$. Solid lines refer to $m^{\prime }=1$, dashed lines to $m^{\prime }=0$, dot-dashed lines to $m^{\prime }=-1$, dotted lines to $m^{\prime }=-2$. In the case ${\mathcal{A}}_1={10}^{-2}{\mathcal{A}}_2$ the error for $m^{\prime }=0$ and $m^{\prime }=-1$ becomes quite large, and for clarity we show results using both a linear and a log scale.

Figure 16

Modulation of the total SNR, Eq. (6.6), induced by the second overtone. We consider an $l=m=2$ perturbation and three different values of the angular momentum $j$ (solid lines: $j=0$, dashed lines: $j=0.8$, dot-dashed lines: $j=0.98$). For concreteness, we assume that the first overtone has an amplitude ${\mathcal{A}}_2={10}^{-1}{\mathcal{A}}_1$. The nearly horizontal (red) lines correspond to the SNR in the absence of modulations, Eq. (6.4).

Figure 17

Scaled errors $(\rho {\sigma }_M,\rho {\sigma }_j)$ for the two-mode waveform (solid lines) versus a one-mode waveform (dashed line). Here we assume ${\mathcal{A}}_1=10{\mathcal{A}}_2$ and consider perturbations with $l=m=2$ (left) and $l=2$, $m=-2$ (right).

Figure 18

Critical SNR ${\rho }_{\mathrm{crit}}$ (top) and ${\rho }_{\mathrm{both}}$ (bottom) required to resolve different combinations of modes. Here we consider the “quasiorthonormal” perturbations of Sec. 6a (either $l\ne l^{\prime }$ or $m\ne m^{\prime }$). Following the treatment in that Section, we always assume $l=m=2$ for the first mode. Different colors/linestyles refer to different values of $l^{\prime }$ and $m^{\prime }$ for the second mode. Black: $l^{\prime }=2$ and $m^{\prime }=1$ (solid), $m^{\prime }=0$ (dotted), $m^{\prime }=-1$, $-2$ (dashed, top to bottom). Blue: $l^{\prime }=3$ and $m^{\prime }=2$ (solid), $m^{\prime }=3$ (dot-dashed). Red: $l^{\prime }=4$ and $m^{\prime }=2$ (solid), $m^{\prime }=4$ (dot-dashed). Different panels refer to different ratios ${\mathcal{A}}_1/{\mathcal{A}}_2$, as indicated.

Figure 19

Modulus of the error in the SNR due to the $\delta$-function approximation, $|({\rho }_{\delta }-{\rho }_{\mathrm{exact}})/{\rho }_{\mathrm{exact}}|$, for modes with $l=2$ and $l=4$, for $j=0.98$ and $D_L=3\mathrm{Gpc}$. For each $l$ we consider the fundamental mode ($n=0$) and the second overtone ($n=2$). The horizontal line marks a 10% deviation from the exact result: for those modes lying below the horizontal line the $\delta$-function approximation can be considered very accurate. The “inverted peaks” are just zero-crossings of $|({\rho }_{\delta }-{\rho }_{\mathrm{exact}})/{\rho }_{\mathrm{exact}}|$.

Figure 20

Analytic approximation to the LISA root noise spectral density curve used in this paper and in Ref. 2 (dashed line) and the curve produced using the LISA Sensitivity Curve Generator [51] (solid line). The SCG curve has been multiplied by a factor of $\sqrt{3/20}$ to obtain an effective non-sky-averaged noise spectral density. The SCG noise curve does not include the extra-galactic white-dwarf confusion noise while the analytical approximation curve does.

Figure 21

Accuracy of the fits of frequency (black) and quality factor (red) for the fundamental mode with $l=m=2$ (left) and $l=2$, $m=0$ (right). Solid lines refer to our fits, dashed lines to the fits proposed in Refs. 5, 36.