Science with the space-based interferometer eLISA: Supermassive black hole binaries

We compare the science capabilities of different eLISA mission designs, including four-link (two-arm) and six-link (three-arm) configurations with different arm lengths, low-frequency noise sensitivities and mission durations. For each of these configurations we consider a few representative massive black hole formation scenarios. These scenarios are chosen to explore two physical mechanisms that greatly affect eLISA rates, namely (i) black hole seeding, and (ii) the delays between the merger of two galaxies and the merger of the black holes hosted by those galaxies. We assess the eLISA parameter estimation accuracy using a Fisher matrix analysis with spin-precessing, inspiral-only waveforms. We quantify the information present in the merger and ringdown by rescaling the inspiral-only Fisher matrix estimates using the signal-to-noise ratio from nonprecessing inspiral-merger-ringdown phenomenological waveforms, and from a reduced set of precessing numerical relativity/post-Newtonian hybrid waveforms. We find that all of the eLISA configurations considered in our study should detect some massive black hole binaries. However, configurations with six links and better low-frequency noise will provide much more information on the origin of black holes at high redshifts and on their accretion history, and they may allow the identification of electromagnetic counterparts to massive black hole mergers.

Figure 1

Analytic fits to the sensitivity curves for different configurations investigated in this paper. In both panels the curves running from the top down are for A1 (blue), A2 (green) and A5 (red) configurations, and all curves have the same high-frequency noise by design. The (black) curve with wiggles at high frequency is the numerical sensitivity for eLISA/NGO, which is shown in both panels for reference. Dashed curves include only instrumental noise, while solid lines represent the total noise (including the contribution from CWD confusion noise). The left panel shows all configurations with pessimistic (N1) acceleration noise levels; the right panel assumes optimistic (N2) acceleration noise levels.

Figure 2

Schematic summary of the model of [56], with the improvements of [57, 58, 59]. Red boxes on the left highlight the elements that heavily affect rates (black hole seeding and delays), for which we consider multiple options in this paper.

Figure 3

Predicted merger rates per unit redshift (left panel) and per unit total redshifted mass $M_z=(m_1+m_2)(1+z)$ (right panel) for the three models described in the text.

Figure 4

Contribution of each halo mass to the total merger rate for models popIII (long-dashed brown lines), Q3-nod (short-dashed green lines) and Q3-d (solid orange lines). Lines in the top panel are proportional to the number of mergers per halo (i.e., removing the Press and Schechter weights), whereas lines in the bottom panel represent the halo contribution to the cosmic merger rate $dN/d\log M_H$. In both panels, thin lines are the results of our MBH population models; thick lines are extrapolations assuming a linear relation between the number of mergers and the halo mass.

Figure 5

SNR gain $\mathcal{R}(\rho )$ as a function of redshifted total mass $M_z$. PhenomC waveforms applied to one realization of the Q3-nod population model are represented by black ($0.5<m_{2z}/m_{1z}<1$) and blue ($0<m_{2z}/m_{1z}<0.5$) dots. The red and green dots are computed using nonspinning PhenomC waveforms at a fixed $M_z$ for decreasing values of $m_{2z}/m_{1z}$ (from top to bottom); red dots are for $m_{2z}/m_{1z}>0.5$ and green dots are for $m_{2z}/m_{1z}<0.5$. This calculation refers to the detector configuration that we labeled N2A1M2L6.

Figure 6

Median improvement in sky localization $\mathcal{R}(\mathrm{\Delta }\mathrm{\Omega })$ (solid lines) and luminosity distance errors $\mathcal{R}(\mathrm{\Delta }{D}_l/D_l)$ (dashed lines) as a function of the SNR gain $\mathcal{R}(\rho )$. Black and red lines are for a six-link and four-link detector configuration, respectively, as indicated in figure. Dotted lines represent linear and quadratic scalings to guide the eye.

Figure 7

Total number of detections (i.e., sources with $\rho >8$, left plot) and total number of detections with $\mathrm{\Delta }\mathrm{\Omega }<10{\mathrm{deg}}^2$ (right plot) assuming a five-year mission (M5). In each plot, the left and right panels are for inspiral and IMR-rescaled waveforms, respectively. The bottom panels represent the Absolute number of detections for different eLISA configurations, while the top panels represent the gain/loss of a given configuration with respect to the standard NGO design, i.e., the ratio [number of sources for $(\mathrm{N}i\mathrm{A}j\mathrm{M}k\mathrm{L}\ell )]/[\text{number of sources with}(\mathrm{N}2\mathrm{A}1\mathrm{M}k\mathrm{L}4)]$. Long-dashed brown lines are for model popIII, solid orange lines for model Q3-d, and short-dashed green lines for model Q3-nod. Thick lines with filled triangles are for six links (L6), while thin lines with open triangles are for four links (L4).

Figure 8

Number of detections with fractional errors of less than 0.01 for both redshifted masses. The left and right panels are for a mission lifetime of two years and five years, respectively. Inspiral-only waveforms have been used in all cases. The bottom panels represent the number of sources as a function of the eLISA configuration, while the top panels represent the gain/loss of a given configuration with respect to NGO, i.e., the ratio [number of sources for $(\mathrm{N}i\mathrm{A}j\mathrm{M}k\mathrm{L}\ell )]/[\text{number of sources with}(\mathrm{N}2\mathrm{A}1\mathrm{M}k\mathrm{L}4)]$. Long-dashed brown lines are for model popIII, solid orange lines for model Q3-d and short-dashed green lines for model Q3-nod. Thick lines with filled triangles are for six links (L6), while thin lines with open triangles are for four links (L4).

Figure 9

Number of detections with Absolute error on the primary MBH spin smaller than 0.01 (left plot), and with Absolute error on the secondary MBH spin smaller than 0.1. In each plot, left and right panels are for a mission lifetime of two years and five years, respectively; inspiral-only waveforms have been used in all cases. The bottom panels represent the number of sources as a function of the eLISA configuration, while the top panels represent the gain/loss of a given configuration with respect to NGO, i.e., the ratio [number of sources for $(\mathrm{N}i\mathrm{A}j\mathrm{M}k\mathrm{L}\ell )]/[\text{number of sources with}(\mathrm{N}2\mathrm{A}1\mathrm{M}k\mathrm{L}4)]$. Long-dashed brown lines are for model popIII, solid orange lines for model Q3-d and short-dashed green lines for model Q3-nod. Thick lines with filled triangles are for six links (L6), while thin lines with open triangles are for four links (L4).

Figure 10

Number of detections such that the Absolute error in the measurement of both misalignment angles, ${\theta }_{{\chi }_1}$ and ${\theta }_{{\chi }_2}$, at the innermost stable circular orbit is less than 10°. The left and right panels are for a mission lifetime of two years and five years, respectively. Inspiral-only waveforms have been used in all cases. The bottom panels represent the number of sources as a function of the eLISA configuration, while the top panels represent the gain/loss of a given configuration with respect to NGO, i.e., the ratio [number of sources for $(\mathrm{N}i\mathrm{A}j\mathrm{M}k\mathrm{L}\ell )]/[\text{number of sources with}(\mathrm{N}2\mathrm{A}1\mathrm{M}k\mathrm{L}4)]$. Long-dashed brown lines are for model popIII, solid orange lines for model Q3-d and short-dashed green lines for model Q3-nod. Thick lines with filled triangles are for six links (L6), while thin lines with open triangles are for four links (L4).

Figure 11

Same as Fig. 9, but for the remnant spin ${\chi }_r$.

Figure 12

(Left plot) Total number of detections at $z<5$ with $\mathrm{\Delta }\mathrm{\Omega }<10{\mathrm{deg}}^2$ and $\mathrm{\Delta }{D}_l/D_l<0.1$. (Right plot) Total number of detections at $z>7$ with $\mathrm{\Delta }{D}_l/D_l<0.3$. In each plot, left and right panels are for inspiral and IMR rescaled waveform, respectively; five years of observations are assumed. The bottom panels represent the number of sources as a function of the eLISA configuration, while the top panels represent the gain/loss of a given configuration with respect to NGO, i.e., the ratio [number of sources for $(\mathrm{N}i\mathrm{A}j\mathrm{M}k\mathrm{L}\ell )]/[\text{number of sources with}(\mathrm{N}2\mathrm{A}1\mathrm{M}k\mathrm{L}4)]$. Long-dashed brown lines are for model popIII, solid orange lines for model Q3-d and short-dashed green lines for model Q3-nod. Thick lines with filled triangles are for six links (L6), while thin lines with open triangles are for four links (L4).

Figure 13

Science loss as a function of specific design choices. In each plot, we compare the results of simulations that only differ by a specific design element (see the description in the main text). Those are number of links $\mathrm{L}4/\mathrm{L}6$ (18 simulations, upper left); low-frequency noise $\mathrm{N}1/\mathrm{N}2$ (18 simulations, upper right); arm length $\mathrm{A}2/\mathrm{A}5$ (12 simulations, lower left); arm length $\mathrm{A}1/\mathrm{A}5$ (12 simulations, lower right). Each of the four plots visualizes the science loss for specific design choices according to four different indicators: total number of Detections (upper left quadrant), Median SNR of detected sources (upper right quadrant), number of low-$z$ sources with good enough sky localization for counterpart searches and good distance determination (lower left quadrant), and number of high-$z$ sources with good distance measurement (lower right quadrant). In each panel we plot the histograms of the indicators over all the simulations, and the mean value (triangles). Blue is for inspiral waveforms only, orange is for IMR-rescaled waveforms.