Effects of different eLISA-like configurations on massive black hole parameter estimation

As the theme for the future L3 Cosmic Vision mission, ESA has recently chosen the “Gravitational Wave Universe.” Within this program, a mission concept called eLISA has been proposed. This observatory has a current initial configuration consisting of four laser links between the three satellites, which are separated by a distance of one million kilometers, constructing a single-channel Michelson interferometer. However, the final configuration for the observatory will not be fixed until the end of this decade. With this in mind, we investigate the effect of different eLISA-like configurations on massive black hole detections. This work compares the results of a Bayesian inference study of 120 massive black hole binaries out to a redshift of $z\sim 13$ for a ${10}^6\mathrm{km}$ arm length eLISA with four and six links, as well as a $2\times {10}^6\mathrm{km}$ arm length observatory with four links. We demonstrate that the original eLISA configuration should allow us to recover the luminosity distance of the source with an error of less than 10% out to a redshift of $z\sim 4$, and a sky error box of $\mathrm{\Delta }\mathrm{\Omega }\le {10}^2{\mathrm{deg}}^2$ out to $z\sim 0.1$. In contrast, both alternative configurations suggest that we should be able to conduct the same parameter recovery with errors of less than 10% in luminosity distance out to $z\sim 12$ and $\mathrm{\Delta }\mathrm{\Omega }\le {10}^2{\mathrm{deg}}^2$ out to $z\sim 0.4$. Using the information from these studies, we also infer that if we were able to construct a 2 Gm, six-link detector, the above values would shift to $z\sim 20$ for luminosity distance and $z\sim 0.9$ for sky error. While the final configuration will also be dependent on both technological and financial considerations, our study suggests that increasing the size of a two-arm detector is a viable alternative to the inclusion of a third arm in a smaller detector. More importantly, this work further suggests no clear scientific loss between either choice.

Figure 1

A comparison of an instrumental noise model for a 1 Gm arm length, four- and six-link configuration (C1,C2) and a 2 Gm arm length, four-link eLISA configuration (C3), assuming a constant laser power and telescope size.

Figure 2

Massive black hole redshift horizon distance (excluding merger and ringdown) as a function of the redshifted total mass for configurations C1 (solid blue line), C2 (dashed red line), and C3 (dot-dashed orange line). Each curve assumes an inspiral-only detection threshold of ${\rho }_{th}=(10,9,8)$, respectively.

Figure 3

A comparison of 95% credible intervals for $D_L$ (top left), $\mathrm{\Delta }\mathrm{\Omega }$ (top right), $M_c$ (middle left), $\mu$ (middle right), $\iota$ (bottom left), and $t_c$ (bottom right) between a C1 (blue) and C2 (orange) eLISA-like observatory. In the cell representing the luminosity distance, the dashed black line represents the true injected values. For ease of interpretation, the mass parameters are represented as credible intervals in percentage error, while the credible intervals for $(\iota ,t_c)$ are centered around the median subtracted true values, and are expressed in radians and seconds, respectively.

Figure 4

A comparison of 95% credible intervals for $D_L$ (top left), $\mathrm{\Delta }\mathrm{\Omega }$ (top right), $M_c$ (middle left), $\mu$ (middle right), $\iota$ (bottom left), and $t_c$ (bottom right) between a C1 (blue line) and C3 (orange line) eLISA-like observatory. In the cell representing the luminosity distance, the dashed black line represents the true injected values. For ease of interpretation, the mass parameters are represented as credible intervals in percentage error, while the credible intervals for $(\iota ,t_c)$ are centered around the median subtracted true values, and are expressed in radians and seconds, respectively.

Figure 5

Based on a regression analysis of the chain medians for each of the 120 sources, we plot of the growth of percentage error in luminosity distance (left) and solid sky angle (right) out to a redshift of $z=20$. In each cell, we plot the configurations C1 (blue solid line), C2 (blue dashed line), and C3 (orange solid line). Assuming the inclusion of a third detector arm has a similar scaling effect in all cases, we can also infer the performance of a 2 Gm, six-link observatory, which we label C4 (orange dashed line).