Parameter estimation of coalescing supermassive black hole binaries with LISA

Laser Interferometer Space Antenna (LISA) will routinely observe coalescences of supermassive black hole (BH) binaries up to very high redshifts. LISA can measure mass parameters of such coalescences to a relative accuracy of ${10}^{-4}–{10}^{-6}$, for sources at a distance of 3 Gpc. The problem of parameter estimation of massive nonspinning binary black holes using post-Newtonian (PN) phasing formula is studied in the context of LISA. Specifically, the performance of the 3.5PN templates is contrasted against its 2PN counterpart using a waveform which is averaged over the LISA pattern functions. The improvement due to the higher order corrections to the phasing formula is examined by calculating the errors in the estimation of mass parameters at each order. The estimation of the mass parameters $\mathcal{M}$ and $\eta$ is significantly enhanced by using the 3.5PN waveform instead of the 2PN one. For an equal mass binary of $2\times {10}^6{M}_{\odot }$ at a luminosity distance of 3 Gpc, the improvement in chirp mass is $\sim 11\%$ and that of $\eta$ is $\sim 39\%$. Estimation of coalescence time $t_c$ worsens by 43%. The improvement is larger for the unequal mass binary mergers. These results are compared to the ones obtained using a nonpattern averaged waveform. The errors depend very much on the location and orientation of the source and general conclusions cannot be drawn without performing Monte Carlo simulations. Finally the effect of the choice of the lower frequency cutoff for LISA on the parameter estimation is studied.

Figure 1

Variation of errors with the total observed mass for different PN order restricted waveforms for LISA. Pattern-averaged waveform is used. The convergence of the results is evident in both cases. Sources are assumed to be at 3 Gpc.

Figure 2

Variation of errors in $t_c$ with the total observed mass for different PN order restricted waveforms for LISA. Pattern-averaged waveform is used. The convergence of the results is evident from the plot. Sources are assumed to be at 3 Gpc.

Figure 3

Variation of errors of chirp mass and $\eta$ with the total observed mass for different PN order restricted waveforms for LISA. Pattern-averaged waveform is used for the two choices of the lower frequency cutoffs $f_1={10}^{-4}\mathrm{Hz}$ and $f_2={10}^{-5}\mathrm{Hz}$. Sources are assumed to be at 3 Gpc.

Figure 4

Variation of the signal-to-noise ratio, relative errors in chirp mass, and relative errors in $\eta$ with the total observed mass of the binary for different choices of location and orientation of the source. Sources are assumed to be at a luminosity distance of 3 Gpc and a nonpattern averaged waveform is used. Angle:1 corresponds to $\{{\overline{\mu }}_L=0.5,{\overline{\mu }}_S=-0.8,{\overline{\varphi }}_L=3,{\overline{\varphi }}_S=1\}$. Angle:2 is $\{{\overline{\mu }}_L=0.2,{\overline{\mu }}_S=-0.6,{\overline{\varphi }}_L=3,{\overline{\varphi }}_S=1\}$ and Angle:3 $\{{\overline{\mu }}_L=0.8,{\overline{\mu }}_S=0.3,{\overline{\varphi }}_L=2,{\overline{\varphi }}_S=5\}$. The errors thus depend very much on the position and orientation of the source in the sky.