Resolving Super Massive Black Holes with LISA Stanislav Babak, 1 Mark Hannam, 2 Sascha Husa, 1 and Bernard Schutz 1 1 Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨ uhlenberg 1, D-14476 Golm bei Potsdam, Germany 2 Physics Department, University College Cork, Cork, Ireland We study the angular resolution of the gravitational wave detector LISA and show that numerical relativity can drastically improve the accuracy of position location for coalescing Super Massive Black Hole (SMBH) binaries. For systems with total redshifted mass above 10 7 M, LISA will mainly see the merger and ring-down of the gravitational wave (GW) signal, which can now be computed numerically using the full Einstein equations. Using numerical waveforms that also include about ten GW cycles of inspiral, we improve inspiral-only position estimates by an order of magnitude. We show that LISA localizes half of all such systems at z = 1 to better than 3 arcminutes and the best 20% to within one arcminute. This will give excellent prospects for identifying the host galaxy. PACS numbers: 04.30.Tv 04.25.dg 95.30.Sf Introduction.— Astronomical observation provides very strong evidence for the existence of a “dark” com- pact massive (10 6 - 10 9 M  ) object in the core of every galaxy for which the central parsec region can be resolved [1]. These objects are believed to be supermassive black holes (SMBH), and are of great interest to researchers in fundamental physics, astrophysics and cosmology. Their formation and the observed correlation between SMBH mass and galaxy morphology (see [2] for an overview) are still open questions. SMBHs probably arise at least in part through the mergers of smaller-mass BHs [3]. These mergers and the mergers of SMBH binaries following col- lisions of galaxies constitute some of the most powerful sources of gravitational waves (GW) predicted by current models. We will be able to detect such events through- out the Universe with LISA [4], a proposed space-borne gravitational-wave observatory, scheduled for launch in 2018+ and designed to be sensitive to GW signals in the range 10 -4 - 0.1 Hz. Realizing the full scientific benefit of the LISA mission will require accurate estimates of the binary’s parame- ters. Precise measurements of the masses, spins and dis- tance will allow us to probe models of SMBH formation. Accurate localization of the source in the sky is crucial to relate gravitational-wave and electromagnetic obser- vations of the coalescence event, and hopefully will allow identification of the host galaxy. Optical observations are required to measure the redshift to the object, while gravitational-wave observations yield precise calibration- free estimates of the distance. Such LISA events with optical counterparts will determine the redshift-distance relationship, which in turn will allow us to map the ge- ometry of the Universe and measure the amount of dark energy. Recently several groups [5, 6, 7, 8] have evaluated the accuracy of parameter estimation using the inspiral part of the GW signal. It was shown that the spin and higher orbital harmonics are necessary to de-correlate the pa- rameters and therefore improve the parameter estima- tion. In this Letter we assess the angular resolution of LISA for SMBH binaries with (red-shifted) masses above 10 7 M  . We expect several such merger signals per year at relatively close distance [3]. For those heavy systems the inspiral signal may be smaller or at least not much larger than the instrumental noise, and the signal will be dominated by the merger and ring-down. We use numerical relativity to compute a waveform which con- tains about ten GW cycles, plus merger and ring-down. We fix the redshifted masses of two non-spinning BHs to 4.44 ×10 6 M  and 8.88 ×10 6 M  and vary the “extrinsic” parameters: sky ecliptic coordinates θ S ,φ S , inclination ι of the orbital angular momentum to the line of sight, polarization angle ψ, fiducial arrival time T 0 which we fix to be the time when the binary separation equals 10 M , orbital phase φ 0 at T 0 , and luminosity distance D L . Data analysis for LISA is based on time-delay in- terferometry (TDI, see [9] for an overview). We con- vert the strain polarizations h + and h × given in the source frame to first generation unequal-arm Michel- son streams X,Y,Z [9, 10] and use two combinations A = (2X - Y - Z )/3, E =(Z - Y )/ √ 3 with uncorre- lated noise. Due to technical difficulties explained below, we do not take into account the third independent combi- nation, which has poor sensitivity to GW at low frequen- cies, and which adds only a few percent to the combined signal-to-noise ratio (SNR). By fixing masses we under- estimate the errors, but at the same time not including the third TDI combination overestimates the error boxes. We mainly concentrate here on the estimation of the lo- calization of the source and usually [8] the directional angles very weakly correlate with masses. Since we use the merger for localizing the hosting galaxy, we cannot produce an early warning for the merger itself; however, we can identify the hosting galaxy by the afterglow [11]. We conduct Monte Carlo simulations by randomly choosing 600 points in the extrinsic parameter space (with fixed masses, and choosing an example distance arXiv:0806.1591v1 [gr-qc] 10 Jun 2008