Measuring parameters of massive black hole binaries with partially aligned spins

Measuring parameters of massive black hole binaries with partially aligned spins

Abstract

The future space-based gravitational wave detector LISA will be able to measure parameters of coalescing massive black hole binaries, often to extremely high accuracy. Previous work has demonstrated that the black hole spins can have a strong impact on the accuracy of parameter measurement. Relativistic spin-induced precession modulates the waveform in a manner which can break degeneracies between parameters, in principle significantly improving how well they are measured. Recent studies have indicated, however, that spin precession may be weak for an important subset of astrophysical binary black holes: those in which the spins are aligned due to interactions with gas. In this paper, we examine how well a binary’s parameters can be measured when its spins are partially aligned and compare results using waveforms that include higher post-Newtonian harmonics to those that are truncated at leading quadrupole order. We find that the weakened precession can substantially degrade parameter estimation. This degradation is particularly devastating for the extrinsic parameters sky position and distance. Absent higher harmonics, LISA typically localizes the sky position of a nearly aligned binary a factor of less accurately than for one in which the spin orientations are random. Our knowledge of a source’s sky position will thus be worst for the gas-rich systems which are most likely to produce electromagnetic counterparts. Fortunately, higher harmonics of the waveform can make up for this degradation. By including harmonics beyond the quadrupole in our waveform model, we find that the accuracy with which most of the binary’s parameters are measured can be substantially improved. In some cases, parameters can be measured as well in partially aligned binaries as they can be when the binary spins are random.

pacs:
04.80.Nn, 04.30.Db, 04.30.Tv

I Introduction

The coalescence of massive black hole binaries is a primary source for the future space-based gravitational wave (GW) detector LISA.1 LISA will be able to detect such sources with extremely high signal-to-noise ratio (SNR) at low redshift (), as well as with moderate, but still reasonable, SNR at extremely high redshift () Baker et al. (2007). Estimated event rates for these sources vary widely based on formation scenarios but tend to predict roughly tens of sources per year, with as a pessimistic estimate and as an optimistic one Sesana et al. (2007). (The actual detection rate will, of course, tell us much about the formation of black hole binaries and the growth of massive black holes in the universe.)

While the detection of gravitational waves from these sources will certainly be interesting for its own sake, attention has turned in recent years to the capabilities of LISA as a true astronomical observatory. Many papers Cutler (1998); Hughes (2002); Vecchio (2004); Berti et al. (2005); Holz and Hughes (2005); Berti et al. (2006); Lang and Hughes (); Arun et al. (2007a, b); Trias and Sintes (2008); Lang and Hughes (2008); Porter and Cornish (2008); Lang and Hughes (2009); Klein et al. (2009); McWilliams et al. (2010); Key and Cornish (2011); McWilliams et al. () have investigated just how well LISA can measure the parameters of the binaries it detects. This is often done using the Fisher-matrix method Finn (1992); Cutler and Flanagan (1994), which essentially measures the local curvature of the posterior probability distribution for parameters in the vicinity of the maximum. Parameters for which the posterior is more strongly curved (i.e., which more strongly affect the waveform) are measured more accurately than those for which the posterior is only weakly curved. Correlations between parameters are also extremely important. When two parameters are strongly correlated, it is difficult to “detangle” the influence of one on the waveform over the other. This means that the accuracy with which both parameters are measured is controlled by the one which is most poorly determined.

Recent studies have considered how well parameters can be measured while doing the actual data analysis problem of removing signals from noise K. A. Arnaud, S. Babak, J. G. Baker, M. J. Benacquista, N. J. Cornish, C. Cutler, S. L. Larson, B. S. Sathyaprakash, M. Vallisneri, A. Vecchio et al. (2006). In effect, these studies simulate the parameter extraction process with enough detail to uncover issues such as multiple extrema of the posterior surfaces which are missed by the simpler (and cruder) Fisher analyses. Such “realistic” studies are typically much more CPU-intensive and cannot easily study parameter measurement issues over a broad swath of astrophysically important parameter space. Both families of studies have substantially advanced our understanding of LISA’s science reach in the past 5 or so years.

Some parameters that are especially interesting are the intrinsic system properties, namely, the masses and spins of the black holes. Masses can be measured extremely well in the best cases [with a relative error of for individual masses and for the chirp mass ]. Spins are not measured quite so well but are still expected to be determined with percent-level accuracy. By measuring these parameters for many systems, one can construct a merger history of black holes, and by extension, their host galaxies, learning much about galaxy formation, black hole formation, AGN feedback, and so forth.

We are also interested in measuring parameters extrinsic to the system, namely, its position on the sky and its luminosity distance. With the position and distance (converted into an approximate redshift), astronomers can search the sky for probable electromagnetic counterparts to the gravitational wave events. Various types of counterparts have been proposed, from signals during the inspiral Armitage and Natarajan (2002), to bright flashes at the time of merger Bode and Phinney (2009) (or even reductions in luminosity O’Neill et al. (2009)), to long-delayed afterglows Milosavljević and Phinney (2005). The different scenarios arise because the behavior of gas around an inspiraling binary system is not well understood. A very different kind of electromagnetic counterpart can be produced by a kicked remnant black hole that triggers a telltale sequence of stellar disruptions Stone and Loeb (2011). (Tidal disruption of stars may also allow us to flag the presence of a binary long before it enters the LISA band, allowing a better understanding of the space density of massive black hole binaries Wegg and Bode ().) If a counterpart can be identified, the electromagnetic information can be combined with the gravitational information to reveal more about the astrophysics of the system. Counterparts may also make it possible for binary black holes to be used as probes of the cosmological distance-redshift relation, since the electromagnetic redshift and gravitational distance are determined independently Holz and Hughes (2005). Unfortunately, finding a counterpart, even if a unique signature does exist, will not be easy. The typical error windows for LISA are tens of arcminutes on a side at the end of inspiral, reduced from several square degrees in the weeks and months before merger. Still, this localization does give large survey telescopes like LSST (field of view square degrees) a chance to study a particular area of the sky with advance warning Tyson (SPIE, Bellingham, WA, 2002).

One particularly important result in the study of LISA’s science capabilities was the discovery that including spin precession effects in the waveform model typically improves the accuracy of parameter measurement Vecchio (2004); Lang and Hughes (). Spin precession arises because of geodetic and gravitomagnetic general relativistic effects Apostolatos et al. (1994); Kidder (1995). The orbital plane of the system also precesses in order to preserve the total angular momentum on time scales shorter than the radiation reaction time. Together, these precessions modulate the amplitude and phase of the waveform, breaking correlations between certain sets of parameters and improving how well the members of those sets can be measured. The greatest improvement is to the measured masses of a binary’s members (accuracy typically improved by 1–2 orders of magnitude). The measured sky position angles and distance to the source are all improved by about half an order of magnitude, reducing the size of the sky position pixel in which one must search for a counterpart by a factor of (or the 3D voxel volume by a factor of ).

The precession of one of the spins in a binary, to 1.5 post-Newtonian order and averaged over an orbit,2 is given by

(1)

where and are the two spins, and are the two masses, is the total mass, is the reduced mass, is the direction of the orbital angular momentum, and is the orbital separation in harmonic coordinates. It is clear that precession is maximal when the spins of the system are orthogonal to the orbital angular momentum and to each other and vanishes when the spins and orbital axis are aligned. In Lang and Hughes () (hereafter Paper I), it was assumed that the relative orientation of the spins and the orbital angular momentum was completely arbitrary. The results of that paper are summarized in a series of histograms describing parameter measurement accuracy when the various angular momentum vectors are allowed to point in any direction.

Recent studies have shown, however, that accreting gas in a system may evolve the spin in such a way that the spins are at least partially aligned with each other and with the orbit Bogdanović et al. (2007); Dotti et al. (2010). The degree of alignment depends on the temperature of the gas: In “hot gas” models, which have polytropic index , the spins align within of the orbital axis. “Cold gas” models with align even more thoroughly, to within Dotti et al. (2010).

Does spin-induced precession, now constrained by initial conditions, still break degeneracies as efficiently as described in Paper I? Any degradation in parameter measurement capability could have a strong effect on the ability to find electromagnetic counterparts. The results of Lang and Hughes (); Lang and Hughes (2008, 2009) may be biased toward gas-free “dry” mergers, severely underestimating localization errors in gaseous “wet” mergers — the very systems which we are most likely to see electromagnetically. The effect of alignment on mass and spin measurements is also interesting (though arguably less so, since even a factor of several degradation for these parameters would still imply excellent accuracy).

The goal of this paper is to answer the question posed above. We do so with a Fisher-matrix analysis of parameter measurement for binaries whose spins are partially aligned according to two wet merger models: hot gas, which aligns the spins and orbit to within , and cold gas, which aligns to within . We demonstrate that this degree of alignment can substantially degrade parameter accuracy but that one can “repair” much of this degradation by using the “full” waveform model, including harmonics beyond the leading quadrupole. In what follows, we will use the terms “gas-free” or “dry” interchangeably with the term “random spins,” “hot gas” interchangeably with the phrase “spins aligned within ,” and “cold gas” interchangeably with the phrase “spins aligned within .” We also sometimes write “ alignment” as shorthand for “spins aligned within ,” and likewise for “ alignment.” (The two distributions contain systems with alignments less than or , although with a bias toward the upper end of the allowed range.)

The outline of the paper is as follows. We begin in Sec. II by describing the operation of our code, including the production of binary black hole waveforms, the LISA response, the noise model, and how we construct the Fisher matrix. We focus on changes from Paper I, leaving detailed description of the theory to that paper.

In Sec. III, we then present results for parameter errors in wet mergers, examining both “hot” and “cold” models. We compare these results to the case of dry mergers, in which spin orientations are chosen to be completely random with respect to each other and to the orbital angular momentum. It should be emphasized that throughout Sec. III, we consider only the leading quadrupole piece of the gravitational waveform (the so-called “restricted” post-Newtonian approximation). As expected, we find that spin alignment largely degrades LISA’s ability to measure parameters. As a rough rule of thumb, we find that extrinsic parameters (sky position angles and luminosity distance) are measured a factor of less accurately for alignment and a factor of less accurately for alignment. In the second case, alignment eliminates most of the advantage gained by adding precession in Paper I. We find that the impact upon measured masses and spins depends strongly on mass ratio, with degradation by a factor at alignment, and a factor at alignment. We find a handful of cases in which partially aligned binaries actually do better than the randomly oriented systems. As we describe in Sec. III, this is due to alignment increasing these systems’ average SNR.

To combat this degradation, we introduce another degeneracy-breaking effect. Much early work in LISA parameter estimation made use of the restricted waveform model, in which only the quadrupole harmonic of the orbital phase was included and only the leading “Newtonian” amplitude term was used with this harmonic (although the phase was constructed to high post-Newtonian order). This was done because the quadrupole harmonic dominates signal power, while the phase is the primary source of information about the signal. However, it has since been shown by several groups that including higher harmonics (and their post-Newtonian amplitudes, making the so-called full waveform model) also breaks degeneracies and reduces parameter errors Arun et al. (2007a, b); Trias and Sintes (2008); Porter and Cornish (2008). The magnitude of the effect is comparable to the improvement seen by including spin precession. Recently, Klein et al. have presented an analysis combining both spin precession and higher harmonics Klein et al. (2009). A similar analysis, based on an earlier version of our own code, was conducted by the LISA Science Team to investigate the science reach of the LISA mission; the results of this study are summarized in Ref. K. G. Arun, S. Babak, E. Berti, N. Cornish, C. Cutler, J. Gair, S. A. Hughes, B. R. Iyer, R. N. Lang, I. Mandel et al. (2009).

In Sec. IV, we replace the leading quadrupole waveform with the full waveform. For the case of random spins, our answers can be compared (with some caveats) to the results of Klein et al. (2009). We also compute the errors for wet, partially aligned binaries with the full waveform. When higher harmonics are included, parameter errors for partially aligned binaries are often no worse, or even better, than for the case of random spins and no higher harmonics. In these particular cases, higher harmonics can more than make up for the degraded impact of spin precession. We find this degree of improvement for the minor axis of the sky position error ellipse and for the luminosity distance in a majority of (mass) cases. The improvement is not quite so good for the major axis: Although higher harmonics can reduce errors by factors of or more, this often does not completely make up for the loss of precession, especially at alignment. Errors in the measured spin behave similarly to the major axis — their measurement is improved, but not enough to fully compensate for the impact of aligned spins. By contrast, we find that higher harmonics always improve mass measurements beyond what can be done with random spins alone. In fact, partial alignment in many cases improves mass measurements, thanks to increased SNR in these cases.

We also briefly take a more detailed look at the relative improvement from spin precession, higher harmonics, and their combination. We confirm previous results that, for extrinsic parameters, the impact of the combined effects is not substantially greater than the impact of each effect alone. For mass errors, the higher harmonics dominate, with precession being almost irrelevant for the full waveform. For spin errors, however, the two effects do seem to be independent, with the combined improvement approximately equal to (or greater than) a simple multiplication of the individual improvements.

We conclude in Sec. V by summarizing our results and discussing additional studies that must be done before the question of LISA parameter estimation is fully understood. Throughout this paper, we use geometrized units in which . A useful conversion factor is that .

Since this paper was originally written, budget constraints have caused a rescoping of the LISA mission, and the mission that eventually flies may differ from the “classic” configuration considered here. We continue to focus our analysis on measurements using LISA Classic for two reasons. First, the design of the rescoped mission is in flux. Until a design is fixed and its associated sensitivity known, we cannot study how well it will make measurements. Second, our goal is to make comparisons with previous studies that were based on the classic design. As such, it is most appropriate for us to use this design as well. We note that our conclusions should be robust in the sense that the general trends we find regarding the impact of spins and higher harmonics will be relevant to any LISA-like design (at least for designs that have five or six links, so both waveform polarizations can be simultaneously measured).

Ii Parameter estimation code

The code used in this paper is a version of the montana-mit code used by the LISA Parameter Estimation Taskforce K. G. Arun, S. Babak, E. Berti, N. Cornish, C. Cutler, J. Gair, S. A. Hughes, B. R. Iyer, R. N. Lang, I. Mandel et al. (2009), updated with some new features and bug fixes.3 In this section, we describe the relevant features of the code, especially how it differs from the code of Paper I Lang and Hughes (). We refer the reader to Paper I for more detailed discussion of the waveform and parameter estimation theory.

ii.1 Massive black hole binary waveform

The waveform from a massive black hole binary coalescence is traditionally divided into three distinct phases: (1) the inspiral of the two holes, which can be described by the post-Newtonian expansion of general relativity; (2) the merger of the two holes into a common event horizon, describable only by full numerical relativistic simulations; and (3) the ringdown of the final hole into the stationary Kerr solution, which can be described by black hole perturbation theory. In this work, we consider only the inspiral, which for LISA sources can last for months to years, accumulating large amounts of SNR and parameter information. Because of this fact, as well as the ease of using the post-Newtonian approximation, inspiral-only waveforms have traditionally been used in most, though not all, LISA parameter estimation studies. Ringdown information was first studied on its own by Berti, Cardoso, and Will Berti et al. (2006). More recently, McWilliams et al. added both the merger and ringdown to the inspiral, albeit for nonspinning binaries with an a priori known mass ratio McWilliams et al. (2010, ). They showed that the merger can add a significant amount of parameter information, about a factor of 3 improvement in measurement accuracy for all parameters but mass. Work in progress will consider the impact of an unknown mass ratio, as well as spins.

The inspiral waveform can be described by 17 parameters: the masses of the black holes, and ; their dimensionless spins, and ; the spin angles at some particular reference time , , , , and ; the orientation angles of the orbital angular momentum at , and ; the eccentricity ; the periastron angle ; the position of the binary on the sky, and ; the luminosity distance ; a reference time (possibly different from ); and a reference phase . In this work, we assume quasicircular orbits, eliminating and and reducing the parameter set to 15. This assumption is also quite common, since radiation reaction has long been expected to circularize binaries Peters (1964). It should be noted, however, that recent studies indicate that gas Armitage and Natarajan (2005); Cuadra et al. (2009) and/or stellar interactions Sesana (2010) may cause binaries to retain a small, but significant, residual eccentricity when they enter the LISA band. Recent work by Key and Cornish Key and Cornish (2011) investigates the impact of this residual eccentricity using a nontrivial extension of our code.

In Paper I, we used the post-Newtonian parameters and as the reference time and phase . These parameters are, respectively, the time and phase when the post-Newtonian frequency formally diverges. However, Paper I made a slight error in determining the post-Newtonian frequency and phase. To understand this error and how to correct it, begin with the time derivative of orbital angular frequency (shown here to second post-Newtonian order)

(2)

where is the reduced mass ratio, is a spin-orbit coupling term, and is a spin-spin coupling term. Exact expressions for and are given in Paper I. Equation (2) must be integrated once to obtain and twice for the orbital phase . When the spins do not precess, this integration can be done analytically to some specified post-Newtonian order. In Paper I, the analytic results were used, but with the time-dependent expressions for and plugged in at the end of the process. This is technically not correct: The time-dependent spins should be inserted into (2), and then that expression should be numerically integrated to produce and . This is not difficult, only requiring two additional differential equations in the Runge-Kutta solver of Paper I. However, it means that and are no longer acceptable references, since the numerical integrator cannot reach infinite frequency. We describe our current approach momentarily.

Another change from the code used in Paper I is in the choice of cutoff frequency for the inspiral. In Paper I, the inspiral was stopped at the frequency of the Schwarzschild innermost stable circular orbit (ISCO), . This assumption is poor for two reasons. First, while is the ISCO for a test particle orbiting a single Schwarzschild hole of mass , the dynamics of the two-hole system are much more complex, and the transition to plunge and merger is not so well-defined. Second, we are considering Kerr black holes, for which even in the point-particle limit the innermost stable orbit can vary from to depending on the spin of the hole, with a concomitantly wide variation in the ISCO frequency. A better solution is to stop the inspiral at the minimum energy circular orbit (MECO), the orbit which minimizes the expression for post-Newtonian energy Buonanno et al. (2003):

(3)

The MECO is known to be a better approximation to the inspiral-plunge transition than the ISCO, and it properly takes spins into account.

Using the MECO gives us a better reference point for our time and phase than the coalescence time and phase and described above. We choose and and then integrate the spin, frequency, and phase evolution equations backwards from the MECO to . The backwards integration provides stability in the Fisher-matrix calculation: We align the waveforms when they are largest, thus making it easier to introduce slight perturbations.

As seen in Eq. (2), we calculate the phase out to second post-Newtonian (2PN) order. (By numerically integrating (2) to obtain and , we specifically are choosing the “TaylorT4” PN approximant Boyle et al. (2007).) We integrate the spin precession equations out to 1.5PN order, which includes 1PN spin-orbit and 1.5PN spin-spin terms. It is worth noting that all of the relevant quantities are known to higher post-Newtonian order. Work in preparation shows that including terms beyond the order we include here only causes a slight quantitative change in the accuracy with which parameters are measured O’Sullivan and Hughes (). In Sec. III, we use the restricted post-Newtonian approximation, in which we only consider the quadrupole term () with its lowest order, Newtonian amplitude. In Sec. IV, we use the full post-Newtonian waveform, which includes all harmonics to 2PN order in amplitude.

ii.2 LISA response and noise

The LISA response used in this paper differs from the response used in Paper I. For signals which do not reach above Hz, we use the same low-frequency approximation used in that paper. In this approximation, we ignore the transfer functions which arise due to the finite arm lengths of the detector. This approximation is very inaccurate above Hz. With the addition of higher harmonics, many signals now reach into this range where the transfer functions become important.

The full LISA detector response is somewhat complicated to model. Three existing codes provide the full response: the LISA Simulator Rubbo et al. (2004), Synthetic LISA Vallisneri (2005), and LISACode Petiteau et al. (2008). Interfacing with one of these codes would significantly slow our analysis, making it difficult to perform large Monte Carlo studies over our parameter space. We seek a simpler response function which includes the finite arm length transfer functions but ignores some of the more complicated issues.

The three LISA spacecraft follow eccentric orbits around the Sun at 1 AU. The individual orbits combine in such a way that the LISA constellation maintains, at first order in orbital eccentricity, an equilateral triangle formation. By going beyond this leading order, one finds that the arm lengths vary by a small amount on monthlong time scales. The variation in LISA arm lengths is the reason for the development of time delay interferometry (TDI) techniques Armstrong et al. (1999) to eliminate laser phase noise, which cancels exactly in equal-arm interferometers like LIGO. In our detector model, we approximate the constellation as having arm lengths that are equal at all times. Our model detector is a “rigid” equilateral triangle.

The other complexity in the full LISA response is that the spacecraft move during the measurement, causing “point-ahead” effects which must be taken into account. We assume instead an “adiabatic” detector, in which for each time that we require the detector response, the detector is considered to be motionless for that time. The spacecraft then adiabatically move to their next position for the next sample point. This rigid, adiabatic approximation is known to be equivalent to the full response up to very high frequency ( mHz) and thus is appropriate for our Fisher-matrix analysis Rubbo et al. (2004).

For the rigid, adiabatic approximation, the code produces Michelson variables , , and as defined in Vallisneri (2005), Eqs. (10) and (11). Note that these are technically not TDI variables. Since we do not have to subtract phase noise, there is no need to include another pass through the interferometer (cf. the “real” TDI variables in Eq. (13) of Vallisneri (2005)). They do contain the same information, though, so we may refer to them as (pseudo, equal-arm) TDI Michelson variables in this paper. From them, we can construct noise-orthogonal TDI variables , , and , defined as

(4)
(5)
(6)

, , and are used to calculate the SNR and the Fisher matrix. Note that, as defined in Eqs. (10) and (11) of Vallisneri (2005), these are fractional-frequency variables. We can convert them to equivalent strain by integrating the signal in the frequency domain and then multiplying by . For the low-frequency case, we use the Michelson signal and the noise-orthogonal signal . These combinations are denoted and in Paper I (which in turn follows the convention of Cutler Cutler (1998)). The low-frequency approximation is constructed so that these signals are already expressed as equivalent strain.

The LISA noise power spectral density comprises two parts, instrumental noise and confusion noise due to unresolved white dwarf binaries in the Galaxy. Instrumental noise consists of both position noise, due to photon shot noise and other effects along the optical path, and acceleration noise, due to proof mass motion. The total instrument noise in the and (strain) channels is given by

(7)

and the (strain) noise is given by

(8)

Here is the LISA arm length, , is the position noise budget, and is the acceleration noise budget. For the low-frequency approximation, we can calculate similar expressions for the two noise-orthogonal channels and then take , although this attention to detail makes little difference for the frequencies of interest. Notice that the position noise and acceleration noise are both assumed to be white, with no frequency dependence. However, because it is expected that LISA’s acceleration noise performance will degrade somewhat from this white form below Hz, we have also added a “pink” acceleration noise term, with a slope of .

Confusion noise is constructed from the residuals of a fit to the Galaxy Crowder and Cornish (2007) in the Mock LISA Data Challenge K. A. Arnaud, S. Babak, J. G. Baker, M. J. Benacquista, N. J. Cornish, C. Cutler, S. L. Larson, B. S. Sathyaprakash, M. Vallisneri, A. Vecchio et al. (2006). An approximate analytic expression for the confusion noise can be found in Key and Cornish (2011), Eq. (10). It is added to instrument noise for the and channels (or the orthogonal low-frequency channels) to obtain the total noise. It is not added to the channel because it occurs only at low frequency, where that channel adds nothing to the analysis.

Finally, although it is not expressed explicitly in (7) and (8), we enforce a low-frequency cutoff of Hz and do not include any contribution from the signal below that frequency in our analysis. (This frequency is the lowest frequency at which LISA is planned to have good sensitivity to gravitational waves; though it will have sensitivity to sources at lower frequencies, the noise characteristics below Hz cannot be guaranteed.)

ii.3 Construction of the Fisher matrix

The Fisher matrix is defined as

(9)

where is the gravitational wave signal, are the 15 parameters which describe it, and

(10)

is a noise-weighted inner product. The inverse of the Fisher matrix is the covariance matrix, which contains squared parameter errors along the diagonal and correlations elsewhere. To calculate the Fisher matrix, we need the waveforms in the frequency domain. In Paper I, we actually did all calculations in the frequency domain by using the stationary phase approximation. This approximation relies on a separation of time scales and is known to be quite good for nonspinning binaries, where the inspiral time scale is much larger than the orbital time scale . However, when precession is included in the waveform, an additional time scale comes into play, with . We have seen that with precession, the stationary phase approximation tends to smooth out sharp features in the Fourier transform, potentially reducing the information content. The problem becomes worse as the impact of precession increases (i.e., with higher spin values, or for highly nonaligned spins and orbit). To avoid introducing any errors due to this approximation, we here calculate our waveforms in the time domain and then perform a fast Fourier transform (FFT) to bring them into the frequency domain.

This approach has two major limitations. First, it is much slower than the stationary phase approach, since we need to calculate many time samples to observe Nyquist sampling requirements and we then need to compute the FFT. Second, the FFT assumes a periodic signal. Because we have a finite signal which looks much different at the end than at the beginning, we must introduce some kind of window in order to taper the signal to zero at the beginning and end. We use a Hann window (actually half a Hann window at each end of the signal). This window substantially reduces “ringing,” or spectral leakage problems. However, it also cuts out part of the signal. This is particularly unfortunate for the strongly chirping inspiral, since much of the signal power is contained in the last few cycles. By windowing the signal, we lose some of this power. This may cause our SNR and errors to be smaller and larger, respectively, than they would be for a “real,” physical signal. The best solution to this problem would be to include the merger and ringdown portions of the signal, allowing it to fade to zero in a physical, not artificial, way. For now, we must simply accept the windowing as part of the definition of the (unphysical) inspiral-only waveform.

Iii Parameter estimation in partially aligned binaries: Only the quadrupole harmonic

Here we describe the parameter estimation capabilities of LISA without including the influence of higher harmonics. In order to consider a wide range of LISA sources, we choose only three parameters explicitly, the two masses of the system and the luminosity distance. For the masses, we consider a variety of systems ranging in total mass from to , with a mass ratio from . On the other hand, we consider only sources at , corresponding to a luminosity distance of Gpc using our choice of cosmological parameters. Errors at other redshifts can be constructed using the errors at . We note that the results at masses and and redshift can be simply related to the results at masses and and redshift . This is because all time scales in the system are derived from the masses. Since time scales are lengthened (frequencies shortened) by the cosmological redshift, a binary at higher redshift behaves like a binary at lower redshift but with a higher mass. The quantity is generally called the redshifted mass, where is the mass measured locally at the rest frame of the binary. (When we quote masses in this paper, we always mean the rest-frame mass, remembering that when put into the waveform formulas of Paper I, they must be multiplied by .) The amplitude of the waves at redshift is decreased by a factor over the corresponding binary (i.e., the binary with the same redshifted mass) at redshift . This increases the errors by over that corresponding signal.

The other 12 parameters of the system are generated essentially at random, with 1000 different Monte Carlo realizations. For example, is chosen from within an assumed three-year mission time, meaning that some early binaries will have abnormally short signals for a given mass. Spin magnitudes are chosen uniformly between 0 and , and is chosen uniformly between 0 and . Cosines of angles are chosen uniformly between and , while longitudinal angles are chosen uniformly between 0 and . In the case of random spins (as in Paper I), the procedure is then complete. In the case of partially aligned spins, the main focus of this paper, we use the randomly generated parameters to integrate the spin precession equations backwards from the MECO to . We assume that any alignment at is solely due to gas. If either of the resulting spin-orbit angles is greater than the model’s restriction ( or ), we randomly select new spin orientation angles (at MECO) and try again. This procedure guarantees that all sources will have the desired amount of alignment at the start of the signal. However, our sample will include some sources () which move out of alignment by MECO. Since these sources generally precess more strongly than the others, they tend to improve the overall distribution of parameter errors, especially for spin magnitude.

Figure 1: Distribution of , the major axis of the sky position error ellipse, for binaries with randomly aligned spins (dotted line), spins restricted to within of the orbital angular momentum (dashed line), and spins restricted to within of the orbital angular momentum (solid line). Here , , and .

Figure 1 shows a histogram of the Monte Carlo results for a binary with and . We show the major axis of the sky position error ellipse, , comparing the cases of randomly aligned spins to spins restricted to be aligned within (for hot gas) and (for cold gas) of the orbital angular momentum. We see that partial alignment of the spins and orbital angular momentum degrades LISA’s localization capability. For the partially aligned cases, the shapes of the histograms resemble the strongly peaked “no precession” results of Paper I more than the roughly flat random-spin histogram. The medians of the distributions also increase: While randomly oriented binaries have a median major axis of 34.8 arcminutes, systems aligned within have a median of 62.3 arcminutes. For alignment, this degrades further to 90.5 arcminutes. This is a factor of 2.6 degradation from the case of random alignment, just short of the factor improvement seen in Paper I when precession is introduced into the waveform model. In essence, by restricting the spin angles to within of the orbital angular momentum, we have eliminated almost all of the advantage gained from including precession effects in the waveform.

Figure 2: Same as Fig. 1, but for the minor axis .

Figure 2 shows results for the minor axis of the sky position error ellipse,4 . The results are similar: The median value of increases from 24.6 arcminutes for random spins to 40.6 arcminutes for alignment and to 58.6 arcminutes for alignment. Together with the results for the major axis, these numbers imply that the total sky position area increases by a factor when binaries have closely aligned angular momentum vectors, strongly impacting the ability of LISA to find electromagnetic counterparts to the GW signal.

No gas Hot gas Cold gas
27.0 16.4 40.7 25.7 53.8 35.1
17.5 11.7 30.1 17.9 53.7 34.8
33.3 19.0 45.9 27.1 63.1 42.4
23.3 18.3 35.9 21.6 61.6 38.4
34.8 24.6 62.3 40.6 90.5 58.6
56.9 37.5 87.7 57.2 105 68.3
39.0 33.6 57.0 36.8 105 68.1
45.5 32.4 83.3 49.0 131 77.9
71.9 43.6 126 75.6 168 106
47.3 40.2 70.7 46.6 132 83.7
67.3 45.3 131 75.7 234 143
160 84.8 281 136 581 323
Table 1: Median sky position major axis and minor axis , in arcminutes, for binaries of various masses at , in the “no gas” (random-spin), “hot gas” ( alignment), and “cold gas” ( alignment) cases.

Table 1 shows the major and minor sky position axes for a range of masses, in the random-spin, hot gas, and cold gas cases. We see that degradation of a factor between the “no gas” and “cold gas” ( alignment) cases occurs rather consistently for different masses and mass ratios.

Figure 3: Same as Fig. 1, but for the fractional error in luminosity distance, .
No gas Hot gas Cold gas
0.0130
0.0101
0.0121
0.0113
0.0101 0.0136
0.0137 0.0175
0.0147
0.0130 0.0191
0.0137 0.0207 0.0279
0.0135 0.0242
0.0129 0.0243 0.0429
0.0441 0.0613 0.0974
Table 2: Same as Table 1, but for the fractional error in luminosity distance, .

The other extrinsic parameter of interest is the luminosity distance . Figure 3 shows the fractional errors in for different degrees of spin alignment. Again, we see that restricting the spin angles dramatically affects measurement: The median of for random spin orientation doubles to when the spins are aligned within and nearly triples to when the spins are aligned within . However, this particular degradation is almost certainly immaterial, at least at low redshift, since the error remains much smaller than the error produced by weak gravitational lensing at . For sources at higher redshift, this degradation may be more important. Table 2 shows luminosity distance errors for different masses. Like the sky position, the degradation is about a factor of for alignment and for alignment. Note that for the larger masses we consider, the degradation pushes the GW distance error to a value comparable to or even larger than the weak lensing error.

Figure 4: Same as Fig. 1, but for the fractional error in mass, .
Figure 5: Same as Fig. 1, but for the fractional error in mass, .

We now turn to the intrinsic parameters of the system, its masses and spins. Figs. 4 and 5 show the errors in the two black hole masses for the three cases we consider. Medians of are for the random-spin case, for a system with hot gas, and for a system with cold gas. For , these numbers are , , and , respectively. The impact of partially aligned spins does not seem to be as strong on the masses as on the sky position; the mass errors change by less than a factor of 2. In Paper I, we looked at precession improvements not in individual masses but in chirp mass and reduced mass, where we saw factors of and improvement, respectively. Clearly, restricting the spin directions does not remove this entire improvement; even a limited amount of precession appears to significantly aid mass determination. We can check this assertion using our new code by running a case with only alignment between the spins and the orbit. We find that the results improve on the results by a factor of 3. By contrast, the sky position and distance errors differ by only .

Table 3 shows the results for different masses. We see a much stronger dependence on mass ratio here than for the extrinsic parameters. For example, while the cold gas degradation is less than a factor of 2 for the (roughly) 3:1 mass ratio case considered in Figs. 4 and 5, it reaches a factor of for the equal-mass case . This is unusual, since precession is known, at least for extrinsic parameters, to have a stronger impact for unequal masses due to increased complexity in the signal. It is possible that the lack of this complexity essentially “gives away” that the masses are equal, making them easier to determine from the extremely well-measured chirp mass.

Interestingly, there are some examples of 10:1 mass ratio systems that break our general trend; in these cases, we find that partially aligned spins actually do better than random spins. This seemingly counterintuitive result can be explained by our choice of the minimum energy circular orbit (MECO) as the waveform cutoff. Binaries with aligned spins have a smaller MECO (with a corresponding high inspiral cutoff frequency) and thus accumulate more SNR than those with spins out of alignment (as many in the random-spin sample will be). Figure 6 shows the SNR for all three cases in a 10:1 binary. We see that the SNR is substantially larger for the partially aligned cases (medians of 2588 and 2592, for and , respectively) than the randomly aligned case (median of 1445). Even though these binaries precess less, the increase in SNR makes up for it in parameter estimation. It is worth noting that this effect could also be of use in detecting and measuring particularly high-mass binaries. For randomly chosen spins, such a binary might be mostly or completely out of the LISA band. However, if the spins are aligned by interactions with gas, the MECO frequency will be pushed into band.

No gas Hot gas Cold gas
0.0284 0.0284
0.0129 0.0128 0.0327 0.0328
0.0197 0.0197 0.0475 0.0475
0.0121 0.0165 0.0132 0.0233 0.0187
0.0239 0.0237 0.0536 0.0533 0.109 0.109
0.0176 0.0118 0.0167 0.0109 0.0205 0.0133
0.0431 0.0336 0.0581 0.0447 0.0924 0.0713
0.381 0.388 0.424 0.423 0.967 0.971
Table 3: Same as Table 1, but for the mass errors and .
Figure 6: Signal-to-noise ratio for binaries with randomly aligned spins (dotted line), spins restricted to within of the orbital angular momentum (dashed line), and spins restricted to within of the orbital angular momentum (solid line). Here , , and .
Figure 7: Same as Fig. 1, but for the error in spin magnitude, .
Figure 8: Same as Fig. 1, but for the error in spin magnitude, .

Finally, we consider how restriction of spin angles affects measurement of spin magnitudes; Figs. 7 and 8 show these results. For , the median varies from for no gas to for hot gas to for cold gas. For , the situation is similar; the medians are (no gas), (hot gas), and (cold gas). Although the spin errors are degraded by partial alignment, the amount of degradation is somewhat curbed by the contribution of binaries which precess away from alignment before MECO. In addition, the errors at alignment are roughly an order of magnitude better than at alignment. Similar to the situation with mass measurements, even a small amount of precession can have a huge impact on measuring spin.

No gas Hot gas Cold gas
0.0217 0.0210 0.0311 0.0310 0.0391 0.0388
0.0155 0.0125 0.0467
0.0321 0.0315 0.0390 0.0382 0.0430 0.0417
0.0355 0.0225 0.0358
0.0148 0.0172 0.0131 0.0511
0.0534 0.0505 0.0655 0.0645 0.0774 0.0753
0.0601 0.0499 0.0659
0.0186 0.0111 0.0264 0.0252 0.0787
0.134 0.125 0.197 0.198 0.234 0.232
0.124 0.180 0.231
0.0266 0.0446 0.0298 0.0852 0.0652 0.191
1.69 1.54 1.28 1.31 1.86 1.87
Table 4: Same as Table 1, but for the spin magnitude errors and .

Table 4 gives spin errors for a broader range of masses. Like the mass errors, there is a strong dependence on mass ratio. In this case, however, the worst degradation occurs not for equal masses, but for 3:1 mass ratios, with factors of up to 4 increases in and factors of up to 6 increases in . The 10:1 cases show some degradation at alignment, but many cases are slightly improved at alignment. As in the case of mass measurements (cf. Table 3), this can be attributed to the increased SNR for aligned binaries.

Iv Parameter estimation in partially aligned binaries: Including higher harmonics

We now move beyond the leading quadrupole waveform to the full waveforms. Post-Newtonian corrections to the waveform amplitude are included up to 2PN order, including both additional quadrupole terms () and subleading (“higher”) harmonics beyond the quadrupole. For example, the barycentric waveform can be written up to 1PN order in the amplitude as

(11)

where , , , and . Here we see both extra harmonics and a 1PN correction to the quadrupole harmonic. Note that the odd harmonics only contribute if ; just like spin precession, higher harmonic corrections are more complex for unequal masses. Further terms (including the polarization) can be found in Blanchet (2006) (albeit with some differences in sign convention).

Figure 9: Distribution of , the major axis of the sky position error ellipse, for binaries with randomly aligned spins (dotted line), spins restricted to within of the orbital angular momentum (dashed line), and spins restricted to within of the orbital angular momentum (solid line). Here , , and . Higher harmonics are now included in the waveform model.

It has been shown that higher harmonic corrections can improve parameter estimation much like spin precession does Arun et al. (2007a, b); Trias and Sintes (2008); Porter and Cornish (2008). In the case of higher harmonics, degeneracies are broken due to the different sky position dependence of each harmonic. However, these studies did not include precession and so could not comment on how the two effects would combine. More recently, both effects were included in a parameter estimation study by Klein et al. (Ref. Klein et al. (2009)). Their results demonstrate that including both precession and higher harmonics improves measurement accuracy, but, at least for extrinsic variables (sky position and distance), the combined improvement is not as drastic as the improvement from each effect on its own. This indicates that at least in some cases, precession and higher harmonics encode similar information. We might therefore expect that in partially aligned binaries for which spin precession exists but is suppressed, the inclusion of higher harmonics may make up for this suppression and restore much, if not all, of the lost parameter estimation capability. In this section, we test that expectation.

Figure 10: Same as Fig. 9, but for the minor axis .

Figure 9 shows the major axis of the sky position error ellipse for the same binaries as Fig. 1, except with higher harmonics now added to the waveform model. The median value of for random-spin, gas-free systems is 21.7 arcminutes. Comparing to the leading quadrupole waveform value of 34.8 arcminutes, we see that higher harmonics do indeed add some additional information not contained in precession. The difference between the two values is a factor , consistent with the results of Klein et al. (2009). For partially aligned systems, the shape of the plot shows that the higher harmonics have had an important effect; both partially aligned histograms look much more like the roughly flat random-spin case than the strongly peaked histograms shown in Fig. 1. The median is 28.2 arcminutes for hot gas and 32.7 arcminutes for cold gas. Both results are great improvements on the leading quadrupole waveform values (factors and , respectively), indicating that the inclusion of higher harmonics has indeed “made up” for the loss of some spin precession. Both results are actually better than the leading quadrupole, gas-free result of 34.8 arcminutes. In this case, a full waveform with a small amount of precession does better than a leading quadrupole waveform with potentially significant precession.

Figure 10 shows the results for the minor axis , with medians 16.1 arcminutes for random spins and 13.7 arcminutes for both and alignment. These are all better than the leading quadrupole, random-spin result of 24.6 arcminutes. More interestingly, we see that when higher harmonics are included, the “cold gas” and “hot gas” errors are smaller than the “no gas” errors. As discussed in the previous section, this is due to the improvement in SNR in the aligned case. Higher harmonics break degeneracies well enough that it is more beneficial to have partially aligned binaries with more SNR and less precession than randomly aligned binaries with less SNR and more precession.

No gas Hot gas Cold gas
21.8 12.3 30.3 14.2 35.6 15.9
14.3 9.67 19.9 10.1 31.0 13.2
26.2 16.2 30.4 18.3 39.1 25.6
14.0 11.9 13.7 9.25 17.6 9.69
21.7 16.1 28.2 13.7 32.7 13.7
48.1 32.0 60.4 37.1 53.0 32.4
29.1 25.8 25.7 20.0 35.8 24.4
36.0 26.8 48.2 27.4 58.2 30.2
63.5 39.8 103 58.1 109 66.5
36.7 32.4 38.9 27.4 54.7 31.3
45.0 32.8 65.1 33.0 82.7 36.5
114 65.6 144 80.1 228 115
Table 5: Median sky position major axis and minor axis , in arcminutes, for binaries of various masses at , in the “no gas” (random-spin), “hot gas” ( alignment), and “cold gas” ( alignment) cases when higher harmonics are included in the waveform model. Bold entries are those that do better than the no gas case when higher harmonics are ignored (i.e. Table 1). Italic entries do worse than that case, but only by 10% or less.

Table 5 shows results for a variety of masses. All show improvement from Table 1; however, the improvement is not always as strong as in the case discussed above (). Bold text indicates cases in which the errors match or improve upon the results from the leading quadrupole waveform for random spins. Italics indicate errors which are worse, but by no more than 10%. Because of statistical issues, these cases could very well be “bold” in a different Monte Carlo run, so we will consider them as such for purposes of summarizing the results. While hot gas ( alignment) systems achieve this particular benchmark for a majority of mass cases, cold gas ( alignment) systems do not. Cold gas systems do, however, meet it for a majority of mass cases if only the minor axis is considered; generally fares better than . Both axes exhibit cases where errors are smaller with alignment than without, the minimum sometimes occurring in a “sweet spot” of alignment and sometimes at alignment. In general, errors are better for larger mass ratios. This is to be expected because both higher harmonics and precession have a more complicated structure for larger mass ratios. Finally, the improvements are worst for the smallest masses, where the higher harmonics (except the terms, which are technically “lower” harmonics) begin to go out of band.

Figure 11: Same as Fig. 9, but for the fractional error in luminosity distance, .
No gas Hot gas Cold gas
0.0115 0.0139 0.0152
0.0121
0.0223 0.0270 0.0302
Table 6: Same as Table 5, but for the fractional error in luminosity distance, .

Figure 11 shows the results for luminosity distance errors . Here the medians are , , and for the no gas, hot gas, and cold gas cases, respectively. Again, these are all better than the leading quadrupole, no gas value of , as we might expect since distance determination is strongly tied to sky position determination. Table 6 gives the results for various masses. Most cases beat the leading quadrupole, no gas values of Table 2 or come within 10%. That is, except for the lowest mass systems, using the full waveform essentially always brings the distance errors for aligned spins back to the level of random spins. In this respect, distance errors are similar to (and even a bit better than) the minor axis of the sky position error ellipse. Finally, as with sky position, some mass cases feature errors which decrease as spins become aligned.

Figure 12: Same as Fig. 9, but for the fractional error in mass, .
Figure 13: Same as Fig. 9, but for the fractional error in mass, .
No gas Hot gas Cold gas