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the final merger of two black holes ( bhs ) having comparable masses will produce an intense burst of gravitational radiation , and is expected to be one of the strongest sources in the gravitational - wave sky . mergers of stellar black holes are key targets for ground - based detectors such as ligo , virgo , and geo600 , and knowledge of the merger waveforms is an important component of improving the detectability of such systems . the space - based lisa detector will observe mergers of massive black holes at high signal - to - noise ratios , allowing tests of general relativity in the strong - field , dynamical regime . today , numerical relativity ( nr ) studies are beginning to progress toward a full description of black - hole binary merger systems . for noneccentric inspirals , this space is spanned by seven parameters : the symmetric mass - ratio @xmath1 , and the six combined components of the black holes spin vectors . considerable study has been focused on the fiducial center point of this parameter space , the case of equal - mass nonspinning black - hole mergers . after the series of breakthroughs that ushered in an era of rapid progress in the field @xcite , several investigations assessing the accuracy of the available equal mass waveforms and applying them to data analysis were conducted @xcite .    in this paper , we undertake a descriptive study of the waveforms generated in the late inspiral and merger of black - hole binaries for the subspace of nonspinning black holes , parametrized only by @xmath2 . our study is based on a series of numerical simulations , discussed in sec . [ sec : simulations ] , covering at least the last @xmath3 orbits of nonspinning black - hole binary mergers with mass ratios extending to 6:1 ( @xmath4 ) . several of the simulations presented here have already been applied in a recent paper , focusing on the development of a faithful analytic waveform model @xcite . here we provide details of these and additional simulations , together with considerable analysis , focused on providing a qualitative and quantitative picture of how the waveforms from nonspinning black - hole mergers depend on @xmath2 . nonspinning black - hole binary merger waveforms were previously examined in ref . @xcite , but our analysis is novel and complementary to that work . our descriptive presentation puts emphasis on the relationships between waveforms from the different mass - ratio cases and different harmonic modes , with references to ref . @xcite where related observations have been made . our approach to describing the inspiral - merger - ringdown transition is particularly distinct , founded in a uniform approach that describes all stages of this process in similar terms , and ultimately suggesting a complementary physical picture . black - hole - binary merger waveforms have been noted for their `` simplicity . '' for the nonspinning systems the simple physics of the coalescence is exposed by a spherical harmonic decomposition of the waveforms . in sec . [ sec : description ] we walk readers through the basic features of the radiation , characterizing amplitude and phase evolution of the multipolar components , and discussing relationships among the simulations representing different mass ratios , and among the multipolar components of each simulation . as we analyze the waveforms we develop a conceptual interpretation of the basic waveform features . in this interpretation we consider the structure of an _ implicit rotating source _ , which could have generated the measured radiation through its rotational motion . this allows a uniform interpretation that applies throughout the coalescence process : inspiral , merger and ringdown .    in sec . [ sec : description2 ] , we examine the strong final burst of radiation beginning @xmath5 before the formation of a common horizon . we quantitatively describe the phasing in terms of an analytic model , based on a continuous , monotonically increasing frequency . we find , in particular , that the peak rate of change in frequency , appropriately scaled , is the same across all @xmath0 modes and mass ratios . we also identify relationships among the mode amplitudes and phases , which are connected to an approximately linear relationship between angular momentum and frequency : @xmath6 . we interpret these relationships in terms of the implicit source .    finally , in sec . [ sec : neweob ] , we demonstrate the utility of what we have learned in our waveform characterization by applying some of the quantitative features we have uncovered in a new variation on the analytic waveform model in @xcite , which was based on the effective - one - body ( eob ) resummation of the post - newtonian(pn ) approximation to inspiral dynamics @xcite . in particular , we provide a distinct late - time waveform model , alternative to the common `` spectroscopic '' model @xcite based on sums of quasinormal mode overtones . we begin with some examples of gravitational strain waveforms as they might be observed by gravitational - wave instruments . in observational work , and pn analysis , it is customary to describe the radiation in terms of gravitational - wave strain , @xmath7 . in representing the strain , it is convenient to combine the two real waveform polarization components , @xmath8 and @xmath9 , into one complex strain waveform , @xmath10 we decompose the strain waveforms measured on a sphere of radius @xmath11 , into spin - weighted spherical harmonic components , @xmath12 . the details of the decomposition , and how the waveform information is extracted from the numerical simulations , are given in appendix  [ appendix : radiation ] . the waveforms in this section are aligned in time and phase so that the maximum strain amplitude occurs at @xmath13 . the remaining figures of this paper will be aligned in a similar way , but with @xmath13 marking the time of peak ( 2,2 ) mode energy flux , @xmath14 ( unless stated otherwise ) . [ fig : waves ] shows waveforms from mergers of nonspinning black holes for various mass ratios , as observed at distance @xmath15 on the rotational / orbital axis of the system . the figure shows @xmath8 for each of the four mass ratios 1:1 , 2:1 , 4:1 , and 6:1 . for these observers the observed waveforms will be circularly polarized , so that @xmath16 is @xmath17 out of phase with @xmath18 . we use units in which @xmath19 and @xmath20 and express both time and spatial distances in terms of the total mass @xmath21 , where @xmath22 .        more typically , the observer will not be located on the system s orbital axis . the left panel of fig . [ fig : waveform_lm ] shows @xmath8 for the 4:1 case . the strain is measured at an azimuthal angle of @xmath23 and various inclinationsthe inclination angle is defined here as the angle between the line - of - sight with respect to the detector and the orbital axis of the binary . this is the same angle referred to as `` inclination '' in the pn / nr literature , and most equations are constructed using that definition . however , the astronomical literature has often defined inclination to be the angle between the line of sight and the orbital plane of the binary , resulting in a @xmath17 inconsistency . ] . the detailed shapes of the waveforms change as the system is reoriented so that the observer moves off the system s rotational axis . for larger inclinations ( closer to being viewed edge - on ) there are notable modulations at half the base gravitational - wave frequency . the right panel of fig . [ fig : waveform_lm ] shows @xmath8 for different mass ratios oriented at an inclination of @xmath17 and an azimuthal angle of @xmath23 . for this orientation , @xmath8 constitutes the full strain waveform . for larger mass ratios , the lower frequency modulation increases .         for gravitational - wave observations of sufficiently strong binary black - hole sources , the types of differences shown in fig . [ fig : waveform_lm ] could be exploited to estimate the inclination and mass - ratio of the source system .    for observational purposes , the combined waveform information encoding all possible source orientations can be conveniently represented in terms of spin - weighted spherical harmonic components [ see eqs.([eqn : psi4lm_def])-([eqn : hlm_def ] ) ] , providing a neat description of the leading waveform features , the multipolar decomposition is even more valuable as a tool for exposing the hallmark simplicity of the merger radiation . the readily apparent simplicity in the waves viewed from the system s orbital axis in fig .  [ fig : waves ] extends to each of the spherical harmonic components . viewed off axis , these components linearly combine to yield the more complex appearance of the waveforms in fig . [ fig : waveform_lm ] . this characterization of the gravitational radiation from a merging black - hole binary in terms of circular polarization was first recognized in the lazarus project studies @xcite . in this picture , the radiation can be represented by a slowly varying amplitude and a polarization angle ; see eq . ( [ eqn : strain_modes ] ) below . this description relies on how the radiation appears to distant observers located on the rotational axis of the system . other observers will typically see elliptically polarized waves , having a generally simple pattern that conforms to the rotational nature of the source . in the equatorial plane , the radiation reduces to the plus polarization , corresponding to the observer seeing no circulation in the source . looking along the negative rotation axis , the observer sees circular polarization with the opposite helicity . each of the spherical harmonic waveform components exhibits circular polarization with steadily varying phase and amplitude , providing a natural framework for developing a practical and intuitive understanding of binary black - hole merger radiation . our basic waveform description , in sec . [ sec : description ] , and the more detailed analysis that follows , is based on this spherical harmonic decomposition . as we describe the waveforms , we will also suggest a heuristic interpretation of what the radiation tells us about the motion and structure of the binary black - hole source . in the weak - field description of radiation from a rotating object , the multipolar waveform components of the gravitational radiation can be associated with dynamics of multipolar moments of the radiating source @xcite . it is useful , in conceptualizing the full coalescence radiation from inspiral through merger and ringdown , to think of the multipolar radiation description as providing information about the motion of a changing source object , described as a sum of several multipolar mass moments . this source object is what we will interpret as an effective rigid rotator radiation source , with a slowly changing structure . we refer to this as the _ implicit rotating source _ ( irs ) .    in the process of coalescence , the source begins as a separated black - hole binary system and ends as a single distorted black hole . for nonspinning binary mergers , numerical and pn results consistently indicate that the radiation is circularly polarized , in the sense first recognized in the lazarus project studies @xcite , not only in the inspiral , but uniformly through the merger and ringdown . in our conceptual source description , this pattern of circular polarization is consistent with radiation generated by rotational motion of each source multipolar moment , where the polarization phase is tied to the instantaneous orientation of the source . similarly , we think of the amplitude of the radiation multipole as related , through some generalization of the quadrupole formula , to the magnitude of the implicit source multipole .    we can write each multipolar component in a specific polar form natural for circularly polarized radiation : @xmath24 each amplitude @xmath25 is expected to be a slowly varying function of time and can be conceptually considered as a function only of the magnitude of the source multipolar moment and its rotational frequency . the additional sign for some @xmath26 cases allows a consistent interpretation for the component phases and the component amplitudes . the waveform phase is given here by @xmath27 , but in most of our analysis we refer to @xmath28 , which we call the _ rotational phase_. in our terms of our implicit source heuristic , the rotational phase for some particular @xmath29 mode can be thought of as the azimuthal orientation of the specified multipolar source component . in the inspiral , where the source can be considered as a separated binary , @xmath28 should coincide with the orbital phase , independent of @xmath29 . in the post - newtonian expansion @xcite , all the @xmath28 defined here agree to at least 2pn order , while the amplitudes @xmath25 remain real and non - negative . the equatorial - plane symmetry of these mergers ensures that @xmath30 and that @xmath31 , so that we need consider only @xmath32 modes for this analysis . the expansion ( [ eqn : strain_modes ] ) is not appropriate for the @xmath33 modes . this points to an important caveat to our implicit rotating source interpretation of the radiation , that it applies to the degree that the radiation is circularly polarized . while not strictly vanishing , the @xmath33 waveform components , and other deviations from circular polarization are generally extremely small , and largely unmeasurable at the resolutions of the simulations we study . for the most part , we will not address deviations from circular polarization , and the consequent limitations of our implicit - rotating - source interpretation in this paper , focusing for now on the dominant features of the radiation . our analysis is based on four simulations , representing mass ratios 1:1 , 2:1 , 4:1 and 6:1 . results from the 1:1 and 4:1 simulations have appeared in previous publications ( @xcite and @xcite , respectively ) . more recently , higher accuracy simulations have been presented by other groups for the 1:1 case @xcite . our older waveform is sufficiently accurate for our present purpose , to examine the general features of the nonspinning merger waveforms . our numerical simulations are carried out with the hahndol evolution code @xcite , which uses finite - differencing methods to solve a 3 + 1 formulation of einstein s equations on a cartesian grid . for initial data we solve the elliptic equation given by brandt and brgmann for conformally flat data in which the black holes are represented by punctures @xcite . this is performed numerically using the multigrid solver amrmg @xcite , which is second - order - accurate but tuned to give truncation errors typically much smaller than those produced by the evolution code . the momentum parameters are chosen according to the 2pn - accurate quasicircular approximation given by kidder @xcite , which has been found to result in low eccentricity . we evolve these data using the moving puncture method @xcite with a modified version of the baumgarte - shapiro - shibata - nakamura equations @xcite . specifically , as suggested in @xcite , we replaced the conformal factor variable @xmath34 with @xmath35 , which vanishes at the punctures . further , we added the constraint - damping terms suggested in @xcite , and the dissipation terms suggested in @xcite . for the gauge we use the specific 1+log lapse and gamma - freezing shift conditions recommended for moving punctures in @xcite . accurate simulations require adequate spatial resolution near the black holes ( length scales @xmath36 ) as well as in the wave zone where the gravitational waves are extracted ( length scales @xmath37 ) . to this end , the grid has multiple refinement levels , determined adaptively near the black holes , but fixed in regions farther away ( typically , @xmath38 ) where the waves are extracted ; all grid refinement is handled within the framework of the software package paramesh @xcite . the adaptive mesh refinement criterion near the black holes is designed to keep the scale of the square root of an invariantly defined curvature component , known as the coulomb scalar @xcite , roughly constant with respect to the grid spacing . interpolation in guard - cells between refinement regions is fifth - order - accurate , coupling with differencing stencils to yield at least fourth - order accuracy in the bulk . spatial derivatives are taken by sixth - order - accurate differencing stencils , with the exception of advection derivatives , which are handled by fifth - order - accurate mesh - adapted differencing for greater stability @xcite . time integration is performed with a fourth - order runge - kutta algorithm . the initial configurations of the simulations we analyze are given in table  [ table : rundata ] . in each case , the initial separation was chosen to be large enough to result in at least five orbits . the finest resolution , @xmath39 , ranged from @xmath40 to @xmath41 , as required to adequately resolve the black hole with the smaller mass in each case . the outer boundary was typically at @xmath42 , far enough away to prevent reflections from reaching the wave - extraction region during the simulation . it is instructive to compare these values with recent predictions for the final mass and spin from analytic and numerical methods . working with a subset of the data supplied here , and also invoking the test - mass limit , @xcite suggest the one - parameter fits : @xmath43 the numerical data from table  [ tab : kerrparams ] also fits well the wholly numerical formula ( 3.17a ) from @xcite , as well as the parametrized formula of @xcite . we note that when discussing the mass @xmath44 of the end - state kerr hole in secs . [ ssec : transfreq ] and later , we have consistently taken the radiation - derived value , @xmath45 , as results from @xcite indicate this has errors up to a factor of 3 less than qnm - based mass . since a far larger proportion of the initial angular momentum than mass is emitted during the inspiral and merger , there is greater uncertainty in the radiation - derived final spin than the final mass . for this reason , our preferred measure of the final dimensionless spin is the qnm - derived value @xmath46 .
we present a detailed descriptive analysis of the gravitational radiation from black - hole binary mergers of nonspinning black holes , based on numerical simulations of systems varying from equal - mass to a 6:1 mass ratio . 
 our primary goal is to present relatively complete information about the waveforms , including all the leading multipolar components , to interested researchers . in our analysis 
 , we pursue the simplest physical description of the dominant features in the radiation , providing an interpretation of the waveforms in terms of an _ implicit rotating source_. this interpretation applies uniformly to the full wave train , from inspiral through ringdown . 
 we emphasize strong relationships among the @xmath0 modes that persist through the full wave train . 
 exploring the structure of the waveforms in more detail , we conduct detailed analytic fitting of the late - time frequency evolution , identifying a key quantitative feature shared by the @xmath0 modes among all mass ratios . 
 we identify relationships , with a simple interpretation in terms of the implicit rotating source , among the evolution of frequency and amplitude , which hold for the late - time radiation . 
 these detailed relationships provide sufficient information about the late - time radiation to yield a predictive model for the late - time waveforms , an alternative to the common practice of modeling by a sum of quasinormal mode overtones . 
 we demonstrate an application of this in a new effective - one - body - based analytic waveform model .