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Laser Interferometer Gravitational-Wave Observatory

BURST Simulations

Burst simulations

The search for gravitational wave bursts with LIGO (from, eg, supernovae, GRB engines, binary black hole mergers and ringdowns, and other very energetic phenomena) is made complicated by the fact that the burst waveforms are not known, or at best, only crudely modeled. The burst search algorithms employed by LIGO aim to detect bursts with a broad range of waveforms; the main tools are to detect non-stationary deviations from typical detector noise, and requiring coincidence between detectors in time, amplitude, frequency band, and waveform.

We are guided by supernova GW burst simulations from Zwerger & Muller (see below), which suggest that such bursts would have durations ranging from a fraction of a ms to 10's or 100's of ms, and with power in the LIGO frequency band.

These burst finding algorithms are developed, trained, tuned and evaluated using the model waveforms described here. Specifically, these waveforms are used:

  • as guides to develop and tune the algorithms;
  • to test the full analysis pipeline, from ingestion of raw data to the generation of candidate coincident burst events;
  • to evaluate the efficiency of the pipeline for detection of such waveforms, as a function of their parameters (h_peak or h_rss, central frequency, duration, etc);
  • to compare the response of the pipeline to signal injected in software with signals injected in hardware (through excitation of the end test mass suspensions);
  • to test our understanding of the dependence on source direction and polarization.

    We endeavor to have a variety of model waveforms with a range of characteristics:

  • longish-duration, small bandwidth (ringdowns)
  • longish-duration, large bandwidth (chirps)
  • short duration, large bandwidth (merger)
  • In-between (Zwerger-Muller or Dimmelmeier SN waveforms)

    Some of these models are ad hoc, with no astrophysical significance (eg, sine-Gaussians and Gaussians). The main virtue of these ad hoc models is that they are easy to describe, and have well-defined durations and frequency bands. In particular, since LIGO sensitivity is a strong function of frequency, waveforms like sine-Gaussians can be used as a "swept-sine" calibration of our burst detection capability, as a function of central frequency of narrow-band waveforms. Typically, ad hoc model waveforms are characterized by their frequency band, duration, and peak or root-sum-square amplitude.

  • Some of these models are based on simulations of astrophysical sources: Zwerger-Muller and Dimmelmeier-Font-Muller stellar core collapse simulations, and binary black hole merger waveforms from the Lazarus project. These waveforms are characterized by astrophysical parameters (supernova spin and adiabatic index, masses of the binary). They are distance-calibrated.

    All waveforms are sampled at the LIGO ADC frequency of 16384 Hz. Most are exactly 1 second (16384 samples) long. Most waveforms are considerably shorter than 1 second, so the waveform is padded with zeros at the end; for some waveforms (such as gaussians and sine-gaussians) the peak is at 0.5 sec, and the waveforms approach 0 at the beginning and end of the 1-second data stretch.

    All waveforms are in units of strain (ie, unitless).

    Remember that these waveforms may occur in sequence, eg, chirp -> merger -> ringdown.

    For each class of waveforms, we supply "LIGO lightweight" files, suitable for ingestion into LDAS, as well as a matlab .mat file. The .mat file is loaded into matlab with the "load" command. It contains a structure with a family of waveforms. For each waveform, there is a "name", an "hoft" (ie, h(t), 16384 samples), "hrms" is the root-sum-square amplitude, "ltim" is the length in time of the waveform as computed by the simulation, "dtim" is the duration over which the magnitude of the waveform is larger than 5% of its peak value, "dfreq" is the frequency band over which the magnitude of the waveform is larger than 5% of its peak value in frequency space.

    We welcome suggestions and pointers for other waveforms to consider. Please contact Alan Weinstein.

    Sine-Gaussians

  • h(t) = h0*exp(-(t-0.5).^2/tau^2).*sin(2*pi*f0*t);
    My apologies for the historical use of tau instead of sigma_t = tau/sqrt(2).
  • Peak amplitude h0 = 1.
  • f0 = [100, 153, 235, 361, 554, 850, 1304, 2000]
  • tau = 2/f0 (Q = sqrt(2)*pi*f0*tau = 8.88) or
    tau = 2/3/f0 (Q = sqrt(2)*pi*f0*tau = 2.96)
  • So, 8 central frequencies and 2 Q values = 16 waveforms
  • Wcat.mat (matlab .mat file, 5 MB) made using the Wcat_gen.m matlab script.

    Q = 8.88 (tau*f0 = 2):

    Upper left: Typical sine-Gaussian with tau = 2/f0.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Q = 3 (tau*f0 = 2/3):

    Upper left: Typical sine-Gaussian with tau = 2/3/f0.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Damped sinusoids

  • h(t) = h0*exp(-t/tau).*sin(2*pi*f0*t);
  • Peak amplitude h0 = 1.
  • f0 = [100, 153, 235, 361, 554, 850, 1304, 2000]
  • tau = 4/f0 (Q = 2*pi*f0*tau = 25) or
    tau = 4/3/f0 (Q = 2*pi*f0*tau = 8)
  • So, 8 central frequencies and 2 Q values = 16 waveforms
  • Wcat.mat (matlab .mat file, 5 MB) made using the Wcat_gen.m matlab script.

    Upper left: Typical damped sinusoid with tau = 4/f0.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Upper left: Typical damped sinusoid with tau = 4/3/f0.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Gaussians

  • h(t) = h0*exp(-(t-0.5).^2/tau^2); My apologies for the historical use of tau instead of sigma_t = tau/sqrt(2).
  • Peak amplitude h0 = 1.
  • tau = [0.00049 0.001, 0.0025, 0.005, 0.0075, 0.01, 0.02, 0.05 ];
  • Wcat.mat (matlab .mat file, 5 MB) made using the Wcat_gen.m matlab script.

    Upper left: Typical Gaussian.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Others to consider:

  • sinc functions
  • box functions
  • Low-order Hermite-Gaussians
  • "Sombrero" functions
  • "Delta" functions
  • N half-cycles of sine waves

    Zwerger-Muller supernova waveforms

    The Muller group at MPA-Garching do hydrodynamical simulations of stellar core collapse with a variety of initial conditions (spin, differential roatation profile, adiabatic index) in 2D and 3D, extracting example waveforms. They have published a catalog of 78 such waveforms with varying initial conditions.
  • These waveforms are distance calibrated. We have placed the supernova at 1 Mpc, optimally oriented. The GW burst amplitude varies with the inclination i of the spin of the progenitor (relative to the line of sight) as sin^2(i). The waveforms have only one polarization. These are 2D axisymmetric simulations.
  • Web pages from Zwerger / Muller group at MPA-Garching: Gravitational Radiation from Newtonian Rotational Core Collapse , in 2D (axisymmetric) and 3D Non-axisymmetric.
  • Zwerger, T., and Muller, E., "Dynamics and gravitational wave signature of axisymmetric rotational core collapse", Astron. Astrophys., 320, 209-227, (1997), http://www.mpa-garching.mpg.de/~ewald/GRAV/tazewm.ps.gz.
  • ZMcat.mat (matlab .mat file, 10 MB) made using the ZMcat_gen.m matlab script.

    Upper left: Typical ZM.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Dimmelmeier-Font-Muller supernova waveforms

    Dimmelmeier et al added the effects of general relativity to the simulation of the core collapse. They started with 26 initial conditions that were simulated by Zwerger-Muller, so that the effects of GR on the waveforms can be immediately apparent.
  • NEW Dimmelmeier-Font-Mueller relativistic waveforms; and here is the paper.
  • These waveforms are distance calibrated. We have placed the supernova at 1 Mpc, optimally oriented. The GW burst amplitude varies with the inclination i of the spin of the progenitor (relative to the line of sight) as sin^2(i).
  • DFMcat.mat (matlab .mat file, 3.4 MB), made using the DFMcat_gen.m matlab script.

    Upper left: Typical DFM.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Burrows-Ott supernova waveforms

    There is a catalog of 72 waveforms from this group: two-dimensional, axisymmetric, purely-hydrodynamic calculations of rotational stellar core collapse with a realistic, finite-temperature nuclear equation of state and realistic massive star progenitor models.
  • These waveforms are distance calibrated. We have placed the supernova at 1 Mpc, optimally oriented. The GW burst amplitude varies with the inclination i of the spin of the progenitor (relative to the line of sight) as sin^2(i). The waveforms have only one polarization. These are 2D axisymmetric simulations.
  • Web page, and article astro-ph/0307472.
  • OBcat.mat (matlab .mat file, 10 MB) made using the OBcat_gen.m matlab script.

    Binary Black hole merger waveforms

    These waveforms are from the Astrophysical GW Source Archive, John Baker at GSFC.
  • Calculations performed using the Lazarus project approach, applying numerical simulation of the vacuum Einstein's equations for the most significantly nonlinear part of the interaction together with close- limit perturabation theory for the late-time dynamics. See (J. Baker, M. Campanelli, C. Lousto, R. Takahashi--astro-ph/0202469) for detailed information about these calculations.
  • We supply 5 waveforms, corresponding to stellar masses of 1, 3, 10, 30, and 100 solar masses (times 2 for the two black holes). The source is 1 Mpc away, optimally oriented.
  • BMcat.mat (matlab .mat file, 0.6 MB) made using the BMcat_gen.m matlab script.

    Upper left: Typical BM.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).
    Right: frequency band dfreq versus duration dtime for this family of waveforms.

    Chirps

  • first post-newtonian approx, 1 Mpc distance, optimal orientation.
  • Eight waveforms, varying the mass of the star (both stars have the same mass): 1, 2, 3, 5, 10, 5, 20, 30 solar masses.
  • Only the last one second before merger. This means, for the lowest-mass binaries, that we are excluding some low-frequency oscillations which are in the LIGO band. For example, for the lowest-mass binary (m1=m2=1 Msol), we start at chirp frequency of 165 Hz.
  • CHcat.mat (matlab .mat file, 1 MB) made using the CHcat_gen.m matlab script.

    Upper left: Typical chirp.
    Lower left: Frequency spectrum for this waveform (blue), and its integral (red).

    ILWD files

  • CIT: http://www.ldas-cit.ligo.caltech.edu/ldas_outgoing/jobs/ilwd/burstsim/
  • MIT: http://www-ldas.mit.edu/ldas_outgoing/jobs/ilwd/burstsim/
  • LLO: http://www.ldas.ligo-la.caltech.edu/ldas_outgoing/jobs/ilwd/burstsim/
  • LHO: http://www.ldas.ligo-wa.caltech.edu/ldas_outgoing/jobs/ilwd/burstsim/

    Relevant links

  • LIGO Burst Sources web page.
  • Astrophysical GW Source Archive, GSFC
  • LIGO home page
  • LSC home page

    URL: http://www.ligo.caltech.edu/~ajw/bursts/burstsim.html
    This page last modified September 2003.
    Questions? Contact Alan Weinstein.