Mismatch between gravitational waves¶

kuibit has a module to compute the mismatch between two gravitational waves (currently, only for the $$l=2$$, $$m=2$$ mode). See, Reference on kuibit.gw_mismatch, for a comprehensive reference.

Warning

You have to read carefully and understand everything in this page to use and interpret the results from gw_mismatch. In some cases, misusing the functions will lead to incorrect results without raising any error. The code has a lot of comments, you are encouraged to read them.

Overlap and mismatch¶

Given $$h_1$$ and $$h_2$$ gravitational-wave strains, one can compute the overlap between the two as

$\mathrm{overlap}(h_1, h_2) = \frac{(h_1, h_2)}{\sqrt{(h_1, h_1)(h_2, h_2)}}$

where

$(h_1, h_2) = 4 \Re \int_{f_min}^{f_max} \frac{\tilde{h}_1(f) \tilde{h}_2^*(f)}{S_n(f)} df$

is the inner product between $$h_1$$ and $$h_2$$, the tildas indicate Fourier transform, and $$S_n(f)$$ is the power spectral density (typically in units of one over Hertz). In case multiple detectors are considered, the inner product is the sum of the inner products of each detector (with their own spectral noise density).

From the overlap, one computes the mismatch between two waves. The mismatch (also known as unfaithfulness) is the overlap marginalized over some unknown quantities (more on this later). If $$h_1$$ is an observed signal and $$h_2$$ a template, the numerical value of mismatch is related to the signal-to-noise ratio needed to experimentally distinguish $$h_1$$ from $$h_2$$.

Typically, the mismatch is defined as

$\textrm{mismatch}(h_1, h_2) = \max_{\phi, t, \psi} \textrm{overlap}(h_1, h_2)$

where the max is taken over polarization angles, phase and time shifts. Again, if multiple detectors are considered, the overlap has to be computed with the network inner product (sum of all the inner products). If only the $$l = 2$$, $$m = 2$$ mode is considered, phase and polarization shifts are degenerate, so one can consider only one of the two. In gw_mismatch, we implement the mismatch computation for a network of detectors restricting to the $$l = 2$$, $$m = 2$$.

Note

The approach implemented here is probably not the best. One can compute the overlap directly form $$\Psi_4$$, since the fixed frequency integration already returns the Fourier transform. This would avoid taking an additional Fourier transform, and would avoid all the problems with windowing and zero-padding. If I had to write this code again, I would follow this approach. Hopefully, the work was not completely useless because the current method is more easily generalizable to cases that are not $$l = 2$$, $$m = 2$$.

network_mismatch_from_psi4¶

network_mismatch_from_psi4() is the function the most general interface and the one you will be most likely to use if you are interested in working with actual LIGO-Virgo data.

Before we start, it is important to stress that computing the mismatch is an optimization problem. As it is always the case, it is difficult to determine if the value found is a local maximum or the absolute one. PostCacuts implements a simple grid search. This is an inefficient but robust method. However, to work well, it is important to provide reasonable limits. For the polarization, we search from 0 to $$2 \pi$$, and we take as input the extremes of the search for time shifts. You should start providing physical values, inspect the result, and refine your search. In should also expand the bounds to make sure that you are localizing the absolute maximum.

Warning

kuibit has no way to determine if the maximum found is the absolute one. It is your job to set the limits of the search in a meaningful way.

To make up for the algorithmic inefficiency, kuibit optionally uses numba to speed up the search. Using numba enables high-resolution searches that would not be possible otherwise. Numba compiles the main mismatch function (_mismatch_core_numerical()) to machine code to achieve native performances. Numba requires a substantial overhead to do this, so for small searches it is not convenient to use it. Therefore, kuibit activates numba only when the size of the parameter space is larger than 2500 elements. If you want to use numba with fewer elements, you can set force_numba to True. This may be faster in some cases (for example, for very long arrays). To use numba, make sure that the package is available (it is installed among the extras).

The limits of the serach are specified by the paramters time_shift_start and time_shift_end for time shifts, and they are always from 0 to $$2 \pi$$ for polarization shifts. The number of points inspected is num_time_shifts and num_polarization_shifts.

network_mismatch_from_psi4() takes the two $$\Psi_4$$ and compute the strains from there. Hence, you have to provide all the quantities you would need for computing the strains: pcut, window_function, and the arguments to the window function. $$\Psi_4$$ are passed as GravitationalWavesOneDet, as they are found in SimDir, when a radius is specified for gravitational waves. window_function can be None, a string (indicating one of the buil-in windows), or a function that implements a custom-window.

Since the operation requires taking Fourier transform, we provide ways to pre-process the strain signals. First, the window function that you provide to compute the strain from $$\Psi_4$$ will be used to window also the strain. Second, the signal is zero padded so that it has a total of num_zero_pad points. num_zero_pad is not the number of zeros added: it is the final length of the signal.

An important quantity you may want to provide is the noise curve associated to the detectors. For this network_mismatch_from_psi4() takes a paramters, noises. This can be None, in which case the mismatch will be computed with no noise. If noises is not None, then, it has to be a Detectors object (Detectors) with each entry being a FrequencySeries with the noise power spectral densities. At the moment, Detectors are set to work with the LIGO and Virgo interferometers. In case you want to disable one of the detectors, set the entry to -1 (see example below). You can also set one entry to None so that its contribution is computed without noise.

In case you want to remove part of the signal from the comparison, you can use the two paramters time_to_keep_after_max and time_removed_beginning. The first sets how much signal to keep after the peak, everything else after that is removed. The second controls how much signal to remove at the very beginning. They are always provided in computational units. You may need to set trim_ends=False if you want to have finer control on how much signal to consider. For a meaningful comparison, it is important that the time limits are set properly, if they are not, the window function may produce incorrect results (because the two series are windowed in physically different ways). Visualize your data to make sure that the comparison is meaningful!

Typically, we perform simulations in some geometrized units, but we want to compare signals using actual noise (in physical units). For this, you can provide the mass scales in solar masses of the two signals. The waves are assumed to be in geometrized units in which M=1. If you provide the mass scales, they are converted in waveform with M=mass_scale_msun * M_sun. Additionally, if you provide a mass scale, you can provide a distance in megaparsec. The signal will be redshifted according to the cosmological redshift corresponding to that distance (assuming standard LCDM). Moreover, you have to provide the sky localization of the event with the paramters right_ascension, declination, and time_utc. In case you want to work with theta and phi, you should use the mismatch_from_strains() function.

A (roughly) complete example would look like:

 mass_scale = 65
CU = unitconv.geom_umass_msun(mass_scale)

pcut1 = 120
pcut2 = 140

fmin = 20
fmax = 512

rex = 110  # Extraction radius

psi1 = simdir.SimDir("folder1").gws[rex]
psi2 = simdir.SimDir("folder2").gws[rex]

distance = 500  # Mpc

# -1 disables Virgo
noises = Detectors(virgo=-1,
hanford=noise_hanford,
livingston=noise_livingston)

return gw_mismatch.network_mismatch_from_psi4(psi1,
psi2,
8,
-70,
"2015-09-14 09:50:45",
pcut1,
pcut2,
0.125,  # tukey alpha
noises=noises,
trim_ends=False,
window_function='tukey',
mass_scale1_msun=mass_scale,
mass_scale2_msun=mass_scale,
distance1=distance,
distance2=distance,
fmin=fmin,
fmax=fmax,
num_time_shifts=1000,
num_polarization_shifts=1000,
time_shift_start=-10 * CU.time,
time_shift_end=10 * CU.time,
time_to_keep_after_max=400,
time_removed_beginning=200)


In case you want to compute the optimal mismatch considering only one detector, you can use the function one_detector_mismatch_from_psi4(), which is similar to network_mismatch_from_psi4() but considers only one detector.

mismatch_from_strains¶

mismatch_from_strains() implements a more low-level interface to compute the mismatch between the waveforms. Internally, this is what is used by network_mismatch_from_psi4().

With mismatch_from_strains() you are responsible of providing valid strain data h1 and h2, as well as antenna_patterns and noises. Here, antenna_patterns and noises are lists where the corresponding index represents the same detector.

If you want to learn how the mismatch computation works, read the comments in the code of this function.