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About the galaxy lists

For each GW trigger (from 2020 onwards) the Swift GW pages include lists of the galaxies within the GW localisation region, with (and ordered by) the probability that each is the host of the GW event. This page explains how those values are produced.

Contents

Probability calculations

The probability calculation associated with each galaxy is based on the work presented in Section 3.2 of Evans et al. 2016 (MNRAS, 462, 1591; hereafter E16)1, modified slightly to report results per galaxy rather than per pixel in the GW skymap. In the description below, equation numbers quoted refer to those from that paper.

The probability that a given galaxy is the host of the GW event is based on a number of factors:

Galaxy luminosity (L)
Assumed to be a proxy for mass, and hence probability of hosting a merger.
PdistOK
The probability that the gravitational wave and galaxy are at the same distance from the Earth. This is defined below, and in eqation 10 of E17.
Completeness
How complete, in terms of luminosity, the galaxy catalogue employed is at the distance to the GW event. Since the distance to the GW event depends on the position, this value is calculated for each galaxy individually.
PGW
The probablity in the GW map (from the LVC) at the position of this galaxy. Note that this is not a point probability, but the probability in the HEALPIX pixel covering the galaxy position; these pixels can be very large, depending on the resolution of the GW sky map.

The basic calculation for the probability in galaxy \(g\) is based on equation 7 of E16 and links these 4 components, along with a normalising factor:

$$\mathcal{P}_g =N \ \mathcal{P}_{\rm GW} \ C_g \ P_{\rm distOK} \ \frac{L}{L_{\rm tot}} $$

The normalising factor is defined in equation 8 of E16, and accounts for the assumption that the GW event occured in (or on the outskirts of) a galaxy, as discussed below.

\( P_{\rm distOK} \), the probability that the GW event and galaxy are at the same distance, is given simply by integrating the product of the probability distribution functions for the galaxy distance and GW distance, as in equation 10 of E16, i.e.

$$ P_{\rm distOK} = \int P_{\rm GW}(D) P_g(D) dD $$

\( P_g(D) dD \) is treated as a Gaussian function, with mean and uncertainty defined by the galaxy catalogue (note that, for 2MPZ, we added a systematic error; see E16, section 3.1 and equation 4). \( P_{\rm GW}(D) \) is taken from equation 2 of Singer et al., 2016.

The normalisation, \( N\), ensures that \( \sum_g{\mathcal{P}_g } = \bar{C} \); that is, the probability that the source is in a known galaxy, sums to the completeness of the galaxy catalogue for the distance of the GW event. This is essential to guarantee that the galaxy probabilities (and the convolved skymap as a whole) are properly normalised. That is, if for example, we know that our galaxy catalogue is 50% complete (in terms of the luminosity in the catalogue, not the number of galaxies) and if our assumption that compact mergers occur in galaxies is true, then if we observed every single galaxy in the catalogue we must, by definition, have a 50% chance of observing the location in which the merger occurred. The use of \( \mathcal{P}_{\rm GW} \) in the calculation of the normalisation and \( \mathcal{P}_g \) is also essential to ensure that the relative weights of the galaxies are correct.

This can be illustrated by two simplified examples. In both cases, assume that the galaxy catalogue employed is exactly 50% complete at the distance of the GW event. In the first example, let us also imagine that there is only a single galaxy in the catalogue. By definition, 50% of the probability must lie within this galaxy, hence \mathcal{P}_g=0.5 (and the sum of the rest of the GW localisation must also be only 0.5). For the second example, imagine that there are 4 catalogued galaxies, of equal luminosity. One of these lies in a high probability part of the GW sky localisation, the other 3 lie outside the 99% contour. In this case, we will still find that \( \sum_g{\mathcal{P}_g } = 0.5 \), however the galaxies will not all have equal probability. The galaxy in the GW error region will have a probability of very nearly 0.5, while the other 4 will have very small probabilities.

It is important to note that the equation \( \sum_g{\mathcal{P}_g } = \bar{C} \) is only true because the normalisation and total luminosity are calculated for all galaxies, not just those within some confidence contour of the GW localisation2

1 An erratum relating to the normalisation was published in 2019 (MNRAS, 484, 2362). The arXiv version is recommended as the correction has been applied inline.

2 This is not quite true: for reasons of computational speed I ignore galaxies in regions where \( \mathcal{P}_{\rm GW} < 10^{-10}\), however this generally means that ≫99.999% of the GW probability is included, so the effect of this is negligible. If, for example, one were to take only galaxies in the 50% or 90% contours of the GW localisation then the normalisation has to be calcualted in a different way (perhaps cite some papers here?)

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Choice of galaxy catalogue

In selecting a galaxy catalogue to use there are three main considerations:

  1. Completeness as a function of distance.
  2. Accuracy of the distance measurements.
  3. Uniformity of luminosity information.

Completeness ideally needs definiting in terms of the probability of hosting a compact merger, which is obviously not a product in most catalogues! However, stellar mass serves as a reasonable proxy and, in turn, luminosity gives an indication of stellar mass. We, in common with other authors (e.g. Gehrels et al., 2016; Arcavi et al., 2017), therefore defined completeness in terms of luminosity.

Accuracy is primarily an issue because, in order to ensure adequate completeness beyond 80 Mpc or so, one is compelled to rely on photometric redshifts. Such measurements are less direct than spectroscopic redshifts, since they must also rely on some understanding or assumption about the galaxy colour and reddening, which brings potential error.

Luminosity is a function of wavelength, this function depending on the galaxy type. It is therefore obviously preferable if a single waveband can be used for all galaxies in the catalogue, to ensure that \( \frac{L}{L_{\rm tot} \) in the above equation is calculated consistently for each galaxy.

Based on these considerations, we determined the 2MPZ catalogue (Bilicki et al., 2014) to be the best option. This catalogue is based entirely on 2MASS observations, and therefore has a common photometric band for all galaxies. While it is a photometric redshift catalogue, many galaxies also have spectroscopic redshifts. This allows us both to select the latter when available, but also to calibrate the quality of the former. As demonstrated in E16 section 3.1, a systematic error (itself a function of reddening) was necessary to ensure that the photometric errors were properly handled. Its completeness is good, compared to the other catalogues explored by E16, as demonstrated in the plot below. This shows the K--band luminosity in the catalogue in a series of distance bins, compared to that expected assuming a Schechter function. The bottom panel shows the ratio of the catalogue value to Schechter function, i.e. the completness. Note that this plot shows completeness in each distance bin and not the cumulative completeness; it is the 2MPZ-only version of the right-hand panel of Fig. 5 of E16.

Completeness of 2MPZ vs distance

The exception to this is for nearby events, D<80 Mpc. In this case 2MPZ is less complete, and we instead use the GWGC catalogue. Which catalogue was used is given on the web page.

At the time of our work (E16) GLADE (Dálya et al., 2018) had not been published. We investigated an early version of the catalogue but found issues (discussed with the authors, and corrected for the published version of GLADE) and therefore chose not to use it. A preliminary analysis of the published catalogue suggests that the completeness of 2MPZ is still superior, however we will conduct a more in-depth analysis at the earliest opportunity to confirm whether we should be using GLADE instead of 2MPZ.

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Using the webpage

By default, the galaxy list shown online will relate to the most recent skymap produced by the LVC. If multiple skymaps exist, a drop-down menu allows you to select which one the galaxies shown relate to.

Before the table are a series of controls which should be self-explanatory; these allow you to select the coordinate style (all coordanates are J2000), and how many galaxies to display - by default only the first 30 matches are shown.

If you wish to filter the results to a specific cone on the sky, this can be done using the ‘Restrict to region,’ box. This may be of use, for example, when there is a possible gamma-ray or neutrino counterpart which reduces the sky localisation from the GW-only area.

You can also choose to return an HTML table, viewed within the webpage, or a comma-separated list for download. The HTML table shows only the main details of each galaxy at first; there is a link, ‘More’ to view full details. The CSV file in contrast contains all details in each row. A brief explanation of each property is given below; this presupposes you have read the documenation above concerning how the probabilities are calculated.

By default, and when a CSV table is selected, galaxies are sorted in order of decreasing probability. Clicking on the headers of the table (in the webpage) will resort the table on that column, a second click will invert the sort order. Note: If the number of lines to return is not set to ‘-1’ = all, then the lines returned will be based upon the selected sort order. e.g. if sorting by RA and requesting the first 30 galaxies, the result will be the first 30 galaxies selected by RA not by probability.

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Description of galaxy properties

ID
The galaxy identifier from the catalogue.
Phost
The probability that this galaxy hosted the GW event, as described above.
RA / Dec
The position of the galaxy, J2000.
Dgal
The catalogued distance to the galaxy, in Mpc, with a 1-σ uncertainty. This was calculated from the redshift, assuming ΩM=0.3, ΩΛ=0.7 and H0 = 70 km s-1 Mpc -1.
DGW
The distance to the GW event, if it is in the direction of the galaxy. The quoted values are μ±σ from the GW skymap.
Shape
The shape of the galaxy (as an ellipse). Quoted values are the major and minor radii in arc minutes, and the position angle as degrees from North, via East.
PGW
The probability in the GW skymap at the RA,Dec of the centre of this galaxy.
L/Ltot
This is \( L*P_{\rm distOK} \) for this galaxy, divided by the sum of this property for all galaxies (see equation 9 of E16). i.e. it reflects the probability of this galaxy hosting a merger compared to the other galaxies in the catalogue, given the GW distance, but not accounting for the RA/Dec dependence of the GW probability.
PdistOK
This is the probability that the GW event and galaxy are at the same distance from Earth, as defined in the equation above.
The completeness of the galaxy catalogue used, given the mean GW distance to the event if it is along the line of sight to this galaxy.

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