# Two-Dimensional Correlation Function of Binary Black Hole Coalescences

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## Abstract

**:**

## 1. Introduction

## 2. Two-Dimensional Correlation Function

## 3. Results

`NSIDE`resolution of 256, corresponding to a pixel angular resolution of ${\theta}_{\mathrm{pix}}\sim {0.23}^{\xb0}\ll {\theta}_{\mathrm{res}}$. Choosing different values of the map resolution affected the final results only by a few percent. We then created the map ${M}_{\mathrm{obs}}(\chi ,\phi )$ in Equation (1) by summing the sky localization error regions of each BBH event and normalizing to the number of detections, such as ${\sum}_{j}{M}_{\mathrm{obs}}\left({p}_{j}\right)=1$, where ${p}_{j}$ denotes the ${j}^{\mathrm{th}}$ pixel. A Mollweide representation of ${M}_{\mathrm{obs}}(\chi ,\phi )$ in Equation (1) is shown in Figure 1.

`map2alm`was used to compute ${a}_{lm}$. The coefficients of the Legendre expansion in Equation (4) were then obtained by summing the $|{a}_{lm}{|}^{2}$ in m. We followed the same procedure to compute the CF from reference maps ${M}_{\mathrm{ref},k}(\chi ,\phi )$ used to test possible angular correlation signatures in the CF.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Heat sky map of the combined O1-O2 LIGO-Virgo detections, ${M}_{\mathrm{obs}}(\chi ,\phi )$. The color scale denotes the probability density of sky localization (yellow to blue: high to low, normalized to ${\sum}_{j}M\left({p}_{j}\right)=0.1$ for better visualization purposes).

**Figure 2.**Example of a synthetic sky map for Model B. The (arbitrary) color scale denotes the probability density of sky localization (yellow to blue: high to low, normalized to ${\sum}_{j}M\left({p}_{j}\right)=0.1$ for better visualization purposes)

**Figure 3.**Top: Comparison of the measured CF (red continuous curve) and the reference CF for a set of maps obtained by randomly distributing the LIGO-Virgo observations in the sky (Model A). The grey-shaded bands denote one- through five-$\sigma $ deviations from the reference CF obtained by averaging over 500 “random” maps. Bottom: The observed CF normalized to the reference CF. The observed CF lies within two-$\sigma $ of ${C}_{\mathrm{ref},A}\left(\theta \right)$.

**Figure 4.**Top: Comparison of the measured CF (red continuous curve) and the CF for the set of synthetic maps of Model B. The grey-shaded bands denote one- through five-$\sigma $ deviations from the CF averaged over 500 synthetic maps. Bottom: Observed z-score of the observed CF for Model B. The observed CF lies within two-$\sigma $ of ${C}_{\mathrm{ref},B}\left(\theta \right)$. The deviation at low angular distances is likely due to the approximation used to simulate the synthetic maps.

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Cavaglià, M.; Modi, A. Two-Dimensional Correlation Function of Binary Black Hole Coalescences. *Universe* **2020**, *6*, 93.
https://doi.org/10.3390/universe6070093

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Cavaglià M, Modi A. Two-Dimensional Correlation Function of Binary Black Hole Coalescences. *Universe*. 2020; 6(7):93.
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Cavaglià, Marco, and Ashini Modi. 2020. "Two-Dimensional Correlation Function of Binary Black Hole Coalescences" *Universe* 6, no. 7: 93.
https://doi.org/10.3390/universe6070093