# Multi-Messenger Constraints on the Hubble Constant through Combination of Gravitational Waves, Gamma-Ray Bursts and Kilonovae from Neutron Star Mergers

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## Abstract

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## 1. Introduction

## 2. The Hubble Constant Tension

## 3. Gravitational Waves as Standard Sirens

## 4. Inclination Constraints from the Gamma-Ray Burst

#### 4.1. Afterglow

#### 4.2. Superluminal Motion

## 5. Inclination Constraints from the Kilonova

#### 5.1. Matter Outflows as Kilonova Engines

#### 5.2. Constraints from Kilonova Spectro-Photometry

#### 5.3. Constraints from Kilonova Polarimetry

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACT | Atacama Cosmology Telescope |

BAO | Baryon Acoustic Oscillations |

BBH | Binary Black Hole |

BH | Black Hole |

BNS | Binary Neutron Star |

CMB | Cosmic Microwave Background |

GRB | Gamma-ray burst |

GTC | Gran Telescopio CANARIAS |

GW | Gravitational Wave |

IFU | Integral Field Unit |

IGWN | International Gravitational-Wave Observatory Network |

IR | Infrared |

KN | Kilonova |

$\mathrm{\Lambda}$CDM | $\mathrm{\Lambda}$ Cold Dark Matter |

LIGO | Laser Interferometer Gravitational-wave Observatory |

LSST | Legacy Survey of Space and Time |

MAAT | Mirror-slicer Array for Astronomical Transients |

MAP | Maximum a posteriori |

NS | Neutron Star |

SNe | Supernovae |

TRGB | Tip of the red giant branch |

UV | Ultraviolet |

VLBI | Very Long Baseline Interferometer |

VLT | Very Large Telescope |

VRO | Vera Rubin Observatory |

WMAP | Wilkinson Microwave Anisotropy Probe |

## Notes

1 | As admitted by the authors, the term was coined by Sterl Phinney and Sean Carroll |

2 | |

3 | Note that the viewing angle ${\theta}_{\mathrm{obs}}$ is measured from the jet axis whereas the inclination i is measured from the axis orthogonal to the binary’s orbital plane. Therefore, this relation between ${\theta}_{\mathrm{obs}}$ and i assumes that the jet axis is orthogonal to the orbital plane. |

4 | These relations are valid for frequencies ${\nu}_{\mathrm{a}},{\nu}_{\mathrm{m}}<\nu <{\nu}_{\mathrm{c}}$ (where ${\nu}_{\mathrm{a}}$ is the self-absorption frequency, ${\nu}_{\mathrm{m}}$ is the synchrotron break frequency and ${\nu}_{\mathrm{c}}$ is the cooling break frequency), a condition that is satisfied from X-ray to radio wavelengths as long as the density of the circum-merger environment is not much higher than the one inferred for GW170817. |

5 | The polarization signal of extragalactic events as supernovae and KNe is the result of integrating over all the contributions coming from different regions of the ejecta. |

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**Figure 1.**Constraint on ${H}_{0}$ from the BNS merger GW170817 [10]. Figure adapted from [10] and using publicly available data from Gravitational Wave Open Science Center (https://www.gw-openscience.org, aeecssed on 3 May 2022). Copyright 2017 LVK. (

**Top**) 2D posterior density of ${H}_{0}$ and the viewing angle ${\theta}_{\mathrm{obs}}$, where $68.3\%$ ($1\sigma $) and $95.4\%$ ($2\sigma $) contours are shown with solid and dotted black lines, respectively. The viewing angle ${\theta}_{\mathrm{obs}}$ is calculated relative to a face-on observer, i.e., ${\theta}_{\mathrm{obs}}=180\xb0-i$, where i is the system inclination obtained from the GW data. (

**Bottom**) Marginalized 1D posterior density for ${H}_{0}$. In both panels, ${H}_{0}$ values inferred from Planck [2] and SHOES [3] are shown with their $1\sigma $ intervals in brown and green, respectively. The inferred ${H}_{0}$ values are reported in the legend.

**Figure 2.**Improved constraints on ${H}_{0}$ from the BNS merger GW170817 through combination of GW and EM data. (

**Top-right**) Posterior distributions on the observer viewing-angle from model fitting of the associated short GRB and KN. Constraints from GRB 170817A are shown for model fits with (cyan [12]) or without (light green [11]; red [13]) information from the jet superluminal motion [95,96]. Constraints from the KN AT 2017gfo are shown for model fits of broad-band photometry (pink [14]) and spectroscopy (orange [17]). The color scheme is the same in the remaining two panels. (

**Top-left**) Same as in Figure 1 but adding improvements to the 2D posterior density contours when the viewing-angle constraints from GRB and KN fitting are used as priors for the inclination in the GW analysis. (

**Bottom**) Marginalized 1D posterior density distributions for ${H}_{0}$ when using the original standard siren approach (black, same as in Figure 1) and when adding constraints on the viewing-angle from EM probes. The inferred ${H}_{0}$ values are reported in the legend.

**Figure 3.**Polarized light from KNe. (

**a**) Sketch illustrating the origin of polarization in KNe. Photons escaping from lanthanide-rich dynamical ejecta around the equatorial plane (in red) are preferentially depolarized by bound–bound line interactions; photons escaping from a lanthanide-free wind (in blue) can be linearly polarized by Thomson scattering. Figure adapted with permission from [155]. Copyright 2018 Bulla. (

**b**) Polarization predictions from [154] for a two-component BNS model. Polarization levels are shown at 7000 Å as a function of viewing angle ${\theta}_{\mathrm{obs}}$ for three different epochs: 1.5 (yellow stars), 2.5 (orange squares) and 3.5 (white diamonds) days from the merger. The $V$—band polarization upper limit derived for AT 2017gfo at 1.5 days is shown with a horizontal dashed line and is consistent with an observer viewing the system from an angle within ${\theta}_{\mathrm{obs}}\sim 70\xb0$ from the jet axis ($\mathrm{cos}{\theta}_{\mathrm{obs}}\gtrsim 0.35$).

**Table 1.**${H}_{0}$ values obtained for GW170817 with the standard siren approach (’GW’), together with improvements using inclination constraints from model fitting of the different EM probes: the associated GRB afterglow light curve with (’GW + GRB lc + motion’) or without (’GW + GRB lc’) constraints on from the jet superluminal motion; and the KN broad-band photometry (’GW + KN photometry’) and spectroscopy (’GW + KN spectroscopy’). The $\Delta {\sigma}_{{H}_{0}}/{\sigma}_{{H}_{0},\mathrm{GW}}=({\sigma}_{{H}_{0},\mathrm{GW}}-{\sigma}_{{H}_{0}})/{\sigma}_{{H}_{0},\mathrm{GW}}$ column shows the percentage improvement in the $68.3\%$ ($1\sigma $) interval. ${H}_{0}$ values derived from CMB and are shown for comparison.

Method | ${\mathit{H}}_{0}$ (km s${}^{-1}$ Mpc${}^{-1}$) | Δ${\mathit{\sigma}}_{{\mathit{H}}_{0}}/{\mathit{\sigma}}_{{\mathit{H}}_{0},\mathbf{GW}}$ (%) | Reference |
---|---|---|---|

GW${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1}$ | ${70.0}_{-8.0}^{+12.0}$ | / | [10] |

GW${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2}$ | ${74.0}_{-8.0}^{+16.0}$ | / | [10] |

GW + GRB lc${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1}$ | ${75.5}_{-7.3}^{+14.0}$ | 10.7 | [11] |

GW + GRB lc${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1}$ | ${69.5}_{-4.2}^{+4.3}$ | 61.0 | [13] |

GW + GRB lc + motion${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2}$ | ${68.1}_{-4.3}^{+4.5}$ | 63.1 | [12] |

GW + KN photometry${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1}$ | ${72.4}_{-7.3}^{+7.9}$ | 34.0 | [14] |

GW + KN spectroscopy${}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1}$ | ${69.6}_{-4.6}^{+6.3}$ | 53.9 | [17] |

Planck (CMB) | $67.4\pm 0.5$ | / | [2] |

SH0ES (SNe Ia) | $73.0\pm 1.0$ | / | [3] |

^{1}Maximum a posteriori (MAP) interval (MAP value and smallest range enclosing 68.3% of the posterior).

^{2}68.3% symmetric interval (median plus the 15.85–84.15% range).

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Bulla, M.; Coughlin, M.W.; Dhawan, S.; Dietrich, T. Multi-Messenger Constraints on the Hubble Constant through Combination of Gravitational Waves, Gamma-Ray Bursts and Kilonovae from Neutron Star Mergers. *Universe* **2022**, *8*, 289.
https://doi.org/10.3390/universe8050289

**AMA Style**

Bulla M, Coughlin MW, Dhawan S, Dietrich T. Multi-Messenger Constraints on the Hubble Constant through Combination of Gravitational Waves, Gamma-Ray Bursts and Kilonovae from Neutron Star Mergers. *Universe*. 2022; 8(5):289.
https://doi.org/10.3390/universe8050289

**Chicago/Turabian Style**

Bulla, Mattia, Michael W. Coughlin, Suhail Dhawan, and Tim Dietrich. 2022. "Multi-Messenger Constraints on the Hubble Constant through Combination of Gravitational Waves, Gamma-Ray Bursts and Kilonovae from Neutron Star Mergers" *Universe* 8, no. 5: 289.
https://doi.org/10.3390/universe8050289