bright_siren¶

The Hubble constant from an event with an identified host galaxy (bright siren), alone or combined with the spectral siren. It takes seconds: no sampling.

gwtc_analysis bright_siren                                            # GW170817 and NGC 4993
gwtc_analysis bright_siren --spectral-posterior hubble_constant_run   # combined with the spectral siren
gwtc_analysis bright_siren --src-name GW190521                        # candidate AGN flare, z = 0.438

Method¶

The GW signal gives the luminosity distance \(d_L\) with no distance ladder; the host gives the Hubble-flow redshift \(z\). For a nearby galaxy, \(z\) comes from its recession velocity minus its peculiar velocity, \(c z = v_r - \langle v_p \rangle\). In flat ΛCDM, with \(\Omega_m = 0.3065\) as in the spectral siren,

\[ d_L = \frac{c}{H_0}\, D(z; \Omega_m). \]

For sources uniform in comoving volume and source-frame time, \(p_\text{pop}(z) \propto \frac{dV_c}{dz} \frac{1}{1+z}\), and with a flat H₀ prior (Chen, Fishbach & Holz 2018 [72]; Mandel, Farr & Gair 2019 [35]):

\[ p(H_0 \mid \text{data}) \propto \frac{1}{\beta(H_0)} \int \mathcal{N}(z_\text{obs};\, z,\, \sigma_z)\; p_\text{pop}(z)\; \mathcal{L}_\text{GW}\big(d_L(z, H_0)\big)\, dz . \]

\(\mathcal{L}_\text{GW}\) is the posterior of \(d_L\) divided by the PE distance prior, read from the file (\(d_L^2\) up to O3, uniform in comoving volume in O4). Over the PE samples \(d_i\), whose redshift at a trial H₀ is \(z_i\), the integral is a kernel sum:

\[ \sum_i \mathcal{N}(z_\text{obs};\, z_i,\, \sigma)\, \frac{p_\text{pop}(z_i)}{\pi_\text{PE}(d_i)\; \partial d_L/\partial z\,(z_i)}, \qquad \sigma = \max(\sigma_z, h), \]

where \(h\) is the kernel width of the \(z_i\) (Silverman's rule). It only matters for a distant host with a spectroscopic redshift, much narrower than the spread of the samples.

The selection term \(\beta(H_0)\) is the fraction of the population that would be detected. At a given redshift a larger H₀ means a smaller distance, so a louder signal. Leaving β out biases H₀: see Validation. --selection:

Value	β(H₀)	Use
`euclidean`	∝ H₀³: GW-limited detection of nearby sources. It cancels the volume factor of the numerator, and with a \(d_L^2\) prior the posterior becomes \(\sum_i \mathcal{N}(v_H;\, H_0 d_i,\, \sigma_v)\), the LVK 2017 formula [27]	z < 0.05
`injections`	the found LVK sensitivity injections of the event's observing run, carried to the source frame at each trial H₀ and reweighted to the population of the event's class: Power Law + Peak for black holes (as in `rates`), uniform 1–2.5 M☉ for neutron stars; the injected isotropic spins	distant events
`auto` (default)	`euclidean` below z = 0.05, `injections` above

Sky position. The distance must be the one at the counterpart's position: distance and sky position are correlated through the antenna pattern. The GWTC-1 GW170817 samples were produced with the sky fixed to AT2017gfo and are used as they are. Other samples are restricted to those within --sky-radius of the counterpart.

Registered events¶

Event	Host	z	Status	Selection (auto)
GW170817	NGC 4993, AT2017gfo	3017 ± 166 km/s (\(v_r\) 3327 ± 72, \(\langle v_p \rangle\) 310 ± 150) [27]	confirmed (kilonova)	euclidean
GW190521	AGN J124942.3+344929, ZTF19abanrhr	0.438 ± 0.0015 [74]	candidate	injections (O3a, BBH)

GW190521 is a candidate. Graham et al. proposed the ZTF flare in an AGN disk as the counterpart [74]. The odds of a common source are only 1 to 12, depending on the waveform model (Ashton et al. 2021 [75]). Its result is H₀ if the flare is the counterpart, and the report says so.

A new event needs one entry in COUNTERPARTS (gwtc_analysis/bright_siren.py): the host, the counterpart's position, the redshift or the velocities, the observing run, the population class, and the PE file (pe_event, read from Zenodo).

Results¶

Event, PE label	Maximum a posteriori, 68% interval	Median, 90% interval
GW170817, LowSpin	69.8 (+23.6 / −8.2)	79.9, 63.0–135.5
GW170817, HighSpin	69.4 (+14.8 / −7.4)	74.8, 62.2–112.2
GW170817, LVK 2017 [27]	70.0 (+12.0 / −8.0)
GW190521 + ZTF19abanrhr, IMRPhenomXPHM	30.4 (+61.0 / −7.0)	63.2, 26.1–129.5

H0 from GW170817 and NGC 4993

GW170817. The maximum a posteriori and the lower side agree with the published value. The upper side is wider because the public GWTC-1 samples come from a later reanalysis (GWTC-1, IMRPhenomPv2_NRTidal), whose distance has a longer low-distance tail than the 2017 analysis (43.8 +2.9 −6.9 Mpc). That tail comes from the distance–inclination degeneracy: an inclined binary is fainter than a face-on one at the same distance. Constraining the inclination with the radio jet narrows H₀ considerably (Hotokezaka et al. 2019 [73]); this mode does not use EM inclination constraints. The full ΛCDM \(d_L\) raises H₀ by 0.8% (0.6 km/s/Mpc) compared with the linear Hubble law of the 2017 paper.

H0 from GW190521 and ZTF19abanrhr

GW190521. The posterior is bimodal because the distance posterior is, in the direction of the flare. At z = 0.438, H₀ ≈ 30 corresponds to \(d_L\) ≈ 5.6 Gpc and H₀ ≈ 75 to 2.25 Gpc; the samples within 3° of the flare have a median of 4.6 Gpc with 16% below 2.2 Gpc. The result depends on the sky selection (median 54.7 within 5°, against 63.2 within 3°) and on the waveform model; only IMRPhenomXPHM has enough samples near the flare in GWTC-2.1. The selection term is computed from 106,376 found O3a injections, with 13,000–16,000 effective injections over the H₀ range.

Combination with the spectral siren¶

--spectral-posterior takes a hubble_constant work directory (its posterior_reweighted.tsv, else posterior.tsv) or any TSV with an H0 column. The two measurements use different events (the spectral siren uses binary black holes; leave the bright-siren event out of it) and the same flat prior, so their posteriors multiply. The spectral-siren density is a Gaussian kernel estimate, reflected at the prior bounds.

Bright siren combined with the spectral siren

GW170817 with the Power Law + Peak spectral siren (GWTC-4.0 setup): H₀ = 71.7 (+22.3 / −8.0) km/s/Mpc (maximum a posteriori, 68%). GW170817 dominates; the broad spectral siren shifts the posterior slightly upwards.

Validation: mock bright sirens¶

The analysis is tested on simulated detections with a known H₀ (tests/test_bright_siren.py).

Sources: uniform in comoving volume up to z = 4, masses uniform in 20–40 M☉, random orientations.
Detection: each source has a network SNR \(\rho = A(\mathcal{M}_\text{det})\, w(\iota) / d_L\) plus unit Gaussian noise, and is detected above 12. The detected events have a median z ≈ 0.5.
Observations: each detected event gets the posterior of \(d_L\) given its observed SNR. It uses a \(d_L^2\) prior and is marginalized over the inclination, which reproduces the real distance–inclination degeneracy. The host redshift has an error of 0.001.
Injections: they are made at H₀ = 70 with broader masses than the population (5–120 M☉), as the LVK ones, so that they cover the population at every trial H₀.

Test	Result
β from the reweighted injections against the detected fraction simulated directly at each H₀ (40–120)	equal within 3% (β varies by more than a factor of 10)
100 events, H₀ = 70: with the selection term	70.9 ± 1.1
the same without the selection term	80.7 ± 2.7: biased by more than 3σ
P–P test, 100 events with H₀ drawn from the prior	uniform with the selection term (KS p = 0.16), not without (p = 10⁻⁴)
nearby sources (horizon z ≈ 0.02)	β from injections ∝ H₀^3.0, the `euclidean` selection

Mock bright sirens: catalog posterior and P–P test

The injections must cover the population at every trial H₀. In a first version of the mock, with injection masses drawn from the population's own 20–40 M☉, β was wrong by up to a factor of 2.5 at the edges of the H₀ range, and H₀ was biased by 3σ. The LVK injections are drawn much more broadly than any population; the report gives the effective number of injections over the H₀ range.

Options¶

Option	Default	Meaning
`--src-name`	`GW170817`	registered event (`GW170817`, `GW190521`)
`--pe-label`	the registry's, else all the labels (LowSpin first)	PE labels; the first one is the main result and the one combined
`--pe-file`	the event's bundle or Zenodo file	another PE file (PESummary layout)
`--pe-cache`	that of `hubble_constant`	where Zenodo PE files are downloaded
`--v-recession V SIGMA`, `--v-peculiar V SIGMA`	the registry's	velocities of a nearby host, km/s
`--redshift Z SIGMA`	the registry's	Hubble-flow redshift of the host, instead of the velocities
`--selection`	`auto`	`euclidean`, `injections` or `auto`
`--sensitivity-release`, `--sensitivity-file`	`gwtc4`	injections of the selection term
`--far-threshold`, `--snr-threshold`	0.25 /yr, 10	found injections (real searches, semi-analytic O1+O2)
`--sky-radius`	3°	for samples not fixed to the counterpart
`--spectral-posterior`	none	spectral-siren posterior to combine with
`--h0-range MIN MAX`	10 200	flat prior; keep the spectral siren's for a combination

The published comparison is shown only with the registry's velocities or redshift.

Outputs¶

--out-report (HTML): the result, the caveat of a candidate counterpart, the inputs, the selection term and the plots;
--out-summary (TSV): maximum a posteriori, 68.3% highest-density interval, median and 90% interval of each analysis, with the distance of each label;
<summary>.posterior.tsv: the posterior densities on an H₀ grid;
--plots-dir: h0_bright_siren_<event>.png, and h0_combined.png with --spectral-posterior.