Dark sirens and the impact of redshift precision

TL;DR

The study assesses how redshift precision and catalogue completeness impact $H_0$ measurements from dark sirens in future GW observing runs. Using a Bayesian dark-siren framework with the MICEcat mock catalogue and redMaGiC photometric data, it compares spectroscopic-like ($m{ m \sigma_z=10^{-4}}$) and photometric-like ($m{ m \sigma_z=10^{-2}}$) redshift errors, evaluates the effect of redshift outliers, and explores incompleteness via uniform sub-sampling. It finds that spectroscopic redshifts improve $H_0$ precision most for small localisation areas and low distance uncertainties, while photometric redshifts with well-controlled outliers do not bias $H_0$, though completeness losses can erode gains from precision redshifts. The results advocate a hybrid redshift catalogue approach—combining spectroscopic and photometric data—to achieve competitive dark-siren $H_0$ measurements in the near term, with full spectroscopic completeness becoming increasingly important for 3G detectors.

Abstract

With the growing number of gravitational wave detections, achieving a competitive measurement of $H_0$ with dark sirens is becoming increasingly feasible. The expansion of the Ligo-Virgo-KAGRA Collaboration into a four detector network will reduce both the localisation area and the luminosity distance uncertainty associated with each gravitational wave event. It is therefore essential to identify and mitigate other major sources of error that could increase the uncertainty in $H_0$. In this work, we explore three scenarios relevant to the dark siren method in future observing runs. First, we demonstrate that there is a precision gain offered by a catalogue of spectroscopic-like redshifts compared to photometric-like redshifts, with the greatest improvements observed in smaller localisation areas. Second, we show that redshift outliers (as occur in realistic photometric redshift catalogues), do not introduce bias into the measurement of $H_0$. Finally, we find that uniformly sub-sampling spectroscopic-like redshift catalogues increases the uncertainty in $H_0$ as the completeness fraction is decreased; at a completeness of 50% the benefit of spectroscopic redshift precision is outweighed by the degradation from incompleteness. In all three scenarios, we obtain unbiased estimates of $H_0$. We conclude that a competitive measurement of $H_0$ using the dark siren method will require a hybrid catalogue of both photometric and spectroscopic redshifts, at least until highly complete spectroscopic catalogues become available. This, however, will come at the cost of a more complex selection function.

Figures (13)

• Figure 1: The fibre hours required on a telescope like the AAT to spectroscopically observe all galaxies individually within a one-square-degree localisation area are shown for five different apparent magnitude limits. These fibre hours represent only the exposure time and therefore exclude configuration time. The top panel presents exposure times for four combinations of signal-to-noise ratio and seeing conditions under dark time, while the bottom panel displays the same for bright time conditions. The numbers above each scatter point indicate the total number of galaxies within the localisation area that fall below the corresponding apparent magnitude limit. Note: the SNR 3, Seeing 2.0" are hidden behind the SNR 6, Seeing 1.0" points.
• Figure 2: The top panel shows the redMaGiC redshift sample of photometric redshifts minus the corresponding spectroscopic redshifts. Outliers are identified by calculating the $p$-value of the $\chi^2$ statistic between the photometric and spectroscopic redshift for each galaxy and removing the galaxy using the $p$-value as a sampling probability. The remaining redshifts are binned, each bin having a mean and standard deviation. The bottom panel shows the standard deviation of each bin, with a best fit line that relates the spectroscopic redshift to the standard deviation.
• Figure 3: The top panel shows the probability of detecting a gravitational wave at LIGO (given 2021 LVK capabilities) as a function of luminosity distance for two fractional luminosity distance uncertainties. In practice, the detection probability depends on the specific sensitivity of the gravitational wave detector. Unless otherwise stated, we simulate only gravitational wave events with a fractional luminosity distance uncertainty of $\sigma_{d_L}/d_L=10\%$ in this analysis. The dashed line at $\hat{d}_L^{\mathrm{thr}}=1550$ Mpc represents the threshold assumed in our analysis, beyond which galaxies with higher observed luminosity distances are considered undetectable (essentially acting as a signal-to-noise cutoff). The bottom panel shows the gravitational wave likelihood for two fractional luminosity distance uncertainties. Gravitational waves can only be drawn from galaxies with true redshifts in the range $z_{\mathrm{lower}}\leq z\leq z_{\rm upper}$.
• Figure 4: Probability of a true host galaxy redshift given an ensemble of measured redshifts for a 50 deg$^2$ localisation area and a photometric-like redshift uncertainty ($\sigma_{z}=10^{-2}$). The redshift bins show the distribution of true redshifts in the localisation area, and the dashed line shows the redshift prior (uniform-in-comoving-volume). The interpolant is given by $p_{\mathrm{CBC}(z)}$ (Equation \ref{['eq:pCBC']}) for a single realisation.
• Figure 5: Schematic of the statistical framework used in this work.