---
abstract: 'Gravitational waves (GW) can be employed as standard sirens that will soon measure the Hubble constant with sufficient precision to weigh in on the $\sim 5\sigma$ Hubble tension. Most GW sources will have no identified electromagnetic counterpart, leading to uncertainty in the redshift of the source, and in turn a degeneracy between host galaxy distance, redshift, and $H_0$. In the case where no electromagnetic counterparts are identified, it has been proposed that a statistical canvassing of candidate GW hosts, found in a large galaxy survey for example, can be used to accurately constrain the Hubble constant. We study and simulate this "galaxy voting" method to compute $H_0$. We find that the Hubble constant posterior is in general biased relative to the true value even when making optimistic assumptions about the statistical properties of the sample. This bias is caused by the fundamental degeneracy between redshift and $H_0$, and is effectively irreducible without accurate information about the redshift distribution from which the GW sources come.'
author:
- Emery Trott
- Dragan Huterer
bibliography:
- 'refs.bib'
title: 'Challenges for the statistical gravitational-wave method to measure the Hubble constant '
---

Introduction
============

Gravitational-wave events are an emerging powerful probe that can be used to measure distances to inspiraling black holes or neutron stars. In the "standard siren" technique @Schutz:1986gp [@Holz:2005df], a combination of the distance measurement to the gravitational wave (GW) event and redshift to the measured electromagnetic (EM) counterpart can be used to determine the luminosity distance to the object. This can be further turned into a measurement of the matter and dark-energy densities, as well as that of the Hubble constant $H_0$. Such a measurement would be particularly useful, as current CMB measurements currently give $H_0 = (67.4\pm0.5){\,{\rm km/s/Mpc}}$ @Aghanim:2018eyx, while low-redshift measurements based on the distance ladder give $H_0 = (74.03\pm1.42){\,{\rm km/s/Mpc}}$ @Riess:2020sih, implying the $H_0$ tension of 4.4$\sigma$. Standard-siren measurements have the potential to shed significant new light on the Hubble tension.

Currently, the main challenge with the standard-siren method is the low number of merger events with a detected EM counterpart. LIGO has detected 50 mergers where at least one of the objects is a black hole, but only one of them, the binary neutron star system GW170817, has an EM counterpart that has been identified and measured @LIGO:2017. The situation is expected to dramatically evolve in the next few years largely due to sensitivity upgrades to LIGO, and we can expect $\sim$100 binary neutron star detections by around 2025 @Chen:2017rfc. Some of these events will be close enough to be able to identify a single EM counterpart, in which case getting a measurement of $H_0$ is relatively straightforward.

However, the majority of distant GW events (which will outnumber nearby events due to volume considerations) will be poorly localized, leading to uncertainty as to which of the tens of thousands of potential host galaxies is the true host. In these cases we may employ a probabilistic method that allows the potential hosts to "vote" on the preferred $H_0$, each host being weighted by its position relative to the center of the localization region @Finn:1994cg [@MacLeod:2007jd; @Messenger:2011gi; @DelPozzo:2012zz]. It has been argued that such an approach — which we heretofore refer to as the *statistical GW method* — leads to an unbiased measurement of the Hubble constant @MacLeod:2007jd [@Nair:2018ign; @Chen:2017rfc; @Gray:2019ksv]. Such a method has already been applied in practice @Soares-Santos:2019irc [@Palmese:2020aof]. The statistical method's constraints on $H_0$ are currently weak, but are forecasted to become much tighter in the future @Chen:2017rfc.

In this paper we examine the statistical standard-siren method in some detail. We point out that there are irreducible degeneracies between the parameters that enter the determination of the Hubble constant. We find that these degeneracies generally lead to large biases in the derived value of $H_0$. This brings into question the overall potential of the statistical method, as currently defined, to deliver reliably accurate measurements of $H_0$ in the coming decade(s).

Statement of the problem {#sec:statement}
========================

The waveform of a GW event enables a measurement of the luminosity distance, which is given by $$d_L(z,H_0) =  \frac{c(1+z)}{H_0} \int_0^z \frac{{\rm d}z'}{\sqrt{(1-\Omega_M)+\Omega_M(1+z')^3}}
    \label{eq:dL}$$ where for simplicity we have assumed a flat $\Lambda$CDM universe. Here $\Omega_M$ is the matter density relative to critical, $z$ is the redshift of the GW host galaxy, and $c$ is the speed of light. At $z\ll 1$, the equation simplifies to $d_L\simeq cz/H_0 + O(z^2)$, where $\Omega_M$ only comes in at the $z^2$ order in the term in angular parentheses; in this limit, the luminosity distance depends only on $z$ and $H_0$, and not on other cosmological parameters. In what follows, we retain the general expression in Eq. (\[eq:dL\]), and fix $\Omega_M=0.3$, but we occasionally come back to the simplified first-order expansion of $d_L(z)$ to make pedagogical points.

Equation (\[eq:dL\]) showcases a fundamental challenge to measuring $H_0$, namely, for a given measurement of $d_L$ there is a perfect degeneracy between $H_0$ and $z$. That is, even with an infinite-precision $d_L$ measurement to a GW event, galaxies at higher redshift than the true host will be reporting a higher-than-true Hubble constant, and vice-versa for galaxies at lower $z$ than the actual host. In the statistical GW method, the host redshift is not known, and one rather marginalizes over redshifts of many thousands of potential GW hosts, each of which is at a higher or lower redshift than the true host(s). Mathematically, individual posteriors on $H_0$ from each of the many thousands of potential-host galaxies are averaged with the hope that they will average in just such a way to leave an unbiased measurement of $H_0$. In this paper, we study the statistical GW method, and show that there is no reason to believe that such an unbiased measurement will be obtained even under optimistic assumptions about some of the physical variables that enter the mathematical modeling of the method.

Methodology {#sec:method}
===========

Let us proceed by reviewing and implementing the standard approach for the statistical GW method @Chen:2017rfc [@Fishbach:2018gjp; @Soares-Santos:2019irc]. The goal is to compute a posterior in $H_0$ given: a set of distances to GW events, a distance uncertainty for each event, a redshift distribution of *potential* GW-event host galaxies, and the uncertainty in the redshift of each potential host galaxy.

First, we can use the GW distances and uncertainties to weight each allowed combination of $z$ and $H_0$ by how likely they are to produce the measured ${{d_{\rm GW}}}$. This will essentially tell us how much we should value the "vote" of each galaxy. Without loss of generality (given that we are presenting difficulties with the method and some of our assumptions likely err on the optimistic side), we model the likelihood of the distance by a Gaussian, $$p({{d_{\rm GW}}}|{d_{\rm L}}(z,H_0))\propto \exp\bigg{[}-\frac{1}{2}\bigg{(}\frac{d_L(z,H_0)-{{d_{\rm GW}}}}{\sigma_{{d_{\rm GW}}}}\bigg{)}^2\bigg{]}
    \label{eq:p_dgw}$$ where ${{d_{\rm GW}}}$ is the maximum-likelihood distance measured by the GW waveform, $\sigma_{{d_{\rm GW}}}$ is its associated uncertainty, and $d_L(z,H_0)$ is a theoretically calculated luminosity distance. We assume an $H_0$ prior of $H_0\in[40,100]$ and a fiducial distance uncertainty of $\sigma_{{d_{\rm GW}}}= 20\%$ (that is, $\sigma_{{d_{\rm GW}}}/{{d_{\rm GW}}}=0.2$). We will vary some of these inputs in the tests that follow.

Second, we can use the galaxy redshift information to tally up the number of "votes" from each redshift. We model the redshift of each $i$th potential host galaxy as a Gaussian with photometric redshift uncertainty $\sigma_z^i$ centered on its maximum likelihood redshift $\bar{z}^i$. We assume a redshift uncertainty $\sigma_z^i = 0.013(1+\bar{z}^i)^3$ with a lower limit of $\sigma_z^i = 0.015$, adopted from Ref. [@Soares-Santos:2019irc]. Once each galaxy's distribution is computed, the total distribution is the sum over all galaxies, $$p(z) \propto \frac{1}{N_{\rm gal}} \frac{r^2(z)}{H(z)} \sum_i^{N_{\rm gal}} \exp \bigg{[}-\frac{1}{2}\bigg{(}\frac{\bar{z}^i-z}{\sigma_z^i}\bigg{)}^2\bigg{]}
    \label{eq:p_z}.$$ Note that we include a $r^2(z)/H(z)$ correction (where $r(z)$ is the comoving distance) to account for the fact that the photometric redshift distribution of each galaxy is not truly Gaussian, but rather favors higher redshifts due to volume considerations.

When computing Eq. (\[eq:p\_z\]) it is important to specify the distribution from which we are drawing our potential host galaxies. In the fiducial case we assume that the radial localization region is volume-limited, which at $z\ll 1$ implies that the number density of galaxies scales as $$dN(z)\propto z^2dz.      
\label{eq:Nz}$$ Hence we re-write Eq. (\[eq:p\_z\]) in the integral form as $$p(z) \propto \frac{r^2(z)}{H(z)} \int_{{z_{\rm min}}}^{{z_{\rm max}}}\exp \bigg{[}-\frac{1}{2}\bigg{(}\frac{z'-z}{\sigma(z')}\bigg{)}^2\bigg{]} \,N(z') dz',
    \label{eq:pz_int}$$ where ${{z_{\rm min}}}$ and ${{z_{\rm max}}}$ are the redshift bounds, within which we consider a galaxy to be a potential host. We henceforth refer to this region ${{z_{\rm min}}}<z<{{z_{\rm max}}}$ as the "redshift window". Also note that we retain a general $N(z)$ in Eq. (\[eq:pz\_int\]) because we will later test the effect of varying $N(z)$.

For specificity and without loss of generality, we assume a redshift window of $0.10\leq z \leq0.15$. We choose this window because we will later test the effect of varying the redshift window, and we want to have space to vary it higher and lower while staying within the low-$z$ approximation regime. There is nothing unique about our fiducial choice here, and we have verified that our results hold when we make alternate choices for the redshift window.

Now we are ready to combine all of our GW and EM information by convolving Eq. (\[eq:p\_dgw\]) with the photo-z distribution $p(z)$ to give us the likelihood $p({{d_{\rm GW}}},{d_{\rm EM}}|H_0)$. A simple application of Bayes' theorem then gives us the Bayesian posterior on $H_0$, $$p(H_0 | {{d_{\rm GW}}},{d_{\rm EM}}) \propto \ \frac{\int p({{d_{\rm GW}}}|{d_{\rm L}}(z,H_0))p(z){\rm d}z}{\int p(z){\rm d}z}.
    \label{eq:posterior}$$ This expression agrees with previous work @Chen:2017rfc assuming the most optimistic detection conditions, which greatly simplifies the denominator. More generally, the denominator (commonly denoted $\beta(H_0)$ in the literature) can depend on the detection probabilities of both the GW events and the potential host galaxies (for a full explanation, we recommend Appendix A of Ref. @Fishbach:2018gjp). Note this important distinction: knowledge of the potential host galaxies that enter in $p(z)$ (which may well be accurately known) is separate from knowledge of the GW source distribution (which is varied in Fig. \[fig:windows\]a).

\[t\] ![image](Images/pH0.pdf){width="0.47\linewidth"} ![image](Images/pH0_windows.pdf){width="0.47\linewidth"}

To test if this posterior is biased or not, we compute the combined posterior of 50 GW events. Each event is drawn from the Eq. (\[eq:Nz\]) distribution within the range $0.10\leq z \leq0.15$. Next, the event redshift is converted into a distance using Eq. (\[eq:dL\]) assuming $H_{0,{\rm true}}=70{\,{\rm km/s/Mpc}}$; this is the $H_0$ we expect to recover if the poster is unbiased. Finally, the distance is re-drawn from a Gaussian centered on the true distance and with width $\sigma_{{d_{\rm GW}}}$ to simulate the "observed" GW distance. Although our fiducial redshift window coincides with the host redshift range (both $0.10\leq z \leq0.15$), we allow the former to vary in principle. Specifically, in some of our tests presented below, we hold the host redshift range fixed and vary the redshift window, with the constraint that the redshift window always completely contains the host redshift range.

For simplicity, we ignore some known real-world complications. We assume that the likelihood in Eq. (\[eq:p\_dgw\]) is Gaussian and, further, characterized by a constant uncertainty per event $\sigma_{{d_{\rm GW}}}$. We assume that the photometric galaxy survey is volume-limited (and not, say, magnitude limited). We also do not distinguish between properties of different galaxies in the survey — their mass or luminosity, for example. We assume that the merger rate that leads to GW events is redshift-independent. Finally, we assume that the angular localization is isotropic across the footprint of the survey. We emphasize that these assumptions in no way affect our conclusions below. Incorporating these details in the analysis analytically or numerically may well be possible, but their inclusion would not help improve the accuracy of the recovery of the Hubble constant that we present below.

Results
=======

The left panel of Fig. \[fig:pH0\_sim\] shows the posterior on the Hubble constant in our fiducial analysis. Grey curves show posteriors from each of 50 individual simulated GW events, and the combined posterior (the product of all of the grey curves) is shown in red. The differences in the grey background curves are due to different randomly generated hosts, drawn as explained near the end of our Methodology. It is already clear that there is no particular reason that the biased individual posteriors will produce an unbiased combined posterior (where "unbiased" means peaking at $H_{0,{\rm true}}$). Indeed, the combined $H_0$ posterior is clearly biased in our fiducial analysis.

The analysis choice with the greatest effect on the $H_0$ posterior is the redshift window ${{z_{\rm min}}}\leq z \leq {{z_{\rm max}}}$ that enters in Eq. (\[eq:pz\_int\]). This effect is shown in the right panel of Fig. \[fig:pH0\_sim\]. The red curve matches the fiducial conditions shown in the red curve in the left panel, and the other four curves match all fiducial conditions except for the redshift window. Note, all of the curves in the right panel are the combined posteriors of the same 50 simulated GW events. Each choice of the redshift window leads to a different "voting pattern" of galaxies, and thus to wildly different Hubble constant posteriors, as expected. Most posteriors are biased high, but it is possible for them to be biased low (as one of our test cases shows).

It is possible to choose an unbiased window for any fixed set of other conditions, but such a window is not evident *a priori*. In other words, one can choose a redshift window that leads to an unbiased posterior (for example, one peaking at a slightly higher $H_0$ than the green curve), but such a choice holds only for that specific redshift distribution, distance uncertainty, and host galaxy range. Furthermore, we can only choose a window in the simulated case that we know is unbiased because we know our $H_{0,{\rm true}}$ input, which is a luxury we are not afforded in reality. Already, Fig. \[fig:pH0\_sim\] presents a great challenge for the statistical GW method.

The $H_0$ posterior is also affected by the choice of distribution from which the GW events are drawn. This effect is shown in the left panel of Fig \[fig:windows\]. Here we have chosen three different scenarios: events chosen from a $z^2$ distribution (that is, $dN(z)\propto z\,dz$); a $z^3$ distribution ($dN(z)\propto z^2\,dz$), and a $z^4$ distribution ($dN(z)\propto z^3\,dz$). We again show individual events' posteriors with faint lines (color-coded for each of the three scenarios shown), and the combined posterior for each scenario with a bold curve. As expected, the biases on the Hubble constant are different for different $N(z)$ because a distribution skewed toward selecting high-z galaxies will (for fixed measurements of $d_L$) prefer higher values of $H_0$, and vice versa. While the biases are quantitatively smaller than when varying the redshift window, they are similarly unknowable, since we do not know the distribution from which GW events (as opposed to the source galaxies which "vote") come from. Other works have included assumptions about mass distributions or star formation rates in their $N(z)$ @Mastrogiovanni:2021wsd [@Farr:2019twy; @Ye:2021klk], but it is unclear whether or not they match the true source distribution and therefore may be susceptible to bias.

\[t\] ![image](Images/pH0_distributions.pdf){width="0.47\linewidth"} ![image](Images/pH0_sigmas.pdf){width="0.47\linewidth"}

Of course, the hope is that, even without finding the electromagnetic counterpart of most GW event hosts, precise 3D localization of GW events will lead to sufficiently narrow windows that the true event redshift is effectively selected, thus leading to an unbiased $H_0$. Better localization corresponds to a smaller per-event error $\sigma_{{d_{\rm GW}}}$, and vice versa. In the right panel of Fig. \[fig:windows\] we show the combined posteriors on $H_0$ with $\sigma_{{d_{\rm GW}}}/{{d_{\rm GW}}}$ of 0.1, 0.2 (our fiducial case), and 0.3. Unsurprisingly, the posteriors show a dependence on this input. The problem here is the same as with the redshift window and $N(z)$ choices: the per-event accuracy will not be accurately known. This problem is arguably less severe than with the redshift window and $N(z)$. For example, one could marginalize over $\sigma_{{d_{\rm GW}}}$ with some prior range. We do not pursue investigating this further.

Discussion and Conclusion
=========================

We demonstrated that the fundamental degeneracy between an event redshift and the Hubble constant $H_0$ makes reliable estimates of the latter very difficult without excellent localization of the event in space. Assuming for definitiveness 50 GW events coming from the $dN(z)\propto z^2dz$ distribution and the range $0.10<z<0.15$, we showed that the individual events' posteriors on $H_0$ are quite broad, and generally lead to the combined posterior that is much narrower, but biased relative to true underlying $H_0$. These biases persist, and generally get worse, when we also vary the distribution from which the GW events are drawn and the per-event distance uncertainty.

This leads us to the main challenge for the statistical GW method: it is not that the fiducial choices lead to a biased $H_0$, but rather that the $H_0$ constraints depend on inputs which will likely not be known to sufficient precision to avoid these biases. The dependencies studied in our Results section all furnish such examples. This is fundamentally easy to understand: assuming for the moment the $z\ll 1$ limit for simplicity, we have $$d_L(z, H_0) \simeq \frac{cz}{H_0}.$$ Therefore, even a very accurate measurement of $d_L$ can lead to a reliable constraint on the Hubble constant only if the redshift is known. With many GW sources, this argument still holds provided the redshift *distribution* from which GW sources come from be accurately known. And the latter requirement is generally not satisfied, nor expected to be in the foreseeable future. This is because the physical phenomena that govern GW events take place on scales some ten orders of magnitude smaller than those well understood by first-principles theory ($\sim$Mpc). This makes a reliable first-principles calculation of the radial distribution of GW events a daunting task. Including the mass of merging objects, something that we have ignored in this paper thus rendering it effectively perfectly known, would only complicate the picture.

The most striking illustration of the difficulty of reliably constraining the Hubble constant is shown in the right panel of Fig. \[fig:pH0\_sim\], where we show the dependence of the posterior on $H_0$ on the redshift window. The constraints are, in general, biased. It is possible to select redshift window that leads to an unbiased $H_0$ posterior — however, such a choice would still rely on correct modeling of the GW source distribution. This distribution is, and likely will be, poorly constrained, absent a rather complete theory of the population synthesis of GW events.

Therefore, the statistical GW method has major challenges to overcome to produce unbiased constraints on the Hubble constant.

Acknowledgments
===============

We would like to thank Jim Annis, Daniel Holz, Keith Riles, Marcelle Soares-Santos, and especially Maya Fishbach and Antonella Palmese for useful conversations and feedback. Our work has been supported in part by NASA under contract 19-ATP19-0058. DH has additionally been supported by DOE under Contract No. DE-FG02-95ER40899 an NSF under contract AST-1812961. DH thanks the Humboldt Foundation for support via the Friedrich Wilhelm Bessel award, and the Max-Planck Institute for Astrophysics for hospitality while a large chunk of this work was carried out.

{'source': 'arXiv_out/2112'}