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Precision of Hubble constant derived using black hole binary absolute distances and statistical redshift information

Chelsea L. MacLeod, Craig J. Hogan

TL;DR

The paper addresses measuring the Hubble constant $H_0$ from gravitational-wave standard sirens when EM counterparts do not provide redshift. It introduces a statistical redshift method that uses galaxy clustering within LISA error boxes and per-event likelihoods $\ln{\cal L}_j(H_0)$, combined as a joint $\ln{\cal L}(H_0)$. Using mock SDSS DR6 catalogs, the study demonstrates that ~20 EMRIs out to $z\approx 0.5$ can yield $H_0$ with precision $<1\%$, enabling direct tests of cosmic acceleration at $z<0.5$. The method provides a low-bias, cross-validated cosmological probe with distinct systematics from traditional distance ladders.

Abstract

Measured gravitational waveforms from black hole binary inspiral events directly determine absolute luminosity distances. To use these data for cosmology, it is necessary to independently obtain redshifts for the events, which may be difficult for those without electromagnetic counterparts. Here it is demonstrated that certainly in principle, and possibly in practice, clustering of galaxies allows extraction of the redshift information from a sample statistically for the purpose of estimating mean cosmological parameters, without identification of host galaxies for individual events. We extract mock galaxy samples from the 6th Data Release of the Sloan Digital Sky Survey resembling those that would be associated with inspiral events of stellar mass black holes falling into massive black holes at redshift z ~ 0.1 to 0.5. A simple statistical procedure is described to estimate a likelihood function for the Hubble constant H_0: each galaxy in a LISA error volume contributes linearly to the log likelihood for the source redshift, and the log likelihood for each source contributes linearly to that of H_0. This procedure is shown to provide an accurate and unbiased estimator of H_0. It is estimated that a precision better than one percent in H_0 may be possible if the rate of such events is sufficiently high, on the order of 20 to z = 0.5.

Precision of Hubble constant derived using black hole binary absolute distances and statistical redshift information

TL;DR

The paper addresses measuring the Hubble constant from gravitational-wave standard sirens when EM counterparts do not provide redshift. It introduces a statistical redshift method that uses galaxy clustering within LISA error boxes and per-event likelihoods , combined as a joint . Using mock SDSS DR6 catalogs, the study demonstrates that ~20 EMRIs out to can yield with precision , enabling direct tests of cosmic acceleration at . The method provides a low-bias, cross-validated cosmological probe with distinct systematics from traditional distance ladders.

Abstract

Measured gravitational waveforms from black hole binary inspiral events directly determine absolute luminosity distances. To use these data for cosmology, it is necessary to independently obtain redshifts for the events, which may be difficult for those without electromagnetic counterparts. Here it is demonstrated that certainly in principle, and possibly in practice, clustering of galaxies allows extraction of the redshift information from a sample statistically for the purpose of estimating mean cosmological parameters, without identification of host galaxies for individual events. We extract mock galaxy samples from the 6th Data Release of the Sloan Digital Sky Survey resembling those that would be associated with inspiral events of stellar mass black holes falling into massive black holes at redshift z ~ 0.1 to 0.5. A simple statistical procedure is described to estimate a likelihood function for the Hubble constant H_0: each galaxy in a LISA error volume contributes linearly to the log likelihood for the source redshift, and the log likelihood for each source contributes linearly to that of H_0. This procedure is shown to provide an accurate and unbiased estimator of H_0. It is estimated that a precision better than one percent in H_0 may be possible if the rate of such events is sufficiently high, on the order of 20 to z = 0.5.

Paper Structure

This paper contains 3 sections, 6 equations, 3 figures, 2 tables.

Table of Contents

  1. Precision cosmology from black hole binaries
  2. Use of statistical redshifts for cosmological parameter estimation from gravitational waveforms
  3. Mock LISA/EMRI host-galaxy samples from SDSS

Figures (3)

  • Figure 1: Simulated LISA error boxes for one synthetic interferometer. In each box the chosen source galaxy lies at the center, indicated by an asterisk, and dots represent all other galaxies in the box. Boxes are scaled to represent events at $z_{LISA} = Az_{SDSS}$, where $A$ is specified above each box, and the pretend LISA angular coordinates, given by $\theta_{LISA} = (1/A)\theta_{SDSS}$, are plotted.
  • Figure 2: Summed histograms (bin size = 0.1) with Gaussian fits overlayed. Top row: one synthetic interferometer case; bottom row: two synthetic interferometers. The true value of $H_0$ was assumed to be 70 km s$^{-1}$ Mpc$^{-1}$. Gaussian fits were generated using a Levenberg-Marquardt technique to iteratively search for a best least-squares fit to $\ln{\cal L}(H_0)$. The starting value for the mean was a random value between 68 and 72, and those for the width and area were both 1. Vertical axes are in arbitrary units.
  • Figure 3: Summed histograms for one synthetic interferometer, scaled to represent events at $z_{LISA} = 2z_{SDSS}$ (top row), $z_{LISA} = 3z_{SDSS}$ (second row), $z_{LISA} = 4z_{SDSS}$ (third row), and $z_{LISA} = 5z_{SDSS}$ (bottom row). Vertical axes are in arbitrary units.