Cosmography Using Strongly Lensed Gravitational Waves from Binary Black Holes

Third generation gravitational wave (GW) detectors are expected to detect millions of binary black hole (BBH) mergers during their operation period. A small fraction of them ($\sim 1\%$) will be strongly lensed by intervening galaxies and clusters, producing multiple observable copies of the GW signals. The expected number of lensed events and the distribution of the time delay between lensed images depend on the cosmology. We develop a Bayesian analysis method for estimating cosmological parameters from the detected number of lensed events and their time delay distribution. The expected constraints are comparable to that obtained from other cosmological measurements, but probing a different redshift regime ($z\sim 10$) that is not explored by other probes.

Figure 1

Left: Expected number of lens pairs for different values of $\overrightarrow{\mathrm{\Omega }}$ in flat $\mathrm{\Lambda }\mathrm{CDM}$ model, assuming a merger rate $R=5\times {10}^5{\mathrm{yr}}^{-1}$ and observation time period $T_{\mathrm{obs}}=10\mathrm{yrs}$. Right: The strong-lensing time-delay distributions for different values of cosmological parameters: Increasing $H_0$ (${\mathrm{\Omega }}_m$) shifts the peak of the distribution towards smaller (larger) time-delay values. Dashed lines show the actual distributions $p(\mathrm{\Delta }t|\overrightarrow{\mathrm{\Omega }})$ while the solid lines show the distribution of time delays observable in a period of 10 yr.

Figure 2

Left panel: Expected posterior distributions (68% and 95% credible regions) of $H_0$ and ${\mathrm{\Omega }}_m$ computed from the time delay distribution and the number of lensed events separately, along with the combined posterior (shown in orange). We assume a BBH merger rate $R=5\times {10}^5{\mathrm{yr}}^{-1}$ and total observation time period $T_{\mathrm{obs}}=10\mathrm{yrs}$. The true cosmology (dashed cross hairs) is recovered within the 68% credible interval (orange shade), with $H_0=67.8\pm 1.1\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and ${\mathrm{\Omega }}_m=0.3142\pm 0.0056$. Right panel: A comparison of the combined posterior obtained from GW lensing with the same obtained from CMB observations by Planck. While the orange contours assume that the ${\sigma }_8$ parameter is well measured from other observations, the gray contour corresponds to 95% credible region of the posterior marginalized over the ${\sigma }_8$ parameter.

Figure 3

Expected posterior distributions of $H_0$ and ${\mathrm{\Omega }}_m$ from a 10-yr observation period, assuming different values for the merger rate $R$ (shown in the legend). A lower merger rate (producing a smaller number of lensed events) will result in less precise estimates of the cosmological parameters.

Figure 4

Expected posterior distributions (68% and 95% credible regions) of ${\mathrm{\Omega }}_m$ and $w_0$ of the $w\mathrm{CDM}$ model computed from the time delay distribution and the number of lensed events (jointly). We assume a BBH merger rate $R=5\times {10}^5{\mathrm{yr}}^{-1}$ and total observation time period $T_{\mathrm{obs}}=10\mathrm{yr}$. The true cosmology (dashed cross hairs) is recovered within the 68% credible interval (orange shade), with $w_0=-1.52_{-0.12}^{+0.16}$ and ${\mathrm{\Omega }}_m=0.203\pm 0.001$.