Inference of cosmological parameters from gravitational waves: Applications to second generation interferometers

The advanced worldwide network of gravitational waves (GW) observatories is scheduled to begin operations within the current decade. Thanks to their improved sensitivity, they promise to yield a number of detections and thus to open new observational windows for astronomy and astrophysics. Among the scientific goals that should be achieved, there is the independent measurement of the value of the cosmological parameters, hence an independent test of the current cosmological paradigm. Because of the importance of such a task, a number of studies have evaluated the capabilities of GW telescopes in this respect. However, since GW do not yield information about the source redshift, different groups have made different assumptions regarding the means through which the GW redshift can be obtained. These different assumptions imply also different methodologies to solve this inference problem. This work presents a formalism based on Bayesian inference developed to facilitate the inclusion of all assumptions and prior information about a GW source within a single data analysis framework. This approach guarantees the minimization of information loss and the possibility of including naturally event-specific knowledge (such as the sky position for a gamma ray burst–GW coincident observation) in the analysis. The workings of the method are applied to a specific example, loosely designed along the lines of the method proposed by Schutz in 1986, in which one uses information from wide-field galaxy surveys as prior information for the location of a GW source. I show that combining the results from few tens of observations from a network of advanced interferometers will constrain the Hubble constant $H_0$ to an accuracy of $\sim 4\%–5\%$ at 95% confidence.

Figure 1

The high-power, zero-detuning noise curve for advanced LIGO and the BNS-optimized advanced Virgo noise curve.

Figure 2

Properties of the 1000 GW events catalog. Top: distribution of the distances $D_L$ calculated from a $\Lambda \mathrm{CDM}$ universe and the galaxy redshift. Center: distribution of the redshifts $z$ of the sources (solid line) compared to the overall SDSS redshift distribution (dashed line). Bottom: distribution of the network signal-to-noise ratio ${\rho }_{\mathrm{network}}$ for the HLV network (solid line), the HLVJ network (dotted line), and the HLVIJ network (dashed line). The distributions for $z$ and for ${\rho }_{\mathrm{network}}$ have been scaled to facilitate the comparison.

Figure 3

Two-dimensional posterior distributions for the sky position of a sample source as observed by the HLV network (left), by the HLVJ network (center), or by the HLVJI network (right). The source signal-to-noise ratios for this particular injection are H:7.1, L:7.5, V:8.5, J:7.1, I:7.0. The remaining parameters are given in Table . In both panels, the cross indicates the location of the GW real host. The (colored) dots indicate the galactic population identified as consistent with the GW event, color coded according to their redshift, while the black dots indicate all the galaxies within the field of view. Left: two-dimensional posterior distribution for $\alpha$ and $\delta$ for the HLV network for which ${\rho }_{\mathrm{network}}\simeq 13.4$. The contours indicate the 95% and 75% confidence intervals. The 95% confidence area is equal to $14.8{\mathrm{deg}}^2$, giving a total number of possible hosts of 600. Center: two-dimensional posterior distribution for $\alpha$ and $\delta$ for the HLVJ network for which ${\rho }_{\mathrm{network}}\simeq 15.1$. The contours indicate the 95% and 75% confidence intervals. The 95% confidence area is equal to $3.9{\mathrm{deg}}^2$, within which the number of possible hosts identified is 339. Right: two-dimensional posterior distribution for $\alpha$ and $\delta$ for the HLVJI network for which ${\rho }_{\mathrm{network}}\simeq 17.7$. The contours indicate the 95% and 75% confidence intervals. The 95% confidence area is equal to $2.2{\mathrm{deg}}^2$, within which the number of possible hosts identified is 230.

Figure 4

Top: joint two-dimensional posterior distributions for the redshift and for $h$ of the same source as in Fig. 3 as observed by the HLV network (left), by the HLVJ network ( center), or by the HLVJI network (right). The star indicates the real value of the redshift and of $h$. Bottom: joint two-dimensional posterior distributions for $h$ and $\cos \iota$. The star indicates the injection value. In all panels, the contours indicate the 95%, 75%, 50%, 25%, and 5% confidence intervals. In all cases the strong correlation between $h$ and $z$ is evident, which is an obvious consequence of Eq. (7), and between $h$ and $\cos \iota$. This last degeneracy is just the translation of the known $D_L-\iota$ degeneracy that ultimately is the limiting factor in the determination of the parameters appearing in the amplitude of the GW. Moreover, the distributions are multimodal, corresponding to the different combinations of $h$, $z$, and $\cos \iota$ that give constant $D_L$. The accuracy of the estimation of the redshift is similar for all networks. However, the increase in resolving power for $\cos \iota$ when more detectors are considered is noticeable.

Figure 5

Evolution of the medians (solid line) and 95% confidence intervals evolution for $\overrightarrow{\Omega }$ as a function of the number of events included in the computation of the joint posterior distributions. Each data point, and relative error bar, is the result of averaging over 20 independent realizations of the GW events catalog. The left column presents the result for the HLV network, the center column present the result for the HLVJ network, while the right column is relative to the HLVJI network. In all figures the (red) dashed line is the injected value. The irregular jumps are due to the finite bin size. The convergence of ${\Omega }_k$ toward the injected value 0 is artificial and due to the prior probability choice for the analysis. See text for a discussion.