Cosmology with the lights off: Standard sirens in the Einstein Telescope era

We explore the prospects for constraining cosmology using gravitational-wave (GW) observations of neutron-star binaries by the proposed Einstein Telescope (ET), exploiting the narrowness of the neutron-star mass function. This builds on our previous work in the context of advanced-era GW detectors. Double neutron-star (DNS) binaries are expected to be one of the first sources detected after “first-light” of Advanced LIGO. DNS systems are expected to be detected at a rate of a few tens per year in the advanced era, but the proposed ET could catalog tens, if not hundreds, of thousands per year. Combining the measured source redshift distributions with GW-network distance determinations will permit not only the precision measurement of background cosmological parameters, but will provide an insight into the astrophysical properties of these DNS systems. Of particular interest will be to probe the distribution of delay times between DNS-binary creation and subsequent merger, as well as the evolution of the star-formation rate density within ET’s detection horizon. Keeping $H_0$, ${\Omega }_{m,0}$ and ${\Omega }_{\Lambda ,0}$ fixed and investigating the precision with which the dark-energy equation-of-state parameters could be recovered, we found that with ${10}^5$ detected DNS binaries, we could constrain these parameters to an accuracy similar to forecasted constraints from future $\mathrm{CMB}+\mathrm{BAO}+\mathrm{SNIa}$ measurements. Furthermore, modeling the merger delay-time distribution as a power-law ($\propto t^{\alpha }$) and the star-formation rate density as a parametrized version of the Porciani and Madau SF2 model, we find that the associated astrophysical parameters are constrained to within $\sim 10\%$. All parameter precisions scaled as $1/\sqrt{N}$, where $N$ is the number of cataloged detections. We also investigated how parameter precisions varied with the intrinsic underlying properties of the Universe and with the distance reach of the network (which is affected, for instance, by the low-frequency cutoff of the detector). We also consider various sources of distance-measurement errors in the third-generation era and how these can be folded into the analysis.

Figure 1

Comparison of the different noise curves for AdLIGO (high-power zero-detuning) [40], the initial Einstein Telescope noise curve based on conventional techniques, ET-B [33], the initial xylophone noise curve, ET-C [34], and the improved, more realistic xylophone noise curve, ET-D [29]. These noise curves are for one 10 km right-angled interferometer.

Figure 2

Merger-rate densities computed for the reference cosmology (solid line) and for cosmological parameters chosen randomly from within the prior range (red crosses).

Figure 3

Marginalized 2D posterior distributions for the reference catalog of 4500 detections. Only those 2D distributions showing correlations between parameters are shown. The reference parameters are ${\mu }_{\mathrm{NS}}=1.35M_{\odot }$, ${\sigma }_{\mathrm{NS}}=0.06M_{\odot }$, $w_0=-1$, $w_a=0$, $\alpha =-1$ and ${\beta }_1={\beta }_2=3.4$.

Figure 4

Marginalized 1D posterior distributions for the reference catalog of 4500 detections. Dotted lines indicate the boundaries of the 95% and 68% confidence intervals. The reference parameters are ${\mu }_{\mathrm{NS}}=1.35M_{\odot }$, ${\sigma }_{\mathrm{NS}}=0.06M_{\odot }$, $w_0=-1$, $w_a=0$, $\alpha =-1$ and ${\beta }_1=3.4$.

Figure 5

95% confidence intervals of the 1D marginalized distributions relative to those of the 4500-event reference catalog, shown as a function of the number of cataloged events. The same intrinsic parameters of the underlying distributions are used to create the mock catalogs. The expected $\sim 1/\sqrt{N}$ relationship is overlaid on the plot.

Figure 6

The variation of measurement precision with different choices of the intrinsic parameters of the underlying distributions. One parameter is varied at a time, and in the interest of testing how the precision of parameter recovery is affected by the underlying distribution of events, all catalogs are generated with the same number of events (4500 to match the reference catalog). Each point in each panel represents the average 95% confidence interval width of 3 realizations of the catalog.

Figure 7

We show the variation of the expected detection rate as the SNR threshold, ${\rho }_0$, is raised. This can also be interpreted as lowering the characteristic distance reach of the network. Since $\rho$ scales as $1/D_L$ [see Eq. (5)], and the difference between the luminosity distance and radial comoving distance becomes smaller at lower redshift, one would expect that at high enough values of ${\rho }_0$, the comoving detection volume (and hence detection rate) would scale as $1/{\rho }_0^3$. This is approximately valid for ${\rho }_0\gtrsim 15$.

Figure 8

The reduction of the characteristic distance reach associated with raising the low-frequency cutoff, $f_l$, of 3rd-generation detectors. This can also be interpreted, via Eq. (28), as raising the network SNR threshold. The figures were produced using the ET-D noise curve [32].

Figure 9

The variation of the parameter measurement precisions with the network SNR threshold. The left panel shows precisions, characterized by the width of the 68% confidence intervals, for the merger-rate density parameters, while the right panel shows precisions, characterized by the 95% confidence intervals used elsewhere, for all other parameters. We use the narrower confidence intervals for the merger-rate parameters to mitigate the effect of poor sampling in the low-$\alpha$ region which was observed in some AM-MCMC chains in this analysis. All catalogs contain the same numbers of events at each threshold value, which, as in the previous subsection, is 4500 to match the reference catalog.

Figure 10

We show the redshift distribution of the DNS merger-rate density for various choices of the power-law index of the delay-time distribution, $\alpha$ (all other parameters are fixed at their reference values). The merger-rate density is relatively featureless beyond $z\sim 2.5$, making it difficult for our analysis (which is insensitive to linear scalings of the merger-rate density) to discriminate between values of $\alpha$ in this range. Overlaid on this figure, we show the redshift distribution of detections for various SNR thresholds.

Figure 11

The redshift dependence of the sensitivity of the luminosity distance to $w_0$. This parameter has a very weak intrinsic impact on $D_L$ at low redshifts, while distance errors from instrumental noise and weak-lensing dominate at higher redshifts. This results in a redshift “sweet spot,” where these effects are minimized for lines of constant SNR. We also plot the individual $\Delta w_0$ values calculated for the reference catalog events.