Luminosity distance uncertainties from gravitational wave detections of binary neutron stars by third generation observatories

A new generation of terrestrial gravitational wave detectors is currently being planned for the next decade, and it is expected to detect most of the coalescences of compact objects in the Universe with masses up to a thousand times the solar mass. Among the several possible applications of current and future detections, we focus on the impact to the measure of the luminosity distance of the sources, which is an invaluable tool for constraining the cosmic expansion history of the Universe. We study two specific detector topologies, triangular and $L$-shaped, by investigating how the topology and relative orientation of up to three detectors can minimize the uncertainty measure of the luminosity distance. While the precision of the distance measurement is correlated with several geometric angles determining the source position and orientation, focusing on the bright standard sirens and assuming a redshift to be measured with high accuracy, we obtain analytic and numerical results for its uncertainty, depending on the type and number of detectors composing a network, as well as on the inclination angle of the binary plane with respect to the wave propagation direction. We also analyze the best relative location and orientation of two third generation detectors to minimize the luminosity distance uncertainty, showing that prior knowledge of the inclination angle distribution plays an important role in the precision recovery of luminosity distance and that a suitably arranged network of detectors can drastically reduce the uncertainty measure, approaching the limit imposed by lensing effects intervening between source and detector at a redshift $z\gtrsim 0.7$.

Figure 1

Expected redshift distribution of bright standard sirens assuming an electromagnetic counterpart is detected by Theseus [37], compared with observed star formation rate [40].

Figure 2

Left: Luminosity distance reach for equal mass, nonspinning system, assuming fundamental $l=m=2$ mode only, for optimally oriented binaries, given the spectral noise density $S_n$ [45, 46] for CE and [ET-D] from [47] for ET. The mass and luminosity distance of GW170817 are highlighted, as well as the mass region where BNSs are expected. Right: Dimensionless noise characteristic strain $h_c$, defined in terms of spectral noise $h_c\equiv \sqrt{f{S}_n}$ for $L$-shaped CE and triangle-shaped ET.

Figure 3

Schematic representation of detector geometry and of radiation frame.

Figure 4

Examples of two-dimensional PDF for $\iota$ vs $d_L$ for a single CE-like, $L$-shaped interferometer (top) and a triangle-shaped one (bottom) for source parameters giving $\epsilon =0.0089$ [75]. Note that the volumetric prior at recovery tends to disfavor $\iota \to \pi /2$ for $L$-shaped detectors.

Figure 5

Values of $\epsilon$ (left) and $\sigma$ (right) for a single ET detector (top, ET location marked with a triangle) and for an ET-CE network (bottom, CE location marked with a diamond).

Figure 6

Left: Distribution of $\sigma$ and $\epsilon$ values, defined in Eq. (20), for a triangle-shaped interferometer. Right: Points in the sky presenting bimodality are confined to the plane of the detector, where blind directions to individual interferometer appear. Operationally, we defined bimodality to be present when the ratio (smaller or equal to 1) of the height of the peaks of the $\iota$ PDF is larger than the PDF value at $\iota =\pi /2$.

Figure 7

Scatter values and averages for relative uncertainty of $d_L$ (top) and absolute one of $\iota$ (bottom) as a function of $\iota$ for various distances for an ET-like detector (left) and for a single CE one (right), for 300 simulations at each distance. Continuous lines are averages over intervals of 0.1 radians in $\iota$. Note the dip in $d_L$ uncertainty for $\iota \to \pi /2$ in the ET case. The points where CE outperforms ET in $\mathrm{\Delta }{d}_L$ are due to the better spectral noise sensitivity of the detector, see Fig. 2, hence higher SNR, see Fig. 8. Most of the recovered $\iota$ for CE present bimodality ($\mathrm{\Delta }\iota \sim 60°$), bimodality that happens far more rarely for ET, as shown by the red dots in the left plot of Fig. 6 compared to the majority of gray points in the bottom right plot here clustering around $\mathrm{\Delta }\iota \sim 60°$. For an ET-like detector, there is no dip in $d_L$ uncertainty for $z\sim 1$ as the SNR decrease for $\iota \to \pi /2$ moves the signal below the $\mathrm{SNR}=8$ threshold, whereas for CE injections at $z=1$ are just above threshold.

Figure 8

SNR as a function of $\iota$ for various distances for a triangle interferometer (left) and for a single $L$-shaped detector (right).

Figure 9

Examples of two-dimensional PDF for $\psi$ vs $d_L$ for a single triangle-shaped interferometer showing that $\psi$ determination accuracy improves as $\iota \to \pi /2$, as expected from Eq. (21), whose $\psi$-dependent term is maximum for $\upsilon =1$, which is $\iota =\pi /2$. The value for $\epsilon$ is the same as in Fig. 4.

Figure 10

The three inclination angle distributions for $\iota$ used in injections, dubbed “isotropic”, “smooth cutoff”, “hard cutoff”.

Figure 11

Error in distance determination averaged over source location, as a function of the angular distance between two 3G detectors. Sources are distributed isotropically in the sky before the ${\mathrm{SNR}}_i>8$ cut in each detector. Source inclinations are distributed according to smooth-cutoff function, see Fig. 10, Bayesian prior for $\iota$ at recovery is isotropic. The bottom right plot refers to two ETs, the others to two CEs with sources respectively at $z=0.1$, 0.55, 1.

Figure 12

Same as in Fig. 11 with source inclinations distributed isotropically on the two-sphere.

Figure 13

Luminosity distance uncertainty of a network of three detectors ($\mathrm{ET}+\mathrm{CE}+\mathrm{ET}$) averaged over source location, depending on the location of the second ET-like detector. Empty circles denote the 90 trial locations of the third detector, while the first ET and the CE detector are denoted, respectively, by a yellow triangle and red diamond. The three figures refer to a redshift $z=0.1$, 0.5, 1, respectively, moving clockwise from top left.

Figure 14

Average distance measure error for various network of ET- and CE-like detectors as a function of redshift. The top graph shows the case of anisotropic distribution of inclination angles with prior at recovery. The bottom one shows the case of isotropic prior at recovery for all cases of $\iota$ injection distribution. See Fig. 10 for $\iota$ injection distributions (hard, smooth, iso).

Figure 15

Quadratic sum of pattern function ${(f_{+}^2+f_{\times }^2)}^{1/2}$ for each of the three component of a triangular interferometer with arms at 60°. The blind spots of each individual interferometer lie in the plane of the detector.

Figure 16

Values of $\sigma$ and $\epsilon$ for various networks of 2G detectors, spectral noise curves used are displayed in Fig. 17.

Figure 17

Left: Characteristic strain $\sqrt{f{S}_n}$ used to generate maps in Fig. 16, from [86]. Right: Examples of TaylorF2 (dashed) and IMRPhenomD (solid) waveforms for total mass $3M_{\odot }$ and equal binary component masses.

Figure 18

Same as in Fig. 14, with uncertainties in luminosity distance of 2G detections from [1, 2, 3] and standard sirens from [80] added.

Figure 19

Histograms representing the dispersions of $\mathrm{\Delta }{d}_L/d_L$ measurements for sample 1,2, and 3 detector case of Fig. 14. Hard cutoff refers to the $\iota$ distributions of the injections as per Fig. 10.

Figure 20

Sky distribution of above threshold events for the three detector network ($\mathrm{ET}+\mathrm{CE}+\mathrm{ET}$) of Sec. 3c2, showing how threshold cut selects source sky regions.