Correlation of gravitational wave background noises and statistical loss for angular averaged sensitivity curves

Gravitational wave backgrounds generate correlated noises to separated detectors. This correlation can induce statistical losses to actual detector networks, compared with idealized noise-independent networks. Assuming that the backgrounds are isotropic, we examine the statistical losses specifically for the angular averaged sensitivity curves, and derive simple expressions that depend on the overlap reduction functions and the strength of the background noises relative to the instrumental noises. For future triangular interferometers such as ET and LISA, we also discuss preferred network geometries to suppress the potential statistical losses.

Figure 1

Contour plot for the effective detector number $G_2(K,\gamma )$ defined for two L-shaped interferometers with the same instrumental noise spectrum. Their background noises have correlation, characterized by the ORF $\gamma$. The parameter $K$ represents the ratio between the background and instrumental noises. We have $1\le G_2(K,\gamma )\le 2$ with $G_2(0,\gamma )=G_2(K,0)=2$.

Figure 2

The orientations of the two effective interferometers $A$ and $E$ made from the symmetric data streams $X$, $Y$, and $Z$. The instrumental noises of $A$ and $E$ have no correlation, and their ORF is ${\gamma }_{AE}=0$.

Figure 3

Adjusted orientations of the effectively L-shaped interferometers $A$, $E$, $A^{\prime }$ and $E^{\prime }$. These four are generated from the two triangular detectors $U_1$ and $U_2$ tangential to a sphere. The dashed line shows the geodesic (great circle) connecting the two triangle on the sphere. The effective interferometers $A$ and $A^{\prime }$ have arms parallel or perpendicular to the geodesic, and $E$ and $E^{\prime }$ are misaligned by 45°. Due to the geometrical symmetry, we have the ORFs ${\gamma }_{AE}={\gamma }_{A^{\prime }{E}^{\prime }}={\gamma }_{A{E}^{\prime }}={\gamma }_{A^{\prime }E}=0$.

Figure 4

Contour plot for the minimum value $G_{\mathrm{\Delta }\min }({\gamma }_{A{A}^{\prime }},{\gamma }_{E{E}^{\prime }})$ of the statistical gain showing the effective number of triangular units. This function is symmetric with respect to the two arguments ${\gamma }_{A{A}^{\prime }}$ and ${\gamma }_{E{E}^{\prime }}$. We have $G_{\mathrm{\Delta }\min }(0,0)=2$ and $G_{\mathrm{\Delta }\min }(1,1)=1$.

Figure 5

Contour plots for the ORFs $|{\gamma }_{A{A}^{\prime }}|$ (upper) and $|{\gamma }_{E{E}^{\prime }}|$ (lower). The interferometers $(A,E,A^{\prime },E^{\prime })$ are virtually aligned by using the geodesic as in Fig. 3. The parameter $\beta$ represents their angular separation measured from the center of the contact sphere. We use the scaled frequency $\eta =f/f_R$ with $f_R\equiv c/(2\pi R)$.

Figure 6

Contour plot for the lower bound $G_{\mathrm{\Delta }\min }({\gamma }_{A{A}^{\prime }},{\gamma }_{E{E}^{\prime }})$ defined in Eq. (43). We have $f_R=7.58\mathrm{Hz}$ for ground based detectors and $f_R=0.28\mathrm{mHz}$ for LISA-like detectors.

Figure 7

Geometrical configuration of two LISA-like triangular units $U_1$ and $U_2$. The two units move nearly on the ecliptic plane at the distance of $\sim 1\mathrm{a}.\mathrm{u}.$ from the Sun with the phase difference $\theta$. The detector planes are inclined to the orbital (ecliptic) plane by 60°. Their envelope (green belt) are tangential to a virtual sphere of radius $R=1.15\mathrm{a}.\mathrm{u}.$. From the center O of the virtual sphere, the two units are separated by the angle $\beta$. The relation between $\theta$ and $\beta$ is given in Eq. (57).

Figure 8

Contour plot of the lower bound $G_{\mathrm{\Delta }\min }({\gamma }_{A{A}^{\prime }},{\gamma }_{E{E}^{\prime }})$ for two LISA-like triangular units. In contrast to Fig. 6, the vertical axis is the orbital phase difference $\theta$, instead of $\beta$ (see Fig. 7) The proposed LISA-Taiji network has $\theta =40°$.

Figure 9

The detector noise spectrum $N_D(f)$ of LISA and the estimated Galactic confusion noise spectrum $N_C(f)$ after 4 yr integration [18]. These spectra are presented for the single L-shaped interferometers (corresponding to the $A$ or $E$ modes) without the angular averaging. The relative noise strength is given by $K(f)=N_C(f)/N_D(f)$.

Figure 10

The statistical gain $G_{\mathrm{\Delta }}(K(f),{\gamma }_{A{A}^{\prime }}(f),{\gamma }_{E{E}^{\prime }}(f))$ for two LISA-like units. The vertical axis represents the orbital phase difference $\theta$.