Correlation analysis of stochastic gravitational wave background around 0.1–1 Hz

We discuss prospects for direct measurement of stochastic gravitational wave background around 0.1–1 Hz with future space missions. It is assumed to use correlation analysis technique with the optimal time-delay-interferometry (TDI) variables for two sets of LISA-type interferometers. The signal to noise for detection of the background and the estimation errors for its basic parameters (amplitude, spectral index) are evaluated for proposed missions.

Figure 1

Three vertices (1, 2, 3) and three arms ($L_1$, $L_2$, $L_3$) of the first unit (solid lines). Configuration of the second unit (short-dashed lines) is obtained by 180° rotation of the first one around its center. Labels for vertices and arms of the second ones are transported with this rotation.

Figure 2

The optimal sensitivities for BBO lite (thin solid curve), BBO standard (solid curve), and BBO grand (thick solid curve) configurations. The sensitivities for ($A$, $E$) modes (long-dashed curve) and $T$ mode (short-dashed curve) are also given for BBO standard configuration.

Figure 3

The normalized overlap reduction function $\gamma (f)$ for a starlike constellation. These curves have a scaling parameter $f_{*}=c/(2\pi L)$ that is determined by the arm length.

Figure 4

Dependence of the signal-to-noise ratio on the lower cutoff frequency $f_{\mathrm{cut}}$.

Figure 5

Error ellipses ($2\sigma$; 86% CL) for two-dimensional parameter estimation with different lower cutoff frequencies $f_{\mathrm{cut}}$ (dashed curve: $f_{\mathrm{cut}}=0$, solid curve: $f_{\mathrm{cut}}=0.2\mathrm{Hz}$, the dotted curve: $f_{\mathrm{cut}}=0.3\mathrm{Hz}$). The fisher matrix approach is used. The central frequency $F$ is set at $F=1\mathrm{Hz}$ that is slightly higher than the optimal sensitivity for measuring the background with above two BBO configurations. The true values of the spectrum ${\Omega }_{\mathrm{GW}}(f)$ are ${\Omega }_{\mathrm{GW},F}={10}^{-16}$ and $n=0$.

Figure 6

Same as Fig. 5, but with the central frequencies $F$ at 0.26 Hz (BBO standard) and 0.25 Hz (BBO grand). The two-dimensional covariance matrices are diagonalized with $f_{\mathrm{cut}}=0.2\mathrm{Hz}$.

Figure 7

Dependence of various quantities to the slope $n$. SNR (solid curve) and two errors $\Delta {\alpha }_1$ (dotted curve) and $\Delta {\alpha }_2$ (short-dashed curve) are normalized by their values at $n=0$. The long-dashed curve represents the correlation coefficient $r$. The central frequencies are $F=0.26\mathrm{Hz}$ (BBO standard) and $F=0.25\mathrm{Hz}$ (BBO grand).