Quest for circular polarization of a gravitational wave background and orbits of laser interferometers in space

We show that an isotropic component of circular polarization of a stochastic gravitational wave background can be explored by contriving configuration of multiple laser interferometers for correlation analysis. For the proposed BBO mission, the circular polarization degree $\Pi$ can be measured down to $\Pi \sim 0.08({\Omega }_{\mathrm{GW}}/{10}^{-15}{)}^{-1}(\mathrm{SNR}/5)$ with slightly ($\sim 10\%$) sacrificing the detection limit for the total intensity ${\Omega }_{\mathrm{GW}}$. This might allow us to detect a signature of parity violation in the very early universe.

Figure 1

Three spacecrafts (SCs) of a LISA-type system (B1) are shown with circles that are put on the $XY$-plane. The barycenter of three SCs is at the origin $(X,Y,Z)=(0,0,0)$. The bottom figure shows the projection of the structures to the $XZ$-plane. At low frequency limit the responses of two TDI modes $A$ and $E$ can be effectively regarded as those of two $L$-shaped interferometers with shown orientations. On the coordinate system $(X,Y,Z)$ fixed to the first system B1, the center of the second system B2 moves on the dotted curves for the specific parameter choice $(\alpha ,d)=(0,0.24)$. A typical snapshot of B1 (solid line) and B2 (dashed line) is shown on the right side. The orientations of two effective $L$-shaped interferometers for $A^{\prime }$ and $E^{\prime }$ modes are also given.

Figure 2

Configuration of two orbital planes for B1 and B2 seen from their node (orbital phase: $\omega =\pi /2$). With these systems we can capture circular polarization $V_{00}$ of gravitational wave background.

Figure 3

Overlap reduction functions for the unpolarized $I$ and circularly polarized $V$ modes for parameters $\omega =0.5$ and $\alpha =0$. The long-dashed curve is result with $d=0$ (traditional choice) for which we have ${\gamma }_{V,A{E}^{\prime }*}={\gamma }_{I,A{E}^{\prime }*}=0$. The solid curves are for $d=1$, and the short-dashed ones are for $d=5$. The functions $|{\gamma }_{I,A{A}^{\prime }*}|$ approach 1 at $\widehat{f}\to 0$, while the functions ${\gamma }_{V,A{E}^{\prime }*}$ and ${\gamma }_{I,A{E}^{\prime }*}$ have asymptotic behaviors $O({\widehat{f}}^1)$ and $O({\widehat{f}}^2)$ respectively. For $\alpha =0$ we have ${\gamma }_{I,A{E}^{\prime }*}=0$ (within numerical errors) at $\omega =0$.

Figure 4

The signal to noise ratios for detecting total intensity $I_{00}$ and circular polarization $V_{00}$ of gravitational wave background from combinations $⟨A{A}^{\prime *}⟩$ and $⟨A{E}^{\prime *}⟩$. The results are normalized with ${\mathrm{SNR}}_I(0)$ for the simple orbital choice $d=0$. The shape of BBO noise curve is used and a flat spectrum ${\Omega }_{\mathrm{GW}}=\mathrm{const}$ is assumed. The polarization degree $\Pi$ is set at $\Pi =1$ for ${\mathrm{SNR}}_V(d)$.