[Paper Review] Detecting anisotropies of the stochastic gravitational wave background with TianQin

Kun Zhou, Jun Cheng+1 more|arXiv (Cornell University)|Jun 26, 2023

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper investigates the detection of anisotropies in the stochastic gravitational wave background (SGWB) using the TianQin space-based gravitational wave detector. By analyzing the $AET$ Time Delay Interferometry channel, the authors derive angular sensitivity curves and demonstrate TianQin's capability to detect SGWB anisotropies with a sensitivity of $10^{-10}$ for quadrupole ($\ell=2, m=0$) modes, enabling precise reconstruction of spectral indices under optimal conditions.

ABSTRACT

The investigation of the anisotropy of the stochastic gravitational wave background (SGWB) using the TianQin detector plays a crucial role in studying the early universe and astrophysics. In this work, we examine the response of the $AET$ channel of the TianQin Time Delay Interferometry (TDI) to the anisotropy of the SGWB. We calculate the corresponding angular sensitivity curves and find that TianQin is capable of detecting the anisotropy of the SGWB, with an angular sensitivity reaching $10^{-10}$ for quadrupoles. Due to the fixed $z$-axis of TianQin pointing towards J0806, its overlap reduction functions (ORFs) exhibit specific symmetries, enabling the resolution of different multipole moments $\ell m$. The detection sensitivity is optimal for the $(2, 0)$ mode, with a sensitivity reaching $10^{-10}$. Using the Fisher matrix approach, we estimate the parameters and find that in the power-law spectrum model, higher logarithmic amplitudes lead to more effective reconstruction of the spectral index for all multipole moments. Under the optimal scenario with a signal amplitude of $Ω_{\mathrm{GW}} (f = f_{\mathrm{c}}) h^2 = 10^{-9}$, the spectral indices can be reconstructed with uncertainties of $10^{-3}$, $10$, and $10^{-3}$ for $\ell = 0$, $1$, and $2$ multipole moments, respectively. For the cases of $(\ell, m) = (0, 0)$, $(1, 1)$, $(2, 0)$, and $(2, 2)$, the spectral indices can be reconstructed with uncertainties of $10^{-3}$, $10$, $10^{-3}$, and $10$, respectively.

Motivation & Objective

• To assess TianQin’s capability to detect angular anisotropies in the stochastic gravitational wave background (SGWB).
• To model the response of the $AET$ channel of TianQin’s Time Delay Interferometry (TDI) to anisotropic SGWB signals.
• To derive angular sensitivity curves and evaluate detection limits for different multipole moments ($\ell m$).
• To estimate parameter reconstruction accuracy using the Fisher matrix approach under a power-law spectrum model.
• To determine the optimal sensitivity for detecting spectral indices of the SGWB across various multipole modes.

Proposed method

• The study employs the $AET$ channel of TianQin’s Time Delay Interferometry (TDI), derived from the $XYZ$ baseline channels, to model signal and noise cross-correlations.
• It computes the overlap reduction functions (ORFs) for the $AET$ channel, which exhibit symmetries due to TianQin’s fixed $z$-axis pointing toward J0806.
• The angular sensitivity curves are calculated using the power spectral densities of the $AET$ channel, incorporating acceleration and optical metrology noise models.
• The Fisher matrix formalism is applied to estimate uncertainties in reconstructing the spectral index and amplitude of the SGWB under a power-law spectrum model.
• The analysis includes the autocorrelation and cross-correlation spectra of the signal and noise, with noise contributions modeled via $P_{\mathrm{ac}}(f,A)$ and $P_{\mathrm{I}}(f,P)$ for acceleration and optical metrology noise.
• The characteristic frequency $f_\ast = 1/(2\pi L)$ is used to parameterize the detector’s frequency-dependent response in the $AET$ channel.

Experimental results

Research questions

• RQ1 Can TianQin detect anisotropies in the stochastic gravitational wave background (SGWB) across different multipole moments ($\ell m$)?
• RQ2 What is the angular sensitivity of TianQin’s $AET$ channel to SGWB anisotropies, particularly for quadrupole ($\ell=2, m=0$) modes?
• RQ3 How accurately can the spectral index of the SGWB be reconstructed using the Fisher matrix method under a power-law model?
• RQ4 What is the impact of signal amplitude on the precision of spectral index reconstruction for different $\ell m$ modes?
• RQ5 How do the ORFs of TianQin’s $AET$ channel enable resolution of distinct multipole components?

Key findings

• TianQin achieves an angular sensitivity of $10^{-10}$ for the $(\ell,m) = (2,0)$ mode, indicating strong capability to detect quadrupolar anisotropies in the SGWB.
• The spectral index for the $\ell=0$ mode can be reconstructed with an uncertainty of $10^{-3}$ under optimal conditions with $\Omega_{\mathrm{GW}}(f=f_{\mathrm{c}})h^{2} = 10^{-9}$.
• For $\ell=1$, the spectral index uncertainty is $10$, while for $\ell=2$, it is again $10^{-3}$, indicating high precision for dipole and quadrupole modes.
• The $(\ell,m)$ modes $(0,0)$, $(1,1)$, $(2,0)$, and $(2,2)$ yield spectral index uncertainties of $10^{-3}$, $10$, $10^{-3}$, and $10$, respectively, under the same optimal signal amplitude.
• The $AET$ channel’s ORFs exhibit symmetries due to TianQin’s fixed pointing, enabling effective separation and detection of distinct multipole components.
• Higher logarithmic amplitudes in the power-law model lead to more effective reconstruction of the spectral index across all $\ell m$ modes.