Low-Frequency Stochastic Gravitational-Wave Background in Gaia DR3 catalogue

Abstract

We investigate the potential to detect low-frequency gravitational waves (GWs) through their imprints on the proper motions of distant quasars observed by the Gaia mission. Using astrometric data from Gaia DR3, we simulate the effect of GWs on the proper motions of quasars, incorporating their actual sky positions and measurement uncertainties. We investigate two data analysis techniques for the extraction and characterization of GW signals from quasar proper motions: Vector Spherical Harmonics (VSH) and angular correlation functions, commonly referred to as Hellings-Downs curves (HDC). Using realistic simulated data, we forecast their sensitivity and accuracy to GWs, and evaluate the impact of systematic errors. From these simulations, we derive an upper limit on the amplitude of a stochastic GW background, constrained by the observational timespan, astrometric precision, and the sky distribution of quasars. Compared to HDC, VSH appears more statistically robust, less prone to selection effects, and with a significantly smaller computational cost, scaling as N. The HDC method is more sensitive for detecting gravitational waves, but its complexity scales as N^2. We find that, with Gaia DR3 proper motion errors, the lower limit for a detectable GW strain is of 10^{-11}, with possible improvements to about 3 x 10^{-12} for the next Gaia Data Release 4 (for the same number of quasars). This limit holds for a stochastic GW spectrum integrated over all frequencies less than half the inverse of the 34-month observational timespan of Gaia DR3, corresponding to approximately 5.6 nHz. We also investigate how different data-restriction and weighting schemes influence the final estimate of the gravitational wave strain.

Figures (11)

• Figure 1: Simulated apparent positional deflections $\delta \bf u$ of sources induced by a GW with strain amplitude $h=10^{-11}$, propagating toward the direction $\alpha=\delta=0^{\circ}$ (top panel) and $\alpha= 0^{\circ}$ and $\delta=90^{\circ}$ (bottom panel). The two GW polarizations are shown: "+" (left panel) and "$\times$" (right panel). The vector length and color encode the magnitude of the displacement $|\delta u|$. The maps use an Aitoff projection in equatorial coordinates.
• Figure 2: Hellings-Down curves for different GW induced displacements. In red the parallel-parallel and perpendicular-perpendicular correlation function, in blue the cross-terms (trivially zero for an isotropic GWB) and in black the radial correlation function as measured by PTA (normalized to $\frac{1}{2}$ due to the non negligible star term for pulsars).
• Figure 3: The top row shows the Hellings–Downs curves for all four proper motion correlation (PMC) components, calculated from noise-free simulated data for a uniform sky distribution of sources (top left panels) and for the actual distribution of 1.5 million quasars from the Gaia-CRF3 catalog (top right panels). The bottom row presents the power spectra of the vector spherical harmonics (VSH) coefficients, $P_\ell$, for the same simulated datasets. For clarity, the toroidal (blue) and spheroidal (green) harmonic points are horizontally offset by $\Delta \ell = 0.1$. The simulations assume a plane gravitational wave with equal “$+$” and “$\times$” polarizations, characterized by a strain amplitude $h_c = 10^{-11}$, propagating in the direction $\alpha = 45^{\circ}$, $\delta = 45^{\circ}$.
• Figure 4: The top row shows the Hellings–Downs curves for all four proper motion correlation (PMC) components, calculated from simulated proper motions for the actual quasar sky distribution, with Gaia-CRF3 error-dependent noise added (top left panel). The top right panel presents the averaged correlation components $C_{\perp\perp}$ and $C_{\parallel\parallel}$ (red), together with the cross terms ($C_{\parallel\perp}$ and $C_{\perp\parallel}$) (blue). Also shown are the fitted Hellings–Downs function $\Gamma(\theta)$ and its corresponding $95\%$ confidence interval (green), derived from these correlation curves. The bottom row shows the power spectra of the vector spherical harmonics (VSH) coefficients $P_l$ for the same simulated dataset (bottom left panel), and for a simulation in which the quasar proper motion uncertainties are reduced by a factor of three, as expected for future Gaia DR4 (bottom right panel). For clarity, the toroidal (blue) and spheroidal (green) harmonic points are horizontally offset by 0.1 in $\ell$. The simulations assume a plane GW with equal “$+$” and “$\times$” polarizations, characterized by a strain amplitude $h_c = 10^{-11}$, propagating in the direction $\alpha = 45^{\circ}, \delta = 45^{\circ}$.
• Figure 5: The top row shows the power spectra of the vector spherical harmonic coefficients (left panel) and the mean toroidal and spheroidal powers (right panel) characterizing systematics in the proper motions of $\sim$1.5 million quasars from the Gaia-CRF3 catalogue. For clarity, the toroidal (blue) and spheroidal (green) harmonic coefficients are horizontally offset by $\Delta\ell = 0.1$. The bottom row displays the correlation components $C_{\perp\perp}$ and $C_{\parallel\parallel}$ together with the cross terms $C_{\parallel\perp}$ and $C_{\perp\parallel}$ (left panel). Also the right panel shows the averaged correlation components and the best-fitting Hellings–Downs correlation function $\Gamma(\theta)$ with its corresponding $95\%$ confidence interval (green), obtained after applying a dipole-mode correction based on the proper motions of $\sim$1.5 million Gaia-CRF3 quasars.