Gravitational waves and small-field astrometry

TL;DR

This paper analyzes the imprint of plane gravitational waves on small-field astrometry, focusing on differential changes in angular separations between sources within a finite FoV. It derives analytic upper bounds for the GW-induced differential signal, showing $|\delta\psi_{\mathrm{AB}}|\le 3\varepsilon\Delta_{\rm max}$ for small separations and revealing a four-petaled sky pattern that depends on the eccentricity $e$ and geometry. The authors validate these results with extensive numerical simulations, mapping the signal across the sky and within FoVs, and demonstrate that the differential signal is highly susceptible to being absorbed by simple plate calibrations. They conclude that, in practice, small-field astrometry is unlikely to detect GWs, and successful GW astrometry will require long baselines and/or all-sky surveys like Gaia.

Abstract

Astrometric observations can, in principle, be used to detect gravitational waves. In this paper we give a practical overview of the gravitational wave effects which can be expected specifically in small-field astrometric data. Particular emphasis is placed on the differential effect between pairs of sources within a finite field of view. We also present several general findings that are not restricted to the small-field case. A detailed theoretical derivation of the general astrometric effect of a plane gravitational wave is provided. Numerical simulations, which underline our theoretical findings, are presented. We find that small-field missions suffer from significant detrimental properties, largely because their relatively small fields only allow the measurement of small differential effects which can be expected to be almost totally absorbed by standard plate calibrations.

Figures (3)

• Figure 1: Resulting normalized angular changes represented by $|\mathcal{F}|$ from numerical simulations using random star pairs and three GWs with fixed parameters. The top row corresponds to a GW signal with an eccentricity of $e=0$, the middle row to $e=0.7$, and the bottom row to $e=1$. The left column presents sky maps showing the maximum absolute normalized change of angular distance, $\max\left(|\mathcal{F}|\right)$, per HEALPix of level 6. The small white dot in the sky maps marks the position of the GW emitter. In the right column, all simulated normalized angular changes are displayed as a function of $\overline{\alpha}_r$. A black line in these plots marks the maximal achievable $|\mathcal{F}|$ according to Eq. \ref{['fig__non_histogram_theta']}. The sky maps use the Hammer-Aitoff projection in equatorial coordinates, with $\alpha = \delta = 0$ at the center, north up, and $\alpha$ increasing from right to left.
• Figure 2: The normalized changes of the angular distance from the numerical simulations, $|\mathcal{F}|$, as a function of $\theta_r$. This plot is valid for all eccentricities $e$. The differences between the three cases shown in Fig. \ref{['fig__skymaps_and_non_histograms']} are negligible. The black line represents the value given by Eq. \ref{['estimate-2-costheta']}.
• Figure 3: Vector field visualizations of the absolute (left column) and differential (right column) astrometric GW effects within an $(\varepsilon\times\varepsilon)$-sized FoV, for different positions on the sky relative to the GW propagation direction, as indicated at the top of each plot with the ($\overline{\alpha}, \theta$) of the center. In the left column, the absolute GW effect is plotted, i.e., the displacement of every point due to the GW, normalized to the maximum shift $\Delta_{\rm max}$ for this GW. The gray arrows in this column are scaled independently to optimize visibility. At the top of each plot in this column, the maximum overall displacement, normalized to the maximal GW effect $\sqrt{(\delta\alpha^*)^2 + (\delta\delta)^2}$ in terms of $\Delta_{\rm max}$, is indicated. The right column shows the differential GW effect referenced to the central point with $(\Delta\alpha^*, \Delta\delta) = (0,0)$. The arrow length in this column is determined by the difference between the displacement at the point and the displacement at the center. The colors in the plots indicate the normalized angular change $|\mathcal{F}|$ with respect to the center point. The asterisk in $\Delta\alpha^*$ means that the difference in right ascension is a true arc, thus: $\Delta\alpha^* \equiv (\Delta\alpha) \cos \delta$. All plots have been created using the gnomonic projection. It should be noted that all arrow lengths are significantly exaggerated for illustrative purposes compared to typical GW signal magnitudes. A detailed discussion can be found in the text.