Time-dependent deflection reconstruction: new technique to search for gravitational waves with the cosmic microwave background

Gravitational waves (GWs) passing through the Earth cause a correlated pattern of time-dependent deflections of the apparent position of astronomical sources. We build upon standard lensing reconstruction techniques to develop a new time-dependent quadratic estimator, providing a novel technique to search for the deflections produced by GWs using observations of the cosmic microwave background (CMB). We find that the time-dependent deflection reconstruction is many orders of magnitude more sensitive than the ordinary static lensing estimator, and it can be employed with the data collected by existing and future CMB surveys, without requiring any modification to the experimental or survey design. We demonstrate that CMB surveys offer sensitivity to GWs across a broad frequency range: while the sensitivity will not be competitive over the frequency range covered by pulsar timing arrays, it does extend coverage to both lower and higher frequencies. Finally, we discuss how our methods can be extended to search for other time-varying signals, and also how it can be applied to surveys at other wavelengths.

💡 Research Summary

The paper introduces a novel method for detecting gravitational waves (GWs) by exploiting the time‑dependent angular deflection they induce on the apparent positions of photons arriving at Earth. Building on the well‑established quadratic estimators used for static CMB lensing, the authors develop a “time‑dependent quadratic estimator” that operates on a three‑dimensional Fourier space comprising two spatial dimensions (ℓ) and one temporal frequency (ω).

The physical motivation is that a stochastic GW background (SGWB) produces both divergence‑type (E‑mode) and curl‑type (B‑mode) deflection fields with comparable power, in contrast to large‑scale‑structure lensing which is purely divergence‑type. Moreover, the GW‑induced deflection peaks at the lowest multipoles (ℓ≈2–3) and varies on timescales comparable to the cadence of modern CMB surveys (hours to days), whereas the primary CMB and foregrounds are essentially static over the multi‑year duration of a survey.

To capture this temporal information, the authors propose splitting the full survey data (duration Δt_survey) into many short‑duration maps (cadence Δt_cadence). Each map T( n̂ , t_j ) is Fourier‑transformed in both sky coordinates and time, yielding T(ℓ, ω). The ω=0 slice contains the usual static CMB and foregrounds, while ω≠0 slices contain only time‑varying signals and instrumental noise. Assuming stationary, white noise in both space and time, the noise power spectrum is N_TT(ℓ, ω)=Δt_survey Δ_T² exp