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

TL;DR

This work introduces a time-dependent deflection reconstruction method to search for gravitational waves using cosmic microwave background data. By performing a Fourier transform in time and constructing a 3D quadratic estimator that correlates the time-averaged CMB with nonzero temporal frequency maps, the authors achieve substantially lower reconstruction noise than traditional static lensing estimators. They derive how the measured time-dependent deflection power maps to the GW energy density $\Omega_{\mathrm{GW}}(f)$ and provide minimum-variance estimators that translate into constraints on the GW spectrum, including a discussion of the GW strain $h_c(f)$ and the relevant frequency coverage. Forecasts for ACT, CMB-S4, and CMB-HD show the method extends sensitivity into the microhertz regime, offering a valuable, complementary probe to pulsar timing arrays and space-based interferometers, with foregrounds turning into useful deflection sources rather than hindrances.

Abstract

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.

Figures (4)

• Figure 1: Illustration of the (exaggerated) effect on the observed CMB due to gravitational waves passing through Earth, modeled here with divergence-type quadrupolar deflection potential $\phi_{l=2,m=\pm2}$. The stretching and squeezing of the CMB hot and cold spots is apparent. While a continuous gravitational wave at a single temporal frequency would cause the deflections to oscillate in time with a fixed spatial distribution in this way, a stochastic background of gravitational waves would produce a random time-dependent pattern of deflections with statistics governed by Eq. \ref{['eq:thetaQrmsl']}.
• Figure 2: Top: A time series of CMB maps constructed from a set of fixed-interval observations. Bottom: A Fourier transform of the time series above, which isolates the primary, constant CMB to the $\omega=0$ slice. The higher frequency modes normally consist only of noise, at the same depth. A time-dependent GW deflection of the CMB maps has been injected in the fourth nonzero frequency slice (significantly exaggerated for visibility), showing the 'echo' generated at that slice (Eq. \ref{['eq:deflectedT']}). Our estimator searches for correlations between this echo and the constant slice.
• Figure 3: Forecasts for reconstructed deflection noise for CMB surveys with configurations like ACT (Left Panel), CMB-S4 (Middle Panel), and CMB-HD (Right Panel). The orange and green lines represent the deflection reconstruction noise using the standard $TT$ and $EB$ static quadratic lensing estimators HuOkamoto:MassReconstruction2002. For our time-dependent deflection reconstruction we plot $\Delta t_\mathrm{survey}^{-1}\mathcal{N}_{L,\Omega}^{dd}$, which represents the deflection reconstruction noise in a frequency band of width of $\Delta t_\mathrm{survey}^{-1}$, shown in red and blue. The red line shows the reconstruction noise assuming no astrophysical foregrounds, while the blue line is the time-dependent deflection noise obtained from our estimator when including astrophysical foregrounds. We focus on reconstruction of only the largest angular scales, since a stochastic background of gravitational waves produces primarily quadrupolar deflection, with power falling rapidly at smaller scales BookFlanagan:AstrometricEffects2011. As can be seen, our time-dependent deflection estimator exhibits significantly lower noise within each frequency band than the noise of standard static lensing quadratic estimator, and including astrophysical foregrounds improves rather than hampers the time-dependent deflection reconstruction.
• Figure 4: Sensitivity of various methods to search for gravitational waves across a wide range of frequencies. The forecasted sensitivity of our new time-dependent deflection reconstruction method applied to current and future CMB surveys is shown in blue. The existing sensitivity from pulsar timing array analysis by NANOGrav NANOGrav:2023hfp is shown in orange. Laser interferometer sensitivity from existing ground-based observations (LIGO+Virgo) KAGRA:2023pio and future space-based observations (LISA) LISA:2024hlh are shown in red. The spectrum of a stochastic background of gravitational waves produced by a population of supermassive black hole binaries, with an amplitude matching the inferred value from NANOGrav observations NANOGrav:2023hfp, is shown as a black dash-dot line. Although the CMB reconstruction is not competitive with pulsar timing in their common frequency band, it does provide sensitivity in the microhertz frequency gap between pulsar timing (NANOGrav) and future space-based laser interferometer (LISA) observations.