Can cross-correlating CMB and galaxy surveys detect the Rees-Sciama effect?

A faint signal from evolving gravitational potentials

The Rees-Sciama effect is the nonlinear counterpart of the integrated Sachs-Wolfe effect. CMB photons crossing a gravitational potential that changes with time receive a small temperature shift, described in the paper as an integral of the potential's time derivative along the line of sight. In the linear regime, a non-zero derivative can arise from accelerated expansion, modified gravity, or spatial curvature. In the nonlinear regime, structure formation drives the evolution, and that is where massive neutrinos enter.

The paper's mechanism is specific: after neutrinos become non-relativistic they free-stream, suppressing the growth of density perturbations below a free-streaming length. This produces a slow decay of the gravitational potential even without background expansion, and the effect grows with the total neutrino mass. In the ISWRS angular power spectrum, this appears as a decrease relative to the massless-neutrino case, deepening as the neutrino mass rises.

The practical problem is brightness. The ISWRS signal is exceedingly faint next to primary CMB anisotropies, and the paper states that direct measurements have not yet been achieved. The proposed workaround is cross-correlation: large-scale structure probes trace the same gravitational potential whose variation generates the effect, so correlating them with CMB temperature maps helps isolate the signal from primary anisotropies, foregrounds, and noise.

Three tracers, four CMB experiments, five masses

The study forecasts cross-angular power spectra between the ISWRS effect and three gravitational-potential tracers: galaxy clustering, cosmic shear, and the CMB-lensing potential. It pairs these tracers with the Simons Observatory, a CMB-S4-like experiment, CMB-HD, and the PICO space mission, combined with Euclid and LSST as the large-scale structure surveys. The cross-spectra are computed with an analytical method the authors developed in earlier work.

Neutrino mass is varied across five νΛCDM cosmologies built on a Planck 2018 baseline, with Mν of 0.06, 0.12, 0.18, 0.24, and 0.30 eV. The forecasts are staged by realism. An ideal case assumes a tracer whose window function perfectly overlaps the ISWRS one, full-sky coverage, and negligible noise. Optimistic cases keep full-sky coverage and drop detector noise and foreground residuals. Realistic cases add primary CMB anisotropies, detector noise, and, in the most complex scenario, residual foregrounds combined with an internal linear combination method.

Two quantities are reported and should not be conflated. The cumulative signal-to-noise ratio measures whether the cross-spectrum can be detected at all. A separate χ²-based quantity measures how well a νΛCDM model can be distinguished from the massless-neutrino case. A high detection significance therefore does not by itself imply a tight constraint on Mν.

The ideal case sets the ceiling, not the forecast

In the ideal case the authors use ISWRS auto-spectra extracted from DEMNUni N-body simulations, because analytical treatments cannot properly represent nonlinear scales. With Mν values of 0.0, 0.17, and 0.30 eV, the effect is detectable at roughly 100σ, rising to about 200σ when delensing is applied to the CMB anisotropy field. Distinguishing a 0.17 eV model from the massless case reaches about 15σ, or about 22σ with delensing.

These numbers describe an idealized configuration with perfect overlap and negligible noise, so they mark an upper bound rather than an expectation for real surveys. The paper uses them to show that the cross-correlation approach has room to work before turning to the tracers and instruments that actual surveys provide.

CMB lensing leads when noise is switched off

Under optimistic assumptions, the three tracers separate. The CMB-lensing cross-correlation reaches a signal-to-noise ratio of about 40 in the lensed case and about 120 with delensing. Galaxy clustering and cosmic shear perform similarly, with detections ranging from roughly 30σ to 80σ when delensing is applied. In this scenario, distinguishing the νΛCDM model from the massless case becomes possible even for the lightest tested mass of 0.06 eV once delensing is included.

The optimistic cosmic-shear result depends on a galaxy number density that gives a large overlap between the ISWRS and cosmic-shear window functions. The paper explicitly contrasts this with the realistic cosmic-shear case, attributing the gap to that overlap and to the higher galaxy number density assumed here. This is a warning that the optimistic ranking is not the final ranking.

CMB-HD lensing survives noise and foregrounds

The realistic forecasts are the paper's central result. Across all tested νΛCDM cosmologies, the best performance comes from cross-correlating the ISWRS effect with the CMB-lensing potential measured by CMB-HD, for both lensed and delensed temperature spectra, and whether the scenario includes only detector noise or also residual foregrounds. The paper reports at least 5σ detections even under conservative assumptions about detector noise and foregrounds.

The explanation the authors give is resolution. CMB-HD lacks the sky coverage of a space mission like PICO, but its high sensitivity at high resolution yields particularly low CMB-lensing reconstruction noise at the scales of interest. Combined with the extended overlap between the CMB-lensing and ISWRS window functions, this makes it a better probe than the alternatives tested. The paper notes this is the first assessment of ISWRS detectability through CMB-lensing cross-correlation.

Galaxy clustering also yields significant detections, with the best results from combining the LSST Optimistic sample with CMB-HD. The authors highlight that optimal weighting, which reshapes the galaxy distribution to increase overlap with the ISWRS window function and reduce noise, raises the signal-to-noise ratio by about 60 percent on average. High-significance galaxy-clustering detections are expected for CMB-S4 and CMB-HD provided residual foregrounds stay under control.

Cosmic shear stalls on unmodelled nonlinearities

Cosmic shear is the weak link. The paper validates its analytical cross-spectrum against DEMNUni maps and finds that the overall shape and large-scale amplitude are recovered, but the sign-inversion position and small-scale amplitude are reproduced less well than for galaxy clustering or CMB lensing. The authors attribute this to cosmic shear's greater sensitivity to nonlinearities and to the tested Halofit models not reconstructing nonlinear scales precisely enough for this cross-correlation.

That mismatch drives the discouraging realistic cosmic-shear results. The abstract states the probe can deliver valuable results only if nonlinearities are accurately modelled, beyond currently available analytical approaches. This is a modelling limitation rather than a statement that the physical correlation is absent.

Detection does not translate into tight neutrino constraints

The paper's answer on neutrino mass is deliberately split from its answer on detection. The CMB-lensing cross-correlation can allow identification of a νΛCDM model with Mν of at least 0.12 eV in a measurement with negligible residual foregrounds, a threshold the authors compare with current estimates pointing to Mν below 0.11 eV at 95 percent confidence. Galaxy clustering and cosmic shear, the two large-scale structure probes, do not provide strong constraining power on Mν.

The overall conclusion is that future CMB experiments and large-scale structure surveys should enable detection of the Rees-Sciama effect, but the paper does not imply significant prospects for imposing new neutrino-mass constraints in the near future. The distinction matters because the same cross-spectra serve both goals: a detection significance answers whether the signal is there, while the model-discrimination statistic answers how finely Mν can be separated.

Delensing and foreground control decide the outcome

The paper's priority ordering is clear. CMB-HD's reconstructed CMB-lensing potential is the leading channel, and the two conditions that most change the forecast are delensing of the CMB temperature data and control of residual foregrounds. Delensing raises signal-to-noise ratios substantially in every optimistic case, and the most complex realistic scenario adds residual foregrounds on top of detector noise, which is where the conservative 5σ claim for CMB-HD lensing is anchored.

For galaxy clustering, the decisive lever is optimal weighting, worth about 60 percent in signal-to-noise on average, plus the galaxy density and sky overlap of the LSST Optimistic sample. For cosmic shear, the decisive lever is nonlinear modelling, which currently falls short. The practical reading is that the detection case rests on CMB-HD lensing with foreground control, while the neutrino-mass case remains gated by whether Mν sits at or above the 0.12 eV threshold the paper identifies.

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