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Characterising the epoch of reionisation using the cross-correlation of the kSZ effect and CMB lensing

This paper demonstrates that cross-correlating the squared kinematic Sunyaev-Zeldovich effect with CMB lensing serves as a sensitive probe of the epoch of reionisation, forecasting that while current-generation experiments like Simons Observatory may only marginally detect the signal, future missions such as CMB-HD could achieve high-significance measurements to constrain reionisation scenarios.

Original authors: Niall MacCrann, Christopher Cain, Aleksandra Kusiak, Fiona McCarthy, Alexander Van Engelen, Darby Kramer, William R. Coulton, Frank J. Qu

Published 2026-08-06

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CC BY 4.0

Original authors: Niall MacCrann, Christopher Cain, Aleksandra Kusiak, Fiona McCarthy, Alexander Van Engelen, Darby Kramer, William R. Coulton, Frank J. Qu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. For the first few hundred thousand years after the Big Bang, this balloon was filled with a thick, opaque fog of gas, hiding everything inside. Then, the first stars and galaxies ignited, blasting out intense ultraviolet light that acted like a cosmic blowtorch. This light began to burn away the fog, turning neutral gas into a transparent soup of charged particles (electrons and protons). This dramatic cleanup crew's work is called the "Epoch of Reionisation." It's a crucial chapter in our cosmic story, but it's incredibly hard to read because the event happened so long ago and the "fog" is now completely gone.

To peek back at this era, scientists use the Cosmic Microwave Background (CMB), which is essentially the afterglow of the Big Bang—a faint, static-like hum of light that fills the entire sky. Think of the CMB as a giant, ancient projector screen. When the "fog" of the early universe was being burned away, the bubbles of ionized gas were moving around. As these bubbles moved, they gave a tiny "kick" to the CMB photons passing through them, slightly shifting their energy. This effect is called the kinematic Sunyaev-Zeldovich (kSZ) effect. It's like a gentle breeze nudging a leaf; the leaf (the light) changes its speed just a tiny bit, but if you look at the whole forest, you can tell the wind was blowing. However, this signal is incredibly faint and gets lost in a sea of other cosmic noise.

Now, imagine trying to see the shape of a hidden object by looking at how it distorts the light behind it. That's what "lensing" is. Massive clumps of matter in the universe act like a funhouse mirror, bending the path of the CMB light as it travels to us. By studying these distortions, we can map where the invisible "clumps" of matter are hiding. This paper asks a clever question: Can we combine the "wind" (the kSZ effect from the early universe) with the "mirror" (the lensing from matter) to get a clearer picture of how the universe was cleaned up?

The authors of this paper, Niall MacCrann and his colleagues, propose a new way to hunt for the fingerprints of the Epoch of Reionisation. Instead of just looking at the kSZ signal alone, they suggest squaring the kSZ temperature map (a mathematical trick that highlights the strength of the "bubbles" regardless of which way they were moving) and then cross-correlating it with the map of the cosmic lensing potential. Think of it like this: if you have a map of where the wind is strongest and a map of where the heavy rocks are, seeing if they line up tells you if the wind is blowing around the rocks. In this cosmic case, they are checking if the bubbles of ionized gas formed around the densest clumps of matter.

Using sophisticated computer simulations called AMBER, the team tested this idea. They found that this cross-correlation does indeed carry information about the reionisation process, specifically how long it lasted and when it happened in the middle. However, the signal is incredibly weak. For a current or near-future experiment like the Simons Observatory, the signal is just barely detectable, with a signal-to-noise ratio of only about 2 to 3. It's like trying to hear a whisper in a crowded stadium; you might catch a word, but you can't be sure.

But the story gets more exciting with a futuristic experiment called CMB-HD. The authors forecast that if we build an instrument with 50 times less noise than current ones, this method could provide a massive signal-to-noise ratio of about 50. That would be like turning that whisper into a shout, allowing scientists to clearly distinguish between different theories of how the universe was reionised.

The paper also tackles the messy reality of trying to measure this. The universe is full of "noise" from other sources, like gas in nearby galaxy clusters or dust from distant galaxies, which can fake the signal. The authors show that these contaminants are a major headache, potentially overwhelming the reionisation signal by a factor of 5 to 10. They propose several ways to clean up the data, such as using only polarized light (which is less affected by some of the noise) and using galaxy surveys to subtract the nearby "fog." They also discuss a tricky bias where the lensing itself can mess up the measurement, but they suggest a "lensing-hardened" method to fix it.

In short, this paper suggests a powerful new tool for peering into the universe's teenage years. While current technology might only give us a blurry glimpse, the authors argue that with the next generation of ultra-sensitive telescopes, we could finally get a sharp, high-definition look at how the first stars cleared the cosmic fog, provided we can successfully filter out the cosmic static that tries to hide it.

Technical Summary: Characterising the epoch of reionisation using the cross-correlation of the kSZ effect and CMB lensing

Problem Statement
The epoch of reionisation (EoR) is characterized by the formation of ionised electron "bubbles" around early galaxies, which generate temperature perturbations in the Cosmic Microwave Background (CMB) via the kinematic Sunyaev-Zeldovich (kSZ) effect. While the kSZ effect is a potential probe of reionisation physics, isolating its signal from other small-scale foregrounds in the CMB temperature is challenging. Previous attempts to constrain reionisation parameters have utilized the kSZ power spectrum and trispectrum (e.g., Reichardt et al. 2021; Raghunathan et al. 2024; MacCrann et al. 2024), but these are often foreground-limited. Furthermore, the standard kSZ temperature fluctuation field, $T_{\text{kSZ}}$ , has a close-to-zero cross-correlation with the CMB lensing potential ( $\phi$ ) because the line-of-sight velocity ( $v_r$ ) can be positive or negative with equal probability for a given matter density. Consequently, there is a need for statistics that can extract reionisation information from cross-correlations with large-scale structure tracers while mitigating sign cancellations and foreground contamination.

Methodology
The authors propose a cross-correlation statistic between the squared kSZ field, denoted as $K(\hat{\mathbf{n}})$ , and the CMB lensing potential, $\phi$ .

The $K$ Statistic: Defined as the square of a high-pass filtered CMB temperature field (Smith & Ferraro 2017; Ferraro & Smith 2018), $K$ removes the dependence on the sign of the radial velocity, $v_r$ . In Fourier space, $K_L$ is constructed via a quadratic estimator involving the product of temperature modes, weighted by a filter function $W_l$ that optimizes the signal-to-noise ratio based on the kSZ power spectrum relative to the total observed power spectrum.
The Cross-Correlation ( $C_L^{K\phi}$ ): The authors investigate the angular cross-power spectrum between $K$ and $\phi$ . Theoretical intuition suggests that while $\langle v_r \phi \rangle \approx 0$ , the cross-correlation $\langle v_r^2 \delta_e \delta_m \rangle$ (where $\delta_e$ is electron overdensity and $\delta_m$ is matter overdensity) is non-zero. This signal arises from the bispectrum $\langle \delta_e \delta_e \delta_m \rangle$ , which is sensitive to the non-Gaussianity of the ionised electron field during reionisation.
Simulations and Forecasts: The signal is measured using full-sky simulations generated by the Abundance Matching Box for the Epoch of Reionisation (AMBER) code. The authors forecast signal-to-noise ( $S/N$ $S / N$ ) ratios for two experimental setups:
1. A "Simons Observatory (SO)-like" experiment (5 $\mu$ K-arcmin white noise, 1.5' beam, 40% sky coverage).
2. A "CMB-HD-like" futuristic experiment (0.5 $\mu$ K-arcmin noise, 0.3' beam).
Bias Mitigation: The paper systematically addresses estimator biases, including the disconnected trispectrum ( $N_0$ bias), low-redshift kSZ contamination, extragalactic foregrounds (tSZ, CIB, radio sources), and lensing-induced bias in the $K$ estimator. Strategies include using polarization-only data for lensing reconstruction, applying harmonic-space Internal Linear Combination (ILC) with foreground deprojection, and employing "lensing-hardened" estimators.

Key Contributions and Results

Signal Detection: Using AMBER simulations with fiducial reionisation parameters ( $z_{\text{mid}}=8, \Delta z_{90}=4$ ), the authors find that the $C_L^{K\phi}$ signal is marginally detectable with an SO-like experiment, yielding an $S/N$ of approximately 2.3. In contrast, a CMB-HD-like experiment could achieve an $S/N \sim 50$ (rising to $\sim 62$ with optimal estimators and low-redshift lensing removal).
Sensitivity to Reionisation Parameters: The signal is sensitive to the duration ( $\Delta z$ ) and midpoint ( $z_{\text{mid}}$ ) of reionisation. For the CMB-HD setup, the authors forecast the ability to rule out extreme reionisation scenarios (e.g., $\Delta z_{90}=2$ or $6$) at high significance ( $>11\sigma$ ). For the SO-like setup, the signal is too weak to strongly discriminate between models, though upper limits could still provide constraints.
Foreground and Bias Analysis:
- Low-Redshift Contamination: The low-redshift kSZ signal (from galaxies and clusters) is found to be 5–10 times larger than the reionisation signal. The authors propose mitigating this by subtracting low-redshift lensing contributions using galaxy surveys or CIB maps.
- Extragalactic Foregrounds: Foregrounds such as the thermal SZ (tSZ) and Cosmic Infrared Background (CIB) introduce significant biases (orders of magnitude larger than the signal) if not mitigated. The authors demonstrate that deprojecting these components in the $K$ estimator, combined with low-redshift lensing cleaning, can reduce biases to order unity.
- Lensing Bias: The $K$ estimator is biased by lensing mode-coupling. The authors show that using a "lensing-hardened" estimator reduces this fractional bias from $\sim 100\%$ to $\sim 10\%$ .
- $N_0$ Bias: Using polarization-only data for the lensing estimator ( $\phi$ ) effectively removes the $N_0$ noise bias and reduces foreground trispectrum biases.

Significance and Claims
The paper claims that the cross-correlation of the squared kSZ field ( $K$ ) with CMB lensing ( $\phi$ ) offers a novel probe of the EoR, complementary to existing statistics like the kSZ trispectrum ( $C_L^{KK}$ ) and cross-correlations with galaxies or patchy screening.

Unique Physics: The statistic probes the $\langle \delta_e \delta_e \delta_m \rangle$ bispectrum, providing sensitivity to the spatial distribution of ionised electrons relative to matter density.
Feasibility: While the signal is challenging to detect with current-generation experiments like the Simons Observatory (where it may only be marginally detected or provide upper limits), the authors argue it is a viable and informative target for next-generation, high-resolution, low-noise experiments like CMB-HD.
Systematics: The work highlights that while significant challenges exist—specifically low-redshift contamination and foreground biases—these can be mitigated through specific estimator choices (polarization-only lensing, lensing-hardening) and foreground cleaning techniques (deprojection, low-redshift subtraction).
Future Outlook: The authors suggest that combining $C_L^{K\phi}$ with the kSZ power spectrum and trispectrum could break degeneracies in reionisation physics parameters, offering a comprehensive view of the EoR when paired with future CMB data.

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