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Dynamical Test of Cosmic Acceleration: $k$ -nearest Neighbor Cross Correlation of Cosmic Microwave Background and Cosmic Infrared Background

This paper presents a dynamical test of cosmic acceleration by detecting a positive cross-correlation between the Cosmic Microwave Background and a foreground-free Cosmic Infrared Background map using the $k$ -nearest neighbor cumulative distribution function, achieving a $4.8\sigma$ significance that improves upon traditional two-point correlation methods and remains consistent with the $\Lambda$ CDM model.

Original authors: Dongkok Kim, Donghui Jeong, Yi-Kuan Chiang, Ho Seong Hwang

Published 2026-07-23

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CC BY 4.0

Original authors: Dongkok Kim, Donghui Jeong, Yi-Kuan Chiang, Ho Seong Hwang

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 a long time, scientists thought this balloon was slowing down its expansion, like a car running out of gas. But in the late 1990s, we discovered something shocking: the balloon isn't just expanding; it's speeding up. This mysterious force pushing the universe apart is called "dark energy." It makes up about 70% of everything in the cosmos, yet we have no idea what it actually is. Is it a constant push, or does it change its mind over time? To find out, we need to watch how the universe's structure grows and changes, not just how far away things are.

One way to watch this growth is by looking at the "afterglow" of the Big Bang, known as the Cosmic Microwave Background (CMB). Think of the CMB as the oldest light in the universe, a faint, static-filled glow that fills the sky. As this light travels across billions of years to reach us, it passes through giant clumps of matter (like galaxies and dark matter). If the universe were just coasting, the light would lose energy going into a gravity well and gain it back coming out, canceling out perfectly. But because dark energy is speeding up the expansion, the gravity wells change shape while the light is passing through. This leaves a tiny, subtle fingerprint on the light—a slight warming or cooling. This phenomenon is called the Integrated Sachs-Wolfe (ISW) effect. Detecting this faint signal is like trying to hear a whisper in a hurricane, but if we can hear it, it tells us exactly how dark energy is behaving.

This paper is a detective story about catching that whisper. The authors, a team of cosmologists, decided to listen for the ISW signal by comparing two different maps of the universe. The first map is the CMB, the ancient light from the Big Bang. The second map is the Cosmic Infrared Background (CIB), which is essentially a glow made by the heat of dusty, star-forming galaxies. The CIB acts like a tracer, showing us where the massive structures of the universe are located. The team used a clever new statistical tool called the "k-nearest neighbor" method. Imagine you are at a crowded party and you want to know if people are standing in groups or scattered randomly. Instead of just measuring the distance between two specific people (the old way), this new method looks at how many people are standing near every person in the room. It's a way of capturing the "clumpiness" of the crowd much more efficiently.

By applying this method to data from the Planck satellite (which mapped the CMB) and a new, ultra-clean map of the CIB built from 600 million galaxies, the team found a match. They discovered a positive correlation: the places where the universe is clumpy (the CIB) line up with the specific temperature shifts in the CMB predicted by the ISW effect. The paper explicitly rules out the idea that this is just random noise or a fluke; the chance of this happening by accident is very low, about 2 in 100 (a significance of 2.3σ when looking at the simplest test). However, when they compared the strength of this signal to what our standard model of the universe (called ΛCDM) predicts, the match was even tighter. The signal was 4.8 times stronger than the expected random noise, which is a very strong confirmation.

The authors also tested if dark energy might be changing over time. They split the galaxy map into two layers: one for nearby galaxies and one for distant ones. They found that the signal was present in both layers and matched the standard model's predictions, suggesting that dark energy is behaving consistently, just like a constant cosmological constant. Interestingly, they found that their new "k-nearest neighbor" method was about 20% better at finding this signal than the traditional methods used in the past. This suggests that by looking at the universe's structure in a more complex, "group-aware" way, we can hear the whispers of dark energy much more clearly. While the paper doesn't claim to have solved the mystery of what dark energy is, it provides a robust, dynamic test confirming that the universe is indeed accelerating in the way our current theories predict, and it offers a powerful new tool for future investigations.

Technical Summary: Dynamical Test of Cosmic Acceleration via kNN-Cross Correlation

Problem and Motivation
The physical origin of cosmic acceleration remains a central open question in modern cosmology. While geometrical probes (e.g., Type Ia supernovae, BAO) constrain the distance-redshift relation, dynamical probes offer independent information by directly tracing the growth of large-scale structure and the evolution of gravitational potentials. The Integrated Sachs–Wolfe (ISW) and Rees–Sciama (RS) effects—collectively termed ISWRS—arise when Cosmic Microwave Background (CMB) photons traverse time-varying gravitational potentials in a dark-energy-dominated universe. Detecting the ISWRS signal is challenging because the induced temperature anisotropies are an order of magnitude smaller than primary CMB fluctuations. Previous detections using cross-correlations with large-scale structure tracers (e.g., SDSS, 2MASS, WISE) have typically required stacking multiple surveys to reach significance levels of $\gtrsim 4\sigma$ , while stacking analyses of voids and clusters have yielded lower significances ( $\lesssim 3\sigma$ ).

Methodology
This study employs a novel statistical approach, the $k$ -nearest neighbor cumulative distribution function (kNN-CDF), to measure the cross-correlation between the CMB and the Cosmic Infrared Background (CIB).

Datasets:
- CIB Tracer: The authors utilize a foreground-cleaned 100 $\mu$ m CIB map reconstructed by Y.-K. Chiang (2023, C23). This map is derived from $\sim$ 600 million Wide-field Infrared Survey Explorer (WISE) galaxies extending to $z \approx 2.5$ , effectively serving as a galaxy–CMB cross-correlation tracer. The map isolates the extragalactic component from Galactic dust using cross-correlations with spectroscopic reference samples.
- CMB Data: Planck Public Data Release 3 (PR3) temperature maps, specifically the "SZ-free" (SMICA-nosz) product, to minimize contamination from the Sunyaev–Zel'dovich effect.
- Mock Data: To quantify significance against the $\Lambda$ CDM model, the authors construct mock CIB and ISWRS maps using the Millennium-XXL (MXXL) N-body simulation. The mock CIB is created by applying a halo infrared emission model (W18) to the simulation's lightcone halos, then scaling and adding noise terms to match the observed CIB power spectrum and probability distribution function (PDF). The ISWRS map is generated by integrating the time derivative of the gravitational potential along the line of sight.
Statistical Framework:
- The kNN-CDF statistic, $\psi$ , measures the joint probability of finding neighbors within a given volume in two discrete or continuous fields. Here, it is applied to top-hat smoothed CMB and CIB maps.
- The analysis uses fixed percentile thresholds ( $T^*, I^* \in \{50\%, 70\%, 90\%\}$ ) and angular smoothing scales ( $\theta \in \{0.1^\circ, \dots, 0.9^\circ\}$ ), resulting in a 45-element data vector.
- Covariance matrices are estimated using 200 random realizations of CIB and CMB maps. The Hartlap factor is applied to correct the inverse covariance matrix.
- Systematic uncertainties are addressed by constructing masks based on CIB reconstruction errors and excluding outlier pixels (fiducial cut: 0% for CMB, 2% for CIB).

Key Contributions and Results

Detection of ISWRS: The study reports a positive cross-correlation between the 100 $\mu$ m CIB and the CMB.
- Rejecting the null hypothesis of no correlation yields a significance of $p = 0.02$ ( $\chi^2_{\text{null}} \approx 66.6$ for 45 degrees of freedom, corresponding to $2.3\sigma$ ).
- When quantifying the amplitude relative to $\Lambda$ CDM mock data including ISWRS effects, the best-fit amplitude is $A_{\text{kNN}} = 0.95 \pm 0.20$ . This corresponds to a detection significance of $4.8\sigma$ .
- Individual angular bins show deviations from the null hypothesis ranging from $1.8\sigma$ to $4.1\sigma$ .
Superiority of kNN-CDF:
- The authors compare their kNN-CDF results with a standard two-point angular power spectrum analysis ( $C_\ell$ ) using the same data and masks.
- The two-point analysis yields a detection significance of $\approx 3.9\sigma$ (linear theory) to $4.0\sigma$ (simulation-based).
- The kNN-CDF method improves the detection significance by approximately 20% ( $4.8\sigma$ vs. $4.0\sigma$ ). The authors attribute this gain to the kNN statistic's ability to capture non-Gaussian, higher-order clustering information, rather than simply accessing a broader range of Fourier modes.
Systematics and Robustness:
- Tests using various CMB component-separation pipelines (SMICA, NILC, SEVEM, Commander, L-GMCA) show that maps with residual thermal SZ contamination (e.g., standard SMICA) yield lower amplitudes, confirming that SZ residuals in dense regions can suppress the signal. The "SZ-free" map provides the most robust result.
- Cross-correlations with Galactic dust maps show no significant contamination.
Tomographic Analysis:
- To probe evolving dark energy, the CIB map is split into two tomographic bins at $z_{\text{cut}} \approx 0.7$ .
- The low-redshift and high-redshift bins yield amplitudes of $A_{\text{low-}z} = 1.41 \pm 0.47$ ( $3.0\sigma$ ) and $A_{\text{high-}z} = 1.66 \pm 0.81$ ( $2.0\sigma$ ), respectively.
- While the uncertainties are too large to definitively distinguish these from the full-map result or the $\Lambda$ CDM prediction, the results are consistent with $\Lambda$ CDM.

Significance and Claims
The paper claims that the kNN-CDF method provides a more powerful dynamical test of cosmic acceleration than traditional two-point correlation functions, achieving a $4.8\sigma$ detection of the ISWRS effect using a single, foreground-cleaned CIB tracer. This detection is consistent with the $\Lambda$ CDM model. The authors emphasize that the improvement in significance stems from the method's sensitivity to non-Gaussian clustering information.

The study concludes that finer tomographic reconstruction of the CIB, particularly using multiband data from future surveys like SPHEREx, would further tighten constraints on evolving dark energy. The combination of CIB reconstruction with kNN-CDF statistics is presented as a powerful framework for future dark energy probes, leveraging the full-sky coverage and high galaxy counts of current and upcoming surveys.

Technical Summary: Dynamical Test of Cosmic Acceleration via kNN-Cross Correlation

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