🔭 astrophysics

Low-redshift constraints on structure growth from CMB lensing tomography

This paper presents low-redshift constraints on structure growth using galaxy clustering and CMB lensing data, finding that while the reconstructed growth history aligns with Planck predictions, the data without external priors favors a matter density ( $\Omega_m$ ) significantly lower than Planck's value, a result potentially influenced by theoretical uncertainties in the hybrid effective field theory bias modeling.

Original authors: Andrea Rubiola, Matteo Zennaro, Carlos García-García, David Alonso, Raul Angulo

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrea Rubiola, Matteo Zennaro, Carlos García-García, David Alonso, Raul Angulo

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

The Big Picture: Mapping the Universe's "Muscle"

Imagine the universe as a giant, invisible sponge. Over billions of years, gravity has been squeezing this sponge, pulling matter together to form clumps (like galaxies) and leaving empty spaces in between. This process is called the growth of structure.

Scientists want to know: How hard is the universe squeezing right now? And more importantly, is it squeezing at the rate we expect based on our current laws of physics?

This paper is a report card on how well the universe is growing its "muscle" (clumping matter) in the recent past (low redshift, roughly the last 4 billion years). The authors are checking if the universe is behaving exactly as predicted by the standard model of cosmology (the "Cosmological Constant" model).

The Tools: A Cosmic "X-Ray" and a "Star Map"

To measure this growth, the team used two main tools:

The Star Map (Galaxy Surveys): They looked at millions of nearby galaxies using data from the 2MPZ and WISE×SuperCOSMOS surveys. Think of this as taking a high-resolution photo of the "clumps" in our cosmic sponge.
The Cosmic X-Ray (CMB Lensing): They used data from the Planck satellite. The Cosmic Microwave Background (CMB) is the afterglow of the Big Bang. As this light travels to us, the gravity of the matter in the universe bends it, like a lens. This bending (lensing) creates a map of where the "invisible" matter is, even if we can't see the galaxies there.

The Trick: By comparing the "Star Map" (where galaxies are) with the "Cosmic X-Ray" (where the gravity is), they can measure how much the universe has grown at specific times. It's like checking the size of a tree by looking at its leaves today and comparing it to the shadow it casts on the ground.

The Method: Slicing the Cake

The universe is huge, and looking at everything at once is messy. So, the authors sliced the universe into three time periods (redshift bins):

Bin 1: The very recent past (closest to us).
Bin 2: A bit further back.
Bin 3: The furthest back they looked in this study.

They used a sophisticated mathematical model called HEFT (Hybrid Effective Field Theory). You can think of HEFT as a very advanced "translator." It translates the messy, complex way galaxies cluster together into a clean language that cosmologists can understand, allowing them to ignore the confusing noise and focus on the signal.

The Findings: Two Different Stories

The paper presents two main results, depending on which "rules" they decide to follow:

Story A: The "Strict Rule-Follower" (Using External Data)
If the authors assume the universe's matter density ( $\Omega_m$ ) is exactly what other scientists (DESI) have measured using a different method (Baryon Acoustic Oscillations, or "standard rulers"), their results are very calm.

Result: The growth of structure matches the predictions of the standard model perfectly.
The Number: They found a value called $S_8$ (a measure of how clumpy the universe is) to be 0.79. This is very close to what the Planck satellite predicted from the early universe.
Analogy: It's like checking your car's speedometer against a GPS. They match perfectly.

Story B: The "Free Spirit" (Letting the Data Speak Alone)
If they don't use the external "standard ruler" data and let their galaxy data decide the matter density on its own, things get interesting.

Result: The data suggests the universe has less matter than the standard model predicts.
The Number: They found the matter density ( $\Omega_m$ ) to be about 0.245, which is significantly lower than the Planck prediction of ~0.315.
The Tension: This is a "2.8 sigma" tension. In science, this is like flipping a coin 10 times and getting 9 heads. It's suspicious, but not impossible. It suggests that either our understanding of how galaxies cluster is slightly off, or the universe is actually less "stuff-filled" than we thought.
Analogy: If you ignore the GPS and only look at the speedometer, the car seems to be driving slower than it should be.

Why the Difference?

The authors explain that the "Free Spirit" result is driven by the shape of the galaxy clustering data. It's like looking at the pattern of footprints in the sand. The pattern suggests fewer people were walking there than expected. However, the authors are cautious. They admit that their mathematical model (HEFT) might have some small uncertainties when extrapolating to these specific conditions, which could be causing this "low matter" result.

The "Growth History"

Even with the tension on matter density, when they reconstructed the history of growth (how fast the universe was clumping over time), it looked very much like the standard model predicted.

Conclusion: The universe is growing at a rate that is "compatible" with the standard model, though there is a tiny hint that it might be growing slightly faster than expected at the very recent past.

The Bottom Line

This paper is a careful check-up on the universe's recent history.

If you trust other measurements: Everything is fine; the universe is growing exactly as the standard model says.
If you trust only this specific data: There is a hint that the universe might have less matter than we thought, but this could be due to the complexity of modeling how galaxies behave.

The authors conclude that while their results are generally in agreement with the standard model, the "low matter" hint is interesting enough to warrant more study with even better data from future telescopes. They are essentially saying, "The universe looks mostly normal, but there's a tiny wrinkle in the fabric we need to smooth out."

Technical Summary: Low-redshift constraints on structure growth from CMB lensing tomography

Problem and Motivation
The growth of matter density fluctuations is a critical cosmological observable, sensitive to the nature of dark energy and the laws of gravity. While various probes exist, including redshift-space distortions and weak lensing, there is a specific gap in constraints at low redshifts ( $z \lesssim 0.3$ ). This regime is crucial for testing potential deviations from a cosmological constant and investigating tensions in the amplitude of matter fluctuations ( $S_8$ ) observed in late-time cosmic shear data compared to Cosmic Microwave Background (CMB) predictions. Previous low-redshift lensing tomography measurements have been limited by low lensing signals and the difficulty of modeling galaxy clustering on small, non-linear scales that project onto large angular scales. This work aims to fill this gap by combining high-density galaxy samples at low redshifts with a sophisticated bias model capable of recovering robust information from relatively small scales ( $k \lesssim 0.7 \, h \, \text{Mpc}^{-1}$ ).

Methodology
The analysis utilizes a hybrid approach combining galaxy clustering and CMB lensing convergence:

Data Sources:
- Galaxy Samples: The study employs the 2MASS Photometric Redshift Survey (2MPZ) and the WISE $\times$ SuperCOSMOS (WIxSC) survey. These are divided into three tomographic bins: "Bin 1" (2MPZ, $z_m \approx 0.074$ ), "Bin 2" (WIxSC, $0.1 \le z_p < 0.25$ , $z_m \approx 0.18$ ), and "Bin 3" (WIxSC, $z_p \ge 0.25$ , $z_m \approx 0.302$ ).
- CMB Lensing: Planck 2018 "Minimum Variance" lensing convergence maps are used, covering a sky fraction overlapping with the galaxy samples ( $f_{sky} \approx 0.66$ ).
Theoretical Modeling:
- Bias Expansion: The galaxy power spectra are modeled using a Hybrid Effective Field Theory (HEFT) approach. This involves a Lagrangian bias expansion up to second order, including operators $\delta$ , $\delta^2$ , $s^2$ , and $\nabla^2\delta$ . The non-linear evolution of these operators is handled via the BACCO N-body simulations and the baccoemu neural-network emulator.
- Statistics: The analysis measures angular galaxy auto-correlations ( $C^{gg}_\ell$ ) and galaxy-CMB lensing cross-correlations ( $C^{g\kappa}_\ell$ ) using the pseudo- $C_\ell$ approach.
- Covariance: The covariance matrix includes Gaussian (disconnected), non-Gaussian (trispectrum via halo model), shot noise, and redshift distribution uncertainties.
- Likelihood and Marginalization: A Gaussian likelihood is employed. To mitigate "volume effects" (biases arising from marginalizing over a large number of nuisance bias parameters), the authors use the Approximate Analytic Marginalization (AAM) technique, which effectively imposes a Jeffreys-like prior on the bias parameters.
Scale Cuts: The analysis is restricted to physical scales $k < 0.42 \, \text{Mpc}^{-1}$ (corresponding to $\ell_{max} \approx 2000$ ) where HEFT is known to be accurate. Large scales ( $\ell < 20$ ) are excluded to avoid systematics in the galaxy auto-correlation.

Key Contributions

Low-Redshift Tomography: The paper provides constraints on structure growth specifically in the $z \lesssim 0.3$ regime, a range less explored by previous lensing tomography studies which typically focus on $z > 0.5$ .
HEFT Application: It demonstrates the application of the HEFT framework to low-redshift photometric data, allowing for the inclusion of smaller scales than traditional perturbative bias models.
Robustness Checks: The study explicitly tests the impact of volume effects by comparing AAM results with "brute-force" marginalization, showing that the latter artificially shrinks uncertainties and shifts posteriors. It also validates results against conservative scale cuts and simplified linear bias models.

Results

Cosmological Constraints ( $S_8$ and $\Omega_m$ ):
- When $\Omega_m$ is left free, the data prefers a value of $\Omega_m = 0.245 \pm 0.024$ , which is $2.8\sigma$ lower than the Planck 2018 constraints. This preference is driven by the broadband shape of the galaxy auto-correlation in the highest redshift bin (Bin 3).
- When imposing a Gaussian prior on $\Omega_m$ based on DESI BAO measurements ( $\Omega_m = 0.2975 \pm 0.0086$ ), the derived parameter $S_8 = \sigma_8 \sqrt{\Omega_m/0.3}$ is found to be $0.79 \pm 0.06$ . This is in reasonable agreement with Planck ( $S_8 = 0.832 \pm 0.013$ ) and other lensing tomography analyses.
- Without the BAO prior, the derived $S_8$ is $0.82 \pm 0.07$ .
Growth History: The reconstructed growth history, expressed as $\sigma_8(z)$ , is compatible with Planck predictions. The data shows a marginal preference for a steeper growth rate at late times, with the lowest and highest redshift measurements lying $1.8\sigma$ and $1.4\sigma$ away from Planck values, respectively, though uncertainties remain large.
Bias Parameters: The inferred higher-order bias parameters ( $b_2, b_s, b_{\nabla^2}$ ) are found to be within $1\sigma$ of the "coevolution relations" predicted by simulations for galaxies and haloes. The analysis notes that the lowest redshift bin (Bin 1) exhibits a higher linear bias ( $b_1$ ) than expected for its redshift, attributed to it probing a different galaxy population (cross-matched with 2MASS) compared to the other bins.
Goodness of Fit: The models provide a good fit to the data, with probability-to-exceed (PTE) values of 9% (free $\Omega_m$ ) and 4.5% (DESI prior), which are considered acceptable given the complexity of the model and the nature of the fluctuations.

Significance and Claims
The paper claims to provide a valuable confirmation of previous $S_8$ constraints from a complementary, lower-redshift range ( $z \lesssim 0.3$ ). It highlights that while the data is consistent with the standard $\Lambda$ CDM model when constrained by external BAO data, the intrinsic preference for a low $\Omega_m$ (driven by the broadband shape of the power spectrum) warrants caution. The authors emphasize that this preference is less robust than standard ruler measurements and may be susceptible to theoretical uncertainties in the non-linear bias model or systematic contamination in galaxy clustering.

The study concludes that while current statistical uncertainties are too large to draw definitive conclusions about deviations from a cosmological constant, the methodology establishes a baseline for future, more precise measurements. The authors suggest that replacing CMB lensing with cosmic shear from current surveys (DES, KiDS, HSC) could improve bounds on $\sigma_8(z)$ in this low-redshift regime, and that future data from photometric surveys (Rubin Observatory) and CMB experiments (Simons Observatory) will be necessary to reach the precision required to test dynamical dark energy models.