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Testing Scale-Dependent Suppression of Structure Growth in the Linear Regime

This paper investigates scale-dependent modified gravity as a solution to the observed suppression of cosmic structure growth, finding that such models provide a viable explanation consistent with Planck CMB data and resolve tensions without exacerbating cosmological discrepancies.

Original authors: Fernanda Oliveira, Miguel A. Sabogal, Felipe Avila, Rafael C. Nunes, Armando Bernui

Published 2026-08-04

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CC BY 4.0

Original authors: Fernanda Oliveira, Miguel A. Sabogal, Felipe Avila, Rafael C. Nunes, Armando Bernui

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, invisible ocean. For decades, scientists have been trying to understand how the "waves" in this ocean—huge clumps of galaxies and dark matter—grow and crash into each other over billions of years. The standard story, known as the $\Lambda$ CDM model, suggests that gravity acts like a universal rulebook: it pulls everything together in exactly the same way, no matter how big or small the clump is. It's like a baker who uses the exact same amount of pressure to knead a tiny cookie dough ball and a massive loaf of bread. But recently, some astronomers noticed something odd. When they looked at the growth of these cosmic structures, the data seemed to whisper that the "pressure" might not be the same everywhere. Maybe gravity gets a little weaker when dealing with the biggest, most distant waves in the ocean. This paper dives into that mystery, asking a simple but profound question: Does the way the universe grows depend on the size of the structure?

The researchers, a team of cosmologists from Brazil, Italy, and beyond, decided to test this idea using a "scale-dependent" model. Think of it like checking if the rules of a video game change when you zoom out to see the whole map versus zooming in on a single character. In their model, they introduced a special "knob" (a parameter they call $A$ ) that allows gravity to behave differently depending on the scale of the cosmic structure. If the knob is turned off, the universe follows the standard rules. If it's turned on, gravity gets a little weaker on the largest scales, slowing down the growth of massive galaxy clusters.

To find out if this knob needs to be turned, the team acted like cosmic detectives, gathering clues from four different sources: the speed at which galaxies are moving apart (measured by cosmic chronometers), the brightness of exploding stars called Type Ia supernovae, the ripples in the oldest light in the universe (the Cosmic Microwave Background), and, most importantly, direct measurements of how fast cosmic structures are growing ( $f\sigma_8$ ). They fed all this data into a powerful computer program that ran millions of simulations to see which version of the universe fit the clues best.

Here is what they found: The standard model, where gravity is the same everywhere, still does a fantastic job. However, when they added the "scale-dependent" knob to the mix, the data showed a slight, intriguing preference for it being turned on. Specifically, they found a 2.2 sigma hint (which is like a "maybe" in the world of science, meaning it's interesting but not a slam-dunk proof) that the growth of cosmic structures is indeed being suppressed on the largest scales. It's as if the universe is saying, "Hey, for the biggest structures, maybe gravity isn't quite as strong as we thought."

Crucially, this new model doesn't break the universe. It still fits the background expansion and the early history of the cosmos perfectly, matching the famous Planck satellite's observations. It also suggests that this "weakening" of gravity is more noticeable in the past, during the era when matter ruled the universe, than it is today. While the paper doesn't claim to have solved the biggest mysteries of cosmology (like the "Hubble Tension" or the "S8 Tension"), it offers a fresh, viable explanation for why the growth of the universe might look a little different than the standard recipe predicts. The authors conclude that while the standard model remains the champion, there is a faint, exciting possibility that the rules of the cosmic game change depending on how big you look, a discovery that future, even more powerful telescopes will need to confirm.

Technical Summary: Testing Scale-Dependent Suppression of Structure Growth in the Linear Regime

Problem Statement
Recent analyses of the growth rate of cosmic structures, specifically the $f\sigma_8(z)$ dataset, have reported a suppression in structure growth that challenges the standard $\Lambda$ CDM model. While the $\Lambda$ CDM framework successfully reproduces background expansion data (Type Ia Supernovae, BAO, CMB), the growth of matter perturbations is often assumed to be scale-independent on sub-horizon scales. However, theoretical considerations suggest that on scales larger than $50 \, h^{-1}\text{Mpc}$ , the linear matter density contrast evolution equation may require a scale-dependent term to accurately reproduce General Relativity (GR) solutions. This paper investigates whether a scale-dependent modification to the growth equation can explain the observed suppression in structure growth without exacerbating existing cosmological tensions, such as the $S_8$ and $H_0$ discrepancies.

Methodology
The authors propose a phenomenological model where the evolution of linear matter density perturbations, $\delta_m$ , is governed by a modified growth equation that includes an explicit dependence on the wavenumber $k$ . The standard growth equation is modified by introducing a scale-dependent function $\xi(t, k)$ , which alters the effective gravitational coupling $G_{\text{eff}}(t, k)$ :
$\ddot{\delta}_m + 2H\dot{\delta}_m - 4\pi G_{\text{eff}}(t, k) \bar{\rho}_m \delta_m = 0$
where $G_{\text{eff}}(t, k) = G / (1 + \xi(t, k))$ and $\xi(t, k) \propto A / (a(t) k^2)$ . The amplitude of this correction is parameterized by a transformed variable $B = \ln(1 + A/10^4)$ , where $B=0$ recovers the standard scale-independent $\Lambda$ CDM limit.

The study employs a joint Markov Chain Monte Carlo (MCMC) analysis using the MontePython sampler and the CLASS Boltzmann solver. The analysis combines four distinct datasets:

$f\sigma_8(z)$ : 35 uncorrelated measurements of the growth rate derived from redshift-space distortions.
Cosmic Chronometers (CC): 31 direct measurements of the Hubble parameter $H(z)$ .
Type Ia Supernovae (SN): 1550 distance modulus measurements from the PantheonPlus sample.
Cosmic Microwave Background (CMB): Planck 2018 legacy data (temperature, polarization, and lensing power spectra).

The parameter space includes the standard six $\Lambda$ CDM parameters plus the new scale-dependent parameter $B$ . The analysis fixes the comoving wavenumber to $k = 0.1 \, h \text{Mpc}^{-1}$ to remain within the linear regime.

Key Results

Scale-Dependent Growth Evidence: When combining all datasets ( $f\sigma_8 + \text{CC} + \text{SN} + \text{CMB}$ ), the analysis yields a non-zero central value for the scale amplitude parameter: $B = 0.36^{+0.14}_{-0.12}$ (corresponding to $A = 4425 \pm 2000$ ). This represents a statistical significance of $2.2\sigma$ for a non-zero scale-dependent correction. In contrast, analyses excluding CMB data only provide upper limits on $A$ .
Redshift Dependence: The scale-dependent model predicts a suppression of structure growth that is more pronounced during the matter-dominated era ( $z \gtrsim 0.5$ ) compared to the dark-energy-dominated epoch. At low redshifts ( $z \lesssim 0.5$ ), the model converges toward the standard $\Lambda$ CDM prediction.
Cosmological Parameters: The inclusion of the scale-dependent parameter does not significantly alter the constraints on the Hubble constant or the matter density. The results are consistent with Planck constraints: $H_0 = 67.38 \pm 0.52 \, \text{km s}^{-1} \text{Mpc}^{-1}$ and $\Omega_{m,0} = 0.3152 \pm 0.0072$ .
$S_8$ Tension: The derived $S_8$ parameter is $0.831 \pm 0.011$ , which is in excellent agreement with the Planck Collaboration value ( $0.832 \pm 0.013$ ). The authors note that while the model offers a mechanism to reduce clustering amplitude, it does not significantly alleviate the current $S_8$ tension beyond what is already observed in standard analyses.
Model Comparison: The scale-dependent model provides a better fit to the data than standard $\Lambda$ CDM, with a total $\Delta\chi^2_{\text{min}} = -6.04$ , driven primarily by the $f\sigma_8$ dataset ( $\Delta\chi^2_{\text{min}} = -6.42$ ). However, Bayesian evidence calculations yield an inconclusive result ( $|\ln B_{ij}| < 1$ ), indicating that the scale-dependent model is equally viable as $\Lambda$ CDM but not statistically preferred when penalizing for the additional free parameter.

Significance and Claims
The paper claims that its primary contribution is the demonstration that scale-dependent growth models provide a viable explanation for the observed clustering of matter without introducing new tensions in background cosmology. The detection of a $2.2\sigma$ preference for scale-dependent growth suggests that the suppression of structure growth at large scales may be a physical reality, potentially hinting at modified gravity or relativistic corrections beyond the Newtonian approximation.

However, the authors maintain a modest stance regarding the statistical significance. They emphasize that the $2.2\sigma$ indication is "mild" and should be interpreted with caution due to potential systematic uncertainties in large-scale structure data and modeling assumptions. The paper concludes that while the standard $\Lambda$ CDM model remains fully consistent with observations, the results motivate the need for future high-precision surveys to probe larger volumes and wider scale ranges to confirm or rule out these subtle scale-dependent effects.

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