Signatures of Modified Gravity Below $\mathcal{O}(10)$ Mpc in a Dynamical Dark Energy Background

This study demonstrates that within a dynamical dark energy (CPL) framework, modified gravity effects capable of suppressing low-redshift structure growth while satisfying CMB constraints must operate on comoving scales smaller than approximately 10 Mpc, a scenario that is moderately preferred over standard $\Lambda$CDM and further strengthens the case for quintom-like dark energy when combined with current cosmological datasets.

The Gist

Imagine the Universe as a giant, expanding balloon. For decades, scientists have been trying to figure out exactly how fast this balloon is inflating and what is pushing it to expand. The standard theory, called ΛCDM, suggests that the balloon is being pushed by a mysterious, invisible force called "Dark Energy" (represented by the Greek letter Lambda, Λ) and that the structure of the universe (galaxies, clusters) grows in a very predictable way, like a well-oiled machine following the rules of gravity as we know them (General Relativity).

However, recent measurements have started to show that the machine might be a bit "off." Specifically, when scientists look at how fast galaxies are clumping together, the data suggests they are growing slower than the standard theory predicts. It's as if the balloon is inflating, but the patterns drawn on its surface aren't forming as quickly as they should.

This paper, written by Yo Toda and Adrià Gómez-Valent, investigates a potential fix for this problem. They ask: What if gravity itself changes depending on how close you are to things?

The "Gravity Filter" Analogy

Think of gravity like a filter on a camera lens.

• Standard Gravity (General Relativity): The lens is perfectly clear everywhere. It sees everything the same way, whether you are looking at a distant mountain or a pebble in your hand.
• Modified Gravity (The Paper's Idea): The lens has a special filter that only kicks in when you look at things that are very close together (small scales).

The authors propose that in our universe, gravity might act normally when looking at huge distances (like between galaxy clusters), but it might get "weaker" or "stronger" when looking at smaller scales (like within a single galaxy cluster).

The "Two-Zone" Strategy

To test this, the authors divided the universe's history into two time zones:

1. The Recent Past (Redshift 0 to 1): The last few billion years.
2. The Distant Past (Redshift 1 to 3): The time before that.

They also divided space into sizes. They asked: "If gravity changes, does it change for everything, or only for things smaller than a specific size?"

They found a very specific "sweet spot." The data suggests that if gravity is going to be different from the standard rules, it must only happen on scales smaller than about 10 million light-years (roughly the size of a large galaxy cluster).

The Analogy: Imagine a rule that says, "Everyone in the city must walk at 3 mph." But then, you discover that inside individual houses, people are actually walking at 2 mph. The rule works for the whole city (large scale), but changes inside the house (small scale). The paper finds that the universe behaves like this: the "house rules" (modified gravity) only apply to small clusters, not the whole city.

Why Not Change the Whole Thing?

You might ask, "Why not just say gravity is different everywhere?"

The authors explain that if they changed gravity for everything (even the huge, distant scales), it would break the picture of the Cosmic Microwave Background (CMB). The CMB is the "baby photo" of the universe, a faint afterglow from when the universe was just a baby.

• The ISW Effect: There is a specific signal in this baby photo (called the Integrated Sachs-Wolfe effect) that acts like a fingerprint. If gravity were different on large scales, this fingerprint would look completely wrong compared to what we see in the photo.
• The Lensing Effect: Gravity also acts like a lens, bending light from the baby photo. If gravity were different everywhere, the "lens" would distort the photo in a way that doesn't match reality.

So, the paper concludes: To fix the "slow growth" problem without ruining the "baby photo," the change in gravity must be hidden on large scales and only appear on small scales (under 10 million light-years).

The "Dynamical Dark Energy" Twist

The authors also considered that the "push" behind the universe's expansion (Dark Energy) might not be a constant force, but something that changes over time (like a car that speeds up and slows down). They call this the CPL model.

When they combined this "changing push" with their "small-scale gravity filter," the results got even better.

• The standard model (ΛCDM) fits the data okay, but has some tension (it's a bit uncomfortable).
• The "Changing Push" model fits better.
• The "Changing Push" + "Small-Scale Gravity Filter" fits the best.

It's like trying to solve a puzzle. The standard pieces fit mostly, but there are gaps. Adding the "small-scale gravity" piece fills those gaps perfectly, making the whole picture much clearer.

The Bottom Line

The paper claims that:

1. Gravity might be "scale-dependent": It behaves normally on huge cosmic scales but might act differently on smaller scales (under 10 million light-years).
2. Dark Energy might be changing: The force driving the universe's expansion might not be constant.
3. Together, they solve the tension: By allowing gravity to change only on small scales and Dark Energy to evolve, the model explains why galaxies are clumping together slower than the standard theory predicts, without breaking the rules of the early universe.

The authors are careful to say this is a "hint" or a "signal" (about 2.6 to 2.8 times more likely than the standard model), not a final proof. But it suggests that to understand the universe's growth, we might need to stop thinking of gravity as a single, unchanging rulebook and start thinking of it as a rulebook that has different chapters for different sizes of cosmic neighborhoods.

Technical Summary

Technical Summary: Signatures of Modified Gravity Below O(10) Mpc in a Dynamical Dark Energy Background

Problem Statement
Current cosmological data, including the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), and Type Ia supernovae (SNIa), suggest that the component driving the Universe's accelerated expansion may be dynamical rather than a cosmological constant ($\Lambda$), with hints of dynamical dark energy (DE) appearing at the $\sim 2.5\text{--}3\sigma$ confidence level. However, the standard $\Lambda$CDM model and even dynamical DE scenarios (such as the quintom model) exhibit tension with redshift-space distortion (RSD) data and weak gravitational lensing observations. These datasets typically favor a lower level of structure growth ($S_8$ or $\sigma_8$) than predicted by the best-fit models. While neutrino mass variations or dark sector interactions can alleviate this tension to some extent, the authors investigate whether scale-dependent modified gravity (MG) effects in the late Universe could simultaneously resolve the growth tension and remain consistent with CMB constraints.

Methodology
The authors employ a phenomenological approach to parametrize departures from General Relativity (GR) without committing to a specific MG theory. They utilize the Chevallier-Polarski-Linder (CPL) parametrization for the background dark energy equation of state, $w(a) = w_0 + w_a(1-a)$, to model a dynamical DE background.

To address MG effects at the perturbation level, they introduce an effective gravitational coupling, $G_{\text{eff}}(k, a) = \mu(k, a)G$, which deviates from Newton's constant $G$. The deviation is modeled using a binning strategy in both redshift and scale:

1. Redshift Binning: Two bins are defined: $0 \le z < 1$ (parameter $\mu_1$) and $1 \le z \le 3$ (parameter $\mu_2$). For $z > 3$, $\mu$ is fixed to 1 (GR).
2. Scale Dependence: The deviation is suppressed at large scales (low $k$) to avoid altering the Integrated Sachs-Wolfe (ISW) effect and CMB lensing excessively. A critical comoving scale $\lambda_c$ is introduced, such that MG effects only manifest below this scale ($k \gg k_c = 1/\lambda_c$). The function $\mu(z, k)$ is smoothed using hyperbolic tangents to ensure continuity.

The analysis is performed using the Monte Carlo Markov Chain (MCMC) code Cobaya coupled with the modified Einstein-Boltzmann solver MGCAMB. The dataset includes:

• Planck PR4 CMB data (high-$\ell$ CamSpec likelihoods and low-$\ell$ Commander/SimAll).
• DESI DR2 BAO distance measurements.
• Pantheon+ (and alternatively DES-Dovekie) SNIa compilation.
• RSD data (using $f\sigma_{12}$ at $R=12$ Mpc to avoid $h$-dependence biases associated with $\sigma_8$).

Key Results
The authors find that to suppress structure growth at low redshifts ($z \lesssim 1$) while satisfying CMB constraints (particularly the late-time ISW effect and lensing), modified gravity effects must be confined to scales smaller than $\lambda_c \sim \mathcal{O}(10)$ Mpc.

• Scale Constraint: When $\lambda_c$ is allowed to vary, the data constrain it to be less than $\sim 28.8$ Mpc (95% CL) for a $\Lambda$CDM background, tightening slightly for a CPL background. Larger values of $\lambda_c$ (e.g., $\lambda_c \to \infty$) are disfavored because they over-suppress power at large scales, conflicting with CMB observations.
• Parameter Constraints: For a fixed $\lambda_c = 10$ Mpc in a $\Lambda$CDM background, the best-fit values are $\mu_1 \approx 0.56$ (indicating suppressed gravity at low $z$) and $\mu_2 \approx 1.12$. In a CPL background, the constraints shift slightly, with $\mu_1 \approx 0.51$ and $\mu_2 \approx 1.08$.
• Model Preference:
  • A CPL background with standard gravity is moderately preferred over $\Lambda$CDM.
  • Including MG effects in the CPL scenario strengthens this preference. The combined CPL+MG model reduces the total $\chi^2$ by approximately 15 units compared to standard $\Lambda$CDM.
  • Using the Akaike Information Criterion (AIC) to penalize additional parameters, the CPL+MG scenario is preferred over standard $\Lambda$CDM at the $2.63\sigma$ confidence level ($\Delta \text{AIC} = 5.45$).
  • Replacing Pantheon+ with the DES-Dovekie SNIa sample further enhances the significance to $2.80\sigma$.
• CPL Parameters: The inclusion of MG effects does not drastically alter the CPL background parameters ($w_0, w_a$), though they shift slightly further into the "quintom" region (where $w$ crosses the phantom divide).

Significance and Claims
The paper claims that scale-dependent modified gravity, acting only on scales smaller than $\sim 10$ Mpc in a dynamical dark energy background, provides a viable mechanism to explain the suppressed growth of structure observed in RSD data without violating CMB constraints.

The authors emphasize that this scale ($\lambda_c \sim 10$ Mpc) is significantly smaller than the characteristic scale ($\sim 100$ Mpc) explored in previous DESI full-shape clustering analyses, highlighting the necessity of probing smaller scales to detect such effects. They conclude that while the statistical significance of the hints for new physics is currently "moderate" (around $2.6\text{--}2.8\sigma$), the results suggest that a combination of dynamical dark energy and small-scale modified gravity is a necessary direction to reconcile the growth tension with the rest of the cosmological data. The findings are consistent with models inspired by massive symmetron theories or transient weak gravity scenarios.