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Late-Time Cosmology and Structure Formation in Quadratic $f(Q)$ Gravity

This paper investigates a quadratic $f(Q)$ gravity model that introduces an $H^4$ term to the Friedmann equation, demonstrating that its weakened effective gravitational coupling can reproduce background cosmological observations while naturally alleviating the $S_8$ tension by suppressing structure growth.

Original authors: G. G. L. Nashed, P. V. Tretyakov, A. Eid

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: G. G. L. Nashed, P. V. Tretyakov, A. Eid

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 decades, scientists have used a standard recipe to describe how this balloon inflates and how the "stuff" inside it (like galaxies) clumps together. This recipe is called $\Lambda$ CDM. It works incredibly well, but recently, measurements have started to show tiny cracks in the recipe. Two main problems have emerged:

The Hubble Tension: Different ways of measuring how fast the balloon is expanding give slightly different answers.
The $S_8$ Tension: When we look at how much the "stuff" inside the balloon has clumped together into galaxies and clusters, it seems less clumpy than our standard recipe predicts.

This paper proposes a new, slightly tweaked recipe called Quadratic $f(Q)$ Gravity. Instead of changing the ingredients (like adding new types of dark energy), the authors suggest changing the cooking instructions (the laws of gravity) themselves.

Here is a simple breakdown of what they did and found:

1. The New Ingredient: A "Speed-Squared" Rule

In standard gravity, the force of gravity depends on how much stuff is there. In this new model, the authors added a "quadratic" term to the gravity equation.

The Analogy: Imagine driving a car.

Standard Gravity ( $\Lambda$ CDM): Your speed is determined only by how hard you press the gas pedal (the amount of matter/energy).
Quadratic $f(Q)$ Gravity: Your speed is determined by the gas pedal plus a bonus rule: if you are already going very fast, the engine gets a little extra boost (or a little extra drag, depending on the setting).

In the universe, this "speed" is the expansion rate ( $H$ ). Because the new rule involves the square of the speed ( $H^2$ ) and then squares that again ( $H^4$ ), it becomes very important when the universe was expanding quickly in the past, but barely matters when things are moving slowly today.

2. The Background: How the Universe Expands

The authors calculated how this new rule changes the expansion history of the universe.

The Result: At low speeds (today), the universe looks almost exactly the same as the standard recipe. It's like driving on a flat road; the extra engine boost doesn't do much.
The Twist: At higher speeds (in the past, around redshift $z \approx 1$ ), the new rule kicks in. It creates a slight "wiggle" in the expansion history.
Why it matters: This wiggle mimics the behavior of "Dynamical Dark Energy"—a fancy term for a mysterious force that changes over time. This helps explain why recent telescope data (like DESI) hints that dark energy might not be constant, without needing to invent a new, invisible substance.

3. The Growth: How Galaxies Clump Together

This is where the paper solves the bigger mystery: the $S_8$ Tension.

The Analogy: Imagine gravity as a magnet trying to pull metal shavings (galaxies) together to form a pile.

Standard Gravity: The magnet is strong and constant. It pulls the shavings together very efficiently, creating big, dense piles.
Quadratic $f(Q)$ Gravity: The authors found that for the model to work, the "magnet" (gravity) actually gets weaker over time in this specific setup.

The Consequence:
Because gravity is slightly weaker in this model, the metal shavings don't clump together as tightly as the standard recipe predicts.

Linear Growth: The rate at which structures grow slows down.
Nonlinear Growth: When a cloud of gas tries to collapse into a galaxy cluster, the "critical threshold" (the point where it finally collapses) becomes harder to reach. It's like trying to build a sandcastle with wet sand that is slightly less sticky; you need more sand to make it hold its shape.

The Solution:
Because the gravity is weaker, the universe ends up with less clumping than the standard model predicts. This perfectly matches the recent observations that show the universe is "smoother" (less clumpy) than we thought. This naturally resolves the $S_8$ tension without needing to fudge the numbers.

4. The Catch: The "Positive" Rule Only

The paper notes that this only works if the new parameter ( $\alpha$ ) is positive.

If $\alpha$ is positive: Gravity gets slightly weaker, clumping is suppressed, and the model fits the data.
If $\alpha$ is negative: Gravity gets stronger. This would cause the universe to clump too much (worse than the problem we are trying to fix) and could lead to mathematical "explosions" (divergences) where the equations break down. So, the universe likely follows the "positive" path.

Summary

The authors propose that gravity isn't quite as simple as Einstein originally wrote it down. By adding a small, speed-dependent "quadratic" correction to the laws of gravity:

They can explain why the universe's expansion history looks slightly different from the standard model at certain times (hinting at dynamical dark energy).
They naturally explain why galaxies aren't clumping together as much as we expected (solving the $S_8$ tension).

It's a "minimal" change—no new particles, no new fields—just a slight adjustment to the geometry of space and time that makes the universe look exactly like the one we are observing today.

Technical Summary: Late-Time Cosmology and Structure Formation in Quadratic f(Q) Gravity

Problem Statement
The standard $\Lambda$ CDM cosmological model, while successful in explaining a wide range of observations, faces persistent observational tensions, specifically the Hubble tension ( $H_0$ ) and the $S_8$ tension (the amplitude of matter fluctuations). Recent data from the Dark Energy Spectroscopic Instrument (DESI) and other large-scale structure surveys suggest mild deviations from $\Lambda$ CDM, hinting at dynamical dark energy (DDE). This paper investigates whether a modified gravity framework, specifically quadratic symmetric teleparallel gravity, can naturally reproduce these deviations and alleviate the $S_8$ tension without invoking a cosmological constant or additional scalar fields.

Methodology
The authors analyze a quadratic extension of $f(Q)$ gravity, defined by the action function $f(Q) = Q + \alpha Q^2 + \beta$ , where $Q$ is the non-metricity scalar, $\alpha$ encodes nonlinear corrections, and $\beta$ acts as an effective cosmological constant. The study proceeds through three distinct levels of analysis:

Background Evolution:
- Assuming a spatially flat FLRW metric in the coincident gauge, the authors derive the modified Friedmann equations. Since $Q \propto H^2$ , the quadratic term introduces an $H^4$ contribution.
- An exact algebraic solution for the normalized Hubble parameter $E(z) = H(z)/H_0$ is derived.
- The model is reconstructed as an effective dark-energy sector to calculate the effective equation of state ( $w_{de}$ ) and total equation of state ( $w_{eff}$ ).
- The background evolution is constrained and compared against $\Lambda$ CDM using Type Ia supernovae, Baryon Acoustic Oscillations (BAO), and cosmic-chronometer data.
Linear Perturbations:
- The study extends to linear cosmological perturbations in the Newtonian gauge.
- The Poisson equation is modified to include a time-dependent effective gravitational coupling, $G_{eff}(z) = G [1 + \frac{2}{3}A E^2(z)]^{-1}$ , where $A = 18\alpha H_0^2$ .
- The linear growth factor $D(z)$ , growth rate $f(z)$ , and the redshift-space distortion observable $f\sigma_8(z)$ are computed using the CLASS Boltzmann code, adapted to include the modified background and perturbation equations.
Nonlinear Structure Formation:
- The analysis moves to the nonlinear regime to assess the impact on halo formation.
- The spherical collapse threshold ( $\delta_c$ ) is recalculated based on the weakened gravitational coupling.
- The halo mass function (HMF) is derived to determine the abundance of massive clusters.
- The parameter $S_8 = \sigma_8 \sqrt{\Omega_m/0.3}$ is evaluated to test the model's ability to resolve the $S_8$ tension.

Key Contributions and Results

Background Dynamics: The quadratic correction scales as $H^4$ , leading to redshift-dependent modifications. For $\alpha > 0$ , the expansion rate $E(z)$ increases relative to $\Lambda$ CDM at intermediate redshifts ( $z \gtrsim 1$ ). The model reconstructs an effective dark-energy equation of state that exhibits dynamical behavior, including phantom crossings and sign changes in effective density, mimicking the DDE hints from DESI data without introducing new fields.
Modified Gravity and Growth: The effective gravitational coupling $G_{eff}$ becomes time-dependent. For $\alpha > 0$ , $G_{eff} < G$ , resulting in a weakened gravitational interaction. This suppression leads to a reduced linear growth factor $D(z)$ , a lower growth rate $f(z)$ , and a decreased $f\sigma_8(z)$ compared to $\Lambda$ CDM.
Nonlinear Suppression and $S_8$ : In the nonlinear regime, the weakened gravity increases the spherical-collapse threshold ( $\delta_c$ ) and suppresses the halo mass function. Consequently, the predicted value of $S_8$ is lower than the Planck- $\Lambda$ CDM prediction. This provides a natural geometric mechanism to alleviate the $S_8$ tension.
Parameter Constraints: The analysis indicates that $\alpha > 0$ is the viable branch, consistent with observational data and capable of resolving tensions. Conversely, $\alpha < 0$ leads to $G_{eff} > G$ , enhancing structure formation and potentially causing unphysical divergences, making it observationally disfavored.
Slow-Roll Dynamics: The study defines Hubble slow-roll parameters ( $\epsilon_H, \eta_H$ ) for the late-time universe. The model exhibits a quasi-de Sitter regime today, with mild deviations at intermediate redshifts consistent with the transition from matter domination to dark-energy domination.

Significance and Claims
The paper claims that the quadratic $f(Q)$ extension offers a viable, theoretically motivated modified-gravity explanation for recent observational hints of dynamical dark energy. Its primary significance lies in its ability to:

Reproduce $\Lambda$ CDM at low redshifts, ensuring consistency with local distance-ladder measurements.
Generate mild deviations at intermediate redshifts that mimic dynamical dark energy, aligning with DESI BAO data.
Naturally alleviate the $S_8$ tension through the suppression of structure formation caused by the time-dependent weakening of gravity ( $G_{eff} < G$ ) for $\alpha > 0$ .

The authors emphasize that these results arise purely from the geometric modification of the non-metricity scalar, avoiding the need for additional dynamical degrees of freedom or fine-tuning. The work complements earlier studies on inflationary dynamics by demonstrating that the same class of nonlinear geometric modifications can also govern late-time cosmic acceleration and structure formation.