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

Abstract: We investigate the cosmological evolution associated with the quadratic symmetric teleparallel gravity framework, ( f(Q)=Q+αQ^{2}+β) where the relation (Q\propto H^{2}) generates an additional (H^{4}) contribution to the Friedmann equation. Using the exact algebraic solution for $H(z)$, we reconstruct the effective dark-energy sector and compare the background evolution with $Λ$CDM using Type Ia supernovae, BAO, and cosmic-chronometer data. At the perturbative level, the model modifies the Poisson equation through a time-dependent effective gravitational coupling $$G_{\textrm eff}(z)=G\big[1+\tfrac{2}{3}A E<sup>{2}(z)\big]<sup>{-1}$$, where $A=18αH_{0}<sup>{2}$. For $α&gt;0$ this produces a weakened gravitational interaction, suppressing the linear growth factor $D(z)$, the growth rate $f(z)$, and the RSD observable $fσ<em>{8}(z)$. In the nonlinear regime, the reduced gravitational strength increases the spherical-collapse threshold and suppresses the halo mass function, leading to a lower predicted value of $S</em>{8}=σ<em>{8}\sqrt{Ω</em>{m}/0.3}$. Thus, the quadratic $f(Q)$ extension can reproduce mild deviations from $Λ$CDM at the background level while naturally alleviating the $S_{8}$ tension, offering a viable modified-gravity explanation for recent observational hints of dynamical dark energy.

• The paper derives a closed-form expansion history in quadratic f(Q) gravity, showing that positive quadratic corrections produce mild intermediate-redshift deviations from ΛCDM while preserving near-standard late-time behavior.
• The paper finds that the time-dependent coupling G_eff/G = [1 + (2/3)AE²(z)]⁻¹ suppresses growth, raises the collapse threshold, and reduces halo abundance for positive α, offering a potential mechanism for easing the S8 tension.
• The paper implements the modified background and Poisson equation in CLASS, but emphasizes that full likelihood analyses, calibrated nonlinear predictions, CMB constraints, and high-redshift structure tests remain necessary to establish viability.

Overview and motivation

This paper analyzes the cosmological consequences of the quadratic extension of symmetric teleparallel ($f(Q)$) gravity,

$f(Q) = Q + \alpha Q^2 + \beta,$

where $Q$ is the non-metricity scalar, $\alpha$ controls the leading nonlinear correction, and $\beta$ acts as an effective cosmological constant (2606.02660). The motivation is twofold. First, persistent observational tensions—most notably the $S_8$ tension between CMB-inferred and weak-lensing/galaxy-clustering measurements of the matter-fluctuation amplitude—suggest that $\Lambda$CDM may be incomplete. Second, DESI BAO measurements have hinted at mild dynamical dark energy (DDE) behavior at intermediate redshifts. Rather than adopting phenomenological dark-energy parameterizations, the authors pursue a modified-gravity explanation in which gravity is encoded entirely in non-metricity, with curvature and torsion vanishing identically.

Because $Q = 6H^2$ for a spatially flat FLRW metric in the coincident gauge, the quadratic correction contributes a term proportional to $H^4$ to the Friedmann equation. This scaling makes the modification negligible at low redshift but increasingly relevant at intermediate redshifts ($z \gtrsim 1$), producing redshift-dependent departures from $f(Q) = Q + \alpha Q^2 + \beta,$0CDM that can mimic DDE without introducing extra dynamical fields. The authors also note that the same $f(Q) = Q + \alpha Q^2 + \beta,$1 term becomes dominant at high energy scales and could drive an inflationary phase, although inflationary dynamics are explicitly deferred to future work.

Field equations and background dynamics

The paper derives the full $f(Q) = Q + \alpha Q^2 + \beta,$2 field equations by varying the action with respect to both the metric and the connection. The metric equations remain second order—a key theoretical advantage over many curvature-based modifications—and reduce to Einstein's equations when $f(Q) = Q + \alpha Q^2 + \beta,$3. The connection equation takes the form $f(Q) = Q + \alpha Q^2 + \beta,$4, where $f(Q) = Q + \alpha Q^2 + \beta,$5 is the non-metricity superpotential.

For the flat FLRW background, the modified Friedmann equation reads

$f(Q) = Q + \alpha Q^2 + \beta,$6

Introducing $f(Q) = Q + \alpha Q^2 + \beta,$7, $f(Q) = Q + \alpha Q^2 + \beta,$8, and $f(Q) = Q + \alpha Q^2 + \beta,$9, this reduces to an algebraic quartic whose physical branch is

$Q$0

The closed-form solution is a notable strength: it permits analytic reconstruction of all background diagnostics. Normalization at $Q$1 fixes $Q$2.

Reinterpreting the modification as an effective dark-energy sector, the authors reconstruct $Q$3, $Q$4, $Q$5, and $Q$6. The main qualitative findings are:

• For $Q$7, the expansion rate exceeds $Q$8CDM at intermediate and high redshift; for $Q$9 it is suppressed.
• Positive $\alpha$0 produces mildly non-monotonic behavior of $\alpha$1 around $\alpha$2, mimicking DDE parameterizations.
• The reconstructed $\alpha$3 exhibits vertical divergences where $\alpha$4 crosses zero. The authors argue these are not pathological: $\alpha$5 is an effective quantity obtained by rewriting the modified Friedmann equation in GR form, and sign changes simply indicate epochs where the modification acts attractively.
• Small $\alpha$6 can generate sharp spikes in $\alpha$7 near $\alpha$8 while $\alpha$9 itself remains finite and well behaved; such features are strongly constrained by distance data.

The model reproduces a matter-to-acceleration transition at $\beta$0, consistent with the observationally favored range $\beta$1–$\beta$2. Present-day Hubble flow parameters are nearly identical to $\beta$3CDM ($\beta$4, $\beta$5 versus $\beta$6), confirming that the late-time dynamics sit close to quasi-de Sitter.

Linear perturbations and growth

At the perturbative level, working in Newtonian gauge with the coincident connection, the Poisson equation acquires a time-dependent effective gravitational coupling:

$\beta$7

Since symmetric teleparallel gravity induces no anisotropic stress at linear order, the potentials satisfy $\beta$8, so lensing and dynamical probes see the same coupling. The growth equation for subhorizon modes follows directly from combining the modified Poisson equation with the standard matter-contrast evolution written in redshift space.

For $\beta$9, $S_8$0 throughout the relevant redshift range, suppressing the linear growth factor $S_8$1, the logarithmic growth rate $S_8$2, and the RSD observable $S_8$3. The suppression is scale-independent but time-varying, with the strongest deviations at intermediate redshifts where the $S_8$4 correction matters most. Negative $S_8$5 enhances growth and is disfavoured by current RSD and weak-lensing data. The authors implement the model in the CLASS Boltzmann solver by replacing the background Friedmann equation with the algebraic solution and substituting $S_8$6 into the Poisson equation—an implementation that enables direct comparison with full-shape clustering data.

An important implication follows immediately: because the same parameter $S_8$7 controls both the background expansion history and the strength of clustering, the model links geometric and growth observables in a correlated way, providing a sharper test than models that modify only one sector.

Nonlinear structure formation and the $S_8$8 tension

In the nonlinear regime, weakened gravity raises the spherical-collapse threshold above the standard value $S_8$9, since top-hat overdensities take longer to collapse when $\Lambda$0 is reduced. Combined with the suppressed linear variance $\Lambda$1, this produces a significant reduction of the halo mass function at fixed mass. Massive cluster counts therefore constitute a sensitive probe of $\Lambda$2.

The observable consequence is a lower predicted value of

$\Lambda$3

arising from two intertwined effects: slower linear growth and reduced nonlinear collapse abundance. The authors present this as a natural mechanism for alleviating the $\Lambda$4 tension—one that requires no additional fields or fine-tuning, since it follows directly from the structure of the theory. They further note that only the $\Lambda$5 branch is cosmologically viable: negative $\Lambda$6 leads to enhanced and potentially divergent $\Lambda$7.

Regarding early structure formation, the authors concede that their numerical analysis covers only $\Lambda$8, but argue this restriction is innocuous because $\Lambda$9 is strongly subdominant to the linear $Q = 6H^2$0 term at higher redshift, where growth reverts to the standard GR behavior $Q = 6H^2$1. On this basis they claim the model permits formation of galaxies observed at $Q = 6H^2$2; however, this claim is asserted rather than demonstrated with explicit high-redshift growth calculations or halo-abundance checks against JWST-era data.

Limitations and open questions

Several caveats bear on the results. The comparison with data uses SNe Ia, BAO, and cosmic chronometers at the background level, plus linear growth diagnostics, but the paper does not report a full joint statistical fit with credible intervals on $Q = 6H^2$3, nor does it quantify the improvement over $Q = 6H^2$4CDM via information criteria despite stating this as a goal in the introduction. The spherical-collapse treatment is approximate—the correction $Q = 6H^2$5 is written schematically as a first-order shift rather than computed numerically—and the halo mass function relies on standard Press–Schechter-type machinery whose calibration in modified gravity is not established here. The claim that the model "naturally alleviates" the $Q = 6H^2$6 tension would require demonstrating that the required suppression magnitude is achievable within the $Q = 6H^2$7 range allowed by background data; this quantitative consistency check remains open. Finally, the inflationary potential of the $Q = 6H^2$8 term, the behavior of the effective dark-energy density sign changes under scrutiny by perturbation-level stability analyses (e.g., absence of ghost/gradient instabilities), and full CMB constraints are explicitly left unexplored.

Conclusion

This work establishes that the minimal quadratic extension of symmetric teleparallel gravity yields a closed-form expansion history closely tracking $Q = 6H^2$9CDM at low redshift while generating mild DDE-like deviations at intermediate redshifts compatible in character with recent DESI hints. Its central result is the time-dependent effective gravitational coupling $H^4$0, which for positive $H^4$1 suppresses linear growth, raises the collapse threshold, lowers the halo mass function, and thereby reduces $H^4$2 through a single geometric parameter. The viability of this mechanism as a quantified resolution of the $H^4$3 tension, its consistency with full CMB and large-scale-structure likelihoods, and its high-redshift behavior await dedicated statistical and numerical follow-up.