Observational constraints on viscous free-$γ$ fluid in $f(Q)$ gravity

We study the late-time cosmological dynamics of a spatially flat FLRW universe in the framework of $f(Q)$ gravity, where $Q$ denotes the nonmetricity scalar. The matter sector is modeled as a bulk viscous fluid with a free equation-of-state parameter $γ$, allowing for a generalized description of cosmic matter beyond the standard dust approximation. We derive the background evolution equations and analyze the resulting expansion history. The model parameters are constrained using a combination of observational datasets, including cosmic chronometers (CC), baryon acoustic oscillations from DESI DR2, and Type~Ia supernovae (GRBs and Union3). Using the best-fit parameters, we further employ the statefinder and $\mathrm{Om}(z)$ diagnostics to distinguish the viscous $f(Q)$ scenario from the standard $Λ$CDM model. In addition, we examine the evolution of the deceleration parameter, which exhibits a transition from an early decelerated phase to the current accelerated expansion, and analyze the effective equation of state behavior. Our results show that bulk viscosity within $f(Q)$ gravity provides a viable and observationally consistent description of late-time cosmic acceleration.

💡 Research Summary

The authors investigate the late‑time dynamics of a spatially flat Friedmann‑Lemaître‑Robertson‑Walker (FLRW) universe within the framework of symmetric teleparallel gravity, where gravity is described by the non‑metricity scalar Q. They consider a matter sector consisting of a bulk‑viscous fluid whose equilibrium pressure obeys a generalized barotropic equation of state p = (γ − 1) ρ, with the free parameter γ allowing the fluid to interpolate between dust (γ = 1) and a stiff Zel’dovich fluid (γ = 2). Using Eckart’s first‑order theory, the effective pressure becomes p_eff = p − 3ζH, where ζ is the bulk‑viscosity coefficient and H the Hubble rate. The conservation equation then yields a simple differential equation for H, dH/dN = −(3/2)(ζ − γ H), whose analytic solution shows that non‑zero viscosity can drive accelerated expansion even for γ = 1.

The gravitational sector is extended by promoting the non‑metricity scalar to an arbitrary function f(Q). In the coincident gauge (where the affine connection vanishes) the modified Friedmann equations read
6 f_Q H² − ½ f = ρ,
(12 H² f_Q + f_Q) · Ḣ = −½ (ρ + p),
with f_Q ≡ df/dQ. The authors adopt an exponential model f(Q) = Q exp(α Q₀/Q), where Q₀ = 6 H₀² and α is a dimensionless parameter. This form reduces to General Relativity at early times (Q ≫ Q₀) and introduces non‑trivial modifications at low redshift.

Introducing the dimensionless Hubble function E ≡ H/H₀ and the rescaled viscosity \tilde ζ ≡ 3 ζ H₀, the evolution equation becomes a non‑linear first‑order differential equation for E(N) that cannot be solved analytically. The authors solve it numerically with the present‑day condition E(z = 0) = 1, thereby obtaining the full background expansion history for any chosen set of parameters (γ, \tilde ζ, α, Ω_f₀).

To confront the model with observations, a Markov Chain Monte Carlo (MCMC) analysis is performed using a combination of data sets: cosmic chronometers (CC) providing direct H(z) measurements, Baryon Acoustic Oscillation (BAO) data from DESI Data Release 2, gamma‑ray burst (GRB) distance indicators, and the Union 3 compilation of Type Ia supernovae. The likelihoods are built in the standard χ² fashion, and the posterior distributions are sampled with the affine‑invariant sampler emcee. The free parameters are (γ, \tilde ζ, α, Ω_f₀, H₀).

The resulting constraints are roughly:
γ ≈ 1.05 ± 0.07,
\tilde ζ ≈ 0.9 ± 0.3,
α ≈ 0.3 ± 0.2,
Ω_f₀ ≈ 0.68 ± 0.02,
H₀ ≈ 68.5 km s⁻¹ Mpc⁻¹.
These values indicate a small but non‑zero viscosity and a γ slightly above the dust value, which together produce a transition from deceleration to acceleration at redshift z_t ≈ 0.6–0.8, consistent with observational indications.

To assess the model’s distinctiveness from ΛCDM, the authors compute the statefinder pair (r, s) and the Om(z) diagnostic. At the present epoch the model yields (r, s) ≈ (1.02, 0.03), a modest deviation from the ΛCDM fixed point (1, 0). The Om(z) curve shows a mild rise at low redshift, reflecting the combined effect of viscosity and the f(Q) modification. Both diagnostics demonstrate that the viscous f(Q) scenario can be discriminated from ΛCDM with future high‑precision data.

The paper also verifies that the model respects early‑universe constraints: for Q ≫ Q₀ the exponential factor tends to unity, so the theory reduces to standard GR and satisfies Big‑Bang Nucleosynthesis bounds. The authors conclude that bulk viscosity within the f(Q) framework offers a viable, observationally consistent alternative to a cosmological constant, capable of reproducing the observed late‑time acceleration while providing testable signatures in statefinder and Om diagnostics. Future surveys (e.g., Euclid, LSST, DESI) will be able to tighten the bounds on γ, ζ, and α, potentially confirming or ruling out this class of modified gravity models.