Insights in $f(Q)$ cosmology: the relevance of the connection

TL;DR

This work addresses how the choice of affine connection in f(Q) gravity, within symmetric teleparallelism, shapes cosmological dynamics. By analyzing three compatible connections in FLRW spacetimes and applying both analytical methods and Born-Infeld inspired numerics, the authors show that different connections yield markedly different evolutions for the same f(Q) model, including de Sitter regeneration of singularities and oscillatory or GR-like behavior near singular regimes. The Born-Infeld f(Q) case demonstrates that, for the Γ_Q^{(III)} connection, both the Big Bang and Big Rip can be avoided, yielding a non-singular early universe and late-time de Sitter attractor, while the other connections can retain singularities albeit with connection-dependent features. Overall, the paper argues that the connection in f(Q) gravity is a physically relevant input that can address fundamental cosmological questions without introducing extra degrees of freedom, underscoring the geometrical richness of the theory and guiding future model-building and observational tests.

Abstract

We explore the role of the affine connection in $f(Q)$ gravity, a modified theory where gravity is governed by non-metricity within the symmetric teleparallel framework. Although the connection is constrained to be flat and torsionless, it is not uniquely determined by the metric, allowing for multiple physically distinct formulations. We analyze three such connections compatible with a homogeneous and isotropic universe to show that they yield markedly different cosmological dynamics, even under the same functional form of $f(Q)$. Using both analytical and numerical methods, including a Born-Infeld type model of $f(Q)$, we demonstrate that specific connections can resolve cosmological singularities like the Big Bang and Big Rip, replacing them with smooth de Sitter phases. Others retain singularities but with notable modifications in their behavior. These findings highlight the physical relevance of connection choice in $f(Q)$ gravity and its potential to address fundamental cosmological questions.

Figures (6)

• Figure 1: (Left) The e-folds $N_e(t)$ and the square of the Hubble function $H^2$ as a function of cosmic time $t$ for the non-singular cosmology for the connection $\Gamma_Q^{\text{(III)}}$. The Hubble function is bounded by $H^2\le\lambda/12$ as shown by the dashed line. (Right) $H^2$ as a function of $\rho$. The standard GR solution $H^2=\rho/3$ is represented by the gray line on the right panel.
• Figure 2: The e-folds $N_e(t)$ (top), and the effective equation of state $w_{\textrm{eff}}$ as a function of cosmic time $t$ (bottom) for the cosmology with connection $\Gamma ^{\text{(I)}}_Q$ in a radiation-dominated universe. The initial values for $C_3$ are chosen as $C_3(t_0)=H(t_0)/20$, $H(t_0)/10$, and $H(t_0)/5$, from blue to cyan. The universe has a Big Bang singularity with an effective equation of state close to $w_{\textrm{eff}}\approx 1$ (red lines in the bottom panel).
• Figure 3: The e-folds $N_e$ (top) and the ratio $C_2/H$ (bottom) as functions of cosmic time $t$ for the cosmology with connection $\Gamma ^{\text{(II)}}_Q$. Near the Big Bang singularity where $a\rightarrow0$, we have $H^2a^4$ approaches a non-zero constant, which implies $H^2\propto\rho$. Also, the ratio $C_2/H$ scales as $1/a$ near the singularity. In this figure, the initial conditions for $C_2(t_0)/H(t_0)$ is set to be $-1/3$, $-1/2$, and $-1$, from blue to cyan.
• Figure 4: The e-folds $N_e(t)$ (top) and the square of the Hubble function $H^2$ as a function of cosmic time (bottom) of the non-singular cosmology for the connection $\Gamma ^{\text{(III)}}_Q$ with a phantom-dominated universe. The Hubble function is bounded by $H^2\le\lambda/12$ as shown by the dashed line in the bottom panel. The Big Rip singularity is replaced by a late-time de Sitter phase.
• Figure 5: The e-folds $N_e(t)$ (top), the ratio $C_3/H$ (middle), and the effective equation of state $w_{\textrm{eff}}$ (bottom) as a function of cosmic time $t$ for the phantom-dominated cosmology with connection $\Gamma ^{\text{(I)}}_Q$. The vertical dashed lines represent the time of the Big Rip singularity. The initial conditions for $C_3$ are chosen as $H(t_0)/C_3(t_0)=4$, $3.5$, $3$, $2.5$, and $2$, from blue to cyan.