Testing Non-Coincident $f(Q)$-gravity with DESI DR2 BAO and GRBs

We consider the $f\left(Q\right)$-theory for the description of dark energy with a non-trivial connection defined in the non-coincident gauge. The resulting field equations form a two-scalar-field, quintom-like gravitational model. For the power-law model $f\left(Q\right)\simeq Q^{\frac{n}{n-1}}$, we construct an analytic expression for the dynamical evolution of dark energy, which depends on the parameter $n$. We constrain this dark energy model using the the baryon acoustic oscillations from DESI DR2, and gamma-ray bursts. The cosmological data provides $n\simeq 0.33$. The $f\left(Q\right)$-model challenges the $\mathrm{\Lambda }$CDM by providing a smaller value for ${\chi }_{\min }^{}$. Although the Akaike Information Criterion indicates that the two cosmological models fit the data from the Pantheon+, the cosmic chronometers and the baryonic acoustic oscillators in a similar way, when the gamma-ray bursts are introduced there is weak evidence in favor of the $f\left(Q\right)$-theory.