How different connections in flat FLRW geometry impact energy conditions in $f(Q)$ theory?

Understand

In this study of the modified $f(Q)$ theory of gravity in the spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime, we explore all the affine connections compatible with the symmetric teleparallel structure; three classes of such connections exist, each involving an unknown time-varying parameter $\gamma(t)$.

• Assuming ordinary barotropic fluid as the matter source, we first derive the Friedmann-like pressure and energy equations.
• Next we impose constraints on the parameters of two separate $f(Q)$ models, $f(Q)=Q+\beta Q^2$ and $f(Q)=Q+\beta\sqrt{-Q}$ from the traditional energy conditions.
• Observational values of some prominent cosmological parameters are used for this purpose, yielding an effective equation of state $\omega^{eff}=-0.82$.