A No-Go Theorem for Curvature Neutrality in f(Q) Cosmology?

Abstract

In the framework of $f(Q)$ gravity, we investigate whether spatial curvature can be made dynamically invisible in the cosmological background equations. We consider open spatial sections, $k<0$, on an explicitly specified homogeneous and isotropic symmetric-teleparallel connection branch. For $k\neq0$, the coincident gauge cannot be imposed simultaneously with the standard curved-FLRW coordinate form. Re-deriving the background equations from the minisuperspace action, we show that curvature enters through $x=H+δ\sqrt{-k}/a$ and appears with inequivalent weights in the energy and pressure equations. For $f\in C^3((0,\infty))$, we prove a branch-specific no-go theorem: strict curvature independence for every scale factor and every $k<0$ requires $f=\mathrm{const}$, which contains no metric kinetic term. STEGR is not curvature-neutral, since its Friedmann equation retains the usual $3k/a^2$ contribution. A weaker cancellation, imposed only on backgrounds satisfying $δ\sqrt{-k}/a=cH$, necessarily produces a coasting expansion and yields $f(Q)=A Q^{(c+2)/2}+B$, or $f(Q)=A\ln Q+B$ when $c=-2$. For the illustrative choice $c=-3$, the solution becomes $f=α_1/\sqrt Q+β_1$. Under the conventional positive-coupling assumption $f_Q>0$, the required source violates the null energy condition. Moreover, adding positive radiation or pressureless matter forces a compensating negative-energy component at sufficiently early times. The weak branch is therefore neither curvature-neutral in an invariant sense nor a viable cosmological model. The principal result is the background-level obstruction itself. Perturbative stability and gravitational-wave propagation require a separate analysis including perturbations of the nontrivial affine connection and are not established in this work.