Existence of nonzero scalar modes under shear-free constraint closure

Determine, for linear scalar perturbations of Friedmann–Lemaître–Robertson–Walker solutions in metric f(R) gravity with a minimally coupled barotropic perfect fluid and imposed propagated shear-freeness, whether any nonzero scalar mode can remain in the time-dependent consistent subspace generated by the shear-free constraint and all of its temporal consistency conditions.

Background

For nonvacuum backgrounds, the scalar perturbations are represented by a four-component gauge-invariant system containing matter-density, expansion, curvature, and curvature-time-derivative gradients. The shear-free condition supplies an additional scalar constraint, and successive time derivatives generate an infinite hierarchy of consistency rows. A permissible perturbation must remain within the intersection of the kernels of all these rows throughout the evolution interval.

The paper emphasizes that satisfying the scalaron’s formal second-order wave equation is not sufficient, because the scalar mode must also satisfy the full propagated shear-free constraint. The authors therefore leave the existence of a nonzero scalar mode unresolved in the general model- and background-dependent case, although they establish a no-go result for fixed nonzero comoving scalar harmonics in the separately analyzed geodesic, spatially flat, expanding de Sitter vacuum setting.