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We propose a new class of $f(R)$ theory where its Weyl gauge symmetry isbroken in the primordial era of the universe. This symmetry forces one to adopta new scalar field, namely a Weyl field and a gauge vector boson. Furthermore,an equivalent form of the Einstein-Hilbert Lagrangian with a non-minimallycoupled scalar field corresponding to the function $f(R)$ is found. Due to thegeometrical feature of the Weyl field, it turns out that the symmetry breakinginduces a non-minimal coupling, which cannot be expected in the standard $f(R)$theories. We explain how this affects the evolution of the universe atcosmological scales. It is shown that there may be a value shift in the Planckconstant and the cosmological constant. This can be regarded as a genuineexemplification of the cosmological backreaction. Furthermore, one also findsnew features in the evolution of perturbational variables and cosmic microwavebackground anisotropy. Moreover, we prove that when a specific $f(R)$ modelinvokes inflation, the amplitude of the primordial gravitational waves affectsthe evolution of scalar perturbation due to the new non-minimal coupling. As acase study, we explain how this can be embodied in the Starobinsky inflation.Finally, we discuss some impacts that this physics can bear and the possibilityof giving a new restriction of the estimation of cosmological variables such asthe gravitational wave amplitude with experiments.

$f(R)$ gravity with broken Weyl gauge symmetry, cosmological backreaction, and its effects on CMB anisotropy