In the context of expanding maps of the circle with an indifferent fixed point, understanding the joint behavior of dynamics and pairs of moduli of continuity $ (ω, Ω) $ may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus $ Ω$ (precisely $ \lim_{x \to 0^+} \sup_{\mathsf d} Ω\big({\mathsf d} x \big) / Ω(\mathsf d) = 0 $) as a sufficient condition for the system to exhibit exponential decay of correlations with respect to the unique equilibrium state associated with a potential having $ ω$ as modulus of continuity. This result is derived from obtaining the spectral gap property for the transfer operator acting on the space of observables with $ Ω$ as modulus of continuity, a property that, as is well known, also ensures the Central Limit Theorem. Examples of application of our results include the Manneville-Pomeau family