Non-screening of the Cosmological Background in K-mouflage modified gravity

We describe the effects of the cosmological background on the K-mouflage screening properties of an astrophysical structure. We show that the K-mouflage screening of the spatial gradients of the scalar field, i.e. the screening of the fifth force, happens inside a dynamically generated screening radius. This radius is smaller than the location where the quasistatic approximation, i.e. where the spatial gradients exceed the time derivative, holds. Even though this quasistatic radius is much smaller than the size of the matter overdensity, spatial gradients remain well described by the quasistatic approximation up to the horizon. Cosmologically we find that the time derivatives can remain dominant at redshifts $z\gtrsim 2$, when the cosmic web shows a faster growth. Despite the existence of K-mouflage screening, we confirm that the values of the scalar field itself are still dominated by the cosmological background, down to the center of the matter overdensity, and that for instance the time drift of Newton's constant due to the large-scale cosmological evolution highly constrains K-mouflage models.