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Our Universe has an arrow of time. In accordance with the second law ofthermodynamics, entropy has been increasing ever since the Big Bang. The factthat matter is in thermal equilibrium in the very early Universe, as indicatedby the cosmic microwave background, has led to the idea that gravitationalentropy must be very low in the beginning. Penrose proposed that gravitationalentropy can be quantified by the Weyl curvature, which increases as structuresformed. A concrete realization of such a measure is the Clifton-Ellis-Tavakolgravitational entropy, which has been shown to be increasing in quite a numberof cosmological models. In this work, we show a counter-example involving aclass of inhomogeneous universes that are supported by a chameleon masslessscalar field and exhibit anisotropic spacetime shearing effects. In fact, inour model the Clifton-Ellis-Tavakol gravitational entropy is increasingalthough the magnitude of the Weyl curvature is decreasing; this is due to thegrowth of the spacetime shear. The topology and the values of the three freeparameters of the model are constrained by imposing a positive energy densityfor the cosmic fluid, and the thermodynamical requirements which follow fromthe cosmological holographic principle and the second law. It is shown that anegative deceleration parameter and a time decreasing Weyl curvatureautomatically follow from those conditions. Thus, we argue that our model canaccount for the formation of some primordial structures, like the Large QuasarGroups, which has required a non-standard evolution of the spatialanisotropies.

Thermodynamics of Shearing Massless Scalar Field Spacetimes is Inconsistent With the Weyl Curvature Hypothesis