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

Abstract: Our Universe has an arrow of time. In accordance with the second law of thermodynamics, entropy has been increasing ever since the Big Bang. The fact that matter is in thermal equilibrium in the very early Universe, as indicated by the cosmic microwave background, has led to the idea that gravitational entropy must be very low in the beginning. Penrose proposed that gravitational entropy can be quantified by the Weyl curvature, which increases as structures formed. A concrete realization of such a measure is the Clifton-Ellis-Tavakol gravitational entropy, which has been shown to be increasing in quite a number of cosmological models. In this work, we show a counter-example involving a class of inhomogeneous universes that are supported by a chameleon massless scalar field and exhibit anisotropic spacetime shearing effects. In fact, in our model the Clifton-Ellis-Tavakol gravitational entropy is increasing although the magnitude of the Weyl curvature is decreasing; this is due to the growth of the spacetime shear. The topology and the values of the three free parameters of the model are constrained by imposing a positive energy density for the cosmic fluid, and the thermodynamical requirements which follow from the cosmological holographic principle and the second law. It is shown that a negative deceleration parameter and a time decreasing Weyl curvature automatically follow from those conditions. Thus, we argue that our model can account for the formation of some primordial structures, like the Large Quasar Groups, which has required a non-standard evolution of the spatial anisotropies.