Weyl-Invariant Selection of Low-Weyl Boundary Data

The Past Hypothesis posits that the Universe began in a state of exceptionally low entropy, matter being near thermal equilibrium while the gravitational degrees of freedom were unusually ordered, and the Weyl Curvature Hypothesis identifies the latter condition with a very small early-time Weyl curvature. Here, an origin for it is proposed within Weyl-invariant (WI) gravity, whose conformalized Einstein–Hilbert (EH) sector of Lagrangian density is supplemented with a quadratic curvature term, −αCμνρσCμνρσ, where Cμνρσ is the Weyl curvature tensor. In the Einstein gauge, the reduced action separates into terms that distinguish one spatial shape from another and terms that do not, the latter canceling from any comparison of configurations. The sign of the Weyl coupling, α>0, is fixed by the spectrum of the theory; α<0 renders the massive spin-2 mode tachyonic. With the sign so fixed, and with the gravitational path integral continued to the Euclidean regime, the Weyl sector of the action is non-negative and vanishes only on conformally flat geometries, so configurations of large Weyl curvature are exponentially suppressed whatever the character of that curvature. The resulting preference for low-Weyl initial configurations is one possible realization of the Weyl Curvature Hypothesis, in which the content of the Past Hypothesis resides as the matter sector is observationally inferred to have been near thermal equilibrium. Its connection to small initial shear is conditional, since conformally flat geometries can in principle be sheared. The quantitative estimates discussed here illustrate the weighting within certain families of configurations. The proposal addresses the low-Weyl gravitational initial conditions relevant to inflation, without—however—requiring an inflationary epoch.