Logarithmic gravity from a very slow roll limit of inflation

Abstract: We study the inflationary no-boundary wavefunction in a ``linear-roll limit,'' in which $G_N \to 0$ while the Hubble scale $H$ and the slope of the inflaton potential are held fixed. Although bulk graviton fluctuations are suppressed in this limit, quantum inflaton fluctuations remain finite. Because reheating occurs on a surface of fixed inflaton value, these fluctuations become fluctuations of the conformal factor of the reheating metric. The probability measure for these fluctuations simplifies drastically and becomes Gaussian, even for large fluctuations. The resulting log-correlated theory at superhorizon scales is a $d$-dimensional analog of Liouville gravity without the exponential potential, which we call Logarithmic Gravity. The quadratic and linear terms are determined by conformally covariant scattering data; in even dimensions we relate them to the critical GJMS operator and Branson $Q$-curvature. From bulk unitarity, we argue that one-loop effects do not generate additional conformal-factor dependence in the physical probability measure, and we verify the cancellation explicitly in $2 + 1$ dimensions. For the no-boundary norm with a rolling inflaton, we show that the noncompact residual conformal symmetries can be gauge fixed with a finite Faddeev-Popov determinant, so they do not force the sphere contribution to vanish. In addition, conditioning on the absence of vacuum decay by bubble nucleation generates the Liouville exponential potential in the probability measure.