Cosmological singularity, conformal anomaly and symmetric polynomials

Abstract: We consider a spacetime singularity at $t = 0$ arising in a Kasner-type metric that solves the gravitational equations modified by quantum effects of a conformal field theory (CFT). The resulting constraints can be solved efficiently when expressed in terms of symmetric polynomials. Focusing first on the trace part of the modified gravitational equation, we determine the corresponding solution surfaces in Kasner-parameter space. The geometry of these surfaces depends sensitively on the ratio $η= A/C$, the quotient of the conformal charges characterizing the underlying CFT. We then fully integrate the conformal anomaly near the singularity for a generic Kasner-type metric and obtain the corresponding stress-energy tensor. Its components are expressed in terms of three symmetric polynomials (of degrees $2$, $3$ and $4$) and depend on seven arbitrary constants, which may be interpreted as parameterizing different choices of the quantum state at the singularity. By imposing a set of constraints we reduce this parameter space to a single free constant. Subsequently, we solve, at leading order near the singularity, the modified gravitational equations. Among the admissible solutions, we identify, in particular, those that develop a curvature singularity while remaining geodesically complete.