Cosmological singularity, conformal anomaly and symmetric polynomials

TL;DR

This work analyzes cosmological singularities in semiclassical gravity with a conformal field theory by examining Kasner-type metrics. It derives the trace constraint from the conformal anomaly and shows that the resulting conditions are tractable when reformulated in symmetric polynomials of the Kasner parameters. The anomaly is integrated explicitly to obtain a full CFT stress tensor, and the leading-order gravitational equations reduce to algebraic constraints (the Master equation) on the Kasner parameters, whose solutions depend on the conformal charges through η = A/C. The isotropic (Starobinsky) case yields a regular spacetime only for a specific quantum state (C0 = 0), while more general Kasner configurations can exhibit curvature singularities yet remain geodesically complete, indicating novel ways quantum effects can modify singularity formation and structure in the early universe.

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.

Figures (8)

• Figure 1: Plot of curve ${\cal M}(a,b)=0$ for value $\eta=3$ (left) and the zoomed-in view of the region of small negative values of $a$ (right).
• Figure 2: Plot of curve ${\cal M}(a,b)=0$ for value $\eta=\frac{1}{3}$.
• Figure 3: Plot of curve ${\cal M}(a,b)=0$ for value $\eta=-\frac{1}{3}$ (left) and the zoomed-in view of the region of positive values of $(a,b)$ (right).
• Figure 4: Surface ${\cal M}(a, b, c)=0$ for $\eta=A/C=1/2$. Left: It contains of two open shells. One for positive values of parameters $(a,b,c)$ and the other for all negative values. The conical point has coordinates $(0,0,0)$ in the parameter space. Right: The three disjoint regions of the surface where each parameter is less than $-1$, in the corresponding spacetime although the Riemann tensor is singular the geodesics remain complete.
• Figure 5: Left: Surface ${\cal M}(a, b, c)=0$ for $\eta=A/C=2$. There appears three additional closed shells in which one of the parameters is negative and two others are positive. Right: Zoomed-in view of such a shell for negative values of $a$ and positive values of $b$ and $c$.