Quantum Signatures of Cosmic Topology: How Casimir Backreaction Transmits Isotropy Violation

Abstract

A finite, scheme-independent Casimir contribution to the stress-energy tensor arises naturally for quantum fields in universes with non-trivial spatial topology. We compute this Casimir stress-energy tensor contribution for a conformally coupled scalar field and for a minimally coupled scalar field. We show that, for the conformally coupled case, the backreaction of this contribution to the Einstein equations during an expanding de Sitter phase drives anisotropic expansion even when the Universe begins in a locally homogeneous and isotropic state. We conclude that quantum imprints of the underlying non-trivial topology inevitably give rise to local departures from homogeneity and isotropy.

Figures (3)

• Figure 1: A two-dimensional torus is obtained by boundary conditions identifying the opposite sides of a rectangular fundamental domain. Similarly, in three-dimensional Euclidean space, each of the 17 non-trivial topologies can be represented by topological identifications imposed on a fundamental domain Riazuelo:2003ud.
• Figure 2: A simply connected space allows fluctuations of arbitrary wavelength on the last scattering surface, the furthest optically observable region with radius $R_{\mathrm{LSS}}$. By contrast, non-trivial topology, such as a three-dimensional cubic torus of size $L$, restricts the admissible modes to a discrete set fixed by its boundary conditions. For the torus, only modes satisfying periodic boundary conditions contribute to the statistics of CMB fluctuations, even when $L>2R_{\mathrm{LSS}}$.
• Figure 3: Anisotropy generated purely by quantum Casimir backreaction in a three-torus with a rectangular fundamental domain. Panels (a) and (b) show, for different values of the fundamental-domain size $L$, the quantities $\delta a_1( N)$ and $\delta a_3(N)$ as functions of the number of $e$-folds $N$, where the dashed lines mark the threshold beyond which the perturbative description is no longer valid. Panel (c) displays the fractional anisotropy between the $11$-component and $33$-component of the first-order corrected metric, starting from zero at $\eta= \eta_{\mathrm{in}}$ (as imposed by the initial conditions) and growing to a maximum of order $\sim 17\%$ by the end of inflation. In all panels we take $\beta_1=\beta_2=1$, $\beta_3=2$, $8 \pi G H^2 = 10^{-10}$, and $\eta_{\mathrm{in}}=- e^{N} 10^{14}\,\mathrm{GeV}^{-1}$. In (a) and (b) we evaluate the corrections at $\eta=\eta_{\mathrm{end}}=- 10^{14}\,\mathrm{GeV}^{-1}$. In (c) we show the limiting case $L=2 R_{\mathrm{LSS}}=4 \cdot 10^{42}\, \mathrm{GeV}^{-1}$ and $N=71.8$, chosen to remain within the perturbative regime.