Radion stabilization in compact hyperbolic extra dimensions

TL;DR

In compact hyperbolic extra-dimensional brane-world models, stabilizing the radion while allowing an acceptable FRW cosmology requires bulk matter that violates the null energy condition. The authors demonstrate this in Einstein gravity, give a d-form example that fails the condition, and suggest quantum Casimir effects as a possible loophole.

Key findings

• Any bulk matter stabilizing the compact hyperbolic radion and yielding acceptable brane cosmology must violate the null energy condition; this holds even allowing FRW expansion on the brane.
• For a Minkowski brane and pure CHM, the required stress tensor gives \(T_{AB}N^A N^B = - (d-1)(d+2)M^{2+d}/(4R_h^2) < 0\) for a null vector along the hyperboloid.
• A specific stabilizing candidate—a cosmological constant plus a d-form flux—also violates the null energy condition, with \(T_{AB}N^A N^B = - (d-1)(d-2)\beta^2/8\, g_{ii}N_iN_i < 0\).
• Casimir energy may help: \(\rho_C \propto R_h^{-p}\) with \(p>0\), and if \(\rho_C<0\), the sign of energy density and pressure is “precisely right” to help stabilize \(H^3/\Gamma\); a numerical study of a minimally coupled scalar on the Thurston manifold found negative Casimir energy density.
• Supersymmetry in hyperbolic backgrounds is generally broken: for even \(d\), all supersymmetries are broken; for odd \(d\), survival depends on \(\Gamma\), and in the special case \(d=3\) no supersymmetries remain.

Methods

• The authors analyze Einstein gravity in \(4+d+n\) dimensions with a factorizable bulk \(\Sigma^{4+d+n}=F^{3+1}\times H^d/\Gamma\times S^n\), where \(H^d/\Gamma\) is a compact hyperbolic manifold and \(S^n\) is an n-sphere.
• They derive both an effective 3+1-dimensional theory with radion fields \(\phi,\psi\) and the full higher-dimensional Einstein equations, using a perfect-fluid ansatz for bulk matter and imposing the null energy condition.
• They illustrate the general result with a d-form flux plus cosmological constant example, then discuss Casimir energy and supersymmetry in compact hyperbolic backgrounds.

Limitations

• The paper states that understanding Casimir energy in the specific highly topologically complex manifolds useful for brane-world models is “extremely useful, but seems calculationally formidable and is beyond the scope of this paper.”
• The authors do not present generic results for \(\rho_C\) in specific CHMs; they only note a numerical study of a minimally coupled scalar on the Thurston manifold.
• The results apply to extra dimensions large enough that general relativity is a valid theory; the paper notes that the assumption of complete GR validity may break down, and that simple Casimir estimates are not a substitute for detailed calculation.