AdS Distance Conjecture (KK scale versus cosmological constant scaling)

Establish the AdS Distance Conjecture scaling relation: In Anti–de Sitter spacetime with Λ < 0, show that as |Λ| → 0 a tower of states becomes light and that, in particular, the Kaluza–Klein mass scale satisfies m_KK ∼ |Λ|^a with exponent a = 1/4.

Background

The authors explicitly invoke the AdS Distance Conjecture to correlate the cosmological constant with the Kaluza–Klein scale, stating a specific scaling exponent a = 1/4. This scaling is central to predicting a micron-sized extra dimension, a key ingredient of the dark-dimension scenario analyzed throughout the paper.

A rigorous proof of the stated scaling, including the precise exponent, would solidify the theoretical foundation for the phenomenological conclusions drawn in the paper, particularly those concerning suppressed Hawking radiation and the viability of five-dimensional rotating primordial black holes as dark matter.

References

Remarkably, this can be adapted to an Anti–de Sitter (AdS) background. The AdS Distance Conjecture postulates that as the cosmological constant Λ <0 approaches zero, a tower of states becomes light, with a mass scale related to |Λ|. In particular, for the Kaluza–Klein (KK) scale m_{KK} one finds that it is connected to the cosmological constant as follows m_{KK} ∼ |Λ|a, a = 1/4.

— 5D Rotating Black Holes as dark matter in Dark Dimension Scenario: Hawking Radiation versus the Memory Burden Effect (2512.10381 - Leontaris et al., 11 Dec 2025) in Section 1, Introduction