Dark Dimension Framework

• Dark Dimension Framework is a theory proposing a mesoscopic extra dimension whose size is dictated by dark energy and UV/IR mixing.
• It employs string theory and T-fold constructions to predict a Kaluza-Klein tower with implications for dark matter, neutrino masses, and black hole phenomena.
• The framework unifies cosmological observations with swampland constraints, offering experimental targets from sub-millimeter tests to collider signals.

The Dark Dimension Framework encompasses a class of theories predicting the existence of a single, mesoscopic extra dimension whose characteristic size is dictated by the observed value of the cosmological constant (i.e., dark energy) rather than by high-scale unification arguments. In its modern incarnation, the framework is motivated by swampland conjectures and UV/IR mixing in quantum gravity, resulting in the emergence of an extra dimension at the micron scale, with significant implications for dark matter, neutrino physics, black hole phenomenology, and the structure of quantum gravity. Implementation occurs in settings ranging from string theory to effective field theories and phenomenological models.

1. Foundational Principles and Motivation

A central realization of the Dark Dimension Framework arises from the intersection of observational cosmology (the smallness of $\Lambda$), theoretical constraints—including the swampland distance conjecture and the absence of stable de Sitter vacua in string theory—and the necessity to explain hierarchies naturally. In this approach, the observed vacuum energy $\Lambda_{\text{dark}}$ is related to a new physics mass gap $m_{\text{gap}}$ (the Kaluza–Klein or KK scale) via the scaling relation

$$\Lambda_{\text{dark}} \sim m_{\text{KK}}^4 \sim (1/R)^4,$$

where $R$ is the radius of the extra dimension (Basile et al., 18 Sep 2024). This relation is a haLLMark of IR/UV mixing emerging from modular invariance of the worldsheet in string theory compactifications and is reinforced by the distance conjecture, which predicts that approaching the infinite distance limit in field space produces an infinite tower of exponentially light states (Montero et al., 2022, Anchordoqui, 2022).

The framework naturally addresses the cosmological hierarchy problem: as $\Lambda$ becomes small, $R$ becomes large, leading to a mesoscopic extra dimension ($R \approx 1\ \mu\text{m}$ for observed $\Lambda$) (Branchina et al., 2023, Noble et al., 2023). The associated KK tower of light states and a lowered gravitational cutoff (the species scale $M_*$) are intrinsic consequences (Anchordoqui et al., 7 May 2024).

2. Realizations in String Theory and T-fold Constructions

A rigorous top-down realization in string theory identifies the dark dimension with an $S^1$ base in compactifications such as T-folds, which generalize purely geometric backgrounds to include T-duality twists. In these compactifications on $T^5 \times S^1$, global properties are determined by monodromies in the T-duality group $\mathrm{O}(5,5;\mathbb{Z})$. The Scherk–Schwarz reduction introduces a twist along $S^1$, generating an effective 4D potential of the form

$V \sim m_{\text{KK}}^4,$

where $m_{\text{KK}}^2 \sim e^{2(\alpha-\beta)\varphi}$ with $\varphi$ the Scherk–Schwarz radion and $\alpha$, $\beta$ fixed by the dimensional reduction (Nian et al., 28 Nov 2024).

Duality twists in the internal space (elliptic or parabolic conjugacy classes) stabilize subsets of the compactification moduli: elliptic twists create Minkowski vacua for complex structure or Kähler moduli, while parabolic twists induce necessary runaway directions aligned with the dark dimension. Upon stabilization of other moduli, the scalar potential asymptotes to a form proportional to $m_{\text{KK}}^4$, as required for the dark dimension scenario (Nian et al., 28 Nov 2024, Basile et al., 18 Sep 2024).

String theory worldsheet methods, such as the modular-invariant torus partition function, further support the scaling $\Lambda_{\text{dark}} \sim m_{\text{KK}}^4$, revealing how the measured smallness of $\Lambda$ forces the presence of an extra low-tension dimension by way of UV/IR mixing (Basile et al., 18 Sep 2024).

3. Swampland Constraints and Strong IR/UV Mixing

Swampland conjectures, especially the distance conjecture and de Sitter conjecture, play a pivotal role: they enforce that a small dark energy must be accompanied by a light KK tower (the infinite tower of states as one moves far in moduli space), placing the universe near an infinite distance boundary of moduli space (Montero et al., 2022, Noble et al., 2023). The resulting species scale,

$M_* \sim m_{\text{KK}}^{1/3} M_{\text{Pl}}^{2/3},$

acts as a low-gravity cutoff, distinct from the 4D Planck mass, and controls strong gravitational coupling, the endpoint of effective field theory, and the onset of possible nonlocal physics.

Matching the vacuum energy computed from a 5D effective field theory to the swampland expectation is subtle. While naive calculations yield $\rho_4 \sim m_\text{KK}^4$, correct treatment of higher-dimensional loop integrals (imposing a 5D cutoff $\vec{p}^2 + (n/R)^2 \leq \Lambda^2$) exposes additional UV-sensitive terms scaling with odd powers of the cutoff $\Lambda$ (Branchina et al., 2023). Cancellation or suppression of these terms—potentially via modular invariance or special symmetries—remains an outstanding requirement for the self-consistency of the framework.

4. Phenomenological Consequences: Dark Matter and Cosmic Structure

Primordial Black Holes (PBHs): In the context of the dark dimension, PBHs evolve distinctively. Black holes with horizon size smaller than $R$ exhibit 5D radiation laws,

$T_H^{\text{(5D)}} \sim 1/r_s,$

slower Hawking evaporation, and increased lifetimes. The allowed mass window for PBHs to constitute all of dark matter is extended down to $M_{\text{BH}} \sim 10^{11}$–$10^{21}$ g for bulk PBHs, facilitated by the rapid escape of PBHs into the bulk extra dimension following recoil from graviton emission (Anchordoqui et al., 28 Mar 2024, Anchordoqui et al., 16 Jul 2024). Once in the bulk, evaporation products (mostly bulk gravitons) are invisible to standard $\gamma$-ray searches, evading constraints and opening the full mass window for PBHs as dark matter.

KK Gravitons and Dynamical Dark Matter: The dark dimension’s Kaluza–Klein gravitons, universally coupled to the brane, form an ensemble of long-lived weakly coupled states. Their cumulative relic abundance is shaped by a balancing between decay widths and abundances, satisfying

$$\Omega_i \cdot \Gamma_i^{-\alpha} \simeq \text{constant},\quad \alpha<0,$$

and yielding a time-evolving total dark matter density $\Omega_\text{tot}(t)$ with $w_\text{eff}(t) \neq 0$ (Dienes et al., 2011). The tower nature naturally regularizes decay and detection signatures, with unique implications for indirect detection (e.g., mono-energetic photon lines from $KK \rightarrow \gamma\gamma$) and early-universe cosmology (e.g., modulation of redshifted 21-cm lines) (Anchordoqui et al., 2022, Anchordoqui et al., 2023).

Axion Sector and Unified Scales: The QCD axion, when localized on the brane as favored by string-theoretic consistency and the Weak Gravity Conjecture for 5D setups, is constrained in decay constant to $f_a \sim 10^9$–$10^{10}$ GeV, yielding $m_a \sim 1$–$10$ meV—a mass scale coincident with that of dark energy ($\Lambda^{1/4}$), the KK gap, and typical neutrino masses in these constructions. Although such axions only form a small fraction of dark matter, this "ladder" of scaling relations offers a unified parametric structure for the dark sector (Gendler et al., 23 Apr 2024).

Neutrino Masses: The lightness of neutrinos is explained via right-handed sterile neutrinos propagating in the bulk, with Dirac masses suppressed by $1/\sqrt{M_5 R}$, setting $m_{\nu_s} \sim 1/R$ and matching the scale of dark energy and the KK tower (Montero et al., 2022, Anchordoqui et al., 2022, Anchordoqui et al., 7 May 2024).

5. Collider and Astrophysical Signatures

The mesoscopic size of the dark dimension produces concrete experimental signatures:

• Modifications to Newton’s law at micron scales, testable in precision tabletop experiments (Basile et al., 18 Sep 2024).
• Ultra-high-energy cosmic ray (UHECR) spectrum: The species scale $M_*$, set by the dark dimension geometry, aligns with the observed energy cutoff in UHECR flux (Greisen–Zatsepin–Kuzmin cutoff), potentially explained by strong gravitational interactions or gravitational diffraction radiation above $M_*$ (Noble et al., 2023, Anchordoqui, 2022).
• Astrophysical photon lines from dark matter decay, notably from long-lived KK gravitons (Anchordoqui et al., 2022).
• Collider signatures: In "little string theory" alternatives with string scale $\sim 10\ \text{TeV}$ (and extremely small string coupling $g_s\sim 10^{-15}$), direct production of stringy resonances or KK states may be possible (Basile et al., 18 Sep 2024).
• Mono-h and mono-Z signals at the LHC: Non-linear EFT frameworks allow for the inclusion of multiple mediators and higher-dimensional operators, supporting systematic diagnosis of new neutral scalars or fermions even when their UV origin is not fully specified (Arcadi et al., 8 Nov 2024, Alanne et al., 2017).

6. Challenges, Open Questions, and Future Directions

A central technical challenge remains the control and suppression of UV-sensitive contributions in calculations of the effective vacuum energy (Branchina et al., 2023). A fully consistent embedding of the dark dimension scenario likely requires modular invariance or explicit string constructions to ensure only the $m_{\text{KK}}^4$ scaling survives and to avoid destabilizing quantum corrections.

Further, realizing consistent stabilization of all moduli (particularly the radion associated with the dark dimension) without introducing large backreactions or destabilizing decompactification represents another open avenue; T-fold compactifications with engineered duality twists provide a promising approach, but a complete moduli fixing in this context is a work in progress (Nian et al., 28 Nov 2024).

Phenomenologically, continued and more refined probes—both astrophysical (UHECRs, $\gamma$-ray observations, high-precision cosmology) and terrestrial (sub-millimeter gravity, collider searches for axions and extra scalars)—are crucial for either confirming or falsifying the key predictions of the framework.

7. Theoretical and Unifying Aspects

The Dark Dimension Framework represents an origin for the "dark sector unification": dark energy, dark matter, axion mass, and neutrino mass scales are all emergent consequences of the same geometric and UV/IR structure, defined by the measured vacuum energy and enforced by swampland/consistency criteria. By placing the universe near a special locus—an infinite-distance limit—in the string landscape, the framework offers a highly constrained, predictive arena in which hierarchies and cosmic coincidences are reinterpreted as geometric consequences rather than arbitrary tunings.

This paradigm posits that a single micron-scale extra dimension may be the key organizing principle underlying a wide variety of low-energy phenomena, and testing this proposal lies at the intersection of theoretical quantum gravity, string phenomenology, cosmology, and precision experiments.