Swampland volume-growth and field-range bounds beyond supersymmetry: deciding the field-excursion-bound vacuum-counting argument

The Field Excursion Bound (FEB) of Herderschee and Wall limits scalar field excursions in spatially flat cosmologies linearly by the number of e-folds of expansion. Combined with Bézout-type counting of vacua, it implies that populating the $\sim 10^{150}$ vacua apparently needed for anthropic fine-tuning requires trans-Planckian excursions and far more than the observed $\sim 60$ e-folds of inflation — unless field space is negatively curved, in which case exponentially growing volume evades the count. Rudelius argued that swampland results may close this loophole: finiteness of quantum-gravity amplitudes bounds moduli-space volume growth polynomially, and the exponentially decaying asymptotic regime of potentials and particle towers begins within $O(M_{\rm Pl})$, which would cap the number of vacua accessible at fixed flux at $(3k)^{n_s}$ — essentially zero for $n_s\lesssim 100$. Both ingredients, however, were derived only in supersymmetric settings, and the synthesis is explicitly speculative. The open problem is to prove, refute, or sharply delimit the extension of these volume-growth and field-range bounds to non-supersymmetric quantum-gravity landscapes and the resulting bound on stable vacua accessible at fixed flux, thereby deciding whether the anthropic many-e-folds requirement survives the negative-curvature loophole.

Background

Scalar fields in theories of gravity often take values in a multi-dimensional 'moduli space' — the space of possible vacuum expectation values, equipped with a field-space metric. Herderschee and Wall (2025) proved a Field Excursion Bound (FEB): along any null geodesic (a light-ray trajectory), the displacement of a canonically normalized scalar field (one whose kinetic term has the standard coefficient) is bounded in terms of the expansion, i.e. the fractional rate of change of the cross-sectional area of a bundle of neighboring light rays. The derivation uses the Raychaudhuri equation (the evolution equation for that expansion) and assumes all other matter satisfies the null energy condition (NEC: $T_{\mu\nu}k^\mu k^\nu\ge 0$ for every null vector $k^\mu$), which focuses the bundle. In a spatially flat Friedmann–Robertson–Walker (FRW) cosmology — a homogeneous, isotropic universe with scale factor $a(t)$, Hubble parameter $H=\dot a/a$, and $\Delta N=\log(a(t_f)/a(t_i))$ the number of e-folds of expansion — their bound takes the form Rudelius dubbed the FLEB:

$$\sqrt{2}\,\kappa_4\,|\Delta\phi|\leq \Delta N+\log\!\big(H(t_i)/H(t_f)\big),\qquad \kappa_4^2=8\pi G=M_{\rm Pl}^{-2}.$$

Unlike the Lyth bound — the well-known lower limit $\Delta\phi\gtrsim\mathcal{O}(1)\sqrt{r}\,\Delta N$ tying field excursions to the tensor-to-scalar ratio $r$ of primordial gravitational waves — the FLEB is an upper limit on how far a scalar can travel, independent of the inflationary potential. Rudelius (2025) gave an alternate proof from the Friedmann equations and showed the bound is saturated only in universes with vanishing acceleration $\ddot a=0$.

Herderschee and Wall applied the FLEB to anthropic vacuum selection (selection of vacua by the requirement that the resulting universe support observers). Fine-tuning of the cosmological constant ($\sim 10^{-120}$) and of the Higgs mass squared ($\sim 10^{-30}$) appears to require at least $n_v\sim 10^{150}$ vacua to expect one as tuned as ours. Holding discrete data such as fluxes (quantized background charges in string compactifications) fixed, so that only $n_s$ scalar fields vary, the vacuum condition $\partial_A V=0$ for a generic quartic potential is a system of $n_s$ cubic equations; by Bézout's theorem (a system of $n$ polynomial equations in $n$ variables has at most $\prod_i \deg f_i$ complex solutions) there are at most $3^{n_s}$ critical points per Planck-radius ball of field space. If field space is flat, reaching $10^{150}$ vacua then requires a trans-Planckian excursion $|\Delta\phi|\gtrsim 10^{150/n_s}/3$ for $n_s\lesssim 100$, and the FLEB converts this into a lower bound on e-folds, $\Delta N\gtrsim (4\sqrt{\pi}/3)\,10^{150/n_s}$, far beyond the $\approx 60$ e-folds observation requires. The authors noted one loophole: if field space is negatively curved, the volume accessible within radius $|\Delta\phi|$ grows exponentially, so many vacua could populate a small proper region.

Rudelius argued that results from the swampland program — the body of conjectures characterizing which effective field theories can couple consistently to quantum gravity — may close this loophole. Three inputs: (1) the Distance Conjecture of Ooguri and Vafa, that toward any point at infinite field distance a tower of states becomes exponentially light; (2) the backreaction calculations of Baume and Palti — in type IIA axion monodromy setups (axion monodromy: a periodic shift-symmetric scalar whose potential grows monotonically over many periods through its coupling to background fluxes), the energy of the axion feeds back on the field-space geometry so that this exponential regime begins within $O(M_{\rm Pl})$ — extended by van de Heisteeg, Vafa, Wiesner, and Wu, who showed, using the species scale (the scale $\Lambda_s\sim M_{\rm Pl}/\sqrt{N}$ at which $N$ light species invalidate a decoupled EFT) and the emergent string conjecture (at infinite-distance limits a string or a Kaluza–Klein tower — the ladder of states carrying quantized momentum in the compact extra dimensions — becomes tensionless), that regions of slowly varying positive potential have diameter $\Delta\phi\leq a\log(1/V)+b$ in Planck units; and (3) the finiteness arguments of Delgado, van de Heisteeg, Raman, Torres, Vafa, and Xu, that finiteness of quantum-gravity amplitudes in fully compactified theories requires the moduli-space volume of a ball of radius $|\Delta\phi|$ to grow no faster than $|\Delta\phi|^{n_s+\epsilon}$ for any $\epsilon>0$ — which would close the negative-curvature loophole. If these hold generally, the number of vacua within excursion range satisfies (Rudelius, Eq. (37)) $n_v\lesssim (3|\Delta\phi|/M_{\rm Pl})^{n_s}$ for $|\Delta\phi|\lesssim M_{\rm Pl}$ and $n_v\lesssim (3k)^{n_s}$ for $|\Delta\phi|\gg M_{\rm Pl}$, with $k$ an order-one radius of the vacuum-rich region; for $n_s\lesssim 100$ this is vastly smaller than $10^{150}$, so at fixed fluxes there are essentially zero anthropically viable vacua across field space, and reaching one requires flux-changing phase transitions or $\Delta N\gg 1$. But Rudelius stresses that this synthesis is speculative: the finiteness argument of (3) was derived assuming supersymmetry (an exact moduli space, $V=0$); the tower-mass results of (2) must be extrapolated from towers to scalar potentials; and equating critical points of the truncated potential with stable vacua becomes increasingly suspect as $n_s$ grows. Whether the bridge survives in realistic, non-supersymmetric landscapes is the open question. (Two adjacent facts support the cosmological side: bubble universes formed in vacuum decay are negatively curved, so the FLEB applies to them, and the higher-derivative violation route requires a superluminal speed of sound — but these do not touch the vacuum-counting bridge at issue here.)

Problem Statement

Determine whether the swampland-based bridge between the Field Excursion Bound, vacuum counting, and swampland asymptotics is valid in non-supersymmetric quantum-gravity landscapes (string compactifications without supersymmetry, or with supersymmetry broken at the compactification scale). Concretely, decide:

(i) Volume growth: does the volume of a ball of radius $|\Delta\phi|$ in the space of scalar fields grow no faster than $|\Delta\phi|^{n_s+\epsilon}$ for every $\epsilon>0$, as the finiteness arguments of Delgado et al. establish for supersymmetric moduli spaces — or can non-supersymmetric field spaces realize exponentially growing (negatively curved) volume at parametrically large field range?

(ii) Reach of the asymptotic regime: does the exponentially decaying asymptotic behavior of scalar potentials and light-tower masses set in within $O(M_{\rm Pl})$ proper distance of any point in field space, as the backreaction analyses of Baume–Palti and the species-scale field-range bounds suggest in supersymmetric settings?

(iii) Vacuum count at fixed flux: as a consequence of (i) and (ii) — or of their failure — is the number of stable vacua accessible by field excursion with the discrete data (fluxes) held fixed bounded as $n_v\lesssim(3k)^{n_s}$ independently of the excursion range (Rudelius, Eq. (37)), implying that the Herderschee–Wall anthropic argument requires flux-changing transitions and/or $\Delta N\gg 1$ e-folds of inflation and that the negatively-curved-field-space loophole does not evade it?

Any admissible class must satisfy a minimum scope: a non-supersymmetric quantum-gravity landscape of the kind fixed above (a string compactification without supersymmetry, or with supersymmetry broken at the compactification scale) with all moduli stabilized and fluxes held fixed; a bottom-up effective theory qualifies only as a controlled representative of such a landscape within its regime of validity, and a class in which an ingredient holds only vacuously or degenerately does not resolve the task. The single objective uniting sub-questions (i)–(iii) is to settle the soundness of the excursion-bound vacuum-counting argument for realistic non-supersymmetric landscapes.

References

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