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A Geometrical Upper Bound on the Inflaton Range

Michele Cicoli, David Ciupke, Christoph Mayrhofer, Pramod Shukla

TL;DR

The authors address the size of the inflaton range in type IIB LVS by focusing on the reduced moduli space of flat directions after leading stabilization. They combine a systematic scan of CY geometries (Kreuzer–Skarke) with analytic control of the Kähler cone to show that, for $h^{1,1}=3$ toric CYs, the reduced moduli space is compact and its volume scales as a logarithm of the CY volume, yielding an upper bound on inflaton excursions. This geometrical bound translates into constraints on inflationary observables, notably limiting the tensor-to-scalar ratio in many LVS constructions, while allowing trans-Planckian ranges primarily in K3-fibred examples. They propose a broader LVS moduli space conjecture and discuss connections to swampland bounds, highlighting the impact on cosmology and moduli stabilization in string compactifications.

Abstract

We argue that in type IIB LVS string models, after including the leading order moduli stabilisation effects, the moduli space for the remaining flat directions is compact due the Calabi-Yau Kähler cone conditions. In cosmological applications, this gives an inflaton field range which is bounded from above, in analogy with recent results from the weak gravity and swampland conjectures. We support our claim by explicitly showing that it holds for all LVS vacua with $h^{1,1} = 3$ obtained from 4-dimensional reflexive polytopes. In particular, we first search for all Calabi-Yau threefolds from the Kreuzer-Skarke list with $h^{1,1}=2$, $3$ and $4$ which allow for LVS vacua, finding several new LVS geometries which were so far unknown. We then focus on the $h^{1,1} = 3$ cases and show that the Kähler cones of all toric hypersurface threefolds force the effective 1-dimensional LVS moduli space to be compact. We find that the moduli space size can generically be trans-Planckian only for K3 fibred examples.

A Geometrical Upper Bound on the Inflaton Range

TL;DR

The authors address the size of the inflaton range in type IIB LVS by focusing on the reduced moduli space of flat directions after leading stabilization. They combine a systematic scan of CY geometries (Kreuzer–Skarke) with analytic control of the Kähler cone to show that, for toric CYs, the reduced moduli space is compact and its volume scales as a logarithm of the CY volume, yielding an upper bound on inflaton excursions. This geometrical bound translates into constraints on inflationary observables, notably limiting the tensor-to-scalar ratio in many LVS constructions, while allowing trans-Planckian ranges primarily in K3-fibred examples. They propose a broader LVS moduli space conjecture and discuss connections to swampland bounds, highlighting the impact on cosmology and moduli stabilization in string compactifications.

Abstract

We argue that in type IIB LVS string models, after including the leading order moduli stabilisation effects, the moduli space for the remaining flat directions is compact due the Calabi-Yau Kähler cone conditions. In cosmological applications, this gives an inflaton field range which is bounded from above, in analogy with recent results from the weak gravity and swampland conjectures. We support our claim by explicitly showing that it holds for all LVS vacua with obtained from 4-dimensional reflexive polytopes. In particular, we first search for all Calabi-Yau threefolds from the Kreuzer-Skarke list with , and which allow for LVS vacua, finding several new LVS geometries which were so far unknown. We then focus on the cases and show that the Kähler cones of all toric hypersurface threefolds force the effective 1-dimensional LVS moduli space to be compact. We find that the moduli space size can generically be trans-Planckian only for K3 fibred examples.

Paper Structure

This paper contains 14 sections, 19 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. LVS vacua with flat directions
  3. General conditions for LVS vacua
  4. Classes of LVS vacua with h11=3
  5. The reduced moduli space
  6. Computation of the Kähler cone
  7. Improved computation of the Kähler cone
  8. Compactness of the reduced moduli space
  9. Scanning results
  10. Analytic proof
  11. Implications
  12. Phenomenological implications
  13. Conceptual implications
  14. Conclusions

Figures (3)

  • Figure 1: Pictorial representation of a Kähler cone with $h^{1,1}=3$ parametrised by the 2-cycle volumes $t^i$. The hypersurfaces of \ref{['eq:hypersurface']} are represented respectively in blue and red. The intersection between these two hypersurfaces inside the cone represents the reduced moduli space $\mathcal{M}_r$.
  • Figure 2: Distribution of volumes in units of $M_p$ of reduced moduli spaces for different subclasses of LVS vacua with $h^{1,1}=3$ for $g_s=0.1$ and $\mathcal{V}_{ 0}=10^3$, $10^4$ and $10^5$. The histograms on the left display the distribution of ${\rm Vol}\,(\mathcal{M}_{A,r})$ whereas the ones on the right correspond to ${\rm Vol}\,(\mathcal{M}_{\cap,r})$.
  • Figure 3: Distribution of volumes in units of $M_p$ of the reduced moduli space $\mathcal{M}_{A,r}$ for all 43 K3 fibred LVS vacua for $g_s=0.1$, $0.2$ and $0.3$, and $\mathcal{V}_{ 0}=10^3$, $10^4$ and $10^5$.