Axion-Scalar Systems and Dynamical Distances

Abstract

We study the cosmology of axion-scalar pairs, coupled by a hyperbolic field-space metric and with a string-motivated rational scalar potential. Borrowing tools from the theory of dynamical systems, we are able to classify all late-time trajectories and extract physical properties of the asymptotic solutions. These results suggest a Dynamical Distance Conjecture: along the physical (possibly non-geodesic) trajectories, towers of states become exponentially light as a function of the traversed field-space distance. We further rule out possible counterexamples with wildly oscillating solutions. The considered axion-scalar systems are realized in F-theory compactifications, where the axion-scalar pair is a complex-structure modulus and four-form fluxes induce the asymptotic potentials. We also provide a complete Hodge-theoretic classification of all one-modulus asymptotic potentials of this type.

Figures (5)

• Figure 1: Schematic depiction of different dynamical trajectories connecting a point $Q$ in the bulk of moduli space $\mathcal{M}$ to a point $P$ on the boundary. The red, solid line is a geodesic between the two points, while the dashed orange and purple lines represent oscillating and non-oscillating trajectories respectively (as described in the text).
• Figure 2: Numerical solution to the system \ref{['eq:sysC']}, with initial conditions specified by $x(0)=-0.4,y(0)=0.3,w(0)=1$. The left panel shows the full evolution from $N=0$ to $N=3.5$, while the right panel zooms in on the interval $[3.3,3.5]$, and only shows $w(N),x(N)$ for clarity. The qualitative features of the plot are described in the text, and can be understood analytically.
• Figure 3: Numerical solution to the system \ref{['eq:sysC']} plotted in phase space, exhibiting the segment-like attractor described in the main text. The left and right panel depict the $y-x$ and $y-w$ planes respectively, with color varying as a function of time. As in the previous figure, we have chosen initial conditions specified by $x(0)=-0.4,y(0)=0.3,w(0)=1$, and the evolutions is between $N=0$ and $N=3.5$.
• Figure 4: Numerical solution of the system \ref{['eq:sysNg']} with the potential \ref{['eq:sp2']}. The initial conditions are specified by $x(0)=0.1,y(0)=0.2,w(0)=1,v(0)=20$, and the evolutions is between $N=0$ and $N=11$. The dotted line denotes the asymptotic value of $T$, equal to $T_m$.
• Figure 5: Numerical solution of the system \ref{['eq:sysNg']} with the potentials \ref{['eq:sp1']}. The initial conditions are specified by $x(0)=0.1,y(0)=0.2,w(0)=1,v(0)=20$, and the evolutions is between $N=0$ and $N=12$. The dotted line denotes the asymptotic values of $T$, equal to $1$.