STOchastic LAttice Simulation of hybrid inflation

Abstract

We investigate the spatial profile of the curvature perturbation generated in multi-waterfall hybrid inflation models, which are known to produce various topological defects. Using the lattice simulation code \acl{STOLAS}, based on the stochastic formalism of inflation, we analyse six cases by varying the number of waterfall fields $n$ and the functional form of the inflaton potential (``Quadratic'' and ``Cubic'' cases). Our statistical analysis shows that the \acp{PDF} and power spectra are broadly consistent with the so-called stochastic-$δN$ algorithm. The ``Cubic'' case also exhibits a characteristic upper bound in the \ac{PDF}, as discovered in our previous work, that suppresses \acl{PBH} formation while potentially affecting halo formation. Furthermore, we employ the Euler characteristic as a topological diagnostic tool to identify the structures of the waterfall fields as well as the curvature perturbation. We find that the topological defects, such as domain walls ($n=1$), cosmic strings ($n=2$), and monopoles ($n=3$), are reconnected during inflation into finer structures by the stochastic noise, making their correlation lengths much smaller than the Hubble scale at the critical point of the waterfall phase transition counterintuitively. The Euler characteristic also implies global structures of the curvature perturbation for $n=1$, though we do not conclude if they are due to the domain wall, because neither the strings ($n=2$) nor monopoles ($n=3$) leave such structures. The global structures of the curvature perturbation will provide a novel probe for the physics of the early universe.

Figures (7)

• Figure 2: PDF of the curvature perturbation corresponding to the density plots in Fig. \ref{['fig: maps']}. The vertical thin line at $\delta\mathcal{N}=0.133$ for "Cubic" $n=1$ and $\delta\mathcal{N}=0.213$ for "Cubic" $n=2$ corresponds to the upper bound on $\delta\mathcal{N}$.
• Figure 3: Power spectrum for each model as the Fourier transform of the density plot in Fig. \ref{['fig: maps']}. The blue dots show the result of STOLAS. The black dashed line shows the analytic formula \ref{['eq:analytic_formula']} with fitting parameters ($N_{\rm water}$, $\mathcal{P}_{\zeta}^{\rm peak}$), whose values are listed in Table \ref{['tab: power']}. We do not show too high $k$ (i.e., high $\mathcal{N}_k$) modes, which are not reliable due to the discreteness of the lattice.
• Figure 4: Snapshots of the waterfall fields below the threshold values \ref{['eq: psi threshold']}. From top row to bottom row, they are "Quadratic" $n=1$, the "Cubic" $n=1$, the "Quadratic" $n=2$, the "Cubic" $n=2$, the "Quadratic" $n=3$, and the "Quadratic" $n=15$. Animation of the snapshots can be found https://github.com/STOchasticLAtticeSimulation/STOLAS_dist/tree/main/hybrid-inflation/Fig4_sample.
• Figure 5: The total Euler characteristic $\chi$\ref{['eq: chi total']} of the waterfall fields below the threshold \ref{['eq: psi threshold']} as a function of the time $N$. The blue and orange dots show the positive and negative values, respectively.
• Figure 6: The Euler characteristic of the curvature perturbation divided by the number of objects in the simulation box, $\chi/\mathscr{N}$, evaluated for each threshold value $\zeta_\mathrm{th}$ shown in the horizontal axis in units of the standard deviation $\sigma_{\zeta} =\sqrt{\overline{\delta\mathcal{N}^2}}$.