Power spectrum of density perturbations in chain inflation

Chain inflation is an alternative to slow roll inflation in which the Universe undergoes a series of transitions between different vacua. The density perturbations (studied in this paper) are seeded by the probabilistic nature of tunneling rather than quantum fluctuations of the inflaton. We find the scalar power spectrum of chain inflation and show that it is fully consistent with a $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. In agreement with some of the previous literature (and disagreement with others), we show that ${10}^4$ phase transitions per $e$-fold are required in order to agree with the amplitude of cosmic microwave background anisotropies within the observed range of scales. Interestingly, the amplitude of perturbations constrains chain inflation to a regime of highly unstable de Sitter spaces, which may be favorable from a quantum gravity perspective since the swampland conjecture on trans-Planckian censorship is automatically satisfied. We provide new analytic estimates for the bounce action and the tunneling rate in periodic potentials which replace the thin-wall approximation in the regime of fast tunneling. Finally, we study model implications and derive an upper limit of $\sim {10}^{10}\mathrm{GeV}$ on the axion decay constant in viable chain inflation with axions.

Figure 1

Illustration of inflation theories. In slow roll inflation, a scalar field slowly moves down its potential. In old inflation, the Universe undergoes a single phase transition. In chain inflation, the Universe tunnels along a chain of vacua with slowly decreasing energy.

Figure 2

During inflation the two comoving scales $r$ and $r^{\prime }$ evolve in the same way, only shifted by the time $\mathrm{\Delta }t$.

Figure 3

Evolution of bubble walls in a past light cone diagram in $1+1$ dimensions. Note that time moves upward in the diagram; the spacetime point of interest is at the top of the diagram and looking at the past light cone means looking downward from this point. In the three panels the tunneling rate increases from left to right. The horizontal axis denotes the position in real space (not the comoving position) in Hubble units. Vacuum transitions are depicted by points, while the lines correspond to bubble walls. The total number of vacuum transitions $N_W$ is obtained by counting the number of walls hitting an observer at $r=0$. The three panels above feature $N_W=\{40,90,182\}$ (from left to right).

Figure 4

Derivative of the mean inflaton field value from our simulation (error bars). Statistical errors are tiny and, therefore, hardly visible in the figure. The dashed line indicates the fit shown in the plot legend. The mean inflaton field value scales linearly with $\mathrm{\Delta }\varphi$ which we set to unity in the figure (to avoid clutter).

Figure 5

Derivative of the variance from our simulation (error bars) expressed by the derivative of the mean inflaton field value. The variance scales quadratically with $\mathrm{\Delta }\varphi$ which we set to unity in the figure.

Figure 6

Typical potential in chain inflation between two minima (${\varphi }_{-}$,$V_{-}$) and (${\varphi }_{+}$,$V_{+}$). The field value of the intermediate maximum is denoted by ${\varphi }_T$.

Figure 7

The rescaled bounce action $\mathcal{S}$ [defined in (33)] as a function of $x=f{\mu }^3/{\mathrm{\Lambda }}^4$. The exact numerical solution evaluated at several $x$ (points) is shown together with the analytic fit indicated in the plot legend. Also shown is $\mathcal{S}$ as obtained in the thin-wall and the triangle approximation.

Figure 8

The Born approximation term [defined in (37)] as a function of $x=f{\mu }^3/{\mathrm{\Lambda }}^4$. The exact numerical solution evaluated at several $x$ (points) is shown together with the analytic fit indicated in the plot legend.

Figure 9

Constraint on the axion decay constant as a function of $x=f{\mu }^3/{\mathrm{\Lambda }}^4$. The blue shaded region is excluded by the combination of CMB and unitarity constraints.