Composite Hybrid Inflation : Primordial Black Holes and Stochastic Gravitational Waves

Abstract

We investigate the production of primordial black holes and gravitational waves in composite hybrid inflation. Starting from an effective chiral Lagrangian with a dilaton and pions, we identify inflation occurring due to the walking dynamics of the theory. A $\mathbb{Z}_2$ symmetry-breaking term in the pion sector induces a shift in the inflaton's trajectory, which leads to a tachyonic instability phase. Curvature perturbations grow exponentially, producing copious primordial black holes and a stochastic gravitational wave background. We show that the primordial black hole mass and the gravitational wave frequency are strongly restricted by the anomalous dimensions of the pion operators, with larger anomalous dimensions giving lighter primordial black holes and higher frequency gravitational waves. In both cases, the associated signatures lie within reach of future gravitational wave observatories.

Figures (17)

• Figure 1: Field trajectory (solid line) in the $(\chi, \phi)$ plane, with the contours representing the potential height. The full trajectory depicts a turn (left), where a zoomed-in figure displays the tachyonic hill responsible for the turn (right).
• Figure 2: Evolution of $\epsilon$ (left) and of the effective scalar masses (right) as a function of $\mathcal{N}$. For both plots, we solve the background evolution equations \ref{['eq:background']} for a specific set of parameters (corresponding to set 2 in Table \ref{['tab:Benchmark']}). In the right plot, the yellow-shaded region shows the stage that the background is dominated by a tachyonic field $\phi$, i.e., $m_{\phi}^2< 0$, while the horizontal dotted line indicates the value of the Hubble parameter.
• Figure 3: E-folding number $N$ during the stages $1$ and $3$, and the total, as a function of some model parameters with fixed $\gamma_m = \gamma_{4f} = 1$. In the left plot, we use $f_\chi=0.5$, $\lambda_{\chi}=1\times 10^{-13}$ and two values of $\delta_2 = 1$ (solid lines) and $0.8$ (dashed lines). In the right plot, we use $f_\chi=0.5$, $\delta_{1}=1.03\times 10^{-4}$ and $\delta_{2}=1$. The blue and red lines show the e-folding number of stage 1 and stage 3, while the black line shows the total. The pink vertical line shows the benchmark set 2 in Table \ref{['tab:Benchmark']}, and the green vertical line shows the parameters for $n_s=0.9668$. The orange shaded area shows the parameter space of the model that gives $n_s\in (0.9668 - 0.0037,0.9668 + 0.0037)$ for $\delta_2=1$.
• Figure 4: The $M_{\text{eff}}^{2} / H^2$ (left) and the $\dot{\theta}^2 / H^2$ (right) for set 2 in Table \ref{['tab:Benchmark']}. The temporary tachyonic stage where $|M_{\text{eff}}^{2}| \gg 9/4 H^2$ induces the isocurvature perturbation growth, and the turn rate $\dot{\theta}^2 \gtrsim H^2$ sources these enhanced isocurvature perturbations to the curvature perturbation.
• Figure 5: The curvature power spectrum $\mathcal{P}_{\mathcal{R}}(k, \mathcal{N})$(blue) and the isocurvature power spectrum $\mathcal{P}_{\mathcal{S}}(k, \mathcal{N})$(orange) for modes $k = e^{-10}k_{\rm inst}$(left) and $k = e^{-15}k_{\rm inst}$(right), where $k_{\rm inst}\equiv e^{\mathcal{N}_{\rm inst}}H(\mathcal{N}_{\rm inst})$. The black dotted vertical line depicts the horizon exit e-folding number $\mathcal{N}_{k}$ , where the blue dotted vertical line corresponds to the start of the tachyonic growth. For the plot, we numerically solve for the perturbations referring to the Transport method Dias:2015rca using the parameters set 2 in Table \ref{['tab:Benchmark']}.