Hybrid Inflation from Fermion Condensate

Abstract

We investigate how inflation can emerge from four-fermion interactions generated by spacetime torsion, eliminating the need for additional scalar fields beyond the Standard Model. We partition fermions in two sectors and introduce two bound fields. In the effective theory approach, once all the fermions have been integrated out, the bound fields serve as the inflaton and the auxiliary field, in analogy to the hybrid inflation and accounting for a waterfall (hybrid) mechanism. The inclusion of an axial chemical potential naturally facilitates the end of reheating. During the waterfall regime, the effective potential governing the fermion condensate supports the formation of non-topological solitons, known as Q-balls, which can be accounted for seeding primordial black holes (PBHs).

Figures (3)

• Figure 1: We plot the normalized effective potential $V(A,B)/\Lambda^4$, with the $B$-axis truncated at $0.2\,\Lambda$ for a clearer view of the minima. The minima of $A$ are highlighted in red, while those of $B$ in blue, the other parameters in the potential being kept constant. With the parameters hence specified, the figure shows that the fermion condensate field $A$ undergoes a phase transition, when the bound field $B$, acting as the inflaton, is rolling approaching zero. The golden color point marks the global minimum of the potential, located at the intersection of the two curves of minima. This observation supports the presence of a waterfall mechanism.
• Figure 2: The value of the right-hand side of the equation of motion for the chemical potential is expressed in terms of shades of colors. The solid curves represent specific values of the normalized number density. Solutions to the gap equation lie on the dashed curve. The intersections between the dashed curves and the solid curves are the solutions allowed, with specific number densities.
• Figure 3: We plot the second derivative of the normalized effective potential for different values of the chemical potential. Regions where the derivative is negative are highlighted in gray, below the dashed horizontal line. When the value of the chemical potential is small, the potential exhibits a region of instability before becoming positive. For larger chemical potential values, the second derivative remains positive throughout, indicating stability. This suggests that at low chemical potential, the effective potential is unstable for small gap field values, but the instability diminishes as the gap field grows.