End of inflation, oscillons, and matter-antimatter asymmetry

The dynamics at the end of inflation can generate an asymmetry between particles and antiparticles of the inflaton field. This asymmetry can be transferred to baryons via decays, generating a baryon asymmetry in our Universe. We explore this idea in detail for a complex inflaton governed by an observationally consistent—“flatter than quadratic”—potential with a weakly broken global $U(1)$ symmetry. We find that most of the inflaton asymmetry is locked in nontopological soliton-like configurations (oscillons) produced copiously at the end of inflation. These solitons eventually decay into baryons and generate the observed matter-antimatter asymmetry for a range of model parameters. Through a combination of three dimensional lattice simulations and a detailed linearized analysis, we show how the inflaton asymmetry depends on the fragmentation, the magnitude of the symmetry breaking term and initial conditions at the end of inflation. We discuss the final decay into baryons, but leave a detailed analysis of the inhomogeneous annihilation, reheating and thermalization to future work. As part of our work, we pay particular attention to generating multifield initial conditions for the field fluctuations (including metric perturbations) at the end of inflation for lattice simulations.

Figure 1

A qualitative picture of the homogenous evolution of the complex inflaton field. During inflation, the symmetry breaking term is suppressed. As a result ${\theta }_i={\theta }_{\text{inf}}=\text{constant}$. Note that the field typically starts spiraling around $|\varphi |\lesssim m_{\mathrm{Pl}}$.

Figure 2

Different components of the power spectra of the fields at the end of inflation (with ${\theta }_i=0.7\times \pi /3$). Inside the horizon, the diagonal components match the Minkowski space power spectrum, whereas the cross spectra are small. Outside the horizon, the perturbation spectra [diagonal spectra (orange) and cross spectra (green)] are much larger than the Minkowski space approximations (dashed line). Starting from Bunch-Davies initial conditions deep inside the horizon during inflation, we evolved the perturbations including metric perturbations self-consistently. Ignoring metric perturbations underestimates the spectra on superhorizon scales.

Figure 3

Floquet charts for field fluctuations: parallel to the motion of the homogeneous field (left) and perpendicular to the motion of the homogeneous field (right). The vertical axis is the amplitude of oscillation of the homogeneous mode (assumed to be in the ${\phi }^1$ directions). Lighter colors correspond to unstable regions. The legend shows the magnitude of the real part of the Floquet exponent: $\Re ({\mu }_k)/m$. Note that the parallel perturbations have a broad, strong instability band near $k\lesssim 0.5m$ which is not present for the perpendicular perturbations.

Figure 4

The homogeneous inflaton condensate starts fragmenting within $\sim 20$ oscillations after the end of inflation. The fragmentation is driven by parametric resonance in the fluctuations along the direction of motion of the field. After the perturbations become nonlinear, localized, long-lived field configurations called oscillons form and dominate the energy density of the inflaton field. The oscillons once formed maintain a fixed size and density, and can be very long lived with lifetimes $\gg m^{-1}$, $H^{-1}$. They are highly overdense regions, the contours in the above plots are drawn at $5\times$ the average density. Most of the inflaton asymmetry is locked in these oscillons although they occupy a small fraction of the volume. The comoving size of the box is comparable to the Hubble horizon at the end of inflation.

Figure 5

Evolution of the inflaton/anti-inflaton asymmetry as a function of time. The asymmetry is zero at the end of inflation ($t=0$). Asymmetry is generated during the explosive dynamics after the end of inflation. After the inflaton fragments into localized solitons ($t\sim 150m^{-1}$), the asymmetry settles down to a constant value. We have not checked the asymmetry for significantly longer time scales due to numerical considerations. Although not shown above, a similar plot for the asymmetry for the homogeneous case continues to show large oscillations and settles down at a much later time $t\ge 10^3{m}^{-1}$.

Figure 6

The ratio of the inflaton asymmetry in regions with twice the average density to the total asymmetry (orange curve is smoothed over a few oscillations). After $t\approx 150$, the overdense regions are composed of localized pseudosolitons (oscillons). Once oscillons are formed, most of the asymmetry is locked inside them with a final value of $A_{\text{osc}}/A_{\text{tot}}\approx 0.7$. A qualitatively similar behavior is found if we consider regions with ten times the average density instead. For that case we get $A_{\text{osc}}/A_{\text{tot}}\approx 0.6$.

Figure 7

Inflaton asymmetry as a function of the initial angle made by the homogeneous inflaton field in the complex plane for different values of $c_3$. The black curve corresponds to the homogeneous case, whereas the orange points are results of lattice simulations. This sinusoidal behavior seen for $c_3=10^{-2}$ is seen for all $c_3\ll 1$. The $\pi /3$ period is related to the form of the symmetry breaking term. When $c_3\lesssim 1$, both the homogeneous and fragmented curves become much more complicated, no longer remaining sinusoidal. However, the $\pi /3$ period is still respected.

Figure 8

Inflaton asymmetry as a function of symmetry breaking parameter, with all other parameters fixed (${\theta }_i=0.7\times \pi /3$, $M=10^{-2}{m}_{\mathrm{Pl}}$). The black points correspond to the homogeneous case, whereas the orange points correspond to the results from a full lattice simulation. For $c_3\ll 1$, in both cases $A_{\varphi }\propto c_3^2$, with the inhomogeneous value always being below the homogeneous one.

Figure 9

Asymmetry as a function of $m_{\mathrm{Pl}}/M$ (with all other parameters fixed). The black points and curve correspond to the homogeneous case, whereas the orange points correspond to the results from lattice simulations. Note that the difference between the homogeneous and lattice case becomes larger and larger as $m_{\mathrm{Pl}}/M$ increases. The ratio $m_{\mathrm{Pl}}/M$ can be interpreted as the fragmentation efficiency parameter [see Eq. (42)]. However, the symmetry breaking term also gets suppressed in the high density regions resulting from fragmentation. Hence both the fragmentation into high density regions and the suppression of asymmetry in high density regions due to the form of the symmetry breaking term determine the decrease in asymmetry as a function of $m_{\mathrm{Pl}}/M$ seen in the above figure.