Nonlinear dynamics of oscillons and transients during preheating after single field inflation

The single field inflation driven by a power-law potential is ruled out by the measurements of cosmic microwave background (CMB) anisotropies Aghanim et al. (2020); Akrami et al. (2020). The observational constraints favor a broad class of potentials characterized by a power-law shape at the bottom and plateaus in region away from the minimum Cai et al. (2015); Cook et al. (2015); Kabir et al. (2019); Martin et al. (2015); German et al. (2024). These class of models has interesting nonlinear dynamics in the post-inflationary evolution Hindmarsh and Salmi (2008); Enqvist et al. (2002); Coleman (1985); Frolov (2008); Amin et al. (2019); Karam et al. (2023). After the end of inflation, the inflaton field oscillates around the minimum of its potential Guth (1981); Mukhanov and Chibisov (1981); Starobinsky (1982); Linde (1983, 1990). The energy transfer from the inflaton field to standard model particles occurs through the oscillating decay process, leading to the subsequent thermalization of the universe and the onset of a radiation-dominated era Albrecht et al. (1982). In traditional preheating scenarios, the fluctuations of the coupling matter field undergo exponential amplification through parametric resonance Kofman et al. (1994, 1997). However in the models with plateaus-like potential, self-resonance occurs and leads to the amplification of the inflaton field itself Amin et al. (2012).

Oscillon is a kind of long-lived, localized excitations of nonlinear scalar fields Kasuya et al. (2003); Amin (2010); Amin et al. (2010). If the inflaton potential has a quadratic bottom and flattens away from the minimum, oscillons are copiously produced as a consequence of self-resonance Kawasaki et al. (2020); Mahbub and Mishra (2023); Sang and Huang (2021). A possible matter-dominated phase exists if oscillons are fully formed in the post-inflationary stage, which has been proved by simulating the equation of state in some models Lozanov and Amin (2017, 2018); Liu et al. (2018). Gravitational waves are generated when the oscillons are forming, because of the time-evolving and inhomogeneous energy density Lozanov and Amin (2019); Liu et al. (2019); Hu et al. (2023); Zhou et al. (2013); Sang and Huang (2019); Hiramatsu et al. (2021); Li et al. (2020); Amin et al. (2018); Antusch et al. (2017); Fu et al. (2019); Jin et al. (2020); Adshead et al. (2023, 2024); El Bourakadi et al. (2021).

Another kind of nonlinear structure was found to be formed during preheating after the $\alpha$-attractor models which are consistent with the CMB observation and attract lot of interest of the community Ueno and Yamamoto (2016); El Bourakadi et al. (2022); Eshaghi et al. (2016). The equation of state and duration to radiation domination were calculated in the $\alpha$-attractor T model for both quadratic minima and nonquadratic minima cases Lozanov and Amin (2017, 2018). If the minimum shallower than quadratic, e.g., $V\propto|\phi|^{2n}$ by $n>1$, the inflaton condensates fragment into transients, which are oscillon-like localised spherical objects but much less stable. The equation of state is similar to radiation ($w\rightarrow 1/3$) in the case of transients formation, as oppose to the oscillon-dominated phase ($w\rightarrow 0$).

In this paper we consider a class of observationally favored inflation models, taking $\alpha$-attractor T model, E model and monodromy-like models as examples. We will emphasize several key questions within these models, including whether symmetric potential in T model and the asymmetric potential in E model lead to distinct nonlinear dynamics, how different parameters in monodromy models affect the the formation of oscillons and trasients, the equation of state during the evolution.

This paper is organized as follows. In Sec. II we introduce the inflation models used in this work. We give a linear Floquet analysis in Sec. III and small amplitude analysis in Sec. IV. The lattice simulation method is introduced in Sec. V, and the simulation results are given in Sec . VI for $\alpha$-attractor models and Sec. VII for monodromy models. In Sec. VIII we summarizes our work along with future prospects. We adopt natural units with $G=\hbar=c=1$ and the reduce Planck mass $m_{\rm pl}=1/\sqrt{8\pi G}$. The metric signature is chosen as $(+---)$, and we employ the Friedmann-Robertson-Walker metric in the form of $\mathrm{d}s^{2}=\mathrm{d}t^{2}-a^{2}\mathrm{d}x^{2}$, where $a(t)$ represents the scale factor.

We consider the single field inflation model characterized by minimal coupling with gravity. The action is expressed as follows

where $R$ represents the Ricci scalar, $\phi$ denotes the inflaton field. We consider the Friedmann-Robertson-Walker (FRW) metric here. The equation of motion for the scalar field and the Friedmann equation have the form

In this work, we consider a class of inflation models that is consistent with the constraint of CMB observations Akrami et al. (2020). The inflaton potential exhibits a plateau at large field values $\left(\phi\gg M\right)$ and a power law form $\propto|\phi|^{2n}$ near small field values ($\phi\ll M$). We chose three distinct parameterized models:

• •

  the $\alpha$-attractor T models Carrasco et al. (2015); Kallosh and Linde (2013); Galante et al. (2015)

  $$V\left(\phi\right)=\Lambda^{4}\tanh^{2n}\left(\frac{\left|\phi\right|}{M}\right),$$

  (4)

• •

  the $\alpha$-attractor E models Carrasco et al. (2015); Kallosh and Linde (2013); Galante et al. (2015)

  $$V\left(\phi\right)=\Lambda^{4}\left|1-\exp\left(-\frac{2\phi}{M}\right)\right|^{2n},$$

  (5)

• •

  monodromy type potentials McAllister et al. (2010, 2014); Silverstein and Westphal (2008)

  $$V\left(\phi\right)=\Lambda^{4}\left[\left(1+\left|\frac{\phi}{M}\right|^{2n}\right)^{\frac{q}{2n}}-1\right].$$

  (6)

The potentials of $\alpha$-attractor T model, E model and monodromy model are shown in Fig. 1. The dotted line is the inflection point of the curve. In all models, the potential has a power-law bottom, and exhibits different asymptotic behavior far away from it. The parameter $M$ is related to the inflection point of the potential. In this work, we set the value of parameter $M=0.01m_{\rm pl}$, ensuring that all of them satisfy the observational boundary requirements $\left(M\ll 10~{}m_{\rm pl}\right)$ and successfully implement the slow-roll process. Parameter $n$ determines its behavior near the bottom in all models. The energy scale parameter $\Lambda$ is eliminated using the amplitude of the scalar perturbations and spectral tilt from the CMB observations.

Both the T models and monodromy models are symmetric, while the E models are asymmetric. In the T models, the potential tends to a constant value at infinity, whereas in E models it reaches an extreme value on one side only. In addition, in monodromy models, there is an extra parameter $q$ that can be utilized to suppress the upward trend of the curve near the large field values. It controls the asymptotic behavior of the potential in infinity. CMB observation restricts this parameter to a maximum value of 1 Akrami et al. (2020). The monodromy type potential goes back to the axion monodromy model if the parameters $n$ and $q$ are 1.

In the following sections, we use the effective masses in the analysis for convenience. The effective masses of the three models are defined as

which makes the approximation of the potential in the form $V=m^{2}\phi^{2n}/2n$ near its minimum Lozanov and Amin (2018). Here we assume $\phi_{0}$ represents the inflaton field at the ending time of slow roll inflation, as well as the starting time of preheating.

We assume a homogeneous background $\bar{\phi}\left(t\right)$ and a small perturbation $\delta\phi\left(x,t\right)$ when the inflaton field oscillates at the bottom of the potential after inflation. The expansion of the universe is neglected in Floquet analysis since the oscillating frequency $\omega\gg H$ Lin (2023a); Lozanov (2019). This implies that the period of oscillation is much shorter than the characteristic time scale of the expansion of universe.

In linear analysis, the fluctuations of each Fourier mode are independent and can be easily solved numerically Amin et al. (2014); Turzyński and Wieczorek (2019); Antusch et al. (2018). The equation of motion for the perturbative field is

which is known as the Mathieu equation (or Hill equation), characterized by its harmonic damping term. According to Floquet’s theorem, the solution of the equation is

where $\mathcal{P}_{k\pm}$ is a periodic function determined by the initial conditions. The Floquet exponent $\mu_{k}$ is a complex number that only depends on the wavenumber $k$. If the real part of the exponent $\Re_{\mu_{k}}\neq 0$, the solution is unstable and exponentially growing, implying that a large number of resonant particles are produced.

Here we take the monodromy model as an example, and perform Floquet analysis with different model parameters. The results of Floquet analysis are shown in Fig. 2. The bright regions indicate the presence of unstable solutions. Due to their continuous regional distribution, they are called the instability bands. The instability bands shown in Fig. 2 are divided into a broad band and several narrow bands.

The wide region in the Floquet plot in the Fig. 2 near the $k=0$ axis is the broad resonance band. The broad resonance band appears in all the parameter combinations we have selected, which represents the main region of the nonperturbative decay channel of inflaton. Compared with other unstable regions, the resonance strength of it is higher and the range is wider. The strong self-interaction with the inflaton background leads to a violation of the adiabatic conditions $\dot{\omega}/\omega^{2}\geq 1$ during each oscillation, resulting in the production of resonant particles in bursts within this band.

There are several narrow resonance bands away from the $k=0$ axis in all cases we have shown. For the $n=1$ cases, as shown in the first column in Fig. 2, the narrow resonance bands represent the other main nonperturbative decay channels in addition to the broad resonance band. However in the case of $n>1$, as the instability zone moves, a series of new narrow resonance bands appear in the region $\phi\sim M$. Their resonance strength is lower and gradually disappears as the wave number $k$ increases. Since preheating occurs after the end of the slow roll inflation, with the amplitude of the oscillations $\phi\sim M$, such narrow bands, especially the first narrow band, dominate the evolution of the preheating.

The parameter $n$ has a significant influence on the region of resonance bands. As the increase of $n$, the region of broad resonance band becomes smaller and some narrow resonance bands even disappear. The parameter $n$ has a weak influence on the resonance strength, causing it to decrease slightly with increasing $n$. As for the parameter $q$, it does not affect the region of resonance bands and has a minimal impact on the resonance strength. Therefore, in monodromy models, the shape of potential bottom, rather than that of the plateau, significantly influences the resonance bands.

In this section we calculate the solution for the inflaton field through the small amplitude analysis which has been well applied to many models Fodor et al. (2006); Farhi et al. (2008); Fodor et al. (2008); Hertzberg (2010). The small amplitude analysis is employed to examine the dynamics of the field before the lattice simulation and identify any peculiar field configurations. For the case of $n=1$, there exists an oscillating solution with a stable field configuration known as oscillons Hasegawa and Hong (2018); Amin and Shirokoff (2010); Amin (2013); Manton and Romańczukiewicz (2023). Here we take the $\alpha$-attractor T Model as an example, performing small amplitude analysis for the oscillon ($n=1$) and transient ($n>1$) cases.

We perform a 3-dimensional (3D) lattice simulation of nonlinear real scalar field dynamics in an expanding universe. The public code GABE¹¹1https://cosmo.kenyon.edu/gabe.html Child et al. (2013) is used in our work. In the simulation, the dynamics of inhomogeneous inflaton field is evolved by Eq. (2), and the expansion of the universe is described by

Here the bracket denotes spatial average over the lattice.

In this work, we focus on the dynamical evolution of the field and the non-adiabatic particles generation at the subhorizon scales. To avoid any influence from gravitational collapse, we have not set an excessively long duration for the simulation. The starting time of the simulation is set around the end of the inflation where the slow-roll approximation is broken. We use a $N=256^{3}$ box in the lattice simulations and the lattice size $L=0.3/H$, which is always smaller than the hubble horizon scale.

The nonlinear growth of the perturbation leads to inhomogeneity in the spatial distribution of the field, which is manifested in the enhanced resonance of the gradient term. We calculate the spatial distribution of energy density and transfer between different components including kinetic energy, potential energy, and gradient terms of the total energy density

Another way to describe the evolutionary process is the equation of state of the universe. We use the spatially averaged equation of state

where $\rho$ and $p$ represent the energy density and pressure of the inflaton field, respectively.

In the following discussion, we will fix the value of parameter $M$ in these three models and consider different parameters $n$ or parameter combinations $\left[n,q\right]$ to simulate and analyze the results.

The $\alpha$-attractor model has a power-law region near the origin ($\phi\ll M$) and a flat plateau far away from the bottom where the preheating process happens. In this section, we show the results of lattice simulation for $\alpha$-attractor model.

In this section, we will show the results of lattice simulation for monodromy model. The shape of the potential is shown in Fig. 1. In the region near the bottom, the potential is given by

While in the region of $\phi\gg M$, the potential is approximated by

The model in the small field region depends on both $n$ and $q$, depend only on $q$ in large field region. The values of parameters using in the simulation is listed in Table. 2.

In this paper, we investigated the nonlinear dynamics of the inflaton field during preheating in the $\alpha$-attractor T model, E model and monodromy model, which are favored by the observations. These models have a power-law form $\left|\phi\right|^{2n}$ near the origin ($\phi\ll M$) and a plateau in the region of $\phi\gg M$. We presented the results of Floquet analysis in monodromy models and found that the potential parameters $n$ have a significant influence on the region of resonance bands, but have a weak influence on the resonance strength. Utilizing the small amplitude analysis method, we derived the analytical expression of the oscillons profile ($n=1$) for the $\alpha$-attractor T model and discussed the method applicability to the transient situations ($n>1$). Through the (3+1) dimensional lattice simulation, we performed a detailed analysis of the dynamic evolution and the nonlinear structure during preheating in all three models. In the $\alpha$-attractor models, symmetric potential in T model and the asymmetric potential in E model lead to similar nonlinear dynamics. We also find that with the increase of the potential parameter $n$ in monodromy model, the lifetime of the transient is largely reduced, but the potential parameter $q$ has limited influence on the lifetime of the transient.

With the different combinations of the parameters $q$ and $n$, we have investigated the influence of models parameters on the preheating. The parameters $q$ is only appear in the monodromy models. It mainly controls the region that $\phi\gg M$. In our simulation, we find that back-reaction process will be later with the parameters $q$ increases, but the lifetime of the transients we calculate in the simulations is almost unchanged. While the parameter $n$ appear in three models only controls the phase of preheating that inflaton condensate fragment into oscillon ($n=1$) or transients ($n>1$). As it increase, the lifetime of the transient from the fragmentation of the condensate to its decay of completely will be shorter. Compare to the parameters $q$, the effect of $n$ is more significant.

Note that such structures, oscillons and transients, can be generated under arbitrary parameter we selected. Therefore, we expect that such structures will exist in all of the models with a wider the parameter range. Besides, the phases of formation and decay of these structures can generate the gravitional waves, and whether difference combinations of parameter will influence their frequency or energy density remains an open question. We will stress these issues in the future work.

We would like to thank Shulan Li, Qingyang Wang and Shoupan Liu for helpful discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12005184, 12175192, and 12005183).