Constraints on real scalar inflation from preheating of LATTICEEASY

The inflationary paradigm has emerged as an elegant solution to several challenges within the standard model of cosmology Guth:1980zm ; Linde:1981mu , capturing the attention of numerous researchers since its proposal. Experimentally, precise measurements of the cosmic microwave background radiation temperature and anisotropic polarization provide strong support for the slow-rolling inflation Planck:2015bpv ; Planck:2018jri . Theoretical investigations have led to the proposal of numerous inflation models, with the single-field inflation model being the simplest and most widely studied. Examples of such models include polynomial potential models  Martin:2013tda ; Martin:2006rs ; Martin:2010kz ; Dai:2014jja , the Starobinsky model Starobinsky:1980te ; Jin:2020tmm ; Mantziris:2022fuu , Higgs Inflation Bezrukov:2007ep ; Gundhi:2018wyz ; Antoniadis:2021axu ; Liu:2022myp ; Barrie:2021mwi , Natural Inflation Freese:1990rb ; Zhang:2018wbn ; Odintsov:2019mlf ; Cheng:2021qmc , Hilltop Inflation  Linde:1983psb ; Bostan:2019wsd ; Kallosh:2019jnl , and others.

The testing of numerous inflation models has become a central challenge in the field of inflationary physics. Unfortunately, direct observations of inflation in the early universe are not possible, and researchers rely on indirect methods to test these models. One common approach is to utilize the scalar spectral index and tensor-to-scalar power ratio, which are predicted by Planck’s observations of the cosmic microwave background (CMB). As the accuracy of Planck’s observations has improved, it has led to the exclusion of many polynomial inflation models Planck:2015bpv ; Planck:2018jri . However, a promising avenue to address this issue is to consider the non-minimal coupling between the scalar inflaton and the Ricci scalar. This framework offers a mitigation scheme that can potentially rescue a significant number of inflation models Choubey:2017hsq ; Lebedev:2021zdh ; SBudhi:2019vln ; Cheng:2022hcm . By incorporating this non-minimal coupling, the problematic aspects of certain inflation models can be alleviated, providing a more favorable agreement with the observational data from Planck and preserving their viability.

Considerable efforts have been devoted to further constraining scalar inflation models by exploring the interconnectedness of various physical phenomena. After inflation ends, the universe enters a reheating stage Albrecht:1982mp ; Kofman:1994rk ; Kofman:1997yn ; Lozanov:2019jxc ; Saha:2020bis , during which the inflaton decays into standard model particles or even dark matter particles, and the universe will be heated. Thus the reheating may provide a platform to study the inflation.

The limitations of reheating in the context of the single-field scalar inflation model have been extensively studied in Ref. Cook:2015vqa . Reheating involves a preheating process during the early stages, as described in works such as Kofman:1994rk ; Kofman:1997yn ; Greene:1997fu ; Dolgov:1989us ; Traschen:1990sw ; Shtanov:1994ce . The preheating phase plays a crucial role as it enables efficient transfer of energy from the inflaton field to the coupled fields. This rapid, non-thermal process is vital for reheating the universe, establishing the initial conditions for the hot Big Bang phase. Without preheating, the transition from the inflationary phase to the radiation-dominated era would occur at a considerably slower pace and with lower efficiency, possibly resulting in discrepancies with observational cosmology. The presence of preheating can be discerned through its influence on the power spectrum of CMB fluctuations and the formation of large-scale structures. Specifically, models with preheating predict distinct non-Gaussianities in the CMB fluctuations, which can be probed by current and future observational missions.

A comprehensive analysis of the interplay between preheating and inflation is presented in Ref. ElBourakadi:2021nyb . Furthermore, the constraints imposed by preheating on $\alpha$-attractor inflation models are investigated by allowing for flexible parameter choices in the properties of the preheating stage, as discussed in Ref. ElBourakadi:2022lqf . The complexity of the preheating process arises from several factors, including the non-perturbative generation of material particles and the rapid energy transfer. This complexity gives rise to highly nonlinear and uncertain physics, making the study of preheating challenging and necessitating careful consideration of its implications on inflationary models.

Many non-perturbative reheating models have been proposed to study the nature of preheating, which include the parametric resonance model Kofman:1997yn ; Cai:2021yvq , the hyperluminal instability model Felder:2000hj ; Enqvist:2005qu ; Dufaux:2008dn , the instantaneous preheating model Felder:1998vq ; Panda:2009ji ; Dimopoulos:2017tud and the holographic preheating cosmological model Cai:2016sdu ; Cai:2016lqa . In our study, we focus on investigating the inflation model by leveraging the properties of preheating. Specifically, we consider the minimal scalar inflation model, which is a simplified version derived from the Polynomial potentials model. To analyze the preheating stage, we employ the LATTICEEASY simulation framework Felder:2000hq , which enables us to replicate and study the evolution of preheating processes.

Specifically, we extensively investigated the evolution of particle number density, scale factor, and energy density during the preheating process using LATTICEEASY simulations. By analyzing the simulated data, we deduced important parameters such as the energy ratio $\gamma$ and the e-folding number of preheating $N_{pre}$. Additionally, we established an analytical relationship between the preheating process and the inflation model under study. By combining this analytical relation with the observed properties of preheating, we were able to derive constraints on the minimal scalar inflation model. These constraints provide valuable insights into the viability and parameter space of the inflation model, thereby enhancing our understanding of its dynamics and predictions.

The rest of this work is organized as follows. In Sec. 2, we briefly review the minimal scalar inflation model and preheating constraints, and study the S-field driven inflation in detail. Then in Sec. 3, we make a simulation for the preheating by applying LATTICEEASY, and discuss the evolution of particle number density, scale factor, and energy density. Numerical analysis and discussion are presented in Sec. 4. We make a summary in Sec. 5.

As the physical properties of the preheating process are highly non-linear and complex, so the effective research method is usually related to lattice simulation. In this section, we will apply LATTICEEASY Felder:2000hq to reconstruct the physics of the preheating and test the scalar inflation model, which simulate the emergence of the Higgs field subsequent to the damping of the S-field through a set of coupled field motion equations.

Consider the following couplings as illustrative examples: $\lambda_{S}=10^{-13}$, $\lambda_{Sh}=2\times 10^{-11}$ and $\lambda_{h}=8\times 10^{-12}$. Firstly, the choice of $\lambda_{Sh}$ directly impacts the generation process of Higgs. When $\lambda_{Sh}\geq 10^{-11}$, the production of Higgs may occur through either the freeze-in or freeze-out mechanism, for which a more in-depth discussion can be found in Ref. Lebedev:2021ixj . Secondly, the parameters $\lambda_{S}$ and $\lambda_{h}$ represent the coupling strengths of the S field and the Higgs field, respectively, and they significantly impact the dynamics of the system. Ref. Lebedev:2021zdh ; Prokopec:1996rr indicates that a strong self-interaction of the Higgs field can lead to a large effective mass term, thereby suppressing resonance phenomena. Therefore, the selection of $\lambda_{S}$ and $\lambda_{h}$ is crucial for the outcomes of our simulations. In our simulation strategy, we deliberately chose smaller values for $\lambda_{S}$ and $\lambda_{h}$. This choice results in a smaller effective mass term, which in turn enhances the resonance phenomena, thereby significantly enhancing the backreaction effect during the preheating process. This approach allows us to deeply explore the significance of backreaction effects during the preheating phase and their impact on particle dynamics under early universe conditions. We will discuss in detail how to use preheating to constrain the inflation model Lebedev:2019ton ; Lebedev:2021zdh .

Fig. 2 show the relationship between $\xi_{s}$ and $N_{k}$ by fixing $\lambda_{S}=10^{-13}$, and the variation of $\xi_{S}$ with respect to $N_{k}$ is remarkably little. However, the value of $\xi_{S}$ does not bring any impact on the constraints of preheating on inflation.

The evolution of the number density of S- and h- particles are shown in Fig. 3 by applying the LATTICEEASY simulation. Note that the Higgs field indeed begins to emerge as the inflaton starts its decay process. However, it is crucial to point out that the decay of the inflaton does not strictly start immediately after inflation ends. This implies that particle production might be observable even before the end of inflation. PIL At the beginning of evolution, particle number density grows exponentially. When the conformal time $\tau$ ($d\tau=\frac{\sqrt{\lambda_{S}}S_{0}}{a}dt$, with $S_{0}=S(\tau=0)$) is about $130$, both particles almost reach the maximum number density, and then slowly change, which indicates that the evolution of the universe transitioned from preheating to reheating.

The physics behind this phenomenon can be explained as follows: when the slow-rolling condition is upset, the inflaton will soon tumble to the lowest point of potential and oscillate periodically around the lowest point, which causes the equation of motion of the h-field to become a Lame equation.

Parameter resonance for the h-field will emerge in some scenarios, which further bring an increase exponentially for the h-particle number, while the h-particle decay produces a significant backreaction for S-particles, and the S-particle number density also increases exponentially.

When the conformal time is about $130$, the amplitude of the S-field oscillation decreases, which leads to the weakening or even disappearance of the parametric resonance effect of the h-field, so that the h-particle number density no longer increases any more, and at the same conformal time, the backreaction effect also disappears, and the S-particle number density does not increase.

The variation of the scale factor over conformal time is shown on the left of Fig. 4. The end conformal time of the preheating obtained by the simulation of particle number density evolution is about $130$, and the corresponding scale factor is $a_{end}\approx 70$, while the value of the scale factor at the beginning of the preheating is $a_{star}=1$. Thus, one can calculate that the preheating e-folding number $N_{pre}=ln[a_{end}/a_{star}]\approx 4.25$.

The evolution of normalized energy density $\rho_{tot}(\tau)/\rho_{tot}(\tau=0)$ with respect to conformal time $\tau$ is shown in the right of Fig. 4, where the green dot is the ended energy of preheating according to the left of Fig. 4. Note that, $\rho_{tot}(\tau=0)=\rho_{end}$ and $\rho_{tot}(\tau=130)=\rho_{pre}$. Therefore, at $\tau=130$, the y-label of Fig. 4 is represented as $\rho_{tot}(\tau=130)/\rho_{tot}(\tau=0)\approx 0.77$, i.e., $\rho_{pre}/\rho_{end}=\gamma\approx 0.77$. This simulation result is quite different from the assumption ($\gamma\approx 10^{3}$ or $10^{6}$) in Ref. ElBourakadi:2021nyb ; ElBourakadi:2022lqf .

Fig. 5 shows the relationship between the e-folding number of preheating $N_{pre}$ and scalar spectral index $n_{s}$. It indicates $N_{pre}$ increases with the state parameter $\omega$. After combining the prediction of $n_{s}$ from Planck with the LATTICEEASY simulation, the model has a feasible parameter space. The feasible range of $\omega$ is $1/4$ to $1$, and the corresponding $n_{s}$ range is $[0.9607,0.9623]$.

The relation between $n_{s}$ and $r$ is shown in Fig. 6. Note that, the blue line between red point and cyan point is the feasible space of the LATTICEEASY simulation prediction obtained from Fig. 5. Therefore, the feasible parameter range of $r$ can be obtained, i.e., $r\thicksim[3.9\times 10^{-4},4.3\times 10^{-4}]$.

To illuminate the, we collect the fixed couplings of the scalar inflation model and the LATTICEEASY predictions in Table. 1. With the same strategy, one can test other model parameters.

In this paper, we focus on studying a non-minimum coupled real scalar inflation model using the preheating phenomenon simulated by LATTICEEASY. During the inflation, the S-field plays a crucial role in driving the expansion of the universe, with its quartic term dominating the dynamics of inflation. Interestingly, we find that the evolutionary behavior of the inflationary potential remains unaffected by the coupling coefficient of the model. Furthermore, the predictions of important quantities derived from the model are also independent of this coupling coefficient.

Consequently, the tensor-to-scalar power ratio $r$ and scalar spectral index $n_{s}$ predicted by the Planck observations do not impose constraints on the coefficients of the model. However, by leveraging the relationship between preheating and inflation, the preheating process simulated using LATTICEEASY provides a valuable avenue to address this issue effectively.

We specifically investigate the constraint of inflation using the simulated preheating process, taking the couplings $\lambda_{S}=10^{-13}$, $\lambda_{Sh}=2\times 10^{-11}$, and $\lambda_{h}=8\times 10^{-12}$ as an illustrative example. By employing LATTICEEASY, we accurately reproduce the preheating dynamics within the scalar inflation model. With the introduction of LATTICEEASY, we have reproduced the preheating process of the scalar inflation model and obtained the particle number density evolution during the preheating process, and further deduced the end conformal time value of preheating. At the same conformal time, the e-folding number of preheating $N_{pre}$ and the energy ratio $\gamma$ are deduced by combining the simulated evolution of the scale factor and energy density, respectively.

By exploiting the relationship between preheating and inflation, we present the variation of $N_{pre}$ with $n_{s}$ in Fig. 5, enabling us to deduce the range of $w$ and $n_{s}$. Further, by exploring the connections between $n_{s}$ and $N_{k}$, as well as $N_{k}$ and $r$, we can obtain constraints on the ranges of $N_{k}$ and $r$. This implies that the preheating simulations conducted with LATTICEEASY can effectively predict $n_{s}$, $r$, and $N_{k}$, allowing for restrictions on scalar inflation models. Furthermore, this strategy can be extended to effectively constrain models involving inflaton dark matter model, as well as the interplay between inflation and dark matter model.