Impact of other scalar fields on oscillons after hilltop inflation
IImpact of other scalar fields on oscillons after hilltop inflation
Stefan Antusch (cid:63) † , Stefano Orani (cid:63) (cid:63) Department of Physics, University of Basel,Klingelbergstr. 82, CH-4056 Basel, Switzerland † Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, D-80805 M¨unchen, Germany
Oscillons are spatially localized and relatively stable field fluctuations which can formafter inflation under suitable conditions. In order to reheat the universe, the fields whichdominate the energy density after inflation have to couple to other degrees of freedom andfinally produce the matter particles present in the universe today. In this study, we use latticesimulations in 2+1 dimensions to investigate how such couplings can affect the formation andstability of oscillons. We focus on models of hilltop inflation, where we have recently shownthat hill crossing oscillons generically form, and consider the coupling to an additional scalarfield which, depending on the value of the coupling parameter, can get resonantly enhancedfrom the inhomogeneous inflaton field. We find that three cases are realized: without aparametric resonance, the additional scalar field has no effects on the oscillons. For a fastand strong parametric resonance of the other scalar field, oscillons are strongly suppressed.For a delayed parametric resonance, on the other hand, the oscillons get imprinted on theother scalar field and their stability is even enhanced compared to the single-field oscillons. Email: [email protected] Email: [email protected] a r X i v : . [ h e p - ph ] A p r Introduction
Inflation provides an attractive framework for explaining the initial conditions of hot bigbang cosmology. During inflation, the universe undergoes a phase of accelerated expansion,driven by a dominating vacuum energy component. Its simplest realization consists of asingle scalar field, the inflaton field, which is slowly rolling down a mild potential slope. Thevacuum energy is then provided by the potential energy of the inflaton field, which, afterinflation, is converted into radiation during the phase of reheating.Reheating is typically subdivided into two stages: an initial non-perturbative stagecalled preheating, and a final stage of perturbative particle decays and thermalization. Oncethermal equilibrium is reached, the epoch of conventional hot big bang cosmology starts.Understanding the details of reheating is necessary in order to connect the inflationary dy-namics to the subsequent evolution of the universe and finally to the underlying particlephysics theory. Furthermore, to derive precise predictions of the primordial spectrum onCMB scales we need to know the exact expansion history of the universe during this phase.Observations from the Planck satellite [1, 2] constrain the possible shapes of the inflatonpotential. Among the viable potentials [3], those with a plateau, such as in hilltop models[4–9], are interesting candidates. In this class of models, inflation happens while the inflaton φ rolls away from the top of a hill and towards a minimum of its potential. Hilltop inflationof this type can be realized with the following inflaton potentials: V ( φ ) (cid:39) V (cid:18) − φ p v p (cid:19) , (1.1)where p ≥ p ≥
6, the model is compatible with the most recent 1 σ Planck bounds. Potentials of the form (1.1) can be found in realistic particle physics models,where they are associated with, e.g., phase transitions breaking a GUT symmetry [4] or aflavour symmetry [10]. The initial conditions for hilltop inflation models can be obtainedwith a preinflation mechanism, by which the position of the inflaton at the top of the hillis dynamically generated. This motivates the introduction of a second scalar field χ , whichforces the inflaton to be at the maximum of its potential (see, for example, [6, 9, 11]).The introduction of other fields besides the inflaton is of course also necessary in order toreheat the universe, since, as already mentioned above, the inflaton has to transfer its energyto other degrees of freedom in order to finally produce the matter particles of the presentuniverse. An interesting possibility in this context is to indentify the scalar field χ introducedas a preinflaton in [9] with a right-handed sneutrino. Via the decays of the sneutrinos intoHiggs(inos) and (s)leptons, the universe can reheat efficiently and at the same time thematter-antimatter asymmetry may be generated via non-thermal leptogenesis.Preheating after hilltop inflation is challenging due to the non-linear nature of the fielddynamics: as the inflaton rolls down the potential and oscillates around the minimum, itsfluctuations get enhanced by “tachonic preheating” as well as “tachyonic oscillations” [12–14]and quickly become non-linear. As has recently been shown in [15], the highly inhomogeneousinflaton field can furthermore excite other scalar fields, such as the χ field, via parametricresonance. Whether this resonant enhancement of the χ fluctuations occurs depends on thevalue of the coupling between φ and χ .A particularly interesting phenomenon which can arise during preheating when the po-tential is shallower than quadratic away from the minimum [17], is the formation of localizedoscillating configurations of the inflaton field φ , called oscillons [16, 17]. After hilltop infla-tion with potential of the form of eq. (1.1), it has recently been shown that “hill crossing”1scillons generically form, with the field initially oscillating between the two minima of thepotential [14]. Eventually, the hill crossing stops (i.e. the bubbles of “wrong vacuum” finallycollapse and do not reform) and the oscillons fluctuate around the “true” minimum towardswhich the inflaton was initially rolling.Oscillons are of course not unique to hilltop models: other inflationary models whereoscillons have been shown to form include hybrid inflation [18] and supersymmetric exten-sions of it [19, 20]. In addition to inflation, oscillons can also arise in high-energy physics,such as Abelian Higgs models [21, 22] and the Standard Model of particle physics [23, 24].They naturally arise in field theories as a remnant of collapse and collisions of false vacuumbubbles [16, 25]. Their role during preheating has been studied using lattice simulationsin a variety of inflationary models and they are known to have a suprisingly long lifetime[26–29], although quantum corrections can accelerate their decay [30]. Possible observationalconsequences of oscillons range from the production of the primordial baryon asymmetry [31]to the generation of gravitational waves during their formation and decay [32]. Furthermore,they can delay thermalization [33] thus having an effect on the primordial curvature pertur-bation. The impact of couplings to other scalar fields on oscillons after hilltop inflation hasbeen investigated in [29] for a potential similar to eq. (1.1), however no effects have beenobserved.In this paper we revisit the question whether couplings between the inflaton and otherfields can affect the formation and stability of oscillons. We consider hilltop inflation with apotential as in eq. (1.1) with p = 6 and focus on the coupling to an additional scalar field χ which, as discussed in [15], can get resonantly enhanced from the inhomogeneous inflationfield. To this end, we use lattice simulations in 2 + 1 dimensions and show that, dependingon the strength of the coupling between φ and χ , and therefore on the occurrence and timingof the parametric resonance of χ , the formation and stability of the oscillons can be eitherenhanced, suppressed or left unaffected. We expect the resonant amplification of χ causedby the inhomogeneous inflaton to also have a strong impact on oscillons in 3 + 1 dimensions.The paper is structured as follows: in section 2, we introduce the hilltop inflation modelunder consideration and summarize its predictions for the inflationary observables. A shortoverview of the dynamics of preheating after hilltop inflation is given in section 3. In section 4,we introduce the method used for the lattice simulations in 2 + 1 dimensions and present ourresults. Finally, in section 5, we conclude with a summary and discussion of the results. The potential V ( φ, χ ) we consider for our analysis contains the hilltop inflation potential ofeq. (1.1) with p = 6 plus an interaction term between the inflaton φ and the additional scalarfield χ : V ( φ, χ ) = V (cid:18) − φ v (cid:19) + λ φ χ (cid:0) φ + χ (cid:1) , (2.1)where v ∝ m pl , V ∝ m and λ ∝ m − . Observable inflation: initial conditions and predictions
For the hilltop inflation model of eq. (2.1), the initial conditions for observable inflation aregenerated dynamically via a preinflation mechanism, as shown in [9]. Starting with large2eld values of φ and χ , a mass for φ is generated from the interaction term, which efficientlydrives φ towards 0. Subsequently, χ is slowly rolling towards 0 during a phase of preinflation.With both φ and χ close to 0, the induced mass term for φ becomes so small that the fieldsare in a diffusion region where quantum fluctuations dominate over classical rolling. Whenthe fields eventually leave the diffusion region and φ starts rolling down the top of the hilltowards one of the minima of the potential at φ = ± v (while χ ≈ χ at this stage is typically very small and we set χ = 0 at the start of observableinflation in what follows. Furthermore, without loss of generality, we choose to roll along φ > V ( φ ) (cid:39) V (cid:18) − φ v (cid:19) . (2.2)In order to solve the flatness and horizon problems, this phase of inflation lasts more than N ∗ ≡ ln( a e /a ∗ ) ∼ e -folds, where a is the scale factor, the subscripts e and ∗ denote,respectively, the end of inflation and the time when perturbations on the scales of the CMBexit the horizon. Inflation ends when φ exits the slow-roll regime and accelerates towardsthe minimum.The scalar spectral index n s and the tensor to scalar ratio r of the CMB pertur-bations are given in terms of the slow-roll parameters, (cid:15) φ ≡ / m ( ∂V /∂φ ) /V and η φ ≡ m (cid:0) ∂ V /∂φ (cid:1) /V , evaluated at N ∗ : n s = 1 − (cid:15) φ ( φ ∗ ) + 2 η φ ( φ ∗ ) (cid:39) −
105 + 4 N ∗ (cid:39) . ,r = 16 (cid:15) φ ( φ ∗ ) (cid:39) × − (cid:18) vm pl (cid:19) , (2.3)where φ ∗ /m pl (cid:39) . v/m pl ) / . The predictions for v < m pl are compatible with the mostrecent Planck bounds n s = 0 . ± .
006 at 68% CL and r < .
09 at 95% CL [1, 2]. Further-more, the amplitude of CMB scalar fluctuations, A s (cid:39) . × − , leads us to fix the valueof V to V = 24 π ε φ ( φ ∗ ) A s m (cid:39) − v m Pl . (2.4) Derivation of the potential from supersymmetry
The potential of eq. (2.1) can be derived from supersymmetry. Its form follows from thesuperpotential W = (cid:112) V ˆ S (cid:32) v − (cid:33) + λ ˆΦ ˆ X , (2.5)where ˆΦ and ˆ X are, respectively, the chiral superfields containing φ = √ χ = √ X ], the real scalar fields which appear in eq. (2.1). As discussed in [15], we set theimaginary components of the fields to zero. The imaginary component of the inflaton can3ffect the dynamics, leading to different inflationary dynamics and prediction for the primor-dial spectrum [34]. However, for large part of parameter space, the inflationary dynamicsreduce to the single-field limit. The imaginary part of X has the same equation of motionas the real part χ , leading us, for simplicity, to consider only χ . In addition, we assumethat the scalar field S contained in ˆ S is initially confined at S = 0 due to a K¨ahler inducedsuper-Hubble mass during inflation [35]. As we have shown in [15], even if S = 0 initially, itsfluctuations are efficiently enhanced during preheating, such that S essentially evolves as φ ,without significantly affecting the behaviour of χ . Based on this, in what follows we do notinclude S . Moreover, we assume negligible K¨ahler corrections to the masses of φ and χ . Thiscan either be by accident or due to symmetry (e.g. a Heisenberg symmetry [8]). Therefore,for the simulations in this paper, we will restrict ourselves to the potential of eq. (2.1).The advantage of the suspersymmetric formulation of the model is that the form of thesuperpotential (2.5) can be justified by imposing a U (1) R and a Z p symmetry. Then, ˆ S hastwo units of U (1) R charge and is a Z p -singlet. ˆΦ is a U (1) R -singlet and has one unit of Z p charge. Finally, the form of the interaction term is fixed by giving ˆ X one unit of U (1) R charge and two units of Z p charge. In this section we give a brief overview of preheating after hilltop inflation models of thetype of eq. (2.1). The dynamics outlined here are discussed in greater details in [14] and[15]. It is convenient for what follows to consider the Fourier decomposition of a field f inD spatial dimensions: f ( t, (cid:126)x ) = (cid:82) d D k (2 π ) D f k ( t ) e − i(cid:126)k · (cid:126)x , where k ≡ | (cid:126)k | . We will first describe thedynamics of preheating after inflation for the single-field scenario and then discuss how theyare affected when λ (cid:54) = 0.As the inflaton leaves the slow-roll regime and accelerates towards φ = v , tachyonicpreheating very efficiently amplifies modes k for which k /a + ∂ V /∂φ <
0. The efficiencyof this amplification grows as v decreases and for v (cid:46) − m pl , the system becomes non-linearalready during this phase.For v (cid:38) − m pl , this phase is followed by a period of tachyonic oscillations of φ around φ = v , when the homogeneous φ periodically enters the tachyonic region where ∂ V /∂φ < k peak ∼ (cid:112) V / /m pl . For v (cid:38) − m pl , Hubbledamping rapidly prevents the inflaton from entering the tachyonic region, stopping the growthof the field’s fluctuations before they become non-linear. However, for 10 − (cid:38) v/m pl (cid:38) − ,the fluctuations become large enough so that (cid:104) δφ (cid:105) ∼ . v , see fig. 1. This growth leadsto the formation of localized field fluctuations, i.e. oscillons, which initially oscillate between − v (cid:46) φ (cid:46) v and eventually settle at the initial minimum at φ = v [14]. These oscillons areseparated by a characteristic distance d osc (cid:39) π/k peak .In [15], it has been shown that a non-zero coupling λ between the inflaton φ and anotherscalar field χ can lead to a parametric resonance of χ after φ has become inhomogeneous andoscillons have formed. In particular, for v = 10 − m pl there is a resonance band at values of λ such that the ratio of the mass of χ to the mass of φ at the global minimum, m χ /m φ , isaround (and somewhat below) 0 .
5. With m φ ≡ ∂ V∂φ (cid:12)(cid:12)(cid:12)(cid:12) min = 72 V v , m χ ≡ ∂ V∂χ (cid:12)(cid:12)(cid:12)(cid:12) min = λ v ⇒ m χ m φ = λ v √ V , (3.1)4 δϕ 〉 / - - - - λ = 〈δϕ 〉 / 〈δχ 〉 / - - - - λ = × - / pl 〈δϕ 〉 / 〈δχ 〉 / - - - - λ = × - / pl 〈δϕ 〉 / 〈δχ 〉 / - - - - λ = - / pl Figure 1 . Variances (cid:104) δφ (cid:105) /v and (cid:104) δχ (cid:105) /v for v = 10 − m pl and λ = 0 (top left), λ = 1 . × − /m pl (top right), λ = 1 . × − /m pl (bottom left) and λ = 10 − /m pl (bottom right) from simulations in2 + 1 dimensions with 1024 points. The behaviour of (cid:104) δφ (cid:105) is qualitatively similar in the four plots:the φ fluctuations grow rapidly when φ oscillates around v until its homogeneous mode decays becauseof non-linear interactions at a ∼ .
5. On the other hand, χ ’s behaviour depends on the value of λ . For λ = 1 . × − /m pl the amplitude of its perturbations redshifts from the initial vacuum fluctuations.For λ = 1 . × − /m pl and λ = 10 − /m pl , the amplitude of χ ’s fluctuations eventually grows dueto a parametric resonance with the inhomogeneous inflaton field [15]. The timing and efficiency ofthe resonance depends on the strength of the coupling: for λ = 1 . × − /m pl , (cid:104) δχ (cid:105) ∼ (cid:104) δφ (cid:105) at a ∼
4, whereas for λ = 10 − /m pl , χ ’s variance grows earlier, exceeding φ ’s at a ∼ (cid:104) δχ (cid:105) ∼ (cid:104) δφ (cid:105) . the resonance band has been found in [15] to lie in the range 0 . (cid:46) m χ /m φ (cid:46) .
5, whichcorresponds to 0 . × − /m pl (cid:46) λ (cid:46) . × − /m pl This resonance can be very efficient,leading to (cid:104) δχ (cid:105) (cid:38) (cid:104) δφ (cid:105) for certain λ s, see fig. 1. When the parametric resonance occurs,the fluctuations of χ are peaked at scales close to k peak .Motivated by these results, in the remainder of this paper we study the formation andevolution of the oscillons in the hilltop inflation model of eq. (2.1), choosing v = 10 − m pl asan example value, as in [15]. We will compare the dynamics of the single-field case λ = 0to three different choices of λ (cid:54) = 0 (see fig. 1): a value that lies outside the resonance band,for which χ ’s fluctuations redshift from their initial vacuum amplitude; and two other valuesthat lie inside the resonance band and which differ regarding the timing and the strengthof the parametric resonance, which, nevertheless, occurs in both cases after the inflaton hasbecome inhomogeneous. We will see in section 4 that a coupling outside the resonance bandhas no significant effect on the evolution of oscillons. On the other hand, values of λ insidethe resonance band have a dramatic effect: depending on the time at which the fluctuations5 /m pl (cid:104) φ (cid:105) i /v (cid:104) ˙ φ (cid:105) i /v (cid:104) χ (cid:105) i /v (cid:104) ˙ χ (cid:105) i /v H i /m pl − .
08 2 . × − . × − D N k uv k ir δx L . × H i . H i − /H i . /H i Table 1 . Initial values of (cid:104) φ (cid:105) and (cid:104) ˙ φ (cid:105) and other parameters of the 2D lattice simulations. of χ grow, the oscillons can be either enhanced or suppressed. In this section we present and analyse the results of lattice simulations of the hilltop inflationmodel of eq. (2.1). The aim is to understand the evolution of the oscillons that form at theend of inflation. To this end, we numerically solve the non-linear equations of motion¨ f ( t, ¯ x ) + 3 H ˙ f ( t, ¯ x ) − a ¯ ∇ f ( t, ¯ x ) + ∂V∂f = 0 , (4.1) H ≡ (cid:104) ρ (cid:105) m = 13 m (cid:42) V + (cid:88) f (cid:18)
12 ˙ f + 12 a (cid:12)(cid:12) ¯ ∇ f (cid:12)(cid:12) (cid:19)(cid:43) , (4.2)where f represents the fields, φ and χ , ¯ ∇ is the gradient with respect to the comovingcoordinates ¯ x and (cid:104) .. (cid:105) denotes the average over space. We use a modified version of theprogram LATTICEEASY [36] to solve these equations on a discrete spacetime lattice.Since oscillons are known to remain stable over a surprisingly large number of oscilla-tions, the lattice simulations have to run for long timescales. This leads us to focus on 2 + 1dimensions with 1024 points. We start the lattice simulations shortly after the end of inflation, when | η φ | > (cid:15) φ < χ = 0 at the end of inflation. However, χ can acquire anon-zero value at the end of inflation and its main effect on the preheating dynamics is toincrease the efficiency of the parametric resonance (see [15]).The fields are initialized in Fourier space as random fields whose norms obey theRayleigh distribution with variance given by the vacuum fluctuations: f k = 1 a | f k | (cid:113) α + α − (cid:16) α + e i πθ + + ikt + α − e i πθ − − ikt (cid:17) , ˙ f k = ika | f k | (cid:113) α + α − (cid:16) α + e i πθ + + ikt − α − e i πθ − − ikt (cid:17) − Hf k , (4.3)where α + , α − , θ + and θ − are random real numbers uniformly distributed between 0 and 1.The fields (4.3) are then Fourier transformed to position space, where a discretizedversion of eqs. (4.2) is solved with lattice spacing δx = 2 π/k uv , where k uv is the ultraviolet6 = × - / pl , ρ slice at = ρ / 〈ρ〉 λ = × - / pl , ρ slice at = = = - - - ρ / 〈ρ〉ρ tail distribution for λ = × - / pl Figure 2 . Above : energy density slices at a = 10 (left) and a = 22 (right) for v = 10 − m pl and λ = 1 . × − /m pl , from a simulation with 1024 points. Oscillons (localized red spots) are clearlyvisible at a = 10 but most of them have decayed by a = 22. The single-field case λ = 0 is qualitativelythe same. Below : tail distribution of the energy density ρ as a function of ρ/ (cid:104) ρ (cid:105) at a = 10 (blue) and a = 22(orange) for λ = 1 . × − /m pl . In grey we show the tail distribution of ρ g = 36 V g /v + λ v g / g and g are two discrete Gaussian random fields defined on a lattice with 1024 points, withzero means and standard deviations chosen such that (cid:104) g (cid:105) / (cid:104) g (cid:105) ∼ (cid:104) δφ (cid:105) / (cid:104) δχ (cid:105) ∼ . The largest ρ goes from ρ > (cid:104) ρ (cid:105) at a = 10 to ρ < (cid:104) ρ (cid:105) at a = 22. Furthermore, at a = 22 the tail distributionis closer to what one would expect from a Gaussian random field. This confirms that most oscillonsdecay by the end of the lattice simulation. For the single-field case λ = 0 we obtain qualitatively thesame results. cutoff. The infrared cutoff is given by k ir = k uv /N , where N is the number of points perspatial dimension. The lattice length is L = 2 π/k ir .In subsection 4.2 we present the numerical results of lattice simulations in 2 spatialdimensions with parameters given in table 1. We study the evolution of the single-field case,corresponding to λ = 0 in eq. (2.1), and of the two-field case with λ = 1 . × − /m pl , λ = 1 . × − /m pl and λ = 10 − /m pl . In this section, we present and discuss the results of four lattice simulations in 2 + 1 dimen-sions with N = 1024 and other parameters, given in table 1. The simulations correspond,respectively, to λ = 0 (single-field case), λ = 1 . × − /m pl , λ = 1 . × − /m pl and7 = × - / pl , ρ slice at = ρ / 〈ρ〉 λ = × - / pl , ρ slice at = = = - - - ρ / 〈ρ〉ρ tail distribution for λ = × - / pl Figure 3 . Above : energy density slices at a = 10 (left) and a = 22 (right) for v = 10 − m pl and λ = 1 . × − /m pl , from a simulation with 1024 points. Oscillons are clearly visible both at a = 10and a = 22. Below : tail distribution of the energy density ρ as a function of ρ/ (cid:104) ρ (cid:105) at a = 10 (blue) and a = 22(orange) for λ = 1 . × − /m pl . In grey we show the tail distribution of ρ g = 36 V g /v + λ v g / g and g are two discrete Gaussian random fields defined on a lattice with 1024 points, withzero means and standard deviations chosen such that (cid:104) g (cid:105) / (cid:104) g (cid:105) ∼ (cid:104) δφ (cid:105) / (cid:104) δχ (cid:105) ∼ /
2. Both a = 10and a = 22 tail distributions spread to O (1000) (cid:104) ρ (cid:105) , far from the Gaussian expectation, indicating thatoscillons are enhanced compared to the non-resonant case χ in fig. 2. λ = 10 − /m pl . The latter two cases are two-field cases where χ ’s fluctuations grow due to aparametric resonance caused by the inhomogeneous φ field.We start by discussing the position space slices of the energy density ρ in 4.2.1 (figs. 2,3 and 4), since they provide the most intuitive illustration of the formation and evolution ofoscillons. Next, in 4.2.2 we discuss the energy density tail distributions T D ρ (same figures,below the ρ slices). They are defined as the probability that ρ is greater than a value (cid:37) ,that is T D ρ ( (cid:37) ) = P ( ρ > (cid:37) ). Finally, in 4.2.3 and fig. 6, we present the time evolution of thequantity ρ . , which we define later and illustrates the evolution of the oscillons, emphasizingthe dependence of their stability on λ . Figs. 2, 3 and 4 show the energy density slices at times a = 10 and a = 22 , together withthe corresponding energy tail distributions, for λ = 1 . × − /m pl , λ = 1 . × − /m pl and8 = - / pl , ρ slice at = ρ / 〈ρ〉 λ = - / pl , ρ slice at = = = - - - ρ / 〈ρ〉ρ tail distribution for λ = - / pl Figure 4 . Above : energy density slices at a = 10 (left) and a = 22 (right) for v = 10 − m pl and λ = 10 − /m pl , from a simulation with 1024 points. No oscillons can be seen both at a = 10 and a = 22. Below : tail distribution of the energy density ρ as a function of ρ/ (cid:104) ρ (cid:105) at a = 10 (blue) and a = 22(orange) for λ = 10 − /m pl . In grey we show the tail distribution of ρ g = 36 V g /v + λ v g / g and g are two discrete Gaussian random fields defined on a lattice with 1024 points,with zero means and standard deviations chosen such that (cid:104) g (cid:105) / (cid:104) g (cid:105) ∼ (cid:104) δφ (cid:105) / (cid:104) δχ (cid:105) ∼ /
5. The taildistributions at a = 10 and a = 22 are both very close to the Gaussian tail distributions, indicatingthat no oscillons are present. λ = 10 − /m pl , respectively. Fig. 5 shows the field values of φ and χ for slices at a = 22 for λ = 1 . × − /m pl .Let us first look at the ρ slices for the coupling outside the resonance band, i.e. λ =1 . × − m pl , in fig. 2. The upper-left slice shows ρ at a = 10. We can clearly see many redspots, which correspond to overdensities with ρ (cid:38) (cid:104) ρ (cid:105) . These spots are localized oscillationsof the inflaton field around the minimum, i.e. oscillons. The upper-right slice shows ρ at a = 22. By then, most of the oscillons present at a = 10 have disappeared and just a fewremain. Comparing the case λ = 1 . × − m pl with the single-field case λ = 0 we find thatthe evolution of the oscillons is qualitatively the same: a coupling λ outside the resonancebands does not significantly affect the formation and stability of the oscillons.For λ = 1 . × − /m pl , at a = 10 the situation is similar to the non-resonant χ case.Indeed, the upper-left slice of fig. 3 contains oscillons, analogously to the corresponding slicewith λ = 1 . × − /m pl . On the other hand, at a = 22 the situation is very different: one9 = × - / pl , ϕ slice at = δ / 〈δ 〉 / - - λ = × - / pl , χ slice at = Figure 5 . Slices of φ (left) and χ (right) at a = 22 for v = 10 − m pl and λ = 1 . × − /m pl , froma simulation with 1024 points. The red spots in both slices show that both φ and χ constribute tothe oscillons. The fluctutaions of the two fields are clearly correlated, although the χ red spots seemlarger than those of φ . Note that not all correlations may be visible since the localized oscillations in φ and χ can be out of phase. can see in the upper-right slice of fig. 3 that the oscillons have not decayed. On the contrary,they seem to have expanded compared to a = 10, i.e. the presence of χ , enhanced by theparametric resonance, makes the oscillons more stable for λ = 1 . × − /m pl . The relativecontributions of φ and χ to the oscillons can be seen in the field slices at a = 22 shownin fig. 5. Both fields contribute to the oscillons, however those the localized χ fluctuationslook somewhat larger. Furthermore, one can see that the fluctuations in the two fields arecorrelated: by looking at the two slices in fig. 5 we can pair many red spots of the φ slice withred spots of the χ slice. It should be noted, however, that it may be that not all correlationsare identified, since the oscillations of φ and χ may be out of phase at the time of shownslices and thus one of the red spots may not be visible.Finally, the ρ slices for λ = 10 − /m pl are shown in fig. 4, where a strong parametricresonance of χ happens earlier. In contrast to the other cases, no oscillons can be seen at a = 10 and a = 22. This is a dramatic difference to the two other cases and emphasizes thestrong dependence of the fate of oscillons on the timing of the parametric resonance. Notethat the fluctuations of χ start growing after the inflaton field has become inhomogeneous:the resonance happens shortly after hill crossing [14] and rapidly destabilizes the oscillons.While we have seen that a delayed parametric resonance of χ (e.g. for λ = 1 . × − /m pl discussed above) enhances the oscillons and makes them more stable, an early and strongparametric resonance of χ has the opposite effect and strongly suppresses the oscillons. In order to quantify the contribution of the oscillons to the energy of the lattice it is usefulto look at the tail distribution of the energy density of the lattice. The tail distribution
T D ρ of a real random variable ρ is defined as the probability that ρ is greater than a value (cid:37) , thatis T D ρ ( (cid:37) ) = P ( ρ > (cid:37) ).The tail distributions of the energy density at times a = 10 and a = 22 can be seen infigs. 2, 3 and 4, for λ = 1 . × − /m pl , λ = 1 . × − /m pl and λ = 10 − /m pl , respec-10ively. In order to compare the tail distributions to a Gaussian expectation, we generatedtwo discrete Gaussian random fields g and g on a lattice with 1024 points, with zeromean and standard deviations chosen such that (cid:104) g (cid:105) / (cid:104) g (cid:105) ∼ (cid:104) δφ (cid:105) / (cid:104) δχ (cid:105) . We then define ρ g = 36 V g /v + λ v g / λ . Of course, we do not expect the fields to beexactly Gaussian. It is nevertheless useful to compare the lattice distributions to the onesfor Gaussian random fields because the latter provide examples of field configurations withsame variances but clearly without oscillons. The comparison allows to distinguish the over-densities from the expected statistical fluctuations of the energy density.In the λ = 1 . × − /m pl case, shown in fig. 2, at a = 10 the energy density taildistribution reaches values of ρ > (cid:104) ρ (cid:105) . This is what one would expect from a latticecontaining many oscillons, with a large fraction of the energy density contained in regionswith ρ (cid:29) (cid:104) ρ (cid:105) . By a = 22, the tail distribution has shrunk, closer to the expectation for aGaussian energy density ρ g . For λ = 0, we obtained the same results. This means that theoscillons start decaying within our simulations when λ lies outside the resonance band (andfor λ = 0), as we can also see from the energy density slices.For λ = 1 . × − /m pl , see fig. 3, both at a = 10 and a = 22 the tail distributionsreach values of order 1000 (cid:104) ρ (cid:105) , much larger than the Gaussian expectation. This confirmswhat we see in the energy density slices: the oscillons do not decay within the simulationwhen λ = 1 . × − /m pl . In this case, the parametric resonance of χ , sourced by theinhomogeneous φ , stabilizes the oscillons.Lastly, fig. 4 shows the tail distributions for λ = 10 − /m pl . Here, the tail distributionsat a = 10 and a = 22 are both very close to the Gaussian expectation. The oscillons havealready decayed at a = 10: in this case, the parametric resonance of χ suppresses the oscillons. Fig. 6 summarizes the results discussed in the previous subsections. It shows the value ρ . for which 10% of the energy is contained in regions with energy density ρ > ρ . for the foursimulations. That is, ρ . is defined by: (cid:88) ¯ x with ρ (¯ x ) >ρ . ρ (¯ x ) = 0 . (cid:88) all ¯ x ρ (¯ x ) , (4.4)With oscillons being localized overdensities of energy, larger values of ρ . indicate more (ormore spatially extended, or more energetic) oscillons.The curves for λ = 1 . × − /m pl (blue in the figure) and λ = 0 (dashed black) trackeach other, emphasizing that a coupling outside the resonance band has no effect on theformation and stability of the oscillons. In both cases, one can see that ρ . reaches ∼ (cid:104) ρ (cid:105) at a ∼
10 before decreasing to ∼ (cid:104) ρ (cid:105) , indicating that most oscillons decay by the end of thesimulation. Therefore, a coupling outside the resonance band has no effect on the oscillons.For λ = 1 . × − /m pl (green), where χ ’s fluctuations are amplified at a ∼ ρ . reaches ∼ (cid:104) ρ (cid:105) at a ∼
10 before settling at ∼ (cid:104) ρ (cid:105) . In this case, the parametric resonanceof χ enhances the oscillons and stabilizes them.For λ = 10 − /m pl (red), where the parametric resonance of χ occurs at a ∼ ρ . rapidly settles at ∼ (cid:104) ρ (cid:105) , close to the Gaussian expectation (dashed gray). In contrast to the λ = 1 . × − /m pl case, the formation of oscillons is strongly suppressed by χ .11
10 15 20125102050 ρ . / 〈 ρ 〉
2D lattice with 1024 points λ = × - / pl λ = × - / pl λ = - / pl λ = Figure 6 . The value ρ . for which 10% of the energy resides regions with ρ > ρ . for v = 10 − m pl and λ = 1 . × − /m pl (blue), λ = 1 . × − /m pl (green), λ = 10 − /m pl (red) and λ = 0(dashed black). The gray dashed line is what one would expect for ρ g ∝ g , where g is a discreteGaussian random field with zero mean, defined on a lattice with 1024 points. The formation ofoscillons drives ρ . / (cid:104) ρ (cid:105) to larger values. For λ = 1 . × − /m pl and λ = 0, the lines track eachother, showing that a coupling outside the resonance band has no effect on the evolution of oscillons.For λ = 1 . × − /m pl one can see that ρ . / (cid:104) ρ (cid:105) increases at the same pace as the single-field andnon-resonant cases until a ∼
4. Afterwards, the growth of the fluctuations of χ drives ρ . / (cid:104) ρ (cid:105) tolarger values for λ = 1 . × − /m pl , reaching a maximum of ∼
50 at a ∼
10 before settling at ∼ λ = 1 . × − /m pl and λ = 0, ρ . / (cid:104) ρ (cid:105) decreases down to ∼ ∼
20 at a ∼
10, indicating that oscillons decay in the single-field and non-resonantcases in 2 + 1 dimensions. Finally, for λ = 10 − /m pl , ρ . / (cid:104) ρ (cid:105) initially grows to values close to 5,corresponding to the initial stages of oscillon formation. However, it subsequently drops to lowervalues, indicating that the oscillons are strongly suppressed in this case. The factor that determines whether the growth of the fluctuations of χ enhances orsuppresses the formation of oscillons is the time at which the growth starts. Indeed, as canbe seen in fig. 1, for λ = 1 . × − /m pl , (cid:104) δχ (cid:105) starts growing around a ∼
4, when the φ oscillons have already oscillated around φ = v many times. Fig. 6 shows this clearly: until a ∼
4, the evolution of ρ . for λ = 1 . × − /m pl is practically the same as for the single-field case λ = 0 and for λ = 1 . × − /m pl , where χ is not amplified. The contributionof the φ oscillons to the energy density until a ∼ (cid:104) δχ (cid:105) ∼ (cid:104) δφ (cid:105) , the two curves diverge and, for λ = 1 . × − /m pl , theoscillons’ contribution to the energy density is enhanced. As can be seen in fig. 5, in thiscase also χ forms oscillons.On the other hand, for λ = 10 − /m pl , the growth of (cid:104) δχ (cid:105) starts earlier, just after thehill crossing and during the formation phase of the oscillons: already at a = 2, (cid:104) δχ (cid:105) (cid:38) (cid:104) δφ (cid:105) .Correspondingly, the value of ρ . decreases and the formation of oscillons is suppressed. In this paper we have investigated how oscillons, which are spatially localized and relativelystable fluctuations of the inflaton field φ that can form after inflation, are affected by thecouplings to other scalar fields. We considered hilltop inflation models of the type studiedin [9, 14, 15], where we have recently shown that hill crossing oscillons generically form [14].Furthermore, in [15] we have shown that when another scalar field χ is coupled to the inflaton,depending on the value of the coupling parameter, the latter can get resonantly enhanced bythe inhomogeneous inflaton field. 12n order to study the effect of χ on the oscillons, we solved the discretized equationsof motion on a lattice with 2 + 1 dimensions for different values of the coupling constant,including the single-field case (which corresponds to zero coupling). The other choices of thecoupling constant λ were guided by the observations of [15]: we chose one value for which theparametric resonance of χ happens around the time at which oscillons form, λ = 10 − /m pl ,a second value for which it happens later and is somewhat less efficient, λ = 1 . × − /m pl ,and a third value λ = 1 . × − /m pl for which no resonant enhancement occurs.In contrast to earlier studies [29] which used a similar potential and observed no ef-fects of other scalar fields on the oscillons, we found that the impact of the coupling to χ can be very strong and that three cases can be realized: for a fast and strong parametricresonance of χ (e.g. for λ = 10 − /m pl ), oscillons are strongly suppressed. For a delayed andsomewhat weaker parametric resonance (e.g. for λ = 1 . × − /m pl ), the inflaton oscillonsget imprinted on the other scalar field and their stability is even enhanced compared to thesingle-field oscillons. Only when no parametric resonance occurs (e.g. for λ = 1 . × − /m pl ),and the χ field stays subdominant, the additional scalar field has no effects on the oscillons.We expect the resonant amplification of χ caused by the inhomogeneous inflaton to also havea strong impact on oscillons in 3 + 1 dimensions.Of course, the other scalar field can be excited for other reasons than parametric res-onance, i.e. it might have an initially large homogeneous mode or could be produced byfast perturbative decays. In such cases, we would also expect potentially strong effects onoscillons. However, the study of such scenarios is beyond the scope of this paper.Another interesting question that we did not address, is the effect of quantum fluctu-ations on the evolution of oscillons after hilltop inflation. It has been shown in [30] thatcouplings to fermions can decrease the classical lifetime of oscillons. This question needsfurther investigation, although, since the effect of the other scalar field starts relatively earlyin the oscillons’ evolution, we expect that the enhancement and suppression of oscillons weobserved will remain even when quantum corrections are taken into account.In summary, we found that the couplings of the inflaton to other scalar fields can havea strong impact on the formation and stability of oscillons, especially when the latter fieldsare resonantly enhanced. Such impact affects the expansion history of the universe, possiblyleading to observable effects from oscillons after inflation. Acknowledgements
This work has been supported by the Swiss National Science Foundation.
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