OOscillons during Dirac-Born-Infeld Preheating
Yu Sang ∗ and Qing-Guo Huang
2, 3, 4, † Center for Gravitation and Cosmology, College of Physical Scienceand Technology, Yangzhou University, Yangzhou 225009, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China School of Fundamental Physics and Mathematical Sciences HangzhouInstitute for Advanced Study, UCAS, Hangzhou 310024, China (Dated: January 1, 2021)Oscillons are long-lived, localized, oscillating nonlinear excitations of a real scalar field which canbe abundantly produced during preheating after inflation. We give the first (3 + 1)-dimensionalsimulation for the oscillon formation during preheating with noncanonical kinetic terms, e.g. theDirac-Born-Infeld form, and find that the formation of oscillons are significantly suppressed by thenoncanonical effect.
Introduction.
An interesting possible observable effectfrom preheating is the long-lived, localized, oscillatingnonlinear excitation of a real scalar field, so-called oscil-lon [1–16]. During preheating after inflation, oscillons areabundantly produced following efficient self-resonance ofinflaton field if the potential is quadratic at the bottomand flattening away from the minimum [10]. The for-mation of oscillons in the postinflationary Universe hassignificant cosmological consequences, e.g., the existenceof an oscillon-dominated phase [12, 17–20] and the pro-duction of gravitational wave background [20–28].Although most of the literature concentrates on os-cillons of a canonical scalar field, it is still interest-ing to probe oscillons in the scenario of noncanonicalscalar field. In particular, brane inflationary modelsbased on string theory have been extensively studiedin the literature [29–33]. The kinetic term for the in-flaton field associated with the motion of a D3-branein higher-dimensional background spacetime naturallytakes the Dirac-Born-Infeld (DBI) form which reducesto the canonical kinetic term in the low velocity limit.During preheating after brane inflation, the inflaton fieldmoves fastly and it is natural to take the full DBI forminto account, and the phenomenological consequences ofthe nonlinear dynamics during preheating after brane in-flation provide a possible way to explore string theory.Preheating with noncanonical kinetic terms has beenstudied in literature. See some related works in [34–39].In particular, lattice simulation on preheating with para-metric resonance has been extended to the case of a DBIinfalton field coupled to a canonical matter field in [38],and the authors showed that the parametric resonancein the matter field and self-resonance in the inflaton fieldare as efficient as in traditional preheating. In [39], theexistence of oscillons in scalar field theories with non-canonical kinetic terms in the small-amplitude limit canbe supported solely by the noncanonical kinetic terms,without any need for nonlinear terms in the potential. Inthis letter, we will give the first fully (3 + 1)-dimensional lattice simulation for the formation of oscillons duringDBI preheating, and show that the noncanonical effectcan significantly suppress the formation of oscillons.
Model and method.
Let’s start with a general form ofLagrangian for the noncanonical inflaton field, S = 12 (cid:90) d x √− g (cid:2) M pl R + 2 P ( X, φ ) (cid:3) , (1)where φ is the inflaton field, X = − g µν ∂ µ φ∂ ν φ ,and M pl = 1 / √ πG is the reduced Planck mass. Ina Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) uni-verse ds = − dt + a ( t ) δ ij dx i dx j , (2)the equation of motion of φ is P ,X ¨ φ + 3 ˙ aa P ,X ˙ φ + ∂ ( P ,X ) ˙ φ − a P ,X ∇ φ − a ∂ i ( P ,X ) ∂ i φ − P ,φ = 0 , (3)where P ,X denotes the derivative with respect to X and P ,φ denotes the derivative with respect to φ .In this letter, based on string theory, we consider theDBI inflation [40, 41] with P ( X, φ ) = − f ( φ ) (cid:16)(cid:112) − Xf ( φ ) − (cid:17) − V ( φ ) , (4)which is motivated by brane inflationary models inwarped compactifications. The Lagrangian is a low en-ergy description of a D3 brane travelling in a warpedthroat. The inflaton φ is a collective coordinate whichmeasures the radial position of the brane in the throat.The potential V ( φ ) describes how the brane is attractedtowards the bottom of the throat. The warp factor f ( φ )is determined by the warped geometry. For an anti-deSitter throat, f ( φ ) takes the form f ( φ ) = λ ( φ + µ ) , (5) a r X i v : . [ h e p - t h ] D ec where µ is a parameter corresponding to the infrared cut-off scale. The equation of motion of φ is given by¨ φ (cid:18) fa ( ∂ i φ ) (cid:19) + 3 ˙ aa ˙ φγ − ∇ φa − f ,φ f + 3 f ,φ Xf + 1 γ (cid:18) f ,φ f + V ,φ (cid:19) + fa (cid:34) (cid:18) ˙ aa ˙ φ − ∇ φa (cid:19) ( ∂ i φ ) + ∇ φ ˙ φ + ∂ i φ∂ j φ∂ j ∂ i φa − φ∂ i φ∂ i ˙ φ (cid:35) = 0 , (6)where the Lorentz factor γ ≡ (1 − Xf ) − / , f ,φ ≡ dfdφ and V ,φ ≡ ∂V∂φ . In this letter, we consider the potentialof inflaton φ as follows V ( φ ) = m M (cid:34)(cid:18) φ M (cid:19) / − (cid:35) , (7)which is motivated by the axion monodromy infation[42, 43]. Self-resonance of a canonical infaton field withsuch a potential leads to copious oscillon generation [12]and gravitational waves are significantly produced whenoscillons are being formed [21, 27].The expansion of the Universe is driven by the energydensity of the inflaton field. The evolution of scale factoris described by the Friedmann equation (cid:18) ˙ aa (cid:19) = 13 M ρ , (8)where ρ is the energy density averaged over the spatialvolume ρ ≡ (cid:104) T (cid:105) = (cid:28) f ( γ − −
1) + γ ˙ φ + V (cid:29) . (9)We use a (3+1)-dimensional numerical simulation tostudy the preheating and oscillon formation during DBIpreheating after inflation. Our code is based on a pub-licly available lattice evolution program, Grid and Bub-ble Evolver (GABE) [38, 44], which has been well-usedin the preheating scenarios [45–49]. Using a second-orderRunge-Kutta method for time integration, GABE storesthe field and field derivative value at the same times dur-ing the time step, which is required by terms such as theproduct of the field with its time derivative. For spatialderivative terms, GABE uses the second-order finite dif-ferencing, and we further update the code to fourth-ordercentered differencing scheme. We use a N = 256 box inthe lattice simulations. The parameters in potential areset to m = 3 . × − M pl and M = 0 . M pl , and pa-rameters in the warp factor f are µ = 5 × M pl , and λ = 3 × . We will show that the space-averaged valueof Lorentz factor γ has a maximum value of 2.075 in thesimulation, which means the inflaton field significantlydeparts from canonical case. We expect the preheating follows a slow-roll inflationwith γ = 1 but the noncanonical terms begin to be im-portant after inflation. We evolve the homogeneous in-flaton field from slow-roll region, which gives a constrainton the field and its derivative. Using this constraint asan initial condition, we numerically solve the equation ofmotion of the homogeneous inflaton field, i.e., the Eq.(6) without the gradient terms, to determine the ini-tial values of inflaton field and its derivative for our lat-tice simulation. The the initial values are φ i ≈ . M pl and ˙ φ i ≈ − . × − M for the noncanonical model.As a contrast, we also consider the canonical case inour simulation, initial values of which are determined as φ i ≈ . M pl and ˙ φ i ≈ − . × − M . Those initialvalues of the inflaton field and its derivatives ensure thesame initial horizons in lattice simulations for both mod-els. As an approximation, the fluctuations of field andits derivative are initialized as the Bunch-Davies vaccum(see Ref. [38] for details). Since preheating is insensitiveto the initial power spectrum [50], we expected the initialconditions of the fluctuations will not affect the resultsof the simulation. Results.
The noncanonical kinetic terms bring termswith γ factor and higher order derivatives into the equa-tion of motion of inflaton field. The effect of these termson the evolution of the field can be illustrated by themean inflaton field value shown in Fig. 1. From Fig. 1, canonicalnoncanonical - ( m - ) ϕ ( M p l ) FIG. 1. Evolution of the mean inflaton field for simulationswith µ = 5 × M pl , λ = 3 × , γ max ≈ .
075 (red solidline) and canonical case γ ≈ we see that the noncanonical kinetic term can signifi-cantly change the evolution of the inflaton field duringDBI preheating. As the inflaton field and its velocityoscillate with time, the relativistic factor γ also exhibitsoscillatory behavior. The space-averaged value of γ ar-rives its maximum value γ max ≈ .
075 at t ≈ . m − when field has largest velocity in our setup. Due to thepresence of noncanonical terms, the motion of inflatonfield approaches a sawtooth pattern, unlike the canoni-cal field with an sinusoidal evolution. As the noncanon-ical terms become efficient, the sawtooth grows sharper,which enhance the resonance. However, the lengthenedperiod of oscillations suppresses resonance and these twoeffects compete with each other. According to Ref. [37],the second effect should dominate over the former one,leading to a less efficient preheating for the noncanonicalinflaton field. The result of self-resonance here is consis-tent with the simulation in Ref. [38].The evolution of variance of the inflaton field is shownin Fig. 2. By comparing the time of occurrence, we see canonicalnoncanonical - - - - - t ( m - ) ( Δ ϕ ) ( M p l ) FIG. 2. Evolution of the variance of the inflaton field forsimulations with µ = 5 × M pl , λ = 3 × , γ max ≈ . γ ≈ the sharp troughs in the evolution in Fig. 2 correspond tothe extreme points of motion of inflaton field in Fig. 1,and the peaks to the zero points, for both noncanonicaland canonical cases, except for the initial part of the firstoscillation period ( t (cid:46) m − ). This implies that inflatonfield with larger velocity tends to be more inhomogeneousin a local temporal region. For example, during the evo-lution of noncanonical inflaton field from t ≈ m − to t ≈ m − , the velocity of inflaton field varies from zeroto its local maximum. Meanwhile, the variance of fieldgrows from 1 . × − M to 1 . × − M . Unlike thecase of inflaton field coupled with a scaler matter field, inwhich self-resonance of noncanonical and canonical infla-ton field have dramatic difference (see Fig. 3 in Ref. [38]),here for the single inflaton preheating with potential (7),we see the variance of φ grows quickly in the early stagefor both noncanonical and canonical cases. The reasonis that the mechanism for self-resonance is provided bythe potential term, and for the case with noncanonicalterms, they suppress the efficiency of self-resonance. Forexample, the noncanonical terms delay the end of self-resonance until approximately t ∼ m − , comparingto the canonical self-resonance ending at t ∼ m − .The oscillon configurations at t = 300 m − are shownin Fig. 3 and Fig. 4. The number of oscillons doesnot change with time in our simulations, correspond-ing to an oscillon dominated phase when oscillon hasbeen fully formed. In both figures, the canonical case isshown in the upper panel, and the noncanonical case with µ = 5 × M pl and λ = 3 × is shown in the lower (a)(b) FIG. 3. The snapshot of the energy density at t = 300 m − for simulations with (a) canonical case γ ≈ µ = 5 × M pl , λ = 3 × , γ max ≈ .
075 (lowerpanel). The energy density isosurface is taken at ρ = 5 (cid:104) ρ (cid:105) . panel. In Fig. 3 the energy density isosurface is taken ata value 5 times the average energy density over lattice.The number of the overdensity region and the volume foreach one in canonical model are much larger than thosein noncanonical cases. Fig. 4 is a two-dimensional slice ofthe energy density rescaled with the average value. Theenergy density in some region is even up to 40 (cid:104) ρ (cid:105) in thecanonical case, but it is suppressed to only 15 (cid:104) ρ (cid:105) in thenoncanonical case. Discussion.
In the canonical case, oscillons can formwhen the potential satisfies the open up condition, i.e.,the potential is quadratic at the bottom, and is shallowerthan quadratic in the field space away from the minimum.For the noncanonical case, the contribution of potentialterms on the nonlinear evolution of field is reduced by thenoncanonical kinetic terms, leading to the suppression ofself-resonance and oscillon formation. Our result is de-rived from numerical simulation using (3+1)-dimensional (a)(b)
FIG. 4. The snapshot of the energy density ρ/ (cid:104) ρ (cid:105) on atwo-dimensional slice at t = 300 m − for simulations with (a)canonical case γ ≈ µ = 5 × M pl , λ = 3 × , γ max ≈ .
075 (lower panel). lattice without any approximation of small amplitude os-cillons. In fact, Ref. [39] studies a rather general form ofscalar field Lagrangians with noncanonical kinetic termsand gives the condition for the existence of oscillons inthe small amplitude limit, and they supposed that theexistence of oscillons can be supported solely by the non-canonical kinetic terms, without any need for nonlinearpotential terms. However the conclusion for small ampli-tude oscillons cannot be generalized to large amplitudecase.The choice of values of parameters in the model needsfurther studied in the further works. As discussed in Ref.[12], the formation of oscillon is sensitive to parameter M in potential (7) but is insensitive to the detailed formof the potential. For the parameters in DBI model, weexpect the suppression of oscillons is more efficient whenthe parameters depart further from the canonical case.The gravitational-wave background during the forma-tion of oscillon has not been simulated in this work. Dueto the presence of the noncanonical effect in the model,we expect the gravitational-wave spectrum should be sig-nificantly different from the canonical model as well. Wehope to figure out this issue in the near future. Acknowledgments.
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