Aspects of Wave Turbulence in Preheating
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Aspects of Wave Turbulence inPreheating
José A. Crespo a and H. P. de Oliveira a, a Universidade do Estado do Rio de JaneiroInstituto de Física - Departamento de Física Teórica,CEP 20550-013. Rio de Janeiro, RJ, Brazil.
E-mail: [email protected], [email protected]
Abstract.
In this work we have studied the nonlinear preheating dynamics of the λφ in-flationary model. It is well established that after a linear stage of preheating characterizedby the parametric resonance, the nonlinear dynamics becomes relevant driving the systemtowards turbulence. Wave turbulence is the appropriated description of this phase since mat-ter distributions are fields instead of usual fluids. Therefore, turbulence develops due to thenonlinear interations of waves, here represented by the small inhomogeneities of the infla-ton field. We present relevant aspects of wave turbulence such as the Kolmogorov-Zakharovspectrum in frequency and wave number domains that indicates that there are a transfer ofenergy through scales. From the power spectrum of the matter energy density we were ableto estimate the temperature of the thermalized system. Corresponding author. ontents
One of the most significant challenges in modern Cosmology is the description of the earlystages of the universe. There is a general acceptance that an inflationary phase [1] charac-terized by a huge expansion of the universe might have occurred that preceded the radiationdominant phase. In this context, the process of reheating plays a crucial role in the transitionof the Universe from the inflationary phase into the radiation phase, and consequently in thecreation of almost all matter constituting the present Universe.The reheating begins at the end of inflation with a stage of parametric resonance witha rapid transfer of energy from the inflaton field into other matter fields, leading to particleproduction and the inflaton decay, far away from thermal equilibrium. Several authors haveexplored [2–6, 8–10] the entire evolution of the inflaton field until the universe has settleddown in a thermalized state characterizing the radiation era. The main feature shared by thesestudies on the later stages of non-perturbative preheating is the ubiquity of turbulence in theprocess towards thermalization. However, the matter constituents at the end of inflation arefields together with their small inhomogeneities rather conventional fluids, which makes moreappropriate to deal with wave turbulence [11] instead hydrodynamic turbulence [13].An appropriate and useful definition of wave turbulence is a state of out-of-equilibriumstatistical mechanics of random nonlinear waves [11]. The most important class of solutionsin wave turbulence are called Kolmogorov-Zakharov (KZ) spectra [12] that correspond toa constant flux of energy through scales, where scaling laws spectra in frequency and wavenumbers are present. In the context of preheating the small inhomogeneities associated to thefields play the role of the waves, some of them are resonant in the first phase of preheating. Thenonlinear interaction of these waves results in the transfer of energy from the homogeneousinflaton field through different scales with the establishment of turbulence in this scenario.In this paper, our main goal is to explore aspects of wave turbulence of late stages ofpreheating exhibiting the KZ spectrum in frequency and wave number. We have considered asingle field inflationary model with quartic potential λφ , where φ is the inflaton field. Thisis the simplest model with a self interacting inflaton field and also the first model in which thenonlinear effects of preheating were studied. Although the observational data have ruled outthis class of inflationary model [14], we claim that the features of wave turbulence are robustand present in more realistic models. We have organized the paper as follows. We introducethe basic equations of the model and the numerical treatment based on spectral methods inthe second Section. In Section 3, we present the numerical results starting from a standard– 1 –erification of the accuracy of the numerical method. In the sequence, we present the relevantaspects of wave turbulence that are the scaling laws associated to the relevant quantities infrequency and wave number. In Section 4, we briefly discussed the effect of backreaction andthe evolution of the equation of state. Finally, in Section 5 we conclude. Let us consider the simplest model of one field preheating that has already been studied inseveral works [6, 8, 15]. We denote φ ( x , t ) as the inflaton field with potential V ( φ ) = λφ evolving in a spatial flat Friedmann-Robertson-Walker universe [2, 3]. The homogeneouscomponent, or simply the homogeneous mode of the inflaton φ ( t ) , is responsible by theinflationary. At the end of inflation, we donote φ (0) = φ e the homogeneous mode at theend of inflation. It will be useful to introduce the conformal time τ defined by a ( τ ) dτ = √ λφ (0) a (0) dt , and the conformal scalar field ϕ ( x , τ ) = φa ( τ ) /φ (0) a (0) , with τ being thescale factor. The evolution equation for the inflaton is written in a simple and convenientway: ϕ ′′ − ∇ ϕ − a ′′ a ϕ + ϕ = 0 , (2.1)where prime indicates derivative with respect to the conformal time τ , with τ = 0 signalizingthe end of inflation. The beginning of preheating is characterized by coherent oscillationsof the inflaton which yields an effective traceless energy-momentum tensor [24]. As a conse-quence, a ( τ ) ∼ τ , therefore, allowing us to set a ′′ = 0 in Eq. (2.1).We have integrated numerically Eq. (2.1) using the collocation or pseudospectral method[18] in a two dimensional square box of size L with periodic boundary conditions, but thereare some other numerical approaches applied to this problem [19]. As in any spectral method,we have approximated the conformal scalar field is approximated as a series with respect to aset of basis functions. According to the boundary conditions, these basis function are Fourierfunctions, and the spectral approximation establishes that, ϕ ( x , t ) = N X l,m = − N a lm ( τ ) ψ lm ( x, y ) . (2.2)In this expression N is the truncation order that limit the number of unknown modes a lm ( τ ) ,the basis function are ψ k = exp (cid:0) πiL k . x (cid:1) , and k = ( l, m ) is the comoving momentum. Noticethat the mode a ( τ ) corresponds to the homogeneous component of the inflaton while theremaining modes account for its small inhomogeneities.We have used the standard Galerkin method to solve this problem previously [6, 8], butwith low truncation order once the equations were expressed solely in terms of the modes.The collocation method adopts a more convenient approach to deal with the nonlinearities byestablishing that the residual equation - the equation arising when the spectral approximation(2.2) is substituted into the evolution equation (2.1) - vanishes at particular points namedcollocation or grid points x k = 2 π k / (2 N +1) . As a consequence, we have Res( x k , τ ) = 0 , with k = ( l, m ) and l, m = 1 , , .., N + 1 , and the resulting equations are expressed schematicallyby, – 2 – ′′ k ( τ ) + X j ω j a j ( τ ) ψ j ( x k ) + ϕ k ( τ ) = 0 , (2.3)where ω j = π L j and ϕ k ( τ ) denotes the value of the scalar field at the collocation point x k . Itbecomes very economical to express the resulting equations using both representations of thescalar field: the spectral representation through the modes a k , and the physical representationthrough values of the scalar field at the collocation points. There are (2 N + 1) independentequations that coincide with the same number of independent modes a k ( τ ) recalling thatthese modes are decomposed into imaginary and real pieces, but not all are independent onthe grounds of the scalar field given by Eq. (2.1) is real [6]. The bridge connecting bothrepresentation consists in the following relation, ϕ k ( τ ) = N X l,m = − N a lm ( τ ) ψ lm ( x k ) . (2.4)Accordingly, the (2 N + 1) values ϕ k ( τ ) are related to an equal number of independentmodes a j ( τ ) . In all numerical simulations we have evolved the dynamical equations (2.3)with a fourth-order Runge-Kuta integrator. We have integrated the dynamical equations (2.3) with the initial conditions ( ϕ k (0) , ϕ ′ k (0)) atthe end of inflation. In accordance to the choice of the conformal scalar field, its homogeneouscomponent is set initially as a (0) = 1 , whereas the other modes, a j (0) , have amplitudes oforder of − . With respect to the associated velocities, we have a ′ (0) = 0 and | a ′ j (0) | ∼O (10 − ) (for more details see Refs. [4, 6, 8, 15]). Figure 1 . Log-linear plot of the average error δE ϕ /E ϕ × over the interval from τ = 0 to τ = 1 , for λ = 10 − . The error exponential decay of the error is a confirmation of the accuracy ofthe numerical method. The parameter L dictates which modes with wave vector k = ( l, m ) undergo an initialphase of parametric resonance by considering the stability/instability chart for the Lamé– 3 – igure 2 . Evolution of the homogeneous component of the inflaton field a ( τ ) and the spatial averageof the variance σ ( τ ) . The qualitative behavior is in agreement with other similar plots, and also fordistinct models. Here we have set λ = 10 − so that the homogeneous mode starts to evolve in anapproximately constant amplitude after τ ≈ , . equation that governs the evolution of the modes a j ( τ ) in the linearized regime [2, 6]. In thiscase, it can be shown that if L = π p l + m ) / , where l, m assumes any integer value inthe interval [ − N, .., N ] , then those modes with | k | = √ l + m will grow exponentially in thefirst stages of preheating. In our numerical simulations, we have chosen these resonant modeswith | k | = √ , or equivalently L = 5 π/ √ .Before proceeding with our numerical study, we present a numerical test that consistsin verifying the conservation of the total energy of the conformal inflaton field, E ϕ ( τ ) , E ϕ ( τ ) = Z D ρ ϕ ( τ, x ) d x , (3.1)where D represents the spatial domain and ρ ϕ = 1 / ϕ ′ + 1 / ∇ ϕ ) + 1 / ϕ . As a matterof fact, this is valid only if a ′′ = 0 in Eq. 2.1. A very useful way of checking the energyconservation is to evolve the relative variation of energy given by δE ϕ ( τ ) /E ϕ (0) × withseveral truncation orders and evaluate the root mean square deviation for each truncationorder. In Fig. 1, we present the results that show a good convergence to a relative deviationof about − .The main stages of the dynamics of the inflaton have been described with details in Refs.[6, 8, 15]. We do not intend to repeat the description of these stages towards turbulence, butillustrate them by displaying in Fig. 2 the long time behavior of the homogeneous componentof the inflaton or the homogeneous mode, a ( τ ) , and σ ( τ ) defined by, σ ( τ ) = (cid:10) ( ϕ − h ϕ i ) (cid:11) = X k | a k | − a , (3.2)where var( ϕ ) = ( ϕ − h ϕ i ) is the variance of the field ϕ , and h .. i = 1 /L R D ..d x is theaverage over the spatial domain. Using expression (2.4) it follows that homogeneous mode is– 4 –he spatial average of the inflaton field h ϕ i = a .In the numerical experiments, we have set λ = 10 − and λ = 10 − . In both cases, thestructure of the time signals of a ( τ ) and σ ( τ ) is essentially the same. The only difference isthe time scale that separates approximately the stages until the turbulent phase establishes.In particular, we call attention to the decay of the homogeneous mode amplitude whichstarts when the resonant modes grow beyond the linear regime. We have verified that theamplitude decays approximately as τ − / in agreement with previous studies [4] until a certaintime, beyond which the amplitude remains approximately constant. For the homogeneousmode of Fig. 2, this occurs at τ ≈ , and, for λ = 10 − , we found τ ≈ , . Weinterpret this stage as representing the regime of stationary turbulence identifying it as thethermalized phase. In what follows we have considered this phase to exhibit some aspects ofwave turbulence.One of the most relevant features of any turbulent signal concerns, not to its detailedstructure, but to some property that is reproducible and related to the statistical descriptionof turbulence. We exhibit this property by constructing a sequence of histograms of σ ( τ ) considering intervals of time, ∆ τ = 100 , , , , about τ = 2 , . As shown in Fig. 3the histograms are identical, and the same axisymmetric Gaussian distribution fits them. Figure 3 . Histograms of σ ( τ ) constructed with the same bin using intervals of time ∆ τ =100 , , , (from left to right and top to bottom), about τ = 2 , . It is clear that the samedistribution is found no matter is the interval of time. This reproducible property of the turbulentsignal indicates its self-similar nature. – 5 –he aspect of wave turbulence relevant for the preheating is the cascade or transfer ofenergy among distinct scales. In particular, for preheating this is crucial to distribute theenergy content from the homogeneous component of the inflaton to all perturbative modes.In this way, turbulence offers a very elegant mechanism to describe the transition from anempty and cold post-inflationary universe to a hot universe dominated by radiation in accor-dance with the hot big bang model. The imprint of such mechanism results in the so-calledKolmogorov-Zakharov spectrum [11] that have the form of power-law in space and time do-mains.We start with the power spectrum of σ ( τ ) in the time domain shown in Fig. 4. Thepower spectrum presents a very rich structure. We have recognized three typical frequenciesdelimiting four regions with scaling laws. The first frequency is ω ≈ . , which is thesmallest natural frequency associated to the modes with | k | = 1 . The second frequency, ω ≈ . , is the initial frequency of the unperturbed homogeneous component of the inflaton, a ( τ ) , and the third frequency, ω ≈ . , is the highest frequency associated to those modeswith | k | max = N √ √ . It is necessary to comment about the first region of lowfrequencies ω ≤ ω min . These frequencies originate from the period bifurcations which takeplace in route to turbulence [6, 8, 20]. In the intervals ω ≤ ω min and ω ≥ ω max , we found that P ( ω ) ∼ ω − . while for ω min ≤ ω ≤ ω max there are two regions for which P ( ω ) ∼ ω − . .These scaling laws in the time domain together with the scaling law in the space domainare a direct consequence of the turbulent process in the late stages of preheating. They arerepresentative of a KZ spectrum for a steady-state turbulent system. Figure 4 . Power spectrum in time domain of the spatial average of variance evaluated for λ = 10 − .The vertical lines indicate, from left to right three values of typical frequencies, ω ≈ . , . and . . These are the minimum frequency, the initial frequency of the homogeneous inflaton field,and the maximum frequency, respectively. Here the minimum and maximum natural frequencies ofthe modes. It is clear the scaling law present in each region. The same structure of power spectrumis obtained for other values of λ and larger intervals of times. Besides the power spectrum in the time domain, we can calculate the power spectrumin the space domain of the variance. It is convenient to expand it with respect to the basisfunctions, – 6 – ar( ϕ ) = ( ϕ − h ϕ i ) = X k b k ( τ ) ψ k ( x ) , (3.3)where the modes b k are associated to the wavenumber vector k ; in particular the th -modeis the spatial average of the variance, b = σ . From the above expansion, we can constructthe power spectrum in wave numbers k = | k | at several times. In Fig. 5 we show the powerspectrum in the space domain evaluated at τ = 15 , with λ = 10 − . The structure ofthe power spectrum almost does not change if calculated at any instant during the turbulentphase, in this case τ ≥ , . We were able to fit the whole spectrum by a curve that containsboth the power law and exponential factors, P ( k ) = 0 . k − . e − . k . . (3.4)This kind of spectrum decay in wave numbers occurs in magnetohydrodynamic turbulence[7]. Figure 5 . Power spectrum P ( k ) of the variance in the space domain evaluated at τ = 15 , andwith λ = 10 − . The continuous line described by Eq. (3.4) fits the whole spectrum. A similarcombination of exponential and power decay can be found in magnetohydrodynamic turbulence. Wehave considered λ = 10 − . As established by the spectral approximation of Eq. (2.2), the classical modes a k = α k + β k are interpreted as c -number amplitudes associated to processes of creation and annihilationof quantum fluctuations of the inflaton field in the mode k . In this case, a relevant quantityis the occupation number of created particles, n k , given by, n k = 1 ω k | ˙ a k | + ω k | a k | (3.5)where ω k = 2 π | k | /L . We have considered λ = 10 − and evaluated the power spectrum of n k with respect to k = | k | at τ = 15 , . The power spectrum showed in Fig. 6 was also foundby Micha and Tkachev [15] using a different numerical approach in a 3D cubic lattice with agrid of points. Accordingly, the scaling law n ( k ) ∼ k − s for small k (straight line) with s ≈ . , is in agreement of the Micha’s result.– 7 –he turbulent phase of preheating produces a thermalized universe. A possible way ofevaluating the temperature of this thermalized phase is to calculate the power spectrum inthe space domain of the energy density a ρ φ ( x , τ ) /λφ e (cf. Eq. (4.3)), whose expansion withrespect to the basis function is, a ρ φ ( x , τ ) λφ e = X k E k ( τ ) ψ k ( x ) . (3.6)With the above expansion, we have evaluated the spectrum of energy E ( k ) in wave numbers k , where the energy E ( k ) is the root mean squared of all energies E k whose corresponding k has modulus k . In Fig. 7 we show the power spectrum at τ = 15 , and corresponding to λ = 10 − . The structure of the power spectrum displays two components separated by a gapof energy can also be found in Ref. [8]. Most of the points lie in the second component thatcorresponds to the energy distribution for large wave-numbers, or small scales, which can beinterpreted as the inertial range.We can understand the second piece of the spectrum as resulting from the energy transferfrom the homogeneous mode to small scale modes as expected in turbulence. The distributionat large wave numbers might correspond to a thermalized system consistent with a distributionof incoherent radiation satisfying the Planck distribution given by, E k = E k e bk − , (3.7)where E and b being constants. The best fit of (3.6) represented by the continuous line has b ≈ . and E ≈ − . . Another interesting feature is that a considerable part of the firstregion of the distribution can be fitted by the above distribution with the same value of b ,but E ≈ − . (cf. Fig. 7). Since the parameter b depends on the temperature of thedistribution, both pieces of the distribution are thermalized with the same temperature. Figure 6 . Power spectrum of the occupation number n ( k ) . The straight line represents the scalinglaw n ( k ) ∼ k − . . In order to extract the value of the temperature of the distribution, it is necessary torecover the physical variables present in the argument of the exponential from the dimension-less coordinates x and the momenta k . In this case we have bk = ~ ck phys /k B T , where k phys – 8 –s physical momentum, k B and T are the Boltzmann constant and temperature, respectively.From the relation k = Lk phys / √ λφ e a , with φ e = 0 . M pl , a = 1 [1] corresponding to thescalar field, and scale factor at the end of inflation, the temperature of the distribution innatural units is estimated as, T = ~ c √ λφ e a k B bL ≈ − M pl ≈ GeV. (3.8)This temperature is consistent with the beginning of the radiation era [21].
Figure 7 . Power spectrum E ( k ) of the energy density of the inflaton field. Two components arepresent separated with a gap of energy. The continuous lines represent the Planck distribution withthe same temperature, but different values of E . This is an evidence that the whole system becamethermalized. Here, the power spectrum was evaluated at at τ = 15 , and λ = 10 − . We have studied the evolution of the self-interacting inflaton field without taking into accountthe backreaction of its perturbative modes. The primary importance of the back-reaction isthe generation of an effective energy-momentum equation. In the present model a perfectfluid emerges with an effective equation of state, w = h p φ ih ρ φ i , (4.1)where h .. i represents the spatial averaging as mentioned before, and the pressure and energydensity associated to the scalar field are, respectively, p φ = λφ e a " (cid:18) ϕ ′ − ϕ a ′ a (cid:19) −
16 ( ∇ ϕ ) + 14 ϕ (4.2) ρ φ = λφ e a " (cid:18) ϕ ′ − ϕ a ′ a (cid:19) + 12 ( ∇ ϕ ) + 14 ϕ . (4.3)– 9 –n order to provide a consistent framework for including the back-reaction, we haveintegrated the following field equations, a ′ = φ e M pl * (cid:18) ϕ ′ − ϕ a ′ a (cid:19) + 12 ( ∇ ϕ ) + 14 ϕ + (4.4) ϕ ′′ − ∇ ϕ − a ′′ a ϕ + ϕ = 0 . (4.5)We have set φ e = 0 . M pl , a (0) = 1 , λ = 10 − and the initial conditions for the conformalscalar field were the same used in the previous Section. We have integrated the field equationsin a squared box of size L = 5 π/ √ . At each time step the integrals resulting from thespace average were calculated using Gauss quadrature formulae [18]. In Fig. 8 we show theevolution of w ( t ) . Notice that at the initial phase of preheating the homogeneous modesdominates producing an effective equation of state of radiation, w ≈ / as expected in thepresent model [24]. At late stages w ( t ) seems to converge to an approximate fixed value w ≈ . . Figure 8 . Evolution of the effective equation of state parameter w (blue line). In this paper we have studied the late stages of preheating of an inflationary model with po-tential V ( φ ) = λφ / . Although the present model is not in agreement, within an acceptablelevel of confidence, with current observational data, the stages towards the turbulence andthermalization seems to be robust and model independent.We have evolved the field equation using collocation method. The complex modes a k ( τ ) in the spectral approximation of the inflaton field (Eq. (2.2) can be viewed as the amplitudesof plane waves of wave number vector k . Turbulence develops with the nonlinear interaction– 10 –f these waves together with the transfer of energy from the homogeneous mode (zeroth mode)to other scales. The KZ spectra is an indicator of the energy transfer in a state of steadyturbulence. Therefore, this mechanism plays a fundamental role in the establishment ofthermalization allowing the universe to enter in the radiation era. We were able to determinethe temperature of the thermalized state by assuming the spectrum of the energy density isconsistent with the Plank distribution.We intend to explore further the interplay between wave turbulence processes in theearly universe. According to the latest observational data most one-field inflationary modelsbecomes unfavorable, but we mention two one-field models worth of investigation. The firstwas proposed by Amin et al [22] characterized by the potential, V ( φ ) = m M α (cid:20)(cid:18) φ M (cid:19) α − (cid:21) . (5.1)The authors have pointed out that choosing α < is in according with the observationaldata, and also it is possible to generate a preheating phase with broad resonance. The secondmodel was proposed by Kallosh and Linde [23], and consists in a class of self interacting singlefield with the same potential treated in this work, but the field is non-minimally coupled withcurvature [2].Finally, another relevant topic is the possibility of the parametric growth of gravitationalperturbations during the preheating [25]. Therefore, it might be possible that the nonlinearphase undergoes what could be called gravitational wave turbulence. Acknowledgements
The authors acknowledge the financial support of the Brazilian agencies CNPq, CAPES andFAPERJ.
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