Recrudescence of massive fermion production by oscillons
PPrepared for submission to JHEP
Recrudescence of massive fermion production byoscillons
Paul M. Saffin,
School of Physics and Astronomy, University Park, University of Nottingham,Nottingham NG7 2RD, United Kingdom
E-mail: [email protected]
Abstract:
We bring together the physics of preheating, following a period of inflation,and the dynamics of non-topological solitons, namely oscillons. We show that the oscillat-ing condensate that makes up an oscillon can be an efficient engine for producing heavyfermions, just as a homogeneous condensate is known for doing the same. This then allowsheavy fermions to be produced when the energy scale of the Universe has dropped belowthe scale naturally associated to the fermions.
Keywords:
Solitons Monopoles and Instantons, Nonperturbative Effects a r X i v : . [ h e p - t h ] A ug ontents The period in the early Universe following an inflationary epoch can be a rather explosivetime. In many models of inflation the Universe transforms from a cold, empty, near deSitter phase into a hot environment, via some sort of resonant behaviour of the fields [1].The basic idea is that the inflaton forms a (nearly) homogeneous oscillating condensate andsome modes of a daughter species, to which the inflaton is coupled, are in resonance withthe oscillations, causing their particle number to grow exponentially. This (p)reheatingscenario has found many uses in early-Universe cosmology, including the production ofgravitational waves [2]; non-thermal phase transitions and the formation of topological [3]and non-topological defects [4][5]; baryogenesis [6].This great enhancement in particle number is not available to fermions, as they mustsatisfy the Pauli exclusion principle, which limits their number occupancy to unity. ThisPauli-blocking was initially believed to mean that a perturbative calculation of fermioneffects would suffice, and that their presence would not significantly affect the inflatondecay [1]. The physics of such a decay process ignores the coherent nature of the inflatonand treats the fermion production as coming from the decay of single inflaton particles,leading to a decay rate of Γ φ → ψψ = ξ m φ π , and a narrow peak in the spectrum centredaround m φ /
2, where ξ is the Yukawa coupling and m φ the inflaton mass [1]. However, theequation of motion that governs the mode functions for the fermion field are, in fact, rather– 1 –imilar to those that appear in bosonic preheating, and while they do indeed exclude thepossibility of the particle number exceeding unity, it is clear that the perturbative approachis insufficient, as noted by [7]. The importance of this departure from the perturbativeresult was stressed in [8], where it was found that fermions could get excited within tensof inflaton oscillations, as opposed to the ∼ predicted by the perturbative calculation.A further distinction between bosonic and fermionic preheating is that the effectivemass of fermions can vanish as the condensate evolves through certain values, making theircreation much easier than bosons (see [9] for a clear comparison of fermion and bosonproduction due to preheating). This effect was utilised for leptogenesis in [10] to producemassive right-handed neutrinos, orders of magnitude heavier than the oscillating inflatonfield. This interesting effect means that fermions may be created in significant numbers,even when the energy scale of the Universe has dropped below the natural energy scale ofthe fermions, leading to a recrudescence of their production.The first studies of fermions coupled to a scalar condensate used a mode functionapproach, taking the condensate to be homogeneous. This was later extended, using tech-niques such as those pioneered in [11], to include inhomogeneities, as well as back-reaction.This led to a better understanding of how the homogeneous-condensate assumption un-derestimates the fermion production rate [12]. Indeed, the decay of oscillons was brieflydescribed in [11], but from a different perspective that we pursue here.The aim of this paper is to study the localized regions of oscillating condensate knownas oscillons [13], and focus on how they emit massive fermions. If we view them as smalllumps of preheating-phase physics, we might expect them to be able to emit heavy fermionsrather efficiently, just as homogeneous preheating does. In this initial study we shallignore the effects of back-reaction from the fermions onto the oscillon, and use techniquesintroduced by Cohen, Coleman, Georgi and Manohar [14], and followed up by [15]. Inthese the authors examined fermion emission from Q-balls [16], and calculated the particleproduction using a Bogoliubov-type analysis. In practical terms, the difference that makesthe calculation more involved for oscillons, is that their amplitude is not constant, but thisis also what leads to the fermion emission not being just a surface effect, unlike for Q-balls[14].We study two models for a spherical, localised, oscillating condensate; one where thecondensate has a broad almost homogeneous core, and one where the condensate profileis Gaussian. In our calculation the condensate is considered as an external source, andthe dynamics of the oscillon itself are not examined, rather we look to see how fermionsreact to the condensate. Nevertheless, it is of interest to see what models produce suchcondensates, and these are described in the appendices.The layout of the paper is as follows: In section 2 we set out the basic sphericalfermion ansatz, and describe the equations of motion, with the relevant mode functions forquantising the fermi field calculated in section 3. In section 4 we describe how to includethe oscillon, with our results presented in 5.1 and 5.2 for the flat-topped and Gaussianoscillons respectively, with these oscillon models being described in appendices A and B.We conclude in section 6. – 2 – Equations of motion
In this we will not assume that the effective fermion mass, µ , is a constant, as the Yukawacoupling between the scalar and fermion will contribute to µ . We shall, however, take it tobe spherically symmetric, as the oscillon that contributes to µ will be taken to be spherical.This allows us, using standard spherical harmonic spinors Ω j,l ± ,M [17], to write down anansatz for the solution to the Dirac equation in terms of functions that depend solely on t and r , Ψ j,M ( t, r ) = (cid:32) f ( t, r )Ω j,l + ,M ( θ, φ ) g ( t, r )Ω j,l − ,M ( θ, φ ) (cid:33) + (cid:32) f ( t, r )Ω j,l − ,M ( θ, φ ) g ( t, r )Ω j,l + ,M ( θ, φ ) (cid:33) . (2.1)This is substituted into the Dirac equation to find the equations of motion for the functions f i ( t, r ) and g i ( t, r ), for example˙ f = iµf + g (cid:48) − j − / r g , (2.2)˙ g = − iµg + f (cid:48) + j + 3 / r f . (2.3) In order to define a vacuum state for the fermion we need to know the mode functions forthe Dirac equation in the absence of an oscillon. For this we consider constant- µ , definiteenergy/wavenumber solutions of the Dirac equation, Ψ ± ( α ) j,M ( t, r ; k ),Ψ +( α ) j,M = e − iωt U ( α ) j,M ( r ; k ) , Ψ − ( α ) j,M = e iωt V ( α ) j,M ( r ; k ) , where U ( α ) and V ( α ) are found to be U (1 , j,M ( r ; k ) = k √ πω (cid:32) √ ω − µ j l ± ( kr ) Ω j,l ± ,M ± i √ ω + µ j l ∓ ( kr ) Ω j,l ∓ ,M (cid:33) , (3.1) V (1 , j,M ( r ; k ) = k √ πω (cid:32) ± i √ ω + µ j l ± ( kr ) Ω j,l ± ,M √ ω − µ j l ∓ ( kr ) Ω j,l ∓ ,M (cid:33) , (3.2) ω = + (cid:112) µ + k . These play the role of the usual U and V plane-wave modes in Minkowski space Carte-sian co-ordinates, with the j l ( kr ) being spherical Bessel functions. The particular factorsappearing (3.1-3.2) are chosen such that (cid:90) d x U ( α ) † j,M ( r ; k ) U ( α ) j (cid:48) ,M (cid:48) ( r ; k (cid:48) ) = δ jj (cid:48) δ MM (cid:48) δ ( k − k (cid:48) ) , with a similar relation for the V , as well as the U being orthogonal to the V under thesame inner product. – 3 –ith the basic wave solutions found and normalized, we may consider quantization,and so we expand a general wave operator asˆΨ( t, r ) = (cid:88) α,j,M (cid:90) dk (cid:110) ˆ b ( α ) ( k, j, M ) e − iωt U ( α ) j,M ( r ; k ) + ˆ d † ( α ) ( k, j, M ) e iωt V ( α ) j,M ( r ; k ) (cid:111) , and find that taking the canonical anti-commutation relation for ˆΨ and its conjugate leadsto (cid:110) ˆ b ( α ) ( k, j, M ) , ˆ b † ( α (cid:48) ) ( k (cid:48) , j (cid:48) , M (cid:48) ) (cid:111) = δ αα (cid:48) δ jj (cid:48) δ MM (cid:48) δ ( k − k (cid:48) ) , with a similar relation for ˆ d . While it certainly proves useful to calculate the above modefunctions, they do not constitute the basis of interest for our problem, rather we look to abasis that utilises incoming and outgoing modes. This we do in the next section. Before we get to the solution in the presence of the oscillon we note from (2.2-2.3) and(2.1) that time evolution couples f to g , and it couples f to g . In terms of the U ( α ) and V ( α ) modes this means U (1) is paired with V (1) , while U (2) is paired with V (2) . So, wemay introduce another basis of functions (the scattering basis) that are also solutions forconstant µ . One of these basis functions is, schematically, χ ∼ e − iωt U (1) ( r, k | h (2) ) + e − iωt R U (1) ( r, k | h (1) ) + e iωt T V (1) ( r, k | h (1) ) . (4.1)This solution has an extra argument for U (1) and V (1) , namely h (1 , , which is to indicatethat the j l ( kr ) of (3.1-3.2) are to be replaced by spherical Bessel functions of the third kind( h (1 , n = j n ± iy n ). This means that they are still solutions to the equations of motion,but are only regular away from the origin. In fact, we will only be interested in this format radii r (cid:29) R osc , which is where we shall make our measurements of particle number.Note that at large radius, h (1) ∼ kr e ikr and so corresponds to an outgoing wave, while h (2) corresponds to an ingoing wave. We now think about wavepackets formed from suchsolutions, rather than the pure frequency modes, and then we see that at early times (4.1)should be thought of as an ingoing U (1) mode, and at late times it is a combination ofoutgoing U (1) and outgoing V (1) , in proportion determined by R and T .Unlike the Q-balls studied in [14], oscillons have varying amplitude, which makesthe situation more involved, as the time dependence cannot be solved by a simple phasedependence. In practise, this means that even if the ingoing wave has a single frequency,the outgoing wave does not, and so contains a spectrum of wavenumbers. The scattering– 4 –asis for large radius, r (cid:29) R osc , is then χ ( α ) ( t, r ; k in , j, M ) = e − iω in t U ( α ) j,M ( r ; k in | h (2) ) (4.2)+ (cid:90) dk e − iωt R ( α ) ( k, k in ) U ( α ) j,M ( r ; k | h (1) ) + (cid:90) dk e iωt T ( α ) ( k, k in ) V ( α ) j,M ( r ; k | h (1) ) ,ζ ( α ) ( t, r ; k in , j, M ) = e iω in t V ( α ) j,M ( r ; k in | h (2) ) (4.3)+ (cid:90) dk e iωt ˜ R ( α ) ( k, k in ) V ( α ) j,M ( r ; k | h (1) ) + (cid:90) dk e − iωt ˜ T ( α ) ( k, k in ) U ( α ) j,M ( r ; k | h (1) ) , and so we expand a general wave operator as,Ψ( t, r ) = (cid:88) α,j,M (cid:90) dk in (cid:110) ˆ b in ( α ) ( k in , j, M ) χ ( α ) ( t, r ; k in , j, M ) + ˆ d † in ( α ) ( k in , j, M ) ζ ( α ) ( t, r ; k in , j, M ) (cid:111) . (4.4)At this point it is useful to understand why the operators have been labelled with thesubscript “ in ”. Thinking about the wave operator in terms of wave packets, we have thatin the far past only the incoming part of the scattering basis survives (the terms dependingon h (2) ) and so the wave operator is indeed a sum of ingoing waves, and the operators( d † in ), b † in acquire the interpretation as creation operators for ingoing (anti-)particles.At late times, the wave operator will be composed solely of outgoing particles, so atlarge radius we would writeΨ( t, r ) = (cid:88) α,j,M (cid:90) dk (cid:110) ˆ b out ( α ) ( k, j, M ) e − iωt U ( α ) j,M ( r ; k | h (1) ) + ˆ d † out ( α ) ( k, j, M ) e iωt V ( α ) j,M ( r ; k | h (1) ) (cid:111) , (4.5)and the operators ( d † out ), b † out are creation operators for outgoing (anti-)particles. However,we know what the late time, large radius form is from (4.4) and (4.2-4.3), so we comparethese (in the late time, large radius limit) with (4.5) to findˆ b out ( α ) ( k, j, M ) = (cid:90) dk in (cid:110) R ( α ) ( k, k in )ˆ b in ( α ) ( k in , j, M ) + ˜ T ( α ) ( k, k in ) ˆ d † in ( α ) ( k in , j, M ) (cid:111) , and a similar relation holds for ˆ d in and ˆ d out . These give the Bogoliubov transformationbetween asymptotic in/out creation and annihilation operators.The number of fermions emitted by the oscillon are found by starting the system inthe vacuum state | (cid:105) , defined by having no incoming fermions or anti-fermions, b in | (cid:105) = 0 , d in | (cid:105) = 0 , (4.6)and we count the number of fermions per unit k -space at wavenumber k , at late times, byevaluating N out ( k ) = (cid:88) α,j,M (cid:104) | b † out ( α ) ( k, j, M ) b out ( α ) ( k, j, M ) | (cid:105) , = (cid:88) α,j (2 j + 1) (cid:90) dk in | ˜ T ( α ) ( k, k in ) | . (4.7)– 5 –his term diverges, but the divergence is to be expected as the oscillon is evolving withoutback-reaction and continues producing particles forever. In practise one is really interestedin ∆ N out ∆ τ , where ∆ N out is the number of fermions produced in time interval ∆ τ . Theeffort that goes into finding the rate of (anti-)particle production, therefore, amounts todetermining ( T ( α ) ), ˜ T ( α ) . This is achieved by noting that in terms of the functions f and g appearing in (2.2-2.3), we may use (4.2-4.3) along with (3.1-3.2) to find the following largeradius behaviour for f ( t, r ) f ( t, r ; k in ) = k in √ ω in − µ √ πω in e − iω in t h (2) l + ( k in r ) (4.8)+ (cid:90) dk k √ ω − µ √ πω R (1) ( k, k in ) e − iωt h (1) l + ( kr ) (cid:90) dk ik √ ω + µ √ πω T (1) ( k, k in ) e iωt h (1) l + ( kr ) , with analogous results for f , g and g . We see, therefore, that if we observe f at agiven radius r = ρ (cid:29) R osc , over some time ∆ τ , then we may perform a temporal Fouriertransform (cid:82) dt e − i Ω t f ( t, r = ρ ; k in ) to find T (1) ( K = (cid:112) Ω − µ , k in ). Having set up theformalism, we now need a condensate to create the fermions, and this we address in thenext section. The scalar sector for the flat-topped oscillons is set out in App. A, and we couple this toa fermion which has free mass m , via a Yukawa term L Y uk = ξφ ¯ ψψ , giving an effectivefermion mass of µ = m + ξφ. (5.1)One of the important observations made in [10][18] is that the production of massivefermions due to an oscillating scalar condensate can be efficient if the effective mass ofthe fermions goes through zero at some point during the condensate cycle; this requires φ max (cid:38) m /ξ . Moreover, the coherent nature of the condensate allows for fermions heavierthan the scalar to be produced [10][18].The parameters we chose were as follows: ξ = 1 . m = 0 . m , λ = 0 . g = 4 λ ,Φ = Φ c − − , which leads to an oscillon of radius ∼ m − and energy ∼ , m ,with period of oscillation T osc ∼ . m − . For the simulation of (2.2-2.3), we used alattice of 40 ,
000 points, with a spatial step of dr = 0 . m , and a temporal step size of dt = dr/ .
0. We should note that the mass of the scalar that forms the condensate is lowerthan the free mass of the fermion.In order to evaluate the emission rate we need the Fourier transform of f evaluatedat a fixed radius, ρ . We give an example of the mode function f ( t, r = 100 m − ) in Fig.1. Here we see that not much happens until m t ∼
50 100 150 200 250 300 350-0.0500.050 50 100 150 200 250 300 350 m t -0.0500.05 Figure 1 : An example of f ( t, r = ρ ) for j = 4 , ρ = 100 m − and k in = m for theflat-topped oscillon. The top figure is the real part of f and the lower is the imaginarypart. -1 k/m -1 Figure 2 : A plot showing ∆ N ( k )∆ τ for the flat-topped oscillon example in the text.activity, before settling down to a periodic motion. The burst around m t ∼ →
150 isdue to the osciillon effectively being suddenly switched on at the start of the simulation.We let this burst pass before we start taking the Fourier transform, for which we use180 < m t < k -space interval emitted per unit time, ∆ N ( k )∆ τ , and this is shown in Fig. 2 - for this examplewe found that the sum in (4.7) had converged by j max = 29 , which is what we use for thefigure. The plot indicates a number of resonance peaks, consistent with the picture onehas for a homogeneous condensate [10]. As well as the spectrum of emitted particles it isof interest to know how many fermions are emitted in total over one cycle, which may befound by performing an integral over k -space of the spectrum,∆ N ∆ τ = (cid:90) dk ∆ N ( k )∆ τ , (5.2)and in our example yields ∆ N ∆ τ (cid:39) m . So, over a single cycle of duration T osc our flat-topped oscillon emits ∼
690 fermions, constituting about 2% of its energy budget per– 7 – -1 k/m -1 Figure 3 : A plot showing ∆ N ( k )∆ τ for the Gaussian oscillon example in the text. m r ξφ m Figure 4 : A plot showing the scalar field profiles for the flat-topped oscillon (dashed line)and the Gaussian oscillon (solid line) examples in the text. Note that they are presentedwith the Yukawa factor included.cycle. – 8 – .2 Gaussian oscillons
The more commonly studied class of oscillons, laid out in App. B, are those with anapproximately Gaussian profile. For our simulations in this case, we take a Yukawa couplingof ξ = 0 .
1, with again m = 0 . m . The parameters given in App. B lead to an oscillonwith period T osc (cid:39) . m − , core amplitude of A B = 24 . m , and radius of R = 6 . m − .Numerical simulations using the same lattice parameters as for the flat-topped oscillonyields the production rate spectrum shown in Fig. 3, and the total production rate ∆ N ∆ τ (cid:39) . m . So, over a single cycle of duration T osc our oscillon emits ∼
19 heavy fermions,constituting about 0.06% of its energy budget. Such a reduction is to be expected giventhat for our parameters the Gaussian oscillon is smaller than the flat-topped oscillon, asmay be seem from Fig. 4. Even though this is a small fraction of the oscillon’s energy, oneshould bear in mind that oscillons can live for many thousands of oscillations.
Based on the observation from preheating studies that a homogeneous oscillating scalarcondensate is able to produce heavy fermions, and do so more efficiently than bosons [10],we have initiated a study into the fermion production due to mini preheating regions,oscillons. The basic formalism for calculating the fermion emission was laid out, and anexample given using an approximate analytic form for the oscillon solution. It was shown,by example, that despite the scalar field being lighter than the free-fermion mass, thecoherent nature of the oscillon allowed fermions to be produced, and in significant numbers,given that oscillons can live for thousands of oscillations. The possibility that such oscillonscould form at ”low” energies and yet produce heavy fermions raises interesting questionsfor baryogenesis, where such heavy fermions may, for example, be heavy right-handedMajorana neutrinos. These would break the B − L symmetry of the standard model,and their decay causes a lepton asymmetry, which leads to a baryon asymmetry due tosphaleron processes. It is also amusing to consider the possibility that we may one daybe able to manipulate a Higgs condensate into performing localised oscillations, therebyproducing particles heavier than the scalar itself.There are still a number of things to be done, one of which is to acquire a morecomplete picture of how the fermion production is affected by the oscillon’s properties,such as its amplitude, size or frequency. For example, if φ max < m /ξ then fermionproduction may cease, as their effective mass would never vanish. Another importantaspect is back-reaction, whereby the fermion dynamics alters the behaviour of the oscillon.This has proved an important effect in cosmological preheating, where is has been notedthat neglecting back-reaction underestimates the number of fermions produced [12]. Thefermion coupling may also change the way that oscillons form, with possible incipientfermion production happening during the formation process. This, however, will requirelarge-scale simulations to gain a full understanding, and is something left for future studies. Acknowledgments:
We would like to thank STFC for financial support under grantST/L000393/1. – 9 –
Scalar sector: flat-topped oscillons
Much has been written about oscillons since their discovery [19] and rediscovery [13].They are localized lumps of oscillating scalar-field condensate, made quasi-stable by non-linearities in the field equations, and may appear, for example, at phase transitions [20].Due them not being perfectly stable, they radiate energy [21] and have a lifetime with asignificant dependence on the spatial dimension [22]. Ultimately, one is interested in treat-ing the fields quantum mechanically, and results in that direction may be found in [23].Their relevance to post-inflationary dynamics is examined in [5], and it is found that in alarge class of models that exhibit preheating, oscillons can come to dominate the energydensity. In this paper we will not concern ourselves too much with the scalar sector, andwill not include the back-reaction of the fermions onto the oscillon, leaving this for a futurestudy. We shall follow [24] and take an analytic profile for the oscillons of the form φ ( t, r ) = Φ (cid:115) λg (cid:115) u u cosh(2 αλx/ √ g ) cos( ωmt ) , which corresponds to a localized, oscillating field distribution. This profile is derived as anoscillon for the following Lagrangian density, L scalar, flat = − ∂ µ φ∂ µ φ − m φ + λ φ − g φ , (A.1)with α = 38 Φ −
524 Φ , α c = (cid:112) / , ω = 1 − α λ m g ,u = (cid:112) − ( α/α c ) , Φ = Φ c √ − u, Φ c = (cid:112) / . Strictly speaking, such a solution is valid only in one spatial dimension in the limit g (cid:29) λ/m . The precise results of fermion emission will, of course, be affected by thedetailed structure of the oscillon, but we are interested in how fermions react to a genericlocalized oscillating lump of scalar condensate, and will leave more detailed studies forfuture work.Unfortunately, the region of parameter space required for flat-topped oscillons in thismodel is not radiatively stable. This is seen by requiring the effective eight-point vertex,brought about by the one-loop diagram with two φ vertices, to be smaller than the tree-level diagram with eight external lines (a φ vertex connecting a φ vertex via a propagator),and leads to a cut-off of Λ = (cid:112) λ/g . For the analytic flat-topped solution to be valid werequired g (cid:29) λ/m or, equivalently, m (cid:29) (cid:112) λ/g , which means the scalar mass has to beabove the cut-off.However, we are only considering the behaviour of fermions in the background of suchan extended object, and have not considered the scalar field as dynamical. As such, theLagrangian that leads to the profile has no bearing on the calculation, and we are simplytaking this profile as a prototype for a localized oscillating condensate, and simply considerit as an external source. – 10 – Scalar sector: Gaussian oscillons
The more common class of oscillons are those with Gaussian-like profiles, and for this wemay consider the following Lagrangian L scalar, Gaussian = − ∂ µ φ∂ µ φ − λ (cid:16) [ φ − η ] − η (cid:17) , (B.1)which is simply the familiar Mexican-hat shaped potential, shifted so that the vacua arelocated at φ vac = (0 , η ). This model has oscillons of the approximate form [25] φ ( t, r ) = A B exp (cid:0) − r /R (cid:1) cos( ω B t ) , (B.2) R (cid:39) . √ λη , ω B (cid:39) . √ λη, A B (cid:39) . η, (B.3)and the energy of such a condensate is E Gaussian (cid:39) . η √ λ . (B.4)The mass of the scalar in this model is m = √ λη , which we fix to be half the fermionvacuum mass, m = m , giving E Gaussian (cid:39) . √ λ m . The coupling constant λ is fixedby requiring the Gaussian oscillon to have a comparable energy to the flat-topped oscillon( E flat − topped (cid:39) , m ), and we take λ = 5 × − , which is safely within the perturbativeregime and leads to E Gaussian (cid:39) , m . References [1] A. D. Dolgov and D. P. Kirilova, Sov. J. Nucl. Phys. , 172 (1990) [Yad. Fiz. , 273 (1990)].J. H. Traschen and R. H. Brandenberger, Phys. Rev. D , 2491 (1990). L. Kofman,A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. , 3195 (1994). Y. Shtanov,J. H. Traschen and R. H. Brandenberger, Phys. Rev. D , 5438 (1995). M. Yoshimura, Prog.Theor. Phys. , 873 (1995). D. I. Kaiser, Phys. Rev. D , 1776 (1996). L. Kofman,A. D. Linde and A. A. Starobinsky, Phys. Rev. D , 3258 (1997). S. Y. Khlebnikov andI. I. Tkachev, Phys. Rev. Lett. , 219 (1996). S. Y. Khlebnikov and I. I. Tkachev, Phys. Lett.B , 80 (1997). S. Y. Khlebnikov and I. I. Tkachev, Phys. Rev. Lett. , 1607 (1997).[2] S. Y. Khlebnikov and I. I. Tkachev, Phys. Rev. D , 653 (1997). B. Bassett, Phys. Rev. D , 3439 (1997). D. Tilley and R. Maartens, Class. Quant. Grav. , 2875 (2000).J. Garcia-Bellido and D. G. Figueroa, Phys. Rev. Lett. , 061302 (2007). J. F. Dufaux,A. Bergman, G. N. Felder, L. Kofman and J. P. Uzan, Phys. Rev. D , 123517 (2007).S. Y. Zhou, E. J. Copeland, R. Easther, H. Finkel, Z. G. Mou and P. M. Saffin, JHEP ,026 (2013)[3] S. Kasuya and M. Kawasaki, Phys. Rev. D , 7597 (1997). I. Tkachev, S. Khlebnikov,L. Kofman and A. D. Linde, Phys. Lett. B , 262 (1998). S. Khlebnikov, L. Kofman,A. D. Linde and I. Tkachev, Phys. Rev. Lett. , 2012 (1998). I. I. Tkachev, Phys. Lett. B , 35 (1996). L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. , 1011(1996).[4] M. A. Amin, arXiv:1006.3075 [astro-ph.CO]. M. A. Amin, R. Easther and H. Finkel, JCAP , 001 (2010). – 11 –
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