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both standard @xcite and inflationary cosmology @xcite suffer from the initial singularity problem and hence can not yield complete descriptions of the very early universe . if one were able to construct a non - singular bouncing cosmology , this problem would obviously disappear . however , in order to have a chance to obtain such a non - singular cosmology , one must either go beyond einstein gravity as a theory of space - time ( see e.g. @xcite for an early construction ) , or else one must make use of matter which violates the  null energy condition " ( see @xcite for a review of both types of approaches ) . interest in non - singular bouncing cosmologies has increased with the realization that they can lead to alternatives to inflationary cosmology as a theory for the origin of structure in the universe . a specific scenario which can arise at the level of homogeneous and isotropic cosmology is the  matter bounce " paradigm which is based on the realization @xcite that vacuum fluctuations which exit the hubble radius during a matter - dominated contracting phase evolve into a scale - invariant spectrum of curvature perturbations on super - hubble scales before the bounce . the key point is that the curvature fluctuation variable @xmath0 grows on super - hubble scales in a contracting phase , whereas it is constant on these large scales in an expanding phase . since long wavelength modes exit the hubble radius earlier than short wavelength ones , they grow for a longer period of time . this provides a mechanism for reddening the initial vacuum spectrum . it turns out that a matter dominated contracting phase provides the specific boost in the power of long wavelength modes which is required in order to transform a vacuum spectrum into a scale - invariant one . studies in the case of various non - singular bounce models @xcite have shown that on wavelengths long compared to the duration of the bounce phase , the spectrum of fluctuations is virtually unchanged during the bounce . thus , a scale - invariant spectrum of curvature fluctuations survives on super - hubble scales at late times . provided that the bounce can occur at energy scales much below the planck scale , non - singular cosmologies solve a key conceptual problem from which inflationary cosmology suffers , namely the trans - planckian " problem for fluctuations @xcite : if the period of inflationary expansion of space lasts for more than @xmath1 , where @xmath2 is the hubble expansion rate during inflation ( in order to solve the key cosmological mysteries it was designed to explain , inflation has to last at least @xmath3 ) , then the physical wavelengths of even the largest - scale fluctuation modes we see today will be even smaller than the planck length at the beginning of inflation and thus in the  zone of ignorance " where the physics on which inflation and the theory of cosmological perturbations are based , namely einstein gravity coupled to semiclassical field theory matter , will break down . in contrast , in a non - singular bouncing cosmology the wavelength of modes which are currently probed by cosmological observations is never much smaller than @xmath4 ( the physical wavelength of the mode which corresponds to our current hubble radius evaluated when the temperature of the universe was @xmath5 ) and hence many orders of magnitude larger than the planck length . thus , the fluctuations never enter the  trans - planckian zone of ignorance " of sub - planck - length wavelengths . possibly the simplest realization of the matter bounce scenario is the quintom bounce " model @xcite and is obtained by considering the matter sector to contain two scalar fields , one of them ( the  ghost field " ) having the  wrong " sign of the kinetic action . the potential of the ghost scalar field also has the opposite sign to that of regular scalar fields such that in the absence of interactions , the ghost field has a classically stable minimum . as has been noticed in @xcite , such a quintom bounce model also arises from the scalar field sector of the  lee - wick " ( lw ) lagrangian @xcite which contains higher derivatives terms . the quintom and lee - wick bouncing cosmologies are obtained in the following way @xcite : we begin in the contracting phase with both the regular and the ghost scalar field oscillating homogeneously in space about their respective vacua . we assume that the energy density is dominated by the regular matter field , and that hence the total energy density is positive . once the amplitude of the regular scalar field exceeds the planck scale , the field oscillations will freeze out and a slow - climb phase will begin during which the energy density of the field only grows slowly ( this is the time reverse of the slow - roll phase in scalar field - driven inflation ) . however , the ghost field continues to oscillate and its energy density ( which is negative ) continues to grow in absolute value . hence , the total energy density drops to zero , at which point the bounce occurs , as has been studied both analytically and numerically in the above - mentioned works . note that the energy density in this bounce model scales as matter until the regular scalar field freezes out .    a major problem of bouncing cosmologies realized with matter which scales as @xmath6 as a function of the scale factor @xmath7 is the potential instability of the homogeneous and isotropic background against the effects of radiation ( which scales as @xmath8 and anisotropic stress which scales as @xmath9 . if we simply add a non - interacting radiation component to the two scalar field system , then unless the initial energy density in radiation is tuned to be extremely small , then the radiation component will become dominant long before the bounce can arise , and will prevent the energy density in the ghost field from ever being able to become important , resulting in a big crunch singularity . similarly , unless the initial energy density in anisotropic stress is very small , it will come to dominate the energy density of the universe long before the bounce is expected . the anisotropies will destabilize the homogeneous background cosmology , and will prevent a bounce . note that at the quantum level , there is an additional severe problem for bounce models obtained with matter fields with ghost - like kinetic terms , namely the quantum instability of the vacuum ( see e.g. @xcite ) . in this paper we will focus on the radiation instability problem . for the purpose of this discussion we will simply assume that anisotropic stress is absent . in a recent paper , two of us studied the possibility that a bounce could arise if radiation is supplemented with lee - wick radiation @xcite . however , we showed that this hope is not realized : the addition of lee - wick radiation does not prevent the big crunch singularity from occurring . in the presence of radiation , the only hope to obtain a bounce is to introduce a coupling between radiation and ghost scalar field matter which could effectively drain energy density from the radiation field and prevent the energy density of radiation from becoming dominant . here we study this possibility . however , at least for the specific lagrangian which we consider , we find that a bounce only emerges for highly fine - tuned phases of the fields and their velocities in the initial conditions . the paper is organized as follows : in section ii , we introduce the model we study , namely the scalar field sector of lee - wick theory coupled to radiation , and write down the general equations of motion . in section iii we set up the equations of motion linearized about the bounce background , treating the entire radiation field as an inhomogeneous fluctuation . in particular , we study the different terms which contribute to the energy - momentum tensor and identify those which could assist in obtaining a non - singular bounce . in section iv we study the solutions of the perturbed equations of motion , and in section v we analyze the evolution of the different terms in the energy - momentum tensor , identifying the conditions which would be required in order to obtain a non - singular bounce . we have also evolved the general equations of motion for the two inhomogeneous scalar field configurations and the classical inhomogeneous radiation field in the homogeneous background cosmology . section vi summarizes some of the numerical results . both the analytical and numerical results confirm that we need unnatural fine - tuning of the initial conditions in order to obtain a non - singular bounce . in the final section we offer some conclusions and discussion . the lee - wick scalar field model coupled to electromagnetic radiation is given by the following lagrangian : [ leewick ] l&=&-_^+(^2 ) ^2-m^2 ^ 2-v ( ) + & & - f _ f^-f(,^2,f_f^ )  , where @xmath10 is the mass of the scalar field @xmath11 , and @xmath12 is its potential . here we adopt the convention that ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2 ) , where @xmath7 is the scale factor of the universe . since it is a higher derivative lagrangian in @xmath11 , the scalar field sector contains an extra degree of freedom with the wrong " sign kinetic term and with a mass set by the scale @xmath13 . we choose @xmath14 , where @xmath15 is the planck mass , since we want the regular scalar field to dominate at low energies , but at the same time we do not want to worry about quantum gravity effects . the second line of the lagrangian ( [ leewick ] ) contains the kinetic term of the radiation as well as the coupling term , where we assumed both for the sake of generality and because of foresight that the radiation field couples not only to the scalar field @xmath11 itself , but also to the higher derivative term . the electromagnetic tensor , @xmath16 , is related to the radiation field @xmath17 through the usual definition f _ _ a_-_a _ , where @xmath18 is the covariant derivative . it is convenient to extract the extra degree of freedom as a separate scalar field . to do this , we use the field redefinitions & & _ 1-_2 , + _ 2 & & ^2 /m^2 . the lagrangian ( [ leewick ] ) then takes on a simpler form : [ leewick2 ] l & = & -__1^_1+__2^_2-m^2 ^2+m^2 ^2 + & & - f _ f^-f(_1,_2,f_f^ )  , where we have chosen the potential to be zero . in this new form , the lagrangian describes two massive scalar fields with one of them ( i.e. , @xmath19 ) behaving like aghost " , and both of them coupled to the radiation field .    the coupling term @xmath20 should in principle be arbitrary , however , in this paper we will take a specific form for convenience . the form will be : [ f1 ] f(_1,_2,f_f^ ) = ( c^2+d^2)f _ f^  , where @xmath21 and @xmath22 are coupling constants which have mass dimension @xmath23 . the interaction terms are non - renormalizable . to make sure that such terms could be thought of as arising from an effective field theory which is consistent at the bounce , we must make sure that the coefficients are chosen such that the contribution of the interaction term to the lagrangian density is smaller than that of the other terms . this must be true even at energy densities at which the bounce occurs in the pure scalar field model . it is easy to see that this condition will be satisfied if the coefficients @xmath21 and @xmath22 are both of the order @xmath24 . it is the purpose of this paper to study the effects which these coupling terms have on the dynamics of the system . we know that in the absence of coupling , i.e. when @xmath25 , a bounce will only occur if the initial radiation energy density is tuned to a very small value compared to the scalar field energy density . this is because the positive definite energy density of radiation will scale as @xmath8 which is faster than that of the scalar fields , in particular the ghost scalar field . generically , it will dominate the energy of the universe after some amount of contraction , it will prevent the ghost scalar field energy density from catching up and will thus prevent a bounce , leading to a big crunch singularity instead . with non - vanishing values of @xmath21 and @xmath22 , however , the scalars are in principle able to drain energy from the radiation .    from the lagrangian ( [ leewick2 ] ) , one can obtain the stress - energy tensor @xmath26 by varying the action with respect to the metric @xmath27 . in the hydrodynamical limit , we can take @xmath26 to be of the form of @xmath28 where @xmath29 and @xmath30 are energy density and pressure , respectively . for now , we consider the general form which is :    [ tmunu ] t_&=&g_l+__1__1-__2__2+(1-c_1 ^ 2-d_2 ^ 2)f_f_^  , + & = & g _   + & & + _ _ 1__1-__2__2+(1-c_1 ^ 2-d_2 ^ 2)f_f_^  . since we will be studying the contribution of plane wave perturbations of the scalar fields and we will treat radiation as a superposition of waves , we kept the space - derivative terms .    by varying the lagrangian with respect to the matter fields @xmath31 , @xmath19 and @xmath17 , we also get the equations of motion for all three fields : [ phi1eom]&&_1-(m^2-f^2)_1=0  , + [ phi2eom]&&_2-(m^2+f^2)_2=0  , + [ amueom]&&(1-c_1 ^ 2-d_2 ^ 2)(_f^+3hf^0 ) + & & -2(c_1__1+d_2__2)f^=0   which will be analyzed in detail in the rest of the paper . since the equations of motion are nonlinear , we can not work in fourier space , and use plane wave solutions . however , we are interested in how initially small amounts of radiation build up and possibly transfer their energy to scalar field fluctuations . we treat radiation as a superposition of fluctuations . therefore it makes sense to linearize our equations about the homogeneous scalar field background . thus , we make the following ansatz for the scalar fields : [ ansatzphi ] _ 1(t , z)&=&_1^(0)(t)+_1^(1)(t , z)+^2 _ 1^(2)(t ) + _ 2(t , z)&=&_2^(0)(t)+_2^(1)(t , z)+^2 _ 2^(2)(t )  , where the expansion parameter @xmath32 is taken to be much smaller than @xmath33 . should be viewed as parameterizing the initial ratio of radiation energy to background scalar field energy . thus , the leading contribution of the radiation field is first order in @xmath32 . via the coupling terms in the lagrangian with coefficients @xmath21 and @xmath22 , the linear radiation field induces linear scalar field inhomogeneities @xmath34 and @xmath35 . these corrections will contain a further suppression factor since @xmath21 and @xmath22 are small coefficients . similarly , the same coupling terms in the lagrangian will lead to a perturbation @xmath36 of the rescaled radiation field @xmath37 which is of linear order in @xmath32 but suppressed by factors of @xmath21 and @xmath22 . ] the first term on the right hand side of each line , i.e. @xmath38 correspond to the background fields , the terms @xmath39 are the fluctuations , and the second order terms @xmath40 describe the back - reaction of the fluctuations on the background and can be computed from the leading second order corrections ( averaged over space ) of the equations of motion .    to simplify the analysis , we describe radiation in terms of plane waves in a fixed direction ( which we take to be the @xmath41 direction ) . without loss of generality we can restrict attention to one polarization mode which we take to be the electric field in the @xmath42 direction and the magnetic field in the @xmath43 direction . in this case , the only non - zero components of the field strength tensor are @xmath44 and @xmath45 . using the temporal gauge where @xmath46 , we find that only the first component of the gauge field is non - zero . for a single wavelength fluctuation we can make the ansatz [ ansatzgauge1 ] a_1(k , t ) = f(t)cos(kz ) ( k , t )  , or , equivalently , [ ansatzgauge2 ] a^1(k , t ) = a(t)^-2 ( k , t )  . since in the linearized equations of motion the fourier modes are independent , we can consider @xmath34 and @xmath35 also to be plane waves propagating in @xmath41 direction , so they depend only on @xmath41 and @xmath47 .    with eqs . ( [ ansatzphi]-[ansatzgauge2 ] ) in hand , we can write down the energy densities of the various fields at each order in perturbation theory . first of all , we insert the above perturbative ansatz for the fields into the stress - energy tensor of the system . from the general expression ( [ tmunu ] ) for @xmath26 we get :    [ tmunu2 ] t_&=&g _   + & & + _ _ 1__1-__2__2+(1-c_1 ^ 2-d_2 ^ 2)f_f_^  . the @xmath48 component of eq . ( [ tmunu2 ] ) denotes the energy density of the system : [ rho ] & = & ( _ 1 ^ 2+_1 ^ 2+m^2_1 ^ 2 ) -(_2 ^ 2+_2 ^ 2+m^2_2 ^ 2 ) + & & + ( 1-c_1 ^ 2-d_2 ^ 2)(+f_0f_0^ )  , so at each level in perturbation theory we have : [ rho0]^(0)&=&(^2+m^2_1^(0)^2 ) -(^2+m^2_2^(0)^2 )  , + [ rho1]^(1)&=&(_1^(0)_1^(1)+m^2_1^(0)_1^(1 ) ) -(_2^(0)_2^(1 ) + & & + m^2_2^(0)_2^(1 ) )  , + [ rho2]^(2)&=&(^2+_1^(0)_1^(2 ) + ^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) + ^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + & & + ( 1-c_1^(0)^2-d_2^(0)^2)(+f_0f_0^ )  , + & = & ( ^2+_1^(0)_1^(2 ) + ^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) + ^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + & & + ( 1-c_1^(0)^2-d_2^(0)^2)(^2 + )  . we can similarly obtain the pressure of the system from the @xmath49 components of eq . ( [ tmunu2 ] ) . note that due to the anisotropy in @xmath26 caused by the gauge field as well as by the anisotropic fluctuations of the scalar fields , the pressures in the three directions are no longer identical . the pressure in each direction can be written as : [ pres ] p_i&=&(_1 ^ 2-_1 ^ 2-m^2_1 ^ 2 ) -(_2 ^ 2-_2 ^ 2-m^2_2 ^ 2 ) + & & -(1-c_1 ^ 2-d_2 ^ 2)(-)+ + & & -  , with no summation over the index @xmath50 . from this formula , we can see that at both zero - th and first order , the pressure is isotropic : p_i^(0)&=&(^2-m^2_1^(0)^2 ) -(^2-m^2_2^(0)^2 )  , + p_i^(1)&=&(_1^(0)_1^(1)-m^2_1^(0)_1^(1 ) ) -(_2^(0)_2^(1 ) + & & -m^2_2^(0)_2^(1 ) )  , while the second order pressure for each direction gives p_i^(2)&=&(^2+_1^(0)_1^(2 ) -^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) -^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + - + & & -(1-c_1^(0)^2-d_2^(0)^2)(- )  , where @xmath51 . we can thus obtain every component of @xmath52 : p_1^(2)&=&(^2+_1^(0)_1^(2 ) -^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) -^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + & & + ( 1-c_1^(0)^2-d_2^(0)^2)(^2- )  , + p_2^(2)&=&(^2+_1^(0)_1^(2 ) -^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) -^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + & & -(1-c_1^(0)^2-d_2^(0)^2)(^2- )  , + p_3^(2)&=&(^2+_1^(0)_1^(2 ) -^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) -^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + & & + ( 1-c_1^(0)^2-d_2^(0)^2)(^2 + )  , and the average is : p_eff^(2)&= &   + & = & ( ^2+_1^(0)_1^(2 ) -^2+m^2_1^(1)^2 + & & + m^2_1^(0)_1^(2))-(^2+_2^(0)_2^(2 ) -^2 + & & + m^2_2^(1)^2+m^2_2^(0)_2^(2 ) ) + & & + ( 1-c_1^(0)^2-d_2^(0)^2)(^2 + )  .    from the above , we can also see that in order to analyze the behavior of the energy density up to second order , we need to know the evolution of scalar fields up to second order as well as that of the gauge field up to first order , while the behavior of the gauge field to second order is not required . it will be useful in the following to separate the contributions to the energy density and pressure in a different way , namely , + i ) the contribution from the background homogeneous part of the scalar fields , _ ^(0 )   p_^h= p_^(0 )  , ii ) that of the scalar field perturbations ( in slight abuse of notation we call this the `` inhomogeneous '' term ) , _ ^inh = _ ^(1)+^2 _ ^(2 )   p_^inh = p_^(1)+^2 p_^(2 )  , iii ) the contribution of the gauge field , _ g & = & ( ^2+^2 )  , + p_g & = & ( ^2+^2 )  , and iv ) the contribution of the coupling term , _ c & = & -(^2+^2)=-_g  , + p_c&=&-(^2+^2)=-p_g  , where in the last equation we define @xmath53 to be the quadratic combination of the two fields : = ( c^(0)^2 + d ^(0)^2)/2 . from the above we can deduce the equation of state parameter for each part : w_^h&= &  , + w_^inh&= & ( - ) + & & + ^2(-- )  , + w_g&=&w_c=  .    from the equations above , we see that for positive values of the constants @xmath21 and @xmath22 , the coupling of the scalar field with the gauge field will give rise to a contribution @xmath54 to the energy density which has the same equation of state but opposite sign to that of the gauge field . therefore the coupling can help drain energy from the gauge field . it is because of this mechanism that we might hope to achieve a cosmological bounce in the presence of radiation . a first indication on whether a bounce might occur can be obtained by considering the scaling of each contribution to the energy density as a function of the scale factor @xmath7 . to find these scalings , we need the time dependence of the linear and quadratic contributions to each field . therefore , we need to solve the matter field equations of motion . in the following subsection , we present the equations for the fields at each order , while the solutions and detailed analysis will be performed in the next sections .      keeping in mind the anstze for @xmath17 , @xmath31 and @xmath19 , their equations of motion at each order can be obtained from ( [ phi1eom ] ) , ( [ phi2eom ] ) and ( [ amueom ] ) : + a ) at zero - th order : [ eqn0 ] \ {    c _ 1^(0 ) + 3h _ 1^(0 ) + m^2_1^(0 ) = 0  , +   + _ 2^(0 ) + 3h_2^(0 ) + m^2_2^(0 ) = 0  . note that there is no equation at this order for @xmath17 because it is of first order in @xmath32 . + b ) at first order : [ eqn1 ] \ {    c _ 1^(1 ) + 3h _ 1^(1 ) + ( + m^2)_1^(1 ) = 0  , +   + _ 2^(1 ) + 3h_2^(1 ) + ( + m^2)_2^(1 ) = 0  , +   + ( 1-c_1^(0)^2 - d_2^(0)^2)(_f^+3hf^0 ) + - 2(c_1^(0)__1^(0 ) + d_2^(0)__2^(0))f^=0  , +    . making use of eqs . ( [ ansatzgauge1 ] ) and ( [ ansatzgauge2 ] ) , the equation for the gauge field can also be rewritten as : [ max ] & & ( 1-c_1^(0)^2 - d_2^(0)^2)(+ h+ ) + & & - 2(c_1^(0)_1^(0 ) + d_2^(0)_2^(0))= 0  . c ) at second order : [ eqn2 ] \ {    c _ 1^(2 ) + m^2_1^(2 ) -<f_f^>_1^(0 ) = 0  , +   + _ 2^(2 ) + m^2_2^(2 ) + < f_f^>_2^(0 ) = 0  . +    . here , pointed parentheses indicate spatial averaging ( since we are only focusing on the zero mode of the second order field fluctuations ) . we also neglected the effect of hubble friction since it does not give an important contribution for the second order fluctuations . in this section we will solve the equations of motion ( [ eqn0 ] ) , ( [ eqn1 ] ) and ( [ eqn2 ] ) to see if and how a bounce will happen . it is usually useful to perform the analysis in the conformal frame where the conformal time @xmath55 is used rather than the cosmic time . additionally , to extract the dependence on the scale factor , it is convenient to use the following two variables : u_1 ( ) a()_1 ( ) ,  u_2 ( ) a()_2 ( )  . hereafter , we will use @xmath56 to denote the @xmath50-th order perturbation of the @xmath57-th scalar field . moreover , for simplicity but without loosing generality , we can parameterize the scale factor @xmath7 as [ para ] a ( ) = a_0t^p = a_0||^  , with p =  , where @xmath58 and @xmath59 are the initial value of the scale factor and the equation of state of the universe , respectively . this is a self - consistent assumption when @xmath59 is nearly a constant . the evolution of @xmath59 in our case will be shown numerically in section vi .      using the parametrization ( [ para ] ) , the equations of motion at zeroth order of the two scalar fields become : [ eqn0new ] \ {    c u_1^(0)+(a_0 ^ 2 m^2 ^-)u_1^(0)=0  , +   + u_2^(0)+(a_0 ^ 2 m^2 ^-)u_2^(0)=0  , +    . where a prime denotes the derivative with respect to conformal time @xmath62 . their solutions are : u_1^(0 ) & ~ & h_((1-p)am|| )  , + u_2^(0 ) & ~ & h_((1-p)am|| )  , where @xmath63 represents the @xmath64-th order hankel function . far away or close to the bounce , i.e. for @xmath65 and @xmath66 , respectively , the approximate solutions are : + 1)oscillations for large values of the scale factor @xmath65 : [ sol0sub1 ] u_1^(0)&~&||^((1-p)a_0m||^ + & & + _ 1^(0 ) )  , + [ sol0sub2 ] u_2^(0)&~&||^((1-p)a_0m||^ + & & + _ 2^(0 ) )  , ( where @xmath67 and @xmath68 are phases set by the initial conditions ) . in terms of the non - rescaled fields @xmath69 one obtains damped ( or anti - damped ) oscillations ( depending on whether we are in an expanding or a contracting period ) [ sol0phisub1 ] _ 1^(0)&~&||^((1-p)a_0m||^ + & & + _ 1^(0 ) )  , + [ sol0phisub2 ] _ 2^(0)&~&||^((1-p)a_0m||^ + & & + _ 2^(0 ) )  . 2)``frozen '' evolution for small values of the scale factor @xmath66 : [ sol0sup1 ] u_1^(0)&~ & + & & +  , + [ sol0sup2 ] u_2^(0)&~ & + & & +  , from which it follows that the non - rescaled fields @xmath69 evolve as [ sol0phisup1 ] _ 1^(0)&~ & + & & +  , + [ sol0phisup2 ] _ 2^(0)&~ & + & & +  , from which we can see that the last term of @xmath70 is a constant mode while the first term is a varying one . depending on the value of @xmath30 ( or equivalently @xmath59 ) the varying mode could be growing ( for @xmath71 or @xmath72 ) , in which case it becomes dominant , or decaying ( for @xmath73 or @xmath74 ) , in which case it becomes subdominant . we can usually neglect the decaying part of the fields . following the steps performed in the last subsection , we can also get the solutions for the first order components of the scalar fields . using the equations ( [ eqn1 ] ) for the first order perturbations we obtain the following equations of motion for @xmath77 and @xmath78 : [ eqn1new ] \ {    c u_1^(1)+(k^2+a_0 ^ 2 m^2 ^-)u_1^(1)=0  , +   + u_2^(1)+(k^2+a_0 ^ 2 m^2 ^-)u_2^(1)=0  . +    .    depending on the value of @xmath79 , we obtain different approximation solutions . for wavenumbers large compared both to the hubble radius and to the mass term , we obtain oscillatory solutions with fixed amplitude . considering now modes which are still sub - hubble ( i.e. @xmath80 ) but for which the mass term dominates over the contribution of the field tension ( i.e. the term involving @xmath79 ) , we can neglect both the @xmath81 term and the term involving @xmath82 . the simplified equation for these modes is : \{[u1 in ]    c u_1^(1 ) + a_0 ^ 2 m^2 ^ u_1^(1 ) = 0  , +   + u_2^(1 )  + a_0 ^ 2 m^2 ^ u_2^(1 ) = 0  , +    . whose solutions are : u_1^(1)&~&h_((1-p)a_0m|| ) + & ~&||^((1-p)a_0m|| + & & + ^(1)_1 )  , + u_2^(1)&~&h_((1-p)a_0m|| ) + & ~&||^((1-p)a_0m|| + & & + ^(1)_2 )  .    for modes outside the hubble radius ( @xmath83 ) , we have : \ {    c u_1^(1 )  + ( a_0 ^ 2 m^2 ^-)u_1^(1 ) = 0  , +   + u_2^(1 ) + ( a_0 ^ 2 m^2 ^-)u_2^(1 ) = 0  , +    . which have the same form as eq . ( [ eqn0new ] ) so their solution will be the same as given in eqs . ( [ sol0sup1 ] ) and ( [ sol0sup2 ] ) . we have thus seen that the first order solutions for the scalar fields scale the same way with @xmath84 as the zeroth order solution . this is because in the small @xmath84 region where the @xmath85 term dominates over the other ones , the equations for first order and zero - th order modes are almost the same . thus , unless the energy density in the @xmath86 modes dominates at the initial time , it will never dominate over the background contribution from the @xmath87 terms . thus we can conclude that the first order fluctuations of scalar fields will not prevent the bounce .      in this section , we will analyze the gauge field @xmath37 which is also considered to be of first order . the equation ( [ max ] ) can directly be transformed to conformal frame as : [ gammaeq ] + k^2-  = 0  . since the coefficients @xmath21 and @xmath22 are small , we can take the last term to be a source term . in a first order born approximation , we can write the total solution as _ 0 + , where @xmath88 is the solution for the homogeneous equation obtained by setting @xmath89 , while @xmath90 is the leading correction term obtained by inserting @xmath88 into the source term ( the last term in ( [ gammaeq ] ) ) .    the zero - th order ( homogeneous ) equation is easily solved and gives [ gamma ] _ 0 ~(k|| + _ )  . for the first order equation , it is convenient to define p ( ) - , so that the equation becomes  + k^2 + p()_0 = 0  , where we neglected the small term @xmath91 . inserting the solution of @xmath88 ( [ gamma ] ) , we get the following equation for @xmath90 : [ maxnew ]  + k^2= -p()_0  . we are interested in the scaling of @xmath36 as a function of time . for this purpose , we need to work out the scaling in time of the source term in ( [ maxnew ] ) . since the solutions for @xmath60 and @xmath61 scale differently in time in the two time intervals discussed in subsection ( [ subseca ] ) , it is necessary to analyze these two intervals separately .    for times obeying @xmath92 , then by differentiating ( [ sol0phisub1 ] ) and ( [ sol0phisub2 ] ) with respect to @xmath62 we have :  & ~ & ||^ + & &  , + _ 2^(0)&~&||^ + & &  . note that @xmath93 is a decaying mode in the contracting phase and thus the last terms inside the square brackets in the above formulae can be neglected compared to the first ones . since p ( ) & = & - + & & -2(c_1^(0)_1^(0)+d_2^(0)_2^(0) ) , then combining all these results we get : ~c_1||^  .    for @xmath66 , then differentiating ( [ sol0phisup1 ] ) and ( [ sol0phisup2 ] ) with respect to @xmath62 we obtain & ~ &  , + _ 2^(0) & ~ &  , when @xmath71 and ~_2^(0)0 when @xmath73 . then we can solve equation ( [ maxnew ] ) to get : & ~ & c_2||^  ,  p > + & ~ & c_3(k||+ _ )  . p < in the above expressions for @xmath90 , @xmath94 , @xmath95 and @xmath96 are complicated prefactors in front of the @xmath62-dependent terms . in summary , we see that the interactions give only a subleading correction @xmath90 to @xmath88 . finally , let us consider the homogeneous component of the second order fluctuations of the scalars , namely , @xmath97 and @xmath98 . if we only consider the energy density up to second order , these second order field perturbations give a contribution through their coupling to the background fields . in the following we find the solutions of ( [ eqn2 ] ) and study the effects of the induced terms in the stress - energy tensor on a possible bounce . given the solution for the gauge field @xmath37 obtained in the last subsection , it is easy to rewrite eqs . ( [ eqn2 ] ) as : [ eqn2new ] \ {    c u_1^(2 ) + a^2m^2u_1^(2 ) = cu_1^(0 )  , +   + u_2^(2 )  + a^2m^2u_2^(2 ) = -du_2^(0 )  , +    . where we made use of the fact that f_f^ = 2(k^2 ^ 2-a^2 ^ 2)/a^4 . equation ( [ eqn2new ] ) has the same form as the zero - th order equation but with a small source term generated by the interaction with the gauge field . this equation can be solved using the born approximation ( details are given in the appendix ) . the general solution is the sum of the general solution of the homogeneous solution plus the solution including the source which has vanishing initial data . the inhomogeneous term is suppressed by the coupling constants @xmath21 and @xmath22 compared to the homogeneous solution , but , as shown in the appendix , it scales as a high power of @xmath99 . via the coupling to the background scalar fields , the above second order terms enter into the expression for the energy density to second order . the signs of the corresponding terms in the energy density are indefinite in the sense that they depend on the phases of the initial field configurations . since it is these terms that dominate the energy density near the bounce , we find that whether a bounce occurs or not depends sensitively on the phases in the initial conditions , and that in fact in the case of many plane wave modes initially excited , a bounce requires very special phase correlations . in the previous section we have solved all of the field equations up to second order in the amplitude of the fluctuations . we have found the scaling in time of each field at each order . now we are ready to look at how all of the terms in the expression for the energy density @xmath100 , @xmath101 and @xmath102 at various orders in perturbation theory ( namely , eqs . ( [ rho0])-([rho2 ] ) ) scale in time . this analysis is straightforward but very important if we are to determine whether a bounce is possible , since in four space - time dimensional classical einstein gravity with flat spatial sections a bounce can only happen when the negative terms in the energy density catch up to the positive contributions @xcite .    in the following we give a table of how each term contained in @xmath29 scales with time as the background cosmology bounce point ( the bounce which is achieved in the absence of radiation and scalar field inhomogeneities ) is approached . we will identify the terms which dominate in this limit . this will give us a good indication under which conditions a bounce can occur . the tables are structured as follows : the first line `` terms '' , indicates which term we are considering , the next set of lines `` behavior '' gives the scaling in time in the various limits and in the two relevant ranges of the parameter @xmath30 which indicates the equation of state , and the last line gives the sign with which the term contributes to the energy density . note that we focus on the growing mode solution to each field ( which is constant for small @xmath62 in the case @xmath103 ) . we give separate tables for terms of zero - th , first and second order in @xmath32 . \a ) for terms contained in @xmath100 : +     @xmath104 & @xmath105 & @xmath106 + behavior &    c @xmath107 + @xmath9 (    [ cols="^ " , ]     & indefinite & indefinite +    these terms to not vanish upon spatial averaging . note that we have expressed the time dependence in terms of the dependence on the scale factor @xmath7 . at this stage , we only need to focus on the exponent of the power - law scaling . the more negative the power is , the more rapidly the term grows in a contracting phase ( since @xmath7 is decreasing with time ) .    as mentioned earlier , the conditions for a bounce to occur in four space - time dimensional classical einstein gravity with flat spatial sections is that the total energy density reaches zero during the contracting phase . thus , there needs to be a negative definite term which starts out small but grows faster than the positive definite terms due to the regular scalar field and regular radiation . in the absence of radiation and scalar field inhomogeneities , it is the contribution to the energy density of the ghost field @xmath108 which plays this role .    from the table we see that there are three kinds of terms : positive definite , negative definite and indefinite ones . the first set contains the kinetic and potential terms of the normal scalar as well as the free energy density of the gauge field , the second set is made up of the kinetic and potential terms of the ghost scalar , while the third set contains terms which arise due to the coupling terms between scalars or between scalars and gauge fields .    looking first at the terms which are independent of the coupling term between the fields , we see from the first line of the `` behavior '' set of lines in the third table that , indeed , in the presence of radiation the energy density in radiation grows faster than that in the two scalar fields , thus preventing a bounce . in the presence of coupling between the fields , however , there are terms which scale with a larger negative power of @xmath7 . the signs of some of them , however , depend on the initial phases for the linear fields @xmath37 , @xmath34 and @xmath35 . note that the signs of the scalar coupling terms are determined by the evolution of each field and thus are hard to be identified in a general analysis . the same is true for the gauge coupling terms ( the last two in the third table ) . however , the coupling terms between the scalar fields and the gauge field ( the third to last in the third table ) can be made negative / positive definite easily by setting the signs of the coefficients @xmath21 and @xmath22 to be both positive / negative . it is reasonable to assume that the contracting phase begins with the regular scalar field dominating the energy density , and that the contribution of the lee - wick scalar is much smaller . for single fourier mode initial conditions of the radiation field , this can be achieved with the appropriate choice of the initial phase ( see example 1 in the following section containing our numerical results ) . however , for multiple initial radiation fourier modes excited any initial phase difference between the modes will produce a contribution with the wrong sign and will thus prevent a bounce ( see example 3 in the following section ) . in the presence of an infinite set of modes , the phase correlations required to obtain a bounce thus appear to have negligible measure in initial condition space . thus , even in the presence of coupling between scalar fields and radiation , the lee - wick bounce is unstable . the bounce , if it exists , will happen at a time which can be chosen to be @xmath109 . its duration ( the time interval lasting from the time the hubble radius stops decreasing in the contracting phase until when it starts expanding in the post - bounce phase ) will be denoted by @xmath110 . since the various components of the energy density scale with different powers of @xmath7 , it is clear that the duration of the bounce will be shorter or equal to the hubble radius @xmath111 ( which gives the time scale on which the ratios of energy densities in different components change ) at the beginning of the bounce phase . for the background bounce model , we have @xmath112 . there are two kinds of bounces according to the duration of the bounce phase i ) if the period @xmath113 , the bounce will go from the time t_b^- ~- ~- to the time t_b^+ ~ ~ with a low speed . we call this a `` slow bounce '' . in this case , the universe will enter the bounce period at the critical time @xmath114 , and only the @xmath115 approximate solutions of the previous tables will be applicable and not ones for the interval @xmath116 . ii ) if the period @xmath117 , the bounce will happen in a very short time with very fast speed . this can be called the `` fast bounce '' . in this case , the universe evolves from the far past ( @xmath118 with @xmath119 ) to @xmath109 , passing through the point @xmath120 , then entering into the region @xmath121 before finally reaching the bounce point . in this case , both of the two approximate solutions of the field evolution will be applied . let us now consider the necessary conditions for a bounce ( as we have indicated above and will see from the numerical analysis , these conditions are not sufficient - in addition to the conditions which follow , appropriate correlations in the initial phases are required ) . we start in the region of time @xmath115 . we study the conditions required to have the terms that might give a bounce grow relative to the other terms during this phase . if the conditions are not satisfied , or the bounce does not happen even if the conditions are satisfied , then a bounce may still occure in the @xmath116 region . the conditions for the terms in the energy which could compensate the positive radiation contribution to become dominant are then studied . if these conditions are not satisfied , either , then a bounce is impossible . a necessary condition for a bounce to be possible requires the growth rate of one of the indefinite sign terms in the third table above exceed all that of all of the positive definite terms . in the @xmath115 region , this requires @xmath122 , which equivalently constrains the equation of state parameter @xmath59 to be in the range @xmath123 . if this condition is satisfied in this region , then a slow bounce may happen depending on the choice of the initial phases . if the condition is not satisfied in the @xmath115 region , the universe may evolve into the @xmath116 region , in which the evolution of the fields are different , and new constraints on @xmath30 and @xmath59 will arise if a bounce is to be possible . following the above logic , we find that the conditions under which a bounce might happen are much looser , namely @xmath124 . to summarize this section : we have identified necessary conditions for a bounce to occur . whether one actually does occur even if the conditions are satisfied depends on the initial phases of the fields . this must be studied numerically . in the following section we will give one example of specially chosen phases for which a bounce is possible . however , when we look at a more general choice of phases , the bounce will not occur . in order to support the analysis in the last section , we performed numerical calculations . such numerical work is necessary because our analytical analysis is only approximate . in particular , we worked in perturbation theory up to order second order in @xmath32 . in addition , even in cases where our analytical analysis would indicate the possibility of a bounce , the perturbative analysis will break down near the bounce point , and there is no assurance that the trends seen in the perturbative analysis will persist . we have numerically solved the full nonlinear equations of motion for the matter fields in the presence of a homogeneous expanding background cosmology . the homogeneous cosmology is obtained numerically by solving the first friedmann equation h^2 = , where @xmath125 is newton s gravitational constant ( related to the planck mass used earlier ) , and @xmath29 is the total energy density , averaged over space .    figures [ 1]-[6 ] are two groups of numerical results with different parameters . in both cases we choose the initial energy density of the gauge fields to be larger than that of the lee - wick scalar , but less than that of the normal scalar . these initial conditions correspond to the situation we are interested in , namely starting in a matter - dominated contracting phase in the presence of some radiation which is sub - dominant . [ 1 ] , [ 2 ] and [ 3 ] show an example with parameters @xmath126 and @xmath127 . we choose initial conditions in which a single fourier mode fluctuation is excited , and in which the phases are chosen as indicated in the figure caption . for these initial phases , we obtain a bounce . in fig . [ 1 ] , we see that the equation of state @xmath59 begins with a value slightly larger than @xmath128 , and then evolves to some nearly fixed value . for the case of our initial condition choice , it appears to be @xmath129 , in the region where the bounce is allowed to happen . at the bounce point , the equation of state will drop to @xmath130 , while after the bounce , the equation of state will rise again to @xmath131 . [ 2 ] is the plot of the scale factor in this case which shows explicitly the occurrence of the bounce . cosmic time @xmath47 ( horizontal axis ) , in the first simulation , a simulation with only a single fourier mode excited and phases chosen as indicated below . we see that @xmath59 drops to @xmath130 , indicating that there is a bounce . the background fields are plotted in dimensionless units by normalizing by the mass @xmath132 while the time axis is displayed in units of @xmath133 . the mass parameters m and m were chosen to be @xmath134 and @xmath135 . the initial conditions were chosen to be @xmath136 , @xmath137 , @xmath138 , @xmath139 , @xmath140 , @xmath141 . the coefficients @xmath21 and @xmath22 are chosen to be @xmath142 and @xmath143 . the wavenumber is @xmath144 . ]    . from the plot we see that the bounce happens at @xmath109 . ] [ 3 ] gives a comparison of the energy densities of some components during the process . initially , we set the energy density of the gauge field @xmath37 to be between the normal scalar and lee - wick scalar . when the evolution of the universe enters into a region with nearly constant @xmath59 , the gauge - coupling component of energy density @xmath54 will grow very fast . it is negative and thus enables the negative part of the energy density to catch up with the positive one , thus allowing the bounce to happen . for the inhomogeneous fluctuation , we choose the wavenumber to be @xmath144 which corresponds to a scale which is observable by cmb and lss experiments .    , @xmath19 and @xmath37 in the system with parameters chosen as in fig . the curves from up to down are : @xmath145 ( black ) , @xmath146 ( red ) , @xmath54 ( dark cyan ) and @xmath147 ( blue ) , respectively . the variables are also normalized with the mass scale @xmath148 . ] figs . [ 4 ] , [ 5 ] and [ 6 ] give the corresponding results in the case when we choose @xmath126 while @xmath149 ( with all initial conditions identical ) . this case seems dangerous because the contribution of the lee - wick scalar to the fluctuation terms could lead to an instability . however , as we have mentioned before , since the effects of the lee - wick scalar are less than that of the normal scalar , it is still possible for the bounce to happen . [ 4 ] shows the equation of state of the system . we can see that the evolution of @xmath59 is about the same as that in fig . [ 1 ] , since the change of the sign of @xmath22 does not alter the result too much . [ 5 ] is the behavior of scale factor in this case while fig . [ 6 ] gives the comparison of the energy densities of all components .     as a function of cosmic time @xmath47 ( horizontal axis ) the behavior that @xmath59 drops to @xmath130 indicating that a bounce takes place . the background fields are plotted in dimensionless units by normalizing by the mass @xmath132 while the time axis is displayed in units of @xmath133 . the mass parameters @xmath10 and @xmath13 were chosen to be @xmath134 and @xmath135 . the initial conditions were chosen to be @xmath136 , @xmath137 , @xmath138 , @xmath139 , @xmath140 , @xmath141 . the coefficients @xmath21 and @xmath22 are chosen to be @xmath142 and @xmath150 . the wavenumber @xmath144 . ]    . from the plot we see that the bounce happens at @xmath109 . ]    , @xmath19 and @xmath37 in the system with parameters chosen as in fig . the curves from up to down are : @xmath145 ( black ) , @xmath146 ( red ) , @xmath54 ( dark cyan ) and @xmath147 ( blue ) , respectively . the variables are also normalized with the mass scale @xmath148 . ] a change in the phase of the initial radiation field velocity will not change the results ( if we keep the other initial conditions fixed ) . on the other hand , if we flip the sign of the initial velocity of one of the two scalar fields , then the sign of the dominant contribution to the energy density as we approach the bounce will flip and this will prevent a bounce . if we use initial conditions containing two excited fourier modes , then we obtain a bounce only if the signs of the initial field velocities are both chosen as in the first run whose results are shown here . different phases for the scalar field velocities of the two modes destroys the possibility of obtaining a bounce . figures [ 7 ] , [ 8 ] and [ 9 ] show the results for the equation of state parameter @xmath59 , the hubble parameter @xmath2 and the contribution of the various components to the total @xmath29 in the case of a simulation in which two fourier modes are excited , with velocities of both scalar fields having opposite signs from those in the previous example . as is obvious , a big crunch singularity occurs .     as a function of cosmic time @xmath47 ( horizontal axis ) . the background fields are plotted in dimensionless units by normalizing by the mass @xmath132 while the time axis is displayed in units of @xmath133 . the mass parameters @xmath10 and @xmath13 were chosen to be @xmath134 and @xmath135 . this plot is the evolution of the system with two fourier modes combined together . the one is of which the wavenumber @xmath144 with initial conditions @xmath136 , @xmath137 , @xmath138 , @xmath139 , @xmath140 , @xmath141 , which if taken alone will give the bounce as has been shown in the previous example . the other is of which the wavenumber @xmath151 with initial conditions of the same initial values of the fields but the opposite signs of the scalar field velocity . from this plot we can see that the combination of the two fourier mode will ( generally ) cause @xmath59 blow up , thus preventing the bounce . this means that the bounce requires special fine - tuning of the initial phases for each fourier mode . the coefficients @xmath21 and @xmath22 are chosen to be @xmath142 and @xmath150 . ]    . from the plot we see that there is a singularity at @xmath109 . ]    , @xmath19 and @xmath37 in the system with parameters chosen as in fig . the curves from up to down are : @xmath145 ( black ) , @xmath146 ( red ) , @xmath54 ( dark cyan ) and @xmath147 ( blue ) , respectively . the variables are also normalized with the mass scale @xmath148 . ] in this paper we analyzed in detail the possibility of obtaining a cosmological bounce in a model which corresponds to the scalar field sector of the lee - wick theory coupled to relativistic radiation . it is known that the scalar field sector of the lee - wick theory in the absence of other fields can yield a cosmological bounce @xcite . in fact , the universe will scale as non - relativistic matter with @xmath152 both before and after the bounce . thus , this model is a possible realization of the  matter bounce " scenario . however , this background is unstable to the introduction of radiation since in the contracting phase the growth of energy density in radiation will exceed that of matter and will lead to a big crunch singularity as has been shown in previous work @xcite , the introduction of a lee - wick partner to radiation does not prevent this instability . in this paper , we introduced an interaction between the radiation field and the scalar fields . the interaction could help drain energy from the radiation field to the lee - wick scalar , and thus could prevent the radiation from growing too fast to destroy the bounce . we analyzed the equations describing the evolution of the three matter fields ( regular scalar field , its lee - wick partner and the radiation field ) on a cosmological background both analytically and numerically . our analytical analysis was perturbative and made use of the second order using born approximation . the expansion parameter is set by the initial amplitude of the gauge field . we solved the equations of motion for each field at each order , and obtained their approximations in different cases . we compared their contributions to the total energy density , and derived necessary conditions for a bounce to happen . to support our analysis , we also performed numerical calculations .    specifically , we investigated initial conditions in which one or two fourier modes of the radiation field and the scalar field fluctuations are excited . we found special initial conditions which indeed lead to a non - singular bounce . changing the sign of the initial scalar field velocity will destroy the bounce solution . in the presence of two fourier modes , we found that a bounce requires identical initial phases for the two modes . for general initial conditions , we conjecture that the measure of such initial conditions which lead to a bounce is very small . we thus find that the addition of coupling terms between the scalar fields and radiation can not save the lee - wick bounce background from the instability problem with respect to the addition of radiation ( nor , for that matter , with respect to scalar field fluctuations ) . the instability problem with respect to anisotropic stress will be even worse . the work at mcgill is supported in part by an nserc discovery grant and by funds from the canada research chair program . rb is recipient of a killam research fellowship . the work at cycu is funded in parts by the national science council of r.o.c . under grant nsc99 - 2112-m-033 - 005-my3 and no . nsc99 - 2811-m-033 - 008 and by the national center for theoretical sciences . the solution for the second order scalar field correction @xmath154 can be determined using the green function method . the general solution of ( [ eqn2new ] ) is the sum of the solution @xmath155 of the homogeneous equation which solves the same initial conditions as @xmath156 and the particular solution @xmath157 wich vanishes at time @xmath158 . the particular solution is given by [ gf ] u ( ) & = & u_a()__i^d()u_b()s( ) + & & -u_b()__i^d()u_a()s( )  , where @xmath159 and @xmath160 are two independent solutions of the homogeneous equation , @xmath161 is the wronskian : ( ) = ( u_a^ u_b - u_b^ u_a ) ^-1 , and @xmath162 is the source inhomogeneity .            we will demonstrate the analysis for the case of @xmath163 . let us consider evolution for a short interval of time starting at some initial time @xmath158 . then , we can neglect the expansion of the universe in the equation of motion and take @xmath164 . we are then interested in how the result scales in @xmath158 . using this trick , the solutions of the homogeneous equation can be taken to be u_a ( ) & = & ( _ m ) + u_b ( ) & = & ( _ m ) and the wronskian is ( ) = - where _ m = .      combining these results we obtain [ gf3 ] u_1^(2 ) & ~ & -(_m)__i^ ||^ + & & ( _ m)(2k||+2_)((1-p)am||+_1^(0 ) ) + & & + ( _ m)__i^ ||^ + & & ( _ m)(2k||+2_)((1-p)am||+_1^(0 ) )  , + u_2^(2 ) & ~ & -(_m)__i^ ||^ + & & ( _ m)(2k||+2_)((1-p)am||+_1^(0 ) ) + & & + ( _ m)__i^ ||^ + & & ( _ m)(2k||+2_)((1-p)am||+_1^(0 ) )  . note that if we only care about their scalings with respect to conformal time @xmath62 or scale factor @xmath7 , the above solutions can be reduced to : u_1,2^(2)||^a^-+  , and for the case of a matter dominated era where @xmath166 , it is straightforward to show that u_1,2^(2)||^-6a^-3  . y.  f.  cai , t.  t.  qiu , r.  brandenberger and x.  m.  zhang , `` a nonsingular cosmology with a scale - invariant spectrum of cosmological perturbations from lee - wick theory , '' phys . d * 80 * ( 2009 ) 023511 [ arxiv:0810.4677 [ hep - th ] ] . s. w. hawking , and g. f. r. ellis , _ the large scale structure of space - time _ , cambridge university press ( 1973 ) ; a.  borde and a.  vilenkin , phys . lett .   * 72 * , 3305 ( 1994 ) . a.  borde and a.  vilenkin , `` eternal inflation and the initial singularity , '' phys . lett .   * 72 * , 3305 ( 1994 ) [ arxiv : gr - 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it was recently realized @xcite that a model constructed from a lee - wick type scalar field theory yields , at the level of homogeneous and isotropic background cosmology , a bouncing cosmology . 
 however , bouncing cosmologies induced by pressure - less matter are in general unstable to the addition of relativistic matter ( i.e. radiation ) . here 
 we study the possibility of obtaining a bouncing cosmology if we add radiation coupled to the lee - wick scalar field . 
 this coupling in principle would allow the energy to flow from radiation to matter , thus providing a drain for the radiation energy . 
 however , we find that it takes an extremely unlikely fine tuning of the initial phases of the field configurations for a sufficient amount of radiative energy to flow into matter . for general initial conditions , 
 the evolution leads to a singularity rather than a smooth bounce .