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the reheating period in the early universe stands out as a challenge to theorists due to the close interrelationship of nonlinear and gravitational phenomena in its unfolding ( see @xcite , henceforth papers 1 - 5  ) . the observation that , due to the high occupation numbers produced during preheating , most of the physics of reheating may be understood in terms of the behavior of nonlinear classical waves@xcite has been the key to sustantial progress . the authors of papers 1 - 5 have undertaken systematic numerical simulations of the behavior of matter fields during reheating , finding that the reheating period is actually composed of three consecutive phases : an early one or preheating , where the dominant effect is the parametric amplification of matter fields out of the dynamical inflaton and gravitational backgrounds @xcite , an intermediate stage where the dominant phenomenon is the redistribution of energy among matter field modes through rescattering ( in the terminology of papers 1 - 5 ) , and a final stage where thermal equilibrium sets in . during the intermediate stage , spectra of occupation numbers for the matter fields reduce to simple power laws both in the infrared and ultraviolet limits . as noted in papers 1 - 5 , this behavior suggests a connection between the physics of reheating and the phenomena of weak turbulence @xcite , but to the best of our knowledge no theoretical prediction for the exponents involved is available . our goal is to provide these theoretical estimates .    in this paper we shall follow this same trend of ideas , by observing that , from the macroscopic point of view , a stochastic ensemble of classical waves may be described by a conserved energy momentum tensor subject to the second law of thermodynamics . there is therefore an equivalent fluid description , consisting of a fluid whose energy momentum tensor and equation of state reproduce the observed ones for the microscopic fluctuations . solving the dynamics of this equivalent fluid yields answers to all relevant questions concerning the behavior of the actual fluctuations . an immediate consequence of energy momentum conservation and the second law is that when velocities are low , the phenomenological fluid may be described within the eckart theory of dissipative fluids @xcite ( for an analysis of the limitations of eckart s theory see @xcite ) . it follows that it obeys a continuity equation and a curved space time navier - stokes one . turbulent  spectra found in numerical simulations correspond to the self - similar solutions discussed long ago by chandrasekhar @xcite . they appear in the discussion of decaying turbulence ( which is the case relevant to cosmology ) , as opposed to turbulence driven by some external means . the chandrasekhar solutions are built on the heisenberg closure hypothesis @xcite ( see @xcite for a general discussion of turbulence ) . they were generalized to friedmann - robertson - walker ( frw ) backgrounds by tomita _ et al _ @xcite . these solutions agree with the kolmogorov 1941 theory in the inertial range @xcite , failing to reproduce observations for very small eddies . fortunately , we are most interested in the opposite limit of very large eddies , where it is trustworthy ( we wish to point out that the applicability of kolmogorov s spectrum to large scale turbulence should not be taken for granted @xcite ) . with minor adjustments , tomita s analysis of turbulence decay in frw space times also provides a solution to the evolution of our equivalent fluid . the rest of the paper is organized as follows . in next section we provide a brief summary of hydrodynamics in flat and expanding universes , in order to set up the language for the rest of the paper , and introduce the self similar solutions . in section iii we proceed to discuss the equivalent fluid description of field fluctuations , and how to extract the particle spectrum therefrom . in section iv we place the self - similar solutions in the context of reheating . we state our main conclusions in the final section . we provide a rough estimate of the shear viscosity during reheating in the appendix . the equations governing the dynamics of a fluid in local thermodynamic equilibrium are the continuity and navier - stokes ones , which , in the case of flat space time , read :    @xmath3    @xmath4    where we have assumed incompressibility , valid when typical velocities are much smaller than the sound velocity ; @xmath5 is the kinematic shear viscosity . the transition from laminar to turbulent motion can be universally described by the dimensionless  reynolds number :    @xmath6    where @xmath7 is a typical velocity and @xmath8 a typical length scale . this number represents the order of magnitude of the ratio of the inertial to the viscous term . low reynolds numbers correspond to laminar motion , while high ones suggest turbulent behavior . in general , the velocity profile displays variations in space and time . this implies that the flow must be described probabilistically . thus , each quantity involved in ( [ contpl]-[nspl ] ) is divided in its mean value and a fluctuation from it ; for example , we write @xmath9 , where @xmath10 stands for the fluctuating part of the velocity . in the case where motion is isotropic , the mean value @xmath11 for the velocity must be zero , since otherwise there would be a preferred direction .    to analyze the system s behavior , we define the two - point one - time correlation function for the velocity :    @xmath12    in the case of homogeneous and isotropic motion , this correlation must be only a function of the time @xmath13 and the distance between @xmath14 and @xmath15 , i.e. @xmath16 , where @xmath17 observe that @xmath18 ( summation over repeated indices must be understood ) is twice the average energy density of the flow at time @xmath13 . from ( [ nspl ] ) we obtain the equation that this correlation must obey , namely :    @xmath19    where @xmath20 and    @xmath21    the tensor @xmath22 comes form the inertia term in navier - stokes equation and , as it involves a product of third order in the velocity , reflects the fact that there is not a close set of equation for the correlations of successive orders but there is a hierarchy of equations instead . the problem of closing that hierarchy is known as the  moment closure problem  @xcite . let us call @xmath23 the fourier transform of @xmath24 . then the energy density becomes    @xmath25 where    @xmath26    is the energy density stored in eddies of size @xmath27 defining @xmath28 as the fourier transform of @xmath22 , we obtain from ( [ ecrij ] ) the equation of balance of the energy spectrum : @xmath29 where @xmath30    the inertia term @xmath31 is the one that contains the mode - mode interaction , and its effect is to drain energy form the more energetic modes -typically the bigger ones- to the ones where there is major viscous dissipation -the smaller ones- .      for a curved space - time , in particular a friedmann - robertson - walker ( frw ) universe with zero spatial curvature ( @xmath32 ) , the generalization of the above arguments has been considered by many authors @xcite . we follow tomita _ et al . _ s analysis @xcite , in which they obtain the solution for the energy spectrum in the case of homogeneous , isotropic and incompressible turbulence .    in a generic space time , we describe fluid flow from the energy density @xmath33 , pressure @xmath34 and four velocity @xmath7 . the symmetries of the frw solution suggest using instead the commoving three velocity @xmath35 if @xmath36 the flow is non relativistic , and if @xmath37 it is incompressible ( @xmath38 ) . later on , we shall also use the physical three velocity @xmath39    the corresponding continuity and navier - stokes equations for a robertson - walker background are obtained by the condition of conservation of the energy - momentum tensor @xcite . for a non relativistic incompressible fluid , with shear viscosity @xmath40 ( but no bulk viscosity ) , these reduce to :    @xmath41    @xmath42   { \bf u=-}\frac{\nabla p}{a^2\left ( p+\rho \right ) } + \frac 1{a^2}\nu \nabla ^2 { \bf u }   \label{nscur}\ ] ]    where we have assumed that @xmath43 depends only on time . for the physical three velocity @xmath44 , the corresponding navier - stokes equation reads : @xmath45 { \bf v=-}\frac{\nabla p}{a\left ( p+\rho \right ) } + \frac 1{a^2}\nu \nabla ^2{\bf v }   \label{nscurv}\ ] ] in obtaining ( [ contcur])-([nscurv ] ) we have neglected possible perturbations to the frw metric . the corresponding equations considering fluctuations in the metric ( @xmath46 ) have been obtained by weinberg @xcite . the continuity equation is not corrected by gravitational perturbations , while in the navier - stokes equation the metric fluctuations appear explicitly only within the shear viscosity term . it can be demonstrated that these terms involving metric fluctuations are negligible for scales that are inside the horizon @xcite . for scales bigger than the hubble radius , since dissipation through viscosity is not effective anyway , we may still use the unperturbed navier - stokes equation . the operation of fourier transforming in the case of a robertson - walker cosmology is done in terms of commoving wave - numbers . in doing so , the following equation for the energy spectrum is obtained : @xmath47 where the relationship between @xmath48 and @xmath23 as well as between @xmath31 and @xmath49 is the same as that for a flat space time , if we define @xmath50 and @xmath22 from correlations of physical quantities , as follows : @xmath51      as we have seen in the previous section , the key element in the description of the flow is the energy spectrum @xmath52 ( eq.([et ] ) ) , which is the solution of the balance equation ( eq.([ecesp ] ) ) . in it , the right hand side contains the viscous dissipation as well as the inertial force @xmath53 . the overall effect of this term is to transfer energy from a given scale to smaller ones through mode - mode coupling ; thus it is natural to model the action of the inertia term as a source of viscous dissipation , where the effective turbulent viscosity for a given mode depends on the motion of all smaller eddies @xcite . by providing closure , that is , writing this effective viscosity in terms of the spectrum itself , a closed evolution equation for @xmath52 is obtained . concretely , heisenberg @xcite proposed the ansatz    @xmath54    where    @xmath55    and @xmath56 is a dimensionless constant . with this hypothesis ( known as the heisenberg hypothesis ) as the solution to the closure problem , chandrasekhar @xcite has obtained the energy spectrum for decaying turbulence , assuming that there is a stage in the decay where the bigger eddies have sufficient amount of energy to maintain an equilibrium distribution , thus requiring that the solution for the spectrum should be self similar . with this consideration into account he obtained an energy spectrum :    @xmath57    where @xmath58 and @xmath59 are initial conditions ( namely , the wave number corresponding to the bigger eddy and its typical time of evolution ) . the function @xmath60 obeys the equation    @xmath61 dx^{\prime } = \left\ { \nu k_0 ^ 2t_0+\int_x^\infty \frac{% \sqrt{f(x^{\prime } ) } } { x^{\prime \ 3/2}}dx^{\prime } \right\ } \int_0^xf(x^{\prime } ) x^{\prime \ 2}dx^{\prime }   \label{ecf}\ ] ]    which predicts a kolmogorov type behavior for an inviscid fluid ( @xmath62 , @xmath63 ) in the ultraviolet limit :    @xmath64    while for nonzero viscosity :    @xmath65    in the infrared limit , @xmath60 has the universal behavior @xmath66 @xmath67 , and thus we find a linear energy spectrum for @xmath68    chandrasekhar s self similar solutions are easily generalized to flows in expanding universes . the dependence on time and wave - number for the self similar energy spectrum is @xcite    @xmath69    where @xmath70 and @xmath71 are respectively the taylor s microscale and an average turbulent velocity , defined as :    @xmath72    the second equation implies the normalization condition    @xmath73    to obtain a self similar flow reducing to eq . ( [ flatspacelim ] ) in the flat space limit , we must require the time evolution laws : @xmath74    the equation which determines de function @xmath75 in ( [ speccur1 ] ) turns out to be    @xmath76 dx^{\prime } = \left\ { \frac 25+a\int_x^\infty \frac{\sqrt{f(x^{\prime } ) } } { x^{\prime \ 3/2}}dx^{\prime } \right\ } \int_0^xf(x^{\prime } ) x^{\prime \ 2}dx^{\prime }   \label{curvedchandra}\ ] ]    where @xmath77 is a constant . this equation has the same structure as in flat space time , eq . ( [ ecf ] ) , which means that assuming heisenberg s hypothesis the spectrum is linear in @xmath78 for length scales much bigger than the taylor s microscale . observe that for flat space time , the proportionality between the integral up to a certain wave number @xmath78 of the inertia and the viscous forces is given by ( [ heis1 ] ) and ( [ heis2 ] ) . in the case of a frw space time , the autosimilar solution required by tomita _ ( [ speccur1 ] ) needs a time dependent @xmath79 in eq . ( [ heis1 ] ) , defined by the analog of ( [ heis2 ] ) with @xmath56 replaced by @xmath80 . let us analyze in more detail the solutions of eq . ( [ curvedchandra ] ) . we assume the normalization eq . ( [ normalization ] ) . by taking the @xmath81 limit in eq . ( [ curvedchandra ] ) we find    @xmath82    taking a derivative of eq . ( [ curvedchandra ] ) we get    @xmath83    where    @xmath84    @xmath85    let us consider first the @xmath86 limit . assume @xmath87 . the left hand side of eq . ( [ differential ] ) tends to a finite limit @xmath88 . on the right hand side , @xmath89 and @xmath90 behave as @xmath91 , so if @xmath92 this side goes to zero . we must therefore have @xmath93 , and    @xmath94    were @xmath95 is some constant .    in the @xmath96 limit , assume again a power law behavior @xmath97 . now @xmath98 , so we must have @xmath99 from eq . ( [ normalization2 ] ) we know that in this limit @xmath100 , so we must have @xmath101 and    @xmath102    taking into account both limiting behaviors and eq . ( [ normalization ] ) , the function @xmath60 may be approximated as    @xmath103 = \frac{x}{\left [ \alpha + \beta x^{4}\right ] ^{2}},\qquad \alpha = \left [ \frac{25a\pi ^{2}}{128}\right ] ^{1/3}\qquad \beta = \frac{2 } { 25a }   \label{fapprox}\ ] ] after establishing the basic necessary notions for the description of hydrodynamic flows , our goal is to associate an equivalent fluid description to field fluctuations , and to derive the particle spectrum therefrom . our first step is to obtain the energy density , pressure and velocity of this fluid as functionals of the quantum state of the field .    for simplicity , we shall consider the theory of a single , self - interacting scalar field @xmath104 , minimally coupled to gravity . the action is    @xmath105 \right\}\ ] ]    where @xmath106 $ ] ia renormalized effective potential . the energy - momentum tensor is associated to the heisenberg operator    @xmath107 \right\}\ ] ]    the macroscopic behavior of the field , however , may be described in terms of a c - number energy - momentum tensor    @xmath108    where    @xmath109    and @xmath110 is a stochastic component with zero mean and self - correlation    @xmath111    in these equations , @xmath112 denotes a stochastic average , while @xmath113 is the average with respect to the quantum state of the field . following landau , we define the fluid four - velocity @xmath114 and energy density @xmath115 as the ( only ) time - like eigenvector of @xmath116 and ( minus ) its corresponding eigenvalue    @xmath117    introducing the pressure @xmath118 as given by the equilibrium equation of state ( our theory does not have a conserved particle number current , and therefore the equation of state is barotropic ) , we may decompose    @xmath119    where @xmath120 and by construction @xmath121 . since @xmath122 vanishes by definition in the equilibrium state , it may be parametrized in terms of deviations from equilibrium . remaining within the so - called _ first order formalism _ @xcite , we may write    @xmath123    where    @xmath124   \label{hmunu}\ ] ]    and @xmath125 and @xmath126 are the _ shear _ and _ bulk _ viscosity coefficients , respectively . let us decompose each quantity in a mean component ( denoted by a @xmath95 subscript ) and a fluctuation ( denoted by a @xmath127 ) . if the quantum state shares the symmetries of the frw background , then @xmath128 . since @xmath129 holds identically ( as oppossed to  in the mean  ) we must have    @xmath130    @xmath131 = 0\ ] ]    the second equation shows that @xmath132 is a higher order fluctuation with respect to @xmath133 . if we remain at linear order , then , we may approximate @xmath134 observing that all mean values are homogeneous and isotropic , we see that @xmath135 is also a higher order fluctuation . we find    @xmath136    and therefore the velocity correlation    @xmath137    this is the key equation linking the quantum and stochastic descriptions . to estimate the velocity correlation , let us assume that , after integrating out the hard modes , the soft modes of interest may be described in terms of quasi free , long lived excitations with an effective mass @xmath138 then    @xmath139    the fluctuations are gaussian to a very good approximation , and therefore    @xmath140    where @xmath141 is the positive frequency propagator    @xmath142    ( @xmath143 stand for derivatives with respect to the first argument , while @xmath144 stand for derivatives with respect to the second argument of @xmath145 ) . let us decompose the soft field into modes    @xmath146    at any time @xmath147 we may introduce positive frequency adiabatic modes defined by @xcite    @xmath148    where    @xmath149    and decompose the mode amplitude @xmath150 into positive and negative frequency components    @xmath151    @xmath152    let us define the spectrum    @xmath153    ( because the quantum state is isotropic , the spectrum depends only on @xmath78 ) and assume that    @xmath154    ( this happens , for example , if the different modes acquire random phases through interaction with an environment ) . then    @xmath155    and    @xmath156    observe that only the vacuum part contributes in the coincidence limit @xmath157 . in the large occupation numbers regime we are interested in , this is negligible , and we get    @xmath158    where    @xmath159    @xmath160    we may now write down the fourier transform of the velocity self correlation    @xmath161    in principle , this is an integral equation relating the spectrum to the energy self - correlation . it may be simplified as follows . for small @xmath162 , the integral reads    @xmath163    while for large @xmath34 we get    @xmath164    since the spectra we are considering fall off much faster than @xmath165 ( or even @xmath166 ) at large @xmath167 we may estimate that the contribution from eq . ( [ smallp ] ) is larger than eq . ( [ largep ] ) ; moreover , extending the integral to the whole momentum space ,    @xmath168    and we get    @xmath169    finally , we may relate the particle spectrum to the turbulent energy spectrum    @xmath170 in the previous sections , we have analyzed on one hand self similar turbulent flows in expanding universes , and on the other hand have given the rule to translate the turbulent energy spectrum into a particle number distribution . we must now show that the foregoing analysis is relevant to plausible models of the reheating period , and use it to predict the likely shape of the final particle spectrum .    at this point , it is convenient to be more precise about the model of reheating we have in mind ( for further details , see papers 1 - 5 ) . we assume that inflation is driven by an inflaton field @xmath171 with an effective potential which may be parameterized as @xmath172 the self coupling @xmath173and at the end of inflation @xmath174 ( in natural units ) . this is the dominant contribution to the energy density , so it fixes the scale for the hubble constant @xmath90 : @xmath175 .    during preheating , a large fraction ( nearly all ) of this energy is transferred to the matter fields . these are described by an effective degree of freedom @xmath104 which self interacts with an effective dimensionless coupling constant @xmath176 for simplicity , we shall model this self interaction as a @xmath177 theory . at times @xmath147 at the end of preheating , matter excitations are distributed with a smooth spectrum @xmath178 , where @xmath78 is a comoving wave number and the characteristic momentum @xmath179 ( meaning that the relevant modes are inside the horizon ) . we wish to find the form of the spectrum as follows from the assumption that its further evolution will be self - similar . let us estimate the energy density in the matter fields at @xmath147 as @xmath180 . the self interaction of the matter fields induces a mass @xmath181 . we shall assume @xmath182 in which case    @xmath183    we shall assume that this is the dominant contribution to the matter self - energy . the condition @xmath184 is equivalent to    @xmath185    ( this equation will be important below ) . observe that    @xmath186    since @xmath187 is already less than one , this is not unduly restrictive on @xmath188 ( see papers 1 - 5)@xmath189 also observe that @xmath190 redshifts with the cosmological evolution , so these estimates are not affected by hubble flow . integrating equation ( [ fluid2quantum ] ) over @xmath78 , we find the mean velocity in the equivalent fluid flow as    @xmath191    where    @xmath192    ( @xmath193 ) is the speed of sound , so the flow may be regarded as incompressible . the shear viscosity of the matter fields may be estimated as ( see appendix ) @xcite    @xmath194    leading to a kinematic viscosity    @xmath195    with a typical velocity of order one , and a typical lenght of order @xmath196 we get the reynolds number @xmath197 ( cfr . ( [ lambdan ] ) ) .    of course , we do not have a microscopic justification for heisenberg s closure , so we shall take @xmath77 in eq . ( [ curvedchandra ] ) as a free parameter . as it turns out , agreement with the numerical results reported by papers 1 - 5 is obtained for @xmath198 which , as expected , corresponds to reynolds numbers of order of one . it s worth to point out that chandrasekhar s solution with the function @xmath60 obeying eq . ( [ curvedchandra ] ) is an exact solution that relies only on heisenberg s closure hypotesis eq . ( [ heis1 ] ) . so if we trust this hypotesis , we can get the spectrum by solving eq . ( [ curvedchandra ] ) for any reynolds number . the distribution of occupation numbers is found from eq . ( [ fluid2quantum ] ) , where the energy spectrum is given by eq . ( [ speccur1 ] ) . in the light field limit @xmath199 we find @xmath200 . ( [ fsmall ] ) implies @xmath201 for @xmath202 and eq . ( [ flarge ] ) implies @xmath203 for @xmath204 . this theoretical prediction for the exponents involved is the main result of this paper .    in fig . 1 we show the full particle spectrum based on the approximation eq . ( [ fapprox ] ) for the function @xmath60 . we have scaled the plot to make it easiest to compare with papers 1 - 5 . momentum is measured in units of @xmath205 ( where @xmath206 is the scale factor ) @xcite , and we have chosen @xmath207 observing that in the ultrarelativistic limit the speed of flow @xmath208 and @xmath209 , integrating the particle density times @xmath210 shows that the total energy density in the flow is @xmath211 . the equivalent black - body temperature is then somewhat less than @xmath212 , which is a reasonable value for the reheating era , and high enough to justify the neglect of all masses . the analysis leading to the chandrasekhar solutions begins from the navier - stokes equations for the fluid , which are equivalent in this context to energy - momentum conservation . therefore our model requires that it be possible to assign to the fluid an independently conserved energy - momentum tensor . this is not exactly the same as requiring that the homogeneous mode has totally decayed , but it does mean that there is no significative energy transfer from the homogeneous mode to the fluid , either through parametric amplification or otherwise . the plots of total particle density and effective masses presented in ref . @xcite suggest that this condition is met early , then follows an intermediate stage dominated by rescattering , and finally the thermalization stage . comparison with the results in papers 1 - 5 is meaningful only in the intermediate phase . decaying turbulence is necessarily a transient phenomenon . as time evolves , we expect the field will eventually thermalize , and the spectrum will get closer to a rayleigh - jeans tail , @xmath213 when masses are negligible . the several plots presented in ref . @xcite , where indexes go from @xmath214 to slightly over @xmath215 , capture the transition from turbulence to equilibrium . since the same plots show that earlier spectra are steeper ( see also fig . 1 in ref . @xcite ) this is in satisfactory agreement with the prediction from self - similar flows . fig . 1 : log - log plot of the particle spectrum , as given by eq . ( [ fluid2quantum ] ) . the energy spectrum is given by ( [ speccur1 ] ) , where the function @xmath60 is given by eq . ( [ fapprox ] ) . we have chosen @xmath216 , the taylor microscale @xmath217 , and have scaled the spectrum to make it easiest to compare against the results presented in @xcite and @xcite . in this paper , we have shown that the self similar flows studied by heisenberg , chandrasekhar and tomita may be used to provide an interpretation of the  turbulent  spectra found in papers 1 - 5 . the hydrodynamic model predicts scale invariant spectra @xmath218 both in the infrared and ultraviolet limits , with @xmath219 in the former , and @xmath220 in the latter regime . agreement with the early time results presented in papers 1 - 5 is satisfactory . the connection of hydrodynamics to the behavior of fluctuations during reheating has interest of its own , as it provides an alternative to brute force quantum field theoretic calculations , and also yields physical insight on the macroscopic behavior of quantum fields in the early universe . the equivalent fluid method may be used to advantage also in other regimes , such as the inflationary period itself @xcite . moreover , it opens up a wealth of new phenomena , such as intermittence @xcite and shocks @xcite , which are not apparent in the customary treatments . we will continue our research in this field , which promises a most rewarding dialogue between cosmology , astrophysics , and nonlinear physics at large . we thank mario castagnino for rekindling our interest in the subject . ec has been partially supported by universidad de buenos aires , conicet , anpcyt under project pict-99 03 - 05229 and fundacin antorchas . mg has been partially supported by national science foundation grant phy97 - 22022 . in order to estimate the viscosity for the matter fields we assume that the correspondence eq . ( [ fluid2quantum ] ) between the fluid and particle spectra allows us to associate a solution of the hydrodynamical evolution equations for the former to a self - similar solution of the kinetic equation for the latter ( cfr . @xcite ) . we may then estimate the transport coefficients by adopting the same methods usually applied in equilibrium @xcite . these are discussed in detail , in the quantum field theory context , in references @xcite . @xmath224 & \sim & g^{2}\int dk_{2}dk_{3}dk_{4}\;\delta \left ( k_{1}+k_{2}-k_{3}-k_{4}\right )   \nonumber \\ & & \left\ { \left ( 1+n_{k1}\right ) \left ( 1+n_{k2}\right ) n_{k3}n_{k4}-\left ( 1+n_{k3}\right ) \left ( 1+n_{k4}\right ) n_{k1}n_{k2}\right\}\end{aligned}\ ] ]                                          there is a vast literature on preheating . some representative works are l. kofman , a. linde and a. starobinsky , phys . lett * 73 , * 3195 ( 1994 ) ; y. shtanov , j. traschen and r. brandenberger , phys . rev . * d51 , * 5438 ( 1995 ) ; d. boyanovsky , h. de vega , r. holman and j. salgado , phys . rev . * d54 , * 7570 ( 1996 ) ; s. ramsey and b. l. hu , phys . rev . * d56 , * 678 ( 1997 ) ; p. greene , l. kovman , a. linde and a. starobinsky , phys . rev . * d56 , * 6175 ( 1997 ) ; b. bassett , d. kaiser and r. maartens , phys . lett . * b455 * , 84 ( 1999 ) ; b. bassett , c. gordon , r. maartens and d. kaiser , phys . rev . * d61 , * 061302 ( 2000 ) ; f. finelli and s. khlebnikov , e - print hep - ph/0107143 . s. jeon , phys . rev . * d47 * , 4586 ( 1993 ) ; s. jeon , phys . rev . * d52 * , 3591 ( 1995 ) ; s.jeon and l.g.yaffe , phys . rev . * d * * 53 * , 5799 ( 1996 ) ; e. calzetta , b. l. hu and s. ramsey , phys . rev . * d61 , * 125013 ( 2000 ) . s. chapman and t. cowling , _ the mathematical theory of non - uniform gases _ ( cambridge university press , london , 1990 ) ; r. liboff , _ kinetic theory _ ( john wiley , new york , 1998 ) ; k. huang , _ statistical mechanics _ ( john wiley , new york , 1987 ) .
we show that the  turbulent  particle spectra found in numerical simulations of the behavior of matter fields during reheating admit a simple interpretation in terms of hydrodynamic models of the reheating period . 
 we predict a particle number spectrum @xmath0 with @xmath1 for @xmath2 principal pacs no 98.80.cq ; additional pacs nos : 04.62.+v , 11.10.wx , 47.27