Uncertainty relations for cosmological particle creation and existence of large fluctuations in reheating
aa r X i v : . [ g r- q c ] D ec Uncertainty relations for cosmological particle creation andexistence of large fluctuations in reheating
Ali Kaya ∗ Bo˜gazi¸ci University, Department of Physics,34342, Bebek, Istanbul, Turkey
We derive an uncertainty relation for the energy density and pressure of aquantum scalar field in a time-dependent, homogeneous and isotropic, classicalbackground, which implies the existence of large fluctuations comparable totheir vacuum expectation values. A similar uncertainty relation is known tohold for the field square since the field can be viewed as a Gaussian randomvariable. We discuss possible implications of these results for the reheatingprocess in scalar field driven inflationary models, where reheating is achievedby the decay of the coherently oscillating inflaton field. Specifically we arguethat the evolution after backreaction can seriously be altered by the existenceof these fluctuations. For example, in one model the coherence of the inflatonoscillations is found to be completely lost in a very short time after backreactionstarts. Therefore we argue that entering a smooth phase in thermal equilibriumis questionable in such models and reheating might destroy the smoothnessattained by inflation.Essay written for the Gravity Research Foundation 2011 Awards for Essays on GravitationReceived Honorable Mention ∗ [email protected] Our universe is very special. The smoothness of the cosmic microwave background (CMB) radiationis the prominent evidence for this assertion. It is remarkable that CMB photons were in thermalequilibrium on scales much larger than the horizon size at the time of decoupling. Therefore, causalityprecludes thermalization of the CMB radiation by local interactions and the problem we are facingis completely different than a box of gas reaching thermal equilibrium. Actually, there is a deeperdifficulty since in the standard cosmological model without inflation, most of the CMB photons didnot have a chance to causally communicate at all.Inflation claims to solve this problem by exponentially enlarging a small causal patch, whichlater becomes our observed universe. However, when inflation ends one does not observe thermalequilibrium right away. Rather, in the scalar field driven models for example, one finds an extremelysmooth, flat universe filled only with a coherently oscillating inflaton field; all rest that existed beforeinflation are red-shifted away by the exponential expansion. Thermal equilibrium must be reachedby the interactions accompanying the decay of the inflaton field.This shows that one still needs to understand the thermalization process in an expanding universe,which is a very non-trivial and difficult problem to study (see [1]). To have a correct picture in thissetup, it is very crucial to determine the state of the universe just before the thermalization starts.Our aim in this essay is to point out some important features of quantum particle creation in aclassical background, which must be taken into account during reheating and before thermalization.The particle creation process during reheating is usually studied using Fourier decomposition andmomentum modes (see e.g. [2]). While it is perfectly legitimate to use momentum modes, or inessence any complete set of modes, one must be careful about a few potential issues. Firstly, causalityrequires the particle creation process to take place independently in each horizon. Thus, in physicallyinterpreting the creation of a globally defined momentum mode one must ensure that locality isnot broken down by hand. Secondly, there is a subtlety in the definition of the number densityof a completely dislocalized momentum mode. Finally, to make sense of local physical quantitiesa suitable regularization must be utilized and this may also change some results obtained in themomentum space. An important example of this kind is the modification of the power spectrumby adiabatic regularization, studied recently in [3]. All these issues can be bypassed if the particlecreation effects are described by giving the vacuum expectation value of the energy-momentumtensor, which we adapt in the following.Consider now a real scalar field χ propagating in a Friedmann-Robertson-Walker metric ds = − dt + a ( dx + dy + dz ) , (1)which has the following action S = − Z √− g h ( ∇ χ ) + M χ i . (2)We assume that the mass parameter M may also depend on time: M = M ( t ), and therefore particlecreation effects occur due to the time dependence of the scale factor a and the externally varyingmass parameter M . For quantization, it is convenient to define X = a / χ , and introduce the Fouriermodes and time-independent creation-annihilation operators as X = 1(2 π ) / Z d k h a k X k e − i k . x + a † k X ∗ k e i k . x i , (3)where [ a k , a † k ′ ] = δ ( k − k ′ ), X k obeys the Wronskian condition X k ˙ X ∗ k − X ∗ k ˙ X k = i and ¨ X k + ω k X k = 0with the frequency defined as ω k = M + k a − H − ˙ H .The ground state of the system is defined as a k | > = 0. It is a straightforward exercise to calculatethe vacuum expectation value of the energy momentum tensor < T µν > = < | T µν | > corresponding to the χ -particles created on the background. By noting that the energy density and the pressureof the χ -particles are given by ρ ( t, x ) = 12 ˙ χ + 12 g ij ( ∂ i χ )( ∂ j χ ) + 12 M χ ,P ( t, x ) = 12 ˙ χ − g ij ( ∂ i χ )( ∂ j χ ) − M χ , (4)one can obtain < ρ > = T + V + G, < P > = T − V − G, (5)where T , V and G are given by T = 12(2 πa ) Z d k | ˙ X k − HX k | ,V = M πa ) Z d k | X k | , (6) G = 12(2 πa ) Z d k k a | X k | . Also, it is of some interest to determine the field variance < χ > = < | χ ( t, x ) | > , which can befound as < χ > = 12(2 πa ) Z d k | X k | . (7)It is clear that T and V can be interpreted as the kinetic and the potential energies, respectively,and G can be viewed as the energy stored in the gradient of the χ -field. In general all the aboveexpectation values diverge and thus a suitable regularization must be applied. After regularization,however, the functions T , V , G and < χ > can be made finite. Note that since the background ishomogenous the vacuum expectation values (5) and (7) do not depend on the spatial coordinates.The above expressions only give the average values of the corresponding physical quantities andone would expect to determine (quantum) fluctuations about these averages. In [4], we explicitlycalculate them. For example, after defining the fluctuation operators δρ = ρ ( t, x ) − < ρ > and δP = P ( t, x ) − < P > , a relatively long but straightforward calculation, which uses (3) and (4),gives < δρ > + < δP > = 4 T + 4 V + 2027 G . (8)From (5) this result shows that the deviations in the energy density and the pressure are always of the order of the average energy density < ρ > . Similarly, although their vacuum expectationvalues vanish one can see that the momentum density and the stress tensor have non-zero variancesproportional to T G and G , respectively [4]. Thus, if T and G are not small compared to V ,the deviations in the momentum density and the stress can also be as large as < ρ > . Eq. (8)can be viewed as an uncertainty relation involving energy density and pressure, where the order ofmagnitude of the uncertainty is fixed by the average energy density.It is easy to see that there also exists order one fluctuations in χ , i.e. the field variance has alarge variance. Defining δχ = χ ( t, x ) − < χ > , one can easily calculate q < ( δχ ) > = √ < χ > . (9) This result directly follows from the fact that χ ( t, x ) is a Gaussian random variable defined at thepoint x and (9) is true for any Gaussian distribution.The fluctuations in (8) and (9), which are actually defined pointwise, should be interpreted as thestatistical averages over different points in space at a given time. However, it is crucial to realizethat the field variables at nearby points are not statistically independent . The (comoving) size ofa typical region containing correlated field variables is given by the correlation length ξ c , whichis the minimum value of | x | such that the two-point function vanishes < χ ( t, x ) χ ( t, > = 0. Thecorrelation length is one of the most important parameters characterizing quantum fluctuations. Onecan imagine the space to be divided into regions of typical size ξ c , and in each region the physicalquantities ρ , P and χ can be thought to be uniformly distributed. The variations given in (8) and(9) characterize changes from one such region to another.It is interesting to compare these findings with [5, 6] where we study the particle creation processusing a complete orthonormal family of localized wave-packets and find out existence of fluctuationsin the number and the energy densities of particles produced in any given volume V . As noted in[5, 6], if the volume V is sufficiently large, the relative deviations of these perturbations fall downlike 1 / √ V . This behavior can easily be understood since the total deviation is due to V /v numberof statistically independent random variables, where v is the volume occupied by each randomvariable, which can be fixed in terms of the correlation length as v = ξ c .One may now wonder if the presence of these fluctuations can have any cosmological significance.Here, we consider the reheating process in a single scalar field model. Assume that near its minimumthe potential of the inflaton field takes the form m φ , where m is the inflaton mass. In that casethe inflaton φ ( t ) and the scale factor of the universe a ( t ) can be determined as φ = Φ( t ) sin( mt ) , Φ( t ) ≃ Ct , a ≃ (cid:18) tt (cid:19) / . (10)The pressureless dust equivalent expansion in (10) can be understood as follows: As the inflatonoscillates about its minimum, the kinetic and the potential energy terms consecutively dominate theenergy-momentum tensor. While the kinetic energy dominance can be described by the equation ofstate P = ρ the potential energy dominance implies P = − ρ . Thus on the average one gets P = 0.We further focus on a model where the inflaton decays to a minimally coupled χ -boson via theinteraction L int = − g φ χ or L int = − σφχ . (11)The first term can give a decay in the broad parametric resonance regime and the second oneis usually considered in the perturbative decay. In either case, at least in the first stage untilbackreaction becomes important the decay process can be described as the χ -particle creation in atime dependent classical background . One can thus apply the above formulas about particle creationwhere the time-dependent mass parameter M can be fixed as M = g φ or M = σφ .In [4], we determine the correlation length ξ c in these two different reheating models correspondingto the decay in the broad parametric resonance regime and in the perturbation theory. In the formercase, ξ c is fixed by the (comoving) momentum scale k ∗ of the first instability band as ξ c ∼ /k ∗ . Onthe other hand, in perturbation theory ξ c is determined up to the scalings due to the expansion of theuniverse as ξ c ∼ /m . In obtaining both results one simply uses the spectrum of particles producedduring the decay in calculating the two-point function. Importantly, in each case the correlationlength turns out be smaller than the Hubble scale.The above formulas can be used until backreaction effects become important. Prior to backreaction,linearity is essential and the momentum modes evolve independently without disturbing locality. However, as the created particles start influencing the background, the linearity is lost and onecan no longer treat momentum modes separately. Namely, to determine how the created particlesalter the evolution of the background, which necessarily obeys local field equations , the effects ofall the modes must be summed up. Therefore, it is crucial to determine how physical quantitiescorresponding to particle creation behave in the real configuration space.Consider, for example, the model given by the first interaction term in (11), which modifies theinflaton field equation as ¨ φ + 3 H ˙ φ + m φ − g ij ( ∂ i ∂ j φ ) + g χ φ = 0 , (12)where we also introduce the gradient term for completeness. The backreaction effects become impor-tant when the χ -field grows so that g χ ∼ m . In the Hartree approximation one replaces χ termin (12) with the expectation value < χ > . Since < χ > does not depend on spatial coordinatesand since initially (i.e. just after the end of inflation) the φ -field is homogenous, only the zero mode continues to exist. Thus, in this approximation one can ignore the spatial derivatives in (12) andthe φ -field remains to be homogenous.However, as noted above in (9) there exists large fluctuations in χ , comparable to its vacuumexpectation value < χ > and thus the ”actual value” of χ varies appreciably on scales larger thanthe correlation length ξ c . To determine the evolution of the inflaton field in the presence of thesefluctuations, one can examine (12) in regions of volume ξ c , in which χ can be taken as uniform,and try to glue these local results suitably. It is clear that if only the zero mode survives even whenthe fluctuations are taken into account, then this should be a good approximation.From (12), the frequency of the oscillations in the i ’th region is given ω i ) = m + g χ i , where χ i denotes the value of χ in that region and we ignore the expansion of the universe since in general m ≫ H during reheating. This shows that due to fluctuations in χ given by (9) the frequenciesalso start to differ as δωω ∼ δχ χ ∼ . (13)In a very short time nearly corresponding to a single average oscillation t ∼ /ω ∼ /m , theoscillations of the inflaton field in different regions become completely out of phase. In that case,it is no longer permissible to ignore spatial derivatives in (12) and the whole reheating dynamicschanges completely.Actually, the story is more complicated since χ is a random variable. Consider, for instance, thesituation when g < χ > is about 5% percent of m or so, and thus backreaction effects can beignored on the average . However, even at that time one can find regions in which g χ ∼ m andthus coherence of the oscillations is lost near such regions. In other words, one cannot actually talkabout a definite time after which backreaction effects become important.The above conclusion is specific to the interaction g φ χ and one may wonder if it also holds forother models in general. For example, when the inflaton decays through the interaction σφχ , thebackreaction term does not modify the frequency of the inflaton oscillations. But this time, thereappears a non-homogenous source term σχ in the inflaton field equation which alters the amplitudeof the oscillations locally. One can see that the fluctuations in χ induce order one changes in theamplitude and again the gradient terms cannot be neglected in the inflaton field equation.Thus, we find out that in reheating models characterized by the interactions (11), the smooth-ness of the inflaton background is completely lost and the subsequent evolution must be determinedby taking into account the field derivatives. One may think that since the length scale of inho-mogeneities is small compared to the cosmologically relevant scales today, they cannot have anycosmological imprints. However, it is not possible to view these fluctuations as small perturbations on a homogenous background. Namely, they are expected to alter the evolution of the background ina very non-trivial way (for instance the gradient terms may start dominating the energy-momentumtensor) and the universe may transform into a state which is completely different than a homogenousand isotropic Robertson-Walker space.One may wonder what can be said for other reheating models in general. Until backreactioneffects become significant, (8) holds in any model of reheating provided that the decay process canbe modeled as the quantum particle creation in a classical background. Therefore, in such modelsthere inevitably exists order one fluctuations in the energy density and the pressure. As is known, insome cases the universe can still be described as a perturbed Robertson-Walker metric even though δρ/ρ > O (1). However, the fluctuations corresponding to the particle creation have interestingproperties. For example, it is not possible to characterize them by an equation of state [4] andconsequently the combination ρ + 3 P , which determine ¨ a , would change locally causing differentregions to accelerate with different rates. Moreover, since the size of each region is subhorizonand there exists large pressure gradients (as shown in [4]), the gravitational collapse can occurexponentially fast as in the case of Jeans instability in Newtonian theory. As a result, one expectsthese fluctuations to grow and the universe may not enter into a smooth phase in thermal equilibrium.In the above discussion we left over some important issues which must actually be studied in real-istic scenarios. It is known that for completeness the metric and the inflaton fluctuations must alsobe incorporated in the linearized field equations which may alter the particle creation characteristics[7]. In that case, however, it is not surprising to see the existence of large fluctuations of these fieldsas well. On the other hand, one may also need to consider the rescattering or the thermalizationeffects, which may work for uniformization. There are also some conceptual problems one shouldsolve. For example, in (12) while φ is a classical field it is not obvious what χ2