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GRAVITATIONALLY INDUCED SCALAR FIELD FLUCTUATIONS

arXiv:hep-ph/9208246v1 25 Aug 1992IFT-15/92GRAVITATIONALLY INDUCED SCALAR FIELD FLUCTUATIONSIN THE RADIATION DOMINATED R-W UNIVERSE∗Zygmunt Lalak1, Krzysztof A. Meissner2, Jacek Pawe lczyk3Institute of Theoretical PhysicsUniversity of WarsawWarsaw, PL 00-681ABSTRACTIt is shown that quantum fluctuations due to a nontrivial gravitational backgroundin the flat radiation dominated universe can play an important cosmological role gen-erating nonvanishing cosmological global charge, e.g. baryon number, asymmetry.

Theexplicit form of the fluctuations at vacuum and at finite temperature is given. Impli-cations for particle physics are discussed.1EMAIL:LALAK@FUW.EDU.PL2EMAIL:MEISSNER@FUW.EDU.PL3EMAIL:PAWELC@FUW.EDU.PL∗Work partially supported by Polish Government Research Grant GR-11.

1. IntroductionSince the early eighties it has been widely recognized that quantum fluctuations ofscalar matter fields may play an important role in cosmology, especially in the contextof the inflationary de Sitter epoch [1].

Actually, the case of the scalar field in the deSitter space has been the most extensively and carefully studied one. The reason isthat in the de Sitter space fluctuations of the light fields (m2/H2I << 1, where HI is thede Sitter Hubble parameter) grow linearly with time assuming finally a significantlylarge value of the order ofH4Im2,which singles out the de Sitter universe.

It is believedthat fluctuations produced at the time of inflation are seen during subsequent stagesof the evolution of the universe as energy density inhomogeneities responsible for theformation of the large scale structure. It is also advocated that those fluctuations setinitial conditions for the classical evolution of fields in subsequent epochs.In contrast to the above, it is usually assumed that gravitationally induced scalarfield fluctuations in spatially flat radiation dominated (RD) and matter dominated(MD) epochs are irrelevant for particle cosmology.

We would like to point out thatthis assumption is not properly discussed in the literature. On one hand, one observesthat in the RD universe the fluctuations (as explained in this letter) decrease in time.On the other hand, they may in principle be large enough to control violation of somesymmetries or to alter the evolution of some fields present in field-theoretical models.We want to stress that this problem becomes particularly important in view of theongoing search for a reliable mechanism for production of the baryon asymmetry inthe Universe, the need of the better understanding of the scenarios for late-time phasetransitions and discussions of the possible lepton number nonconservation.In this letter we would like to address explicitly the problem of quantum fluctuationsof the massive scalar field during the RD epoch.

This epoch covers most of the historyof the universe, and the temperature range from, say, 1014 GeV down to 10 eV. Onthat energy scale one can find a lot of interesting phenomena in popular extensionsof the standard model such as its supersymmetric version or string inspired models,what, in our opinion, justifies the research reported in this work.The paper is organized as follows.

In Section 1 we set our notation and subsequentlyevaluate fluctuations of a massive scalar field in the RD flat Robertson-Walker spaceat vacuum and at finite temperature. In Section 2 we apply our formulae to a generalfield theoretical model with particular attention paid to two specific examples resem-1

bling the Affleck-Dine model [2] and the so called spontaneous baryogenesis scenario[3]. Finally, in the last Section we review our results and present conclusions.1.

Scalar field fluctuations in the RD universeThe RD Universe is the solution to the Einstein’s equations with the energy-momentum tensor in the form T µν = diag(−ρ, p, p, p). Tracelessness of the T µν impliesequation of state for the content of the RD Universe: ρ = 3p.

In this letter we assumea flat RD space endowed with the metric gµν = diag(−1, a2(t), a2(t), a2(t)) where a(t)is the RW scale factor given by a(t) ≡(u)1/2 with u defined as u ≡t/t0, t0 being thebeginning of the RD epoch.We couple a massive scalar field to gravity in the minimal way ( note that here thecurvature scalar R vanishes identically )S[φ] =Zd4x√−g−gµν∂µφ∂νφ −m2φ2. (1)As usual in this type of analysis we assume that there is no “back-reaction” of thescalar field on the metric, cf.

[4]. The equation resulting from eq.

(1) is d2du2 + 32uddu + k2t20u+ m2t20!φk(u) = 0(2)where φk(u) is the spatial Fourier transform of the field φ(⃗x, t),φ(⃗x, t) =Zd3k(2π)32kφk(u)ei⃗k⃗xa†k + h.c.(3)with a† and its hermitian conjugate denoting standard creation and anihilation opera-tors respectively.A general solution of the equation (2) is given by confluent hypergeometric functionsφk(u)=A1(k, m)2ikt0e−imut01F1 3/4 + ik2t02m , 3/2, 2imt0u!+A2(k, m) 1√ue−imut01F1 1/4 + ik2t02m , 1/2, 2imt0u!. (4)The two undetermined coefficients A1 and A2 , which may in principle depend on bothm and k, are not independent if one takes into account quantization condition imposed2

on a field φ[φ(⃗x, t), ∂tφ(⃗y, t)]=i√−gδ(3)(⃗x −⃗y)(5)hak, a†k′i=(2π)32kδ(3)(⃗k −⃗k′)(6)From above equations and decomposition (3) we get a normalization conditionIm(¯φk(u)∂tφk(u)) =k√−g(7)which translates into the constraint on A1 and A2 ( we confine them to be real )A1(k, m)A2(k, m) = 1(8)In most general case there are several ways of fixing both coefficients. One possibilityis to use initial conditions set at the timelike surface t = t0 for φ and ∂tφ.

This isthe proper procedure if one knows for example the explicit solution for φ in the epochpreceding the RD one. We do not assume such a detailed knowledge, hence we usean alternative approach instead.

We demand that the “correct” mode functions wechoose, which will define our Fock space, should approach at short distances (k →∞)the massless positive frequency solution,φk(u) →1u1/2e2ikt0√u(9)In this way we obtain the asymptotic behaviour of both coefficientsA1,2(k, m) →1(10)Here we assume that A1 = A2 = 1, what completes the definition of our Fock space.We set out to calculate the fluctuations of the field φ i.e. < 0|φ2|0 >.

This quantityis badly divergent and needs renormalization. We define the renormalized fluctuationsas the difference between RD and the Minkowski space fluctuations, hence the relevantobject to look at is the difference< φ2 >R≡< φ2 > −< φ2 >M(11)As we shall see, this definition gives the finite result.

It may easily be checked thatin terms of Fourier modes φk(u) the renormalized fluctuations (11) are given by theformula< φ2 >R=Zd3k(2π)312k|φk(u)|2 −12u3/2qk2/u + m2,(12)3

Using (4) we can write down the explicit formula< φ2 >R=14πt2Z ∞0dy"y1F1 14 + iy22mt, 12, 2imt!+ 2iy 1F1 34 + iy22mt, 32, 2imt!2−y2√y2 + m2t2#(13)where y = kt/√u. Unfortunately, the above expression cannot be evaluated in its mostgeneral form.

However, it is possible to write down the systematic expansion of themode functions (4) and the integral (12) in terms of mt. Using such an expansion wewill be able to discuss reliably fluctuations in the regime of small mass and to controlthe passage to the massless limit.

For the region mt > 1 we will have to rely on thenumerical calculations.In the case mt < 1, the relevant expansion of modes is given by (cf. [5])φk(u) = 1u∞Xn=0hp(−1/2)n(2imt)jn−1(2y) + ip(1/2)n(2imt)jn(2y)i1(2y)n−1(14)The jn is the n-th spherical Bessel function and coefficient p(µ)ncan be read from∞Xn=0p(µ)n (z)wn = ez/2(coth(2w)−1/2w) zwsinh(zw)!1−µ(15)One easily finds that|φk(u)|2 = 1u"1 + (mt)2 sin2(2y)8y4−12y2!+ O((mt)4)#(16)On the basis of the expansion we see that < φ2 > is the ultraviolet-finite quantity.

Itis also infrared finite, since the modes are perfectly regular functions for k →0 (i.e.y →0).In the region mt < 1, with help of the expansion (16), we get the following formulafor the leading behaviour:< φ2 >= m28π2−ln(mt) + (3/2 −γ −ln2) + O((mt)2)(17)where γ is the Euler constant. One should note that the fluctuations vanish as mapproaches zero and grow with m if we keep mt constant.

This agrees with the earlierresult for an exactly massless field reported in ref. [6].The interesting feature of4

the formula (17) is its non-analyticity in mt and the appearance of the logarithmicsingularity at t = 0 which is related to the singularity of the RD Universe at t = 0.In the regime mt > 1 the numerical study we have performed shows that the integralin (13), which is solely a function of mt, has an oscillating behaviour with the periodclose to π/m and amplitude rising with increasing mt. One can easily check that theintegral (13) is converging rather quickly, hence in a given interval of the variable mtone can cut-offthe integration from above at a numerically determined value Λ. Giventhis observation, it is straightforward to find an algebraic approximation for the integralas a function of mt.

Using the expansion of the Kummer hypergeometric function interms of the modified Bessel functions of the second kind (cf. [7])1F1(a, b, x)=e12 xΓ(b −a −12)(14x)a−b+ 12×∞Xn=0(2b −2a −1)n(b −2a)n(b −a −12 + n)n!

(b)n×(−1)nIb−a−12+n(12x)(18)and expanding the definite integrals in powers of Λ2/mt one obtains in the regionmt > Λ2 the expression of the form< φ2 >R = 1t2√mt(a sin(2mt) + b cos(2mt) + c + o(Λ2/mt))(19)where the coefficients a, b, c area =1(√2π)3Z Λ20dxe−2πx Re(f1(x)f ∗2(x))(20)b =12(√2π)3Z Λ20dxe−2πx (|f1(x)|2 −|f2(x)|2)(21)c =12(√2π)3Z Λ20dxe−2πx (|f1(x)|2 + |f2(x)|2) −13√2π2Λ3(22)with functions f1, f2 defined as followsf1(x)=Γ(−14 −ix)∞Xn=0(−1/2 −2ix)n(1/4 −ix + n)(−2ix)nn! ( 12)ncos π/2(n + 1/4 −ix)+4√xΓ(14 −ix)∞Xn=0(1/2 −2ix)n(1/4 −ix + n)(−2ix)nn!

( 32)ncos π/2(n + 3/4 −ix)(23)5

f2(x)=Γ(−14 −ix)∞Xn=0(−1/2 −2ix)n(1/4 −ix + n)(−2ix)nn! ( 12)nsin π/2(n + 1/4 −ix)+4√xΓ(14 −ix)∞Xn=0(1/2 −2ix)n(1/4 −ix + n)(−2ix)nn!

( 32)nsin π/2(n + 3/4 −ix)(24)where an ≡an−1(a+n−1), a0 ≡1. For the reasonable value of Λ = 3.5 the least squaresfit to numerical data in the interval (4.0, 17.0) gives a = −5.9 10−4, b = 4.2 10−3, c =1.5 10−3.

We note the leading dependence of the < φ2 >R on m in this range of mt:the fluctuations grow proportionally to the square root of the mass.One should also note that the fluctuations < φ2 >R as defined here, see (11), (12),are not positive definite. Remarkably, it can easily be shown, that if one perturbsthe definitions of the functions A1,2(k, m) allowing for the appropriate dependenceon the ratiok2m2, which amounts to the modification of our Hilbert space, then theregions of an oscillating behaviour with negative values of < φ2 >R are pushed towardsincreasing values of the argument mt.

Since in this letter we do not discuss any specificmodification of the Hilbert space beyond the simple and most natural definitions (10)and (11), the non-positiveness of < φ2 >R demands special care when one considersphysical applications of the present result, as we do in the next Section. The point isthat a physically meaningful quantity which has the interpretation of the dispersionsquared should be strictly positive definite.

However, we do not rely on the oscillatingbehaviour of < φ2 >R in the discussion of applications of the present result beingonly interested in the overall time dependence √m/t3/2. This leading time and massdependence in the region mt > 1 we believe to be universal, hence we just neglect thescheme-dependent oscillating contributions in what follows.Up to now we have been calculating curved space vacuum expectation value ofφ2.

However, if we were to take into account that the Universe is “hot”, i.e. it isin fact in a mixed state to which many-particle states may contribute significantly,we should better calculate a thermal average of φ2, with finite temperature effectsincluded.

Assuming thermal equilibrium of the content of the Universe we have< φ2 > |T≥0 =Zd3k(2π)32k|φk|2(1 + 2nk)(25)where φk are modes given by (4), (10), and nk is the occupation number for the particles6

with the comoving momentum k. As nk we takenk =1exp( 1Tqk2/u + m2) −1=1exp(qk2/T 2R + m2/T 2) −1(26)which is correct for sufficiently large k in view of the choice (9). The (25) is againdivergent.However, as usual in finite-temperature calculations, it may be dividedinto T = 0 part and the temperature correction, among which only the former is UVdivergent.

Hence, we can use the renormalization procedure (11) to get meaningfulresults even at T > 0< φ2 > |renormalized,T≥0 →< φ2 >R+ < φ2 >TR(27)where< φ2 >TR= 2Zd3k(2π)32k|φk|21exp(qk2/T 2R + m2/T 2) −1(28)This expression may be approximated analytically in two limiting cases: a) mT >> 1,and b) mT << 1. In the case b) one easily gets< φ2 >TR= T 212(29)exactly as in the flat Minkowski case.

In the case a) one can see that in the regionwhich dominates the integral, k < mTRT , the second term in (4) is unimportant. Hencewe obtain< φ2 >TR= Γ(5/4)6.643/2π3(g∗)3/4 1H2I(TR/T)4 m5/2M3/2PT 3e−m/T cos2(mt −38π)(30)which is exponentially suppressed.2.

Implications for particle physics in the expanding UniverseLet us consider a global U(1) symmetry realized in a single complex scalar fieldmodel. If Q is the charge of the field χ, the Noether current associated with thatsymmetry isjµ = iQ{¯χ∂µχ −χ∂µ ¯χ}(31)7

(we put Q≡1 in what follows) and the conservation law for jµ in the expanding Universereads∂µ(a3(t)jµ) = −ia3(t){¯χ∂V∂¯χ −χ∂V∂χ }(32)One can see that a symmetry is broken once the rhs of (32) is nonvanishing. One cansee also that when a symmetry is broken explicitly, the net cosmological charge densitygets generated according to the formulaa−3 ddt(j0a3(t)) ≈−i{¯χ∂V∂¯χ −χ∂V∂χ }(33)Let us assume that the term violating the symmetry isδV =λ2nΛ2nφ2n(34)where φ ≡Re(χ) (in this section we assume the absence of derivative couplings, theywill be discussed later).Suppose that the initial conditions and the shape of thepotential are such that the Im(χ) and its fluctuations are negligible when comparedwith Re(χ) at any time t (this situation may be easily realized in the Affleck-Dinemodel, cf.

[8]). Hencea−3 ddt(j0a3(t)) ≈i λΛ2nφ2n(35)We can see that the magnitude of the symmetry violation is proportional to a couplingλ, inverse powers of some scale Λ if n > 2, and to some power of the scalar field φ.In this sense one can say that, λ and Λ being fixed in a given theory, it is the φ whatdetermines the amount of symmetry breaking.

Here the quantum fluctuations of thefield φ come into play. In the quasiclassical picture one can describe the evolution of thequantum field, lets call it Φ, writing it down as the superposition of the quasiclassicalfield φ which obeys essentially classical (perhaps perturbatively corrected) equation ofmotion and quantum fluctuations δφ, the dispersion squared of which we identify as< φ2 >R.

If the potential drives the quasiclassical field to zero, then it may happenthat the magnitude of the symmetry breaking term is determined by the dispersion ofδφ. Using < φ >2n = (< φ2 >R)n one gets an estimatea−3 ddt(j0a3(t)) = i λΛ2n(< φ2 >R)n(36)Of course, whether this term is really a dominant one or not, it depends on the relativemagnitude and time-dependence of < φ2 >R and the classical part of the field.

Weshall investigate this issue later in this work.8

At this point let us discuss the explicit form of < φ2 >R as a function of the tem-perature. We know that the time dependence for this quantity is given by (17) if t < 1mand by (19) if t >1m.

Let us say assume that the RD epoch starts at t = t0. Thenmt ≈mH(t0)a2(t)a2(t0), where H(t0) is of the order of the Hubble parameter during inflation,H(t0) ≈HI ≈1014 GeV.

Assuming then an adiabatic expansion in the RD epoch weget< φ2 >R ≈m22 ln(HIT 2mT 2R)(37)as long as T 2 >> T 2RmHI where TR is the reheating temperature after inflationTR = ( 454π3g∗)14min[(HIMP)12, (ΓIMP)12](38)(here g∗is the number of relativistic degrees of freedom at of reheating and ΓI is thetotal decay width of the inflaton field – a field which drives the transition from the deSitter to RD epoch). Let us take for simplicity the case of a “good” reheating whichcorresponds to TR ≈HI.

This gives the conditionT > T∗= (mHI)12(39)If we take m ≤102 GeV, HI ≈1014 GeV then we get T∗≤108 GeV. We note thatfor a really soft potential with m ≈10−21 eV, one has T∗≈10 eV which means thatin that case the regime where < φ2 >R changes only logarithmically extends over thewhole RD epoch.

For T < T∗we have instead of (37)< φ2 >R = O(1)g∗M2PT 4(40)(g∗is the number of relativistic degrees of freedom at temperature T).Now, let us investigate the evolution of the classical field φ.For simplicity weassume that this evolution is dominated by the mass term in the potential, which isusually a good assumption at least in the perturbative regime. In this case the generalsolution to the equation of motion isφ =1( mt2 )14 [CJ1/4(mt) + DJ−1/4(mt)](41)If we set initial conditions at t0 such that z0 = mt << 1, then C = Γ(5/4)φ0 + 2∂tφ0z0mand D = −4Γ(3/4)∂tφ0z03/2m23/2where φ0 = φ(mt0), ∂tφ0 = ∂tφ(mt0).

This gives the φ at9

late times, mt > 1, in the formφ ≈1√π(mt2 )−3/4[Γ(5/4)φ0 + 2z0∂tφ0m] cos(mt −3π/8)(42)From previous analysis we have learnt that for T < T∗the < φ2 >R falls offas√mtt2 ,hence we conclude that <φ2>Rφ2|T

Moreover, ingeneral the field φ has some additional couplings to light particles, which facilitate decayof the field φ with the decay width Γφ. This changes the behaviour of the classical fieldφ, namely φ2 →exp(−Γt) φ2.

Actually, as pointed out by several authors in the contextof the Affleck-Dine mechanism (which corresponds to our toy model when n=2) the Γφshould be large in order to avoid an unobservable excess of the net charge producedduring symmetry violation [8]. We want to stress that in such a case, the < φ2 >R,decaying accordingly to the power law, dominates the divergence of the Nother currentand the net cosmological charge density even at the late times.Let us check whether at T > T∗the < φ2 >R may be significantly large.

From (37)and (42) we obtain, averaging over oscillations,< φ2 >Rφ2|T>T∗≈2m2(Γ(5/4)φ0 + 2z0∂tφ0m)2(43)That means that fluctuations are important as long as the conditionφ0 + z0∂tφ0m< m(44)is fulfilled. One can see that even if m << HI and φ0 ∼HI, which happens to bethe case if the initial conditions at t = t0 are produced by large quantum fluctuationsin the preceding de Sitter epoch [8], the condition (44) may be fulfilled provided that∂tφ0 is sufficiently large and negative.Finally, let us consider models where a massive scalar φ is derivatively coupled toother particle species.

This situation corresponds for instance to models possessingpseudogoldstone bosons with nonvanishing masses. The relevant scenario is similarto that of the “spontaneous baryogenesis” described in ref.

[5]. If a Lagrangian has acoupling of the form Lφ = −1f φ∂µjµ (f being some, presumably large, mass scale) where∂µjµ is a divergence of a current corresponding to some explicitly broken symmetry, the10

baryon number symmetry for instance. Then, as we have shown, there are fluctuationsin the field φ with dispersionq< φ2 >R.

We may represent them as the effective termin the LagrangianLδφ = −1fq< φ2 >R∂µjµ(45)Up to the total divergence (45) is equivalent toLδφ = 1f ∂0q< φ2 >Rj0(46)This produces an effective chemical potential µ = −1f ∂0q< φ2 >R for the charge den-sity j0 which means a nonzero cosmological charge density generated in thermal equi-librium. Explicitly, cf.

[9],j0 ≈−1f ∂tq< φ2 >RT 2(47)or charge to entropy ratioj0/s ≈−1fg∗T ∂tq< φ2 >R(48)where g∗is the number of relativistic degrees of freedom at temperature T. The aboveestimate gives in the case of our toy modelj0/s ≈m4g∗fT1tqln(1/mt)(49)for T > T∗andj0/s ≈O(10−2)g∗fT1t7/4 m1/4(50)for T < T∗. One can see that both expressions fall offas time elapses, and that thedecrease at T < T∗is faster than at T > T∗, essentially ∼T 5/2 below T∗and ∼Tabove.

If there is no phase transition in the model before the end of RD epoch, thenthe final charge to entropy ratio produced will be equal to (49) or (50) taken at the“decoupling” temperature TD. This is the temperature at which symmetry violatinginteractions fall offfrom equilibrium or the one which corresponds to the end of RDstage, when the shape of the fluctuations changes qualitatively i.e.

at TD ≈Tf closeto 10 eV. As previously, the numerical values predicted depend on various details ofa model under investigation.

For example, let us take TD = 10 eV and g∗= 100.Then if we require the charge-to-entropy ratio to be equal to 10−10, as it should befor the baryonic charge, then we get the condition m = f 4 × 10−77GeV, which gives11

m = 10−17GeV for f = 1015GeV and m = 1GeV for f = 1019GeV.3. ConclusionsIn this letter we have found explicit expressions for a massive scalar field fluctua-tions in the flat radiation dominated universe.

It turns out that in the region of smallmt, i.e. shortly after the beginning of the RD epoch or for very light fields, the fluc-tuations decrease with time only logarithmically and are proportional to the squareof the mass of the field in question.

For large mt, i.e. very late or for a heavy field,the time dependence is stronger, 1/t3/2, but the mass dependence becomes weaker –proportional to the square root of m. As far as finite temperatures are concerned,we have concluded that the “radiation-dominated” background modifies Minkowskispace results rather weakly.

At low temperatures, i.e. at large ratios m/T, the ther-mal contribution is exponentially suppressed the suppression becoming stronger as thetemperature decreases with time.

At high temperatures the result coincides essentiallywith that of Minkowski space. In general, fluctuations vanish when one takes the limitm →0.Given all that we argue that the fluctuations may still play a significant role inparticle physics models, which has been illustrated in the second part of the work.Within the family of models we discuss, the case when our parameter n equals 2corresponds precisely to the Affleck-Dine model, and the higher n terms are oftenencountered in the important class of string inspired models.

Hence we conclude thatfluctuations we have described constitute the phenomenon which is relevant in a verygeneral situation when some cosmological charge density, first of all the baryonic chargedensity, is supposed to be generated during the radiation dominated epoch. We alsonote that although the inflationary scenario is widely accepted, our results do not relyon the existence of the de Sitter epoch preceding the RD stage in the early Universe.In conclusion, we have demonstrated that quantum fluctuations due to a nontriv-ial gravitational background during radiation dominated epoch in the evolution of theUniverse do in fact exist and may have observable consequences for cosmology of real-istic particle models.12

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D28, 679 (1983);A.D.Linde, Phys. Lett.

116B, 335 (1982). [2] I.Affleck,M.Dine, Nucl.

Phys. B249, 361 (1985).

[3] A.Cohen,D.Kaplan, Phys. Lett.

199B, 251 (1987). [4] N.Birrel,P.C.Davies, Quantum Fields in Curved Space, Cambridge University Press,Cambridge 1984.

[5] H.Buchholz, Die Konfluente Hypergeometrische Funktion, Springer-Verlag, Berlin1953. [6] C.Pathinayake,L.H.Ford, Phys.

Rev. D37, 2099 (1988).

[7] M.Abramowitz,I.Stegun, Handbook of Mathematical Functions, Dover Publica-tions, Inc., New York. [8] A.D.Dolgov, Yukawa Institute preprint YITP/K-940, 1991.

[9] E.W.Kolb,M.S.Turner, The Early Universe, Addison-Wesley Publishing Company,1990.13

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