Graviton production in non-inflationary cosmology
aa r X i v : . [ a s t r o - ph . C O ] F e b Graviton production in non-inflationary cosmology
Ruth Durrer and Massimiliano Rinaldi
Universit´e de Gen`eve, D´epartment de Physique Th´eorique,24 quai Ernest Ansermet, CH–1211 Gen`eve 4, Switzerland (Dated: November 1, 2018)We discuss the creation of massless particles in a Universe, which transits from a radiation-dominatedera to any other expansion law. We calculate in detail the generation of gravitons during the tran-sition to a matter-dominated era. We show that the resulting gravitons generated in the standardradiation/matter transition are negligible. We use our result to constrain one or more previousmatter-dominated era, or any other expansion law, which may have taken place in the early Uni-verse. We also derive a general formula for the modification of a generic initial graviton spectrumby an early matter dominated era.
PACS numbers: 98.80Cq,04.50+hKeywords: Cosmology, radiation-matter transition, graviton production
I. INTRODUCTION
One of the most interesting aspect of inflation is thatit leads to the generation of a scale-invariant spectrumof scalar perturbations [1] and of gravitational waves [2](see also [3, 4]). The origin of these perturbations isthe quantum generation of field correlations in a time-dependent background (scalar field modes for the scalarperturbations and gravitons for the tensor perturba-tions). Since this generation takes place mainly on super-horizon scales, it is not correct to talk of ’particles’. How-ever, long after inflation, when the perturbations re-enterthe horizon, the particle concept, e.g. for gravitons be-comes meaningful and we can calculate, e.g. the energydensity of the gravitons which have been generated dur-ing inflation.It is natural to ask whether particle production takesplace also in an ordinary expanding but non-inflationaryFriedmann Universe. The answer is that particle produc-tion (or more generically the quantum generation of fieldcorrelations) can indeed take place after inflation, butif there is no inflationary phase to start with, the initialvacuum state is in general not known, and the productionrate cannot be computed. Examples where particle cre-ation taking place after inflation (or after pre-big bang)modifies the final spectrum are given in Refs. [5, 6, 7, 8].Especially in Ref. [7] it has been studied how inflationaryperturbations are modified if the subsequent expansion isnot standard radiation but some other expansion law.In general, the vacuum state, and hence the particleconcept, is well defined only if the spacetime is static orvery slowly varying [9]. Let us consider a mode of fixed(comoving) frequency k in a Friedmann universe. Theabove condition then corresponds to k/ H ≫
1, where H = aH is the comoving Hubble scale. In this sense, thescale (wavelength) under consideration must be “insidethe horizon”. However, the production of a particle witha given energy k can only take place if the energy scaleof expansion is larger or of the order of the energy ofthe associated mode, i.e. k < ∼ H . Therefore, having awell defined initial vacuum state, and subsequent particle creation, usually requires a decreasing comoving Hubblerate. This is verified only during inflation or during acollapsing Friedmann Universe, like in the pre-big bangor in bouncing models.However, one important exception to this general ruleexists, and it is the subject of the present paper: ina radiation-dominated Friedmann background, masslessperturbations do not couple to the expansion of the Uni-verse, and evolve like in ordinary Minkowski space. Thishas already been realized and studied to some extent inRef. [10]. In a radiation-dominated Universe we there-fore can provide vacuum initial conditions for all modesof a massless field, including super-horizon modes. Thus,when the expansion law changes, e.g. from radiation-to matter-dominated, the massless modes couple to theexpansion of the Universe, and those with k/ H < z eq ≃ Notation:
We work in a spatially flat Friedmann Uni-verse, and we denote conformal time by t , so that ds = a ( t ) (cid:0) − dt + δ ij dx i dx j (cid:1) . An over-dot denotes the derivative with respect to theconformal time. We use natural units c = ~ = 1, exceptfor Newton’s constant G , which is related to the reducedPlanck mass by 8 πG = m − p . We normalize the scalefactor, so that a = a ( t ) = 1 at the present time. II. GRAVITON CREATION IN COSMOLOGY
We now consider tensor perturbations of the Fried-mann metric, namely ds = a ( t ) (cid:2) − dt + ( δ ij + 2 h ij ) dx i dx j (cid:3) , where h ij is a transverse and traceless tensor. In Fourierspace we have h ij ( k , t ) = h + ( k, t ) e (+) ij (ˆ k ) + h − ( k, t ) e ( − ) ij (ˆ k ) , (1)where e ( ± ) ij (ˆ k ) denote the positive and negative helicitypolarization tensors, and k i e ( ± ) ij = 0. In a perfect fluidbackground, i.e. if there are no anisotropic stresses, bothamplitudes satisfy the same wave equation, (cid:3) h = ¨ h + 2 ˙ aa ˙ h + k h = 0 , (2)where h ≡ h ± . This equation of motion is obtained whenexpanding the gravitational action in a Friedman uni-verse to second order in h , S + δS = − m p Z d x p − ( g + δg )( R + δR ) . (3)A brief calculation shows that the lowest (second order)contribution to δS can be written in Minkowski-spacecanonical form, δS (2) = − Z d x (cid:18) ∂ µ φ∂ µ φ − ¨ aa φ (cid:19) = Z d x L , (4)if we rescale h as h ( x , t ) = 1 √ m p a ( t ) φ ( x , t ) . (5) Eq. (4) is the action of a canonical scalar field with time-dependent effective squared mass m ( t ) = − ¨ a/a . If theexpansion of the Universe is slow enough (compared tothe frequency of the mode under consideration), then theeffective mass is negligible, and the theory describes amassless scalar field in Minkowski space, which can bequantized according to the usual procedure: we first pro-mote the field to an operatorˆ φ ( x , t ) = Z d k (2 π ) h e i k · x χ k ( t )ˆ b k + e − i k · x χ ∗ k ( t )ˆ b † k i , (6)then we impose the commutation rules[ b k , b † k ′ ] = (2 π ) δ ( k − k ′ ) , [ b k , b k ′ ] = [ b † k , b † k ′ ] = 0 . The field equations derived from the action (4) lead tothe mode equation¨ χ k + (cid:20) k − ¨ aa (cid:21) χ k = 0 . (7)Within linearized gravity we can therefore quantize themetric fluctuations, provided the Universe expands adi-abatically, by making use of the above rescaling of theamplitude h .In particular, we now assume that the Universe isinitially radiation-dominated, so that a ( t ) = t , and¨ a/a = 0, and φ represents exactly a massless scalar fieldin Minkowski space. We consider the vacuum initial con-ditions for the modes χ k ( t ) as given by χ k ( t ) = 1 √ k e − ikt . (8)More general initial conditions will be considered at theend of Sec. IV. The field normalization is determined bythe Klein-Gordon norm iχ ∗ ↔ ∂ χ ≡ i ( χ ∗ ∂ χ − χ∂ χ ∗ ) = 1 . (9)Then, the field operator ˆ φ and its canonically conjugatemomentum, ˆΠ = ∂ L /∂ ( ∂ ˆ φ ) satisfy the canonical com-mutation relations. The operators ˆ b k define the in vac-uum by ˆ b k | in i = 0 ∀ k . In the following, this is the initialvacuum, void of particles by construction.Suppose now that at a time t = t , the Universechanges abruptly from radiation-dominated to anotherexpansion law. Then, the effective squared mass nolonger vanishes, and the initial modes, with kt < χ and ˙ χ match at t = t , and these conditions determine the Bogoliubovcoefficients, which relate the new modes χ out and opera-tors ˆ b k , out to the old ones, χ in and ˆ b k , in by [9] χ out ( t ) = αχ in ( t ) + βχ ∗ in ( t ) , (10)ˆ b out = α ∗ ˆ b in − β ∗ ˆ b † in . (11)With these relations, we can compute the number densityof the particles[13] created at the transition [9] N ( k, t ) = h in | ˆ b † k , out ˆ b k , out | in i = | β | . (12)Thus, the energy density ρ ≡ h T i can be written as ρ h = 1 a Z d k (2 π ) kN ( k, t ) = 12 π a Z dkk | β | , (13)which implies the usual formula[14] dρ h d log k (cid:12)(cid:12)(cid:12) tot = k | β | π a , (14)where we have multiplied Eq. (13) by a factor 2 to takeinto account both polarizations. Note also that k denotescomoving momenta/energy so that we had to divide by a to arrive at the physical energy density. The secondquantity of interest is the power spectrum P h ( k, t ), de-fined by 4 π Z dkk P h ( k, t ) = h in | h ( t, x ) | in i . (15)Using Eqs. (5) and (6), we obtain P h ( k, t ) = k | β | (2 π ) m p a , (16)where, again, we have multiplied by 2 to account for bothpolarizations.Note that for all this it is not important that we con-sider a spin 2 graviton. The exactly same mode equa-tion is obtained for a scalar field and also for a fermionfield. In the latter case, the commutation relations haveto be replaced by the corresponding anti-commutationrelations. III. FROM RADIATION TO MATTER ERA
Before discussing a transition from the radiation-dominated era to the matter era, let us consider the tran-sition from radiation to some generic power law expan-sion phase, a ∝ t q with q = 1 at some time t . In the newera ¨ a/a = q ( q − /t = 0. Note that only if q > q < m ( t ) = − q ( q − /t is negative, we will havesignificant particle production. Since the expansion lawis related to the equation of state parameter w = P/ρ via [4] q = 21 + 3 w , (17)this requires w < / χ is given entirely by the negative frequency modes,Eq. (8). This means that we consider the situation where there are no significant gravity waves present from an ear-lier inflationary epoch. Here, we really want to study theproduction due solely to the radiation/matter transition.The general solution of the mode equation (7) in the newera are the spherical Hankel functions [11] of order q − χ k ( t ) = α √ k zh (2) q − ( z ) + β √ k zh (1) q − ( z ) (18)where z = kt . Note that inside the horizon, i.e. for z ≫ zh (2) q − ( z ) ∝ exp( − iz ) corresponds to the negativefrequency modes while zh (1) q − ( z ) ∝ exp( iz ) correspondsto positive frequency modes. We match χ and ˙ χ at t = t to the radiation-dominated vacuum solution (8). A briefcalculation yields the coefficients ( z ≡ kt ) α = − i e − iz h ( iz + q ) h (1) q − ( z ) − z h (1) q ( z ) i , (19) β = − i e − iz h ( iz + q ) h (2) q − ( z ) − z h (2) q ( z ) i . (20)This instantaneous matching condition is good enoughfor frequencies for which the transition is rapid, i.e. z ≪
1. In fact, for frequencies with z >
1, the transition isadiabatic and no particle creation will take place. Thiscan also been seen when considering the limits of theabove result for large z . Then α → β →
0, butstrictly speaking the above approximations are not validin this regime where no particle creation takes place. Wetherefore concentrate on z ≪ q = 2. Then we have to considerspherical Hankel functions of order 1 and the solution isgiven by χ k ( t ) = α √ k z − iz e − iz + β √ k z + iz e iz . (21)The matching at t now yields α = 1 + iz − z , β = − z e − iz . (22)We want to evaluate the quantum field ˆ φ at late time,when z ≫ k under consideration is sub-horizon. Then, the solution (21) is again the Minkowskisolution, χ k ( t ) ≃ α √ k e − iz + β √ k e iz . (23)The number of gravitons generated during the matterera (before z ≫
1) is, see Eq. (12) N ( k, t ) = | β | . (24)The graviton power spectrum is given by Eq. (16), andthe energy density by Eq. (13).Using that ρ rad a ≡ ρ rad ( t ) a = m p H a and | β | = z − /
4, we obtain d Ω h ( k ) d log k = 2Ω rad π k m p H a | β | = Ω rad π (cid:18) H m p (cid:19) = Ω rad g eff × (cid:18) T m p (cid:19) . (25)For the second equal sign we have used that H = 1 /t = a H , which is strictly true only in the radiation era, inthe matter era we have H = 2 /t and at the transition avalue between 1 and 2 would probably be more accurate.But within our approximation of an instant transition, wedo not bother about such factors. For the last equal signwe used H = ρ rad / (6 m p ) with ρ rad = g eff π T where g eff = N B + N F is the effective number of degrees offreedom.For a generic transition we obtain | β | = z − q / d Ω h ( k ) d log k ≃ Ω rad (cid:18) T m p (cid:19) z − q . (26)This spectrum is blue (i.e. growing with k ) if q < z < q > → matter transition. According toEq. (17), this requires − / < w <
0, a slightly negativepressure, but still non-inflationary expansion.In the standard radiation to matter transition whenthree species of left handed neutrinos and the photon arethe only relativistic degrees of freedom, we have g eff =29 /
4. For this transition T = T eq ≃ . ≃ . × − m p , hence the result (25) is completely negligible. IV. MORE THAN ONE RADIATION-MATTERTRANSITION
We now consider an early matter dominated era. Atsome high temperature T ≫ T eq , corresponding to a co-moving time t , a massive particle may start to dominatethe Universe and render it matter-dominated. At somelater time t , corresponding to temperature T , this mas-sive particle decays and the Universe becomes radiation-dominated again, until the usual radiation–matter tran-sition, which takes place at T eq ≡ T . We want to deter-mine the gravitational wave spectrum and the spectraldensity parameter d Ω h /d log( k ) as functions of T and T .Let us first again start with the vacuum state in theradiation eta before t . When, we just obtain the results(22) for the Bogoliubov coefficients α and β after the first transition. To evaluate the matching conditions atthe second transition, matter to radiation, we set χ = α √ k e − ikt + β √ k e ikt , t ≥ t . Matching χ and ˙ χ at t we can relate the new coefficients α and β to α and β . A brief calculation gives α = α (cid:18) − iz − z (cid:19) + β z e iz = α f ( z ) + β g ( z ) , (27) β = β (cid:18) iz − z (cid:19) + α z e − iz (28)= β ¯ f ( z ) + α ¯ g ( z ) , (29)or in matrix notation (cid:18) α β (cid:19) = M ( z ) (cid:18) α β (cid:19) , with (30) M ( z ) = (cid:18) f ( z ) g ( z )¯ g ( z ) ¯ f ( z ) (cid:19) (31) M − ( z ) = (cid:18) ¯ f ( z ) − g ( z ) − ¯ g ( z ) f ( z ) (cid:19) . (32)The fact that M ∈ Sl (2 , C ), i.e., | f ( z ) | − | g ( z ) | = 1 en-sures that the normalization condition (9) which trans-lates to the condition | α | − | β | = 1 for the Bogolioubovcoefficients of a free field, is maintained at the transition.Finally, the matching at the usual radiation–matter tran-sition yields α = α (cid:18) iz − z (cid:19) − β z e iz ,β = β (cid:18) − iz − z (cid:19) − α z e − iz , (33) (cid:18) α β (cid:19) = M − ( z ) (cid:18) α β (cid:19) (34) (cid:18) α β (cid:19) = M − ( z ) M ( z ) M − ( z ) (cid:18) (cid:19) . (35)To obtain the power spectrum and energy density inthis case, we simply have to replace | β | in Eqs. (16) and(14) by | β | . In Fig. 1 we plot | β | as a function of z for different choices of t . The instantaneous transi-tion approximation breaks down for z >
1, hence onlythe left side of the vertical line is physical. For the rightside one would have to solve the mode equation numeri-cally, but since we know that particle production is sup-pressed for these frequencies, we do not consider them.We concentrate on z ≤
1. For these wave numbers, also z < z < z < FIG. 1: The Bogoliubov coefficient | β ( k ) | for t = 1, and t = 2000, with various values of t . Only the k-values left ofthe vertical dashed line satisfy kt <
1. They show clearly a k − slope and the amplitude is well approximated by (38). This allows the following approximations, α ≃ β ≃ − z , z ≪ α ≃ − β ≃ iz z , z ≪ α ≃ β ≃ − z z z , z ≪ . (38)To obtain the results (37) and (38) we have to expand theexact expression (27) to fourth order and (33) to secondorder, but we consider only the largest term in the resultgiven above, using also z < z < z . Therefore, in theapproximate expression (37), where we have neglecteda term proportional to z /z , one no longer sees that α → β → t → t and hence z → z . Inthis case there is no intermediate matter-dominated eraand therefore no particle creation, hence β = 0. Thiscan be seen from the exact expression given in Eq. (27).Within these approximations, Eqs. (16) and (14) leadto P h ( k ) = 1(2 π ) (cid:18) t m p at t k (cid:19) , kt < d Ω h d log k = Ω rad π k | β | a ρ rad ( t )= Ω rad g eff ( T )18 × (cid:18) T m p (cid:19) (cid:18) T eq T (cid:19) , kt eq < . (40)This result can be generalized to several, say N , interme-diate radiation → matter transitions at times t n − and back to radiation at time t n , 1 ≤ n ≤ N , with the result | β N +1 | ≃ kt ) (cid:18) T · · · T N − T eq T · · · T N (cid:19) . (41)Hence, each return to the radiation-dominated era atsome intermediate temperature T n leads to a suppres-sion factor ( T n +1 /T n ) , where T n +1 denotes the tem-perature at the start of the next matter era.On large scales, kt eq <
1, the energy density spectrumis flat. The best constraints on an intermediate radiation-dominated era therefore come from the largest scales, i.e.from observations of the cosmic microwave background(CMB) as we shall discuss in the next section.We now briefly consider the case when the initial con-ditions differ from the vacuum case, Eq. (8). We assumean arbitrary initial state of the field given by χ k ( t ) = α √ k e − ikt + β √ k e ikt , (42)together with the normalization condition which ensuresthat the field is canonically normalized, | α | − | β | = 1.The same calculations as above now yield (cid:18) α β (cid:19) = M − ( z ) M ( z ) M − ( z ) (cid:18) α β (cid:19) , (43)where M ( z ) is the matrix giving the transition from mat-ter to radiation defined in Eq. (31).Expanding this in z , z and z , using z < z < z < β depends only | α + β | . However, if thephase of α and β are nearly opposite, i.e., α ≃ − β ,and if | α | and therefore also | β | are much larger than 1,a correction proportional to | α − β | becomes important.More precisely, the last of Eqs. (36) now is replaced by β ≃ z z z (cid:20) − z ( α + β ) + 2 i ( α − β ) (cid:21) . (44)If α = 1 and β = 0, the second term can be neglectedwith respect to the first one and we reproduce the pre-vious result (38). As we see from this equation, a largephase difference between α and β changes not only theamplitude but also the slope of the spectrum. Of coursein concrete examples, like for a previous inflationary pe-riod, see Ref. [7], the coefficients α and β also dependon the wavenumber.In Fig. 2 we show the dependence of | β | on | α | fordifferent values of the relative phase between α and β (top panel) and as a function of the relative phase for dif-ferent values of | α | . The difference of | β | between thecase where α and β are perfectly in phase and of op-posite phase is of the order of 1 /z , if | a | is significantlylarger than 1. This is already evident from Eq. (44). Inthe Fig. 2 we have chosen z = 0 .
1, a unrealistically highvalue, in order to have a better visibility of the phasedependence which then changes | β | only by one orderof magnitude. FIG. 2: In the top panel we show | β | as a function of α for α and β in phase (top, solid, black line), of opposite phase(lowest, dashed, red line) and with a phase difference of 0 . π (middle, dot-dashed, blue line).In the bottom panel we show | β | as a function of the phasedifference ψ for | α | = 80 (top line) | α | = 40 (middle line)and | α | = 10 (lowest line). We have chosen z = 0 . z / ( z z )) . In conclusion, in the case of a non-vacuum initial state, | α | significantly larger than 1, graviton production is en-hanced typically by a factor of order | α | ∼ | β | whichis the number of initial particles. Hence in addition tothe spontaneous creation we now also have induced par-ticle creation which is proportional to the initial particlenumber and much larger than the spontaneous creationif the particle number is large. An interesting point isthat the phase shift between α and β can significantlyaffect the final spectrum. V. DISCUSSION AND CONCLUSIONS
The fact that the observed CMB anisotropies are of theorder of 10 − yields a strong limit on gravitational waveswith wave numbers of the order of the present Hubblescale, see e.g. [12]. d Ω h d log k (cid:12)(cid:12)(cid:12)(cid:12) k = H < − . (45)With Ω rad ≃ − , and Eq. (40), this implies the limit (cid:18) T m p (cid:19) T eq T < − . (46)We know that during nucleosynthesis the Universe wasradiation-dominated, hence T ≥ . T eq ∼ T < m p for thevalue T ≃ . T . Hence even though the production ofgravitons during an intermediate matter era is of princi-pal interest, we cannot derive stringent limits on T and T . On the other hand, for values of T and T close tothe maximal respectively minimal value, T ∼ m p and T ∼ . q >
2, i.e. − / < w = P/ρ <
0. Accordingto Eq. (26), in the general case, the particle number is ofthe order of | β | ≃ z − q , so that d Ω h d log k ≃ Ω rad ( kt ) − q (cid:18) T m p (cid:19) (cid:18) T eq T (cid:19) , z ≪ . (47)As above, T denotes the temperature at which the Uni-verse returns to the radiation-dominated state, hencenucleosynthesis requires T > . q >
2, the spectrum becomes red and, at k ∼ H , thelimit can become quite interesting. Hence, graviton pro-duction can significantly limit a (non-inflationary) phasewith negative pressure in the early universe. For k = H ,using t = H − = 1 / ( a H ), we obtain( H t ) − ≃ (cid:18) g eff ( T )10 (cid:19) / (cid:18) T T (cid:19) , where T ≃ . × − eV is the present temperature ofthe Universe. This can be a significant factor for largevalues of T . Inserting this expression in Eq. (47), thelimit (45) yields (cid:18) g eff ( T )10 (cid:19) q − (cid:18) T T (cid:19) q − (cid:18) T m p (cid:19) (cid:18) T eq T (cid:19) < ∼ − . (48)For example, for w = − /
21, ( q = 7 / T T (cid:18) T m p (cid:19) < ∼ − . (49)E.g. for T = 1MeV this implies T < GeV. Forsmaller values of w the limit becomes more stringent.In this paper we have shown that there is cosmo-logical particle production of massless modes wheneverthe expansion law is not radiation-dominated so that¨ a/a = 0. This term acts like a time-dependent massand leads to the production of modes with comovingenergy k < | ¨ a/a | ≃ H , hence with physical energy ω = k/a < ∼ H . One readily sees that particle production issignificant only if q <
1, i.e. the squared mass − ¨ a/a < − / < w < Acknowledgment:
We thank John Barrow for valuablecomments. This work is supported by the Swiss NationalScience Foundation. [1] V. F. Mukhanov and G. V. Chibisov, Sov. Phys. JETP (1982) 258 [Zh. Eksp. Teor. Fiz. (1982) 475].[2] A. A. Starobinsky, JETP Lett. (1979) 682 [Pisma Zh.Eksp. Teor. Fiz. (1979) 719].[3] V. Mukhannov, Physical Foundations of Cosmology ,Cambridge University Press (2005).[4] R. Durrer,
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