Preheating of Fermions

Patrick B. Greene Department of Physics and Astronomy, University of Hawaii, 2505 Correa Rd., Honolulu HI 96822, USA    Lev Kofman Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr., Honolulu, HI 96822, USA
July 16, 2016

In inflationary cosmology, the particles constituting the Universe are created after inflation in the process of reheating due to their interaction with the oscillating inflaton field. In the bosonic sector, the leading channel of particle production is the non-perturbative regime of parametric resonance, preheating, during which bosons are created exponentially fast. Pauli blocking prohibits the unbounded creation of fermions. For this reason, it has been silently assumed that the creation of fermions can be treated with perturbation theory for the decay of individual inflatons. We consider the production of fermions interacting with the coherently oscillating inflatons. We find that the actual particle production occurs in a regime of the parametric excitation of fermions , leading to preheating of fermions. Fermion preheating differs significantly from the perturbative expectation. It turns out that the number density of fermions varies periodically with time. The total number of fermions quickly saturates to an average value within a broad range of momenta ∝q1/4proportional-toabsentsuperscriptπ‘ž14\propto q^{1/4} , where qπ‘žq is the usual resonance parameter. The resonant excitation of fermions may affect the transfer inflaton energy, estimations of the reheating temperature, and the abundance of superheavy fermions and gravitinos. Back in the bosonic sector, outside of the parametric resonance bands there is an additional effect of parametric excitation of bosons with bounded occupation numberin the momentum range ∝q1/4proportional-toabsentsuperscriptπ‘ž14\propto q^{1/4} .

PACS: 98.80.Cq Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β UH-IfA-98-44 Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β 
††preprint: UH-IfA-98-44, hep-ph/xxxxxx, June, 1998



In the inflationary scenario, the Universe initially expands quasi-exponentially in a vacuum-like state with vanishing temperature, entropy, and particle number densities. Consider a simple chaotic inflation during which all energy is contained in the inflaton field Ο•italic-Ο•\phi which is slowly rolling down to the minimum of its effective potential V⁒(Ο•)𝑉italic-Ο•V(\phi) . When inflation ends at Ο•βˆΌMpsimilar-toitalic-Ο•subscript𝑀𝑝\phi\sim M_{p} , the inflaton field begins to oscillate near the minimum of V⁒(Ο•)𝑉italic-Ο•V(\phi) with a very large amplitude, ϕ≃110⁒Mpsimilar-to-or-equalsitalic-Ο•110subscript𝑀𝑝\phi\simeq{1\over 10}M_{p} . This scalar field can be considered as a coherent superposition of Ο•italic-Ο•\phi -quasi-particles with zero momenta, i.e. inflatons at rest.

In this scenario, all the particles constituting the Universe are created due to their interactions with the inflatons. For a toy model describing the interaction between inflatons and other Bose particles Ο‡πœ’\chi , we may consider the term g2⁒ϕ2⁒χ2superscript𝑔2superscriptitalic-Ο•2superscriptπœ’2g^{2}\phi^{2}\chi^{2} or g2⁒σ⁒ϕ⁒χ2superscript𝑔2𝜎italic-Ο•superscriptπœ’2g^{2}\sigma\phi\chi^{2} in the Lagrangian. For Fermi particles Οˆπœ“\psi we can include a Yukawa coupling hβ’ΟˆΒ―β’Ο•β’Οˆβ„ŽΒ―πœ“italic-Ο•πœ“h\bar{\psi}\phi\psi to the inflaton. If we assume for simplicity that the bare masses of the fields Ο‡πœ’\chi and Οˆπœ“\psi are very small, the inflatons will decay into ultra-relativistic particles. If the creation of these particles is sufficiently slow, the particles will simultaneously interact with each other and come to a state of thermal equilibrium at the reheating temperature Trsubscriptπ‘‡π‘ŸT_{r} . The theory of this gradual reheating, developed long ago with the first models of inflation (1), was based on the perturbative theory with respect to the coupling constants g2superscript𝑔2g^{2} and hβ„Žh . Consider a quantum Bose field Ο‡πœ’\chi with the eigenfunctions Ο‡k⁒(t)⁒e+i⁒𝐀⋅𝐱subscriptπœ’π‘˜π‘‘superscript𝑒⋅𝑖𝐀𝐱\chi_{k}(t)\,e^{+i{{\bf k}}\cdot{{\bf x}}} for comoving momenta 𝐀𝐀{\bf k} . The temporal part of the eigenfunction obeys an oscillator-like equation with a time-dependent frequency: Ξ©k2⁒(t)=k2a2+g2⁒ϕ⁒(t)2superscriptsubscriptΞ©π‘˜2𝑑superscriptπ‘˜2superscriptπ‘Ž2superscript𝑔2italic-Ο•superscript𝑑2\Omega_{k}^{2}(t)={{k^{2}}\over a^{2}}+g^{2}\phi(t)^{2} or g2⁒σ⁒ϕ⁒(t)superscript𝑔2𝜎italic-ϕ𝑑g^{2}\sigma\phi(t) . The iterative solution to the equation of motion for Ο‡k⁒(t)subscriptπœ’π‘˜π‘‘\chi_{k}(t) gives the perturbation theory result for the particles’ occupation number: nk∝g4β‰ͺ1proportional-tosubscriptπ‘›π‘˜superscript𝑔4much-less-than1n_{k}\propto g^{4}\ll 1 . The resulting rate for the three-legs process ϕ→χ⁒χ→italic-Ο•πœ’πœ’\phi\to\chi\chi is given by Γϕ→χ⁒χ=g4⁒σ28⁒π⁒mΟ•subscriptΞ“β†’italic-Ο•πœ’πœ’superscript𝑔4superscript𝜎28πœ‹subscriptπ‘šitalic-Ο•\Gamma_{\phi\to\chi\chi}={g^{4}\sigma^{2}\over 8\pi m_{\phi}} . By the same methods, one finds for the process Ο•β†’Οˆβ’Οˆβ†’italic-Ο•πœ“πœ“\phi\to\psi\psi a rate Ξ“Ο•β†’Οˆβ’Οˆ=h2⁒mΟ•8⁒πsubscriptΞ“β†’italic-Ο•πœ“πœ“superscriptβ„Ž2subscriptπ‘šitalic-Ο•8πœ‹\Gamma_{\phi\to\psi\psi}={h^{2}m_{\phi}\over 8\pi} . Each of the created quanta carries away half of an inflaton’s energy, mΟ•2subscriptπ‘šitalic-Ο•2m_{\phi}\over 2 . This perturbative result can be interpreted as individual inflatons decaying independently of each other into pairs of Ο‡πœ’\chi or Οˆπœ“\psi -particles.

However, it was recognized that reheating can begin with a stage of boson production in a regime of parametric resonance (2). Indeed, in the equation for the eigenmode Ο‡k⁒(t)subscriptπœ’π‘˜π‘‘\chi_{k}(t) , inflatons act not as individual particles but as a coherently oscillating field ϕ⁒(t)italic-ϕ𝑑\phi(t) . The smallness of the coupling g2superscript𝑔2g^{2} alone does not necessarily correspond to small occupation number nksubscriptπ‘›π‘˜n_{k} . The oscillating effective frequency Ξ©k⁒(t)subscriptΞ©π‘˜π‘‘\Omega_{k}(t) results in the parametric amplification of the modes: Ο‡k∼eΞΌk⁒tsimilar-tosubscriptπœ’π‘˜superscript𝑒subscriptπœ‡π‘˜π‘‘\chi_{k}\sim e^{\mu_{k}t} . In this case, the energy of the inflatons is explosively transferred to the bose field. This process of bosonic preheating occurs very rapidly and far from thermal equilibrium.

For fermions, the Pauli exclusion principle prohibits the occupation number from exceeding 1/2121/2 . For this reason, it has been assumed that fermions are created in the perturbative regime with the production rate Ξ“Ο•β†’Οˆβ’ΟˆsubscriptΞ“β†’italic-Ο•πœ“πœ“\Gamma_{\phi\to\psi\psi} and occupation number nk∝h2β‰ͺ1proportional-tosubscriptπ‘›π‘˜superscriptβ„Ž2much-less-than1n_{k}\propto h^{2}\ll 1 , as described in the classical papers (1).

Let us, however, consider the Dirac equation for a massless quantum Fermi field ψ⁒(t,xβ†’)πœ“π‘‘β†’π‘₯\psi(t,\vec{x}) :

[iβ’Ξ³ΞΌβ’βˆ‡ΞΌ-h⁒ϕ⁒(t)]⁒ψ=0.delimited-[]𝑖superscriptπ›Ύπœ‡subscriptβˆ‡πœ‡β„Žitalic-Ο•π‘‘πœ“0\left[i\gamma^{\mu}\nabla_{\mu}-h\phi(t)\right]\psi=0\ . (1)

Here, the inflatons producing fermions also act not as individual particles but as a coherently oscillating field ϕ⁒(t)italic-ϕ𝑑\phi(t) . Therefore, we should check whether coherently acting inflatons produce the same number of fermions as an ensemble of individually decaying inflatons. It turns out that the actual production of fermions from the oscillating inflaton field dramatically differs from the perturbative approximation.

We begin our investigation with the model 14⁒λ⁒ϕ4+hβ’ΟˆΒ―β’Ο•β’Οˆ14πœ†superscriptitalic-Ο•4β„ŽΒ―πœ“italic-Ο•πœ“{1\over 4}\lambda\phi^{4}+h\bar{\psi}\phi\psi . A special feature of this theory is that the problem of fermion production by the inflaton Ο•italic-Ο•\phi in an expanding universe can be completely reduced to a similar problem in Minkowski space-time. Indeed, let us perform a conformal transformation of the involved fields, φ≑aβ’Ο•πœ‘π‘Žitalic-Ο•\varphi\equiv a\phi and Ψ≑a3/2⁒ψΨsuperscriptπ‘Ž32πœ“\Psi\equiv a^{3/2}\psi , and use a conformal time variable, τ≑λ⁒φ~2⁒∫d⁒ta⁒(t)πœπœ†superscript~πœ‘2π‘‘π‘‘π‘Žπ‘‘\tau\equiv\sqrt{\lambda\tilde{\varphi}^{2}}\int{dt\over a(t)} . Then (without decay) the amplitude of the transformed field Ο†πœ‘\varphi is constant, Ο†~~πœ‘\tilde{\varphi} . The background inflaton solution is given by an elliptic function φ⁒(Ο„)β‰ˆΟ†~⁒c⁒n⁒(Ο„,12)πœ‘πœ~πœ‘π‘π‘›πœ12\varphi(\tau)\approx\tilde{\varphi}~{}cn\left(\tau,{1\over\sqrt{2}}\right) and the period of the background oscillations Tβ‰ˆ7.416𝑇7.416T\approx 7.416 in this model (4).

The equation for the eigenfunctions of the quantum fluctuations in this theory can be reduced to a second-order equation for an auxillary field X⁒(Ο„,xβ†’)π‘‹πœβ†’π‘₯X(\tau,\vec{x}) , so that Ξ¨=[iβ’Ξ³ΞΌβ’βˆ‡ΞΌ+h⁒φ]⁒XΞ¨delimited-[]𝑖superscriptπ›Ύπœ‡subscriptβˆ‡πœ‡β„Žπœ‘π‘‹\Psi=\left[i\gamma^{\mu}\nabla_{\mu}+h\varphi\right]X . The eigenmodes of the auxillary field have the form Xk⁒(Ο„)⁒e+i⁒𝐀⋅𝐱⁒Rrsubscriptπ‘‹π‘˜πœsuperscript𝑒⋅𝑖𝐀𝐱subscriptπ‘…π‘ŸX_{k}(\tau)e^{+i{{\bf k}}\cdot{{\bf x}}}R_{r} , where the Rrsubscriptπ‘…π‘ŸR_{r} are eigenvectors of the Dirac matrix Ξ³0superscript𝛾0\gamma^{0} with eigenvalue +11+1 . The temporal part of the eigenmode obeys an oscillator-like equation with a complex frequency which depends periodically on time

XΒ¨k+(ΞΊ2+q⁒f2-i⁒q⁒fΛ™)⁒Xk=0.subscriptΒ¨π‘‹π‘˜superscriptπœ…2π‘žsuperscript𝑓2π‘–π‘žΛ™π‘“subscriptπ‘‹π‘˜0\ddot{X}_{k}+{\left(\kappa^{2}+qf^{2}-i\sqrt{q}\dot{f}\right)}X_{k}=0\ . (2)

The comoving momentum kπ‘˜k enters the equation in the combination k2λ⁒φ~2≑κ2superscriptπ‘˜2πœ†superscript~πœ‘2superscriptπœ…2{k^{2}\over\lambda\tilde{\varphi}^{2}}\equiv\kappa^{2} ; therefore, the natural units of momentum are λ⁒φ~πœ†~πœ‘\sqrt{\lambda}\tilde{\varphi} . The background oscillations enter in the form f⁒(Ο„)=c⁒n⁒(Ο„,12)π‘“πœπ‘π‘›πœ12f(\tau)=cn\left(\tau,{1\over\sqrt{2}}\right) having unit amplitude. The combination of the coupling parameters h2λ≑qsuperscriptβ„Ž2πœ†π‘ž{h^{2}\over\lambda}\equiv q ultimately defines the solutions to Eq. ( 2).

Equation ( 2) will be the master equation in our investigation. We choose the vacuum positive-frequency initial condition Xk⁒(Ο„0)=𝒩k⁒e-i⁒κ⁒τsubscriptπ‘‹π‘˜subscript𝜏0subscriptπ’©π‘˜superscriptπ‘’π‘–πœ…πœX_{k}(\tau_{0})={\cal N}_{k}e^{-i\kappa\tau} . The normalization factor is 𝒩k=(2⁒Ωk⁒(Ξ©k+q⁒f))-1/2subscriptπ’©π‘˜superscript2subscriptΞ©π‘˜subscriptΞ©π‘˜π‘žπ‘“12{\cal N}_{k}=\left(2\Omega_{k}(\Omega_{k}+\sqrt{q}f)\right)^{-1/2} with the real part of the effective frequency given by Ξ©k2≑κ2+q⁒f⁒(Ο„)2superscriptsubscriptΞ©π‘˜2superscriptπœ…2π‘žπ‘“superscript𝜏2\Omega_{k}^{2}\equiv\kappa^{2}+qf(\tau)^{2} . Notice that the problem of the production of bose particles Ο‡πœ’\chi in the theory 14⁒λ⁒ϕ4+12⁒g2⁒ϕ2⁒χ214πœ†superscriptitalic-Ο•412superscript𝑔2superscriptitalic-Ο•2superscriptπœ’2{1\over 4}\lambda\phi^{4}+{1\over 2}g^{2}\phi^{2}\chi^{2} is reduced to an equation similar to ( 2) but without the imaginary part of the frequency and with the resonance parameter q=g2Ξ»π‘žsuperscript𝑔2πœ†q={g^{2}\over\lambda} (4).

Using the standard ΨΨ\Psi -field operator expansion, we can express the comoving occupation number of particles in a given spin state through the solution of Eq. ( 2) (see, e.g., (5)for details)

nk⁒(Ο„)=12-ΞΊ2Ξ©kβ’πΌπ‘šβ’(Xk⁒XΛ™k*)-q⁒f2⁒Ωk.subscriptπ‘›π‘˜πœ12superscriptπœ…2subscriptΞ©π‘˜πΌπ‘šsubscriptπ‘‹π‘˜superscriptsubscriptΛ™π‘‹π‘˜π‘žπ‘“2subscriptΞ©π‘˜n_{k}(\tau)={1\over 2}-{\kappa^{2}\over\Omega_{k}}{\it Im}\left(X_{k}\dot{X}_{% k}^{*}\right)-{\sqrt{q}f\over{2\Omega_{k}}}\ . (3)

Certainly nk≀1/2subscriptπ‘›π‘˜12n_{k}\leq 1/2 . The total comoving number density of particles and antiparticles in both spin states is 4(2⁒π)3⁒∫d3⁒k⁒nk4superscript2πœ‹3superscript𝑑3π‘˜subscriptπ‘›π‘˜{4\over(2\pi)^{3}}\int d^{3}k~{}n_{k} . Hence, the energy density of created fermions is ϡψ=12⁒π3⁒∫d3⁒k⁒Ωk⁒nksubscriptitalic-Ο΅πœ“12superscriptπœ‹3superscript𝑑3π‘˜subscriptΞ©π‘˜subscriptπ‘›π‘˜\epsilon_{\psi}={1\over 2\pi^{3}}\int d^{3}k~{}\Omega_{k}~{}n_{k} .

Eq. ( 2) can be solved numerically and the occupation number of particles can be found with ( 3). The results for nk⁒(Ο„)subscriptπ‘›π‘˜πœn_{k}(\tau) with different parameters qπ‘žq are plotted in Fig. 1.

FigureΒ 1: The occupation number nksubscriptπ‘›π‘˜n_{k} in λ⁒ϕ4πœ†superscriptitalic-Ο•4\lambda\phi^{4} -inflation as a function of time Ο„πœ\tau (in units of T𝑇T ) for q≑h2Ξ»=10-4π‘žsuperscriptβ„Ž2πœ†superscript104q\equiv{h^{2}\over\lambda}=10^{-4} (lower), 111 (middle on right), and 100100100 (upper on right) and ΞΊ2=0.18,1.11superscriptπœ…20.181.11\kappa^{2}=0.18,1.11 , and 11.911.911.9 , respectively. The period of the modulation πνkπœ‹subscriptπœˆπ‘˜\pi\over{\nu_{k}} (see Eq. ( 4)) is about 888888 , 202020 and 222222 (in units of T𝑇T ) accordingly.

The occupation number exhibits high frequency (period <T2absent𝑇2<{T\over 2} ) oscillations which are modulated by a long period behavior. If we average the occupation number over these high frequency oscillations, nΒ―k⁒(Ο„)=1Tβ’βˆ«Ο„(Ο„+T)𝑑τ⁒nk⁒(Ο„)subscriptΒ―π‘›π‘˜πœ1𝑇superscriptsubscriptπœπœπ‘‡differential-d𝜏subscriptπ‘›π‘˜πœ\bar{n}_{k}(\tau)={1\over T}\int_{\tau}^{(\tau+T)}d\tau n_{k}(\tau) , we can write the smoothed occupation number of fermions in a factorized form

nΒ―k⁒(Ο„)=Fk⁒sin2⁑νk⁒τ.subscriptΒ―π‘›π‘˜πœsubscriptπΉπ‘˜superscript2subscriptπœˆπ‘˜πœ\bar{n}_{k}(\tau)=F_{k}\sin^{2}\nu_{k}\tau\ . (4)

It is remarkable that, for all qπ‘žq , the occupation number of fermions evolves periodically with time (6). The spectrum (envelope) function FksubscriptπΉπ‘˜F_{k} and the period πνkπœ‹subscriptπœˆπ‘˜{\pi\over\nu_{k}} depend on the parameter qπ‘žq . Typically πνk∼10-100similar-toπœ‹subscriptπœˆπ‘˜10100{\pi\over\nu_{k}}\sim 10-100 , see Fig. 5. Therefore, the fermionic modes get fully exited around their time-average value Fk2subscriptπΉπ‘˜2F_{k}\over 2 within just tens of inflaton oscillations! This is very different from the result of the perturbative approximation which gives the timing for excitation βˆΌΞ“Ο•β†’Οˆβ’Οˆ-1similar-toabsentsuperscriptsubscriptΞ“β†’italic-Ο•πœ“πœ“1\sim\Gamma_{\phi\to\psi\psi}^{-1} which is about 1014superscript101410^{14} inflaton oscillations (3). Notice that for q≫1much-greater-thanπ‘ž1q\gg 1 , the number of fermions is almost constant between two successive zeros of the inflaton field. It jumps in a step-like manner at instances when ϕ⁒(t)italic-ϕ𝑑\phi(t) crosses zero. This is a familiar picture for the creation of bosons explained in (3); (4). However, contrary to the bosonic resonance which is unbounded, the fermionic resonance reaches its peak value Fk≀1/2subscriptπΉπ‘˜12F_{k}\leq 1/2 after a time Ο€2⁒νkπœ‹2subscriptπœˆπ‘˜\pi\over{2\nu_{k}} and then falls back to zero, where it begins the cycle again.

Fortunately, there is a semi-analytic shortcut for the calculation of FksubscriptπΉπ‘˜F_{k} and Ξ½ksubscriptπœˆπ‘˜\nu_{k} (7). We found that the envelope function is given by the expression

Fk=1sin2⁑νk⁒T⁒κ22⁒Ωk2⁒(πΌπ‘šβ’Xk(1)⁒(T))2.subscriptπΉπ‘˜1superscript2subscriptπœˆπ‘˜π‘‡superscriptπœ…22superscriptsubscriptΞ©π‘˜2superscriptπΌπ‘šsuperscriptsubscriptπ‘‹π‘˜1𝑇2F_{k}={1\over{\sin^{2}\nu_{k}T}}\,{\kappa^{2}\over{2\Omega_{k}^{2}}}\,\left({% \it Im}X_{k}^{(1)}(T)\right)^{2}\ . (5)

Here, Xk(1)⁒(T)superscriptsubscriptπ‘‹π‘˜1𝑇X_{k}^{(1)}(T) is the value of the first fundamental solution of Eq. ( 2) (defined by the intial conditions Xk(1)⁒(0)=1superscriptsubscriptπ‘‹π‘˜101X_{k}^{(1)}(0)=1 , XΛ™k(1)⁒(0)=0superscriptsubscriptΛ™π‘‹π‘˜100\dot{X}_{k}^{(1)}(0)=0 ) taken after one full oscillation; it is a function of kπ‘˜k which depends on the parameter qπ‘žq . The modulation frequency Ξ½ksubscriptπœˆπ‘˜\nu_{k} is given by the relation cos⁑νk⁒T=-𝑅𝑒⁒Xk(1)⁒(T)subscriptπœˆπ‘˜π‘‡π‘…π‘’superscriptsubscriptπ‘‹π‘˜1𝑇\cos\nu_{k}T=-{\it Re}X_{k}^{(1)}(T) . Therefore, to find FksubscriptπΉπ‘˜F_{k} and Ξ½ksubscriptπœˆπ‘˜\nu_{k} , one need only calculate the complex value Xk(1)⁒(T)superscriptsubscriptπ‘‹π‘˜1𝑇X_{k}^{(1)}(T) instead of performing a full numerical integration of Eq. ( 2).

FigureΒ 2: The envelope functions FksubscriptπΉπ‘˜F_{k} showing the bands of fermion resonance excitation in λ⁒ϕ4πœ†superscriptitalic-Ο•4\lambda\phi^{4} -inflation for q≑h2Ξ»=10-4,10-2formulae-sequenceπ‘žsuperscriptβ„Ž2πœ†superscript104superscript102q\equiv{h^{2}\over\lambda}=10^{-4},10^{-2} , and (the narrowest to broadest band, respectively). The band in the case q=10-2π‘žsuperscript102q=10^{-2} already deviates considerably from the perturbative expectation.

We calculated Xk(1)⁒(T)superscriptsubscriptπ‘‹π‘˜1𝑇X_{k}^{(1)}(T) numerically and constructed the envelope function FksubscriptπΉπ‘˜F_{k} plotted in Fig. 2. The spectrum of nksubscriptπ‘›π‘˜n_{k} is very different from what is expected from the perturbative calculations, where it is narrowly peaked around ΞΊ=Ο€Tβ‰ˆ0.424πœ…πœ‹π‘‡0.424\kappa={\pi\over T}\approx 0.424 with the width Ξ“Ο•β†’Οˆβ’Οˆ-1superscriptsubscriptΞ“β†’italic-Ο•πœ“πœ“1\Gamma_{\phi\to\psi\psi}^{-1} . In Figs. 3- 4we show, using ( 5), how the fermionic resonance bands are filled after 101010 background oscillations.

The function Ξ½ksubscriptπœˆπ‘˜\nu_{k} gives us the time scale for fermion excitation. In Fig. 5we plot the period of modulation πνkπœ‹subscriptπœˆπ‘˜\pi\over\nu_{k} as a function of kπ‘˜k . This function is peaked where FksubscriptπΉπ‘˜F_{k} is peaked, i.e. the peaks of the resonance curve are the last to fill. For q≳1greater-than-or-equivalent-toπ‘ž1q\gtrsim 1 , the period for modes in the principle peak is about 20⁒T20𝑇20T . When qβ‰ͺ1much-less-thanπ‘ž1q\ll 1 , modes in the principle peak grow more slowly, with a period of approximately Tqπ‘‡π‘ž{T\over\sqrt{q}} . Interestingly, the period of modes out of the FksubscriptπΉπ‘˜F_{k} -peaks is significantly shorter. For this reason, the integrated number of fermions is quickly saturated to its time-average value.

FigureΒ 3: The resonance excitation band of fermions in λ⁒ϕ4πœ†superscriptitalic-Ο•4\lambda\phi^{4} -inflation for q≑h2Ξ»=1.0π‘žsuperscriptβ„Ž2πœ†1.0q\equiv{h^{2}\over\lambda}=1.0 . The heavy line shows the envelope FksubscriptπΉπ‘˜F_{k} of maximum occupation number calculated from Eq. ( 5). The light curve is the actual occupation number nksubscriptπ‘›π‘˜n_{k} of each mode, calculated from Eq. ( 4), after 101010 background oscillations.

FigureΒ 4: As in Fig. 3, the resonance excitation band in λ⁒ϕ4πœ†superscriptitalic-Ο•4\lambda\phi^{4} -inflation for q≑h2Ξ»=100.0π‘žsuperscriptβ„Ž2πœ†100.0q\equiv{h^{2}\over\lambda}=100.0 . The heavy line shows the envelope FksubscriptπΉπ‘˜F_{k} . The light curve is the actual occupation number of each mode nksubscriptπ‘›π‘˜n_{k} after 101010 background oscillations.

The huge differences in the spectrum and rate of excitation of fermions produced by the coherently oscillating background field vs. those in the naive perturbative calculations are due to the interference of the fermion fluctuations. In this respect we can talk about the parametric excitation of fermions , where qπ‘žq plays the role of the resonance parameter, and the frequency Ξ½ksubscriptπœˆπ‘˜\nu_{k} can be viewed as the fermionic conterpart of the bosonic characteristic exponent ΞΌksubscriptπœ‡π‘˜\mu_{k} 11Notice that, for the resonant bands in the bosonic case, the modulating factor in Eq.Β (4) is replaced by sinh2⁑μk⁒tsuperscript2subscriptπœ‡π‘˜π‘‘{\sinh^{2}\mu_{k}t}. This leads to the familiar estimation nk∼e2⁒μk⁒tsimilar-tosubscriptπ‘›π‘˜superscript𝑒2subscriptπœ‡π‘˜π‘‘n_{k}\sim e^{2\mu_{k}t} for the number of bosons.. The amplitude of nksubscriptπ‘›π‘˜n_{k} shows a distinct sequence of resonance bands whose widths sharply narrow with increasing zone number. This is again reminiscent of the usual bosonic resonance. For qβ‰ͺ1much-less-thanπ‘ž1q\ll 1 , the resonance bands are located at ΞΊ=Ο€T⁒(2⁒l+1)πœ…πœ‹π‘‡2𝑙1\kappa={\pi\over T}(2l+1) , where l=0,1,2,𝑙012l=0,1,2, ….

FigureΒ 5: The log\log of the period of modulation πνkπœ‹subscriptπœˆπ‘˜{\pi\over\nu_{k}} (in units of T𝑇T ) as a function of ΞΊ2superscriptπœ…2\kappa^{2} for q≑h2Ξ»=10-4,10-2formulae-sequenceπ‘žsuperscriptβ„Ž2πœ†superscript104superscript102q\equiv{h^{2}\over\lambda}=10^{-4},10^{-2} , and for the light, dotted, and heavy curves respectively.

An important issue is the width of the parametric excitation of fermions. From Fig. 2we notice that for any qπ‘žq the first, broadest zone has the width Ξ”β’ΞΊβˆΌq1/4similar-toΞ”πœ…superscriptπ‘ž14\Delta\kappa\sim q^{1/4} . This can be understood from the condition for non-adiabatic excitation of particles, Ξ©Λ™kβ‰₯Ξ©k2subscriptΛ™Ξ©π‘˜superscriptsubscriptΞ©π‘˜2\dot{\Omega}_{k}\geq\Omega_{k}^{2} , which gives us Δ⁒k≀q1/4⁒λ⁒φ~Ξ”π‘˜superscriptπ‘ž14πœ†~πœ‘\Delta k\leq q^{1/4}\sqrt{\lambda}\tilde{\varphi} . This is similar to how the result was elaborated for the broad bosonic resonance (3). There are also very narrow zones with higher momenta, which are also quickly saturated. This may be important for the production of superheavy fermions with a mass mψ∼q1/4⁒mΟ•similar-tosubscriptπ‘šπœ“superscriptπ‘ž14subscriptπ‘šitalic-Ο•m_{\psi}\sim q^{1/4}m_{\phi} , where the effective inflaton mass mϕ≃1013similar-to-or-equalssubscriptπ‘šitalic-Ο•superscript1013m_{\phi}\simeq 10^{13} Gev.

Let us estimate the energy that will be accumulated in created fermions. Averaging over short-period oscillations, we have

ϡ¯ψ=2Ο€2β’βˆ«π‘‘k⁒k2⁒Fk⁒sin2⁑νk⁒τ.subscriptΒ―italic-Ο΅πœ“2superscriptπœ‹2differential-dπ‘˜superscriptπ‘˜2subscriptπΉπ‘˜superscript2subscriptπœˆπ‘˜πœ\bar{\epsilon}_{\psi}={2\over\pi^{2}}\int dk\,k^{2}\,F_{k}\,\sin^{2}\nu_{k}% \tau\ . (6)

For the most interesting case of the broad resonance excitation, q≫1much-greater-thanπ‘ž1q\gg 1 , we get ϡψ∼0.1⁒h2⁒q1/4⁒ϡϕsimilar-tosubscriptitalic-Ο΅πœ“0.1superscriptβ„Ž2superscriptπ‘ž14subscriptitalic-Ο΅italic-Ο•\epsilon_{\psi}\sim 0.1h^{2}q^{1/4}\epsilon_{\phi} , where the inflaton energy is ϡϕ=14⁒λ⁒φ~4subscriptitalic-Ο΅italic-Ο•14πœ†superscript~πœ‘4\epsilon_{\phi}={1\over 4}\lambda\tilde{\varphi}^{4} . In chaotic 14⁒λ⁒ϕ414πœ†superscriptitalic-Ο•4{1\over 4}\lambda\phi^{4} -inflation, λ≃10-13similar-to-or-equalsπœ†superscript1013\lambda\simeq 10^{-13} . If the resonance parameter qπ‘žq is large but the coupling parameter is small, h≀0.1β„Ž0.1h\leq 0.1 , then only a small fraction of the inflaton energy will be transferred into fermions. However, if we push the parameters to h≳0.3greater-than-or-equivalent-toβ„Ž0.3h\gtrsim 0.3 ( q≳1012greater-than-or-equivalent-toπ‘žsuperscript1012q\gtrsim 10^{12} ), the inflaton energy can be transferred into fermions. This is quite an extreme case, because it requires SUSY cancellation of the radiative corrections ∼h4⁒ϕ4similar-toabsentsuperscriptβ„Ž4superscriptitalic-Ο•4\sim h^{4}\phi^{4} . However, in the hybrid inflation scenario, for example, there is more freedom to chose the model parameters which control the efficiency of preheating (8). There may be a range of parameters for which the inflaton energy is fully converted into fermions even before the bosonic resonance develops.

Let us now turn to the model 12⁒m2⁒ϕ2+hβ’ΟˆΒ―β’Ο•β’Οˆ12superscriptπ‘š2superscriptitalic-Ο•2β„ŽΒ―πœ“italic-Ο•πœ“{1\over 2}m^{2}\phi^{2}+h\bar{\psi}\phi\psi , where mπ‘šm is the inflaton mass. In this inflationary model the inflaton evolution is simply described by harmonic oscillations, Ο•β‰ˆΞ¦β’(t)⁒sin⁑m⁒titalic-Ο•Ξ¦π‘‘π‘šπ‘‘\phi\approx\Phi(t)\sin mt , with decreasing amplitude, Φ⁒(t)=Mp3⁒π⁒m⁒tΦ𝑑subscript𝑀𝑝3πœ‹π‘šπ‘‘\Phi(t)={M_{p}\over{\sqrt{3\pi}mt}} . However, the problem of fermion production in this model cannot be reduced to a problem in Minkowski space-time; the expansion of the universe is essential. Let us assume, for a moment, that there is no expansion and the amplitude Φ⁒(t)=c⁒o⁒n⁒s⁒tΞ¦π‘‘π‘π‘œπ‘›π‘ π‘‘\Phi(t)=const . Then the master equation would be Eq. ( 2) where now the resonance parameter q≑h2⁒Φ2m2π‘žsuperscriptβ„Ž2superscriptΞ¦2superscriptπ‘š2q\equiv{h^{2}{\Phi^{2}}\over{m^{2}}} and ΞΊ2≑k2m2superscriptπœ…2superscriptπ‘˜2superscriptπ‘š2\kappa^{2}\equiv{{k^{2}}\over{m^{2}}} (c.f. the bosonic resonance (3)). The result for the occupation number of fermions in this case can also be found with Eq. ( 5) and is plotted in Fig. 6.

FigureΒ 6: The resonance band in m2⁒ϕ2superscriptπ‘š2superscriptitalic-Ο•2m^{2}\phi^{2} -inflation with q≑h2⁒Φ2m2=10-4,10-2formulae-sequenceπ‘žsuperscriptβ„Ž2superscriptΞ¦2superscriptπ‘š2superscript104superscript102q\equiv{h^{2}{\Phi^{2}}\over{m^{2}}}=10^{-4},10^{-2} , and for the narrowest to broadest band, respectively. The band in the case q=10-2π‘žsuperscript102q=10^{-2} already deviates considerably from the perturbative expectation.

Without expansion of the universe, the qualitative result is similar to that for the conformal theory. However, if we take the expansion into account, we can expect an adiabatic change in the structure of the fermion parametric excitation. This is similar to the erasure of differences between resonant and non-resonant modes and the onset of stochastic resonance in the bosonic case (3). Therefore, we conjecture that, for the massive inflaton, the parametric excitation of fermions fills a Fermi-sphere of radius Δ⁒k∼q1/4⁒msimilar-toΞ”π‘˜superscriptπ‘ž14π‘š\Delta k\sim q^{1/4}m without distinct resonance zones.

If expansion redshifts the fermion modes fast enough, it can prevent them from being resonantly excited. We found that for q0≀1subscriptπ‘ž01q_{0}\leq 1 or h≀10-6β„Žsuperscript106h\leq 10^{-6} we return to the perturbation theory result for the rate of fermion pair production 1a4⁒dd⁒t⁒(a4⁒nψ¯⁒ψ)β‰ˆΞ“Ο•β†’Οˆβ’Οˆβ’nΟ•1superscriptπ‘Ž4𝑑𝑑𝑑superscriptπ‘Ž4subscriptπ‘›Β―πœ“πœ“subscriptΞ“β†’italic-Ο•πœ“πœ“subscript𝑛italic-Ο•{1\over a^{4}}{d\over dt}(a^{4}n_{\bar{\psi}\psi})\approx\Gamma_{\phi\to\psi% \psi}n_{\phi} , where the comoving number density of inflatons is nΟ•=m⁒Φ2⁒a3/2subscript𝑛italic-Ο•π‘šsuperscriptΞ¦2superscriptπ‘Ž32n_{\phi}=m\Phi^{2}a^{3}/2 . This is to be interpreted as the creation of a pair of fermions, ΟˆΒ―Β―πœ“\bar{\psi} and Οˆπœ“\psi , from the decay of a single inflaton at the instant of resonance between a given momentum kπ‘˜k and inflaton mass mπ‘šm , k=12⁒m⁒a⁒(t)π‘˜12π‘šπ‘Žπ‘‘k={1\over 2}ma(t) .

In this paper we studied the production of fermions by an oscillating inflaton field, and found that fermions are created in a regime of parametric excitation, which is very different from the perturbative regime. This can be important to many interesting cosmological applications including the channel of inflaton energy transfer, the estimation of the reheating temperature, the production of superheavy fermions, the production of gravitinos in the supergravity inflationary models (9), etc. It also elucidates another feature of bosonic preheating, missed in the previous studies. Indeed, in the resonance bands bosons are exponentially unstable, nk∼e2⁒μk⁒tsimilar-tosubscriptπ‘›π‘˜superscript𝑒2subscriptπœ‡π‘˜π‘‘n_{k}\sim e^{2\mu_{k}t} . However, outside of the resonance bands ΞΌksubscriptπœ‡π‘˜\mu_{k} is imaginary and we will have bounded excitation of bosonic fluctuations nk∼O⁒(1)similar-tosubscriptπ‘›π‘˜π‘‚1n_{k}\sim O(1) given by Eq. ( 4). How this will alter the preheating scenario and its lattice simulation, based on classical approximation of fluctuations (10), shall be seen.

The authors are grateful to Andrei Linde for fruitful discussions, and organizers of the CAPP98 Workshop at CERN, June 7-12, where the results of the paper were reported 22See for transparencies.. This work was supported by NSF grant AST95-29-225. After this work was completed, we learned about related report (11).


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