3.3 Reheating after inflation

We discuss the dynamics of reheating and the resulting particle production in the Jordan frame for the model (3.6View Equation). The inflationary period is followed by a reheating phase in which the second derivative ¨ R can no longer be neglected in Eq. (3.8View Equation). Introducing ˆ 3∕2 R = a R, we have
¨ ( 3 3 ) Rˆ+ M 2 − -H2 − -H˙ Rˆ = 0. (3.28 ) 4 2
Since M 2 ≫ {H2, | ˙H |} during reheating, the solution to Eq. (3.28View Equation) is given by that of the harmonic oscillator with a frequency M. Hence the Ricci scalar exhibits a damped oscillation around R = 0:
R ∝ a−3∕2sin (M t). (3.29 )

Let us estimate the evolution of the Hubble parameter and the scale factor during reheating in more detail. If we neglect the r.h.s. of Eq. (3.7View Equation), we get the solution H (t) = const × cos2(M t∕2). Setting 2 H (t) = f(t)cos (M t∕2) to derive the solution of Eq. (3.7View Equation), we obtain [426Jump To The Next Citation Point]

1 f(t) = ------------------------------------------, (3.30 ) C + (3∕4 )(t − tos) + 3∕(4M )sin [M (t − tos)]
where tos is the time at the onset of reheating. The constant C is determined by matching Eq. (3.30View Equation) with the slow-roll inflationary solution ˙H = − M 2∕6 at t = tos. Then we get C = 3∕M and
[ ]−1 [ ] H (t) = -3-+ 3(t − tos) +-3--sin M (t − tos) cos2 M-(t − tos) . (3.31 ) M 4 4M 2
Taking the time average of oscillations in the regime M (t − tos) ≫ 1, it follows that −1 ⟨H ⟩ ≃ (2∕3 )(t − tos). This corresponds to the cosmic evolution during the matter-dominated epoch, i.e., 2∕3 ⟨a⟩ ∝ (t − tos). The gravitational effect of coherent oscillations of scalarons with mass M is similar to that of a pressureless perfect fluid. During reheating the Ricci scalar is approximately given by R ≃ 6 ˙H, i.e.
[ 3 3 3 ]− 1 R ≃ − 3 ---+ -(t − tos) + ----sinM (t − tos) M sin [M (t − tos)]. (3.32 ) M 4 4M
In the regime M (t − t ) ≫ 1 os this behaves as
4M R ≃ − ------ sin [M (t − tos)]. (3.33 ) t − tos

In order to study particle production during reheating, we consider a scalar field χ with mass m χ. We also introduce a nonminimal coupling 2 (1∕2 )ξRχ between the field χ and the Ricci scalar R [88Jump To The Next Citation Point]. Then the action is given by

∫ [ ] S = d4x√ −-g f(R-)− 1gμν∂ μχ∂νχ − 1m2 χ2 − 1-ξR χ2 , (3.34 ) 2κ2 2 2 χ 2
where 2 2 f(R ) = R + R ∕(6M ). Taking the variation of this action with respect to χ gives
□ χ − m2 χ − ξR χ = 0. (3.35 ) χ
We decompose the quantum field χ in terms of the Heisenberg representation:
∫ --1---- 3 ( −ik⋅x † ∗ ik⋅x) χ(t,x ) = (2π)3∕2 d k ˆakχk(t)e + ˆakχ k(t)e , (3.36 )
where ˆak and ˆa†k are annihilation and creation operators, respectively. The field χ can be quantized in curved spacetime by generalizing the basic formalism of quantum field theory in the flat spacetime. See the book [88Jump To The Next Citation Point] for the detail of quantum field theory in curved spacetime. Then each Fourier mode χ (t) k obeys the following equation of motion
( 2 ) ¨χk + 3H ˙χk + k--+ m2 + ξR χk = 0, (3.37 ) a2 χ
where k = |k| is a comoving wavenumber. Introducing a new field uk = aχk and conformal time ∫ −1 η = a dt, we obtain
d2u [ ( 1) ] ---2k + k2 + m2χa2 + ξ − -- a2R uk = 0, (3.38 ) dη 6
where the conformal coupling correspond to ξ = 1∕6. This result states that, even though ξ = 0 (that is, the field is minimally coupled to gravity), R still gives a contribution to the effective mass of u k. In the following we first review the reheating scenario based on a minimally coupled massless field (ξ = 0 and m χ = 0). This corresponds to the gravitational particle production in the perturbative regime [565606426Jump To The Next Citation Point]. We then study the case in which the nonminimal coupling |ξ| is larger than the order of 1. In this case the non-adiabatic particle production preheating [584353Jump To The Next Citation Point538354Jump To The Next Citation Point] can occur via parametric resonance.

3.3.1 Case: ξ = 0 and m χ = 0

In this case there is no explicit coupling among the fields χ and R. Hence the χ particles are produced only gravitationally. In fact, Eq. (3.38View Equation) reduces to

2 d-uk-+ k2uk = U uk, (3.39 ) dη2
where U = a2R ∕6. Since U is of the order of (aH )2, one has k2 ≫ U for the mode deep inside the Hubble radius. Initially we choose the field in the vacuum state with the positive-frequency solution [88Jump To The Next Citation Point]: (i) − ikη √ --- uk = e ∕ 2k. The presence of the time-dependent term U(η) leads to the creation of the particle χ. We can write the solution of Eq. (3.39View Equation) iteratively, as [626]
1 ∫ η uk (η ) = u(ik)+ -- U (η′)sin [k(η − η′)]uk (η′)dη ′. (3.40 ) k 0

After the universe enters the radiation-dominated epoch, the term U becomes small so that the flat-space solution is recovered. The choice of decomposition of χ into ˆak and † ˆak is not unique. In curved spacetime it is possible to choose another decomposition in term of new ladder operators 𝒜ˆk and 𝒜ˆ† k, which can be written in terms of ˆak and ˆa† k, such as ˆ ∗ † 𝒜k = αkˆak + β kˆa−k. Provided that ∗ βk ⁄= 0, even though ˆak |0⟩ = 0, we have ˆ 𝒜k |0⟩ ⁄= 0. Hence the vacuum in one basis is not the vacuum in the new basis, and according to the new basis, the particles are created. The Bogoliubov coefficient describing the particle production is

i ∫ ∞ ′ βk = − --- U (η′)e−2ikηdη ′. (3.41 ) 2k 0
The typical wavenumber in the η-coordinate is given by k, whereas in the t-coordinate it is k∕a. Then the energy density per unit comoving volume in the η-coordinate is [426Jump To The Next Citation Point]
∫ --1--- ∞ 2 2 ρη = (2π)3 4πk dk ⋅ k|βk| ∫ ∞0 ∫ ∞ ∫ ∞ = -1-- d ηU (η ) dη ′U (η′) dk ⋅ ke2ik(η′−η) 8π2 0 0 0 1 ∫ ∞ dU ∫ ∞ U(η′) = ----2 d η---- dη ′′----, (3.42 ) 32π 0 d η 0 η − η
where in the last equality we have used the fact that the term U approaches 0 in the early and late times.

During the oscillating phase of the Ricci scalar the time-dependence of U is given by ∫ η U = I(η)sin( 0 ωd ¯η), where I(η) = ca(η)1∕2 and ω = M a (c is a constant). When we evaluate the term dU ∕d η in Eq. (3.42View Equation), the time-dependence of I(η) can be neglected. Differentiating Eq. (3.42View Equation) in terms of η and taking the limit ∫ ηωd ¯η ≫ 1 0, it follows that

( ) dρη ω 2 2 ∫ η ----≃ ----I (η )cos ωd ¯η , (3.43 ) d η 32π 0
where we used the relation limk → ∞ sin (kx )∕x = π δ(x). Shifting the phase of the oscillating factor by π ∕2, we obtain
d ρη M U2 M a4R2 -dt- ≃ -32π--= 1152-π-. (3.44 )
The proper energy density of the field χ is given by ρ χ = (ρ η∕a)∕a3 = ρη∕a4. Taking into account g∗ relativistic degrees of freedom, the total radiation density is
g g ∫ t M a4R2 ρM = -∗4ρη = --∗4 -------dt, (3.45 ) a a tos 1152π
which obeys the following equation
g∗M R2 ˙ρM + 4H ρM = -------. (3.46 ) 1152 π
Comparing this with the continuity equation (2.17View Equation) we obtain the pressure of the created particles, as
1- g∗M-R2-- PM = 3 ρM − 3456πH . (3.47 )
Now the dynamical equations are given by Eqs. (2.15View Equation) and (2.16View Equation) with the energy density (3.45View Equation) and the pressure (3.47View Equation).

In the regime M (t − tos) ≫ 1 the evolution of the scale factor is given by 2∕3 a ≃ a0(t − tos), and hence

4 H2 ≃ --------2-, (3.48 ) 9(t − tos)
where we have neglected the backreaction of created particles. Meanwhile the integration of Eq. (3.45View Equation) gives
g∗M 3 1 ρM ≃ 240π-t-−-t- , (3.49 ) os
where we have used the averaged relation ⟨R2 ⟩ ≃ 8M 2∕(t − tos)2 [which comes from Eq. (3.33View Equation)]. The energy density ρM evolves slowly compared to H2 and finally it becomes a dominant contribution to the total energy density (3H2 ≃ 8π ρM ∕m2 pl) at the time tf ≃ tos + 40m2 ∕(g∗M 3) pl. In [426Jump To The Next Citation Point] it was found that the transition from the oscillating phase to the radiation-dominated epoch occurs slower compared to the estimation given above. Since the epoch of the transient matter-dominated era is about one order of magnitude longer than the analytic estimation [426Jump To The Next Citation Point], we take the value tf ≃ tos + 400m2pl∕ (g∗M 3) to estimate the reheating temperature Tr. Since the particle energy density ρM (tf) is converted to the radiation energy density ρ = g π2T 4∕30 r ∗ r, the reheating temperature can be estimated as4
( M )3 ∕2 Tr ≲ 3 × 1017g1∗∕4 ---- GeV. (3.50 ) mpl
As we will see in Section 7, the WMAP normalization of the CMB temperature anisotropies determines the mass scale to be M ≃ 3 × 10− 6m pl. Taking the value g = 100 ∗, we have T ≲ 5 × 109 GeV r. For t > tf the universe enters the radiation-dominated epoch characterized by 1∕2 a ∝ t, R = 0, and ρr ∝ t−2.

3.3.2 Case: |ξ| ≳ 1

If |ξ| is larger than the order of unity, one can expect the explosive particle production called preheating prior to the perturbative regime discussed above. Originally the dynamics of such gravitational preheating was studied in [70592] for a massive chaotic inflation model in Einstein gravity. Later this was extended to the f (R) model (3.6View Equation[591Jump To The Next Citation Point].

Introducing a new field Xk = a3∕2χk, Eq. (3.37View Equation) reads

( ) k2 2 9 2 3 ¨Xk + -2-+ m χ + ξR − -H − --˙H Xk = 0. (3.51 ) a 4 2
As long as |ξ| is larger than the order of unity, the last two terms in the bracket of Eq. (3.51View Equation) can be neglected relative to ξR. Since the Ricci scalar is given by Eq. (3.33View Equation) in the regime M (t − tos) ≫ 1, it follows that
[k2 4M ξ ] X¨k + --2 + m2χ − ------sin{M (t − tos)} Xk ≃ 0. (3.52 ) a t − tos

The oscillating term gives rise to parametric amplification of the particle χ k. In order to see this we introduce the variable z defined by M (t − tos) = 2z ± π ∕2, where the plus and minus signs correspond to the cases ξ > 0 and ξ < 0 respectively. Then Eq. (3.52View Equation) reduces to the Mathieu equation

-d2- dz2 Xk + [Ak − 2q cos(2z)]Xk ≃ 0, (3.53 )
where
4k2 4m2 8|ξ| Ak = ------+ ---χ, q = ----------. (3.54 ) a2M 2 M 2 M (t − tos)
The strength of parametric resonance depends on the parameters Ak and q. This can be described by a stability-instability chart of the Mathieu equation [419353Jump To The Next Citation Point591Jump To The Next Citation Point]. In the Minkowski spacetime the parameters Ak and q are constant. If Ak and q are in an instability band, then the perturbation X k grows exponentially with a growth index μ k, i.e., X ∝ eμkz k. In the regime q ≪ 1 the resonance occurs only in narrow bands around 2 Ak = ℓ, where ℓ = 1,2,..., with the maximum growth index μk = q∕2 [353]. Meanwhile, for large q (≫ 1), a broad resonance can occur for a wide range of parameter space and momentum modes [354Jump To The Next Citation Point].

In the expanding cosmological background both A k and q vary in time. Initially the field X k is in the broad resonance regime (q ≫ 1) for |ξ| ≫ 1, but it gradually enters the narrow resonance regime (q ≲ 1). Since the field passes many instability and stability bands, the growth index μk stochastically changes with the cosmic expansion. The non-adiabaticity of the change of the frequency ω2k = k2∕a2 + m2χ − 4M ξ sin{M (t − tos)}∕(t − tos) can be estimated by the quantity

| | ||˙ωk-|| ----|k2∕a2-+-2M-ξ-cos{M-(t-−-tos)}∕(t −-tos)|---- rna ≡ |ω2 | = M |k2∕a2 + m2 − 4M ξ sin{M (t − tos)}∕(t − tos)|3∕2, (3.55 ) k χ
where the non-adiabatic regime corresponds to rna ≳ 1. For small k and m χ we have rna ≫ 1 around M (t − tos) = nπ, where n are positive integers. This corresponds to the time at which the Ricci scalar vanishes. Hence, each time R crosses 0 during its oscillation, the non-adiabatic particle production occurs most efficiently. The presence of the mass term m χ tends to suppress the non-adiabaticity parameter rna, but still it is possible to satisfy the condition rna ≳ 1 around R = 0.

For the model (3.6View Equation) it was shown in [591] that massless χ particles are resonantly amplified for |ξ| ≳ 3. Massive particles with m χ of the order of M can be created for |ξ| ≳ 10. Note that in the preheating scenario based on the model 2 2 2 2 2 V(ϕ, χ) = (1∕2)m ϕϕ + (1∕2)g ϕ χ the parameter q decreases more rapidly (2 q ∝ 1 ∕t) than that in the model (3.6View Equation[354Jump To The Next Citation Point]. Hence, in our geometric preheating scenario, we do not require very large initial values of q [such as q > 𝒪 (103)] to lead to the efficient parametric resonance.

While the above discussion is based on the linear analysis, non-linear effects (such as the mode-mode coupling of perturbations) can be important at the late stage of preheating (see, e.g., [354342]). Also the energy density of created particles affects the background cosmological dynamics, which works as a backreaction to the Ricci scalar. The process of the subsequent perturbative reheating stage can be affected by the explosive particle production during preheating. It will be of interest to take into account all these effects and study how the thermalization is reached at the end of reheating. This certainly requires the detailed numerical investigation of lattice simulations, as developed in [255254].

At the end of this section we should mention a number of interesting works about gravitational baryogenesis based on the interaction ∫ √ --- (1∕M 2∗) d4x − gJ μ∂μR between the baryon number current J μ and the Ricci scalar R (M ∗ is the cut-off scale characterizing the effective theory) [179376514]. This interaction can give rise to an equilibrium baryon asymmetry which is observationally acceptable, even for the gravitational Lagrangian n f(R ) = R with n close to 1. It will be of interest to extend the analysis to more general f (R) gravity models.


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