Effective action approach for preheating

We present a semiclassical non-perturbative approach for calculating the preheating process at the end of inflation. Our method involves integrating out the decayed particles within the path integral framework and subsequently determining world-line instanton solutions in the effective action. This enables us to obtain the effective action of the inflaton, with its imaginary part linked to the phenomenon of particle creation driven by coherent inflaton field oscillations. Additionally, we utilize the Bogoliubov transformation to investigate the evolution of particle density within the medium after multiple inflaton oscillations. We apply our approach to various final state particles, including scalar fields, tachyonic fields, and gauge fields. The non-perturbative approach provides analytical results for preheating that are in accord with previous methods.


Introduction
Compelling evidence from cosmological observations points out the existence of an inflationary phase [1][2][3][4][5], a critical element in the history of the Universe.Inflation refers to a primordial epoch at the beginning of the universe, characterized by a slowly varying vacuum energy.This energy is typically attributed to a scalar field known as the inflaton, which accelerates the expansion of the Universe, leading to a supercooled phase by the exponential growth.
In parallel, we have confidence that the Universe underwent a thermal history from electron-positron annihilation with temperature about MeV.To link the supercooled inflation phase with the thermal history, a crucial intermediary step named reheating [6][7][8][9][10][11][12][13] becomes apparent.Reheating entails introducing interactions between the inflaton field and other fields, including standard model particles.We must have these couplings to terminate inflation by facilitating the decay of the inflaton field, thereby generating entropy in the Universe.The decayed particles would reach thermal equilibrium through mutual interactions, initiating the thermal history of the Universe.
However, the conventional notion of perturbative decay does not offer a comprehensive description of the reheating process.This is particularly true when dealing with the decay of a coherent oscillating field at the end of inflation, which differs fundamentally from the decay of individual particles.One intriguing possibility arises when the decayed particles are bosons, leading to Bose-enhanced production rates.More notably, a sizeable coupling between the inflaton and the decayed particles can give rise to non-perturbative particle production via parametric resonances.These enhancement effects typically occur before the perturbative decay, known as preheating.The concept of preheating was initially proposed in [9], and further detailed studies are given in [11][12][13] (see [14] for a review).
Future experiments are opening new windows to explore the critical and phenomenologically rich reheating period, though observing reheating or preheating directly is challenging.
In this paper, we concentrate on the theoretical aspects of preheating and present an alternative perspective on preheating by employing an effective action method.Preheating typically involves the generation of particles when the oscillating inflaton field crosses the minimum of its potential.During this crossing, the frequencies of the newly created particles change non-adiabatically, leading to resonant particle production.The production can be comprehended through the examination of the mode functions of the decayed particles, which can be solved either numerically or by employing the WKB method (as discussed in [13]).Analysis of the mode functions reveals that resonant production occurs within a narrow or broad range of momentum space contingent upon the coupling strength between the inflaton field and the daughter particle.A small coupling results in a narrow instability band where the momentum of decayed particles is close to the mass of the inflaton, assuming the bare mass of the decayed particles is significantly smaller.Resonance becomes much more efficient when the coupling is substantial, introducing significant quantum corrections to the Lagrangian.This is a non-perturbative process involving multi-particle scatterings.Consequently, it becomes imperative to sum all relevant processes within the coherent inflaton background.An essential tool for this purpose is the quantum effective action of the inflaton field, derived by integrating out the decayed fields, which incorporates these non-perturbative quantum corrections.This, in turn, allows us to deduce the resonantly produced number density of the decayed particles, serving as the motivation to explore the one-loop effective action method to gain insight into the non-perturbative preheating process.
Additionally, our application of the effective action method to inflaton decay is inspired by a related phenomenon, the Schwinger mechanism [59], which elucidates the nonperturbative pair production of electrons and positrons in a strong electric field.Both preheating and Schwinger pair production are related to particle creation in some backgrounds: one occurs in the presence of an electric field, and the other in an oscillating inflaton background.In the context of the Schwinger mechanism, one can use the WKB method to solve the mode functions of decayed particles [60] or employ the effective action method to derive the vacuum persistence rate [61][62][63][64][65].The two methods complement each other.In the case of preheating, the literature already contains references to the WKB method (see e.g.[13,66]).In this work, we embark on the exploration of the effective action method for understanding non-perturbative inflaton decays.This method has the potential to apply to various particle production processes in cosmology.
The structure of this paper is organized as follows.Section 2 describes inflaton perturbative decay through the effective action method, serving as a preliminary exercise.In section 3, we calculate the effective action by utilizing instanton solutions to illustrate the non-perturbative particle creation during the preheating period and introduce the Bogoliubov transformation to account for the evolution of particle number density.In section 4, we apply our approach to other models, including tachyonic resonance and gauge preheating.We conclude with a summary and discussion of our findings in section 5. Appendix includes the numerical and analytic methods for calculating the effective action in the presence of a periodic potential.

Reheating -perturbative inflaton decay
In this section, we delve into the analysis of reheating, specifically focusing on perturbative inflaton decay.We employ the effective action method as a preliminary example, which establishes the equivalence between the perturbative decay rate of a coherent oscillating inflaton and that of a particle with the same mass and coupling.
Let us consider a reheating model where an inflaton ϕ is a real scalar field with mass M ϕ .It interacts with another real scalar field χ with mass m χ , and the interaction is characterized by a trilinear coupling g.We can decompose the action into three parts: the free action of the inflaton, S 0 [ϕ], the free action of the χ field, S 0 [ϕ], and the interaction part, S I [ϕ, χ].The action takes the following form (2.1) where the D'Alembert operator □ = ∂ µ ∂ µ .The rate at which an inflaton particle decays at rest can be deduced using a standard S-matrix approach, In the case of reheating, when we neglect the expansion of the universe and the backreaction of inflaton decays, the classical solution of a homogeneous inflaton field with the maximum amplitude ϕ 0 is given by the field equation derived from the free inflaton action S 0 [ϕ], ϕ(t) = ϕ 0 sin(M ϕ t) . (2.3) Here, the inflaton is coherently oscillating, and the effective action approach is a suitable tool to consider the decay of a background field.We expand the interaction action to the second power, and then integrate the χ fields using the path integration, leading to the effective action of ϕ, where ⟨S I ⟩ = Dχe iS 0 [χ] S I and ⟨S 2 I ⟩ is similar.We then treat the inflaton field ϕ as a classical background to eliminate Dϕ and obtain the form of the effective action, where The vacuum persistence probability is calculated using the S-matrix amplitude with the effective action |⟨ϕ|e iS ef f |ϕ⟩| 2 = e −2 Im S ef f (ϕ) , which is related to the coherent inflaton decay rate.The leading imaginary part of the effective action is given by second order in g, and we use the background ϕ(x) given in eq. ( 2.3) to write the imaginary part of the effective action, (2.7) Here, we use the imaginary part of i∆ 2 (2.8) The vacuum persistence probability leads to the inflaton background's total decay rate per unit time per unit volume, The number density of the inflaton field is given by This result is consistent with the number density times the particle decay rate, i.e., R(ϕ → χχ) = n ϕ Γ(ϕ → χχ).Hence, the inflaton decay rate deduced from the effective action approach is equivalent to the particle decay rate given in eq.(2.2).

Parametric Resonance
In the period of the end of inflation, non-perturbative effects come into play when the inflaton condensation oscillates near the minimum of its potential.Two well-established mechanisms are of particular interest: parametric resonance and tachyonic instability.In this section, we will delve into the phenomenon of parametric resonance using the effective action approach and then explore the tachyonic instability in the following section.We begin by considering an inflaton model in which the inflaton field couples with a scalar field χ via a quartic interaction term, represented as g 2 2 ϕ 2 χ 2 .The action for this model takes the following form, 1 By dimensional regularization, one can show that where µ is the renormalization scale.
Given the oscillation behavior of the inflaton condensation as described in eq. ( 2.3), the field equation for χ can be mapped to the Mathieu equation, which manifests certain instability regions in momentum space (k-space).Depending on the strength of the coupling, when the coupling is relatively small (g ≪ m/ϕ 0 ), resonance occurs in a narrow instability band.Conversely, in the case of large coupling (g > m/ϕ 0 ), the resonance effect spans a broader band, leading to more efficient inflaton decays.The broad band resonance is non-perturbative.
It can be understood that the rate of the process involving the annihilation of multiple inflatons into χ particles (nϕ → 2χ) is comparable to or even exceeds the rate of the process 2ϕ → 2χ.Consequently, in this regime, the decay is inherently non-perturbative, resulting in the opening of broad resonance bands.Note that our primary focus here is on the effects of broad resonance.We have checked that the narrow resonance using the effective action method gives an inflaton background decay rate with the Bose enhancement, consistent with the narrow resonance study in [13,66].
Since the non-perturbative inflaton decay yields two χ particles as final states, we evaluate this process using the one-loop effective action approach by integrating out the χ fields, Subsequently, with the effective action of ϕ in hand, we employ the Bogoliubov transformation to investigate the stimulated particle production and gain insights into the evolution of the number density of χ.

One-loop effective action
The oscillating background field ϕ(t) = ϕ 0 (t) sin(M ϕ t) imparts a time-dependent frequency ω(t) to the χ particle.The adiabatic condition ω(t)/ω 2 (t) is significantly violated when the oscillating inflaton passes the minimum of the potential.At the minimum, the field value vanishes at t j = j π/M ϕ , where j is an integer.However, for majority of time, ω(t)/ω 2 (t) ≪ 1, ensuring the conservation of the number of χ particles when inflaton is not near the potential minimum.Therefore, it is sufficient to investigate particle creation when ϕ(t) is close to 0, and ϕ 2 (t) takes an approximate form, This approximation simplifies the calculation of the effective action method and yields analytic results.Additionally, in a flat space, ϕ 0 is a constant, while in an expanding universe, the time-dependence of ϕ 0 (t) needs to be taken into account.Equation (3.3) remains a valid approximation as long as the variation of ϕ 0 (t) is small compared to the inflaton mass M ϕ .Since the time duration of preheating is small compared to the Hubble time scale, these assumptions are valid and self-consistent during the preheating process [13,66].
There is one more complication in the χ particle production rate.The rate is momentumdependent, and due to medium effects, the rates for different modes vary with time according to their occupation number.When integrating out all χ particle modes in the path integration, we will attain the vacuum persistence probability, which is not a useful quantity to understand the momentum-dependent preheating process.Instead, we factorize the χ field into k modes and derive the decay probability for individual k mode via the effective action method.
Since the interaction terms are quadratic in the field χ, different k modes are not coupled together.This is evident from the action, For convenience, we discretize the momentum k, and then integrate out the χ field mode by mode in the Feynman path integration, where N is a k-and t-independent normalization constant.The path integral gives the effective action of the inflaton field with the summation over all k, We compute the effective action for a given momentum k by introducing the Schwinger proper time s, where Ĥ = ∂2 t + g 2 ϕ 2 (t).Here, we sum over a one-particle Hilbert space spanned by |t⟩ via the integration dt.To obtain the effective action, we first compute the amplitude of the particle propagating to the same state after the proper time s.Several approaches can be used to calculate the time t integration in eq.(3.8).The world-line instanton approach is presented here.This approach is used in this section on parametric resonance and the next section on tachyonic instability.An alternative method of summing over all the eigenstates is presented after the instanton approach.

Instanton approach
The instanton approach leverages the Schwinger proper time and finds the semi-classical instanton solutions to the amplitude for quantum mechanical states evolving in the Schwinger time.This approach has been successfully applied to the problems involving electron-positron pair production in the presence of electromagnetic backgrounds [61][62][63][64][65].
Starting from the effective action formula given in eq.(3.8), we rotate both time t and Schwinger time s into Euclidean ones, t → iτ and s → −is. 2 The expectation value of the evolution operator is then evaluated by summing over the histories of τ (s), Here τ cl represents a stationary point of S, signifying a classical trajectory of instanton.The prime symbol indicates differentiation with respect to τ , and S 0 denotes the classical action of the instanton, For the approximation of the inflaton field ϕ(t) oscillating near the minimum, we have a quadratic potential, where we define ω ≡ gM ϕ ϕ 0 . (3.12) We solve the equation of motion for τ (s ′ ) with the boundary condition τ (0) = τ (s) = τ 0 , and find the classical orbit, τ cl (s ′ ) = τ 0 cos ωs cos(2ωs ′ − ωs) . (3.13) Hence, when τ cl (s ′ ) is inserted into eq.(3.10), the classical action takes the following form, The determinant in eq.(3.9) can be evaluated using a classical action S ab with the boundary τ (0) = τ a and τ (s Alternatively, the determinant can be evaluated using the Gelfand-Yaglom formula [67].The normalization factor N is attained for the free case V (τ ) = 0, which gives N = 4π.According to eqs.(3.9), (3.14) and (3.15), the imaginary part of the effective action of eq.(3.8) is therefore In the second line, we integrate the imaginary time τ .We evaluate the integration of s by applying the residual theorem, but the integration is divergent at s = 0, and the pole at this point is not a simple pole.We regulate the divergence by subtracting the pole at s = 0.After regularization, we integrate the Schwinger time s, and find that, where we sum over all positive integers of n.Therefore, during half an oscillation period, vacuum persistence probability is This quantity is the amplitude square, directly corresponding to a probability.Also, k represents the momentum of χ particle.It's important to note that k and −k correspond to the same final states due to pair particle production and momentum conservation.In the final equation above, we combine the contributions of k and −k to account for the pair production of these modes.This equation reveals the probability of creating k-mode after the first oscillation, Note that P vac + k P k ≃ 1 if P k are small quantities.But the probability of creating k-mode P k in the above equation is more accurate, and it does not require that P k is small, which will become clear from the Bogoliubov transformation in section 3.2.
A natural question arises regarding the applicability of the instanton approach without making the near potential minimum approximation.In the appendix, the instanton method is applied to a periodic potential without this approximation, and the effective action is numerically calculated by truncating the Fock space.

Eigenstates
With the approximation of the inflaton field ϕ(t) as described in eq. ( 3.3), the Hamiltonian Ĥ takes on a form analogous to harmonic oscillators with an imaginary frequency.The time integration can be recast as a summation of all the energy eigenstates, giving We regulate the divergence by straightforwardly excluding the pole at s = 0, as shown eq.(3.17).Then, the imaginary part of the effective action is expressed as We calculate this integration by rotating s → −is and applying the residual theorem, resulting in the same outcome as shown in eq.(3.17).Consequently, we obtain the same probability for χ particle creation.

Bogoliubov transformation
The probabilities for particle creation in vacuum (eq.(3.19)) and the vacuum persistence (eq.(3.18)) are only applicable during the initial oscillation of the inflaton.After a few oscillations, the number of χ particles grows exponentially due to the Bose enhancement.Thus, we must consider the medium effects on the inflaton's parametric decay.Still, we assume that the parametric resonance timescale is much shorter than the Hubble time, and the χ particles have neither thermalized nor decayed.Under these conditions, we can elucidate the growth in the number of χ particles through the Bogoliubov transformation.Interestingly, this growth rate relies on the parametric vacuum decay rate, as previously calculated, and the number of inflaton oscillations.With the aid of the Bogoliubov transformation, we map the annihilation and creation operators in one oscillation to the ones in the previous oscillation, where T = π/M ϕ represents half of an inflaton oscillation period.The Bogoliubov coefficients, µ k and ν k , satisfy the relation, Considering that the χ particles are pair-produced, and scatterings among χ particles are at a later stage, we decompose the vacuum state at time t as a superposition of states at t + T , taking into account the produced pairs, This decomposition aligns with the definition of the vacuum state at time t, where a k (t)|0; t⟩ = 0, and the prefactor derives from its normalization.Let's consider the initial oscillation, during which the χ state remains in the vacuum.The vacuum persistence probability, as determined by the Bogoliubov transformation, should coincide with eq.(3.18), This yields the absolute value of ν k , which justifies that P k in eq.(3.19) is an accurate result.To assess stimulated particle pair production in a medium with a non-zero average occupation number, we introduce the density matrix ρ, where θ k represents an arbitrary phase.We consider n k to be large so that all the traces of the density matrix in the above equation gives an approximation of the same result, n k .
When n k is substantial, the occupation number undergoes exponential growth, This result aligns with previous findings, which were obtained using the WKB method [13,66].The outcome is a function of ν k , as governed by the vacuum persistence probability via the effective action method.The phase θ k , which can be evaluated using the WKB method, becomes larger than π and rather complicated time-dependent in an expanding universe.Consequently, it can be approximated as a random phase, introducing stochasticity to the χ particle number growth.This allows us to average the random phase, which yields the evolution of number density.The final result relies on both the vacuum persistent probability and the number of inflaton oscillations.

Tachyonic preheating
Following [72], we introduce a bosonic trilinear coupling between the inflaton ϕ and its daughter particle χ as a realization of tachyonic preheating, Similarly, we assume that the preheating timescale is significantly shorter than the Hubble time, enabling us to disregard the expansion of the universe.Then the field equation for χ k takes the form With the inflaton background evolving as ϕ 0 (t) = ϕ 0 sin(M ϕ t), the field equation above can be reduced to the canonical Mathieu equation.When the coupling is large, with gϕ 0 > M 2 ϕ , we enter the broad resonance region.However, modes with negative ω 2 k undergo tachyonic instability, which cannot be approximated as waves scattering from a parabolic potential, as in the broad resonance case.Interestingly, the effective action doesn't tell the difference between both cases and treats them as a whole.
Our strategy parallels the one employed in the previous section.In the tachyonic resonance model, we first integrate out χ k modes using the path integral to determine the effective action of the inflaton, as in eqs.(3.6) and (3.8).Second, we evaluate the Schwinger time evolution by finding the world-line instanton solution shown in eq.(3.9).One minor distinction from the parametric resonance calculation is that here, we do not perform a rotation of the time t into the Euclidean time.However, we have verified that using both real-time and Euclidean time yields the same result.In the tachyonic resonance model, the classical action of the instanton is given as One of the saddle point solutions occurs at the peak of the potential V (t), with ṫ = 0, representing the mode with the largest negative mass square.Here, we approximate the sine function near the potential's peak as a quadratic term, where j is an integer.The saddle-point solution yields the classical action, with ω = . Combing the classical action and the determinant, which aligns with with the one in broad resonance case in eq.(3.15), we obtain the Schwinger time evolution operator, ⟨t|e − Ĥs |t⟩ ≃ e −S 0 det(− 1 2 e ωt 2 tan ωs+gϕ 0 s .(4.6) By putting these solutions together and conducting a similar integration as in eqs.(3.16) and (3.17), we determine that the imaginary part of the effective action takes the form This imaginary part of the effective action gives the probability for χ particle creation after the first oscillation, ) .(4.8) It's noteworthy that the prefactor in the exponent, equal to π, is in close agreement with the numerical value found in [72], which is approximately 3.4.The time evolution of the number density n k can be obtained through the Bogoliubov transformation as presented in section 3.2, with the phase θ k remaining indeterminate in this context.

Gauge preheating
In this preheating model, we consider a massive or massless gauge boson, denoted as A µ , which couples to the inflaton via a Chern-Simons operator.This coupling arises due to an approximate shift symmetry of inflaton.The Lagrangian for this model can be expressed as Here the Chern-Simons coupling involves the field strength F µν = ∂ µ A ν − ∂ ν A µ and its dual F µν = 1 2 ϵ µνρσ F ρσ , and the coupling strength is represented by a dimensional quantity 1 Λ .A massive gauge boson with a constraint ∂ µ A µ = 0 can be decomposed into helicity basis components: two transverse modes, h = ±1, and one longitudinal mode, h = 0. Consequently, the gauge boson action takes the following form A significant particle production occurs around the minimal of − k Λ φ, indicating that the A + k modes predominantly produced when ϕ = 0 and φ > 0. Similarly, the A − k modes predominantly produced when ϕ = 0 and φ < 0. As the longitudinal mode does not couple to the inflaton and the action is equivalent to a free theory, the longitudinal mode production is not expected during the process.
The world-line instanton solution in this model also possesses a saddle point at the peak of the potential, We still employ the quadratic potential approximation for this case.When the inflaton crosses the origin for the first time with φ > 0, the probability that the A + k modes are produced is given as ) . (4.12) The same result applies to A − k modes, but the production occurs when φ < 0, crossing the minimum of the potential.The particle creation probability is found to be numerically consistent with previous literature [76].

Conclusion
In this paper, we have employed the effective action approach to investigate the non-perturbative inflaton decays in an oscillating inflaton background.By focusing on the imaginary part of the effective action, we obtain the probability of the inflaton decaying to its coupled particle.This approach allows for the examination of both perturbative decays, such as the reheating process and narrow resonances in preheating, and non-perturbative decays.Here, we concentrate on non-perturbative particle production through broad parametric resonance or tachyonic instability.
To arrive at our results, we have integrated out decayed fields in the Path Integral, employed the world-line instanton method to identify the semiclassical solutions of the effective action and subsequently determine the particle decay probabilities in the vacuum.Moreover, we have extended our analysis to establish a connection between particle creation rates in the vacuum and those in a medium.This extension has facilitated an understanding of how the number density of produced particles evolves during the preheating process.
Our approach can address both parametric resonance and tachyonic instability within the same framework.In the case of tachyonic instability, we have explored two models.One is tachyonic preheating, in which the inflaton possesses a trilinear coupling with another scalar and can have both tachyonic instability and broad resonance effects.The other model is gauge preheating, where the inflaton interacts with gauge bosons via Chern-Simons terms.Notably, all our analytic findings are consistent with prior literature.
Crucially, the effective action approach offers an alternative perspective on the preheating process and serves as a valuable complement to the well-established WKB method.The WKB computation relies on satisfying adiabatic conditions.When inflaton crosses its potential minimum, the adiabatic condition is violated.Then, we solve the mode functions before and after passing the minimum using the WKB method, connect the solutions by analytic continuation.On the other hand, the effective action method determines semiclassical solutions near the potential's minimum, where particle production predominantly occurs.This distinction clarifies that these two methods cover different phases of the inflaton oscillation period.It also explains that the effective action method does not capture the phase accumulating during the oscillating period.
Our work stresses that effective action is a powerful tool for investigating non-perturbative particle production processes.Its applicability extends to various domains within cosmology and gravity, opening up alternative ways for computation and offering new perspectives on these subjects.As we look to the future, there is significant potential for applying this approach to scenarios such as other well-motivated preheating models, particle production near black holes, in an expanding universe, or during bubble nucleation.