Imprints of Schwinger Effect on Primordial Spectra

We study the Schwinger effect during inflation and its imprints on the primordial power spectrum and bispectrum. The produced charged particles by Schwinger effect during inflation can leave a unique angular dependence on the primordial spectra.

angular dependence is different from the other mechanisms that produce the angular dependence on the primordial power spectrum. For example, Bianchi universes such as anisotropic inflation generated by a vector field background [44][45][46][47][48], Galileons [49] or higher-order of curvature terms [50][51][52] can produce an angular dependence cos 2 θ (See [53,54] for reviews and many more related works therein); inflation with a massive spin-1 field can produce the angular dependence P 1 (cos θ). Moreover, since the magnitude of non-Gaussianities is directly proportional to the number of particles produced during inflation, the bispectrum has an angular dependence as more charged particles are produced in the direction parallel with the direction of the electric field. This paper is organized as follows, in Section II, we introduce the model we are considering. In Section III, we derive the geodesic equation of a charged scalar particle. In Section IV, we give the primordial power spectrum. In Sectrion V, we give the bispectrum. In Section VI, we give the result of loop corrections to the bispectrum. We give a conclusion in Section VII.

II. MODEL
We consider QED coupled to a pair of charged scalar σ and σ * in four dimensional de Sitter space.
where D µ ≡ ∂ µ − ieA µ is the covariant derivative. F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic tensor. The FRW metric is where τ is the conformal time. We neglect the backreaction of the produced charged scalar particles on the electric field and the FRW background in this work. We consider a constant electric force in the z direction, Thus the equation of motion for the σ field is σ + 2 a a σ − ∂ i ∂ i σ − 2ieA z ∂ z σ + e 2 A 2 z σ + a 2 m 2 σ = 0 .
We quantize the σ field in the following way where a k and a † k are annihilation and creation operators of the positively charged scalar particle, and b k and b † k are annihilation and creation operators of the negatively charged scalar particle. They satisfy the commutation relations [a k , We introduce the variables where κ is imaginary. The real part of the parameter κ characterizes the magnitude of the electric field projected to the direction of the trajectory of the negative charged particle. In this work, we focus on the parameter regime where e 2 E 2 /H 4 + m 2 /H 2 > 9/4, thus µ is imaginary. Our work can be easily generalized to the e 2 E 2 /H 4 + m 2 /H 2 < 9/4 case. The real part of the parameter µ can be understood as the effective mass of a charged particle in de Sitter space in Hubble units with correction 9/4 coming from the curved space time and e 2 E 2 /H 2 from the electric field. The mode function satisfies the equation There are two solutions, which are given by the Whittaker functions W κ,µ (z) and M κ,µ (z). Since in the sub-horizon limit |z| → ∞, the solution must approach to the Minkowski solution, we obtain the mode function In the late time limit, the mode function behaves as The coefficients α k and β k satisfies the normalization condition The Bogoliubov coefficients can be obtained as The qualitative feature of |α k | 2 , |α k ||β k |, |β k | 2 are plotted in FIG. 1. We set m = 3H/2. We see that as E increases, both |α k | 2 and |β k | 2 first decrease exponentially and then eventually approach to a constant value, with |α k | 2 approaching to 2 and |β k | 2 approaching to 1. The number of charged particles being produced with charge e and wave number k per comoving three volumn d 3 k/(2π) 3 is The particle production rate can be also be calculated by the instanton method. In the classical limit |µ| ± iκ 1, these two methods give the same results. The full derivation of the classical action S E is given in Appendix A. In the classical limit, the particle production rate is given approximately by where l ≡ eE/mH characterizes the relative magnitude of the electric field and mass.m 2 = m 2 − 9H 2 /4 is the effective mass of a neutral particle in de Sitter space. S + is the action corresponding to the action of the process that the charged particles are produced but moving to the direction that increases the electric potential energy of itself, whereas S − corresponds to the action of the process that the charged particles are produced and moving to the direction that decreases the electric potential energy. In the terminology of the instanton method, S + corresponds to upward tunneling and S − corresponds to the downward tunneling. It is always the S − that gives the dominant contribution.
There are two interesting limits that we can discuss this problem quite intuitively. The first limit is the weak electric field limit, where l 1. The classical actions S ± in (16) can be approximated as The first term can be understood as the usual Boltzman factor coming from the production of neutral massive particles of effective massm. From the point of view of a geodesic observer, de Sitter space is associated with a thermal bath with the Hawking temperature T = H/(2π). The second term is understood as the chemical potential from the electric field. This chemical potential can assist the production of charged particles along the direction of decreasing potential. Since in this limit, the electric field is very weak, the dominant contribution comes from the first term. Hence, although the particles are charged, they are mainly produced gravitationally due to the expansion of the universe. In [55][56][57], other examples of the chemical potential is also discussed in the context of fermion production in inflation. The other limit is the large electric field limit, where l 1. In this limit, the electric field is so strong that the modes do not feel the curvature of the spacetime. The classical actions S ± can be approximated as If we consider the charged particle pairs moving along the z direction, the second term dominates. As we can see, it also reproduces the flat spacetime result. In order to study the time scale of mass production of charged particles, it is useful to consider the WKB approximation of solution (11).
where w k is the effective frequency given by Then the adiabatic parameter is evaluated as It is around the time that the quantityẇ k /w 2 k approaches its maximum. This means that most particles are produced at this time scale. Now we observe the production of the particle via the Schwinger effect during inflation. One may think about the mechanism in the context of quasi-single field inflation. The charged particles can decay into the primordial curvature perturbations, thus leaving an imprint on the primordial power spectrum and bispectrum. ζ is the primordial curvature perturbation. The second order action of the primordial curvature perturbation can be written down following the procedure in [58,59] Quantizing it in the following way where c † k , c k are the creation and annihilation operators satisfying the usual commutation relations The mode function satisfies the following equation of motion To the lowest order in slow roll parameter, the solution is We consider the following coupling between the primordial curvature perturbation and the positive charged scalar fields The coupling between the inflaton and the negative charged scalar fields are and the coupling between the primordial curvature perturbation and the positive and negative charged scalar fields are where c 2 , c 3 , c * 2 , c * 3 , c 2 and c 3 are some constants coming from the background of the σ or σ * field. Here (28) and (29) do not conserve the charge of σ. They correspond to cases where the phase symmetry of σ is broken, for example, in an Abelian Higgs model (see [60] for discussion of a similar case). In this case, tree level contribution dominates correction to the spectra. Equation (30) corresponds to the case where charge is conserved. In this case, loop diagrams has to be computed. One may worry that these background values contribute to the mass of the gauge field and it won't be able to support a long range force. But now we are considering very small coefficients. In order to calculate the primordial spectrums, we used the Schwinger-Keldysh formalism (For the application in quasi-single field inflation, see [61]). Now we derive the four types of free propagators for both the curvature perturbation and the massive charged scalar fields. For the curvature perturbation sector, the generating functional can be written as The four propagators can be generated using Fourier transforming it into momentum space gives The four types of propagators are given as the following For charged massive scalar pairs, we need to introduce two more sources J * + and J * − to source the complex conjugate of the σ field The four propagators can be generated using Then we have The four types of propagators in the Schwinger-Keldysh formalism are

III. THE GEODESIC EQUATION
To understand the charged particles motion in inflation, we solve the geodesic equation as an intuitive understanding. Following from our metric in (2), the following geodesic equation for a massive charged particle can be written down following the standard procedure (see text books [62,63]).
where the connection is Here, we would like to observe the change in the physical velocity of the massive charged particle. We would like to solve the following geodesic equation: The initial condition we would choose is x (t 0 ) = 0, which means that the particles are produced at zero velocity. This is the choice following from the instanton method. The solution to this equation subjected to the initial condition is The velocity of the particle is At first, the force from electric field dominates over the Hubble friction, thus the particle starts to accelerate. Later, due to the Hubble friction, the particle begins to decelerate. The maximum velocity occurs in the subhorizon.

IV. POWER SPECTRUM
In this section, we study the power spectrum of our model. It can be evaluated using the Schwinger-Keldysh formalism [61]  The comoving velocity of the particle with respect to the physical time t. We take the Hubble parameter to be 1 in making the plot. We choose the initial time to be t0 = −30 and t f = 30 to ensure 60 e-folds. We here plot the evolution of the velocity in the subhorizon level. Initially, the velocity of the particle is zero. Through the acceleration from the electric field, the velocity of the particle increases and the direction depends on the charge of the particle. For stronger electric fields,the maximum velocity attained is much larger. After some time, the particle starts to decelerate and its velocity slowly decreases to zero due to the Hubble expansion of the universe. We can see the velocity has decreased to zero at subhorizon level. Hence, the change in the frequency of the oscillatory signal in the bispectrum can't be observed as the bispectrum is imprinted at late times.
where G(k, τ 1 , τ 2 ) and D(k, τ 1 , τ 2 ) are defined by (34) and (38) respectively. Here, we need to do a sum over of all the + and − modes. The means the momentum conserving delta function (2π) 3 δ (3) (k + k ) is stripped from the two point function.
There are two contributions, the first is The indefinite integral can be integrated directly, which yields where G is the Meijer function defined through the Gamma function in the following way At x = 0, the integral I gives 0. At x → ∞, it gives The second contributions is This integral is difficult to evaluate, thus we integrate it numerically. However, in the limit of large mass and small electric field, an analytical result is possible by integrating out the charged massive scalar fields. We present the results in Appendix B 1.
The power spectrum P ζ is obtained as From here and the following, when making the plot, we set c 2 = c 3 = M pl = = H = 1. We plot the angular dependence of the power spectrum in FIG. 3. The produced charged particles can leave non trivial angular dependence on the power spectrum. The power spectrum grows exponentially as the quantity k z /k increases. This signature is understood as the production of virtual particles increases exponentially when the momentum of the positive charged massive scalar particle is aligned with the electric field E whereas the momentum of the negative charged massive scalar particle is opposite to the direction of the electric field E. This signature is a unique signature which cannot be generated by other mechanisms to the knowledge we know of. We also plot the dependence of the power spectrum on electric field strength in FIG. 4 and FIG. 3: At different electric field strength, the power spectrum has angular dependence. We set the mass of massive field is 3H/2. When the orientation is much close to field's orientation, the power spectrum is much larger. At the orientation perpendicular to the field's orientation, although the electric field is strong, Schwinger effect is weak and the effective mass is large enough to suppress the power spectrum, hence, the power spectrum can be small with strong electric field. At the orientation parallel to the field's orientation, the Schwinger effect is strong enough to get a large power spectrum.

V. BISPECTRUM
In this section, we study the bispectrum of this model. For the bispectrum, using the Schwinger-Keldysh formalism [61], the bispectrum can be expressed as There are three contributions where ζ k1 ζ k2 ζ k3 (1) , ζ k1 ζ k2 ζ k3 (2) and ζ k1 ζ k2 ζ k3 (3) are computed as We can define the bispectrum in the form of dimensionless shape function S(k 1 , k 2 , k 3 ) [57], defined as, where P is the power spectrum for the curvature perturbation without the correction coming from massive fields.
The bispectrum can be evaluated using numerical integration. We plot the angular dependence of the amplified bispectrum shape function (k 1 /k 3 ) × S(k 1 , k 2 , k 3 ) as a function of k 3z /k 3 in FIG. 7. The bispectrum increases exponentially with increasing absolute value of k 3z /k 3 . When k 3z /k 3 is positive, the main contribution comes from the When we measure the non-Gaussianity, we should fix the ratio of the long wavelength momentum and the short wavelength momentum k1/k3. In the meanwhile, we measure the angular dependence of the non-Gaussianity for different k3z/k3. k3z is the magnitude of the long wavelength momentum projected onto the z direction. This figure shows the amplified shape function S(k1, k2, k3) × k1/k3 as a function of the angular dependence of the soft momentum k3z/k3. In making this figure, we set m = 3H/2 and k1/k3 = 100. When orientation is much close to field's orientation, the amplitude of the bispectrum is much larger. At the orientation perpendicular to the field's orientation, although the electric field is strong, Schwinger effect is weak and the effective mass is large enough to suppress the bispectrum. At the orientation parallel to the field's orientation, the Schwinger effect is strong enough to generate a bispectrum of large amplitude.
positive charged particles whereas when k 3z /k 3 is negative, the main contribution comes from the negative charged particles. We plot the clock signals of the squeeze limit of the bispectrum with fixed effective mass µ in FIG. 8 and with fixed  mass m in FIG. 9. In both cases, we see that the clock signal is less obvious when the strength of the electric field increases. In the case of fixing the effective mass µ case, the absolute value of the clock signal increases with increasing electric field strength due to the enhanced particle production rate. However, the relative amplitude between the clock signal and the contribution of the non-oscillating part coming from some local process decreases. This is because the contribution of the non-oscillating part increases faster than the clock signal when the electric field strength increases. For the fixed mass m case, when the electric field strength increases, the amplitude of the clock signal relative to the non-oscillating part decreases more dramatically compared with the fixed effective mass case. This is because the electric field strength contributes to the effective mass of the charged massive scalar particles. The energy needed to produce a particle increases accordingly. The combination of these two effect causes the amplitude of the clock signal relative to the non-oscillating part barely observable starting from κ = 4i. The analytical expression of the clock 10 100 1000 0 FIG. 8: The figure shows the clock signal from the bispectrum. Here we set the momentum to be in the z-direction and hence we set k3z/k3 = 1. We can see the suppression of the clock signal as the electric field increases. The frequency of the clock signal remains unchanged as the change in velocity of the produced particles occurs in the subhorizon and remains stationary during horizon crossing. Hence, no change in frequency of the oscillatory signal would be seen. However, the amplitude of the oscillatory signal would increase as the magnitude of the electric field strength increase. signal in the large mass and small electric field strength can also be obtained by standard procedure. The derivation is shown explicitly in Appendix C. We are interested in the squeezed limit where k 1 ∼ k 2 k 3 . In this limit, the shape function is with the prefactor given by (1 + sin (πµ)) .
We can see that in the large mass limit, all the Γ functions contribute a factor of e −2π|µ| and sin(πµ) would give a contribution of e π|µ| . Hence, the Boltzmann suppression factor e −π|µ| is recovered in this limit. At the end of this section, we would like to compare several mechanisms that can generate large clock signals even if the mass of the σ field are large. There are a few categories of mechanisms listed as the following. • The presence of a new scale. In [64], non-adiabatic production of very heavy fields is studied. The signatures of this model can be large due to the existence of another scaleφ with φ as the inflaton.
• Finite temperature effect. In [65], the clock signal of the quasi-single field inflation is studied in the context of warm inflation. The particle production rate can be unsuppressed when the effective mass of the particle is changed due to the finite temperature effect.
• The presence of chemical potential. In [55][56][57], the effect of the chemical potential is studied. The chemical potential can assist the production of the massive particles during inflation thus leaving a less suppressed clock signal. The mechanism we studied here also belongs to this category. However, our studies shows that although it is promising to generate a larger clock signal, one may worry that the contribution from the non-oscillating part will also increase.
• Non-trivial sound speed. The non-trivial sound speed of the massive field is studied in [66]. The magnitude of the clock signal can also be larger than expected when the ratio of sound speed of the massive field and the inflaton is less than one. In [67,68], the non-trivial sound speed of the inflaton is studied. It is shown in [68] that when the sound speed of the inflaton is close to zero, there will also be a change in the suppression factor of the clock signal.

VI. LOOP CORRECTION TO BISPECTRUM
In this section, we investigate loop corrections coming from the extra massive fields to the primordial non-Gaussianities. The technique of dealing loop correction in quasi-single field inflation can be found in [43,[69][70][71][72]. The non-oscillatory part of the diagram is usually UV divergent and we need a systematic way of regularization and renormalization following [73][74][75][76]. Luckily, the clock signal is free from UV divergence and we can evaluate it easily.
Using the Schwinger-Keldysh formalism, the bispectrum corresponding FIG. 10 can be obtained as where p and q are the loop momentum that satisfies the constraint p + q = k 3 . After evaluation, we get with the prefactor given by and with the definition of the shape function S(k 1 , k 2 , k 3 ) in (53) and taking the limit k 1 = k 2 k 3 , we can obtain the expression for the shape function We can see that from the loop diagram, the massless curvature modes resonates with two pairs of massive fields and generate two sets of clock signals. The final clock signal would be contributed from the interference of these two clock signals and hence has a doubled frequency of the frequency of the tree level diagram. The total Boltzmann suppression factor is of e −2πµ where all the Γ functions contribute e −4πµ in total and the sin functions contribute a factor of e 2πµ . The suppression for the loop correction is the square of the tree-level case due to the excitation of the two massive fields in the loop diagram.

VII. CONCLUSION AND OUTLOOK
In this work, we consider the imprints of the Schwinger effect on the primordial power spectrum and bispectrum. Both the power spectrum and bispectrum obtained an angular dependence due to the fact that the electric field can assist the production of charged massive particles by adding a chemical potential to them. This angular dependence differs from other models that can too cause an angular dependence on the primoridial power spectrum and bispectrum.
As a result, the production rate aligned or opposite to the direction of the charged particles gets enhanced. On the other hand, the production rate perpendicular to the direction of the electric field is suppresed due to the contribution of the electric field strength to the effective mass of the charged scalar particles.
There are many interesting possibilities to explore. We list a few of them and hope to address some of these possibilities in the future.
• The influence of the primordial magnetic field on the power spectrum and bispectrum in the context of quasi-single field inflation. The pair production of the charged scalar field in the presence of a constant electric and magnetic field is studied in [77]. We would like to couple the charged particles with the inflaton and see what kind of signature these particles would imprint on the primordial power spectrum and bispectrum of the curvature perturbations. These signatures on the power spectrum and bispectrum will provide supporting evidences to the existence of the primordial magnetic field.
• The backreation of the produced particles on the primordial electric and magnetic fields as the particles being produced will weaken the primodial electric and magnetic field. This effect is studied in (1+1)D in [78] and general dimension in [79,80]. We would like to estimate the actual magnitude of the clock signal generated when taking into account the backreaction effect.
• The signature from other fields produced by Schwinger effect during inflation. Schwinger effect not only produces charged scalar particles, but charged fermions too [81]. The production rate is very similar. However, the production of fermions may lead to other types signatures on the power spectrum and bispectrum.
• Other sources like SU (2) gauge fields [82,83]. In this case, the production rate is suppressed as the interaction strength increases. However, some signal of cosmological collider type can be generated by the spin-2 field which is required by this type of model.
Then we use F µν F µν = 2E 2 , we have Here, V is the volume of the inside region and˜ = |g| dx µ ∧ dx ν . We can also simplify the Euclidean action by According to the calculation of the surface and the volume of N-spherical worldsheets, we can obtain the Euclidean action.
Here, R 0 is the radius of N -spherical worldsheet and θ 0 is the polar angle on the d-sphere of radius R 0 . We then extremize Euclidean action with respect to θ 0 Then the radius of the Euclidean worldsheet would be given by When we consider the particle production, the dimension of the worldsheet is 1, and we would obtain the following Euclidean action for the instantons The first layer of integral is evaluated to Inserting it into the second integral The two point function of the primordial curvature perturbations would be given by Since this method automatically removes the contribution that is exponentially suppressed in the large µ limit, we don't need to consider the non-time ordered integral. The power spectrum can be written as Using the technique very similar to [94], we obtained the expression which is suitable for all c 2 < |µ| 2 .

Bispectrum
The large mass limit of the bispectrum oin the quasi-single field model with neutral scalar particles is obtained in [95]. After integrating out the massive field, an equilateral non-Gaussianity is obtained. This part contains no clock signal, however, it is the dominant contribution in the large mass limit since it is only suppressed by 1/|µ| 2 whereas the clock signal is supressed by exp(−π|µ|) in the large mass limit. In this section, we would like to derive the bispectrum by integrating out the charged massive scalar particles.
We can get the third term of the bispectrum in the large mass limit in the similar way.