Schwinger effect in de Sitter space

We consider Schwinger pair production in 1+1 dimensional de Sitter space, filled with a constant electric field $E$. This can be thought of as a model for describing false vacuum decay beyond the semiclassical approximation, where pairs of a quantum field $\phi$ of mass $m$ and charge $e$ play the role of vacuum bubbles. We find that the adiabatic"in"vacuum associated with the flat chart develops a space-like expectation value for the current $J$, which manifestly breaks the de Sitter invariance of the background fields. We derive a simple expression for $J(E)$, showing that both"upward"and"downward"tunneling contribute to the build-up of the current. For heavy fields, with $m^2\gg eE,H^2$, the current is exponentially suppressed, in agreement with the results of semiclassical instanton methods. Here $H$ is the inverse de Sitter radius. On the other hand, light fields with $ m \ll H$ lead to a phenomenon of infrared hyperconductivity, where a very small electric field $mH \lesssim eE \ll H^2$ leads to a very large current $J \sim H^3 /E$. We also show that all Hadamard states for $\phi$ necessarily break de Sitter invariance. Finally, we comment on the role of initial conditions, and"persistence of memory"effects.


Introduction
False vacuum decay through bubble nucleation [1][2][3][4] has received considerable attention in field theory and cosmology over the past few decades. Interest in this subject has recently been revived in the context of the eternally inflating multiverse [5][6][7]. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Various suggestions have also been made for possible observational signatures of this scenario, involving the dynamics of bubble formation [8][9][10] or subsequent bubble collisions [11][12][13][14].
Although the mechanism for vacuum decay by quantum tunneling seems to be reasonably well understood, some aspects of it require further exploration. A particularly puzzling issue which has only recently been addressed [15,16], concerns the rest frame of bubble nucleation. If the false vacuum is locally Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. In this setup, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. Refs. [15,16] concentrated on the case of a constant electric field in flat space. In this case, it was shown that the adiabatic in-vacuum for a charged scalar field φ (defined in terms of modes which are positive frequency in the remote past) is Lorentz invariant (LI). Then, by using various detector models, it was shown that particles and antiparticles tend to nucleate preferentially at rest in the detector's frame.
On the other hand, the Lorentz invariant vacuum corresponds to a somewhat idealized situation which is not too realistic. If the electric field is switched on at some initial time t 0 , then in the limit t 0 → −∞ the number of pairs which have been produced per unit volume is infinite for any finite value of t. Because of that, the LI vacuum contains an infinite density of charged particles, the two point function for φ does not have the Hadamard form, and the expectation value of the current is ill defined [15]. For a more realistic case, where we keep t 0 finite, the number density of produced pairs is finite, leading to a space-like expectation value for the current of the form J µ = (0, J), with (see, e.g., [17,18] and references therein): Here, e is the electric charge and is the pair production rate per unit volume (see e.g. Ref. [19] and references therein), where m is the mass of the charge carrier and E is the electric field. 1 In principle, the breaking of Lorentz invariance by the initial hypersurface at t = t 0 could have some influence on the frame of nucleation. However, it was argued in [16] that such influence becomes irrelevant in the asymptotic future, when the proper time τ which the detector has spent in the false vacuum exceeds the size r 0 of the instanton which contributes to the decay rate Pairs would then nucleate in the detector's rest frame to very good approximation, essentially as if the system were in the Lorentz invariant vacuum.
In this paper, we shall study the Schwinger effect in de Sitter space (dS), which is more relevant to the inflationary context. One of our goals will be to clarify the role of initial conditions. As we shall see, in the presence of an electric field, all Hadamard vacua for charged particles have the property that they break dS invariance. The symmetry breaking can be attributed to initial conditions, whose influence persists for arbitrarily late times. This is related to the "persistence of memory" effect first discussed in Ref. [11].
The Schwinger process in 1+1 dS space has previously been considered in [20]. The distribution in the number density of particles created by the electric and gravitational fields was calculated by using the method of Bogoliubov coefficients, and it was shown that in the semiclassical limit the result agrees with instanton computations [21]. This applies both to "downward" tunneling, where the initial false vacuum is more energetic than the new vacuum, and to "upward" tunneling, where the new vacuum is more energetic than the initial one. Upward tunneling [22] is possible during inflation because energy is not conserved on scales larger than the horizon size. This is relevant in determining the frequency at which different vacua in the landscape are visited by a hypothetical observer as a result of vacuum transitions. Perfect ergodicity in the frequency of such visits would imply the absence of a thermodynamic arrow of time, and so the precise rate of upward transitions seems to be important at a fundamental level (see Ref. [23] for a recent discussion of this issue). Since a rigorous justification of instanton methods in dS is still lacking, particularly for upward tunneling, the results of [20] provide valuable evidence for the quantitative accuracy of this approach.
The calculation of particle creation done in [20] is based on the existence of an adiabatic "out" vacuum in the asymptotic future. This, in turn, requires the mass of the particles to be much larger than the Hubble rate m H. We can go beyond this regime by considering the expectation value of the current, J(E), which is generated by the electric field as a result of the Schwinger process. As we shall see, this observable receives distinct contributions both from upward and downward tunneling, and it is well defined regardless of the existence of adiabatic asymptotic regions. Besides, the investigation of the behaviour of the current as a function of the applied electric field seems worth pursuing in its own right, and we shall see that the vacuum shows an interesting phenomenon of infrared hyperconductivity (with possible relevance for cosmology).
We start in Section 2, by reviewing pair production by an electric field in 1+1 dimensional dS. The semiclassical limit, relevant for comparison with bubble nucleation, is discussed in Section 3. Based on the semiclassical picture, we give a heuristic derivation of the current in Section 4. Surprisingly, this coincides with the exact result for the renormalized expectation value of the current, which is calculated in Section 5. The non-vanishing expectation value of the current in the "in" vacuum manifestly breaks dS invariance. Thisis in contrast with the case of flat space, where, as mentioned above, the adiabatic "in" vacuum is Poincaré invariant (and non-Hadamard), and the expectation value of the current is ill defined. We then analyze the conductivity of vacuum in different regimes, characterized by the mass m of the charge carriers and the strength of the electric field E. In Section 6 we consider the question of dS invariance in more general terms, showing that it is broken in any Hadamard vacuum. In Section 7 we discuss some aspects of the persistence of memory of initial conditions. We argue that, unlike in the case of flat space, the influence of the initial hypersurface in determining the rest frame of nucleation is a persistent feature in de Sitter space. Our conclusions are summarized in Section 8. Appendix A discusses the semiclassical trajectories of charged particles in dS, Appendix B contains the calculation of the current from a momentum integral of special functions, and Appendix C deals with particle detectors.

Schwinger effect in dS
Consider a 1+1 dimensional de Sitter space with a constant electric field E. The field strength is given by Here µν is the Levi-Civita symbol, with 01 = 1, and g is the determinant of the dS metric g µν . The symmetry of this background is SO(2,1), rather than the full de Sitter group O(2,1), since the field strength is not invariant under parity. This distinction, however, is not too relevant for our purposes, and for brevity we shall simply refer to SO(2,1) symmetry as dS invariance. An important feature of this background is that it does not possess any preferred rest frame. Following [20], let us consider a charged scalar field φ with action given by In the flat chart, the dS metric reads and the gauge potential leading to (2.1) can be taken as Owing to the spatial homogeneity of (2.3) and (2.4), we can expand the field operator as and then the equation of motion reduces to where primes indicate derivatives with respect to η. The canonical commutation relations , with the modes φ k satisfying the standard Klein-Gordon normalization condition The "in" vacuum corresponds to the choice of modes where W λ,σ are Whittaker functions, with indices given by λ = ieE/H 2 , and In the case of heavy particles, for which m 2 H 2 , σ is purely imaginary. In this case we adopt the convention 2 σ = i|σ|. (2.11) From the asymptotic expansion of W λ,σ (z) ∼ e −z/2 z λ for large |z|, the modes (2.9) behave as and so they are positive frequency with respect to conformal time in the asymptotic past. Such "in" vacuum can be used in order to calculate pair production rates by the method of Bogoliubov coefficients. The "out" vacuum can be defined (for heavy particles with m H) by using the Whittaker function M λ,σ , in terms of which the mode functions are given by , and so in the asymptotic future we have which is positive frequency with respect to cosmological time t. This is related to conformal time through a(t) = e Ht = −(Hη) −1 . (2.15) The "in" and "out" modes are related by the Bogoliubov coefficients α k and β k : These can be read off from the linear relation from which we easily obtain [20] |β k | 2 = |β ± | 2 = e −π(|σ|±|λ|) cosh π(|σ| ∓ |λ|) sinh 2π|σ| . (2.18) Here, the upper sign corresponds to k > 0, and the lower sign to k < 0. In flat space, an electric field causes particles and antiparticles in a pair to nucleate at a distance d = 2r 0 from each other. The distance d is determined by the balance between the potential energy and rest mass energy eEd = 2m, where e is the electric charge. If E > 0, this balance requires that the particle with positive charge should be to the right of the particle with negative charge (i.e., towards increasing values of x). We may call this the "screening" orientation, since the charges would then tend to reduce the value of the electric field in between them. In the language of false vacuum decay, this corresponds to a "downward" transition, reducing the value of the vacuum energy density. Here, we shall treat the electric field as an external source, which will be unaffected by the nucleation of pairs, but we shall still refer to the materialization of pairs with the screening orientation as "downward" tunneling. In de Sitter space, pairs can also nucleate with the "anti-screening" orientation, since energy need not be conserved on scales somewhat bigger than H −1 . This corresponds to "upward" tunneling [22,24]. Fig. 1 illustrates the semiclassical trajectories of two nucleating pairs. Downward tunneling corresponds to the excitation of modes with k < 0, while upward tunneling corresponds to the excitation of modes with k > 0.
As shown originally by Voloshin, Kobzarev and Okun in field theory can decay by quantum tunneling. The proces of true vacuum bubbles of a critical size r 0 . In the semiclassi initially at rest, and then expands with constant proper accel of the false vacuum, however, indicates that bubbles will not in which to nucleate. This observation seems to suggest [2] t unit volume should include an integral over the Lorentz gro As shown originally by Voloshin, Kobzarev and Okun [2], a metasta in field theory can decay by quantum tunneling. The process occurs loca of true vacuum bubbles of a critical size r 0 . In the semiclassical picture, a initially at rest, and then expands with constant proper acceleration r −1 0 . L of the false vacuum, however, indicates that bubbles will not have any pre in which to nucleate. This observation seems to suggest [2] that the total unit volume should include an integral over the Lorentz group, in order t -1 - As shown originally by Voloshin, Kobzarev in field theory can decay by quantum tunneling. of true vacuum bubbles of a critical size r 0 . In th initially at rest, and then expands with constant p of the false vacuum, however, indicates that bubb in which to nucleate. This observation seems to s unit volume should include an integral over the As shown originally by Voloshin, Kobzarev and Okun [2], a metastable false vac in field theory can decay by quantum tunneling. The process occurs locally, by nuclea of true vacuum bubbles of a critical size r 0 . In the semiclassical picture, a critical bubb initially at rest, and then expands with constant proper acceleration r −1 0 . Lorentz invari of the false vacuum, however, indicates that bubbles will not have any preferred rest fr in which to nucleate. This observation seems to suggest [2] that the total rate of decay unit volume should include an integral over the Lorentz group, in order to account fo As shown originally by Voloshi in field theory can decay by quantu of true vacuum bubbles of a critical initially at rest, and then expands wi of the false vacuum, however, indica in which to nucleate. This observati Figure 1. Diagram illustrating the nucleation of charged pairs in a 1+1 dimensional de Sitter space with a constant electric field E. The white region corresponds to the patch which is covered by the flat chart, with coordinates (η, x). The adiabatic "in" state in the flat chart does not contain any particles at η → −∞, which can be thought of as the hypersurface of initial conditions. The shaded part of the diagram is irrelevant for our discussion. If the electric field E > 0 points in the positive x direction, pairs can nucleate with the usual "screening" orientation (red) or the "antiscreening" orientation (blue). The former corresponds to downward tunneling, and the latter to upward tunneling.

Semiclassical limit
We may now elaborate on the semiclassical description of pair creation. For In flat space, pair creation is entirely due to the electric field, but in an expanding background, such as dS, pairs can be produced even if the electric field vanishes. In order to characterize the relative importance of these two effects, we introduce the parameter where,m In this notation, Eq. (3.2) reads In this Section we shall only be concerned with the semiclassical limit, where S ± is large. A necessary condition is that the mass be large compared with H. In this regime will be real.
For 1 pairs are mainly produced by the cosmological expansion, and we have . (3.6) The first term corresponds to the Boltzmann factor for non-relativistic massive particles at the Gibbons-Hawking temperature, while the second can be though of as a small correction due to the electric field. In the opposite limit, 1, we have .
The result (3.7) for S + corresponds to upward tunneling, where the separation of the particles in a pair at the time of production is comparable to the horizon size, while S − reduces to the standard semiclassical instanton action for the Schwinger process in flat space. Note that S + ∼ 2 S − S − , so upward tunneling is highly suppressed compared to downward tunneling in this limit.
In order to estimate the time at which the pairs in a given mode k are excited out of the vacuum, we may adopt the criterion that this occurs when the violation of adiabaticity in the corresponding mode is maximal. To analyze this issue, it is convenient to introduce The mode equation (2.6) can now be rewritten as Here dots indicate derivative with respect to proper time t, defined in (2.15), and The frequencies w k approach a constant in the asymptotic future, leading to a well defined notion of "out" particles. Let us now show that the adiabatic condition is well satisfied (at all times) in the semiclassical parameter range given by (3.1). Tiny deviations from perfect adiabaticity will lead to the exponentially suppressed expectation values (3.2) for the out particle numbers. The violation of adiabaticity is largest at the extrema of f k . (3.14) -7 -which can be seen as the intersection of a parabola with a hyperbola. This has three real solutions for z. For 1, these are given by The subindices in z ± refer to the fact that, according to (3.12), the sign of z coincides with the sign of k. The corresponding maximum values of the adiabaticity parameters are given by 1, as advertised in (3.13). Particle creation in mode k occurs around the time t k when f k is maximum. Using (3.15-3.16) in (3.12), we are led to the estimate k .
For 1 two of the roots of (3.14) are given by There is a third root at z = z 3 ≈ −1/2, which is negative just like z ≈ z − . This is also an extremum of f k (t) for k < 0. However, the adiabaticity parameter is suppressed with respect to f − by a factor of 2 . Hence, for k < 0 the main departure from adiabaticity occurs at z ≈ z − . Using (3.19) in (3.12), we find that the time of particle creation is given by for the time t k at which pair creation occurs in mode k [20]. The number distribution of created pairs per unit co-moving volume is given by Since |β k | depends only on the sign of k, the distribution is flat both for positive and negative k, but discontinuous at k = 0. Also, at any finite value of t, the distribution is cut-off at |k| ∼ a(t)|σ|H, since according to (3.21), modes with a higher value of |k| have not yet been excited. The distribution (3.22) can be compared with the distribution which is obtained by means of instanton techniques [21]. The use of instanton methods in dS is not as rigorously 3 The width of the peaks of the adiabaticity parameter can be estimated by calculating the second derivatives of f . The two peaks given in (3.16) and (3.19) have widths of order (∆z/z) ∼ S −1/2 ± 1, so they are very sharp in the semiclassical limit. On the other hand, the double peaks given in Eq. (3.15) have a width (∆z/z) ∼ −1 S −1/2 ± 1. Here, however, the important parameter is not so much the width of the individual peaks but the separation between them. This is given by ∆z ∼ −1 , which corresponds to a time difference ∆t ∼ r0, where r0 = m/(eE) is the instanton radius. This is in concordance with the case of flat space [16]. justified as it is in flat space. Nonetheless, it was found in [20,21] that the results of instanton and Bogoliubov methods agree in the semiclassical limit (i.e., when |σ| ± |λ| 1) not just in the exponential dependence, but also in the one loop prefactor. This result applies both to downward (k < 0), and upward transitions (k > 0).
For a charged particle of momentum k, the physical momentum with respect to the co-moving observers is given by Note that all particles approach a terminal value of the physical momentum at late times a → ∞, which is positive for particles and negative for antiparticles. This can be interpreted as the momentum which is gained by a charged particle subject to a constant electric field during a Hubble time. Additional time does not increase the physical momentum relative to the co-moving frame, since momentum is also depleted due to Hubble friction. A particle with k < 0 has p < 0 at early times and p > 0 at late times, with a turning point at For such particles, the terminal velocity is approached from below, so |p| < p ∞ at all times. A particle with k > 0 always has p > p ∞ > 0, and the terminal velocity is approached from above, without any turning points (see Fig. 1).
Finally, let us comment on a puzzling aspect concerning the time of pair creation. For large electric field ( 1) and downward tunneling (k < 0), according to Eqs.
This seems to be at odds with the fact that, in the absence of the electric field, an inertial Unruh detector coupled to φ will reach thermal equilibriun at the temperature T = (2π) −1 H, as if it were immersed in a thermal bath. Note that a true thermal bath of heavy particles in flat space has a root mean squared value of the momentum given by which is much smaller than (3.26). A related observation is that the momentum distribution of φ particles, given by (3.22), is not thermal at all. Rather, as mentioned above, it is completely flat, with a cut-off at the physical momentum of order p c . Nonetheless, as we shall see in the following Sections, if instead of using an Unruh detector we use an amperemeter that measures the average current flowing in response to a small electric field, the result is consistent with a flat distribution of the form (3.22), with a cut-off of the form (3.21).

Semiclassical current
In this Section, we give a heuristic derivation of the current based on the semiclassical picture.
The current due to semiclassical particles after pair creation is given by is the diferential number density of carriers and v is their velocity. Separating this into two components, J pairs = J + pairs + J − pairs , corresponding to k > 0 and k < 0 respectively, we have The physical momentum is given by p ± = ±(k/a) + |λ|H, and the upper limit of integration is taken from (3.21), Here we are using the notation k ± c to denote the absolute value |k c | of the cut-off momentum, which in principle can be different for upward or downward tunneling. The uncertainty in the cutoff is of the order of the width of the peak in the adiabaticity parameter 4 . We do not need to be too precise about the value of the momentum cutoff k c , but it will be important to know that it scales with a(t).
If we take equal values for the momentum cutoff k − c = k + c , then the current due to semiclassical particles takes the form For 1 we have |λ| |σ| and Note that, since downward tunneling is more likely than upward tunneling, |β − | 2 > |β + | 2 , the current due to semiclassical pairs (4.5) actually runs opposite to the electric field, which is somewhat counterintuitive.
On the other hand, we should take into consideration that the total semiclassical current is the sum of two contributions where J vac is the vacuum current which links the two members of a pair as they are created out of the vacuum. This is a space-like current which is necessary for local charge conservation, and can be visualized as a line (which may perhaps be rather thick) connecting the negative charge with the positive charge at the moment of creation. For any given pair, which we may label with an index i, the vacuum current can be written as Here, x µ i (s) parametrizes the locus where the current is non-vanishing, which for simplicity we take to be one-dimensional thin line. 5 As a crude approximation, we can take x µ i (s) to be on a t = t i = const. line, where t i is the moment when the i-th pair is created. In this case, we have Here, V t indicates a 2-volume of infinitesimal thickness ∆t in the temporal direction, and arbitrarily large extent in the spatial direction x, and (∆x) i is the spatial coordinate separation between the positive and negative charges in the pair. Note that is just the physical distance between the particle and antiparticle in the pair. Since the electric field and the Hubble rate are constant, this physical distance will be the same for all pairs of the same kind, and we immediately find Here N ± are the number of pairs with the anti-screening or screening orientation, given by and in the last step we have used that k ± c ∝ a(t). We show in Appendix A that on the semiclassical trajectory, the following relation holds: Using this equation in (4.11), and substituting the result in (4.7), with J pairs given by (4.4), we have This expression explicitly shows the two distinct contributions from upward and downward tunneling (which are comparable for |λ| 1). It should be noted that in order to derive (4.14) we did not need to specify the precise cutoff values of k ± c , but only had to assume that the cutoff of the flat distribution (3.22) is at a fixed value of k/a. This is, of course, consistent with the estimate (3.21) for the time of pair creation, which was based on the analysis of the peak in the adiabaticity parameter. It is nice that the result for the current is robust against the uncertainties in the location of this peak, but this also means that this observable carries little information about the value of the momentum of the particles at the time of nucleation. Let us now compare the semiclassical expression (4.14) to the quantum expectation value of the current. As we shall see, the agreement turns out to be impressive.
It was pointed out in [20] that the flat chart "in" vacuum is Hadamard. What is meant by this is that in the coincidence limit the two point function has the same divergences as a neutral field in the Bunch-Davies vacuum (BD), while it is finite when the two points are separated. In 1+1 dimensions, the divergence is actually the same as the logarithmic divergence in flat space. Let us first review the argument showing that the state is Hadamard. For later use, we introduce the gauge invariant two point functions [15] where brackets indicate expectation value in the "in" vacuum. On an equal time slice, we obtain From (2.12), we find that for fixed η and large |k|, The leading term is the same as for the Bunch-Davies modes, and so the integral in (5.3) leads to the standard logarithmic divergence. Here, we have done point splitting on an equal time slice, but it is easy to check that the conclusion is the same if we split the points in an arbitrary direction. Next, let us consider the current. This is given by where D µ = ∂ µ − ieA µ . Its expectation value can be computed as Using (5.3), we have Also, it can be checked by direct substitution of (2.5) into (5.5) that the charge density vanishes J 0 = 0. Using the asymptotic expansion of the Whittaker functions for large argument, we have where the upper and lower signs correspond to k > 0 and k < 0, respectively. Substituting this into (5.8), we find that there is a linear divergence in momentum which is independent of the mass m, and no logarithmic divergence.
The divergence can be renormalized by means of a Pauli-Villars (PV) subtraction, involving a field of large mass M , which we will send to infinity after momentum integration, Here, φ k,M (η) are positive frequency modes of the "in" vacuum for the field of mass M . The momentum integral is finite for any value of M , and we choose the limits of integration to be symmetric around k = 0 for later convenience. For a field of large mass, we can use the WKB form for the mode function φ k,M , This approximation becomes exact in the limit M → ∞, and we can safely substitute |φ k,M | 2 by |φ W KB k,M | 2 in Eq. (5.10). Note also that the contribution of the PV field to the current is actually independent of M when we use W KB mode functions, Substituting (5.12) into (5.10) and using (2.8-2.9), we can rewrite the renormalized current as The first term is the contribution of the PV fields, and in the second term we have introduced x = |kη| as the variable of integration. The second term is actually finite if we perform the sum over positive and negative k (i.e. the sum over ±) before doing the integral, and so we can safely remove the cut-off Λ. With some ingenuity, the integral on the right hand side of Eq. (5.13) can be computed analytically. This is done in Appendix B, where we show that Hσ sin(2πσ) sinh(2π|λ|). (5.14) Surprisingly, this agrees exactly with the semiclassical expression which we derived in Section 4, as can be seen by using Eq. (2.18) for the Bogoliubov coefficients into Eq. (4.14).
In Fig. 2 we plot the value of the current J as a function of the electric field E, for different values of the mass. Let us now comment on the qualitative features of the current in different mass ranges.

Linear response (m 2 = H 2 /4):
It follows from (5.14) that for m 2 = H 2 /4 the current is exactly linear in the electric field: Such linear response is reminiscent of the behaviour of currents due to massless charge carriers in flat space.  [1,2] has received considerable attention in field theory and cosmology over the past few decades. A revival of interest in this subject has been triggered by the study of vacuum transitions in the eternally inflating multiverse. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Also, it may lead to direct observational consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns the rest frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. Refs. [3,4] concentrated on the case of a constant electric field in flat space. First, it was shown that the adiabatic in-vacuum for a charged scalar field φ (defined in terms of modes which are positive frequency in the remote past) is Lorentz invariant [3], and therefore such initial condition does not select any preferred frame. Then, by using various detector models, -1 -   [1,2] has received considerable attention in field theory and cosmology over the past few decades. A revival of interest in this subject has been triggered by the study of vacuum transitions in the eternally inflating multiverse. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Also, it may lead to direct observational consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns the rest frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. Refs. [3,4] concentrated on the case of a constant electric field in flat space. First, it was shown that the adiabatic in-vacuum for a charged scalar field φ (defined in terms of modes which are positive frequency in the remote past) is Lorentz invariant [3], and therefore such -1 - False vacuum decay through bubble nucleation [1,2] has received considerable att in field theory and cosmology over the past few decades. A revival of interest in this su has been triggered by the study of vacuum transitions in the eternally inflating multi Bubble nucleation may play an important role in determining the large scale structure multiverse and the distribution of vacua within it. Also, it may lead to direct observa consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns th frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, wha that determines the frame in which bubbles of the new vacuum nucleate at rest? In prin we may expect this to be partially determined by the hypersurface of initial conditions, the false vacuum is prepared, and partially by the state of motion of the detectors should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair product 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated quantum mechanically, and not just semiclassically as is customary with vacuum bu Refs. [3,4] False vacuum decay through bubble nucleation [1,2] has received considerable attention in field theory and cosmology over the past few decades. A revival of interest in this subject has been triggered by the study of vacuum transitions in the eternally inflating multiverse. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Also, it may lead to direct observational consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns the rest frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. Refs. [3,4] False vacuum decay through bubble nucleation [1,2] has received considerable attention in field theory and cosmology over the past few decades. A revival of interest in this subject has been triggered by the study of vacuum transitions in the eternally inflating multiverse. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Also, it may lead to direct observational consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns the rest frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. Refs. [3,4] False vacuum decay through bubble nucleation [1,2] has received considerable attention in field theory and cosmology over the past few decades. A revival of interest in this subject has been triggered by the study of vacuum transitions in the eternally inflating multiverse. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Also, it may lead to direct observational consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns the rest frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. Refs. [3,4] concentrated on the case of a constant electric field in flat space. First, it was -1 -

IR hyperconductivity
False vacuum decay through bubble nucleation [1,2] has received considerable attention in field theory and cosmology over the past few decades. A revival of interest in this subject has been triggered by the study of vacuum transitions in the eternally inflating multiverse. Bubble nucleation may play an important role in determining the large scale structure of the multiverse and the distribution of vacua within it. Also, it may lead to direct observational consequences.
A puzzling issue which has only been addressed very recently [3,4], concerns the rest frame of bubble nucleation. If the false vacuum is approximately Lorentz invariant, what is it that determines the frame in which bubbles of the new vacuum nucleate at rest? In principle, we may expect this to be partially determined by the hypersurface of initial conditions, where the false vacuum is prepared, and partially by the state of motion of the detectors which should be used in order to probe the process of bubble formation.
A convenient framework for investigating this question is Schwinger pair production in 1+1 dimensions. The advantage is that, in this case, the nucleated pairs can be treated fully quantum mechanically, and not just semiclassically as is customary with vacuum bubbles. The Schwinger pair creation rate for massless carriers in flat space is given by leading to a current which grows in time, at a constant rate which is proportional to the electric field J = e 2 Et/π. In dS, we expect the current to be diluted by the expansion of the universe, and so the linear growth in time will be cut off. Naively replacing t with the expansion time H −1 leads to (5.15). More precisely, we may observe that the number density n of charged pairs satisfies dn dt = Γ − Hn. For massless (or highly relativistic) carriers, the current is which agrees with (5.15) provided that we use the pair production rate (5.16). The exact linearity in eE seems nonetheless somewhat coincidental, particularly since in 1+1 dimensions the conformal value of the mass is m 2 = 0, while Eq. (5.15) holds for m 2 = H 2 /4. The latter value of the mass corresponds to the boundary where long wavelength modes behave as critically damped oscillators. For smaller values of the mass, infrared contributions to the current become important, as we shall now explain.

IR hyperconductivity (m 2 H 2 ):
A striking property of the regime m 2 H 2 is that for eE H 2 the current is dominated by infrared contributions, rather than newly created pairs. This leads to a current of the form J ≈ 1 2π This behaviour is illustrated in Fig. 2, for m = 0 and m = 0.1H. The local maximum is at eE ∼ mH, with J ∼ H 2 /m, and the current grows unbounded for small electric field in the limit m → 0. Actually, for m/H eE/H 2 1 the current is inversely proportional to the applied electric field, much in contrast with Ohm's law. Note that the current is also independent of the electric charge in this limit.
To understand the origin of (5.20), we first note that for small z, the Whittaker function has the behaviour where we have introduced Next, from (5.13) we see that the infrared contribution to the current comes from the first term in round brackets inside the integrand, and can be expressed as where here we have introduced Using (5.22), and ignoring numerical coefficients, we can estimate which substituted into (5.24) leads to (5.20).
An alternative heuristic derivation of (5.20) is the following. From the wave equation in the long wavelength limit it is easy to show that the non-decaying mode behaves as φ ∝ e − Ht . This means that in the absence of pair creation, the number of pairs would slowly dilute as n ∝ φ 2 ∝ e −2 Ht . Including pair creation at the rate Γ per unit time and volume, we get dn dt = Γ − 2 Hn, (5.27) which has the stationary solution n = Γ/(2 H). This leads to which coincides with (5.20) if we use Γ ≈ eE/(2π), which is the pair production rate for massless charge carriers in flat space.
Since the infrared contribution can be very large for small mass and electric field, we will refer to this peculiar behaviour as infrared hyperconductivity. In general, the conductivity, defined as the ratio J/E, is larger for m 2 < H 2 /4 than it is for the case with m 2 = H 2 /4, for all values of eE. Only for eE H 2 do we recover the linear response J ≈ eE/(πH).

Heavy pairs (m H):
In general, the current is suppressed as we increase the mass. We can distinguish two cases, according to the value of .

Cosmological pair production ( 1 m/H)
In this regime, the semiclassical action is given by Pair production is exponentially suppressed, and so is the renormalized current. For very small electric field, |λ| 1, the current is given by The presence of a Boltzmann suppression factor at the Gibbons-Hawking temperature T = H/2π may naively suggest that gravitational particle production creates a hot plasma of charged particles, which are then set in motion by the electric field, leading to a current. However, this interpertation would be rather imprecise. We will come back to this issue in Section 8.

Pair production by the electric field ( 1)
In this limit, upward tunneling is very suppressed with respect to downward tunneling. For 1 m/H the classical action is large and the acceleration time is much smaller than the Hubble time. In this case, an expression of the form (5.19) should be valid, where now Γ is the flat space pair creation rate for massive particles, given in (1.2), This is indeed in agreement with Eq. (4.14) in the same limit. When the electric field is sufficiently large, l m/H, the semiclassical action for tunneling, S − = −πm 2 /eE, is small and pair production is unsuppressed. This is illustrated in the bottom curve in Fig. 2, which shows that the current responds linearly to the electric field in this regime. In this sense, Eq. (5.31) can be extrapolated to very large electric field.

Hadamard vacua and dS invariance
We saw in Section 5 that the "in" vacuum in the flat chart breaks dS invariance. We may ask whether this is due to a bad choice of the quantum state, or whether this feature is general and should be expected on physical grounds. After all, pair production induces the growth of a current. In this Section, we shall make this intuitive expectation more rigorous by showing that in any Hadamard vacuum dS invariance is broken.
It will be useful to think of 1+1 dimensional dS space as the hypersurface η AB X A X B = H −2 , (A, B = 0, 1, 2), (6.1) embedded in 2+1 dimensional Minkowski space with metric η AB = diag(−1, 1, 1). If X A and Y A are the coordinates of two points on this hypersurface, the variable is dS invariant. If X A and Y A are spacelike separated, then ζ = Hd is real, and d is the geodesic distance between the two points in dS. If they are time-like separated, then ζ is purely imaginary and |d| is the proper time separation along the geodesic connecting the two points. The dS metric can be written as where ζ = 0 corresponds to some arbitrarily chosen base point X A . The electric field can be written in terms of the gauge potential where stands for the covariant d'Alembertian and we use the convention τ Z = 1 for the Levi-Civita symbol. For a constant electric field, we can choose σ = σ(Z), with Up to an irrelevant additive constant, the general solution of this equation is In order to have a regular gauge potential in the coincidence limit, Z = 1, we choose C = 0. Let us now consider the two point functions G ± , defined in (5.1) and (5.2). Note that a Wilson line is inserted between the two points in order to make G ± gauge invariant. If this is calculated along the geodesic which links the points x and y, this specification of the path is dS invariant. Now, by using the covariant gauge (6.4) with σ = σ(Z), it is clear that the Wilson line vanishes y x A µ dx µ = 0. (6.8) The reason is that A Z = 0, while along a geodesic dx µ = δ µ Z dZ . On the other hand, it is important to note that at the base point x (corresponding to Z = 1), the value of τ is completely undefined, while A τ = −EH −1 (1 − Z) + 2HC will only vanish at Z = 1 provided that we choose C = 0. In other words, the Wilson line is only well defined for this choice of the integration constant in (6.7).
Using (6.8), we see that in the covariant gauge, and with the dS invariant specification of the path, G ± coincides with the Wightman function. This satisfies the standard wave equation for a charged field: where derivatives are with respect to the second argument, y. Let us now look for a dS invariant solution to (6.9), of the form where from now on we drop the ± superscripts. Noting that ∇ µ A µ = 0, A µ ∂ µ G(Z) = 0 and Eq, (6.9) reduces to It should be noted that this equation is gauge invariant. 6 To determine the behaviour of G in the coincidence limit, we look for solutions in a power series in the vicinity of Z = 1, The indicial equation α 2 = 0 has a double root, and so there is a regular solution and a logarithmically divergent solution. This is the expected behaviour for a two dimensional Green's function. However, we may also look at the behaviour of the solutions when the point y is close to the antipodal point of x, corresponding to Z = −1. These can be expanded as (6.14) In this case the indicial equation gives β = ±ieE/H 2 , and therefore the two point function necessarily has a branch cut singularity. This "infrared" singularity is reminiscent of the case of a massless neutral field in dS, where the solutions of the second order equation for a dS invariant two point function are also singular at the antipodal point. 7 In that case, it is known [25] that there is no dS invariant Fock vacuum, and we expect a similar situation in the present case. Since a dS invariant two point function necessarily includes singularities of a type which is different from the Hadamard form, we conclude that there are no dS invariant Hadamard vacua for charged particles in the presence of an electric field.

Persistence of memory
The current which we have obtained in Section 2 selects a preferred time direction, which is orthogonal to the frame in which the charge density vanishes J 0 = 0. An observer which is boosted with respect to t µ will observe a non-vanishing charge density J 0 = 0. Since the proper magnitude of the current tends to a constant, any effect of the preferred time direction will persist undiminished arbitrarily far into the future.
Of course, this also happens in the case of flat space, where the current has the form (1.1). But, while in flat space we are used to the fact that initial conditions can have a lasting effect, this may seem more surprising in an inflationary context. It is well known that a long period of inflation erases certain features of the initial conditions. For instance, unwanted relics are exponentially diluted away, and cosmological perturbations in the initial hypersurface (of unknown but possibly sizable amplitude) are stretched away to unobservably large distances. While this is true, there are certain observables for which the influence of the initial hypersurface persists after an arbitrarily large period of inflation [11], and the current which we have discussed in this paper belongs to this category. The current is made out of positively charged particles accelerating towards the right, and negatively charged particles accelerating towards the left. If we are in the rest frame of initial conditions, the number of particles or antiparticles which will hit us from the left or from the right is the same. However, if we move towards the right, we are more likely to be hit by a charged particle which is coming from that direction.
The discussion of Ref. [11] considered a simplified model of bubble nucleation, where the size of the bubbles at the time of nucleation was taken to be infinitessimally small. Here, we shall discuss a finite size effect, which has to do with the persistent influence of initial conditions in determining the frame of bubble nucleation 8 . Before moving into the case of de Sitter, let us first briefly recall the situation in flat space.

Flat space
It was found in Refs. [15,16] that, in the Lorentz invariant "in" vacuum, the frame of nucleation is very strongly correlated with the state of motion of the detector. Semiclassically, the trajectory of the two charges in a pair is given by the two branches of a hyperbola where 3) The trajectory (7.2) has contracting and expanding phases, before and after the turning point at t = 0. In the frame of nucleation, which we may denote byS, the trajectory of the charged particles has the same formx 2 −t 2 = r 2 0 but only the expanding phaset > 0 is physical: the particle and antiparticle nucleate at rest att = 0, and subsequently accelerate away from each other. In the frame of a detector consisting of a single particle, and moving at some speed v relative to the frame of nucleation, the trajectory of the charged particles would again have the form x 2 − t 2 = r 2 0 , but the physical half of it (witht > 0) would now correspond to t > −vx . For v = 0, some of the contracting phase, with t < 0, would be visible to the detector. What was found, however, is that the detector only sees the expanding phase, with t > 0, and therefore both frames must coincide to very good accuracy. Quantitatively, the relative speed between the detector and the frame of nucleation was found to be bounded by [16] ∆v ∼ S −1/3 1. Here, S = πm 2 /eE 1 is the action of the instanton which describes pair creation. The correlation between both frames is therefore very strong, 9 at least in the case where the system is in the Lorentz invariant (LI) "in" vacuum.
Suppose now the false vacuum is prepared at time t = 0, say, by turning on a constant electric field. This determines a preferred frame, S, which we call the frame of initial conditions, and so the system is no longer Lorentz invariant. After some transient behaviour, pairs will be produced at the Schwinger rate, for times [16] t τ nuc ∼ r 0 . (7.5) Here, τ nuc is the time it takes for a given pair to be excited out of the vacuum. This can be estimated to be of the same order as the size of the instanton, r 0 , given in (7.3). Let S be the frame of a detector, moving at speed v d relative to S. In the new frame, the false vacuum region t > 0 corresponds to For definiteness, let us choose v d > 0, with the detector following the world line x = 0. The particle and antiparticle in a pair that nucleates at rest with respect to S will be initially at the locations where x 0 is the midpoint between the two charges. If the detector interacts with, say, the positively charged particle at x = x + = 0 shortly after the time of nucleation, then the location of the negatively charged particle is at x ≈ −2r 0 . According to (7.6), for the negatively charged particle to be in the false vacuum, we must have τ 2v d r 0 . (7.8) Here, is the amount of proper time which the detector has spent in the false vacuum, with γ d = (1 − v 2 d ) −1/2 . It follows from (7.5) and (7.8) that if the detector has spent a short proper time τ in the false vacuum, τ r 0 , (7.10) then this detector will feel the influence of initial conditions. Indeed, if the detector is nonrelativistic, so that τ ∼ t, then (7.5) is violated and there is not enough time for the electric field to produce a pair out of a vacuum fluctuation. On the other hand, if the detector is relativistic, there may be enough time, t r 0 , but then Eq. (7.10) is incompatible with (7.8), which tells us that the pairs will not be seen to nucleate in the rest frame of the detector. In both situations, the frame of initial conditions will have an appreciable effect. 9 The bound (7.4) coincides with the minimum quantum uncertainty in the velocity of a non-relativistic charged particle embedded in a constant electric field. A velocity of order ∆v is reached after a time interval of order ∆t ∼ S r0 past the turning point. If the interaction of the nucleated pair with the detector takes place in the vicinity of the turning point, the semiclassical description does not apply. But even in this case, it was found [16] that there is a strong asymmetry in the momentum transferred from the nucleated particles to the detector, in the direction of expansion after the turning point, consistent with the detector seeing only the pairs moving away from each other .
Conversely, if the detector spends a large proper time τ r 0 (7.11) in the constant electric field E, much larger than the size of the instanton, then we do not expect the initial hypersurface to play much of a role in determining the frame of nucleation [16]. The condition (7.11) is trivially satisfied at sufficiently late times, for any given velocity v d of the detector, so we do not expect any influence of the initial conditions to survive in the asymptotic future. This is in agreement with the results which are obtained by using the Lorentz invariant "in" vacuum. In that case, the electric field is switched on at past infinity, and the frame of nucleation is entirely determined by the state of motion of the detector [15,16].

de Sitter
Let us now consider the case of de Sitter. For simplicity, we focus on the case where the initial conditions are imposed on an equal time slice in the flat chart, η = η 0 . A case of particular interest is the "in" vacuum, which we can think of as a limiting case where η 0 → −∞.
The embedding coordinates introduced in Eq. (6.1) are related to the flat chart coordinates (η, x) by 14) The trajectory of a charged pair can be obtained by intersecting the hyperboloid (6.1) with the plane [21] Here with (7.17) In (7.15), the plus sign corresponds to upward tunneling, and the minus sign to downward tunneling. The intersection of (6.1) and (7.15) leads to hyperbolas in the (X 0 , X 1 ) plane, of the form The two branches of (7.18) correspond to the worldlines of the two charges in the pair. In terms of the flat chart coordinates, these worldlines are given by The center of symmetry of the trajectory is at the point X 0 = X 1 = 0, and X 2 = H −1 . In the flat chart coordinates, this corresponds to the spacetime point We may refer to this center of symmetry as the "nucleation event", although strictly speaking nucleation takes up an extended region of spacetime. By using SO(2,1) transformations, the trajectory of any nucleated pair can be brought to the "standard" form (7.19), where the center of symmetry is at (7.20), so without loss of generality we shall restrict attention to this semiclassical trajectory. The physical momentum of each one of the particles in the pair is given by where the superindex ± refers to the solution for upward or downward tunneling, respectively. Comparing this semiclassical expression for the momentum with Eq. (3.23) we find that the trajectories (7.19) correspond to modes with k ≈ ±k σ ≡ ±H|σ|, (7.22) where we have used |σ| 1. According to (3.21), the time at which these modes are excited corresponds to a(η kσ ) = 1. This suggests that the semiclassical trajectory (7.19) should be restricted to η > η kσ = −H −1 .

(7.23)
This would correspond to a pair nucleating on an equal time hypersurface, with the particle and antiparticle materializing at the same value of η.
For the flat chart "in" vacuum state, which breaks dS invariance, the η = const. hypersurfaces correspond to the preferred frame which is determined by the initial conditions. As mentioned at the beginning of this Section, it is reasonable to expect that such initial conditions may have some influence in determining the rest frame of nucleation. We saw that, in flat space, initial conditions may have some impact, but this fades away in the asymptotic future. Let us now show that things can be quite different in dS space.
First, we note that for an inertial detector which is not co-moving, the proper time which has been spent in the flat chart of dS is bounded by [26] τ ≤ 1 2H ln Here, γ d is the relativistic factor of the detector relative to the co-moving observers. Consider now the situation where m 2 H 2 e 2 E 2 . In this case is approximately equal to the flat space value r 0 for the radius of the hyperbola, with It follows that for a highly relativistic detector, with Hr 0 (7.27) we have τ r 0 . (7.28) As we saw in the case of flat space, a detector which has been in the false vacuum for such a short amount of time can feel the influence of initial conditions. Pairs are not necessarily expected to nucleate in its rest frame. A second point to note is that, in dS, we should be specific about what we mean by pairs nucleating at rest in a given frame. For a pair nucleating on the hypersurface η = −H −1 , the physical momentum of the particles at the time of nucleation, relative to the co-moving observers, is given by (7.21) with a = 1. This is non-vanishing, both for upward and for downward tunneling. Because of that, instead of asking whether the pair nucleates at rest in the frame of the detector, it may be more pertinent to ask whether a boost in the detector's worldline is accompanied by a corresponding boost in the hypersurface on which the two charges in the pair are seen to nucleate simultaneously.
For a pair nucleating on the η = −H −1 hypersurface, the temporal coordinate of the particles on the hyperbola (7.18) at the time of nucleation is given by Boosts in the X 1 direction will change initial value X 0 initial , without changing the form of the semiclassical trajectory (7.18). Introducing the boost parameter φ 1 through the relation a boost of velocity v = tanh φ 1 in the (X 0 , X 1 ) plane will bring the initial time in the trajectory of one of the charges to the value X 0 = 0. By further increasing the boost parameter, this initial time will go into negative values of X 0 . However, in the flat chart, there is a minimum value of X 0 on the trajectory of the pair, given by Lower values of X 0 are outside of the flat chart. The boost parameter φ 2 which is needed to bring the initial time of the particle trajectory from X 0 = 0 to X 0 = −w ± 0 is given by The maximum boost which can be applied to the pair which nucleates on the η = H −1 hypersurface without having one of the particles in the pair start its worldline outside of the flat chart is given by For HR 0 1, for downward tunneling, and γ max ∼ 1 for the case of upward tunneling. Ignoring upward tunneling, we conclude that a detector with relativistic factor γ d γ max ∼ 1 HR 0 (7.37) with respect to the co-moving frame, cannot detect pairs whose nucleation hypersurface is boosted by the same relativistic factor relative to the η = const. hypersurface. Including upward tunneling, the same conclusion applies for γ d 1. In this sense, fast moving detectors feel the influence of the hypersurface of initial conditions in which the false vacuum has been prepared. Note that this conclusion is in agreement with our earlier expectation, which was based on Eqs. (7.27) and (7.28).
Unlike the case of flat space, here the influence of initial conditions in determining the frame of nucleation persists arbitrarily far into the future. Note that here we have considered pairs nucleating at η = −H −1 , but since the surface of initial conditions is η → −∞, our analysis, and the estimate of γ max given in (7.37) is independent of the time at which nucleation occurs.

Summary and discussion
Vacuum transitions in an inflating multiverse may proceed by quantum tunneling. A simple model where such transitions can be analyzed beyond the semiclassical approximation is the Schwinger process in 1+1 dimensions.
In de Sitter space, both the electric and gravitational fields can pull pairs out of the vacuum. Particles and antiparticles are subsequently accelerated by the electric field and diluted by cosmic expansion. This results in a stationary spacelike electric current of proper magnitude J, given by Eq. (5.14), as Here J φ = J/e is the charge number current, and Throughout this paper the electric and gravitational fields have been treated as external sources. In the situation where the electric field is dynamical, Gauss's law requires a discontinuity ∆E = e accross the position of the charges. The decoupling limit where the electric field can be treated as external corresponds to e/E → 0 while keeping eE fixed. 10 Within this limit, Eq. (8.1) is valid in the full range of parameters m, eE, and H = 0. 11 A non-vanishing current breaks dS invariance. Since the background is invariant, this has to be attributed to the choice of quantum state. In our case, the current is parallel to the equal time slicing in the flat chart, which is used in order to define the "in" vacuum. 12 More generally, we have shown that it is not possible to choose a dS invariant Hadamard quantum state for E = 0.
The semiclassical regime corresponds to imaginary values of σ, with |σ ± λ| 1, where λ = ieE/H 2 . In this case, the distribution of created particles is given by [20], where β ± are the Bogoliubov coefficients, given in Eq. (2.18), and the double sign refers to positive and negative values of the co-moving momentum k, respectively. These, in turn, correspond to upward and downward transitions. The distribution (8.3) is discontinuous at k = 0, but it is otherwise flat, with a UV cut-off which depends on time, Here a(t) is the scale factor, and p ± c is the physical momentum of the particles at the time of pair creation 13 It was shown in Refs. [20,21] that Eq. (8.3) is in agreement with the distribution of particles which can be calculated with instanton methods. Interestingly, the exact expression for the renormalized expectation value of the current, Eq. (8.1), can also be obtained from a simple semiclassical computation, where we add the contributions from all individual pairs in the distribution (8.3). Aside from the current flowing along the semiclassical trajectories, a vacuum current has to be included, connecting the particle and antiparticle at the time of pair creation, so that charge is locally conserved. It turns out that the contribution from the vacuum current is comparable to that from the semiclassical trajectories, and the sum of these two is insensitive to the precise value which we adopt for p ± c [as long as we take k c ∝ a(t), as in (8.4)]. The total current takes then the form which manifestly shows the separate contributions from downward and upward transitions. This expression reproduces (8.1) exactly once we substitute the Bogoliubov coefficients given by (2.18). The conductivity of the vacuum in different regimes can be readily analyzed from (8.1). For m 2 = H 2 /4, we find a linear response J φ = eE/(πH), with resistivity proportional to the expansion rate H. This particular value of the mass corresponds to the case where (in the absence of the electric field) long wavelength modes behave as critically damped oscillators.
For smaller values of the mass, m 2 H 2 /4, infrared effects are important and the conductivity can be very large. In fact, for mH eE H 2 , the current behaves in inverse proportion to the electric field, J φ ≈ H 3 /(2πeE), much in contrast with Ohm's law. This phenomenon is due to the infrared behaviour of the two point function in dS, and is expected to be present also in 3+1 dimensions, for light fields with m H. Infrared hyperconductivity may have important consequences for cosmology (e.g. in scenarios where magnetic fields are generated during inflation). We leave this as a subject for future research. 13 The time of pair creation is estimated as the time when the violation of adiabaticity in the evolution of the mode functions is maximum.
accelerating towards the left. If we are in the rest frame of initial conditions, the number of particles or antiparticles which will hit us from the left or from the right is the same. However, if we move towards the right, we are more likely to be hit by a charged particle which is coming from that direction. This is the persistence of memory effect first discussed in [11]. Here, we have argued that initial conditions will also have a persistent influence in determining the frame of nucleation of new pairs. The reason is simple. Denoting by R 0 the size of the instanton, the frame of nucleation cannot be boosted by a relativistic factor larger than γ (HR 0 ) −1 relative to the frame of initial conditions, without having one of the particles in the pair intersect the hypersurface of initial conditions. distance is of order d ∼ 2m/eE 2H −1 for p ≈ 0. Again, the time when the physical distance between the particle and antiparticle in a pair is near its minimum is in agreement with our earlier estimate (3.21) for the time of pair creation.

B Momentum integral
Here, we calculate the integral in Eq. (5.13), which gives the expectation value of the current in the "in" vacuum: The integral is convergent, given that λ = i|λ| is purely imaginary, but for the manipulations in the following it is necessary to temporarily insert a factor e − x to make each term converge individually.