Conformally-flat gravitational analogues to the Schwinger effect

Conformally-flat gra vitational analogues to the Sc h winger effect S. A. F ranc hino-Vi ˜ nas, 1, 2, 3 F. D. Mazzitelli, 4, 5 and S. Pla 6 1 Dep artamento de F ´ ısic a, F acultad de Ciencias Exactas Universidad Nacional de L a Plata, C.C. 67 (1900), L a Plata, Ar gentina 2 CONICET, Go doy Cruz 2290, 1425 Buenos A ir es, Ar gentina 3 Universit´ e de T ours, Universit´ e d’Orl´ eans, CNRS, Institut Denis Poisson, UMR 7013, T ours, 37200, F r anc e 4 Centr o At´ omic o Barilo che, Comisi´ on Nacional de Ener g ´ ıa A t´ omic a, R8402AGP Barilo che, Ar gentina 5 Instituto Balseir o, Universidad Nacional de Cuyo, R8402AGP Barilo che, Ar gentina. 6 Physik-Dep artment, T echnische Universit¨ at M¨ unchen, James-F r anck-Str., 85748 Gar ching, Germany W e study particle creation for scalar fields in conformally flat spacetimes using resummed heat- k ernel tec hniques. W e make use of an analogy betw een quantum scalar fields in conformally flat spacetimes and scalar field theories with a Y uk a wa coupling in Mink o wski space. The correspondence holds exactly at the level of the effective action and includes nonconformal curv ature couplings. This framework provides access to particle creation at strong curv ature. In a radiation dominated univ erse, the particle pro duction rates in arbitrary dimensions are indep endently confirmed through explicit calculations of the Bogoliub o v co efficien ts. W e also find new exact gravitational analogues of the Sch winger effect in quantum field theory in curved spacetime. I. INTR ODUCTION In tense fields are b eliev ed to b e ubiquitous in our uni- v erse, arising for example in the early universe and in the vicinity of singularities. In such scenarios, the de- scription of physical phenomena frequently necessitates nonp erturbativ e techniques capable of encompassing the strong-field c haracter of the bac kgrounds. The archet ype of such effects is pair creation, which, to cite a few ex- amples, has clear implications for cosmology [ 1 ] and for the ev aporation of blac k holes [ 2 ]. Recen tly , in Refs. [ 3 , 4 ] the authors of the present letter ha ve developed a heat-kernel approac h to obtain (par- tially) resummed effective actions, b y appropriately in- tegrating out scalar and spinor fields in the presence of electromagnetic and Y uk a w a backgrounds. Our analyt- ical approach is v alid for an arbitrary n umber of space- time dimensions and general backgrounds; moreov er, as long as a “high-intensit y” feature is present in the lat- ter, the results condense into compact formulae. This is in contrast with other av ailable analytic nonperturba- tiv e approaches, for which only certain solv able cases are tractable, e.g. when using the Bogoliub o v tec hnique [ 5 ] and the analytic worldline instan tons [ 6 , 7 ]. In general cases, it seems unav oidable to try to resort to numer- ics, suc h as the numeric worldline instanton metho d [ 8 ], whic h are nev ertheless so far restricted to simple cases. Our heat-kernel results can b e readily used to com- pute, in the presence of background fields, the transition probabilit y relev an t for the pair creation pro cess, i.e. the v acuum p ersistence probabilit y . Indeed, in the in-out for- malism, the v acuum is respectively describ ed by the | in ⟩ and | out ⟩ states at early and late times, so that suc h a probabilit y is tightly linked to the effective action Γ through the formula |⟨ out | in ⟩| 2 = | e iΓ | 2 =: e − P , (1) where P signals the instability of the v acuum and can b e in terpreted as the (total, integrated) probability of pair creation. In this letter, our goal is to sho w that the heat-k ernel outcomes of Refs. [ 3 , 4 ] ha ve a wider range of appli- cabilit y than primarily though t, including ev en gravi- tational scenarios. In order to do so, w e are going to app eal to analogue systems. In brief, we will employ the fact that for a scalar field in a conformally flat ge- ometry , the W eyl-rescaled action in conformal time co- incides with the action of a scalar field in Minko wski spacetime with a p osition-dep enden t p oten tial (or effec- tiv e mass), so that our resummed heat-kernel results for Y uk a w a backgrounds apply . F urthermore, we iden tify a class of FLR W metrics for whic h the v acuum p ersistence probabilit y tak es a Sc hwinger-lik e form, closely parallel- ing Scalar Quan tum Electrodynamics (SQED) in a con- stan t electric field. Analogies b etw een quan tum fields in curved spacetime and gauge or scalar systems hav e of course b een devel- op ed in the past b y means of other techniques. F or in- stance, in Refs. [ 9 , 10 ], a massive, conformally coupled scalar field w as considered in a v ariety of FLR W space- times. The corresponding modes ha ve a time-dep enden t frequency whic h w as in terpreted in terms of a (to b e determined) bac kground electromagnetic field and the pair creation probabilit y was computed in conformal time thanks to the Bogoliubov method. Instead, the approach of Refs. [ 11 ] is based on the use of a different time v ari- able, which is neither the cosmological nor the conformal time [ 12 ]. In a giv en time-dependent metric, they employ the Bogoliubov method to compute the pair pro duction probabilit y of a massless and minimally coupled scalar field. Con trary to those cases, our analogies will in principle b e applicable b ey ond FLR W universes and are going to b e at the level of the effective action, i.e. not just at the lev el of the single mo des. A sp ecial emphasis will b e giv en to radiation dominated univ erses, for which w e will in parallel derive the results in the Bogoliub o v formalism 2 and for arbitrary dimensions. Moreov er, w e will highligh t the existence of new analogue cosmological ev olutions for cases in which pair creation is induced b y a nonconformal coupling to the curv ature. Let us also remark that, in the context of the heat- k ernel techniques, the Barvinsky–Vilko visky expansion enables one to obtain the pair creation probabilit y for arbitrary configurations whenev er the curv atures are small [ 13 , 14 ]. Instead, the expressions av ailable in the literature for large curv atures, as far as w e kno w, are con- siderably more restricted in their range of application to pair creation [ 5 ]. The metho d developed in the follo wing will thus contribute to partially filling this gap, since it is inheren tly linked to an expansion for strong curv atures and to a nonp erturbativ e effect of pair creation, to which w e are going to generically refer as “Sch winger effect.” As a last comment b efore the computations, note that w e are going to consider pseudo-Riemannian metrics with a mostly minus signature; we will define the Riemann tensor from the Christoffel symbols Γ µ ν ρ as R µ ν αβ := ∂ β Γ µ ν α + · · · , while the Ricci tensor corresp onds to the con traction R µν := R ρ µρν . I I. QUANTUM SCALAR FIELD IN CONF ORMALL Y FLA T METRICS Let us start by considering an arbitrary d -dimensional Loren tzian spacetime, with a given metric g µν . In this univ erse, define the action for a free quantum, massive, real scalar field ϕ ; it can b e written as S := 1 2 Z d d x p | g |  ( ∇ ϕ ) 2 − ( m 2 + ξ R ) ϕ 2  , (2) where we hav e introduced a nonminimal coupling to the Ricci scalar R (with co efficien t ξ ), m is the mass of the field, ∇ is the cov ariant deriv ativ e compatible with the metric and, as customarily , g stands for the determinant of g µν . In the following w e are going to fo cus on conformally flat spacetimes, whose line elemen ts can be describ ed in terms of a single scalar function Ω. In conformal time τ , the line element can b e written as d s 2 = Ω 2 ( τ , x )  d τ 2 − d x 2  , (3) where we hav e split the time component, τ , from the ( d − 1) spatial ones, x . Imp ortantly , this class of metrics encompasses the FLR W univ erses, for whic h Ω b ecomes a time-dep endent scale factor a ( τ ). This can also b e particularly useful for d = 2, since in tw o dimensions ev ery manifold is conformally flat. In tro ducing also a W eyl rescaled field, φ := Ω ( d − 2) / 2 ϕ , the classical action reads S [ φ ] = 1 2 Z d τ d ( d − 1) x  ( ∂ φ ) 2 − Ω 2  m 2 + ( ξ − ξ d ) R  φ 2  , (4) where ∂ µ denotes partial deriv ativ es with resp ect to τ and x , ξ d := ( d − 2) 4( d − 1) is the so-called conformal v alue for the nonminimal coupling and R Ω 2 = 2( d − 1)Ω − 1 ∂ 2 Ω + ( d − 1)( d − 4)Ω − 2 ( ∂ Ω) 2 . (5) Observing Eq. ( 4 ), o ne immediately recognizes that the action corresp onds th us to that of a scalar field in Mink owski spacetime with a Y uk aw a p otential V ( τ , x ) := m 2 Ω 2 + ( ξ − ξ d ) R Ω 2 . (6) T o analyze the quantum effects arising in this model, w e can, therefore, consider the effective action Γ for this field, given by Γ = S + 1 2 log Det Q , (7) where we ha ve defined the so-called op erator of quantum fluctuations, Q := ∂ 2 + Ω 2  m 2 + ( ξ − ξ d ) R  . (8) F or the atten tive reader, a tec hnical comment related to the quan tization pro cess is in order: in the corresp ond- ing functional integration, the W eyl rescaling of the field w ould imply an Ω-dep endent Jacobian. How ever, it will exactly cancel with the factor that is necessary to guaran- tee inv ariance under diffeomorphisms [ 15 ]. Alternativ ely , one c an imp ose a finite renormalization on the energy- momen tum tensor, as in Refs. [ 16 , 17 ], which will lead to the same cancellation. A. Resummed heat-k ernel approac h Eq. ( 8 ) implies that we can study the effectiv e action of the system in the language of Ref. [ 4 ]. W e can thus define the heat k ernel K asso ciated to Q as the solution of the equations [ ∂ s − Q ] K ( x, x ′ ; s ) = 0 , K ( x, x ′ , 0 + ) = δ ( x − x ′ ) , (9) where s is an auxiliary parameter whic h is called the prop ertime. Imp ortan t to our discussion, Ref. [ 4 ] has prov ed that the heat kernel for this type of op erators admits a re- summed expression from which nonp erturbativ e aspects of pair creation can b e studied, with the only assump- tion that the p oten tial is intense enough; in particular, it could depend on both time and space co ordinates. Defin- ing γ 2 αβ := 2 V ,αβ , the diagonal of the heat kernel tak es the form 3 K ( x, x ; s ) = 1 (4 π s ) d/ 2 e − sV + ∂ α V [ γ − 3 ( γ s − 2 tanh( γ s/ 2)) ] αβ ∂ β V det 1 / 2  ( γ s ) − 1 sinh( γ s )  W ( x, x ; s ) , (10) where the prefactor r esums al l the invariants built from V , V ,α and V ,αβ . On the other hand, W contains the information on higher deriv atives; in particular, if higher deriv ativ es v anish, W ( x, x ; s ) = 1. It is worth emphasiz- ing that substituting the p otential from Eq. ( 6 ) into Eq. ( 10 ) leads to a resummed expression for the heat k ernel, v alid for arbitrary conformally flat metrics and exact for quadratic p oten tials. T o further simplify the discussion and the subsequen t comparison with Bogoliub o v metho ds, consider a radia- tion dominated universe, whic h corresp onds to a metric whose conformal factor is a linear function in the confor- mal time 1 m 2 Ω 2 ( τ , x ) → m 2 a 2 ( τ ) = b 2 0 τ 2 , −∞ < τ < ∞ . (11) F or the moment, w e will also set ξ = ξ d . F rom Eq. ( 10 ) w e thus obtain an exact result for the diagonal of the heat kernel, which in Lorentzian signature 2 reads K ( x, x ; s ) = 1 (4 π s ) d/ 2 s b 0 s cos( b 0 s ) sin( b 0 s ) e − b 0 τ 2 tan( b 0 s ) . (12) Imp ortan tly , the effective action can be directly com- puted from a prop ertime integral of the trace of the hea t k ernel; in our case, this reduces to computing Γ = Z d d x (4 π ) d/ 2 Z ∞ 0 d s s 1+ d 2 √ b 0 s e − b 0 τ 2 tan( b 0 s ) p cos( b 0 s ) sin( b 0 s ) = √ π V 0 (4 π ) d/ 2 Z ∞ 0 d s s d +1 2 1 sin( b 0 s ) , (13) where V 0 has been introduced to denote the spatial vol- ume of the manifold. Actually , the in tegral in Eq. ( 13 ) is ill-defined, since its integrand has an infinite n umber of poles, which are placed at s = π n/b 0 , with n = 0 , 1 , · · · . An appropri- ate w ay to give meaning to it is to recall that it comes 1 In the computation of particle production, an extension of the τ > 0 universe to negative conformal times is habitually p er- formed. In our case, we are extending the radiation dominated universe in a mirror-lik e wa y; the interested reader can find in Refs. [ 18 , 19 ] other alternatives that ha ve been discussed in the literature. 2 The heat-k ernel expansions obtained in Ref. [ 4 ] are derived for an Euclidean spacetime. The Lorentzian coun terpart is obtained by performing a Wick rotation, which is assumed in the following, keeping a small imaginary part in the prop ertime s . from a Wick rotation, which giv es a precise prescription to circumv ent the p oles and generates a non v anishing imaginary contribution to the effective action. Let us discuss the different contributions induced by the p oles. The one at n = 0 is related to ultraviolet div ergences and corresp onds to the renormalization of the theory . Instead, the remaining p oles are not affected b y the renormalization process; fo cusing just on their imaginary con tributions and recalling the definition in Eq. ( 1 ) for the v acuum p ersistence probabilit y , it can b e sho wn that P 2 := Im Γ = V 0 2(2 π ) d − 1 b ( d − 1) / 2 0 ∞ X n =1 ( − 1) n +1 n ( d +1) / 2 , (14) whic h can b e written in closed form in terms of a Rie- mann zeta function ζ R ( · ): P 2 = − V 0 2(2 π ) d − 1 b ( d − 1) / 2 0  1 − 2 (1 − d ) / 2  ζ R  d + 1 2  . (15) It is imp ortan t to note that P is the total pair creation probabilit y , i.e. it is already integrated o ver the entire spacetime. The result in Eq. ( 14 ), ev aluated for d = 4, agrees for example with Ref. [ 9 ]; the agreement with the Bogoliub o v metho d in an arbitrary n um b er of dimensions will b e shown in the following section. B. Bogoliub o v coefficients In the simple case that we ha ve considered, one can cross chec k the obtained result by computing the Bogoli- ub o v coefficients linking the in and out v acua. A compre- hensiv e review on this method can be found, for example, in Refs. [ 5 , 20 ]. The first step consists in expanding the quan tized field in F ourier mo des, which are adapted to the underlying homogeneity of the metric in the spatial co ordinates: φ ( τ , x ) = Z d 3 k p 2(2 π ) 3  B k e i kx φ k ( τ ) + B † k e − i kx φ ∗ k ( τ )  . (16) In doing so, we hav e introduced the creation and annihi- lation op erators, B † k and B k , which satisfy the canonical comm utation relations as a consequence of the comm uta- tors b et ween φ and its conjugate momentum. The field equation for the W eyl-rescaled mo des can b e directly de- riv ed from Eq. ( 4 ), φ ′′ k + ω 2 k φ k = 0 , ω 2 k := k 2 + m 2 a 2 + ( ξ − ξ d ) Ra 2 , (17) 4 where a prime denotes deriv ativ es with respect to the conformal time and their corresponding normalization condition is given b y φ k φ ′∗ k − φ ′ k φ ∗ k = 2i . (18) The general solution to Eq. ( 17 ), for ξ = ξ d , can b e compactly expressed as φ k ( τ ) = C k , 1 S k ( τ ) + C k , 2 S ∗ k ( − τ ) , (19) where the function S k is essen tially a parab olic cylinder function D ν ( z ), S k ( τ ) :=  2 b 0  1 / 4 D − 1 2 − 2i κ  e i π 4 p 2 b 0 τ  , (20) and we hav e in tro duced the rescaled, dimensionless squared momentum κ := k 2 4 b 0 . In this context, we can naturally define the v ac- uum states at τ → ±∞ . Indeed, for early or late times, the expansion of the universe slows down and one can naturally define an (infinite order) adiabatic v ac- uum | 0 ± ⟩ . First of all, in the late asymptotic region ( τ → + ∞ ) and according to the adiabatic choice, the preferred (p ositive-frequency) solution for the late-time mo des φ (+) k reads [ 19 , 21 ] φ (+) k ( τ → + ∞ ) ∼ e − i R τ ω k ( u )d u p ω k ( τ ) ∼ e − i b 0 2 τ 2 − i κ ln ( 2 b 0 τ 2 ) √ b 0 τ , (21) where ω k ( τ ) = p k 2 + m 2 a 2 ( τ ). If we imp ose this late- time b ehavior in the general solution ( 19 ), the coefficients are completely determined to b e C + k , 1 = e − πκ 2 +i π 8 , C + k , 2 = 0 . (22) Afterw ards, expanding the field in terms of φ (+) k and its complex conjugate, we can define the F o c k space corre- sp onding to the accompanying annihilation and creation op erators, respectively B (+) k and B (+) , † k , with the v acuum | 0 + ⟩ defined as the state containing no B (+) k particles. Analogously , the early-time adiabatic v acuum | 0 − ⟩ is determined by the adiabatic early-time mo des φ ( − ) k , whic h on their turn satisfy the asymptotic condition φ ( − ) k ( τ → −∞ ) ∼ e − i R τ ω k ( u ) du p ω k ( τ ) ∼ e i b 0 2 τ 2 +i κ log ( 2 b 0 τ 2 ) √ − b 0 τ ; (23) a solution satisfying suc h constrain t is obtained from Eq. ( 19 ) by c ho osing the following co efficien ts: C − k , 1 = 0 , C − k , 2 = e − πκ 2 − i π 8 . (24) A t late times, we can expand the mo des φ ( − ) k in terms of φ (+) k and its conjugate, since they form a basis: φ ( − ) k ( τ → ∞ ) = α k φ (+) k ( τ ) + β k  φ (+) k  ∗ ( τ ) . (25) This relation tigh tly links b oth type of mo des, b eing α k and β k the so-called Bogoliub ov co efficients, which ex- plicitly dep end on the momenta of the mo des inv olv ed. F or the present spacetime, we find α k = e − 3 π κ √ 2 π Γ  1 2 + 2i κ   1 + e 4 π κ  , β k = − i e − 2 π κ , (26) whic h satisfy the Bogoliub o v consistency condition | α k | 2 − | β k | 2 = 1. No w let us prepare our system suc h that at early times | in ⟩ = | 0 − ⟩ . At late times, an observer will naturally define particles through the late-time F o ck s pace, i.e. | out ⟩ = | 0 + ⟩ , so the v acuum persistence probability cor- resp onds to   ⟨ 0 − | 0 + ⟩   2 = exp  − V 0 Z d d − 1 k (2 π ) d − 1 log | α k | 2  , (27) where it should b e recalled that V 0 is the spatial volume. One can readily compute | α k | 2 = 1 + e − 4 π κ , either from its definition or from the consistency Bogoliubov relation; matc hing to Eq. ( 1 ), the probability of pair creation can b e seen to b e in agreemen t with Eq. ( 14 ). A t this p oin t we can mak e an interesting remark re- lated to the pair pro duction rate in Eq. ( 14 ). This for- m ula is similar but not equal to Sch winger’s result for a constant and homogeneous electric background E in SQED; indeed, for massiv e fields of mass m SQED and c harge e , it establishes that the pair creation probability in tegrated during a p eriod of time T 0 is P SQED V 0 = 2 π T 0  eE 4 π 2  d/ 2 ∞ X n =1 ( − 1) n +1 n d/ 2 e − nπ m 2 SQED / ( eE ) . (28) The reason for suc h a difference can b e traced bac k, for example, to the equation for the mo des in the Bogoliub ov approac h. In SQED the corresp onding frequencies are ω 2 k , SQED = m 2 SQED + k 2 ⊥ +( k ∥ + eE 0 t ) 2 , where we ha ve split the momentum comp onents according to whether they are parallel ( k ∥ ) or perp endicular ( k ⊥ ) to the p olarization of the electric field. On the other hand, for a radiation dominated universe the frequencies are ω 2 k , rad = k 2 ⊥ + k 2 ∥ + b 2 0 τ 2 . This implies that the corresp onding Bogoliub o v co efficien ts are the same only when k ∥ = 0 and m SQED = 0, as long as we identify eE ≡ b 0 ; additionally , in SQED, the integration o ver k ∥ giv es the time interv al and not a further p o wer of n . It is w orth noticing that, con trary to the situation in the Sch winger effect, the pair creation probability in a radiation dominated universe do es not display an exp o- nen tial suppression with the mass, even if the field is massiv e. I II. OTHER GRA VIT A TIONAL ANALOGUES Our heat kernel master formula ( 10 ) is rather v ersa- tile, since it is only link ed to the op erator Q in Eq. ( 8 ). 5 Indeed, we hav e already sho wn that results in a gravi- tational setup can b e studied by analyzing an analogous problem with a spacetime-dependent mass. There is a further type of analogy that is immediate but, somehow, has not b een pursued in the literature b efore: the case in whic h, for a massless field, the non- minimal coupling to the curv ature is resp onsible for pair creation. W e will refer to pair creation in this scenario as curv ature-induced; its relev ance has been previously considered, for instance , in the context of generation of dark matter during reheating, in which Ra 2 is oscillat- ing [ 22 – 24 ]. The condition to find a curv ature analogue of the Sc hwinger effect is Ra 2 ∝ τ 2 + c , (29) with c a real constant. T o b e more precise, the radiation dominated universe ab o ve is equiv alen t to mo dels with c = 0; on the other hand, as w e are going to see in Case I I b elo w, a non-v anishing c provides a straightforw ard generalization in which c plays the role of a mass term in the equation for the modes φ k . Let us analyze these alternativ es in detail. Case I. If c = 0 and d = 4, we can determine the analogous scale factor by solving the differen tial equation ( ξ − ξ 4 ) Ra 2 = 6( ξ − ξ 4 ) a ′′ a =: 6( ξ − ξ 4 ) b 2 0 τ 2 , (30) where we inserted a factor 6( ξ − ξ 4 ) into the last equality for conv enience. The general solution to this differen tial equation can b e obtained as a sp ecial case of Eq. ( 19 ): a ( τ ) = c − D − 1 / 2 ( p 2 b 0 τ ) + c + D − 1 / 2 ( − p 2 b 0 τ ) . (31) The function D − 1 / 2 ( − √ 2 b 0 τ ) is a positive and monotoni- cally increasing function, whose asymptotic expansion for large τ > 0 can be read from Eq. ( 23 ). In the general case where both co efficients c ± are non-zero, the scale factor describ es a bouncing univ erse. In cosmological time, de- fined as t := R τ 0 a ( τ 1 )d τ 1 , one can obtain the large-time b eha viour of the scale factor, a ( t ) ≈ √ 2 b 0 t q log  √ b 0 t  . If instead one of the coefficients c ± v anishes, the univ erse either collapses or shows a big bang, b oth at a finite cos- mological time. As already said, in this case the pair creation probabil- it y can be obtained from Eq. ( 14 ) b y a simple rescaling of b 0 . Curiously , the parab olic cylinder functions play a dual role here: on the one hand, they determine the function a ( τ ), while, on the other hand, they are crucial to compute the modes of the field. Case I I. W e can enlarge now our discussion to mo d- els whose v acuum persistence probability is no longer de- scrib ed b y expression ( 14 ). F or example, we can consider a situation in which Ra 2 is a p olynomial quadratic in τ . A simple choice in an arbitrary num ber of dimensions d > 2 is a Gaussian scale factor, a ( τ ) = a 0 exp  − ατ 2 / 2  , where a 0 and α are real parameters that resp ectiv ely go v- ern its in tensit y and its time dependence. F or this simple geometry , the Ricci scalar is given by R = 2( d − 1) α a 2 0  − 1 + d − 2 2 ατ 2  e ατ 2 , (32) so that R a 2 ( τ ) = 2( d − 1) α  d − 2 2 ατ 2 − 1  . Note that, in order to apply our metho ds, the cou- pling to the curv ature should b e such that ξ − ξ d > 0, so that the effectiv e p otential is confining (and not unsta- ble) for large v alues of τ ; we shall also restrict α ≤ 0, so that the quantum field is not tach yonic. In this universe, where the cosmological time can b e written in terms of the imaginary error function Erfi, t a 0 =  π 2 | α |  1 / 2 Erfi r | α | 2 τ ! , t ∈ R , (33) the scale factor describes once again a b ouncing univ erse: it reaches a minim um a ( τ = 0) = a 0 and expands slow er than an exp onen tial. Indeed, using the asymptotic ex- pansion for the imaginary error function, one can show that, in cosmological time, a ( t ) ≃ p 2 | α | t r log  p | α | t  for large v alues of t , which is essentially the same b eha v- ior found in the previous Case I. F or the sak e of complete- ness, since as far as we know this universe has not been widely studied in the literature b efore, let us note that the null, the weak and the dominant energy conditions are violated in the region τ 2 < 1 / | α | , where the transi- tion from a contracting to an expanding universe tak es place, while the strong energy one is alw a ys violated. The diagonal of the heat kernel and the effectiv e action corresp onding to this universe can b e readily computed using Eq. ( 10 ). The result is Γ = √ π V 0 (4 π ) d/ 2 Z ∞ 0 d s s d +1 2 e − ˜ m 2 s sin(˜ as ) , (34) where the effective mass and frequency are resp ectiv ely giv en by ˜ m 2 : = 2( d − 1)( ξ − ξ d ) | α | , (35) ˜ a 2 : = ( d − 1)( d − 2)( ξ − ξ d ) α 2 . (36) In this case, if ξ > ξ d , the curv ature itself en tails an exp ected exponential suppression in the creation of pairs, P 2 = V 0 2(2 π ) d − 1 ˜ a ( d − 1) / 2 ∞ X n =1 ( − 1) n +1 n ( d +1) / 2 e − ˜ m 2 nπ / ˜ a . (37) F rom this formula one can readily see that, for a large- dimensional spacetime, the pair creation could be greatly enhanced b y the prefactor ˜ a ( d − 1) / 2 ∼ ( d − 1) ( d − 1) / 2 , since the quotient ˜ m 2 / ˜ a ∼ d 0 and thus the exp onential count- ing the num ber of created pairs do es not coun teract the effect. 6 IV. DISCUSSION In this letter we hav e built on the results of Ref. [ 4 ], where resummed effectiv e actions for Y uk a wa p otentials ha ve b een obtained, with a t wo-fold aspiration. First, we hav e examined the applicabilit y of our heat- k ernel techniques in a situation other than the well- kno wn homogeneous electric field. As a test-b ed, w e ha v e analyzed the pair creation probability in a radiation dom- inated universe with an arbitrary num ber of dimensions, whic h was sho wn to agree with the computation in terms of the corresp onding Bogoliub o v co efficien ts. This agree- men t is non-trivial, inasm uch as the heat-kernel deriv a- tions are computed in Riemannian metrics and require a subsequen t Wick rotation; in fact, in spite of recent ad- v ances [ 25 ], a complete pro of of the mathematical v alidit y of Wick rotations is still lac king. Second, w e hav e p oin ted out the generality of our results. In effect, they are not limited to Y uk a wa se- tups: they include electromagnetic and gra vitational bac kgrounds, among them the familiar FLR W metrics. In cases where an intensit y scale is dominant, our method can b e directly used, without the need to solv e an equa- tion for the mo des, as would b e the case in the Bogoli- ub o v approach. Note that the range of applicabilit y of our method in- cludes scenarios with arbitrary conformally flat space- time backgrounds and, thus, it could b e used to ana- lyze pair creation ev en in static universes. According to our view, how ev er, the only wa y to such effect to b e presen t, is through an instability at the classical level. This can b e intuited, for instance, by analyzing the ef- fectiv e p oten tial arising for the Sch winger effect in the space-dep enden t gauge [ 26 ] or the tach yonic instabilities discussed in Ref. [ 27 ] (see also [ 28 ]). Extensions to con- formal factors that dep end on b oth space and time co or- dinates, which easily b ecome un tractable under the Bo- goliub o v approac h, seem feasible with the aid of Eq. ( 10 ); these ideas and p ossible applications to inflation scenar- ios [ 29 – 31 ] are being explored. Bey ond the use of heat kernel tec hniques, we ha ve highligh ted the existence of gra vitational analogues of the Sch winger effect for massless, nonconformally cou- pled scalar fields, for cosmological ev olutions in whic h Ra 2 is a quadratic function of conformal time. F urther examples could be obtained by c ho osing the scale factor to satisfy the differen tial equation Ra 2 = A + B tanh( ρτ ) + C tanh 2 ( ρτ ) , (38) whose solutions for a can be written in terms of hyper- geometric functions [ 9 , 20 ]. Coming back to our results for pair creation probabil- it y , it is curious to observe that the radiation dominated univ erse inv olv es an effective horizon, akin to the discus- sion in Ref. [ 32 ]. More in detail, the singularity at τ = 0 is such that a massiv e particle following a geo desic with a nontrivial momentum would reach the sp eed of light if extrap olated to τ = 0, where a v anishes, creating thus a sort of “particle horizon.” This fact can b e heuristically understo od as an isolation of regions and ma y induce one to believe that it is a necessary condition for pair creation to happ en. How ev er, metrics with a particle horizon seem to b e just a sub class of those admitting pair creation; in- deed, it is not hard to see that some of the scale factors giv en by the RHS of Eq. ( 38 ) are everywhere nonv anish- ing and, nevertheless, m ultiparticle states are excited. In an y case, obtaining resummed expressions in a broader class of scenarios is still required. F or example, Ref. [ 5 ] shows that the Euler–Heisenberg-like heat kernel for massless spinors in gra vitational backgrounds, K F ( x, x ; τ ) = i  det  R µν ab Σ ab 4 π i sinh ( τ R µν ab Σ ab )  1 / 2 , (39) b eing Σ αβ := 1 4 ( γ α γ β − γ β γ α ) and γ µ the Dirac matrices, correctly repro duces the (gravitational) axial anomaly . This suggests a potential resummation candidate for the contributions arising from the gamma matrices. Condensed-matter analogues may provide further hints for identifying similar resummation structures. General expansions of effective actions for large curv a- tures in arbitrary spacetimes are also missing and might shed light on the connection b et w een p erturbative ap- proac hes and Hawking radiation; work along these lines is currently b eing pursued. A CKNOWLEDGMENTS The authors ackno wledge useful discussions with C. Garc ´ ıa-Perez and V. Vitagliano. SAF thanks the mem b ers of the Institut Denis Poisson, esp ecially M. Cherno dub, for their warm hospitalit y . The research activities of SAF and FDM hav e been carried out in the framew ork of Pro ject PIP 11220200101426CO, CON- ICET. SAF ac knowledges support from the INFN Re- searc h Pro ject QGSKY and Pro ject 11/X748 of UNLP . The authors would like to ac knowledge the con tribution of the COST Action CA23130. The authors also extend their appreciation to the Italian National Group of Math- ematical Physics (GNFM, INdAM) for its support. The authors lastly ac knowledge fruitful discussions and fund- ing from the workshops “New T rends in First Quan tisa- tion: Field Theory , Gravit y and Quantum Computing” (Heraeus Stiftung). [1] E. W. Kolb and A. J. Long, Cosmological gravita- tional particle pro duction and its implications for cos- mological relics, Rev. Mo d. Phys. 96 , 045005 (2024) , 7 arXiv:2312.09042 [astro-ph.CO] . [2] S. W. Hawking, Black hole explosions, Nature 248 , 30 (1974) . [3] S. A. F ranchino-Vi˜ nas, C. Garc ´ ıa-P ´ erez, F. D. 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