Adiabatic regularization with a Yukawa interaction
AADIABATIC REGULARIZATION WITH A YUKAWA INTERACTION
Adrian del Rio, ∗ Antonio Ferreiro, † Jose Navarro-Salas, ‡ and Francisco Torrenti § Departamento de Fisica Teorica and IFIC,Centro Mixto Universidad de Valencia-CSIC. Facultad de Fisica,Universidad de Valencia, Burjassot-46100, Valencia, Spain. Instituto de Fisica Teorica UAM/CSIC, UniversidadAutonoma de Madrid, Cantoblanco, 28049 Madrid, Spain. (Dated: May 24, 2017)We extend the adiabatic regularization method for an expanding universe to include theYukawa interaction between quantized Dirac fermions and a homogeneous background scalarfield. We give explicit expressions for the renormalized expectation values of the stress-energy tensor (cid:104) T µν (cid:105) and the bilinear (cid:104) ¯ ψψ (cid:105) in a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. These are basic ingredients in the semiclassical field equationsof fermionic matter in curved spacetime interacting with a background scalar field. Theultraviolet subtracting terms of the adiabatic regularization can be naturally interpretedas coming from appropriate counterterms of the background fields. We fix the requiredcovariant counterterms. To test our approach we determine the contribution of the Yukawainteraction to the conformal anomaly in the massless limit and show its consistency with theheat-kernel method using the effective action. Keywords: quantum field theory in curved spacetime, adiabatic regularization, Yukawainteraction, semiclassical gravity, cosmology, inflation, preheating.
PACS numbers: 04.62.+, 11.10.Gh, 98.80.k, 98.80.Cq
I. INTRODUCTION
A major problem in the theory of quantized fields in curved spacetimes [1–4] is the computationof the expectation values of the stress-energy tensor components. These calculations are ratherconvoluted, as they involve products of fields at coincident spacetime points, which are ultraviolet(UV) divergent even for free fields. In cosmological scenarios, this is connected to the fundamentalphenomenon of particle creation by time-dependent backgrounds [5, 6]. A nonadiabatic expansionof the Universe generically induces particle creation of both bosonic and fermionic species, which ∗ [email protected] † antonio.ferreiro@ific.uv.es ‡ jnavarro@ific.uv.es § [email protected] a r X i v : . [ g r- q c ] M a y leads to new UV divergences in their quadratic expectation values, not present in the absence ofexpansion.A very efficient renormalization method, specifically constructed to deal with the UV diver-gences of a free field in an expanding universe, is adiabatic regularization. Originally, this tech-nique was introduced to tame the divergences of the mean particle number of a scalar field in aFriedmann-Lemaitre-Robertson-Walker (FLRW) universe [5], and was later extended to get rid ofthe divergences of the stress-energy tensor [7–9]. The key ingredient of the adiabatic scheme isthe asymptotic expansion of the field modes, in which increasingly higher-order terms in the ex-pansion involve increasingly higher-order time derivatives of the metric (the scale factor). Due todimensional reasons, this is equivalent to an UV asymptotic expansion in momenta. This way, onecan expand adiabatically the integrand of the unrenormalized bilinear, identify the UV-divergentterms, and subtract them directly to obtain a finite, covariant expre ssion. The renormalized ex-pectation value is hence expressed as a finite integral in momentum space, depending exclusivelyon the mode functions defining the quantum state. The particular form of the adiabatic expansiondepends on the spin of the quantized field. For free scalar fields, the well-known WKB expansionprovides an adequate solution (see, for instance, [1, 3, 4]). For spin-1/2 fields, however, the adi-abatic expansion takes a different form [10, 11] (see also [12]). The adiabatic method has beenproven to be equivalent to the DeWitt-Schwinger point-splitting scheme [13, 14] for both scalarfields [9, 15], and spin-1 / ? ]). In certain models, the Higgs decay may alsolead to the reheating of the Universe [30]. In this work, we will not focus on a particular scenario,but consider arbitrary time-dependent scale factors and background fields. The main objectiveis to provide well-motivated and rigorous expressions for the renormalized expectation values ofthe fermion stress-energy tensor (cid:104) T µν (cid:105) and the bilinear (cid:104) ¯ ψψ (cid:105) . In the semiclassical equations ofmotion, these are the quantities that incorporate the backreaction of the created matter ontothe background fields. To check the validity of the adiabatic method, we will also compute thecontribution of the Yukawa interaction to the conformal anomaly in the massless limit, and checkits consistency with the heat-kernel method using the effective action.The paper is organized as follows. In Sec. II we give a general overview of the problem,introducing all necessary notation and equations of motion of the system. In Sec. III we developthe adiabatic expansion of the fermion field modes, subject to a Yukawa interaction with a scalarbackground field. In Sec. IV we derive general expressions for the renormalized expectationvalues of the stress-energy tensor (cid:104) T µν (cid:105) and the bilinear (cid:104) ¯ ψψ (cid:105) . In Sec. V we further analyze theadiabatic regularization program by determining the covariant counterterms associated to the UVdivergences. In Sec. VI we apply the method to calculate the conformal anomaly, and includea discussion concerning the ambiguity of the coefficients of the anomaly on the renormalizationscheme. We also show that our results are compatible with the heat-kernel method. Finally, inSec. VII we summarize our results and conclude. The paper is accompanied by three appendixes.In Appendix A we compute the conformal anomaly of a quantized scalar field coupled to a scalarbackground field. In Appendix B we apply adiabatic regularization to a simple, analytically solvableexample. Finally, in Appendix C we gather the terms of the fermionic adiabatic expansion up tofourth order.In this work, we take the FLRW metric as ds = dt − a ( t ) d(cid:126)x , and we use the Dirac-Paulirepresentation for the Dirac gamma matrices, γ = (cid:18) I − I (cid:19) , (cid:126)γ = (cid:18) (cid:126)σ − (cid:126)σ (cid:19) , with (cid:126)σ the usualPauli matrices. We also assume natural units (cid:126) = 1 = c . II. SEMICLASSICAL EQUATIONS FOR A QUANTIZED DIRAC MATTER FIELDWITH YUKAWA COUPLING
We consider the theory defined by the action functional S = S [ g µν , Φ , ψ, ∇ ψ ], where ψ representsa Dirac field, Φ is a scalar field, and g µν stands for the spacetime metric. We decompose the actionas S = S g + S m , where S m is the matter sector S m = (cid:90) d x √− g (cid:26) i ψγ µ ∇ µ ψ − ( ∇ µ ¯ ψ ) γ µ ψ )] − m ¯ ψψ − g Y Φ ¯ ψψ (cid:27) , (1)and S g is the gravity-scalar sector, which will be presented in the next subsection. Here, γ µ ( x )are the spacetime-dependent Dirac matrices satisfying the anticommutation relations { γ µ , γ ν } =2 g µν , related to the usual Minkowski ones by the vierbein field V aµ ( x ) defined through g µν ( x ) = V aµ ( x ) V bν ( x ) η ab . On the other hand, ∇ µ ≡ ∂ µ − Γ µ is the covariant derivative associated to thespin connection Γ µ , m is the mass of the Dirac field, and g Y is the dimensionless coupling constantof the Yukawa interaction. In (1), both the metric g µν ( x ) and the scalar field Φ( x ) are regardedas classical external fields. The Dirac spinor ψ ( x ) will be our quantized field living in a curvedspacetime and possessing a Yukawa coupling to the classical field Φ. The Dirac equation is( iγ µ ∇ µ − m − g Y Φ) ψ = 0 , (2)and the stress-energy tensor is given by [4] T mµν := 2 √− g δS m δg µν = V νa det V δS m δV µa = i (cid:104) ¯ ψγ ( µ ∇ ν ) ψ − ( ∇ ( µ ¯ ψ ) γ ν ) ψ (cid:105) . (3)The presence of the Yukawa interaction with the external field Φ modifies the standard conservationequation. We have, instead, ∇ µ T µνm = g Y ¯ ψψ ∇ ν Φ . (4)These equations can be easily seen as the consequence of the invariance of the action functional S under spacetime diffeomorphisms δx µ = (cid:15) µ ( x ): δ Φ = (cid:15) µ ∇ µ Φ, δg µν = 2 ∇ ( µ (cid:15) ν ) . One gets ∇ µ T µνm + 1 √− g δS m δ Φ ∇ ν Φ = 0 , (5)which reproduces (4). We will assume that the quantum theory fully respects this symmetry.Therefore, we demand ∇ µ (cid:104) T µνm (cid:105) = g Y (cid:104) ¯ ψψ (cid:105)∇ ν Φ . (6) A. Adding the gravity-scalar sector
The complete theory, including the gravity-scalar sector in the action, can be described by S = S g + S m = 116 πG (cid:90) d x √− gR + (cid:90) d x √− g (cid:26) g µν ∇ µ Φ ∇ ν Φ − V (Φ) (cid:27) + S m , (7)where S m is the action for the matter sector given in (1). We will reconsider the form of the actionin Sec. V, in view of the counterterms required to cancel the UV divergences of the quantizedDirac field. However, let us work for the moment with the action (7). The Einstein equations arethen G µν + 8 πG ( ∇ µ Φ ∇ ν Φ − g µν ∇ ρ Φ ∇ ρ Φ + g µν V (Φ)) = − πGT µνm , (8)and the equation for the scalar field is (cid:50) Φ + ∂V∂
Φ = − g Y ¯ ψψ . (9)The semiclassical equations are obtained from (8) and (9) by replacing T µνm and ¯ ψψ by the corre-sponding (renormalized) vacuum expectation values (cid:104) T µνm (cid:105) ren and (cid:104) ¯ ψψ (cid:105) ren , G µν + 8 πG ( ∇ µ Φ ∇ ν Φ − g µν ∇ ρ Φ ∇ ρ Φ + g µν V (Φ)) = − πG (cid:104) T µνm (cid:105) ren , (10) (cid:50) Φ + ∂V∂
Φ = − g Y (cid:104) ¯ ψψ (cid:105) ren . (11)These equations are consistent with the Bianchi identities ∇ µ G µν = 0 , since ∇ µ ( ∇ µ Φ ∇ ν Φ − g µν ∇ ρ Φ ∇ ρ Φ + g µν V (Φ)) = ( (cid:50) Φ + ∂V∂
Φ ) ∇ ν Φ , (12)and, from (6) and (11), we have ∇ µ (cid:104) T µνm (cid:105) ren = g Y (cid:104) ¯ ψψ (cid:105) ren ∇ ν Φ = − ( (cid:50) Φ + ∂V∂
Φ ) ∇ ν Φ . (13)When the spacetime is an expanding universe [ ds = dt − a ( t ) d(cid:126)x ], and Φ is a homogeneousscalar field Φ = Φ( t ) (e.g. an inflaton), Eqs. (10) and (11) describe the backreaction on themetric-inflaton system due to matter particle production and vacuum polarization codified in therenormalized vacuum expectation values (cid:104) T µνm (cid:105) ren and (cid:104) ¯ ψψ (cid:105) ren . It is then important to elaboratean efficient method to compute these quantities in this cosmological setting. III. ADIABATIC EXPANSION FOR A DIRAC FIELD WITH YUKAWA COUPLING
In a spatially flat FLRW spacetime, the time-dependent gamma matrices are related with theMinkowskian ones by γ ( t ) = γ and γ i ( t ) = γ i /a ( t ), and the components of the spin connectionsare Γ = 0 and Γ i = ( ˙ a/ γ γ i . The Dirac equation with the Yukawa interaction iγ µ ∇ µ ψ − mψ = g Y Φ ψ , taking Φ as a homogenous scalar field Φ = Φ( t ), is then (cid:18) ∂ + 32 ˙ aa + 1 a γ (cid:126)γ (cid:126) ∇ + i ( m + s ( t )) γ (cid:19) ψ = 0 , (14)where we have defined s ( t ) ≡ g Y Φ( t ). If we expand the field ψ as ψ = (cid:82) d (cid:126)k (2 π ) / ψ (cid:126)k ( t ) e i(cid:126)k(cid:126)x , and wesubstitute it into (14), we obtain the following differential equation for ψ (cid:126)k : (cid:32) ∂ t + 3 ˙ a a + iγ (cid:126)γ (cid:126)ka + iγ ( m + s ( t )) (cid:33) ψ (cid:126)k = 0 . (15)In order to solve this equation, it is convenient to write the Dirac field in terms of two two-component spinors of the generic form ψ (cid:126)k,λ ( t ) = a / ( t ) h Ik ( t ) ξ λ ( (cid:126)k ) h IIk ( t ) (cid:126)σ(cid:126)kk ξ λ ( (cid:126)k ) , (16)where ξ λ with λ = ± ξ † λ ξ λ (cid:48) = δ λ,λ (cid:48) ), eigenvectors ofthe helicity operator (cid:126)σ(cid:126)k k ξ λ = λ ξ λ . The explicit forms of ξ +1 and ξ − are ξ +1 ( (cid:126)k ) = 1 (cid:112) k ( k + k ) k + k k + ik , ξ − ( (cid:126)k ) = 1 (cid:112) k ( k + k ) − k + ik k + k , (17)where (cid:126)k = ( k , k , k ) and | (cid:126)k | = k . The time-dependent functions h Ik and h IIk satisfy the first-ordercoupled equations h IIk = iak (cid:18) ∂h Ik ∂t + i ( m + s ( t )) h Ik (cid:19) , h Ik = iak (cid:18) ∂h IIk ∂t − i ( m + s ( t )) h IIk (cid:19) . (18)Given a particular solution { h Ik ( t ), h IIk ( t ) } to Eqs. (18), one can construct the modes u (cid:126)k,λ ( t ) = e i(cid:126)k(cid:126)x √ (2 π ) a ( t ) h Ik ( t ) ξ λ ( (cid:126)k ) h IIk ( t ) (cid:126)σ(cid:126)kk ξ λ ( (cid:126)k ) . (19)Equation (19) will be a solution of positive-frequency type in the adiabatic regime. A solutionof negative-frequency type can be obtained by applying a charge conjugate transformation Cψ = − iγ ψ ∗ (we follow here the convention in [31]) v (cid:126)k,λ ( t ) = Cu (cid:126)k,λ ( t ) = e − i(cid:126)k(cid:126)x √ (2 π ) a ( t ) h II ∗ k ( t ) ξ − λ ( (cid:126)k ) h I ∗ k ( t ) (cid:126)σ(cid:126)kk ξ − λ ( (cid:126)k ) . (20)The Dirac inner product is defined as ( ψ , ψ ) = (cid:82) d xa ψ † ψ . The normalization condition forthe above four-spinors, ( u (cid:126)kλ , v (cid:126)k (cid:48) λ (cid:48) ) = 0, ( u (cid:126)kλ , u (cid:126)k (cid:48) λ (cid:48) ) = ( v (cid:126)kλ , v (cid:126)k (cid:48) λ (cid:48) ) = δ λλ (cid:48) δ (3) ( (cid:126)k − (cid:126)k (cid:48) ), reduces to | h Ik | + | h IIk | = 1 . (21)Since the Dirac scalar product is preserved by the cosmological evolution, the normalization con-dition (21) holds at any time. This ensures also the standard anticommutation relations for thecreation and annihilation operators [ { B (cid:126)k,λ , B † (cid:126)k (cid:48) ,λ (cid:48) } = δ ( (cid:126)k − (cid:126)k (cid:48) ) δ λλ (cid:48) , { B (cid:126)k,λ , B (cid:126)k (cid:48) ,λ (cid:48) } = 0, and similarlyfor the D (cid:126)k,λ , D † (cid:126)k (cid:48) ,λ (cid:48) operators], defined by the Fourier expansion of the Dirac field operator ψ ( x ) = (cid:90) d (cid:126)k (cid:88) λ (cid:104) B (cid:126)kλ u (cid:126)kλ ( x ) + D † (cid:126)kλ v (cid:126)kλ ( x ) (cid:105) . (22) A. Adiabatic expansion
We now compute the adiabatic expansion of a Dirac field living in a FLRW spacetime, andpossessing a Yukawa interaction term with a classical background field. We know that, in theadiabatic limit, and in the absence of interaction, the natural solution of the field modes h Ik and h IIk is h Ik ( t ) = (cid:115) ω ( t ) + m ω ( t ) e − i (cid:82) t ω ( t (cid:48) ) dt (cid:48) , h IIk ( t ) = (cid:115) ω ( t ) − m ω ( t ) e − i (cid:82) t ω ( t (cid:48) ) dt (cid:48) , (23)where ω = (cid:113) k a + m is the frequency of the field mode. This will constitute the zeroth-orderterm of the adiabatic expansion. Mimicking the ansatz introduced in [10], we write the h Ik and h IIk functions as h Ik ( t ) = (cid:115) ω ( t ) + m ω ( t ) e − i (cid:82) t Ω( t (cid:48) ) dt (cid:48) F ( t ) , h IIk ( t ) = (cid:115) ω ( t ) − m ω ( t ) e − i (cid:82) t Ω( t (cid:48) ) dt (cid:48) G ( t ) , (24)where Ω( t ), F ( t ), and G ( t ) are time-dependent functions, which we expand adiabatically asΩ = ω + ω (1) + ω (2) + ω (3) + ω (4) + . . . ,F = 1 + F (1) + F (2) + F (3) + F (4) + . . . ,G = 1 + G (1) + G (2) + G (3) + G (4) + . . . . (25)Here, F ( n ) , G ( n ) , and ω ( n ) are terms of n th adiabatic order (we explain exactly what we meanby that below). By substituting (24) into the equations of motion (18) and the normalizationcondition (21), we obtain the following system of three equations,( ω − m ) G = Ω F + i ˙ F + iF dωdt (cid:18) ω + m − ω (cid:19) − ( m + s ) F , ( ω + m ) F = Ω G + i ˙ G + iG dωdt (cid:18) ω − m − ω (cid:19) + ( m + s ) G , ( ω + m ) F F ∗ + ( ω − m ) GG ∗ = 2 ω . (26)To obtain the expressions for Ω ( n ) , F ( n ) , and G ( n ) , we introduce the adiabatic expansions (25) into(26), and solve order by order. As usual, we consider ˙ a of adiabatic order one, ¨ a of adiabatic ordertwo, and so on. On the other hand, we consider the interaction term s ( t ) of adiabatic order one,so that the zeroth-order term in (24) recovers the free field solution in the adiabatic limit, definedin (23). Similarly, time derivatives of the interaction increase the adiabatic order, so that ˙ s is oforder two, ¨ s of order three, and so on. With this, a generic expression f ( n ) of adiabatic order n (e.g. f ( n ) = F ( n ) , G ( n ) , Ω ( n ) ) will be written as a sum of all possible products of n th adiabaticorder formed by s , a , and their time derivatives. For example, functions of adiabatic orders oneand two will be written respectively as f (1) = α s + α ˙ a ,f (2) = β s + β ˙ s + β ¨ a + β ˙ a + β ˙ as , (27)with α n ≡ α n ( m, k, a ) and β n ≡ β n ( m, k, a ). The assignment of s as adiabatic order one isconsistent with the scaling dimension of the scalar field, as it possesses the same dimensions as ˙ a .
1. First adiabatic order
By keeping only terms of first adiabatic order in (26), the system of three equations gives( ω − m ) G (1) = ( ω − m ) F (1) + ω (1) − s + i dωdt (cid:18) ω + m − ω (cid:19) , ( ω + m ) F (1) = ( ω + m ) G (1) + ω (1) + s + i dωdt (cid:18) ω − m − ω (cid:19) , ( ω + m )( F (1) + F (1) ∗ ) + ( ω − m )( G (1) + G (1) ∗ ) = 0 . (28)We now treat independently the real and imaginary parts by writing F (1) = f (1) x + if (1) y and G (1) = g (1) x + ig (1) y . We obtain for the real part( ω − m )( g (1) x − f (1) x ) = ω (1) − s , ( ω + m )( g (1) x − f (1) x ) = − ω (1) − s , ( ω + m ) f (1) x + ( ω − m ) g (1) x = 0 , (29)which has as solutions f (1) x = s ω − ms ω , g (1) x = − s ω − ms ω , ω (1) = msω . (30)On the other hand, the imaginary part of the system gives( ω − m )( g (1) y − f (1) y ) = 12 dωdt (cid:18) ω + m − ω (cid:19) , ( ω + m )( g (1) y − f (1) y ) = − dωdt (cid:18) ω − m − ω (cid:19) . (31)These two equations are not independent. The obtained solution for g (1) y and f (1) y is f (1) y = A − m ˙ a aω , g (1) y = A , (32)where A is an arbitrary first-order adiabatic function. We will choose the simplest solution f (1) y = − m ˙ a ω a , g (1) y = m ˙ a ω a , (33)obeying the condition F (1) ( m, s ) = G (1) ( − m, − s ). Therefore, the adiabatic expansion will alsopreserve the symmetries of Eqs. (18) with respect to the change ( m, s ) → ( − m, − s ). We havechecked that physical expectation values are independent to any potential ambiguity in this kindof choice.0
2. Second adiabatic order
In the same way, the second-order terms of (26) give( ω − m ) G (2) = ( ω − m ) F (2) + ( ω (1) − s ) F (1) + ω (2) + i ˙ F (1) + i F (1) dωdt (cid:18) ω + m − ω (cid:19) , ( ω + m ) F (2) = ( ω + m ) G (2) + ( ω (1) + s ) G (1) + ω (2) + i ˙ G (1) + i G (1) dωdt (cid:18) ω − m − ω (cid:19) , ( ω + m )( F (2) + F (1) F (1) ∗ + F (2) ∗ ) + ( ω − m )( G (2) + G (1) G (1) ∗ + G (2) ∗ ) = 0 , (34)where the first-order terms have already been deduced above. Taking the real part of these equa-tions, we obtain( ω − m )( g (2) x − f (2) x ) = ( ω (1) − s ) f (1) x + ω (2) − ˙ f (1) y − f (1) y dωdt (cid:18) ω + m − ω (cid:19) , ( ω + m )( g (2) x − f (2) x ) = − ( ω (1) + s ) g (1) x − ω (2) + ˙ g (1) y + g (1) y dωdt (cid:18) ω − m − ω (cid:19) , ( ω + m )(2 f (2) x + ( f (1) x ) + ( f (1) y ) ) + ( ω − m )(2 g (2) x + ( g (1) x ) + ( g (1) y ) ) = 0 , (35)which has as solutions f (2) x = m ¨ a aω − m ¨ a aω − m ˙ a a ω + 5 m ˙ a a ω + 3 m ˙ a a ω − m ˙ a a ω + 5 m s ω − ms ω − s ω ,ω (2) = − m s ω + s ω + 5 m ˙ a a ω − m ˙ a a ω − m ¨ a aω , (36)and g (2) x ( m, s ) = f (2) x ( − m, − s ). On the other hand, taking the imaginary part of the equations, wehave ( ω − m )( g (2) y − f (2) y ) = ( ω (1) − s ) f (1) y + ˙ f (1) x + f (1) x dωdt (cid:18) ω + m − ω (cid:19) , ( ω + m )( g (2) y − f (2) y ) = − ( ω (1) + s ) g (1) y − ˙ g (1) x − g (1) x dωdt (cid:18) ω − m − ω (cid:19) . (37)As before, this system contains an arbitrariness in its solution, f (2) y = B + 5 m s ˙ a aω − s ˙ a aω − ˙ s ω , g (2) y = B , (38)where now B is a linear combination of second-order adiabatic terms. By imposing again thecondition F (2) ( m, s ) = G (2) ( − m, − s ), one finds f (2) y = 5 m s ˙ a aω − s ˙ a aω − ˙ s ω , (39)and g (2) y ( m, s ) = f (2) y ( − m, − s ).1
3. Third and fourth adiabatic order
The same procedure can be repeated for all orders. The real part of the expansion is totallydetermined by the system of equations (26), while every imaginary part contains an arbitrarinessthat can be solved by fixing the condition F ( n ) ( m, s ) = G ( n ) ( − m, − s ). The third- and fourth-orderterms of the expansion are explicitly written in Appendix C. IV. RENORMALIZATION OF THE STRESS-ENERGY TENSOR (cid:104) T µν (cid:105) AND THEBILINEAR (cid:104) ¯ ψψ (cid:105) The classical stress-energy tensor in a FLRW spacetime has two independent components. Fora Dirac field, they are (no sum on i ), T = i (cid:18) ¯ ψγ ∂ψ∂t − ∂ ¯ ψ∂t γ ψ (cid:19) , T ii = i a (cid:18) ¯ ψγ i ∂ψ∂x i − ∂ ¯ ψ∂x i γ i ψ (cid:19) . (40)We define the vacuum state | (cid:105) as B (cid:126)k,λ | (cid:105) ≡ D (cid:126)k,λ | (cid:105) ≡
0, and denote any expectation value onthis vacuum as e.g. (cid:104) T µν (cid:105) ≡ (cid:104) | T µν | (cid:105) . In the quantum theory, the vacuum expectation values ofthe stress-energy tensor take the form (see for example [11]) (cid:104) T (cid:105) = 12 π a (cid:90) ∞ dkk ρ k ( t ) , ρ k ( t ) ≡ i (cid:18) h Ik ∂h I ∗ k ∂t + h IIk ∂h II ∗ k ∂t (cid:19) , (41)and (cid:104) T ii (cid:105) = 12 π a (cid:90) ∞ dkk p k ( t ) , p k ( t ) ≡ − k a ( h Ik h II ∗ k + h I ∗ k h IIk ) . (42)The above formal expressions contain quartic, quadratic, and logarithmic UV divergences, whichturn out to be independent of the particular quantum state. These divergences are similar tothose described in [32]. To characterize them, one plugs in (41)-(42) the adiabatic expansion of h Ik and h IIk , given in Eq. (24). We shall see that, in the presence of a Yukawa interaction, alladiabatic orders up to the fourth one generate UV divergences. This is different to what happensin the case of a free field, where the divergences only appear at zeroth and second adiabatic orders[11]. In general, adiabatic renormalization proceeds by subtracting those adiabatic terms from theintegrand of the expectation values, producing a formal finite quantity. There are two importantconsiderations regarding these subtractions. First, they must refer to all contributions of a givenadiabatic term of fixed (adiabatic) order, othe rwise general covariance is not maintained. Andsecond, one subtracts only the minimum number of terms required to get a finite result [1].We now proceed to calculate the renormalized expressions for the energy density and pressure.2
A. Renormalized energy density
We start by performing the adiabatic expansion of the energy density in momentum space (42) ρ k = ρ (0) k + ρ (1) k + ρ (2) k + ρ (3) k + ρ (4) k + . . . , (43)where ρ ( n ) k is of n th adiabatic order. The adiabatic terms producing UV divergences (after inte-gration in momenta) are ρ (0) k = − ω , (44) ρ (1) k = − msω , (45) ρ (2) k = − ˙ a m a ω + ˙ a m a ω + m s ω − s ω , (46) ρ (3) k = 5 ˙ a m s a ω − a m s a ω + ˙ a ms a ω − ˙ am ˙ s aω + ˙ am ˙ s aω − m s ω + ms ω , (47) ρ (4) k = 105 ˙ a m a ω −
91 ˙ a m a ω + 81 ˙ a m a ω − ˙ a m a ω − a m ¨ a a ω + 5 ˙ a m ¨ a a ω − a m ¨ a a ω (48) −
35 ˙ a m s a ω + 15 ˙ a m s a ω − m ¨ a a ω −
27 ˙ a m s a ω + m ¨ a a ω + ˙ a s a ω + ˙ am a (3) a ω − ˙ am a (3) a ω + 5 ˙ am s ˙ s aω − am s ˙ saω + ˙ as ˙ s aω + 5 m s ω − m s ω − m ˙ s ω + s ω + ˙ s ω , where we have used the notation a (3) ≡ d a/dt , a (4) ≡ d a/dt , etc.We note that if we turn off the Yukawa coupling, we recover the results obtained in [11]. TheYukawa interaction produces new contributions and, in particular, we have now non-zero terms atfirst and third adiabatic orders. The physical meaning of them will be given later on. Note herethat in the UV limit, ρ (0) k ∼ k , ( ρ (1) k + ρ (2) k ) ∼ k − , and ( ρ (3) k + ρ (4) k ) ∼ k − . This indicates thatsubtracting the zeroth-order term will cancel the natural quartic divergence of the stress-energytensor, subtracting up to second order will cancel also the quadratic divergence, and subtracting upto fourth order will cancel the logarithmic divergence. Therefore, defining the adiabatic subtractionterms as (cid:104) T (cid:105) Ad ≡ π a (cid:90) ∞ dkk ( ρ (0) k + ρ (1) k + ρ (2) k + ρ (3) k + ρ (4) k ) ≡ π a (cid:90) ∞ dkk ρ (0 − k , (49)the renormalized 00 component of the stress-energy tensor is (cid:104) T (cid:105) ren ≡ (cid:104) T (cid:105) − (cid:104) T (cid:105) Ad = 12 π a (cid:90) ∞ dkk ( ρ k − ρ (0 − k ) . (50)This integral is, by construction, finite.3 B. Renormalized pressure
The method proceeds in the same way for the pressure. The renormalized ii component of thestress-energy tensor is given by (cid:104) T ii (cid:105) ren ≡ (cid:104) T ii (cid:105) − (cid:104) T ii (cid:105) Ad = 12 π a (cid:90) ∞ dkk ( p k − p (0 − k ) , (51)where p (0 − k ≡ p (0) k + p (1) k + p (2) k + p (3) k + p (4) k , and (cid:104) T ii (cid:105) Ad ≡ π a (cid:90) ∞ dkk p (0 − k . (52)The corresponding adiabatic terms for the pressure are p (0) k = − ω m ω , (53) p (1) k = 2 ms ω − m s ω , (54) p (2) k = − a m a ω + ˙ a m a ω − ˙ a m a ω + m ¨ a aω − m ¨ a aω + m s ω − m s ω + s ω , (55) p (3) k = −
35 ˙ a m s a ω − a m sa ω + 9 ˙ a m s a ω − ˙ a ms a ω − m s ¨ a aω − am ˙ s aω + 7 m s ¨ a aω + 7 ˙ am ˙ s aω − ms ¨ a aω − ˙ am ˙ s aω − m s ω + 8 m s ω + m ¨ s ω − ms ω − m ¨ s ω , (56) p (4) k = 385 ˙ a m a ω −
791 ˙ a m a ω + 1477 ˙ a m a ω − m a (4) aω −
263 ˙ a m a ω + m a (4) aω + ˙ a m a ω −
77 ˙ a m ¨ a a ω + 203 ˙ a m ¨ a a ω −
191 ˙ a m ¨ a a ω + ˙ a m ¨ a a ω −
105 ˙ a m s a ω + 665 ˙ a m s a ω + 7 m ¨ a a ω −
145 ˙ a m s a ω − m ¨ a a ω + 29 ˙ a m s a ω + 3 m ¨ a a ω − ˙ a s a ω + 7 ˙ am a (3) a ω − am a (3) a ω + ˙ am a (3) a ω + 35 m s ¨ a aω + 35 ˙ am s ˙ s aω − m s ¨ aaω −
10 ˙ am s ˙ saω + 9 m s ¨ a aω + 9 ˙ am s ˙ s aω − s ¨ a aω − ˙ as ˙ s aω + 35 m s ω − m s ω − m s ¨ s ω − m ˙ s ω + 11 m s ω + m s ¨ sω + m ˙ s ω − s ω − s ¨ s ω + ˙ s ω . (57)As before, we see that in the UV limit, p (0) k ∼ k , ( p (1) k + p (2) k ) ∼ k − , and ( p (3) k + p (4) k ) ∼ k − . Sub-tracting the zeroth-order term eliminates the quartic divergence, subtracting up to second orderremoves the quadratic divergence, and subtracting up to fourth order removes the logarithmic di-vergence. If the Yukawa interaction is removed, we recover again the results in [11]. The interactionproduces also nonzero contributions to the first and third adiabatic orders.4 C. Renormalization of (cid:104) ¯ ψψ (cid:105) We are also interested in computing the renormalized expectation value (cid:104) ¯ ψψ (cid:105) ren . The formal(unrenormalized) expression for this quantity is (cid:104) ¯ ψψ (cid:105) = − π a (cid:90) ∞ dkk (cid:104) ¯ ψψ (cid:105) k , (cid:104) ¯ ψψ (cid:105) k ≡ | h Ik | − | h IIk | . (58)We define the corresponding terms in the adiabatic expansion as (cid:104) ¯ ψψ (cid:105) k = (cid:104) ¯ ψψ (cid:105) (0) k + (cid:104) ¯ ψψ (cid:105) (1) k + (cid:104) ¯ ψψ (cid:105) (2) k + (cid:104) ¯ ψψ (cid:105) (3) k + ... . Due to the Yukawa interaction, ultraviolet divergences arrive till the thirdadiabatic order. In general, we have (cid:104) ¯ ψψ (cid:105) ( n ) k = ω + m ω (cid:0) | F | (cid:1) ( n ) − ω − m ω (cid:0) | G | (cid:1) ( n ) . (59)From here, we obtain (cid:104) ¯ ψψ (cid:105) (0) k = mω , (60) (cid:104) ¯ ψψ (cid:105) (1) k = sω − m sω , (61) (cid:104) ¯ ψψ (cid:105) (2) k = − a m a ω + 7 ˙ a m a ω − ˙ a m a ω + m ¨ a aω − m ¨ a aω + 3 m s ω − ms ω , (62) (cid:104) ¯ ψψ (cid:105) (3) k = 35 ˙ a m s a ω −
15 ˙ a m s a ω + 27 ˙ a m s a ω − ˙ a s a ω − m s ¨ a aω − am ˙ s aω + 3 m s ¨ a aω + 2 ˙ am ˙ saω − s ¨ a aω − a ˙ s aω − m s ω + 3 m s ω + m ¨ s ω − s ω − ¨ s ω . (63)The adiabatic prescription leads then to (cid:104) ¯ ψψ (cid:105) ren = (cid:104) ¯ ψψ (cid:105) − (cid:104) ¯ ψψ (cid:105) Ad = − π a (cid:90) ∞ dkk ( (cid:104) ¯ ψψ (cid:105) k − (cid:104) ¯ ψψ (cid:105) (0 − k ) . (64)In this case, we observe that in the UV limit, ( (cid:104) ¯ ψψ (cid:105) (0) k + (cid:104) ¯ ψψ (cid:105) (1) k ) ∼ k − , and ( (cid:104) ¯ ψψ (cid:105) (2) k + (cid:104) ¯ ψψ (cid:105) (3) k ) ∼ k − ). Subtracting up to first order eliminates the quadratic divergence, and up to third orderremoves the logarithmic one.Our results can be generically implemented together with numerical methods to compute therenormalized expectation values (cid:104) T µν (cid:105) ren and (cid:104) ¯ ψψ (cid:105) ren . On the other hand, we would like to brieflycomment that a higher-order adiabatic expansion also serves to generate asymptotic analyticalexpressions for the renormalized stress-energy tensor in some special situations. This happensin spacetime regions where the relevant modes always evolve adiabatically. For instance, if weapproximate the form of the exact modes { h Ik , h IIk } by their higher-order adiabatic expansion, wecan find in a very straightforward way an analytic approximation for the renormalized quantitiesin the adiabatic regime, as in the example given in Appendix B. Outside the adiabatic regime one5should use numerical methods to find the exact modes and plug them in the generic renormalizedexpressions obtained above. V. ULTRAVIOLET DIVERGENCES AND RENORMALIZATION COUNTERTERMS
The ultraviolet divergent terms of the adiabatic subtractions can be univocally related to par-ticular counterterms in a Lagrangian density including the background gravity-scalar sector. Bywriting L = L m + √− g (cid:34) g µν ∇ µ Φ ∇ ν Φ − (cid:88) i =1 λ i i ! Φ i − ξ R Φ − ξ R Φ − πG Λ + 116 πG R (cid:35) + √− g (cid:34) δZg µν ∇ µ Φ ∇ ν Φ − (cid:88) i =1 δλ i i ! Φ i − δξ R Φ − δξ R Φ − π δ Λ + 116 π δG − R (cid:35) , (65)the equations of motion for the scalar field are(1 + δZ ) (cid:50) Φ + ( λ + δλ ) + ( λ + δλ )Φ + ( λ + δλ ) 12 Φ + 13! ( λ + δλ )Φ + ( ξ + δξ ) R + ( ξ + δξ ) R Φ = − g Y (cid:104) ¯ ψψ (cid:105) . (66)From (64), we can write the identity (cid:104) ¯ ψψ (cid:105) = (cid:104) ¯ ψψ (cid:105) ren + 1 π a (cid:90) ∞ dkk ( (cid:104) ¯ ψψ (cid:105) (0) k + (cid:104) ¯ ψψ (cid:105) (1) k + (cid:104) ¯ ψψ (cid:105) (2) k + (cid:104) ¯ ψψ (cid:105) (3) k ) , (67)where (cid:104) ¯ ψψ (cid:105) ren is finite and the remaining integrals at the right-hand side of (67) are the adiabaticsubtraction terms. As we shall see, the ultraviolet divergences of the adiabatic subtraction termscan be removed by counterterms of the form: δZ (cid:50) Φ, δλ , δλ Φ, δλ Φ , δλ Φ , δξ R , and δξ R Φ.To deal with the UV-divergent subtraction terms we use dimensional regularization [8]. We cancheck that the (covariantly) regulated divergences take the same form as the above covariantcounterterms. For (cid:104) ¯ ψψ (cid:105) (0) we have ( n denotes the spacetime dimension) (cid:104) ¯ ψψ (cid:105) (0) = − π a (cid:90) ∞ dkk (cid:18) − mω ( t ) (cid:19) → − π a (cid:90) ∞ dkk n − (cid:18) − mω ( t ) (cid:19) = m π ( n −
4) + ... (68)where we will retain only the poles at n = 4. This divergence can be absorbed by δλ . Additionally,we also have (cid:104) ¯ ψψ (cid:105) (1) = − π a (cid:90) ∞ dkk n − (cid:18) − s ( t ) k ω ( t ) a ( t ) (cid:19) = 3 g Y m π ( n −
4) Φ( t ) + · · · . (69)This divergence of adiabatic order one can be absorbed by δλ . The divergences of adiabatic ordertwo (cid:104) ¯ ψψ (cid:105) (2) = − m π ( n − R + 3 mg Y π ( n −
4) Φ ( t ) + · · · (70)6can also be eliminated by δξ and δλ . Finally, the three divergences of adiabatic order three (cid:104) ¯ ψψ (cid:105) (3) = g Y π ( n − (cid:50) Φ( t ) + g Y π ( n − R Φ( t ) + g Y π ( n −
4) Φ ( t ) + · · · (71)are absorbed by δZ , δξ and δλ .On the other hand, the tensorial equations are18 π (cid:18) G + δG − (cid:19) G µν + 18 π (cid:18) Λ G + δ Λ (cid:19) g µν + (1 + δZ )( ∇ µ Φ ∇ ν Φ − g µν ∇ ρ Φ ∇ ρ Φ)+ g µν (cid:88) i =1 ( λ i + δλ i ) i ! Φ i − (cid:88) i =1 ξ i + δξ i i ! ( G µν Φ i − g µν (cid:50) Φ i + ∇ µ ∇ ν Φ i ) = −(cid:104) T µνm (cid:105) , (72)and we find similar cancellations. However, two extra divergences appear. Focusing, for simplicity,at zeroth adiabatic order, we have − π a (cid:90) ∞ dkk n − ρ (0) k ≈ m π n − , − π a (cid:90) ∞ dkk n − a p (0) k ≈ − m a π n − . (73)At first adiabatic order we encounter the following divergences − π a (cid:90) ∞ dkk n − ρ (1) k ≈ m s π n − , − π a (cid:90) ∞ dkk n − a p (1) k ≈ − m a s π n − . (74)At second adiabatic order we find these divergences − π a (cid:90) ∞ dkk n − ρ (2) k ≈ m π n − a a + 3 m π n − s , − π a (cid:90) ∞ dkk n − a p (2) k ≈ − m a π n − (cid:18) aa + ˙ a a (cid:19) − a m s π n − . (75)At third adiabatic order we get the following divergences ( H ≡ ˙ a/a ) − π a (cid:90) ∞ dkk n − ρ (3) k ≈ m π n − (cid:2) H s + 3 H ˙ s + 6 s (cid:3) , (76) − π a (cid:90) ∞ dkk n − a p (3) k ≈ − ma π n − (cid:20) ¨ s + 2 H ˙ s + (cid:18) H + 2 ¨ aa (cid:19) s + 6 s (cid:21) . Finally, at fourth adiabatic order the divergences are − π a (cid:90) ∞ dkk n − ρ (4) k ≈ π n − (cid:2) H s + s + 2 H ˙ ss + ˙ s (cid:3) , (77) − π a (cid:90) ∞ dkk n − a p (4) k ≈ − a π n − (cid:20) s + (cid:18) H + 2 ¨ aa (cid:19) s − ˙ s Hs ˙ s + 23 s ¨ s (cid:21) . (cid:104) T µν (cid:105) (0) Ad ≈ m π ( n − g µν , (78) (cid:104) T µν (cid:105) (1) Ad ≈ g Y Φ m π ( n − g µν , (79) (cid:104) T µν (cid:105) (2) Ad ≈ g Y Φ m π ( n − g µν − m π ( n − G µν , (80) (cid:104) T µν (cid:105) (3) Ad ≈ − mg Y π ( n − (cid:2) G µν Φ − (cid:50) Φ g µν + ∇ µ ∇ ν Φ − g Y Φ g µν (cid:3) , (81) (cid:104) T µν (cid:105) (4) Ad ≈ − g Y π ( n − (cid:20) G µν Φ − g µν (cid:50) Φ + ∇ µ ∇ ν Φ − ∇ µ Φ ∇ ν Φ − g µν ∇ ρ Φ ∇ ρ Φ) − g Y Φ g µν (cid:21) , (82)and can be consistently removed (including also the divergences for (cid:104) ¯ ψψ (cid:105) ) by the renormalizationparameters δ Λ = − m π ( n − , δG − = m π ( n − , δZ = − g Y π ( n − , (83) δλ = − m g Y π ( n − , δλ = − m g Y π ( n − , δλ = − mg Y π ( n − , δλ = − g Y π ( n − , (84) δξ = − mg Y π ( n − , δξ = − g Y π ( n − . (85)We remark that the set of needed counterterms is all possible counterterms having couplingswith non-negative mass dimension, up to Newton’s coupling constant. This is also in agreementwith the results in perturbative Quantum Field Theory in flat spacetime. The renormalizabilityof the Yukawa interaction g Y ϕ ¯ ψψ of a quantized massive scalar field ϕ with a massive quantizedDirac field ψ requires us to add terms of the form λZ λ ϕ , κZ κ ϕ , and also a term linear in ϕ [33].The presence of a curved background would require us to add the terms ξ Rϕ and ξ Rϕ . We notethat a term of the form ξ Rϕ is required by renormalization for a purely quantized scalar field ϕ if a self-interaction term of the form λ ϕ appears in the bare Lagrangian density [34, 35]. Herewe have found that the Yukawa interaction demands the presence of the renormalized terms ξ Rϕ and ξ Rϕ (as well as the terms λ i ϕ i ), even if they are not present in the bare Lagrangian density.Similar counterterms have been identified in the approach in Ref. [36].Therefore, the tentative semiclassical equations presented in Sec. II should be reconsidered toinclude the above-required counterterms. In terms of the renormalized parameters we have18 πG ( G µν + Λ g µν ) + ( ∇ µ Φ ∇ ν Φ − g µν ∇ ρ Φ ∇ ρ Φ + V (Φ) g µν )8 − (cid:88) i =1 ξ i i ! ( G µν Φ i − g µν (cid:50) Φ i + ∇ µ ∇ ν Φ i ) = −(cid:104) T µνm (cid:105) ren , (86)and (cid:50) Φ + ∂V∂
Φ + ξ R + ξ R Φ = − g Y (cid:104) ¯ ψψ (cid:105) ren , (87)where the potential V (Φ) should contain the terms V (Φ) = λ Φ + λ + λ
3! Φ + λ
4! Φ . (88)Obviously, additional terms, not required by renormalization, can be added to the potential if oneadopts an effective field theory viewpoint. Some of the renormalized parameters (Λ, ξ , λ , · · · )could take, by fine-tuning, zero values. We do not consider these issues in this work. VI. CONFORMAL ANOMALY
In this section we will analyze the massless limit of the theory and work out the conformalanomaly. In the massless limit the classical action of the theory enjoys invariance under theconformal transformations g µν ( x ) → Ω ( x ) g µν ( x ) , Φ( x ) → Ω − ( x )Φ( x ) , (89)with ψ ( x ) → Ω − / ( x ) ψ ( x ) , ¯ ψ ( x ) → Ω − / ( x ) ¯ ψ ( x ) . (90)Variation of the action yields the identity g µν T mµν + Φ 1 √− g δS m δ Φ = 0 , (91)which, in our case, turns out to be g µν T µν − g Y Φ ¯ ψψ = 0. At the quantum level the theory will loseits conformal invariance as a consequence of renormalization [which respects general covarianceand hence (6)] and generates an anomaly g µν (cid:104) T mµν (cid:105) ren − g Y Φ (cid:104) ¯ ψψ (cid:105) ren = C f (cid:54) = 0 . (92) C f is independent of the quantum state and depends only on local quantities of the external fields.To calculate the conformal anomaly in the adiabatic regularization method, we have to startwith a massive field and take the massless limit at the end of the calculation. Therefore, C f = g µν (cid:104) T mµν (cid:105) ren − g Y Φ (cid:104) ¯ ψψ (cid:105) ren = lim m → m ( (cid:104) ¯ ψψ (cid:105) ren − (cid:104) ¯ ψψ (cid:105) (4) ) . (93)9Since the divergences of the stress-energy tensor have terms of fourth adiabatic order, the adiabaticsubtractions for (cid:104) ¯ ψψ (cid:105) should also include them. The fourth-order subtraction term, which producesa nonzero finite contribution when m →
0, is codified in (cid:104) ¯ ψψ (cid:105) (4) . The term m (cid:104) ¯ ψψ (cid:105) ren vanisheswhen m →
0. The remaining piece produces the anomaly [recall (58)-(59)] C f = − lim m → mπ a (cid:90) ∞ dkk (cid:18) − ( ω + m )2 ω [ F (4) + F (4) ∗ + F (1) F (3) ∗ + F (1) ∗ F (3) + | F (2) | ]+ ( ω − m )2 ω [ G (4) + G (4) ∗ + G (1) G (3) ∗ + G (1) ∗ G (3) + | G (2) | ] (cid:19) . (94)Applying the adiabatic expansion computed in Sec. III and doing the integrals we obtain C f = a (4) π a + s ¨ a π a + ¨ a π a + 3 s ˙ s ˙ a π a + s ˙ a π a + 3 ˙ aa (3) π a − ˙ a ¨ a π a + s ¨ s π + ˙ s π + s π . (95)Since C f is a scalar, we must be able to rewrite the above result as a linear combination of covariantscalar terms made out of the metric, the Riemann tensor, covariant derivatives, and the externalscalar field Φ. Our result is C f = 12880 π (cid:20) − (cid:18) R αβ R αβ − R (cid:19) + 6 (cid:50) R (cid:21) + g Y π (cid:20) ∇ µ Φ ∇ µ Φ + 2Φ (cid:50)
Φ + 16 Φ R + g Y Φ (cid:21) . (96)The same result is obtained by using the results of Sec. IV. C f can be reexpressed as [recall(50)-(51)] C f = lim m → − π a (cid:90) ∞ dkk (cid:16) ρ (0 − k − p (0 − k − s ( t ) + m ) (cid:104) ¯ ψψ (cid:105) (0 − k (cid:17) . (97)Performing the integrals we get exactly (95) and hence (96).In Appendix A we have computed the conformal anomaly for a massless scalar field φ withconformal coupling to the scalar curvature ξ = 1 /
6, and with a Yukawa-type interaction of theform g Y Φ φ . Adiabatic regularization predicts the following conformal anomaly C s = 12880 π (cid:20) (cid:50) R − (cid:18) R µν R µν − R (cid:19)(cid:21) − h π (Φ (cid:50) Φ + ∇ µ Φ ∇ µ Φ + 3 h ) . (98)In the absence of Yukawa interaction ( h = 0, g Y = 0) we reproduce the well-known traceanomaly for both scalar and spin-1 / , /
2, or 1 in terms of threecoefficients g µν (cid:104) T µν (cid:105) ren = aC µνρσ C µνρσ + bG + c (cid:3) R , (99)where C µνρσ is the Weyl tensor and G = R µνρσ R µνρσ − R µν R µν + R is proportional to the Eulerdensity. The coefficients a and b are independent of the renormalization scheme and are given by0[37, 38] a = 1120(4 π ) ( N s + 6 N f + 12 N v ) ,b = − π ) ( N s + 11 N f + 62 N v ) , (100)where N s is the number of real scalar fields, N f is the number of Dirac fields, and N v is thenumber of vector fields. Our results with g Y = 0 fit the values in (100). [We note that in theFLRW spacetime of adiabatic regularization the Weyl tensor vanishes identically]. In contrast, thecoefficient c depends in general on the particular renormalization scheme [39]. A local countertermproportional to R in the action can modify the coefficient c . For instance, for vector fields thepoint splitting and the dimensional regularization method predict different values for c .When the Yukawa interaction is added, the general form of the conformal anomaly is g µν (cid:10) T mµν (cid:11) ren + Φ 1 √− g (cid:104) δS m δ Φ (cid:105) ren = aC µνρσ C µνρσ + bG + c (cid:3) R + d g Y ∇ µ Φ ∇ µ Φ + e g Y Φ (cid:50) Φ + f g Y Φ R + g g Y Φ . (101)Now, the coefficients f and g are independent of the renormalization scheme but d and e are not.The finite Lagrangian counterterms required by the renormalizability of the Yukawa interactionobtained in previous sections, δZ g µν ∇ µ Φ ∇ ν Φ − δξ R Φ − δλ
4! Φ , (102)might alter the values of the coefficients d and e , but not the coefficients f and g . Note that, due toclassical conformal invariance, one should consider only those counterterms having dimensionlesscoupling parameters. Therefore, our results for the f and g coefficients are f = 13(4 π ) N f , g = − π ) (cid:18) N s − N f (cid:19) . (103)Finally, to show explicitly that the above coefficients are independent of the renormalizationscheme, we will compute them using the heat-kernel method given in [1], by means of the one-loopeffective action. A. Consistency with the heat-kernel results
The conformal anomaly for a field φ j ( x ) obeying the second-order wave equation (cid:2) δ ij g µν ∇ µ ∇ ν + Q ij ( x ) (cid:3) φ j = 0 , (104)1is given by C = ± π ) tr E ( x ) , (105)where E ( x ) is the second Seeley-DeWitt coefficient. The minus sign is for bosons and the plussign is for fermions. These coefficients are local, scalar functions of Q ( x ) and the curvature tensor. E is given by E = (cid:18) − (cid:50) R + 172 R − R µν R µν + 1180 R µνρσ R µνρσ (cid:19) I + 112 W µν W µν + 12 Q − RQ + 16 (cid:50) Q , (106)where W µν = [ ∇ µ , ∇ ν ]. For a single massless scalar field with ξ = 1 / h φ Φ we have Q = 16 R + h Φ , (107)and W µν = 0. For a spatially flat FLRW universe we get C = 12880 π (cid:20) (cid:50) R − (cid:18) R µν R µν − R (cid:19)(cid:21) − h π (Φ (cid:50) Φ + ∇ µ Φ ∇ µ Φ + 3 h ) , (108)in full agreement with the result (98) obtained using adiabatic regularization.For a single massless Dirac field with a Yukawa interaction we have [( iγ µ ∇ µ − g Y Φ) ψ = 0] Q = (cid:18) R + g Y Φ (cid:19) I + ig Y γ µ ∇ µ Φ , (109)and W µν = − iR αβµν Σ αβ = − R αβµν [ γ α , γ β ] . (110)Using the properties of the trace of products of gamma matrices, we get C = 12880 π (cid:20) − (cid:18) R αβ R αβ − R (cid:19) + 6 (cid:50) R (cid:21) + g Y π (cid:20) − ∇ µ Φ ∇ µ Φ + 23 Φ (cid:50)
Φ + 16 Φ R + g Y Φ (cid:21) . (111)The above result reproduces the coefficients f and g obtained from adiabatic regularization. Wenote that there is a mismatch in the coefficients d and e . These are, however, the coefficients thatmight depend on the renormalization scheme. VII. SUMMARY
When a quantum field is coupled to a classical, nonadiabatic time-dependent background, itgets excited, and undergoes a regime of particle creation. In this case, new UV-divergent terms2appear in the expectation values of its quadratic products, which must be appropriately removedto obtain a physical, finite quantity. In cosmological scenarios, adiabatic regularization provides anappropriate solution to this challenge: by means of an adiabatic expansion of the field modes, onecan identify the covariant UV-divergent terms of the corresponding bilinear, and subtract themdirectly from the unrenormalized quantity. The background may be the expansion of the Universeitself, as in the case of inflation, or a classical homogeneous scalar field. The adiabatic scheme canbe applied in both situations, for both bosonic and fermionic species.In this work, we have developed the adiabatic regularization method for spin-1/2 fields inan expanding universe, coupled to a classical background scalar field with a Yukawa interactionterm. The results of this work are a natural generalization of the studies initiated in [10, 11], andbroaden significantly the range of applicability of the adiabatic method. We have computed theadiabatic expansion of the spin-1/2 field modes up to fourth adiabatic order, and used it to obtainexpressions for the renormalized expectation values of the stress-energy tensor (cid:104) T µν (cid:105) ren and thebilinear (cid:104) ¯ ψψ (cid:105) ren . These quantities are fundamental ingredients in the study of the semiclassicalequations of fermionic matter interacting with a background field, as they codify the backreactioneffects from the created matter on the metric/background fields. Therefore, it is essential todevelop an efficient renormalization scheme to correctly quantify the effects of this backreaction.All expressions obtained are generic, depending only on the background scalar field and scalefactor time-dependent functions. This constitutes probably the major advantage of the adiabaticrenormalization scheme. We leave the method prepared to perform numerical computations infuture investigations.Finally, we have tested the overall theoretical construction of the adiabatic scheme by justifyingthe method in terms of renormalization of coupling constants, as well as by computing the conformalanomaly. Our calculation of the conformal anomaly with the Yukawa interaction has been provedto be fully consistent with the generic results obtained via the one-loop effective action. Therefore,by considering such a system, we have also improved our general understanding of quantum fieldtheory in curved spacetimes. ACKNOWLEDGMENTS
F.T. thanks Daniel G. Figueroa and Juan Garcia-Bellido for useful discussions. This work issupported by the Grants. No. FIS2014-57387-C3-1-P , No. FPA2015-68048-C3-3-P, No. MPNS ofCOST Action No. CA15117, and the Severo Ochoa Programs SEV-2014-0398 and SEV-2012-0249.3A.d.R. is supported by the FPU Ph.D. fellowship FPU13/04948, A. F. is supported by the SeveroOchoa Ph.D. fellowship SEV-2014-0398-16-1, and F.T. is supported by the Severo Ochoa Ph.D.fellowship SVP-2013-067697.
Appendix A: Scalar field with a Yukawa-type coupling
In this appendix we compute the conformal anomaly of a quantized real scalar field φ , coupledto another background scalar Φ with a Yukawa-type interaction. This result will be used in Sec.VI. The interaction term can be chosen of the form g Φ φ or h Φ φ . Although the adiabaticregularization can be equally applied in both cases, we will focus on the latter case, since thecoupling constant h is dimensionless and the classical theory inherits the conformal invariance.Therefore, the action functional of the scalar matter field is given by S m = (cid:90) d x √− g
12 ( g µν ∇ µ φ ∇ ν φ − m φ − ξRφ − h Φ φ ) . (A1)As before, the scalar field lives in a spatially flat FLRW metric ds = dt − a ( t ) d(cid:126)x , and weassume that the external field is homogeneous Φ = Φ( t ). In this case, the equation of motion is( (cid:50) + m + s ( t ) + ξR ) φ = 0 , (A2)where we have introduced the notation s ( t ) ≡ h Φ( t ), similar to the one used for the spin-1 / φ ( x ) = 1 (cid:112) πa ) (cid:90) d (cid:126)k [ A (cid:126)k f (cid:126)k ( x ) + A † (cid:126)k f ∗ (cid:126)k ( x )] , (A3)where f (cid:126)k ( x ) = e i(cid:126)k(cid:126)x h k ( t ), and A † (cid:126)k and A (cid:126)k are the usual creation and annihilation operators. Sub-stituting (A3) into (A2) we find d dt h k ( t ) + (cid:2) ω k ( t ) + s ( t ) + σ ( t ) (cid:3) h k ( t ) = 0 , (A4)where σ ( t ) = (6 ξ − )( ˙ a a ) + (6 ξ − )( ¨ aa ), and ω k ( t ) = (cid:113) k a ( t ) + m . The adiabatic expansion forthe scalar field modes is based on the usual WKB ansatz h k ( t ) = 1 √ W k e − i (cid:82) t W k ( t (cid:48) ) dt (cid:48) , W k ( t ) = ω k + ω (1) + ω (2) + · · · , (A5)which satisfies automatically the Wronskian condition h k ˙ h ∗ k − h ∗ k ˙ h k = 2 i . One can substitute theansatz into Eq. (A4), and solve order by order to obtain the different terms of the expansion. Thefunction W k ( t ) obeys the differential equation W k = ( ω + s + σ ) W k + 34 ˙ W k −
12 ¨ W k W k . (A6)4Note that here, s ( t ) ≡ h Φ is assumed of adiabatic order one as in the fermionic case. One obtainssystematically ω ( odd ) = 0 for all terms of odd order in the expansion. At second adiabatic orderone gets ω (2) = 12 ω ( s + σ ) + 3 ˙ ω ω − ¨ ω ω = − m ¨ a aω + 3 ξ ¨ aaω − ¨ a aω + 5 m ˙ a a ω − m ˙ a a ω + 3 ξ ˙ a a ω − ˙ a a ω + s ω , (A7)and at fourth adiabatic order, the result is ω (4) = 2( s + σ ) ωω (2) + 3 /
2( ˙ ω (2) ˙ ω ) − / (cid:2) ¨ ω (2) ω + ¨ ωω (2) (cid:3) − sω ( ω (2) ) ω = − a m a ω + 221¨ a ˙ a m a ω + 221 ˙ a m a ω − aa (3) m a ω − s ˙ a m a ω − a m a ω − ξ ˙ a ¨ am a ω −
111 ˙ a ¨ am a ω − ξ ˙ a m a ω −
69 ˙ a m a ω + 9¨ a ξm a ω + 15 ˙ aa (3) ξm a ω + 18¨ a ˙ a ξm a ω + 9 ˙ a ξm a ω + 5 s ˙ a ˙ sm aω + 3¨ as m aω + a (4) m aω + 2 s ˙ a m a ω + ¨ a m a ω + ˙ aa (3) m a ω −
15 ˙ a ¨ am a ω − ˙ a m a ω + 3¨ a ˙ a ξ a ω + 3 ˙ a ξ a ω + a (4) aω − ˙ s ω − s ¨ s ω − s ω − ξs ¨ a aω − s ˙ a ˙ s aω − ξa (4) aω − ξs ˙ a a ω − ξ ¨ a a ω − s ˙ a a ω − ξ ¨ a a ω + ¨ a a ω − ξ ˙ aa (3) a ω + 5 ˙ aa (3) a ω − ξ ˙ a ¨ aa ω + ¨ a ˙ a a ω − ξ ˙ a a ω − ˙ a a ω . (A8)Expressions for the subtraction terms in conformal time have been obtained in [20]. Here we willbriefly sketch the renormalization counterterms associated to the UV divergences of the stress-energy tensor and the variance (cid:104) φ (cid:105) . We follow a strategy similar to the one used in Sec. V. TheLagrangian density with the required renormalization counterterms is L = L m + √− g (cid:20) g µν ∇ µ Φ ∇ ν Φ − m − λ
4! Φ − ξ R Φ − πG Λ + 116 πG R + αR (cid:21) + √− g (cid:20) δZg µν ∇ µ Φ ∇ ν Φ − δm − δλ
4! Φ − δξ R Φ − π δ Λ + 116 π δG − R + δαR (cid:21) . (A9)One can check that the above counterterms are enough to absorb all the UV divergences thatemerge in the quantization of the scalar field. We note that, due to the symmetry Φ → − Φ of thematter Lagrangian, counterterms of the form R Φ, Φ, Φ are absent. However, a higher-derivativeterm of the form R is now necessary, which did not appear for the Dirac field in a FLRW spacetime.We assume the conformal coupling to the curvature ξ = 1 /
6. For a massive field we have g µν T µν − h Φ φ = m φ . (A10)5Classical conformal invariance is obtained when m = 0. In adiabatic regularization the conformalanomaly is computed by taking the massless limit C s = g µν (cid:104) T µν (cid:105) − h Φ (cid:104) φ (cid:105) = − lim m → m (cid:104) φ (cid:105) (4) = − lim m → m (4 πa ) − (cid:90) ∞ dkk ( W − k ( t )) (4) , where ( W − k ( t )) (4) = ω − ( ω (2) ) − ω − ω (4) is the fourth-order term in the adiabatic expansionof W − k . Note that here, (cid:104) φ (cid:105) (4) is evaluated including fourth-order adiabatic subtractions. Thisis different to the physical vacuum expectation value (cid:104) φ (cid:105) ren , which has to be evaluated withsubtractions only up to second order. This is why only the purely fourth-order adiabatic piececontributes to the anomaly. The explicit expression of ( W − k ) (4) for arbitrary ξ is( W − k ) (4) ( t ) = + 1155 ˙ a m a ω −
231 ˙ a ¨ am a ω −
231 ˙ a m a ω + 105 ξ ˙ a m a ω + 63 ˙ a m a ω + 35 s ˙ a m a ω + 105¨ aξ ˙ a m a ω + 105¨ a ˙ a m a ω + 7 a (3) ˙ am a ω + 21¨ a m a ω + 3 ˙ a m a ω + 27¨ a ˙ a m a ω − as ˙ sm aω − s ¨ am aω − a (4) m aω − a s m a ω − ξ ¨ a m a ω −
15 ˙ aξa (3) m a ω + 3¨ a m a ω − ˙ aa (3) m a ω −
45 ˙ a ξ ¨ am a ω −
15 ˙ a ξm a ω + 27 ξ ˙ a a ω + 3 ˙ a a ω + 9 s ξ ˙ a a ω + 27¨ aξ ˙ a a ω + 3¨ a ˙ a a ω + ˙ s ω + 15 a (3) ξ ˙ a a ω + 5 s ˙ a ˙ s aω + s ¨ s ω + 3 s ω − s ¨ a aω + 9 s ¨ aξ aω + 3 a (4) ξ aω − a (4) aω + 27¨ a ξ a ω − ˙ a s a ω − ξ ¨ a a ω − aa (3) a ω −
27 ˙ a ξ ¨ a a ω − a ξ a ω . (A11)The integral in comoving momenta is finite and independent of the mass. Assuming now ξ = 1 / C s = a (4) π a + ¨ a π a − s ˙ a ˙ s π a + a (3) ˙ a π a − ˙ a ¨ a π a − s ¨ s π − ˙ s π − s π . (A12)We can rewrite the expression in terms of covariant scalar terms as C s = 12880 π (cid:26) (cid:50) R − (cid:18) R µν R µν − R (cid:19)(cid:27) − h π (Φ (cid:50) Φ + ∇ µ Φ ∇ µ Φ + 3 h ) , (A13)which is the result given in Eq. (98). Appendix B: A simple example
In this appendix we consider a simple mathematical example to illustrate how the adiabaticmethod works. We compute the bilinear (cid:104) ¯ ψψ (cid:105) ren of a Dirac field, coupled to a background scalarfield evolving in Minkowski spacetime [ a ( t ) = 1] as s ( t ) = g Y Φ( t ) = µ/t . (B1)6For convenience, we have absorbed the Yukawa coupling g Y in the dimensionless constant µ . Toavoid the mathematical instability at t →
0, we will only consider times in the range −∞ < t < t → −∞ we have s, ˙ s · · · →
0, so that the system is adiabatic initially, and there is no ambiguity when imposinginitial conditions to the field modes. And third, as we shall see, the system behaves in such a waythat, as long as we are well before the instability, (cid:104) ¯ ψψ (cid:105) ren can be approximated by the fourth orderin its adiabatic expansion, giving a final renormalized bilinear that can be easily integrated.It is useful to define a new dimensionless time z ≡ mt and momenta κ ≡ k/m . The fieldequations (18) for h Ik and h IIk in terms of these variables become h IIk = iκ (cid:20) ∂h Ik ∂z + i (cid:16) µz (cid:17) h Ik (cid:21) , h Ik = iκ (cid:20) ∂h IIk ∂z − i (cid:16) µz (cid:17) h IIk (cid:21) , (B2)and from these, we obtain the second-order uncoupled equations d h Ik dz + (cid:18) κ + 2 µz + µ ( µ − i ) z (cid:19) h Ik = 0 , d h IIk dz + (cid:18) κ + 2 µz + µ ( µ + i ) z (cid:19) h IIk = 0 . (B3)Let us also define a dimensionless frequency ω κ ≡ √ κ + 1, so that ω = √ k + m = mω κ . Thegeneral solution for h Ik ( t ) is a linear combination of the first and second kind Whittaker functions M α,λ (2 iω κ t ) and W α,λ (2 iω κ t ), where α ≡ − iµ √ κ +1 and λ ≡ − i (2 µ − i ). The solution for h IIk ( t )is similar, with the change λ → λ ≡ − i (2 µ + i ), so we have h Ik = A Ik M α,λ (2 iω κ z ) + B Ik W α,λ (2 iω κ z ) ,h IIk = A IIk M α,λ (2 iω κ z ) + B IIk W α,λ (2 iω κ z ) . (B4)Note that h Ik and h IIk must obey the constraint (21), so there is only 1 degree of freedom in thefermion solution, which is determined when imposing the initial conditions. To fix the constantsin the linear combinations, we impose the adiabatic behavior (23) at z → −∞ , getting A Ik = A IIk = 0 , B Ik = (cid:114) ω κ + 12 ω κ e µπ ωκ , B IIk = (cid:114) ω κ − ω κ e µπ ωκ . (B5)The final solution is then h Ik = (cid:114) ω κ + 12 ω κ e µπ ωκ W α,λ (2 iω κ z ) , h IIk = (cid:114) ω κ − ω κ e µπ ωκ W α,λ (2 iω κ z ) . (B6)The renormalized expectation value (cid:104) ¯ ψψ (cid:105) ren is given, from (64), by (cid:104) ¯ ψψ (cid:105) ren = − m π (cid:90) ∞ dκκ (cid:18) | h Ik | − | h IIk | − ω κ − µω κ z + µω κ z + 3 µ ω κ z − µ ω κ z + µ + µ ω κ z − µ + 6 µ ω κ z + 5 µ ω κ z (cid:19) , (B7)7 Figure 1. The red line shows m |(cid:104) ¯ ψψ (cid:105) ren | as a function of time for µ = 1 given by Eq. (B7). For mt (cid:46) − . (cid:104) ¯ ψψ (cid:105) ren < mt (cid:38) − . (cid:104) ¯ ψψ (cid:105) ren > where we have that the adiabatic contributions of order n go as (cid:104) ¯ ψψ (cid:105) ( n ) ∝ c n ( µ, κ ) z − n , with c n time-independent functions of µ and κ . The above integral is finite, as one can easily check fromthe asymptotic expansion of the Whittaker function W α,λ ( x ).We can compute analytically the leading term at z → −∞ by performing the adiabatic expan-sion of (cid:104) ¯ ψψ (cid:105) up to fourth order, and subtracting from it the zeroth, first, second, and third orders.Therefore, the leading behavior at very early times is (cid:104) ¯ ψψ (cid:105) ren ∼ (cid:104) ¯ ψψ (cid:105) (4) ≡ − π a (cid:90) ∞ dkk (cid:16)(cid:0) | h Ik | (cid:1) (4) − (cid:0) | h IIk | (cid:1) (4) (cid:17) = − π a (cid:90) ∞ dkk (cid:18) ( ω − m )2 ω [ G (4) + G (4) ∗ + G (1) G (3) ∗ + G (1) ∗ G (3) + | G (2) | ] − ( ω + m )2 ω [ F (4) + F (4) ∗ + F (1) F (3) ∗ + F (1) ∗ F (3) + | F (2) | ] (cid:19) . (B8)Computing the integral, we finally get (cid:104) ¯ ψψ (cid:105) (4) = − a (4) π am + ˙ a ¨ a π a m − ˙ a s π a m − ¨ a π a m − aa (3) π a m − s ¨ a π am − a ˙ ss π am − s π m − s ¨ s π m − ˙ s π m . (B9)Substituting (B1) in this expression, and setting a = 1, we finally obtain (cid:104) ¯ ψψ (cid:105) (4) = − m µ ( µ + 5)8 π z , (B10)8where we have written the solution in terms of z . In Fig. 1 we show m |(cid:104) ¯ ψψ (cid:105)| ren as a function oftime, comparing the exact result (B7) with the approximation (B10). At very early times z → −∞ we have, as expected, (cid:104) ¯ ψψ (cid:105) ren ∼
0. We observe that the approximation holds quite well, exceptwhen the instability is approached.
Appendix C: Adiabatic expansion
In this appendix, we provide the terms of the adiabatic expansion of the spin-1/2 field modesup to fourth order. Although the first- and second-order terms have already been written inSection III, we copy them here for convenience. As introduced in Eqs. (24) and (25), the adiabaticexpansion takes the form h Ik ( t ) = (cid:114) ω + m ω e − i (cid:82) t ( ω + ω (1) + ω (2) + ω (3) + ω (4) + ... ) dt (cid:48) (1 + F (1) + F (2) + F (3) + F (4) + . . . ) ,h IIk ( t ) = (cid:114) ω − m ω e − i (cid:82) t ( ω + ω (1) + ω (2) + ω (3) + ω (4) + ... ) dt (cid:48) (1 + G (1) + G (2) + G (3) + G (4) + . . . ) . (C1)The terms G ( n ) can be obtained from F ( n ) with the relation G ( n ) ( m, s ) = F ( n ) ( − m, − s ), so we donot explicitly write them here. We denote by f ( n ) x and f ( n ) y to the real and imaginary parts of F ( n ) respectively, so that F ( n ) = f ( n ) x + if ( n ) y .The first-order terms are f (1) x = s ω − ms ω , (C2) f (1) y = − m ˙ a ω a , (C3) ω (1) = msω . (C4)The second-order terms are f (2) x = m ¨ a aω − m ¨ a aω − m ˙ a a ω + 5 m ˙ a a ω + 3 m ˙ a a ω − m ˙ a a ω + 5 m s ω − ms ω − s ω , (C5) f (2) y = 5 m s ˙ a aω − s ˙ a aω − ˙ s ω , (C6) ω (2) = − m s ω + s ω + 5 m ˙ a a ω − m ˙ a a ω − m ¨ a aω . (C7)The third-order terms are f (3) x = − m s ω + 11 m s ω + 7 ms ω − s ω + 65 m s ˙ a a ω − m s ˙ a a ω − m s ˙ a a ω + 93 m s ˙ a a ω + ms ˙ a a ω − s ˙ a a ω − m ˙ a ˙ s aω + 5 m ˙ a ˙ s aω + 5 m ˙ a ˙ s aω − a ˙ s aω − m s ¨ a aω + m s ¨ a aω + 3 ms ¨ a aω − s ¨ a aω + m ¨ s ω − ¨ s ω , (C8)9 f (3) y = − m s ˙ a aω + 31 ms ˙ a aω + 65 m ˙ a a ω − m ˙ a a ω + m ˙ a a ω + 5 ms ˙ s ω − m ˙ a ¨ a a ω + m ˙ a ¨ a a ω + ma (3) aω , (C9) ω (3) = m s ω − ms ω − m s ˙ a a ω + 13 m s ˙ a a ω − ms ˙ a a ω + 5 m ˙ a ˙ s aω − m ˙ a ˙ s aω + 3 m s ¨ a aω − ms ¨ a aω − m ¨ s ω . (C10)Finally, the fourth-order terms are f (4) x = 2285 ˙ a m a ω −
565 ˙ a m a ω − a m a ω − s ˙ a m a ω −
457 ˙ a ¨ am a ω + 2611 ˙ a m a ω + 965 s ˙ a m a ω + 113 ˙ a ¨ am a ω + 2371 ˙ a m a ω + 2441 s ˙ a m a ω + 41¨ a m a ω + 65 s ˙ a ˙ sm aω + 725 ˙ a ¨ am a ω + 117 s ¨ am aω + 7 ˙ aa (3) m a ω + 195 s m ω −
333 ˙ a m a ω − s ˙ a m a ω − a m a ω − s ˙ a ˙ sm aω −
749 ˙ a ¨ am a ω − s ¨ am aω − aa (3) m a ω − s m ω − a m a ω − s ˙ a m a ω − s m ω − a m a ω − s ˙ a ˙ sm aω −
19 ˙ a ¨ am a ω − s ¨ am aω − s ¨ sm ω −
13 ˙ aa (3) m a ω − a (4) m aω − s m ω + ˙ a m a ω + 111 s ˙ a m a ω + 5 ˙ s m ω + ¨ a m a ω + 89 s ˙ a ˙ sm aω + 11 ˙ a ¨ am a ω + 49 s ¨ am aω + s ¨ sm ω + 7 ˙ aa (3) m a ω + a (4) m aω + 9 s m ω + s ˙ a a ω − ˙ s ω + s ˙ a ˙ s aω + s ¨ a aω + s ¨ s ω + 11 s ω , (C11) f (4) y = 195 m s ˙ a aω − m s ˙ a aω + 11 s ˙ a aω − m s ˙ a a ω + 2571 m s ˙ a a ω − m s ˙ a a ω + s ˙ a a ω − m s ˙ s ω + 11 s ˙ s ω + 195 m ˙ a ˙ s a ω − m ˙ a ˙ s a ω + 7 ˙ a ˙ s a ω + 247 m s ˙ a ¨ a a ω − m s ˙ a ¨ a a ω + s ˙ a ¨ a a ω − m ˙ s ¨ a aω + ˙ s ¨ a aω − m ˙ a ¨ s aω + 3 ˙ a ¨ s aω − m sa (3) aω + sa (3) aω + s (3) ω , (C12) ω (4) = − m s ω + 3 m s ω − s ω + 175 m s ˙ a a ω − m s ˙ a a ω + 79 m s ˙ a a ω − s ˙ a a ω − m ˙ a a ω + 337 m ˙ a a ω − m ˙ a a ω + 3 m ˙ a a ω − m s ˙ a ˙ s aω + 23 m s ˙ a ˙ s aω − s ˙ a ˙ s aω + 5 m ˙ s ω − m s ¨ a aω + 25 m s ¨ a aω − s ¨ a aω + 221 m ˙ a ¨ a a ω − m ˙ a ¨ a a ω + 13 m ˙ a ¨ a a ω − m ¨ a a ω + m ¨ a a ω + 3 m s ¨ s ω − s ¨ s ω − m ˙ aa (3) a ω + 15 m ˙ aa (3) a ω + m a (4) aω . (C13) [1] L. Parker and D. J. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity ,Cambridge University Press, Cambridge, England (2009). [2] R. M. Wald, Quantum Field Theory in Curved Space-time and Black Hole Thermodynamics , Universityof Chicago Press, Chicago, (1994).[3] S. Fulling,
Aspects of Quantum Field Theory in Curved Space-Time , Cambridge University Press,Cambridge, England (1989).[4] N. D. Birrell and P. C. W. Davies,
Quantum Fields in Curved Space , Cambridge University Press,Cambridge, England (1982).[5] L. Parker,
The creation of particles in an expanding universe , Ph.D. thesis, Harvard University (1966).For an historical overview see: L. Parker,
J. Phys. Conf. Ser. no.1, 012001 (2015);
J. Phys. A ,374023 (2012).[6] L. Parker, Phys. Rev. Lett. , 562 (1968); Phys. Rev. D , 1057 (1969);
Phys. Rev. D , 346 (1971).[7] L. Parker and S. A. Fulling, Phys. Rev. D , 341 (1974); S. A. Fulling and L. Parker, Ann. Phys. (N.Y.) , 176 (1974); S. A. Fulling, L. Parker and B. L. Hu, Phys. Rev. D , 3905 (1974).[8] T. S. Bunch, J. Phys. A , 1297 (1980).[9] P. R. Anderson and L. Parker, Phys. Rev. D , 2963 (1987).[10] A. Landete, J. Navarro-Salas and F. Torrenti, Phys Rev. D , 061501(R) (2013); Phys. Rev. D ,044030 (2014).[11] A. del Rio, J. Navarro-Salas and F. Torrenti, Phys. Rev. D , 084017 (2014).[12] S. Ghosh, Phys. Rev. D , 124075 (2015); Phys. Rev. D , 044032 (2016).[13] S. M. Christensen, Phys. Rev. D , 2490 (1976).[14] S. M. Christensen, Phys. Rev. D , 946 (1978).[15] N. D. Birrell, Proc. R. Soc. B , 513 (1978).[16] A. del Rio and J. Navarro-Salas,
Phys. Rev. D , 064031 (2015).[17] P. R. Anderson, Phys. Rev. D , 1302 (1985); Phys. Rev. D , 1567 (1986); P. R. Anderson andW. Eaker, Phys. Rev. D , 024003 (1999); S. Habib, C. Molina-Paris and E. Mottola, Phys. Rev. D , 024010 (1999); J. D. Bates and P. R. Anderson, Phys. Rev. D , 024018 (2010).[18] B. L. Hu and L. Parker, Phys. Lett. A , 217 (1977); Phys. Rev. D , 933 (1978).[19] P. R. Anderson and W. Eaker, Phys. Rev. D , 024003 (1999).[20] C. Molina-Paris, P. R. Anderson and S. A. Ramsey, Phys. Rev. D , 127501 (2000).[21] L. Parker, arXiv:hep-th/0702216. F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi, Phys. Rev. D ,103528 (2007). I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker, Phys. Rev. Lett. , 061301(2009);
Phys. Rev. D , 043514, (2010); Phys. Rev. D Phys. Rev. D , 065024 (2009). G. Marozzi, M. Rinaldi and R. Durrer, Phys. Rev.D , 105017 (2011). M. Bastero-Gil, A. Berera, N. Mahajan and R. Rangarajan, Phys. Rev. D ,087302 (2013). A. del Rio and J. Navarro-Salas, Phys. Rev. D JCAP no.06, 019 (2015). A. L. Alinea,
JCAP no.10, 027(2016). D. G. Wang, Y. Zhang and J. W. Chen,
Phys. Rev. D , 044033 (2016).[22] R. P. Woodard, Int. J. Mod. Phys. D , 1430020 (2014). [23] T. Markkanen and A. Tranberg, JCAP , 045 (2013). T. Markkanen and A. Rajantie, JHEP ,133 (2017).[24] L. Kofman, A. Linde and A. Starobinsky,
Phys. Rev. Lett. , 3195 (1994); Phys. Rev. D , 3258(1997).[25] P. R. Anderson, C. Molina-Paris, D. Evanich and G. B. Cook, Phys. Rev. D , 083514 (2008).[26] P. B. Greene and L. Kofman, Phys. Lett. B
Phys. Rev. D , 123516 (2000); J. Baacke,K. Heitmann and C. Patzold, Phys. Rev. D , 125013 (1998); G. F. Giudice, M. Peloso, A. Riotto andI. Tkachev, JHEP , 014 (1999); M. Peloso and L. Sorbo,
JHEP , 016 (2000); J. Garcia-Bellidoand E. Ruiz-Morales,
Phys. Lett. B , 193-202 (2002).[27] D. G. Figueroa,
JHEP , 145 (2014).[28] K. Enqvist, T. Meriniemi and S. Nurmi,
JCAP , 057 (2013).[29] D. G. Figueroa, J. Garcia-Bellido and F. Torrenti,
Phys. Rev. D , 083511 (2015); K. Enqvist,S. Nurmi, S. Rusak and D. Weir, JCAP , 057 (2016).[30] J. Garcia-Bellido, D. G. Figueroa and J. Rubio,
Phys. Rev. D , 063531 (2009); F. L. Bezrukov,D. Gorbunov and M. Shaposhnikov, JCAP , 029 (2009); D. G. Figueroa, and C. T. Byrnes,
Phys. Lett. B , 272-277 (2017).[31] M. E. Peskin and D. V. Schroeder,
An Introduction to Quantum Field Theory , Reading, MA: Addison-Wesley, (1995).[32] R. Utiyama and B. S. DeWitt,
J. Math. Phys. , 608 (1962).[33] M. Srednicki, Quantum Field Theory , Cambridge University Press, Cambridge, England (2007).[34] T. S. Bunch and L. Parker,
Phys. Rev. D , 2499 (1979).[35] T. S. Bunch, Ann. Phys.
NY, , 118 (1981).[36] J. Baacke and C. Patzold,
Phys. Rev. D , 084008 (2000).[37] M. J. Duff, Class. Quantum Grav. , 1387 (1994).[38] S. W. Hawking, T. Hertog and H. S. Reall, Phys. Rev. D , 083504 (2001).[39] R. M. Wald, Phys. Rev. D17