Inflaton Effective Potential from Photons for General ε
aa r X i v : . [ g r- q c ] J a n UFIFT-QG-20-06
Inflaton Effective Potential from Photons for General ǫ S. Katuwal ∗ , S. P. Miao ⋆ and R. P. Woodard † Department of Physics, University of Florida,Gainesville, FL 32611, UNITED STATES Department of Physics, National Cheng Kung University,No. 1 University Road, Tainan City 70101, TAIWAN
ABSTRACTWe accurately approximate the contribution that photons make to the ef-fective potential of a charged inflaton for inflationary geometries with anarbitrary first slow roll parameter ǫ . We find a small, nonlocal contributionand a numerically larger, local part. The local part involves first and secondderivatives of ǫ , coming exclusively from the constrained part of the elec-tromagnetic field which carries the long range interaction. This causes theeffective potential induced by electromagnetism to respond more strongly togeometrical evolution than for either scalars, which have no derivatives, orspin one half particles, which have only one derivative. For ǫ = 0 our finalresult agrees with that of Allen [1] on de Sitter background, while the flatspace limit agrees with the classic result of Coleman and Weinberg [2].PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g ∗ e-mail: sanjib.katuwal@ufl.edu ⋆ email: [email protected] † e-mail: [email protected]fl.edu Introduction
No one knows what caused primordial inflation but the data [3] are consistentwith a minimally coupled, complex scalar inflaton ϕ , L = − ∂ µ ϕ∂ ν ϕ ∗ g µν √− g − V ( ϕϕ ∗ ) √− g . (1)If the inflaton couples only to gravity the loop corrections to its effectivepotential come only from quantum gravity and are suppressed by powers ofthe loop-counting parameter GH < ∼ − , where G is Newton’s constantand H is the Hubble parameter during inflation. In that case the classicalevolution suffers little disturbance but reheating is very slow.Efficient reheating requires coupling the inflaton to normal matter suchas electromagnetism with a non-infinitesimal charge q , L = − (cid:16) ∂ µ − iqA µ (cid:17) ϕ (cid:16) ∂ ν + iqA ν (cid:17) ϕ ∗ g µν √− g − V ( ϕϕ ∗ ) √− g − F µν F ρσ g µρ g νσ √− g . (2)But the price of efficient reheating is significant one loop corrections to theinflaton effective potential [4]. For large fields these corrections approach theColeman-Weinberg form of flat space ∆ V −→ π ( q ϕϕ ∗ ) ln( q ϕϕ ∗ /s ),where s is the renormalization scale [2]. However, cosmological Coleman-Weinberg potentials generally depend in a complicated way on the geometryof inflation [5], ds = a h − dη + d~x · d~x i = ⇒ H ≡ ∂ aa , ǫ ( t ) ≡ − ∂ HaH . (3)For the special case of de Sitter (with constant H and ǫ = 0) the result takesthe form [1, 6, 7],∆ V (cid:12)(cid:12)(cid:12) ǫ =0 = 3 H π (cid:26) b (cid:16) q ϕϕ ∗ H (cid:17) + (cid:16) q ϕϕ ∗ H (cid:17) ln (cid:16) H s (cid:17) + 12 (cid:16) q ϕϕ ∗ H (cid:17) ln (cid:16) H s (cid:17)(cid:27) , (4)where the function b ( z ) (whose z and z terms depend on renormalizationconventions) is, b ( z ) = (cid:16) − γ (cid:17) z + (cid:16) −
32 + γ (cid:17) z + Z z dx (1+ x ) (cid:20) ψ (cid:16)
32 + 12 √ − x (cid:17) + ψ (cid:16) − √ − x (cid:17)(cid:27) . (5)1osmological Coleman-Weinberg potentials are problematic because theymake large corrections which cannot be completely subtracted using allowedlocal counterterms [5]. The classical evolution of inflation is subject to unac-ceptable modifications when partial subtractions are restricted to just func-tions of the inflaton [8], or functions of the inflaton and the Ricci scalar [7].No other local subtractions are permitted [9] but it has been suggested thatan acceptably small distortion of classical inflation might result from cancella-tions between the effective potentials induced by fermions and by bosons [10].The purpose of this paper is facilitate study of this scheme by developing anaccurate approximation for extending the de Sitter results (4-5) to a generalcosmological geometry (3).As before on flat space [2], and on de Sitter background [6], we define thederivative of the one loop effective potential through the equation,∆ V ′ ( ϕϕ ∗ ) = δξR + 12 δλϕϕ ∗ + q g µν i h µ ∆ ν i ( x ; x ) . (6)Here i [ µ ∆ ν ]( x ; x ′ ) is the massive photon propagator in Lorentz gauge [11], h νµ − R νµ − M δ νµ i i h ν ∆ ρ i ( x ; x ′ ) = g µρ iδ D ( x − x ′ ) √− g + ∂ µ ∂ ′ ρ i ∆ t ( x ; x ′ ) , (7)where νµ is the covariant vector d’Alembertian, M ≡ q ϕϕ ∗ is the photonmass-squared, and i ∆ t ( x ; x ′ ) is the propagator of a massless, minimally cou-pled (MMC) scalar. We regulate the ultraviolet by working in D spacetimedimensions.In section 2 we express the photon propagator as an exact spatial Fouriermode sum involving massive temporal and spatially transverse vectors, alongwith gradients of the MMC scalar. Section 3 begins by converting the variousmode equations to a dimensionless form, then these are approximated. Eachapproximation is checked against explicit numerical evolution, both for thesimple quadratic potential, which is excluded by the lower bound on thetensor-to-scalar ratio [12], and for a plateau potential [13] that is in goodagreement with all data. In section 4 our approximations are applied torelation (6) to compute the one loop effective potential. This consists of alocal part which depends on the instantaneous geometry and a numericallysmaller nonlocal part which depends on the past geometry. Exact expressionsare obtained, as well as expansions in the large field and small field regimes.Our conclusions are given in section 5.2 Photon Mode Sum
The purpose of this section is to express the Lorentz gauge propagator for amassive photon as a spatial Fourier mode sum. We begin by expressing theright hand side of the propagator equation (7) as mode sum. Then the varioustransverse vector modes are introduced. Next these modes are combined soas to enforce the propagator equation. The section closes by checking the deSitter and flat space correspondence limits.
If we exploit Lorentz gauge, the µ = 0 component of (7) reads, − a h − ∂ + ( D − ∂ aH + a M i i h ∆ ρ i ( x ; x ′ )= − δ ρ iδ D ( x − x ′ ) a D − + ∂ ∂ ′ ρ i ∆ t ( x ; x ′ ) , (8)where ∂ ≡ η µν ∂ µ ∂ ν is the flat space d’Alembertian. The µ = m componentof equation (7) reads, − a (cid:26)h − ∂ + ( D − aH∂ + a M i i h m ∆ ρ i ( x ; x ′ ) + 2 aH∂ m i h ∆ ρ i ( x ; x ′ ) (cid:27) = δ mρ iδ D ( x − x ′ ) a D − + ∂ m ∂ ′ ρ i ∆ t ( x ; x ′ ) . (9)We begin by writing the right hand sides of expressions (8) and (9) as Fouriermode sums.The MMC scalar propagator i ∆ t ( x ; x ′ ) can be expressed as a Fouriermode sum over functions t ( η, k ) whose wave equation and Wronskian are, h ∂ + ( D − aH∂ + k i t ( η, k ) = 0 , t · ∂ t ∗ − ∂ t · t ∗ = ia D − . (10)Although no closed form solution exists to the t ( η, k ) wave equation for ageneral scale factor, relations (10) do define a unique solution when combinedwith the early time asymptotic form, k ≫ aH = ⇒ t ( η, k ) −→ e − ikη √ ka D − . (11)3p to infrared corrections [14], which are irrelevant owing to the derivativesin expressions (7) and (8), the Fourier mode sum for i ∆ t ( x ; x ′ ) is, i ∆ t ( x ; x ′ ) = Z d D − k (2 π ) D − (cid:26) θ (∆ η ) t ( η, k ) t ∗ ( η ′ , k ) e i~k · ∆ ~x + θ ( − ∆ η ) t ∗ ( η, k ) t ( η ′ , k ) e − i~k · ∆ ~x (cid:27) , (12)where ∆ η ≡ η − η ′ and ∆ ~x ≡ ~x − ~x ′ . Acting ∂ ∂ ′ ρ on (12) produces aterm proportional to δ ρ δ (∆ η ), which the Wronskian (10) and the change ofvariable ~k → − ~k allows us to recognize as a D -dimensional delta function, ∂ ∂ ′ ρ i ∆ t ( x ; x ′ ) = Z d D − k (2 π ) D − (cid:26) δ ρ δ (∆ η ) h t · ∂ t ∗ − ∂ t · t ∗ i e i~k · ∆ ~x + θ (∆ η ) ∂ ∂ ′ ρ × h t ( η, k ) t ∗ ( η ′ , k ) e i~k · ∆ ~x i + θ ( − ∆ η ) ∂ ∂ ′ ρ h t ∗ ( η, k ) t ( η ′ , k ) e − i~k · ∆ ~x i(cid:27) , (13)= δ ρ iδ D ( x − x ′ ) a D − + Z d D − k (2 π ) D − (cid:26) θ (∆ η ) T ( x, ~k ) T ∗ ρ ( x ′ , ~k )+ θ ( − ∆ η ) T ∗ ( x, ~k ) T ρ ( x ′ , ~k ) (cid:27) . (14)Here we define T µ ( x, ~k ) ≡ ∂ µ [ t ( η, k ) e i~k · ~x ].Substituting (14) in the right hand side of (8) gives, − a h − ∂ + ( D − ∂ aH + a M i i h ∆ ρ i ( x ; x ′ )= Z d D − k (2 π ) D − (cid:26) θ (∆ η ) T ( x, ~k ) T ∗ ρ ( x ′ , ~k ) + θ ( − ∆ η ) T ∗ ( x, ~k ) T ρ ( x ′ , ~k ) (cid:27) . (15)The corresponding expression for (9) is, − a (cid:26)h − ∂ + ( D − aH∂ + a M i i h m ∆ ρ i ( x ; x ′ ) + 2 aH∂ m i h ∆ ρ i ( x ; x ′ ) (cid:27) = Z d D − k (2 π ) D − (cid:26) δ mρ iδ (∆ η ) e i~k · ∆ ~x a D − + θ (∆ η ) T m ( x, ~k ) T ∗ ρ ( x ′ , ~k )+ θ ( − ∆ η ) T ∗ m ( x, ~k ) T ρ ( x ′ , ~k ) (cid:27) . (16)The right hand sides of (15) and (16) are the Fourier mode sums that willguide us in constructing the photon propagator.4 .2 Transverse Vector Mode Functions In the cosmological geometry (3) a transverse (Lorentz gauge) vector field F µ ( x ) obeys,0 = D µ F µ ( x ) = 1 a h − (cid:16) ∂ +( D − aH (cid:17) F + ∂ i F i i ≡ a h −D F + ∂ i F i i . (17)We seek to express the photon propagator as a Fourier mode sum over alinear combination of transverse vector mode functions. Expressions (15-16)imply that one of these must be the gradient of a MMC scalar plane wave, T µ ( x, ~k ) ≡ ∂ µ h t ( η, k ) e i~k · ~x i . (18)Its transversality follows from the MMC mode equation (10), −D T + ∂ i T i = − h ∂ +( D − aH∂ + k i t ( η, k ) e i~k · ~x = 0 . (19)In D spacetime dimensions there are D − V µ ( x, ~k, λ, M ) ≡ ǫ µ ( ~k, λ ) × v ( η, k ) e i~k · ~x , ǫ = 0 = k i ǫ i . (20)The polarization vectors ǫ µ ( ~k, λ ) are the same as those of flat space, and theirpolarization sum is, X λ ǫ µ ( ~k, λ ) ǫ ∗ ρ ( ~k, λ ) = (cid:18) δ mr − b k m b k r (cid:19) ≡ Π µρ ( ~k ) . (21)The wave equation and Wronskian of v ( η, k ) are, h ∂ +( D − aH∂ + k + a M i v ( η, k ) = 0 , v · ∂ v ∗ − ∂ v · v ∗ = ia D − . (22)Relations (22) define a unique solution when coupled with the form forasymptotically early times, k ≫ n aH, aM o = ⇒ v ( η, k ) −→ ae − ikη √ ka D − . (23)5he spatially transverse vector modes V µ ( x, ~k, λ, M ) represent dynamicalphotons. There is also a single temporal-longitudinal mode which repre-sents the constrained part of the electromagnetic field. It is a combina-tion of T µ ( x, ~k ) with a transverse vector formed from the µ = 0 component u ( η, k, M ) of a massive vector, h ∂ + ( D − ∂ aH + k + a M i u ( η, k, M ) = 0 , u · ∂ u ∗ − ∂ u · u ∗ = ia D − . (24)Relations (24) define a unique solution when combined with the early timeasymptotic form, k ≫ n aH, aM o = ⇒ u ( η, k ) −→ e − ikη √ ka D − . (25)One converts u ( η, k, M ) to a transverse vector U µ ( x, ~k, M ), U µ ( x, ~k, M ) ≡ ∂ µ h u ( η, k ) e i~k · ~x i , (26)where the differential operator ∂ µ has the 3 + 1 decomposition, ∂ ≡ √−∇ −→ k , ∂ i ≡ − ∂ i D√−∇ −→ − i b k i D . (27) We have seen that the photon propagator i [ ρ ∆ ρ ]( x ; x ′ ) is the spatial Fourierintegral of contributions from the three transverse vector modes, each havingthe general form of constants times, F µρ ( x ; x ′ ) = θ (∆ η ) F µ ( x ) F ∗ ρ ( x ′ ) + θ ( − ∆ η ) F ∗ µ ( x ) F ρ ( x ′ ) , F µ ∈ n T µ , U µ , V µ o . (28)We might anticipate that the spatially transverse modes contribute with unitamplitude but the MMC scalar and temporal photon modes must be multi-plied by the square of an inverse mass to even have the correct dimensions.The multiplicative factors are chosen to enforce the propagator equation (7).To check the temporal components (15) of the propagator equation wemust compute, − a h − ∂ + ( D − ∂ aH + a M i F ρ ( x ; x ′ ) . (29)6o check the spatial components (16) we need, − a h − ∂ + ( D − aH∂ + a M i F mρ ( x ; x ′ ) − a × aH∂ m F ρ ( x ; x ′ ) . (30)The factors of ∂ in the differential operators of (29-30) can act on the thetafunctions or on the mode functions. When all derivatives act on the MMCcontribution, the result is − M times the original mode function, − a h − ∂ + ( D − ∂ aH + a M i T ( x ) = − M T ( x ) , (31) − a h − ∂ + ( D − aH∂ + a M i T m ( x ) − a × aH∂ m T ( x ) = − M T m ( x ) . (32)This suggests that the MMC contribution enters the mode sum with a mul-tiplicative factor of − M − . No further information comes from acting thefull differential operators on the other modes, − a h − ∂ + ( D − ∂ aH + a M i U ( x ) = 0 , (33) − a h − ∂ + ( D − aH∂ + a M i U m ( x ) − a × aH∂ m U ( x ) = 0 , (34) − a h − ∂ + ( D − ∂ aH + a M i V ( x ) = 0 , (35) − a h − ∂ + ( D − aH∂ + a M i V m ( x ) − a × aH∂ m V ( x ) = 0 . (36)It remains to check what happens when one or two factors of ∂ fromthe differential operators in (29-30) act on the factors of θ ( ± ∆ η ). A singleconformal time derivative gives, ∂ F µρ ( x ; x ′ ) = θ (∆ η ) ∂ F µ F ∗ ρ + θ ( − ∆ η ) ∂ F ∗ µ F ρ + δ (∆ η ) h F µ F ∗ ρ − F ∗ µ F ρ i . (37)If we change the Fourier integration variable ~k to − ~k in the second of thedelta function terms, the result for the MMC modes is, T µ T ∗ ρ − T ∗ µ T ρ (cid:12)(cid:12)(cid:12) ~k →− ~k = (cid:18) [ ∂ t∂ t ∗ − ∂ t ∗ ∂ t ] − ik r [ ∂ t t ∗ − ∂ t ∗ t ] ik m [ t ∂t ∗ − t ∗ ∂ t ] k m k r [ t t ∗ − t ∗ t ] (cid:19) e i~k · ∆ ~x . (38)= (cid:18) − k r − k m (cid:19) e i~k · ∆ ~x a D − . (39)7he temporal photon modes make exactly the same contribution, U µ U ∗ ρ − U ∗ µ U ρ (cid:12)(cid:12)(cid:12) ~k →− ~k = (cid:18) k [ u u ∗ − u ∗ u ] ik r [ u D u ∗ − u ∗ D u ] − ik m [ D u u ∗ − D u ∗ u ] b k m b k r [ D u D u ∗ − D u ∗ D u ] (cid:19) e i~k · ∆ ~x . (40)= (cid:18) − k r − k m (cid:19) e i~k · ∆ ~x a D − . (41)Canceling (41) against (39) — whose multiplicative coefficient is − M − —fixes the multiplicative coefficient for the temporal photons as + M − . Thedelta function term in (37) vanishes for the spatially transverse modes.We turn now to second derivative which come from − ∂ = ∂ − ∇ , ∂ F µρ ( x ; x ′ ) = θ (∆ η ) ∂ F µ ( x ) F ∗ ρ ( x ′ ) + θ ( − ∆ η ) ∂ F ∗ µ ( x ) F ρ ( x ′ )+ δ (∆ η ) h ∂ F µ F ∗ ρ − ∂ F ∗ µ F ρ i + ∂ (cid:26) δ (∆ η ) h F µ F ∗ ρ − F ∗ µ F ρ i(cid:27) . (42)We have already arranged for the cancellation of the final term in (42). Forthe new delta function term the MMC modes give, ∂ T µ T ∗ ρ − ∂ T ∗ µ T ρ (cid:12)(cid:12)(cid:12) ~k →− ~k = (cid:18) [ ∂ t ∂ t ∗ − ∂ t ∗ ∂ t ] − ik r [ ∂ t t ∗ − ∂ t ∗ t ] ik m [ ∂ t ∂ t ∗ − ∂ t ∗ ∂ t ] k m k r [ ∂ t t ∗ − ∂ t ∗ t ] (cid:19) e i~k · ∆ ~x , (43)= − i (cid:18) k ik r ( D − aH k m k r (cid:19) e i~k · ∆ ~x a D − , (44)where we have used ∂ t = − [( D − aH∂ + k ] t . The corresponding contri-bution for the temporal modes is, ∂ U µ U ∗ ρ − ∂ U ∗ µ U ρ (cid:12)(cid:12)(cid:12) ~k →− ~k = (cid:18) k [ ∂ u u ∗ − ∂ u ∗ u ] ik r [ ∂ u D u ∗ − ∂ u ∗ D u ] − ik m [ ∂ D u u ∗ − ∂ D u ∗ u ] b k r b k m [ ∂ D u D u ∗ − ∂ D u ∗ D u ] (cid:19) e i~k · ∆ ~x , (45)= − i (cid:18) k ik r ( D − aH b k m b k r ( k + a M ) (cid:19) e i~k · ∆ ~x a D − , (46)8here we have used ∂ D u = − ( k + a M ) u . And each of the spatiallytransverse modes gives, ∂ V µ V ∗ ρ − ∂ V ∗ µ V ρ (cid:12)(cid:12)(cid:12) ~k →− ~k = (cid:18) ǫ m ǫ ∗ r [ ∂ v v ∗ − ∂ v ∗ v ] (cid:19) e i~k · ∆ ~x , (47)= − i (cid:18) ǫ m ǫ ∗ r (cid:19) e i~k · ∆ ~x a D − . (48)The second conformal time derivatives in both expression (29) and thecorresponding spatial relation (30) come in the form − a × ∂ . Including themultiplicative factors, we see that the temporal delta functions which areinduced consist of a M times (44) minus the same factor times (46), plusthe polarization sum (21) over (48), iM (cid:18) k ik r ( D − aH k m k r (cid:19) e i~k · ∆ ~x a D − iM (cid:18) k ik r ( D − aH b k m b k r ( k + a M ) (cid:19) e i~k · ∆ ~x a D − i (cid:18) δ mr − b k m b k r (cid:19) e i~k · ∆ ~x a D − = − i (cid:18) δ mr (cid:19) e i~k · ∆ ~x a D − . (49)With − M times expressions (31-32) we see that the propagator equations(15-16) are obeyed by the Fourier mode sum, i h µ ∆ ρ i ( x ; x ′ ) = Z d D − k (2 π ) D − (cid:26) θ (∆ η ) (cid:20) U µ ( x, ~k, M ) U ∗ ρ ( x ′ , ~k, M ) − T µ ( x, ~k ) T ∗ ρ ( x ′ , ~k ) M +Π µρ ( ~k ) v ( η, k ) v ∗ ( η ′ , k ) e i~k · ∆ ~x (cid:21) + θ ( − ∆ η ) (cid:20) U ∗ µ ( x, ~k, M ) U ρ ( x ′ , ~k, M ) M − T ∗ µ ( x, ~k ) T ρ ( x ′ , ~k ) M + Π µρ ( ~k ) v ∗ ( η, k ) v ( η ′ , k ) e − i~k · ∆ ~x (cid:21)(cid:27) . (50)Note that the U µ ( x, ~k, M ) and T µ ( x, ~k ) modes combine to form a vectorintegrated propagator analogous to the scalar ones introduced in [15].The photon propagator can also be expressed as the sum of three bi-vectordifferential operator acting on a scalar propagator, i h µ ∆ ρ i ( x ; x ′ ) = 1 M h − η µρ + Π µρ i iδ D ( x − x ′ ) a D − + 1 M h ∂ µ ∂ ′ ρ i ∆ u ( x ; x ′ ) − ∂ µ ∂ ′ ρ i ∆ t ( x ; x ′ ) i + Π µρ i ∆ v ( x ; x ′ ) . (51)9he Fourier mode sum for the MMC scalar propagator i ∆ t ( x ; x ′ ) was givenin expression (12). The mode sum for the temporal propagator i ∆ u ( x ; x ′ )comes from replacing t ( η, k ) with u ( η, k ) in (12), and the mode sum for thetransverse spatial propagator i ∆ v ( x ; x ′ ) is obtained by replacing t ( η, k ) with v ( η, k ). The resulting lowest order (free) field strength correlators are, D Ω (cid:12)(cid:12)(cid:12) T ∗ h F j ( x ) F ℓ ( x ′ ) i(cid:12)(cid:12)(cid:12) Ω E = ∂ j ∂ ℓ ∇ iδ D ( x − x ′ ) a D − + a a ′ M ∂ j ∂ ℓ ∇ i ∆ u ( x ; x ′ ) + Π jℓ ∂ ∂ ′ i ∆ v ( x ; x ′ ) , (52) D Ω (cid:12)(cid:12)(cid:12) T ∗ h F j ( x ) F kℓ ( x ′ ) i(cid:12)(cid:12)(cid:12) Ω E = h δ jk ∂ ℓ − δ jℓ ∂ k i ∂ i ∆ v ( x ; x ′ ) , (53) D Ω (cid:12)(cid:12)(cid:12) T ∗ h F ij ( x ) F kℓ ( x ′ ) i(cid:12)(cid:12)(cid:12) Ω E = − h δ ik ∂ j ∂ ℓ − δ kj ∂ ℓ ∂ i + δ jℓ ∂ i ∂ k − δ ℓi ∂ k ∂ j i i ∆ v ( x ; x ′ ) . (54)The T ∗ -ordering symbol in these correlators indicates that the derivatives informing the field strength tensor, F µν ( x ) ≡ ∂ µ A ν ( x ) − ∂ ν A µ ( x ), are takenoutside the time-ordering symbol.An important simplification is, T µ ( x, ~k ) = − i lim M → U µ ( x, ~k, M ) . (55)Comparing equations (31) with (33), and (32) with (34), shows that bothsides of relation (55) obey the same wave equation for M = 0. That they areidentical follows from t ( η, k ) and u ( η, k ) having the same asymptotic forms(11) and (25). Relation (55) is of great importance because it guaranteesthat the propagator has no M pole. In the limit of ǫ = 0 the mode functions have closed form solutions, t ( η, k ) −→ e iπ ( ν A + ) r π Ha D − × H (1) ν A (cid:16) kHa (cid:17) , (56) u ( η, k, M ) −→ e iπ ( ν b + ) r π Ha D − × H (1) ν b (cid:16) kHa (cid:17) , (57) In the phase factors for u ( η, k, M ) and v ( η, k, M ) one must regard ν b as a real number,even if M > ( D − H . ( η, k, M ) −→ e iπ ( ν b + ) r π Ha D − × H (1) ν b (cid:16) kHa (cid:17) , (58)where the indices are, ν A ≡ (cid:16) D − (cid:17) , ν b ≡ r(cid:16) D − (cid:17) − M H . (59)The Fourier mode sums for the three propagators can be mostly expressedin terms of the de Sitter length function y ( x ; x ′ ), y ( x ; x ′ ) ≡ (cid:13)(cid:13)(cid:13) ~x − ~x ′ (cid:13)(cid:13)(cid:13) − (cid:16) | η − η ′ |− iε (cid:17) . (60)The de Sitter limit of the temporal photon propagator is a Hypergeometricfunction, i ∆ u ( x ; x ′ ) −→ H D − (4 π ) D Γ( ν a + ν b )Γ( ν A − ν b )Γ( D ) F (cid:16) ν A + ν b , ν A − ν b , D − y (cid:17) ≡ b ( y ) . (61)The de Sitter limit of the spatially transverse photon propagator is closelyrelated, i ∆ v ( x ; x ′ ) −→ aa ′ b ( y ) . (62)However, infrared divergences break de Sitter invariance in the MMC scalarpropagator [16–18]. The result for the noncoincident propagator takes theform [19, 20], i ∆ t ( x ; x ′ ) −→ A ( y ) + H D − (4 π ) D Γ( D − D ) ln( aa ′ ) , (63)where we only need derivatives of the function A ( y ) [21], A ′ ( y ) = 12 (2 − y ) B ′ ( y ) −
12 ( D − B ( y ) , (64) B ( y ) ≡ Γ( D − D ) F (cid:16) D − , , D − y (cid:17) . (65)It is useful to note that the functions B ( y ) and b ( y ) obey,0 = (4 y − y ) B ′′ ( y ) + D (2 − y ) B ′ ( y ) − ( D − B ( y ) , (66)0 = (4 y − y ) b ′′ ( y ) + D (2 − y ) b ′ ( y ) − ( D − b ( y ) − M H b ( y ) . (67)11 direct computation of the photon propagator on de Sitter backgroundgives [11], i h µ ∆ ρ i ( x ; x ′ ) −→ − ∂ y∂x µ ∂x ′ ρ h (4 y − y ) ∂∂y + ( D − − y ) ih b ′ ( y ) − B ′ ( y )2 M i + ∂y∂x µ ∂y∂x ′ ρ h (2 − y ) ∂∂y − ( D − ih b ′ ( y ) − B ′ ( y )2 M i . (68)To see that the de Sitter limit of our mode sum (51) agrees with (68) wesubstitute the de Sitter limits (63), (61) and (62) and make some tediousreorganizations. This is simplest for the MMC contribution, δ µ δ ρ iδ D ( x − x ′ ) M a D − − ∂ µ ∂ ′ ρ i ∆ t ( x ; x ′ ) M −→ − ∂ y∂x µ ∂x ′ ρ A ′ M − ∂y∂x µ ∂y∂x ′ ρ A ′′ M , (69)= − ∂ y∂x µ ∂x ′ ρ h (2 − y ) B ′ − ( D − B M i − ∂y∂x µ ∂y∂x ′ ρ h (2 − y ) B ′′ − ( D − B ′ M i , (70)= ∂ y∂x µ ∂x ′ ρ h (4 y − y ) B ′′ +( D − − y ) B ′ M i − ∂y∂x µ ∂y∂x ′ ρ h (2 − y ) B ′′ − ( D − B ′ M i . (71)Each tensor component of the temporal photon contribution requires aseparate treatment. The case of µ = 0 = ρ gives, ∂ ∂ ′ i ∆ u ( x ; x ′ ) M −→ −∇ b ( y ) M = −∇ y b ′ M − ∂ i y ∂ i y b ′′ M (72)= aa ′ H M (cid:26) − D − b ′ + 4 h − y − aa ′ − a ′ a i b ′′ (cid:27) , (73)= aa ′ H M (cid:26)h − (2 − y ) + 2 (cid:16) aa ′ + a ′ a (cid:17)ih − (4 y − y ) b ′′ − ( D − − y ) b ′ i + h − y + y − − y ) (cid:16) aa ′ + a ′ a (cid:17)ih (2 − y ) b ′′ − ( D − b ′ i(cid:27) , (74)= − ∂ y∂x ∂x ′ h (4 y − y ) ∂∂y + ( D − − y ) i b ′ M + ∂y∂x ∂y∂x ′ h (2 − y ) ∂∂y − ( D − i b ′ M . (75)For µ = 0 and ρ = r we have, ∂ ∂ ′ r i ∆ u ( x ; x ′ ) M −→ ∂ r D ′ b ( y ) M = ∂ r D ′ y b ′ M + ∂ r y ∂ ′ y b ′′ M (76)12 aa ′ H ∆ x r M (cid:26) D − b ′ − − y ) b ′′ + 4 aa ′ b ′′ (cid:27) , (77)= a a ′ H ∆ x r M (cid:26)h (4 y − y ) b ′′ + ( D − − y ) b ′ i + h − y − a ′ a ih (2 − y ) b ′′ − ( D − b ′ i(cid:27) , (78)= − ∂ y∂x ∂x ′ r h (4 y − y ) ∂∂y + ( D − − y ) i b ′ M + ∂y∂x ∂y∂x ′ r h (2 − y ) ∂∂y − ( D − i b ′ M . (79)And the result for µ = m and ρ = 0 is, ∂ m ∂ ′ i ∆ u ( x ; x ′ ) M −→ − ∂ m D b ( y ) M = − ∂ m D y b ′ M − ∂ m y ∂ y b ′′ M (80)= − a a ′ H ∆ x m M (cid:26) D − b ′ − − y ) b ′′ + 4 a ′ a b ′′ (cid:27) , (81)= − aa ′ H ∆ x m M (cid:26)h (4 y − y ) b ′′ + ( D − − y ) b ′ i + h − y − aa ′ ih (2 − y ) b ′′ − ( D − b ′ i(cid:27) , (82)= − ∂ y∂x m ∂x ′ h (4 y − y ) ∂∂y + ( D − − y ) i b ′ M + ∂y∂x m ∂y∂x ′ h (2 − y ) ∂∂y − ( D − i b ′ M . (83)The case of µ = m and ρ = r requires the most intricate analysis. Itbegins with the observation, ∂ m ∂ r ∇ iδ D ( x − x ′ ) a D − + ∂ m ∂ ′ r i ∆ u ( x ; x ′ ) M −→ ∂ m ∂ r ∇ DD ′ b ( y ) M . (84)This component combines with the contribution from spatially transversephotons, Π mr i ∆ v ( x ; x ′ ) −→ (cid:16) δ mr − ∂ m ∂ r ∇ (cid:17) aa ′ b ( y ) . (85)The ∂ m ∂ r / ∇ terms from expressions (84) and (85) give, DD ′ b ( y ) − aa ′ M b ( y ) = aa ′ H (cid:26)h − y + y − − y ) (cid:16) aa ′ + a ′ a (cid:17)i b ′′ h − (2 D − − y ) + 2( D − (cid:16) aa ′ + a ′ a (cid:17)i b ′ + h ( D − − M H i b (cid:27) , (86)= aa ′ H (cid:26) − y ) b ′′ − D − − y ) b ′ + ( D − D − b +2 (cid:16) aa ′ + a ′ a (cid:17)h − (2 − y ) b ′′ + ( D − b ′ i(cid:27) , (87)= 12 ∇ I h − (2 − y ) b ′ + ( D − b i , (88)where I [ f ( y )] represents the indefinite integral of f ( y ) with respect to y .Substituting relation (88) in (84) and (85) gives, ∂ m ∂ r ∇ iδ D ( x − x ′ ) a D − + ∂ m ∂ ′ r i ∆ u ( x ; x ′ ) M + Π mr i ∆ v ( x ; x ′ ) −→ aa ′ δ mr b ( y ) + ∂ m ∂ r M I h − (2 − y ) b ′ + ( D − b i , (89)= aa ′ H M (cid:26) δ mr h (4 y − y ) b ′′ + ( D − − y ) b ′ i +2 aa ′ H ∆ x m ∆ x r h − (2 − y ) b ′′ + ( D − b ′ i(cid:27) , (90)= − ∂ y∂x m ∂x ′ r h (4 y − y ) ∂∂y + ( D − − y ) i b ′ M + ∂y∂x m ∂y∂x ′ r h (2 − y ) ∂∂y − ( D − i b ′ M . (91)This completes our demonstration that the de Sitter limit of our propagatoragrees with the direct calculation (68). It should also be noted that taking H → The results of the previous section are exact but they rely upon mode func-tions t ( η, k ), u ( η, k, M ) and v ( η, k, M ) for which no explicit solution is knownin a general cosmological geometry (3). The purpose of this section is to de-velop approximations for the amplitudes (norm-squares) of these mode func-tions. We begin converting all the dependent and independent variables to14imensionless form. Then approximations are developed for each of the threeamplitudes, checked against numerical evolution for the inflationary geom-etry of a simple quadratic potential which reproduces the scalar amplitudeand spectral index but gives too large a value for the tensor-to-scalar ratio.The section closes by demonstrating that our approximations remain validfor the plateau potentials which agree with current data. Time scales vary so much during cosmology that it is desirable to changethe independent variable from conformal time η to the number of e-foldingssince the start of inflation n , n ≡ ln h a ( η ) a i i = ⇒ ∂ = aH∂ n , ∂ = a H h ∂ n + (1 − ǫ ) ∂ n i . (92)We convert the wave number k and the mass M to dimensionless parametersusing factors of 8 πG , κ ≡ √ πG k , µ ≡ √ πG M . (93)And the dimensionless Hubble parameter, inflaton and classical potential are, χ ( n ) ≡ √ πG H ( η ) , ψ ( n ) ≡ √ πG ϕ ( η ) , U ( ψψ ∗ ) ≡ (8 πG ) V ( ϕϕ ∗ ) . (94)The first slow roll parameter is already dimensionless and we consider it tobe a function of n , ǫ ( n ) ≡ − χ ′ χ . (95)In terms of these dimensionless variables the nontrivial Einstein equationsare, 12 ( D − D − χ = χ ψ ′ ψ ′∗ + U ( ψψ ∗ ) , (96) −
12 ( D − (cid:16) D − − ǫ (cid:17) χ = χ ψ ′ ψ ′∗ − U ( ψψ ∗ ) . (97)The dimensionless inflaton evolution equation is, χ h ψ ′′ + ( D − − ǫ ) ψ ′ i + ψ U ′ ( ψψ ∗ ) = 0 . (98)15his can be expressed entirely in terms of ψ and its derivatives, ψ ′′ + (cid:16) D − − ψ ′ ψ ′∗ D − (cid:17)(cid:20) ψ ′ + ( D − U ′ ( ψψ ∗ ) ψ U ( ψψ ∗ ) (cid:21) = 0 . (99)Although our analytic approximations apply for any model of inflation,comparing them with exact numerical results of course requires an explicitmodel of inflation. It is simplest to carry out most of the analysis using aquadratic model with U ( ψ ) = c ψψ ∗ . Applying the slow roll approximationgives analytic expressions for the scalar, the dimensionless Hubble parameterand the first slow roll parameter, ψ ( n ) ≃ q ψ − n , χ ( n ) ≃ c √ q ψ − n , ǫ ( n ) ≃ ψ − n , (100)Note also that χ ( n ) ≃ χ p − n/ψ . By starting from ψ = 10 . c = 7 . × − makesthis model consistent with the observed values of the scalar spectral indexand the scalar amplitude [12], but the model’s tensor-to-scalar ratio is aboutthree times larger than the 95% confidence upper limit. Although we exploitthe simple slow roll results (100) of this phenomenologically excluded modelto develop approximations, the section closes with a demonstration that ouranalytic approximations continue to apply for viable models.We define the dimensionless MMC scalar amplitude, T ( n, κ ) ≡ ln h | t ( η, k ) | √ πG i . (101)Following the procedure of [22–24] we convert the mode equation and Wron-skian (10) into the nonlinear relation, T ′′ + 12 T ′ + ( D − − ǫ ) T ′ + 2 κ e − n χ − e − D − n − T χ = 0 . (102)The asymptotic relation (11) implies the initial conditions needed for equa-tion (102) to produce a unique solution, T (0 , κ ) = − ln(2 κ ) , T ′ (0 , κ ) = − ( D − . (103)The temporal photon and spatially transverse photon amplitudes are de-fined analogously, U ( n, κ, µ ) ≡ ln h | u ( η, k, M ) | √ πG i , V ( n, κ, µ ) ≡ ln h | v ( η, k, M ) | √ πG i . (104)16pplying the same procedure [22–24] to the temporal photon mode equationand Wronskian (24) gives, U ′′ + 12 U ′ + ( D − − ǫ ) U ′ + 2 κ e − n χ + 2( D − − ǫ ) + 2 µ χ − e − D − n − U χ = 0 . (105)And the initial conditions follow from (25), U (0 , κ, µ ) = − ln(2 κ ) , U ′ (0 , κ, µ ) = − ( D − . (106)The analogous transformation of the spatially transverse photon mode equa-tion and Wronskian (22) produces, V ′′ + 12 V ′ + ( D − − ǫ ) V ′ + 2 κ e − n χ + 2 µ χ − e − D − n − V χ = 0 . (107)The initial conditions associated with (23) are, V (0 , κ, µ ) = − ln(2 κ ) , V ′ (0 , κ, µ ) = − ( D − . (108) The MMCS amplitude is controlled by the relation between the physical wavenumber κe − n and the Hubble parameter χ ( n ). In the sub-horizon regime of κ > χ ( n ) e n the amplitude falls off roughly like T ( n, κ ) ≃ − ln(2 κ ) − ( D − n ,whereas it approaches a constant in the super-horizon regime of κ < χ ( n ) e n .(The e-folding of first horizon crossing is n κ such that κ = χ ( n κ ) e n κ .) Figure 1shows that both the sub-horizon regime, and also the initial phases of thesuper-horizon regime, are well described by the constant ǫ solution [24], T ( n, κ ) ≡ ln (cid:20) π z ( n, κ )2 κe ( D − n (cid:12)(cid:12)(cid:12) H (1) ν t ( n ) (cid:16) z ( n, κ ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) . (109)Here the ratio z ( n, κ ) and the MMCS index ν t ( n ) are, z ( n, κ ) ≡ κe − n [1 − ǫ ( n )] χ ( n ) , ν t ( n ) ≡ (cid:16) D − − ǫ ( n )1 − ǫ ( n ) (cid:17) . (110)17 umericalSolutionUVApproximation
10 20 30 40 50 n - - - - (cid:1) ( n, (cid:2) ) (cid:0) = χ (a) n κ ≃ . NumericalSolutionUVApproximation
10 20 30 40 50 n - - - ( n, κ ) κ = χ (b) n κ ≃ . NumericalSolutionUVApproximation
10 20 30 40 50 n - - - - ( n, κ ) κ = χ (c) n κ ≃ . Figure 1:
Plots the massless, minimally coupled scalar amplitude T ( n, κ ) (in solid green)and the (black dashed) ultraviolet approximation (109) versus the e-folding n for threedifferent values of κ . Of course expression (109) is an approximation to the exact result. Be-cause we propose to use this to compute the divergent coincidence limit ofthe propagator it is important to see how well T ( n, κ ) captures the ultravi-olet behavior of T ( n, κ ). Because (109) is exact for constant first slow rollparameter, the deviation must involve derivatives of ǫ ( n ). It turns out to falloff like κ − [24], T ( n, κ ) − T ( n, κ ) = (cid:16) D − (cid:17)h ( D + 5 − ǫ ) ǫ ′ + ǫ ′′ i(cid:16) χe n κ (cid:17) + O (cid:18)(cid:16) χe n κ (cid:17) (cid:19) . (111)We will see in section 4 that this suffices for an exact description of theultraviolet.The discrepancy between T ( n, κ ) and T ( n, κ ) that is evident at late timesin Figure 1 is due to evolution of the first slow roll parameter ǫ ( n ). Figure 2shows that the asymptotic late time phase is captured with great accuracyby the form, T ( n, κ ) = ln (cid:20) χ ( n κ )2 κ × C (cid:16) ǫ ( n κ ) (cid:17)(cid:21) , (112)where the nearly unit correction factor C ( ǫ ) is, C ( ǫ ) ≡ π Γ (cid:16)
12 + 11 − ǫ (cid:17) [2(1 − ǫ )] − ǫ . (113)18 umericalSolution LateTimeApproximation n - - - - - ( n, κ ) κ = χ (a) n κ ≃ . NumericalSolutionLateTimeApproximation
10 11 12 13 14 15 n - - - (cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) - (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19) - (cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25) - (cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) ( n, κ ) κ = χ (b) n κ ≃ . NumericalSolutionLateTimeApproximation
10 11 12 13 14 15 n - - - - - - - ( n, κ ) κ = χ (c) n κ ≃ . Figure 2:
Plots the massless, minimally coupled scalar amplitude T ( n, κ ) (in solid green)and the (black dashed) late time approximation (112) versus the e-folding n for threedifferent values of κ . Expression (112) is exact for constant ǫ ( n ). When the first slow roll parame-ter evolves there are very small nonlocal corrections whose form is known [25]but whose net contribution is negligible for smooth potentials. The temporal photon amplitude is very similar to the massive scalar whichwas the subject of a previous study [26]. Like that system, the functionalform of the amplitude is controlled by two key events:1. First horizon crossing at n κ such that κe − n κ = χ ( n κ ); and2. Mass domination at n µ such that µ = χ ( n µ ). The ultraviolet is well approximated by the form that applies for constant ǫ ( n ) and µ ∝ χ ( n ) [27], U ( n, κ, µ ) ≡ ln (cid:20) π z ( n, κ )2 κe ( D − n (cid:12)(cid:12)(cid:12) H (1) ν u ( n,µ ) (cid:16) z ( n, κ ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) , (114)where the temporal index is, ν u ( n, µ ) ≡ (cid:16) D − ǫ ( n )1 − ǫ ( n ) (cid:17) − µ [1 − ǫ ( n )] χ ( n ) . (115)Figure 3 shows that the ultraviolet approximation is excellent when matterdomination comes either before or after inflation. The quadratic slow roll approximation (100) gives n µ ≃ ψ [1 − (2 µ/χ ) ]. umericalSolution UVApproximation
10 20 30 40 50 n - - - ( n, κ , μ ) μ = χ (a) n µ < NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - - - - ( n, κ , μ ) μ = χ (b) n µ < NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - - ( n, κ , μ ) μ = χ (c) n µ > Figure 3:
Plots the temporal amplitude U ( n, κ, µ ) and the ultraviolet approximation(114) versus the e-folding n for κ = 3800 χ (with n κ ≃ .
3) and three different values of µ with outside the range of inflation. The ultraviolet regime is κe − n ≫ { χ ( n ) , µ } . To see how well the ultravi-olet approximation captures this regime we substitute the difference into theexact evolution equation (105) and expand in powers of e n χ ( n ) /κ to find [26], U ( n, κ, µ ) − U ( n, κ, µ ) = (cid:26)(cid:16) ǫ − ǫ (cid:17) µ χ + (cid:16) D − (cid:17)h ( D − ǫ ) ǫ ′ − ǫ ′′ i(cid:27)(cid:16) χe n κ (cid:17) + O (cid:18)(cid:16) χe n κ (cid:17) (cid:19) . (116)This is suffices to give an exact result for the ultraviolet so we that can takethe unregulated limit of D = 4 for the approximations which pertain for n > n κ .The various terms in equation (105) behave differently before and afterfirst horizon crossing. Evolution before first horizon crossing is controlled bythe 4th and 7th terms,2 κ e − n χ − e − D − n − U χ ≃ ⇒ U ≃ − ln(2 κ ) − ( D − n . (117)After first horizon crossing these terms rapidly redshift into insignificance.We can take the unregulated limit ( D = 4), and equation (105) becomes, U ′′ + 12 U ′ + (3 − ǫ ) U ′ + 4(1 − ǫ ) + 2 µ χ ≃ . (118)This is a nonlinear, first order equation for U ′ . Following [26] we make theansatz, U ′ ≃ α + β tanh( γ ) . (119)20ubstituting (119) in (118) gives, (cid:16) Eqn . (cid:17) = α ′ + 12 α + 12 β + (3 − ǫ ) α + 4(1 − ǫ ) + 2 µ χ + h (3 − ǫ + α ) β + β ′ i tanh( γ ) + β (cid:16) γ ′ − β (cid:17) sech ( γ ) . (120)Ansatz (119) does not quite solve (118), but the following choices reduce theresidue to terms of order ǫ × tanh( γ ), α = − , β = 14 + ǫ − µ χ , γ ′ = 12 β . (121)Figures 4 and 5 show how U ( n, κ, µ ) behaves when mass dominationcomes after first horizon crossing and before the end of inflation. NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - - - - ( n, κ , μ ) μ = χ NumericalSolution
PhaseII
A" n - - - - - ( n, κ , μ ) μ = χ NumericalSolution
PhaseIII :;<=>?@BCDEFG
30 35 40 45 50 n - HIJ - KLM - NO - ( n, κ , μ ) μ = χ Figure 4:
Plots the temporal amplitude U ( n, χ , . χ ) and the three approxima-tions: (114), (123) and (124). For κ = 3800 χ horizon crossing occurs at n κ ≃ .
3; for µ = 0 . χ mass domination occurs at n µ ≃ . First comes a phase of slow decline followed by a period of oscillations. From(119) with (121) we see that these phases are controlled by a “frequency”defined as, ω u ( n, µ ) ≡
14 + ǫ ( n )2 − µ χ ( n ) ≡ − Ω u ( n, µ ) . (122)During the phase of slow decline ω u ( n, µ ) >
0. Integrating (119) with (121)for this case gives, U ( n, κ, µ ) = U − n − n ) + 2 ln (cid:20) cosh (cid:16)Z nn dn ′ ω u ( n ′ , µ ) (cid:17) + (cid:16) U ′ ω u ( n , µ ) (cid:17) sinh (cid:16)Z nn dn ′ ω u ( n ′ , µ ) (cid:17)(cid:21) , (123)21here n ≡ n κ + 4. The oscillatory phase is characterized by ω u ( n, µ ) < U ( n, κ, µ ) = U − n − n ) + 2 ln (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) cos (cid:16)Z nn dn ′ Ω u ( n ′ , µ ) (cid:17) + (cid:16) U ′ u ( n , µ ) (cid:17) sin (cid:16)Z nn dn ′ Ω u ( n ′ , µ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (124)where n ≡ n µ + 4. Figures 4 and 5 show that these approximations areexcellent. NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - - - - ( n, κ , μ ) μ = χ NumericalSolution
PhaseII
PQRSTUVWXYZ[\]^ _‘
20 25 30 35 n - ab - cd - - - - - ( n, κ , μ ) μ = χ NumericalSolution
PhaseIII efghijklmnopq
30 35 40 45 50 n - rst - uv - - ( n, κ , μ ) μ = χ Figure 5:
Plots the temporal amplitude U ( n, χ , . χ ) and the three approxima-tions: (114), (123) and (124). For κ = 3800 χ horizon crossing occurs at n κ ≃ .
3; for µ = 0 . χ mass domination occurs at n µ ≃ . It is worth noting that the approximations (123) and (124) depend on κ principally through the integration constants U ≡ U ( n , κ, µ ) and U ≡U ( n , κ, µ ). Figure 6 shows the difference U ( n, χ , µ ) − U ( n, χ , µ )for the same two choices of µ in Figures 4 and 5. One can see that thedifference freezes into a constant after first horizon crossing to better thanfive significant figures!
10 20 30 40 50 n - - Δ ( n, μ ) μ = χ , κ = χ , κ = χ
10 20 30 40 50 n Δ ( n, μ ) μ = χ , κ = χ , κ = χ
35 40 45 50 n wxyz{|}~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:146)(cid:147)(cid:148)(cid:149)(cid:150)(cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156)(cid:157)(cid:158)(cid:159)(cid:160) Δ ( n, μ ) μ = χ , κ ¡ = χ , κ = χ Figure 6:
Plots the difference of the temporal amplitude ∆
U ≡ U ( n, κ , µ ) − U ( n, κ , µ )for κ = 400 χ and κ = 3800 χ with µ chosen so that all three approximations (114),(123) and (124) are necessary. .4 Spatially Transverse Photons The general considerations for the amplitude of spatially transverse photonsare similar to those for temporal photons. Before first horizon crossing it isthe 4th and last terms of equation (107) which control the evolution,2 κ e − n χ − e − D − n − V χ ≃ ⇒ V ≃ − ln(2 κ ) − ( D − n . (125)A more accurate approximation is, V ( n, κ, µ ) ≡ ln (cid:20) π z ( n, κ )2 κe ( D − n (cid:12)(cid:12)(cid:12) H (1) ν v ( n,µ ) (cid:16) z ( n, κ ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) , (126)where z ( n, κ ) is the same as (110) and the transverse index is, ν v ( n, µ ) ≡ (cid:16) D − − ǫ ( n )1 − ǫ ( n ) (cid:17) − µ [1 − ǫ ( n )] χ ( n ) . (127)Note the slight (order ǫ ) difference between ν u ( n, µ ) and ν v ( n, µ ). Figure 7shows that (126) is excellent up to several e-foldings after first horizon cross-ing, and throughout inflation for n µ < NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - ( n, κ , μ ) μ = χ (a) n µ < NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - ( n, κ , μ ) μ = χ (b) n µ < NumericalSolution
UVApproximation
10 20 30 40 50 n - - - ( n, κ , μ ) μ = χ (c) n µ > Figure 7:
Plots the transverse amplitude V ( n, κ, µ ) and the ultraviolet approximation(126) versus the e-folding n for κ = 3800 χ (with n κ ≃ .
3) and three different values of µ with outside the range of inflation. Expression (126) also models the ultraviolet to high precision, V ( n, κ, µ ) − V ( n, κ, µ ) = (cid:26)(cid:16) ǫ − ǫ (cid:17) µ χ + (cid:16) D − (cid:17)h ( D + 3 − ǫ ) ǫ ′ + ǫ ′′ i(cid:27)(cid:16) χe n κ (cid:17) + O (cid:18)(cid:16) χe n κ (cid:17) (cid:19) . (128)23igure 8 shows V ( n, κ, µ ) for the case where n µ happens after first horizoncrossing and before the end of inflation. One sees the same phases of slowdecline after first horizon crossing, followed by oscillations. NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - ( n, κ , μ ) μ = χ NumericalSolution
PhaseII ¢£⁄¥ƒ§¤'“«‹›fifl(cid:176) –† ‡· (cid:181)¶ •‚ n - - - - - „” ( n, κ , μ ) μ = χ NumericalSolution
PhaseIII »…‰(cid:190)¿(cid:192)`´ˆ˜¯˘˙
25 30 35 40 45 50 n - - - - ¨(cid:201) ( n, κ , μ ) μ = χ Figure 8:
Plots the transverse amplitude V ( n, χ , . χ ) and the three approxima-tions: (126), (131) and (132). For κ = 3800 χ horizon crossing occurs at n κ ≃ .
3; for µ = 0 . χ mass domination occurs at n µ ≃ . The second and third phases can be understood by noting that the two termsof expression (125) redshift into insignificance after first horizon crossing. Wecan also set D = 4 so that equation (107) degenerates to, V ′′ + 12 V ′ + (1 − ǫ ) V ′ + 2 µ χ ≃ . (129)The same ansatz (119) applies to this regime, with the parameter choices, α = − , β = 14 − µ χ ≡ ω v ≡ − Ω v , γ ′ = 12 β . (130)Just as there was an order ǫ difference between the temporal and transverseindices — expressions (110) and (127), respectively — so too there is an order ǫ difference between ω u ( n, µ ) and ω v ( n, κ ).Integrating (119) with (130) for ω v ( , µ ) > V ( n, κ, µ ) = V − ( n − n ) + 2 ln (cid:20) cosh (cid:16)Z nn dn ′ ω v ( n ′ , µ ) (cid:17) + (cid:16) V ′ ω v ( n , µ ) (cid:17) sinh (cid:16)Z nn dn ′ ω v ( n ′ , µ ) (cid:17)(cid:21) , (131)where n ≡ n κ + 4. Integrating (119) with (130) for ω v ( n, µ ) < V ( n, κ, µ ) = V − ( n − n ) + 2 ln (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) cos (cid:16)Z nn dn ′ Ω v ( n ′ , µ ) (cid:17) + (cid:16) V ′ v ( n , µ ) (cid:17) sin (cid:16)Z nn dn ′ Ω v ( n ′ , µ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (132)24here n ≡ n µ + 4. Figures 8 and 9 demonstrate that the (131) and (132)approximations are excellent. NumericalSolution
UVApproximation
10 20 30 40 50 n - - - - ( n, κ , μ ) μ = χ NumericalSolution
PhaseII ˚¸(cid:204)˝˛ˇ—(cid:209)(cid:210)(cid:211)(cid:212)(cid:213)(cid:214)(cid:215)(cid:216) (cid:217)(cid:218)
20 25 30 n - - - - - - (cid:219)(cid:220) ( n, κ , μ ) μ = χ NumericalSolution
PhaseIII (cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)ŁØ
35 40 45 50 n - - - - - Œº - (cid:236)(cid:237) - ( n, κ , μ ) μ = χ Figure 9:
Plots the transverse amplitude V ( n, χ , . χ ) and the three approxima-tions: (126), (131) and (132). For κ = 3800 χ horizon crossing occurs at n κ ≃ .
3; for µ = 0 . χ mass domination occurs at n µ ≃ . Finally, we note that from Figure 10 that V ′ ( n, κ, µ ) is nearly independentof κ after first horizon crossing.
10 20 30 40 50 n - Δ ( n, μ ) μ = χ , κ = χ , κ = χ
10 20 30 40 50 n Δ ( n, μ ) μ = χ , κ = χ , κ = χ
35 40 45 50 n (cid:238)(cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łøœß(cid:252)(cid:253)(cid:254)(cid:255)1(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16) Δ ( n, μ ) μ = χ , κ (cid:17) = χ , κ = χ Figure 10:
Plots the difference of the transverse amplitude ∆
V ≡ V ( n, κ , µ ) −V ( n, κ , µ )for κ = 400 χ and κ = 3800 χ with µ chosen so that all three approximations (126),(131) and (132) are necessary. One consequence for the (131) and (132) approximations is that only theintegration constants V and V depend on κ . We chose the quadratic dimensionless potential U ( ψψ ∗ ) = c ψψ ∗ for detailedstudies because it gives simple, analytic expressions (100) in the slow rollapproximation for the dimensionless Hubble parameter χ ( n ) and the firstslow roll parameter ǫ ( n ). Setting c ≃ . × − makes this model consistentwith the observed values for the scalar amplitude and the scalar spectralindex [12]. On the other hand, the model’s large prediction of r ≃ . R + R model [13]. Expressing the dimensionless potentialfor this model in our notation gives [28], U ( ψψ ∗ ) = 34 M (cid:16) − e − √ | ψ | (cid:17) , M = 1 . × − . (133)Somewhat over 50 e-foldings of inflation result if one starts from ψ = 4 . M = 1 . × − makes the model consistent with observation[12]. Figure 11 shows why r = 16 ǫ is so small for this model: its dimensionlessHubble parameter χ ( n ) is nearly constant. || (cid:18) || × - × - × - × - U (|| (cid:19) ||)
10 20 30 40 50 n1. × - × - × - × - × - × - (cid:21) χ ( n )
10 20 30 40 50 n0.0050.0100.015 ϵ ( n ) Figure 11:
Potential and geometry for the Einstein-frame representation of Starobinsky’soriginal model of inflation [13]. The left shows the dimensionless potential U ( ψψ ∗ ) (133);the middle plot gives the dimensionless Hubble parameter χ ( n ) and the right hand plotdepicts the first slow roll parameter ǫ ( n ). Inflation was assumed to start from ψ = 4 . Numerical Solution
UV Approximation
10 20 30 40 50 n - - ( n, κ ) κ = χ
10 20 30 40 50 n - - - - - - ω ( n ) μ= χ , (cid:26) = χ Figure 12:
The left hand plot shows the amplitude T ( n, κ ) of the massless, minimallycoupled scalar for κ = 3800 χ , which corresponds to n κ ≃ .
3. The right hand graphshows the frequency ω u ( n, µ ) ≃ ω v ( n, µ ) for µ = 0 . χ which passes through zero at n µ ≃ χ ( n )being so nearly constant is to increase the range over which the ultravioletapproximations pertain. The left hand plot of Figure 12 shows this for theMMCS amplitude T ( n, κ ). Because ǫ ( n ) is so small, the temporal and trans-verse frequencies are nearly equal ω u ( n, µ ) ≃ ω v ( n, µ ) and nearly constant.The right hand plot of Figure 12 shows this for a carefully chosen value of µ = 0 . χ which causes mass domination to occur during inflation. Forthis case we can just see the second and third phases occur in Figure 13. Numerical Solution
UV Approximation
10 20 30 40 50 n - - - - - - - ( n, κ , μ ) μ = (cid:27)(cid:28)(cid:29)(cid:30)(cid:31) χ , κ = χ Numerical Solution
UV Approximation
10 20 30 40 50 n - - - ( n, κ , μ ) μ = !" χ , κ = χ Figure 13:
Plots of the temporal amplitude U ( n, κ, µ ) (left) and the spatially trans-verse amplitude V ( n, κ, µ ) (right) versus n for the Starobinsky potential (133). For eachamplitude κ = 3800 χ (which implies n κ ≃ .
3) and µ = 0 . χ (which implies n µ ≃ The purpose of this section is to evaluate the one photon loop contributionto the inflaton effective potential defined by equation (6). We begin byderiving some exact results for the trace of the coincident propagator, andwe recall that T ( n, κ ) can be obtained from U ( n, κ, ǫ ( n ). The section closeswith a discussion of the nonlocal part of the effective potential which derivesfrom the late time approximations (123), (124), (131) and (132).27 .1 Trace of the Coincident Photon Propagator At coincidence the mixed time-space components of the photon mode sumvanish, and factors of b k m b k n average to δ mn / ( D − i h µ ∆ ν i ( x ; x ) = Z d D − k (2 π ) D − (cid:26) M (cid:18) k uu ∗ δ mn D − D u D u ∗ (cid:19) − M (cid:18) ∂ t∂ t ∗ δ mn D − k tt ∗ (cid:19) + (cid:18) D − D − ) δ mn vv ∗ (cid:19)(cid:27) . (134)Its trace is, g µν i h µ ∆ ν i ( x ; x ) = Z d D − k (2 π ) D − (cid:26) D u D u ∗ − k uu ∗ + ∂ t∂ t ∗ − k tt ∗ a M + ( D − vv ∗ a (cid:27) . (135)Relation (55) allows us to replace the MMCS mode function t ( η, k ) with themassless limit of the temporal mode function u ( η, k ) ≡ u ( η, k, ∂ t∂ t ∗ = k u u ∗ , k tt ∗ = D u D u ∗ . (136)Substituting (136) in (135) gives, g µν i h µ ∆ ν i ( x ; x ) = Z d D − k (2 π ) D − (cid:26) D u D u ∗ −D u D u ∗ − k ( uu ∗ − u u ∗ ) a M + ( D − vv ∗ a (cid:27) . (137)This second form (137) is very important because it demonstrates the absenceof any 1 /M pole as an exact relation, before any approximations are made.The mode equation for temporal photons implies, D u D u ∗ = a H h u ′ u ′∗ + ( D − uu ∗ ) ′ + ( D − uu ∗ i , (138)= ( k + a M ) uu ∗ + a H (cid:16) ∂ n + D − − ǫ (cid:17)(cid:16) ∂ n +2 D − (cid:17) ( uu ∗ ) . (139)Using relations (139) and (137) allows us to express the trace of the coincidentphoton propagator in terms of three coincident scalar propagators, g µν i h µ ∆ ν i ( x ; x ) = i ∆ u ( x ; x ) + ( D − a i ∆ v ( x ; x )+ H M (cid:16) ∂ n + D − − ǫ (cid:17)(cid:16) ∂ n +2 D − (cid:17)h i ∆ u ( x ; x ) − i ∆ u ( x ; x ) i . (140)28he disappearance of any factors of k from the Fourier mode sums in (140),coupled with the ultraviolet expansions (116) and (128), means that the phase1 approximations U ( n, κ, µ ) and V ( n, κ, µ ) exactly reproduce the ultravioletdivergence structures.Two of the scalar propagators in expression (140) are, i ∆ u ( x ; x ′ ) ≡ Z d D − k (2 π ) D − (cid:26) θ (∆ η ) u ( η, k, M ) u ∗ ( η ′ , k, M ) e i~k · ∆ ~x + θ ( − ∆ η ) u ∗ ( η, k, M ) u ( η ′ , k, M ) e − i~k · ∆ ~x (cid:27) , (141) i ∆ v ( x ; x ′ ) ≡ Z d D − k (2 π ) D − (cid:26) θ (∆ η ) v ( η, k, M ) v ∗ ( η ′ , k, M ) e i~k · ∆ ~x + θ ( − ∆ η ) v ∗ ( η, k, M ) v ( η ′ , k, M ) e − i~k · ∆ ~x (cid:27) . (142)The third scalar propagator i ∆ u ( x ; x ′ ) is just the M → i ∆ u ( x ; x ′ ).The coincidence limits of each propagator can be expressed in terms of thecorresponding amplitude, i ∆ u ( x ; x ) √ πG = Z d D − k (2 π ) D − e U ( n,κ,µ ) , i ∆ v ( x ; x ) √ πG = Z d D − k (2 π ) D − e V ( n,κ,µ ) . (143)Expression(140) is exact but not immediately useful because we lack ex-plicit expressions for the coincident propagators (143). It is at this stage thatwe must resort to the analytic approximations developed in section 3. Recallthat the phase 1 approximation is valid until roughly 4 e-foldings after hori-zon crossing. If one instead thinks of this as a condition on the dimensionlesswave number κ ≡ √ πG k at fixed n , it means that κ > κ n − , where we de-fine κ n as the dimensionless wave number which experiences horizon crossingat e-folding n . Taking as an example the temporal photon contribution wecan write, e U ( n,κ,µ ) ≃ θ (cid:16) κ − κ n − (cid:17) e U ( n,κ,µ ) + θ (cid:16) κ n − − κ (cid:17) e U , ( n,κ,µ ) , (144)= e U ( n,κ,µ ) + θ (cid:16) κ n − − κ (cid:17)(cid:20) e U , ( n,κ,µ ) − e U ( n,κ,µ ) (cid:21) . (145)Substituting the approximation (145) into expression (143) allows us to write, i ∆ u ( x ; x ) ≃ L u ( n ) + N u ( n ) , (146)29here we define the local ( L ) and nonlocal ( N ) contributions as, L u ( n ) ≡ √ πG Z d D − k (2 π ) D − e U ( n,κ,µ ) , (147) N u ( n ) ≡ √ πG Z d k (2 π ) θ (cid:16) κ n − − κ (cid:17)(cid:20) e U , ( n,κ,µ ) − e U ( n,κ,µ ) (cid:21) . (148)Note that we have taken the unregulated limit ( D = 4) in expression (148)because it is ultraviolet finite. The same considerations apply as well forthe coincident spatially transverse photon propagator i ∆ v ( x ; x ′ ), and for themassless limit of the temporal photon propagator i ∆ u ( x ; x ). The local contribution for each of the coincident propagators (143) comesfrom using the phase 1 approximation (147). For the temporal modes theamplitude is approximated by expression (114), whereupon we change vari-ables to z using k = (1 − ǫ ) Haz , and then employ integral 6 .
574 L u ( n ) = [(1 − ǫ ) H ] D − (4 π ) D × Γ( D − + ν u )Γ( D − − ν u )Γ( + ν u )Γ( − ν u ) × Γ (cid:16) − D (cid:17) . (149)Recall that the index ν u ( n, µ ) is defined in expression (115). Of course themassless limit is, L u ( n ) = [(1 − ǫ ) H ] D − (4 π ) D × Γ( D − + ν u )Γ( D − − ν u )Γ( + ν u )Γ( − ν u ) × Γ (cid:16) − D (cid:17) , (150)where the index is, ν u ( n ) ≡ ν u ( n,
0) = 12 (cid:16) D − ǫ ( n )1 − ǫ ( n ) (cid:17) . (151)The phase 1 approximation (126) for the transverse amplitude contains twoextra scale factors which serve to exactly cancel the inverse scale factorsthat are evident in the transverse contribution to the trace of the coincidentphoton propagator (140). Hence we have, L v ( n ) a = [(1 − ǫ ) H ] D − (4 π ) D × Γ( D − + ν v )Γ( D − − ν v )Γ( + ν v )Γ( − ν v ) × Γ (cid:16) − D (cid:17) , (152)30here the transverse index ν v ( n, µ ) is given in (127).Each of the local contributions (149), (150) and (152) is proportional tothe same divergent Gamma function,Γ (cid:16) − D (cid:17) = 2 D − O (cid:16) ( D − (cid:17) . (153)Each also contains a similar ratio of Gamma functions,Γ( D − + ν )Γ( D − − ν )Γ( + ν )Γ( − ν ) = h(cid:16) D − (cid:17) − ν i × Γ( D − + ν )Γ( D − − ν )Γ( + ν )Γ( − ν ) , (154)= h(cid:16) D − (cid:17) − ν i(cid:26) h ψ (cid:16)
12 + ν (cid:17) + ψ (cid:16) − ν (cid:17)i(cid:16) D − (cid:17) + O (cid:16) ( D − (cid:17)(cid:27) . (155)These considerations allow us to break up each of the three terms in (140) intoa potentially divergent part plus a manifestly finite part. For i ∆ u ( x ; x ) → L u ( n ) this decomposition is, L u = [(1 − ǫ ) H ] D − (4 π ) D (cid:20) M − ( D − H (cid:16) ( D − ǫ −
12 ( D − ǫ (cid:17)(cid:21) Γ (cid:16) − D (cid:17) + 116 π h M − ǫH ih ψ (cid:16)
12 + ν u (cid:17) + ψ (cid:16) − ν u (cid:17)i + O ( D − . (156)For ( D − i ∆ v ( x ; x ) → ( D − L v ( n ) we have,( D − L v = [(1 − ǫ ) H ] D − (4 π ) D (cid:20) ( D − M − ( D − D − H (cid:16) ( D − ǫ − ( D − ǫ (cid:17)(cid:21) Γ (cid:16) − D (cid:17) + 2 M π h ψ (cid:16)
12 + ν v (cid:17) + ψ (cid:16) − ν v (cid:17)i + O ( D − . (157)And the final term in (140) — the one with derivatives — becomes, H M (cid:16) ∂ n + D − − ǫ (cid:17)(cid:16) ∂ n +2 D − (cid:17)h L u − L u i = H (cid:16) ∂ n + D − − ǫ (cid:17) × (cid:16) ∂ n +2 D − (cid:17) [(1 − ǫ ) H ] D − (4 π ) D Γ (cid:16) − D (cid:17) + H π (cid:16) ∂ n +3 − ǫ (cid:17)(cid:16) ∂ n +4 (cid:17) × (cid:26) ψ (cid:16)
12 + ν u (cid:17) + ψ (cid:16) − ν u (cid:17) − ǫH M h ψ (cid:16)
12 + ν u (cid:17) − ψ (cid:16)
12 + ν u (cid:17) + ψ (cid:16) − ν u (cid:17) − ψ (cid:16) − ν u (cid:17)i(cid:27) + O ( D − . (158)31ote that the difference ψ ( ± ν u ) − ψ ( ± ν u ) is of order M so expression(158) has no 1 /M pole. Note also that the 1 /ǫ pole in ψ ( − ν u ) = ψ ( − ǫ − ǫ )is canceled by an explicit multiplicative factor of ǫ .The potentially divergent terms (the ones proportional to Γ(1 − D )) inexpressions (156), (157 and (158) sum to give,(156) div + (157) div + (158) div = [(1 − ǫ ) H ] D − (4 π ) D h ( D − M + 12 R i Γ (cid:16) − D (cid:17) + H π h − ǫ + 4 ǫ − ǫ ′ − (6 ǫ ′ + ǫ ′′ )1 − ǫ − (cid:16) ǫ ′ − ǫ (cid:17) i + O ( D − , (159)where we recall that the D -dimensional Ricci scalar is R = ( D − D − ǫ ) H .Comparison with expression (6) for ∆ V ′ ( ϕϕ ∗ ) reveals that we can absorb thedivergences with the following counterterms, δξ = − Γ(1 − D ) s D − (4 π ) D × q , δλ = − Γ(1 − D ) s D − (4 π ) D × D − q , (160)where s is the renormalization scale. Up to finite renormalizations, thesechoices agree with previous results [5–7], in the same gauge and using thesame regularization, on de Sitter background.Substituting expressions (156), (157), (158) and (160) into the definition(6) of ∆ V ′ ( ϕϕ ∗ ) and taking the unregulated limit gives the local contribution,∆ V ′ L ( ϕϕ ∗ ) = q H π (cid:26) (6 M + R )2 H ln h (1 − ǫ ) H s i +3 − ǫ +4 ǫ − ǫ ′ − (6 ǫ ′ + ǫ ′′ )1 − ǫ − (cid:16) ǫ ′ − ǫ (cid:17) + M H (cid:20) ψ (cid:16)
12 + ν u (cid:17) + ψ (cid:16) − ν u (cid:17) + 2 ψ (cid:16)
12 + ν v (cid:17) + 2 ψ (cid:16) − ν v (cid:17)(cid:21) + 12 h ( ∂ n +3 − ǫ )( ∂ n +4) − ǫ ih ψ (cid:16)
12 + ν u (cid:17) + ψ (cid:16) − ν u (cid:17)i − ( ∂ n +3 − ǫ )( ∂ n +4) × ǫH M h ψ (cid:16)
12 + ν u (cid:17) − ψ (cid:16) − ǫ (cid:17) + ψ (cid:16) − ν u (cid:17) − ψ (cid:16) − ǫ − ǫ (cid:17)i(cid:27) . (161)It is worth noting that there are no singularities at ǫ = 1, or when either1 / (1 − ǫ ) or − ǫ/ (1 − ǫ ) become non-positive integers [14]. The effectivepotential is obtained by integrating (161) with respect to ϕϕ ∗ . The result isbest expressed using the variable z ≡ q ϕϕ ∗ /H ,∆ V L = H π (cid:26)h z + Rz H i ln h (1 − ǫ ) H s i + h − ǫ +4 ǫ − ǫ ′ − (6 ǫ ′ + ǫ ′′ )1 − ǫ i z ǫ ′ z (1 − ǫ ) + 2 Z z dx x (cid:20) ψ (cid:16)
12 + α (cid:17) + ψ (cid:16) − α (cid:17) +2 ψ (cid:16)
12 + β (cid:17) +2 ψ (cid:16) − β (cid:17)(cid:21) + 12 h ( ∂ n +3 − ǫ )( ∂ n +4 − ǫ ) − ǫ iZ z dx h ψ (cid:16)
12 + α ( x ) (cid:17) + ψ (cid:16) − α ( x ) (cid:17)i − ( ∂ n +3 − ǫ )( ∂ n +4 − ǫ ) Z z dxǫ x h ψ (cid:16)
12 + α ( x ) (cid:17) − ψ (cid:16) − ǫ (cid:17) + ψ (cid:16) − α ( x ) (cid:17) − ψ (cid:16) − ǫ − ǫ (cid:17)i(cid:27) , (162)where the x -dependent indices are, α ( x ) ≡ s
14 + ǫ − x (1 − ǫ ) , β ( x ) ≡ s − x (1 − ǫ ) . (163)Note that the term inside the square brackets on the last line of (162) vanishesfor x = 0, so the integrand is well defined at x = 0. Expression (162) depends principally on the quantity z = q ϕϕ ∗ /H . Duringinflation z is typically quite large, whereas it touches 0 after the end ofinflation. Figure 14 shows this for the quadratic potential, and the resultsare similar for the Starobinsky potential (133). It is therefore desirable toexpand the potential ∆ V L ( ϕϕ ∗ ) for large z and for small z . n | ψ ( n )| n × × × × z ( n ) Figure 14:
Plots of the dimensionless inflaton field ψ ( n ) and the ratio z ≡ q ψ /χ afterthe end of inflation for the quadratic potential. Here we chose q = . The large field regime follows from the large argument expansion of thedigamma function, ψ ( x ) = ln( x ) − x − x + 1120 x − x + O (cid:16) x (cid:17) . (164)33ubstituting (164) in (162), and performing the various integrals gives,∆ V L = H π (cid:26) z ln (cid:16) q ϕϕ ∗ s (cid:17) − z + Rz H ln (cid:16) q ϕϕ ∗ s (cid:17) − (4+8 ǫ − ǫ ) z − ǫ ′ z − h ǫ (1 − ǫ )(2 − ǫ ) + 78 (1 − ǫ ) ǫ ′ + 18 ǫ ′′ i ln (2 z ) + O (cid:16) ln( z ) (cid:17)(cid:27) . (165)The leading contribution of (165) agrees with the famous flat space result ofColeman and Weinberg [2],∆ V −→ q ϕϕ ∗ ) π ln (cid:16) q ϕϕ ∗ s (cid:17) . (166)The first three terms of (165) could be subtracted using allowed countertermsof the form F ( ϕϕ ∗ , R ) [9]. A prominent feature of the remaining terms is thepresence of derivatives of the first slow roll parameter. These derivatives aretypically very small during inflation but Figure 15 shows that they can bequite large after the end of inflation. HIJK LMNO PQRS TUVW n ϵ ( n ) XYZ[ \]^_ ‘abc defg n - - - ϵ h ( n ) ijkl mnop qrst uvwx n - - ϵ y{ ( n ) Figure 15:
Plots of the first slow roll and its derivatives after the end of inflation for thequadratic potential.
The small field expansion derives from expanding the digamma functionsin expression (162) in powers of x , ψ (cid:16)
12 + α ( x ) (cid:17) = ψ (cid:16) − ǫ (cid:17) − ψ ′ (cid:16) − ǫ (cid:17) x − ǫ + O ( x ) , (167) ψ (cid:16) − α ( x ) (cid:17) = ψ (cid:16) − ǫ − ǫ (cid:17) + ψ ′ (cid:16) − ǫ − ǫ (cid:17) x − ǫ + O ( x ) , (168) ψ (cid:16)
12 + β ( x ) (cid:17) = − γ − π x (1 − ǫ ) + O ( x ) , (169) ψ (cid:16) − β ( x ) (cid:17) = − (1 − ǫ ) x + 1 − γ + h π i x (1 − ǫ ) + O ( x ) . (170)34he result is,∆ V L = H π (cid:26)(cid:20) R H ln h (1 − ǫ ) H s i +1 − ǫ +2 ǫ − ǫ ′ − (6 ǫ ′ + ǫ ′′ )1 − ǫ − ǫ ′ (1 − ǫ ) (cid:21) z + 12 h ( ∂ n +3 − ǫ )( ∂ n +4 − ǫ ) − ǫ ih ψ (cid:16) − ǫ (cid:17) + ψ (cid:16) − ǫ − ǫ (cid:17)i z + 12 ( ∂ n +3 − ǫ )( ∂ n +4 − ǫ ) h ψ ′ (cid:16) − ǫ (cid:17) − ψ ′ (cid:16) − ǫ − ǫ (cid:17)i ǫz − ǫ + O ( z ) (cid:27) . (171)Note that the 1 /ǫ pole from ψ ( − ǫ − ǫ ) on the penultimate line of (171) cancelsagainst the double pole from ψ ′ ( − ǫ − ǫ ) on the last line. The nonlocal contribution to the effective potential is obtained by substitut-ing the nonlocal contribution (148) to each coincident propagator in (140),and then into expression (6),∆ V ′ N ( ϕϕ ∗ ) = q N u ( n ) + 2 q e − n N v ( n )+ q H M (cid:16) ∂ n +3 − ǫ (cid:17)(cid:16) ∂ n +4 (cid:17)h N u ( n ) − N u ( n ) i . (172)The nonlocal contributions to the various propagators are, N u ( n ) = Z κ n − dκ κ π G (cid:20) e U , ( n,κ,µ ) − e U ( n,κ,µ ) (cid:21) , (173) N u ( n ) = Z κ n − dκ κ π G (cid:20) e U ( n,κ, − e U ( n,κ, (cid:21) , (174) N v ( n ) = Z κ n − dκ κ π G (cid:20) e V , ( n,κ,µ ) − e V ( n,κ,µ ) (cid:21) . (175)The nonlocal nature of these contributions derives from the integration over κ , which can be converted to an integration over n κ , κ ≡ e n κ χ ( n κ ) = ⇒ dκκ = h − ǫ ( n κ ) i dn κ . (176)After this is done, any factors of κ depend on the earlier geometry.A number of approximations result in huge simplification. First, notefrom Figures 4 and 5 that the ultraviolet approximation (114) for U ( n, κ, µ )35s typically more negative than the late time approximations (123) and (124).Figures 8 and 9 show that the same rule applies to V ( n, κ, µ ). Hence we canwrite, N u ( n ) ≃ Z κ n − dκ κ π G e U , ( n,κ,µ ) , N v ( n ) ≃ Z κ n − dκ κ π G e V , ( n,κ,µ ) . (177)Second, because the temporal and transverse frequencies are nearly equal,we can write, ω u ( n, µ ) ≃ ω v ( n, µ ) = ⇒ U ( n, κ, µ ) ≃ V ( n, κ, µ ) − n . (178)When the mass vanishes there is so little difference between the ultravioletapproximation (114) and its late time extension (123) that we can ignore thiscontribution, N u ( n ) ≃
0. Next, Figures 6 and 10 imply that the late timeapproximations for U ( n, κ, µ ) and V ( n, κ, µ ) inherit their κ dependence fromthe ultraviolet approximation at n ≃ n κ + 4, which is itself independent of µ , n > n κ + 4 = ⇒ U , ( n, κ, µ ) ≃ U ( n κ +4 , κ,
0) + f , ( n, µ ) , (179)where f , ( n, µ ) can be read off from expressions (123) and (124) by omittingthe κ -dependent integration constants. Finally, we can use the slow roll form(112) for the amplitude reached after first horizon crossing and before themass dominates, e U ( n κ +4 ,κ, ≃ χ ( n κ )2 κ × C (cid:16) ǫ ( n κ ) (cid:17) . (180)Putting it all together gives,∆ V N ( ϕϕ ∗ ) ≃ q Z n − dn κ [1 − ǫ ( n κ )] χ ( n κ ) C ( n κ ))32 π G × e f , ( n,µ ) + q χ ( n )2 µ ( ∂ n +3 − ǫ )( ∂ n +4) Z n − dn κ [1 − ǫ ( n κ )] χ ( n κ ) C ( ǫ ( n κ ))32 π G × e f , ( n,µ ) . (181) In section 2 we derived an exact, dimensionally regulated, Fourier mode sum(50) for the Lorentz gauge propagator of a massive photon on an arbitrary36osmological background (3). Our result is expressed in terms of mode func-tions t ( η, k ), u ( η, k, M ) and v ( η, k, M ) whose defining relations are (10), (24)and (22), which respectively represent massless minimally coupled scalars,massive temporal photons, and massive spatially transverse photons. Thephoton propagator can also be expressed as a sum (51) of bi-vector differ-ential operators acting on the scalar propagators i ∆ t ( x ; x ′ ), i ∆ u ( x ; x ′ ) and i ∆ v ( x ; x ′ ) associated with the three mode functions. Because Lorentz gaugeis an exact gauge there should be no linearization instability, even on deSitter, such as occurs for Feynman gauge [30, 31].In section 3 we converted to a dimensionless form with time representedby the number of e-foldings n since the beginning of inflation, and the wavenumber, mass and Hubble parameter all expressed in reduced Planck units, κ ≡ √ πG k , µ ≡ √ πG M and χ ( n ) ≡ √ πG H ( η ). Analytic approxi-mations were derived for the amplitudes T ( n, κ ), U ( n, κ, µ ) and V ( n, κ, µ )associated with each of the mode functions. Which approximation to use iscontrolled by first horizon crossing at κ = e n κ χ ( n κ ) and mass dominationat µ = χ ( n µ ). Until shortly after first horizon crossing we employ the ul-traviolet approximations (109), (114) and (126). After first horizon crossingand before mass domination the appropriate approximations are (112), (123)and (131). And after mass domination (which T ( n, κ ) never experiences) theamplitudes are well approximated by (124) and (132). The validity of theseapproximations was checked against explicit numerical solutions for inflationdriven by the simple quadratic model, and by the phenomenologically favoredplateau model (133).In section 4 we applied our approximations to compute the effective po-tential induced by photons coupled to a charged inflaton. Our result consistsof a part (162) which depends locally on the geometry (3) and a numericallysmaller part (181) which depends on the past history. The local part wasexpanded both for the case of large field strength (165), and for small fieldstrength (171). The existence of the second, nonlocal contribution, was con-jectured on the basis of indirect arguments [5] that have now been explicitlyconfirmed. Another conjecture that has been confirmed is the rough validityof extrapolating de Sitter results [1, 6] from the constant Hubble parameterof de Sitter background to the time dependent one of a general cosmolog-ical background (3). However, we now have good approximations for thedependence on the first slow roll parameter ǫ ( n ).Our most important result is probably the fact that electromagnetic cor-rections to the effective potential depend upon first and second derivatives of37he first slow roll parameter. One consequence is that the effective potentialfrom electromagnetism responds more strongly to changes in the geometrythan for scalars [26] or spin one half fermions [32]. This can be very impor-tant during reheating (see Figure 15); it might also be significant if featuresoccur during inflation. Another consequence is that there cannot be perfectcancellation between the positive effective potentials induced by bosons andthe negative potentials induced by fermions [10]. Note that the derivativesof ǫ come exclusively from the constrained part of the photon propagator —the t ( η, k ) and u ( η, k ) modes — which is responsible for long range electro-magnetic interactions. Dynamical photons — the v ( η, k ) modes — produceno derivatives at all. These statements can be seen from expression (140),which is exact, independent of any approximation.We close with a speculation based on the correlation between the spin ofthe field and the number of derivatives it induces in the effective potential:scalars produce no derivatives [26], spin one half fermions induce one deriva-tive [32], and this paper has shown that spin one vectors give two derivatives.It would be interesting to see if the progression continues for gravitinos (whichought to induce three derivatives) and gravitons (which would induce fourderivatives). Of course gravitons do not acquire a mass through coupling to ascalar inflaton, but they do respond to it, and the mode equations have beenderived in a simple gauge [33, 34]. Until now it was not possible to do muchwith this system because it can only be solved exactly for the case of con-stant ǫ ( n ), however, we now have a reliable approximation scheme that canbe used for arbitrary ǫ ( n ). Further, we have a worthy object of study in thegraviton 1-point function, which defines how quantum 0-point fluctuationsback-react to change the classical geometry. At one loop order it consists ofthe same sort of coincident propagator we have studied in this paper. On deSitter background the result is just a constant times the de Sitter metric [35],which must be absorbed into a renormalization of the cosmological constantif “ H ” is to represent the true Hubble parameter. Now suppose that thegraviton propagator for general first slow roll parameter consists of a localpart with up to 4th derivatives of ǫ ( n ) plus a nonlocal part. That sort ofresult could not be absorbed into any counterterm. So perhaps there is oneloop back-reaction after all [36], and de Sitter represents a case of unstableequilibrium? 38 cknowledgements This work was partially supported by Taiwan MOST grants 108-2112-M-006-004 and 107-2119-M-006-014; by NSF grants PHY-1806218 and PHY-1912484; and by the Institute for Fundamental Theory at the University ofFlorida.
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