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Gravitational waves from fermion production Recent citations - Magnetogenesis from a rotating scalar: à la scalar chiral magnetic effect during axion inflation Kohei Kamada and Chang Sub Shin

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This content was downloaded from IP address 128.119.168.112 on 25/06/2020 at 14:12 JCAP10(2019)018 hysics P icle t b,c ar strop d [email protected], , https://doi.org/10.1088/1475-7516/2019/10/018 Marco Peloso, a [email protected] and Lorenzo Sorbo d [email protected] Lauren Pearce, , osmology and A and Cosmology a 1904.10483 inflation, primordial gravitational waves (theory), cosmological perturbation

We present analytic results for the gravitational wave power spectrum induced in , [email protected] rnal of rnal

ou An IOP and SISSA journal An IOP and 2019 IOP Publishing Ltd and Sissa Medialab INFN, Sezione di Padova, via Marzolo 8, I-35131 Padova,Amherst Italy Center for Fundamental Interactions, Department of Physics, University ofAmherst, Massachusetts, MA 01003, U.S.A. E-mail: [email protected] Department of Physics, University ofUrbana, Illinois Illinois at 61801, Urbana-Champaign, U.S.A. Dipartimento di Fisica evia Astronomia Marzolo G. 8, Galilei, I-35131, Universit`adegli Studi Padova, di Italy Padova, b c d a c models where the inflatonsuch is a coupled coupling to createsing a helically gravitational fermionic polarized waves pseudocurrent. fermions, is WeWe the parametrically also show polarized suppressed show that component with that although of the respecttion amplitude the to cannot of result- the exceed the non-polarized thatfluctuations. gravitational one. wave generated signal by associated the to standard this produc- mechanism of amplification of vacuum Peter Adshead, Michael A. Roberts Gravitational waves from fermion production during axion inflation J Keywords: ArXiv ePrint: Abstract. theory, axions Received May 17, 2019 Revised September 6, 2019 Accepted September 25, 2019 Published October 7, 2019 JCAP10(2019)018 3 9 4 6 5 8 8 3 11 10 11 13 16 17 22 14 −r/8, as = t n 2]. As well as resolving -mode polarization of the B – 1 – 0.06 when the consistency relation . r 0.001 [7]. ∼ r σ 4]. The inflationary scenario also generically predicts primordial gravita- 6], which tightens to 3.2.2 Cubic loop 3.2.1 Quartic loops Slow-roll inflation requires a flat potential; in axion (or natural) inflation this flatness 0.09 [5, 2.3 Constraints 2.4 Explicit form of the fermion action, and fermion-GW3.1 interactions Quantization 3.2 Fermion loop-corrections to the gravitational wave power spectrum 3.3 Scaling of our result 2.2 The action in ADM form 2.1 Starting action . 4 Conclusion A Computation of the fermion-gravitational waveB interactions Interaction Hamiltonian C Details of the quartic loopD computation Details of the cubic loop computation 3 Fermion contributions to the tensor power spectrum 1 Introduction There is strong observational evidence supportingthrough the a hypothesis rapid that period the early of universe accelerated went expansion dubbed inflation [1, the horizon, flatness, curvature,ple and gravitino/monopole explanation problems, for inflationfluctuations the provides observed a [3, sim- red-tilted, approximatelytional waves, Gaussian which and can be adiabaticcosmic measured microwave density or background. constrained through Currentr the measurements restrict the tensor-to-scalar ratio to is a natural consequencepossible solution of to an the approximatein strong string shift CP theory, problem symmetry and [9], monodromy [8].candidates, axions, [10–14] giving or or Originally alignment pseudoscalar, rise [15–18] motivated fields to effectsexperiments. as are vacuum can ubiquitous a primordial make them gravitational inflaton waves within the reach of future Contents 11 Introduction 2 Fermion-graviton interactions during axion inflation appropriate for vacuum fluctuations fromare slow-roll expected inflation, to is reach imposed. Future experiments JCAP10(2019)018 than (1.1) 1 39], and the are produced in H ≈ m must couple derivatively ϕ X, in reference [44]. Because fermion 5 [50]. As the fermion-axion coupling γ is the usual gauge-field field-strength µ H . n-point functions that are generically n/2 F
¯ i Xγ  2 ϕ/f ϕ f µ hδϕ ∂ ϕ/f + – 2 – ˜ F i ∼ regime the effect of this coupling leads to a fermion energy n F H , the energy density can be parametrically larger f ϕ hδϕ  = ˙ ϕ/f ϕ/f ∼ , and thus is negligible. , respectively. Here ∆L 4 k 49]). During axion inflation, however, this conclusion does not X H is a scale known as the axion decay constant. This coupling of the 22], magnetic field production–28], [23 large non-Gaussianity [29– f . As was noticed in reference [43] and confirmed in reference [44], the produced is its dual, and , which is too small to produce observable effects (with the possible exception of H ˜ F 4 H Given the rich phenomenology of the scalar perturbations found in reference [44], it Due to the approximate shift symmetry, the axion inflaton The fermionic coupling has not attracted as much attention, first being studied by one of We have previously studied the regime ˙  On the contrary, in the opposite ˙ , as first noticed in reference [43]. In reference [44] we also identified a regime in which 1 ∼ 4 ˙ ϕ/f fermions have a helicityhelicity asymmetry, asymmetry which raises was the possibility useda that for chiral the component. spectrum leptogenesis of in Sourced sourced production reference gravitational of waves [51]. has gravitational waves This in this context wasdensity previously that is much smaller than is important to characterize how fermions source gravitational waves in the regime where to matter fields. At mass-dimension five, the possible couplings are to gauge fields and fermions is increased, eventually one entersinflaton a zero-mode regime is of controlled strong byinflation particle backreaction studied production; where in this the is evolution reference the ofremains [20]. fermionic the small analogue We still expect of steep holds that the in above the argument regime that of non-Gaussianity strong backreaction. us in reference [43], and,cannot more undergo recently, in the references same [44–46].one exponential Due amplification hand, to as massless Pauli the blocking, fermionsthrough gauge fermions the are fields. expansion conformal of Furthermore, the and on Universeso the ]. [47 therefore that, in cannot On the the be absence other hand, of created very the gravitationally heavy coupling fermions (1.1), decouple, only fermions with mass tensor, axion to gauge fields leadsble to phenomenological exponentially consequences large (see gauge field [19]thermalized amplification, for inflation with a [21, several review). possi- These include steep inflationgeneration [ 20], of primordial black holes [40–42]. sizable quantities. With onlyis one scale in the problem, the Hubble scale, the energy density modes can be populated upH to the sourced contributionthe to non-Gaussianity the is power beneathcontrast spectrum current to dominates observational the bounds. thedifference analogous vacuum can effect This contribution, be in behavior understood yet one systems is to by with in populate noting strong a striking large thatrestricted bosonic number by phenomenologically of Pauli particle exclusion, interesting fermion production. their modes resultsthe sum and, require is central since This uncorrelated the limit and occupation becomes theorem.per of increasingly Gaussian these, Bosonic mode by modes systems, which is conversely, add allowrelated for coherently to the large leading two-point occupation to function numbers by sourced 31], chiral gravitational wave production32–37], [ instantaneous preheating [38, super-heavy dark matter [48, follow due to the presence of the additional scale ˙ JCAP10(2019)018 ) 2 Ψ (2.1) (2.2) (γ 3.3; we . of mass O  X X ) generates below). To 2  1 Ψ 5 γ (γ µ 0. It is therefore O ϕγ → µ ∂ m 1 f − becomes small. We review ) interactions lead to seven m 2 m Ψ − 2 µ (γ 3.2.1, while the D with a shift-symmetric coupling to O µ ϕ iγ 3, we use these interactions to compute  X, 2, we introduce our theory, and working ¯ f ϕ X 5 iγ 56]. The e 3.2.2. We discuss our results in section and Ψ respectively denote, schematically, the is derivatively coupled to a fermion (ϕ) + γ – 3 – V ϕ Ψ = − → ϕ ν X ϕ∂ µ ∂ is the reduced Planck mass. µν 2 g 8πG + ) vertices, where 2 √ R Ψ 2 2 Pl 2 0, whereas in this work we use the basis introduced in reference [44], in = 1/ (γ M O  Pl → M m −g √ and minimally coupled to gravity, so that our action, in mostly minus convention, x 4 X in reference [53]. However, the fermion basis used in that work was leading to d 2 Z = 1, and As discussed in reference [44] (in particular, see appendix B of that work for details), This paper is organized as follows. In section c = Gravitational wave production by non-chiral fermions has also been studied in reference [52]. 2 = S the use ofinteraction, these but fields it obscures makes the apparent fact the that shift-symmetric such interaction nature vanishes of as the inflaton-fermion a fermion reads convenient to redefine the fermion according to 2 Fermion-graviton interactions during axionThe inflation aim of thisin work a is model to where computem, the a as amplitude described pseudoscalar of in inflaton source the the gravitational tensor action (2.1) modes wave below. power sourcedpossible. by at At The fermions leading one-loop. order firstby in topology At two perturbation — one-loop, cubic theory, the order fermions diagrams cubicsecond vertices of consisting loop topology two of — — topologies one the isvertex quartic gravitational are a consisting wave loop two-vertex and of — diagram a is two thatfind fermion gravitational a the is bilinear. one-vertex waves required and generated diagram The interactions, generated a wefluctuations. by fermion therefore a bilinear need quartic-order to (see expand figure the full action to2.1 quartic order in Starting action We consider a theory containing a pseudoscalar inflaton studied pathologies as which perturbation theory remains valid as the fermion mass this basis while introducing our model in section2. in the Arnowitt-Deser-Misner (ADM)equations to [54] second formalism, order. weto We solve then fourth the use order these gravitational in solutions constraint to fluctuations. obtain the From interaction this Lagrangian interaction Lagrangian, we obtain one a two-vertex loop, which we evaluate in section vertex and seven one-loop one-vertex diagrams, which we evaluate in section show that the chirallycontribution asymmetric to contribution is the subdominant,our tensor-to-scalar and calculations that ratio can the is be~ total found beneath sourced in the the vacuum various component. appendices. We Details work of in natural units where tensor and the fermion degreeseight of loop freedom. diagrams In in section the in-in formalism, [55 JCAP10(2019)018 . i k Ψ δ  (2.4) (2.6) (2.7) (2.8) (2.3) (2.5) = ab Ψ Σ jk ,  3}. The Ψ, ac h a, so that # ab η  2, ij 5  Σ = γ h (ϕ) m ac , b  η N V j e  ∈ {1,  f N i − 2ϕ k b ij e ϕ  N h j m . m c 3}, and finally roman i i i, j, k sin ≡ e ϕ∂ ∇ i i 2, , ik K ∂ ) (3) im m N ij = − Γ + h dτ , and the spin connection is j 2 1 m   ∈ {1, (3) k N B + − f i 2ϕ ,K + b , γ 2 3}, capital roman letters are 4D e  NK are represented with primes. The j  2 ϕ A , so that k i a, b, c N  π γ 2, ∂ τ ij  2 N i cos − j c h 1, 4 1 )(dx k , e + b ∇ m F  e dτ =  0 i L − 2 1 (3) 2 . ∂ x Ψ N ∈ {0, +  K − 4 AB ) k d  j 0b + c − Ψ i e Σ N  i Z i AB ij – 4 – 1 2 µ, ν 5 b Σ e ∇ γ K (dx = + ik  ij ij (3) F 0b h K µAB S f Σ K − 2ϕ i ω − − b + 0 ij 1 2 k  e 2 ∂ h 3}. In this paper we will also use lower case roman letters
+ R   dτ ij sin µ 2, 2  is the vierbein. (3) i ∂ K k 1  1, N ν + j  2N is the shift. For the background metric we choose N A µ  0, 2 i 0 N = Pl e h]. 2 C are raised and lowered with γ 2 , in these coordinates reads f − N e M and 2ϕ ∈ { ≡ − − F C " ds N k det [  ϕ S i ij a h j iγ 2 A) ∂ ∂  K √ j cos , where Ne ¯ Ψ  + a −N N 0 ¯ Bν γ Ψ A, B, C x N ∂ i, j, k, . . . e −g − 3   µ 0 0 ] = ¯ √ Ψ g ϕ ∇ Nm i x ν ¯ dτ d 4 Ψγ − + ≡ i A d is the lapse and e Z ( ϕ 3 Z π N = = a In these coordinates, the action for the purely bosonic sector of the theory (involving = B = Ψ AB S denotes conformal time. Derivatives with respect to F S µ L letters from the middlegenerator of of the local alphabet Lorentz as transformations is spatial Σ spacetime indices, Finally, det [ gravity and the inflaton) becomes where τ spatial indices, 2.2 The actionBecause in certain ADM components form ues of depend the on metricgravitationally-mediated the are fermion-graviton fermion constrained couplings bilinears degreesare are (as eliminated of well generated from freedom as the whenpose action. whose the these the In val- other constraints metric order dynamical to usingthe degrees perform constrained the this of degrees ADM analysis, of freedom), formalism.are it freedom algebraic is enter The constraints. convenient the key to The theory decom- advantage metric algebraically; of in their this ADM equations form formulation of reads is motion that ω The fermionic action, where (see appendix which puts the fermion action in the form where greek lettersLorentz are indices spacetime indices from the start of the alphabet as spatial Lorentz indices where JCAP10(2019)018 Ψ (2.9) 5 is the (2.11) (2.12) (2.10) γ c and the . ab  η ¯ Ψγ N . Ψ 2 ϕ j 5 x π Ψ abc i γ  5 2 x c γ 1 ij bj 0 δ e ¯ 2N Ψγ i ¯ =1 Ψγ ∇ − abc 3 i,j )  ¯ ΨΨ component of the (3) , . 2 abc j P j  b   a K γγ e j ≡ e c i i and the dynamical scalar ... e a 4 1 − ... i x . e i ji 6 ij F  x ∇ + γ a j −1], and the spatial vielbeins S K kj (3) Ψ)+ ∇ ij mj + γ , 0 bj γ 1 8 −1, γ (3) ] = B e Ψ (K 4 1 i ¯ im S ij Ψ  + a i 2 γ Pl 5 e −1, 2 ∂ = 2 1 ki )+ γ 1 4 M 1 simply multiply lower-order constraint γ − S  ¯ K + ΨΨ and quartic 1 2 − − Ψ − γ ij i ij f n 2ϕ ∂ h γ V Ψ) 0 1 +  − a γ + −  ij γ = diag[1, and ¯ – 5 – Ψ ij ϕ sin ak ( ¯ δ j Ψ n i (K i i 2  j AB ∂ a δ 2 + ϕ∂ η + i a ∇ ] = 1. Similarly, the spatial vielbeins are expanded − = ∂  ϕ 3 γ i ij (3) = Ψ ∂ ki f i h 2 2ϕ γ Pl ϕ ij ∂ 1 2 2 1 ) π  a e M γ 1 γ − N + ak (e cos ¯ Ψ 2 R denotes the three dimensional covariant derivative, =  ( a δ a i i ) (3) a ¯ i Ψ is the “flat” three-dimensional Levi-Civita tensor, with the conven- = = e 2; terms of order ∇ N 2 Pl i i 2 m j 2 ij − a (3) abc e M h . The total action is the sum ∇ n − +  schematically denotes the graviton. As is well known, in order to deter- δS ij n-th order in the fluctuations, the solutions to the constraint equations γ = (3) h ( δ = j δS δN j b ∇ = 0, implies that det [e e i (3) a ) is the scale factor. The transverse-traceless nature of the tensor modes, 0 = e ij,j − = +1. enter in the action, eqs. (2.5) and (2.8), without time derivatives (in the case of the i γ ab a(τ δ i δS = 123 δN N Our goal is to determine the effective cubic We work in the spatially flat gauge where det[h  ij Repeated lower roman indices are summed with the Kronecker delta: γ 3 0 = ij equations [57]. Thus, tosolutions obtain for the the action constraints upinvariance (the means to lapse that fourth the and order fermion shift) in fields up the only to begin fluctuations, quadratic to we contribute order. require to Note the constraint that equations Lorentz where δ Lagrangian, where mine the action to are required at order in terms of the tensor perturbations as while the equation of motion for the shift is the momentum constraint In these expressions, spatial part of thesatisfy Minkowski metric, 2.3 Constraints When written in termsshift of the ADMlapse, spatial decomposition, derivatives one are also caning missing see from Euler-Lagrange the that equations action). the areHamiltonian This lapse constraint constraints. implies that The the correspond- equation of motion for the lapse is the In these expressions, tion fluctuation degrees of freedomtensor perturbations are of in the the metric fluctuations as of the inflaton. We parametrize the JCAP10(2019)018 ; the (2.14) (2.15) (2.16) (2.13) (2) −1. We θ , = .  123 0   are transverse,  Ψ
 0 2) 0 0 , δϕ Ψ 0 ki 0 ϕ (1 i H ¯ )γ Ψ)γ β  jk ¯ Ψ)γ γ (∆ , 2 0 i 0 − , for the free gravitons and H a (∂ a (∂ ),δϕ(τ, and ~x we treat the 02 0 2 to second order (eqs. (2.14), P ...,  − − ϕ  i , (2) ∆Ψ F M 0 + i` Ψ ) + 0 Ψ S N 0 2 0 H ≡ )γ  ∂ (τ (2) i ¯ + − 0 `i Ψγ 0 β  γ Ψ) ϕ and ¯ Ψ)γ j interactions, and the quartic action = 5 ¯ (2) γ + Ψγ ∂ H γ , ,
S N k (∆ 2 Pl (1) γ (1)  i ia +(  2 H − = β ¯ ki Ψγ 2 Ψ Pl ∆θ 4M ( ... ia + i (ϕ) )γ A. (2) ∆Ψ ∂ + + V O 0 4M 0 S  jk )ϕ(τ, = ~x is its inverse, and we note that ··· 2 γ 0 ijk i` i − a (2)  ¯ Ψγ – 6 –  +  )γ 1 2 −1  α (∂ + qk `i  δϕ , Ψ − (2) + γ γ 2 0 H 2 1 2 j j
P 0 0 θ 2 2 i Pl ∂ ϕ − ia Ψ (1) ∂ (∂ 0 0 M ) 0  ¯  Ψ)γ α  ϕ 0 j i` kq + 4M H ∂ γ 2 (∆ Pl 2 )γ j ¯ + Ψ)γ (1) 1 + 1  describing the ∂ j `i 1 − M θ  8H = 0 γ i` i 3 (∂ interactions. k a ∂ +(  (3) F )γ
∆Ψ − (1) 0 ij (∂ S = `i 2 0 = =  γ −1 Ψ γ γ 2 i k j j 0 ij 2 N ¯ ∂ ∂ Ψγ N γ H )∆ j  Ψ 0 (∂  0  ∂ j , α −1 ¯ (ϕ Ψγ O ∂ 1 −1 ∆ V 16H i ∆ j 1 = 0  is the spatial Laplacian, ∆ ∂ 8H 2 1 H − 2 i 2 2 2    Pl Pl Pl a (1) ∂ a ia i β M M M ∂ −1 −1 −1 − + − = 0. We also expand the inflaton as = ∆ = ∆ = ∆ i β To facilitate a perturbative solution, we expand the lapse and shift functions as To zeroth order, the Hamiltonian constraint reduces to the Friedmann equation describing the (2) (2) (2) j 2) θ β α , (4) F (1 i S fermions, the cubic action have used the lineardetails equation of of this motion calculation for are the given fermion in to2.4 appendix simplify the solution Explicit for formWe of insert the the solutions fermion to(2.15), the action, constraint and and equations (2.16)) fermion-GW for into interactions thefluctuations. action, eq. This2.5) ( gives + the (2.7), quadratic and action then expand order by order in the at quadratic order. We thereforetively, solve to the second above order constraints in in eqs. the (2.9) fluctuations, and before) (2.10 plugging perturba- them back into the original action. while the momentum constraint is automatically satisfied. At first order, we obtain [55] Since we are notdrop interested these in from the now perturbations on. sourced Ignoring by inflaton fluctuations fluctuations, the of second the order inflaton, constraints we read where ∆ = where the superscript denotes the order of the expansion, and where ∂ fermions as first order quanties, so that fermion bilinears are of second order. JCAP10(2019)018 ψ . 5 γ 0 (2.18) (2.17) (2.20) (2.22) (2.19) (2.21) (2.23) (k) ¯ ψγ λ ad )
 , Π = +1) and kb ψ ,  0 γ λ a ψ iλk (4) i 5 ¯ ψγ (∂ i L ), ¯ ∂ ψγ x kc refer to the three (τ 4 γ −  λ k d (k) = t ψ (4) 3 abc i f Z 2ϕ  λ cd L ∂ Pl 0 , Π  1 7 16 2 b M , , i=1 X ¯ ) k (4) 2
ψγ . √ +  sin (3) L ψ ≡ ψ i abc ij 0 7) , Ψ. 5 L a(τ arise from integrating out the γ γ γ (4) k x ) c (4) 1 ima 3/2 4 refer to the other four lines (one ∂ L ¯ (4) 7 ,  )+2 ψ L d a 0 ) = ij ¯ x L + ψ ψγ
4 γ (4) i 5 , ≡ (∆ (τ Z k ψ d γ L abc λ 0 k = 1,... λ,−λ ∂ ¯ c − ψ  (4) 6 ψψ −1. δ ≡ Z ¯ L (i − jb , γ ¯ ψγ  ψγ ψ = γ ≡ , ∆ψ ( 0 ij , , j ∆ 0 f a 0 aj γ 2ϕ ∂ ) (4) 5 ik·x ∂ − γ i 0 ij 123
e (3) (4) i ¯ L  ψγ  γ (k) = 2 1 aic 0 L L kj , 16  0 ¯  ∆ψ ψ γ γ
λ ij 2 x x  (k) cos + (4) 4 – 7 – 0 3 3 ) ij 0 kj kn
d d L λ ij x a γ γ kj ¯ ψ i ψγ 4 γ ma a x γ ∂ Z Z i d (k)Π )Π kn 4 ∂ γ d − i ¯ − − λ ij ψγ −1 (τ Z ∂ 0 m kj k λ Π k ψ ∆ Z ∂ γ ∂ 2 γ Pl
−1
i 8 ) = ) = 2 a − − , 0 ij ∆ M m ∂ (τ (τ 0 − ik ψ γ 3/2 a k ∂
) k γ 3 0 ij γ = j ψ ∂ = d (3) γ int −2 0 ∂ a (4) ,i int (−k) + (2π γ + H ∆ (2) γ (3) 0 0 F jk H λ ij ¯ ψγ S ¯ ∂ γ  jk S ψ Z 0 γ 2 − i Pl − γ bk 0 ∂ λ ik γ M X ψ ¯ γ jk ψ 0 2 V j ab i is canonically normalized. The sum is over the right-handed ( γ (k) = Π ∂ −1) tensor polarizations, with the polarization tensors satisfying γ i  H is the Hubble parameter and ¯ λ k ) = ∂ ψγ i x 4 0 jk = 16 −λ ij 4 d − (γ ( a/a 1 x aH aH i i (x, τ Z −1 = Π 4  8 = ˙ ij d i ∗ ∆ 16 4 32 γ = H Z − − − + (k) (2) At cubic order we find The quadratic action for the gravitons reads To proceed, we expand the tensors in Fourier space as F = λ ij S Π (4) F S where and some straightforward, but lengthy, algebra leads to the quartic order action terms in the first lineterm of per eq. line). (2.20), while Note the that remaining the interactions non-dynamical constraints (the secondand order quartic Lagrangian parts densities of we the find lapse the and interaction Hamiltonian shift). densities From the cubic where we have rescaled the fermion field according to while the quadratic action for the fermions is where the field t left-handed (λ JCAP10(2019)018  k (3.5) (3.4) (3.1) (3.2) (3.3) ψ 5 (2.24) (2.25) γ k ¯ ψ  , f  2ϕ  † r, −k (k) are explicitly c r ) , sin , χ (τ  T ) 0 1 r −k ima (τ V  + r k . + = ¯ k
U B. ) are solutions of the Euler- 0 r k ψ − , 2. Finally, we end this section b k . C k (τ ¯ χ ) # ¯ ψ − , r (τ , −k  ik·x
 ) = k 0 r λ k V k e t  k (τ f ) U δ 2ϕ 1 0 r 0  k − λ −k (τ  t rr  k k δ =±  ψ ) and X ,V r = cos 00 δ = a (τ 0 a +  } r 3/2 k ¯ k χ † λλ ) = ) ma 0 − 3 0 U δ r k (k) d 2 , (τ (k) − – 8 – r r k r = (2π k , c k χ ¯ χ χ  r k i ψ ) ) † ) Z 0 0 z
− {c λ k k k a , ψ (τ (τ λ k · k r k = t r k ) = a + , a 0 λ,† −k u σ } λ k rv ∂ a † r iγ a 0 (k 0  ) h r k , the quadratic action is λ −k (x, τ + + 2 t k (τ 2k ψ 0 1 0 , b k ψ √ ∂ r k ∂ p 0 λ,∗ k  γ {b ), and the spinors ≡ k ) = and + t 3 (τ k d ¯ λ k λ k (τ λ k ψ (k) i a r k Z r )  U k is the charge-conjugation operator, and the spinors χ 1 2 (τ 3 2 d λ k λ γ X 0 Z iγ + " ) = t dt = (τ λ k Z t C ˆ = Inserting eqs. (2.22) and (2.24) in the interaction Hamiltonians eq. (2.21), we obtain (2) S and the fermionic operators satisfy anti-commutation relations In terms of the fields t where the Fourier space Hamiltonian densities we report in appendix and we note that the kinetic terms are canonically normalized. 3 Fermion contributions toIn the this section, tensor making power usecompute of spectrum the the interaction fermion Hamiltonians derived contributionAfter in to quantizing the the previous the free section, gravitational theory, we quartic we wave introduce loops two-point the generated correlation in-in by formalism function. by the and showing compute interactions how the derived cubic simple in and scaling section arguments concur3.1 with our results. Quantization We canonically quantize the theory by expanding the fields into modes given by We also Fourier transform the fermions according to where the creation-annihilation operatorslations for the tensor modes satisfy the commutation re- The mode functionsLagrange t equations of motion that followthe from 4-component the fermionic action in spinors eq. into (2.25). helicity We states further decompose JCAP10(2019)018 ) γ kτ , (τ n (3.7) (3.8) (3.9) (3.6) ˆ (3.10) (3.11) O ≡ − x ... (−2ix) 1 2 quartic loop ˆ . O +4ξ (k) are helicity 2 iE µ r ), with ) χ in √ N (τ . x/x int 44]  , , +2irξ, i 0 2 1 1 i i log ( ,H − → ∞. 0 1 i 2ξ (x) (x) W x  r r − d d . ,... = 1. We discuss these diagrams in i /f /f −πrξ 5 /f ) 0 0 e 1 in 0 −ikτ . ϕ (τ e , γ = 2, with [43, N −irϕ −irϕ 0 = 2 −i µ e int   | ˙ dτ ϕ i r i 2fH 0 − σ kτ −1 |v ,H )/f ) = i diagram with two vertices generated by the N ) ≡ τ − + (τ (x) + e (x) (x 0 1, one for each of the seven vertices generated (τ 0 1 2 r r r (k). We use the Dirac representation for the −σ Z | s s ϕ r  r – 9 – (0)  n , ξ |u ... /f /f ˆ O 0 0 2k , d are the Pauli matrices. Note that = 1 1 rkχ m H i i √ irϕ irϕ dτ ... σ cubic loop e e ≡ τ 2 identity matrix. h h , γ (0) 1 µ × ) = Z ˆ (k) = O (−2ix)  2x 2x 1 1 (τ h 2 N r ) λ k √ √ χ k, and 0 t −1 ... result in two classes of diagrams: there are seven +4ξ σ 2 is some reference time. With these approximations, the mode · (−i 2 µ 0 1 ) = k Dhh in √ =0  ∞ τ (x (x) = X r r N × denotes the 2 = v u = 1 0 γ +2irξ, i E 1 2 ) , where (τ -component of W z in n 2.19), illustrated in the right panel of figure ) denotes the Whittaker W-function and ˆ O (z −πrξ ... is given by is the 4 1 µ, λ ≡ −kτ τ z ˆ O ) = e W k in D x (x In these expressions, r 4 s The interactions in section the next two subsections. which satisfy the normalization condition at time where In the same approximation, the tensor mode functions read In both cases, thethe integration appropriate constants Bunch-Davies have vacuum solution been at chosen early so3.2 times, that the solutions Fermion match loop-corrections onto The to interaction the Hamiltonians derived gravitational abovetions wave allow power from us the spectrum to produced computeputed fermions the using to leading the the order in-in contribu- two-point formalism, function of where the the graviton. correlation function These of are an com- operator functions for the fermion field are given by where eigenspinors which satisfy matrices, and To obtain solutions totionary the spacetime classical as mode deconstant Sitter equations, speed space we in and approximate cosmic take the time. the background This evolution infla- implies of the inflaton to be rolling at a cubic action ( diagrams, illustrated in the leftby panel the of quartic figure action), (2.20 and one JCAP10(2019)018 ). 2 , 7 ( (3.13) (3.12) (3.14) O , which , where ,5 = . We then 1 (4) int + 2 p  2 H · = 1,..., q ,  2 = and ,λ , and 1 1 2 1 λ q , p δ , i (4) ,3 int , ) γ 1 2 H , so that iE p p , ) 1 ) = 0 · 1 ,2 1 p +  (τ p (4) int 1 → ∞. As we have discussed H (p (4) (0, ,int i δ C, we obtain the leading contri- 3 1. We note that this contribution ξ Ψ Ψ 2 ),H  give finite and unambiguous results. µ (τ , f ) 2 2 ) τ −1 λ p 2 1 3 1 −kτ q ) t p − (τ 4 Pl . 1 1 γ 1 ξ λ p t log(−p (q are given in eq. (B.2). Remarkably, all these  4 – 10 – Dh 9π M )δ 1. The seven terms in the quartic action lead to 1 1 H 1 p 8 (4) ,i int dτ  , τ 2 H γ µ q C. Here we summarize the main issues one encounters ' − Z is a scalar, it depends only on − 2 Ψ ) f 1 /0” that needs to be regularized. To deal with this limit we 58]: these undetermined quantities are schematically given by (q a(τ quartic 2i f 1. E 2 2 Pl ) | 2  M (τ q ξ 0. Since 2 2 − 1 and for superhorizon modes λ p − γ ξ → 1 = )ˆ γ  |q (τ  (4) i 1 1 λ 1 p E ) ˆ γ approaches zero in a direction that is orthogonal to D  (τ  2 2 µ λ p γ is an external momentum. We regularize eq. (3.13) by setting . X ) γ 1 vertices (τ p 1 1 Secondly, many integrals contributing to the graviton two-point function are divergent in After long calculations, which we outline in appendix First, several of the terms in the interaction Hamiltonian contain the nonlocal operator λ p , the inverse of the Laplacian., lead When to a evaluating undetermined eq. “0 (3.12) one often encounters the γ D −1 −1 impose that the ultraviolet. We deal withcutoff these Λ divergences and as by subtracting we all did the in terms [44], that by are introducing divergent a as ultraviolet Λ where we eventually send With this convention, all the operators containing ∆ in [44], we expectsubtraction the in result of the this limit procedure to be equivalentbution to from that the obtained quartic by adiabatic diagrams where the interaction Hamiltonians 3.2.1 Quartic loops We begin with the left diagram of figure Figure 1.graviton The two diagrams that contribute at leading order to the two-point function ofseven the quartic contributions to the graviton spectrum of the form follow the prescription given in [ ∆ expectation value of quantities evaluated∆ at vanishing momentum that, when acted upon by diagrams can beresults, computed are exactly. presentedwhen in The performing appendix this details calculation. ofsum At of the the the end calculation, quartic of this loops as section in we well the present limit as the expression the of the exact in the limit is parity-even. Parity-odd terms are associated to the operators JCAP10(2019)018 , and iE D. (3.18) (3.17) (3.16) (3.15) ) A 2 (τ (3) int term. Since 2 ), ,H A , τ i 1 2 ) 1 1 we are interested ,λ 1 (τ and from the parity- λ 1. There is a single  δ (3) int ) ξ log(−p 2 (4) ,2 int . 3 p ξ 2 H ),H + 2 ,λ 1 (τ 1 µ 2 λ 2 2 and yields δ λ p (p ,λ ) δ 1 ) t ). Explicit expressions for 2 λ 2 p B ξ (4) (τ ,3 int )δ 1 2 1 3 1 2 + H λ p µ p p t 1 3 1 4 Pl + p A) and a part that behaves like negative (p Dhh 4 δ 1 M 4 Pl 2 λ t H (p M P dτ δ 4 3 1 2 3 1 τ 1 H p – 11 – λ 2π Z , although it is only present in the (1) error. Next, since the functions appearing in × 1 1 ' = τ O dτ )i τ , with respect to the parity-even component. → (0.1) (τ −odd 2 Z 2 2 /ξ τ λ p 2 ) γ ∼ O ) parity E a(τ 2 D. (τ (3) ) 1 1 2 E Pl λ p ) (τ 2 2 M hγ λ (τ p 2 2 γ − λ p )ˆ γ = )ˆ (τ vanish identically after angular integrations, so that the only parity-odd 1 1 λ (τ p (3) 1 1 ˆ ,5 γ E λ p D ) (4) ˆ int γ ) is given by eq. (B.1). Unlike those appearing in the quartic loops, the integrals D (τ H 2 2 (τ λ p γ (3) int ) H (τ Once the above approximations are in place, the integrals can be computed analytically. We start by setting the external momenta to zero; as discussed in [44] we expect this 1 1 λ p γ can be found in appendix D this term corresponds tothis positive component frequency, “vacuum is only” subtracted, modes, we we are subtract left them. with Once the final result which has the same parametric dependence as3.3 the contribution from Scaling the of quarticThe loop. our gravitational wave result power spectrum is related to the two point function as frequency (whose coefficient is denoted schematically by where in the cubic loopthis are diagram, prohibitively however, difficult are to very similar evaluateto to exactly. the those The which spectrum appeared expressions of in appearingsame scalar the in sequence perturbations cubic loop of considered contribution approximations in developedoutline reference in [44]. these that approximations; Therefore, work the we to details apply the of the present calculation. the calculation Here are we presented in appendix in. These approximateschematically expressions denote contain the a coefficient part of that this part behaves by likeB positive frequency (we We find a divergence in the limit 3.2.2 Cubic loop Next we consider the cubic loop, shown on the right side of figure the integrals arevariables, rapidly so that oscillating, the we Whittakerreal functions perform appearing argument in and a the are Wick fermionWhittaker exponentially mode rotation functions functions increasing as now on or have linearattention the decreasing. given combinations time to Next, of integration the we monomialseqs. branch approximate (D.5) times cuts. and those (D.6), exponentials, The and with we explicit have special form verified their of validity in those the approximations regime are given in contribution to the tensor power spectrum is given by approximation to generate at most a contain the Levi-Civita symbol. However, theodd contributions part from of which is sub-leading, by a factor 1 contribution to this diagram, given by JCAP10(2019)018 , i p ¯ ψ ψ Γp ψ h (3.19) (3.20) (10%) ¯ ψ O ij , where γ ∼ p ψ Γ ¯ ψ , for momenta up 2 ij 10, the parity odd /ξ γ , each fermionic line 2 & (which appear in both µ /ξ can always be brought ∼ 2 i ξ sensitivity of Stage-4 [7] 7 µ ψ . (We note for comparison 6, 0 ξ −3 5, γ . denotes the imaginary part), which contain no dependence 5 (4) 4, ' ). 10 ) τ L τ = 1 ¯ ' (4) 3 1 ψ γ corresponds to the ratio between h L r −kτ 2 Pl and and log(−p /M sector scale as log(−p i (1%), irrespective of the amplitude of 2 3 ψ O 2 ξ (4) 2 i . These were considered in reference [44].) H ¯ 2 ˆ ψψ L p H 3 2 µ Pl i ξ 3 γγ µ/ξ ξ 4 4 2 Pl M ¯ 2 µ ψ γ H . h M µ – 12 – /ξ 2 0.1 µ for momenta up to .1% of the full tensor spectrum, which sends the level and the ' (0.01) /ξ 2 ¯ ψψ t µ γ 'O P λ δ t vacuum t P δP contributes (see eqs. (C.3) and (C.4) — here 1 i -matrices (exceptions are
. Their ratio can be written as γ ψ 2 Pl 0 ¯ /M ψ γ 2 k ∂ H is ultimately zero, due to the asymmetry of the Levi-Civita symbol. A numeri- i . Moreover, the fermion part of the operators in with respect to the parity even one. Therefore, since i, oscillate instead with amplitude 0.1 h= ξ ψ ψ 5 5 ∼ /ξ ' γ i µ, of the diagrams that are relevant for the scalar (bi)spectrum. Here we apply analo- ¯ ˆ ψ γ p We now compare this sourced gravitational wave signal to the vacuum contribution, Our first observation is that the cubic interaction (2.19) gives a contribution More specifically, if we require that the energy in fermions does not exceed Finally, in reference [44] we determined the overall scaling, as a function of the parameter The contribution from the produced fermions, using the results eqs. (3.14) and (3.17) h 0.06, already puts the sourced component below the i −kτ . and vacuum t ¯ ψ γ P As discussed in reference [44], the quantity schematically denotes a quantity that scalesbination as of fermion the momentum Dirac and Γ denotes some com- the cubic and quartic gravitationalto vertices) oscillate with amplitude to a form and Since all fermion bilinears inin the the diagrams of figure gous arguments to the diagrams that led to the result in eq.3.19) ( for the tensor spectrum. cal evaluation shows that the combinations ξ the energy density inmuch fermions smaller and than the unity. totalcannot We energy produce thus density conclude tensor in that modescontrast the to an that Universe, the axion-like which dominate scenario inflaton must over in coupled be the which to an spectrum fermions axionic ofof inflaton vacuum is the fluctuations, coupled background in toamplitude energy, gauge of fields. then the thethe correction tensor tensor-to-scalar induced spectrum ratio. by isr This the at constraint, sourced most along componentCMB with of experiments. to the current More the to observational importantly, both constraint a the first sourcedobservations approximation, and to parity-even the tell and vacuumtensors from contributions scale are, contain one invariant, a component sofactor distinctive from that parity-odd 1 the it component, other. wouldcomponent such be would a While difficult make component theoretically up for of less is the parity than suppressed sourced violation 0 by in this a model well below the threshold of detectability. whereas each quartic interaction in eq. (2.20) gives aon contribution the fermion momentum,h but only on the graviton momentum). The contribution from which can also be seenthat to the scale as bilinears which appear in the diagrams involving fluctuations of the inflaton, we derived in the previous subsections, is JCAP10(2019)018 . . . 3 3 3 D, ξ ξ ξ and , as 4 2 ∼ µ µ
(4) 3 ) (one L ∼ ∼ /ξ ¯ ΨΨ 2 ) 3 γ µ ). The first 4 C. The right and (ξ O × (µ ) (4) 2 — the factor of O L (ξ 2 ξ × 2 ) µ /ξ ), and 2 2 ∼ µ ) (µ 3 , gives a contribution O (ξ can, however, be understood k 3 , of the model. µ d × ξ (1), , giving an overall scaling O (1) 3 ), although in the quartic diagrams and ξ 2 × ) term which arises from interference. µ (µ ) 2 O . We thus obtained the /ξ are integrated out via the corresponding (µ γ 2 1. The left diagram is technically simpler; i O N (1)+ are respectively vanishing (as a consequence of O and – 13 – (4) 3 , giving an overall scaling ( L N 3 ξ with the result obtained by the direct (albeit approx- and 2 (two vertices) times p, and therefore each vertex gives an additional power of µ 2 we presented some simple scaling arguments that correctly (4) 2 ξ L D. The different scaling with 3.3 with respect to the parity-even part. /ξ , which we did indeed find. As mentioned above, however, this is −1 t (one vertex) times P δ ξ ) originates from the fact that this vertex does not contain a power of the 6= ξ 53]. +1 t interactions, which we used to compute the contribution to the gravitational P δ
(two fermion lines) times 43–45, 2 ¯ (1) contribution is always divergent and therefore is removed by regularization. The ΨΨ ) Given the parity violating nature of the system, one can expect a parity violating tensor For the cubic gravitational diagram, a naive implementation of these scalings would read We recall that each interaction Hamiltonian (with the exception of Once we apply these scalings to the quartic diagrams, we have a scaling ( The present work is a direct continuation of our previous paper [44], that focused on the Our main conclusion is that, in contrast to the scenario in which the axion inflaton is O γγ /ξ 2 . Furthermore, every fermionic loop integral, which goes as subdominant by a factor 1 4 Conclusion Axion, or natural inflation is aof class of the models potential for slow-roll is inflationThis where shift protected the symmetry required from means flatness radiative thatlowest any corrections dimension axion-matter by couplings couplings an must ofeq. be approximate (1.1). via an shift derivatives, axion These and symmetry. the couplings inflatonliterature are to (see typically gauge [19] employedcan fields for for lead reheating and a to in fermions review) a thesescarce are has rich models. [ phenomenology given shown The during by that recent inflation. the Analogous coupling studies of for fermions the are axion more to gauge fields spectrum (µ (1) instead of ( fermion momentum. These resultsappendixC. are in agreement with the direct calculations presented in This would disagree by a factor of the cubic diagram, with two fermionis lines, again has removed terms by of renormalization, order leaving the imate) calculation of appendix the symmetries of the operator) and scaling as (µ as follows. Each fermion line in fact scales as fermion line) times The quartic diagrams involving ξ noted above) also carries a power of characterization of the scalar perturbations sourcedthe by computation the of fermions, the and complementsthe sourced it gravitational nondynamical with waves. metric We perturbations worked in the ADM basis, in which energy and momentum constraints.fermionic We field have Ψ solved these and constraints in perturbatively the in gravitational the waves O wave spectrum from the diagrams shown in figure after regularizing it we were able to evaluate it exactly, as described in appendix diagram is much moreanalogous involved, to necessitating those the made approximations forwere discussed the very involved, cubic in in diagram appendix section incapture reference the [44]. scaling Although of the the computations result with the parameters, coupled to vector fields, the gravitational waves sourced by the fermions cannot be greater JCAP10(2019)018 (A.2) (A.3) (A.4) (A.1) (A.5) ,  . These are j ν j b B N e N e ab . Using eq. (A.1) µ − η , A µν e bc 1 g 0 N η AB  m η , c = e
i = a e Bσ Aµ e µν  i is the inverse of the metric in g σµ , e N ν  m AB j η j ∇ N + Γ , N a ν B . e e (3) − , . Starting from (A.1), we can also write  bc Bν µ A i ij e + η F e a 1 0 0 N k µ h e L c ∂ =  i e km = x a 4 j m ν K e = b , d a µν i N A e i ik e g i µ e a N a Z ki A e – 14 – e  = m ab = ab Γ η , e −Nh = δ F Bν
 µ e (3) S  ij µ A j + , − b e K ∇ e j bc bc 0 bc ν η η satisfy ab η N A k k η i k e c c c a i − e e e e b i 0 = e ki ∂ ∂ N i k k ab N a a ∂ η AB i −K e e µ ω N = = = =  0b 0b ab ab i i 0 0 = ω ω ω ω µ B e are the usual Christoffel symbols associated with the metric AB η σµ ν = In terms of the vielbein, the spin connections are obtained from Aµ e eq. (2.4). and one can indeed verify that the product where Γ and (A.2), we find the components of the spin connection in ADM coordinates Inserting these in the actioncoordinates (2.3) we obtain an expanded form of the fermion action in ADM than the vacuum gravitationalreference waves. [44], where This the conclusiondynamic strong holds of backreaction the also of zero in theour mode the fermion studies of regime, degrees for the of inflationary studied inflaton, freedom model and in controls we building. plan the to further exploreAcknowledgments the implications of The work of P.A. and L.P.SC0015655. was supported L.P. by gratefully the acknowledges US support Departmentof from of a Illinois Energy Fortner through Fellowship at at grant DE- the Urbana-Champaign.and University support L.P. thanks through the Nationaland Aspen Science L.S. Center is Foundation partially grant for supported PHY-1607611. Physics by for The the hospitality US-NSF work grant of PHY-1820675. A M.R. Computation of theIn fermion-gravitational this wave interactions appendixwriting we down discuss the the vielbeins steps for that the lead line from element eq. in (2.3) eq. to (2.4), eq. (2.8). We begin by where the spatial components given by JCAP10(2019)018 Ψ ,   Ψ  ab (A.9) (A.8) (A.6) (A.7) Ψ (A.14) (A.13) (A.11) (A.12) (A.10) Σ ki )   0 ac , 0 ab η f  Σ  )(γ 2ϕ m ac 0 nc jk b  η . γ γ e i ,   in  i  sin k (∂ γ b ) 5 e 1 4 1 2 ic , N ... γ m iγ m + c k  + + e ∂ − 0 ` ic ∇ i γ ik ... )  `j dk 0 (3)  γ m 0 γ + − i` Γ f bc − )(γ γ 2ϕ ji η m (3) id `i γ 1 8 k  γ γ , . + a kj j k i + 2N γ bd b ∂  cos (∂ e η 1 8 NK ij  ck  k 1 4 ) )+ γ  ac i Ψ ∂ ¯ + Ψ 1 2 (γ i η kd 5 − − c N ) γ γ 1 4 k ki c m e i d b − b ki γ ∂  ∂ e − γ N 1 2 ij 0 )+ 2 1
c + ¯ ck Ψγ δ , ∂ jk c ∂ γ  Ψ )  + mi +  0 , a N ) − γ − a k i  γ 0b c ki d ij c i ∂ Ψ δ kc δ ∂ e qk Σ (∂ N  i  γ k ¯ a (∂ γ Ψ)γ 1 2 d 5 jab i b – 15 – δ bc j a cm 1 4  e γ ∂ η ak ∂ + γ (∂ 4 1 ik (∂ )  −1 dk − 0b − a δ a 1 − K kq 2 −2 − (γ f Σ ki γ 2ϕ a i (2) 2aN j
Ψ − = = γ j 1 8 b a c k  e ∂ Ψ − − ∂ ∂ ki ∂ 0 kj

∆β a γ )+ γ   ij dk +( 1 2 2 1 sin 0 2 1 jd , 2 id ci  i e γ ij ¯ − ¯ K γ γ k (γ − Ψγ Ψ)γ ab ) c e + j 0 j cj  ak 1 8 η 1 4 ∂ N γ ij i 2  0 N 2 (∂ (γ Pl δ +  − a δ −  γ k a a ij )+ f − ci − ic M δ 2ϕ ) jd 0 − 2 γ = id N γ N γ k  γ d 1 2 i Ψ + i −1 (γ c a j H N a 0 cj ∂  a ∂ )+2a (∂ e + ∂ γ ]/4. The full action is the sum of the bosonic part in eq. (2.5) and the 0 (2) 0 cos + 2 1 2 ci − Ne Pl B =  + α 4 δ a 2 (ϕ  j N ic 0 bd i ¯ ¯ M Ψγ γ Ψ  , γ a ∂ a γ ∂ η bd V   d ∂ e A bc + η ac 0 i H 2  ¯ η (2) Ψ (∂ η ac i Nm  2 Pl (2) α 1 4 8 1 ¯ Ψγ −1 2 a + − N = [γ i aH M − a −η + ∆θ ( − 3 = = = = 2 Pl AB −4a a We are interested in the interactions between the fermions and the tensor modes of the 0b 0b ab ab M i i 0 = 2 = 0 0 = ω ω ω ω F 4H L while the quadratic part of the momentum constraint reads which leads to the followingorder components in of tensors) the spin connection (expanded here to quadratic These relations, along with the expansionthe of constraint the equations lapse and) (2.9 shiftquadratic and in in eqs. (2.10). field), (2.13 are fluctuations The inserted is part into of the Hamiltonian constraint equation where and Σ fermionic part in eq.). (A.5 metric, and so we expand the spatial part of the vielbein as JCAP10(2019)018 . ) 3 (B.1) k (A.15) + , 2 and cubic  k ki  (2) )
−  0 S 1 Ψ Ψ 0 0 )(γ (k jk (3) ¯ γ ¯ Ψ)γ Ψ)γ i δ 0 j (∂ ) (∆ (∂ 3 − − − , k i` ) Ψ  0 + 0  ∆Ψ 2 ∂ 0 Ψ 0 )(γ 0 , `i )(k ¯ Ψγ ¯ 1 Ψγ γ   j ¯ Ψ)γ ∂ (k H We have also made use of the linear Ψ) H λ cj (∆ 5 2 Pl +( 2 Pl 5 ia Π γ ia − 3 ki k k γ is its inverse. In deriving these solutions, 4M 4M ψ ¯ jk Ψγ c ∆Ψ + ) − ( 0 −1 0 γ  i  2 γ ∂ i` i k ) qk ¯ Ψγ 0 ¯ ψ γ  ijk (∂ 1 – 16 – j   λ k ∂  )(γ 1 2 H t 1 2 ) i `i Ψ 2 , consisting of a quadratic (free) part, Pl − γ k 0 − ia kq γ j
3  γ 3/2 d j i` Ψ 4M (∂ ) ∂ ) 0 ¯  Ψ)γ 0 + and j 3 i=1  +( ∂ (2π , which describe the interactions of a fermion bilinear with (∆ )(γ ψ ¯ i` Ψ)γ Q ij ) j `i 1 − ) (4) 0 F 0 8H γ S Z (∂ k (γ  )(γ ∆Ψ − (∂ ij λ `i 0  ) −1 X Ψ γ 0 and k j j ) ¯ ∂ ∂ Ψγ (γ )∆ j  0 1 (∂  0 (3) F  ∂ a(τ S j −1 ¯ (ϕ Ψγ ∂ 1 −1 Pl ∆ V is the spatial Laplacian, and ∆ 16H i ∆ j 1 i  ∂ 1 2 8H 1 H − 2M ∂ 2 i 2 2 2    Pl Pl Pl a √ ∂ a ia 2 M M M −1 −1 −1 − + − − = = ∆ = ∆ = ∆ We are now ready to evaluate the action, eq. (A.6), on the constraint surface and Our solutions for the part of the second order constraints that is quadratic in tensors differs from the (3) int (2) (2) (2) j 5 θ β α H results presented in reference]; [59 however, this does not affect the conclusions of that work. and quartic parts, one and two gravitationaljust waves, described respectively. is very long Thethat and action considerably complicated. that simplify However, results inthe the practice from variation there result. the of are a procedure theof number Because action of motion with the facts before respect constraint substitutingnumber to equations of in the terms, are their lapse and solutions. derived leaves and the from shift, This result one results we can report in use above theB in their cancellation eqs. equations of (2.19) a and Interaction). (2.20 large Hamiltonian In this appendix weobtained write the by explicit inserting formsand the of (2.20). the decompositions interaction For (2.22) Hamiltonian the terms and cubic eq. term,) (2.24 (2.21), we into find the actions in eqs. (2.19) eliminate the non-dynamical lapse andinto shift. the full Inserting action, the eq. solutionsaction (2.5) to for + the (2.8), the constraints we (A.15) dynamical expand fields, the result to quartic order. This results in an where ∆ = we have disregarded fluctuationsinteractions of between gravitational the waves inflaton andintroduces fermions. field terms The because that inclusion we of arein are inflaton quadratic the only fluctuations in first interested the order inorder inflaton perturbation the equation fluctuations, to of as the motionat well lapse for fifth as and the order terms shift. fermion. in quadratic fluctuations This and induces are corrections thus to irrelevant the here. action that begin Eqs. (A.13) andA.14) ( can be solved to find the quadratic order lapse and shift JCAP10(2019)018 ) + (B.2) ) 1 2 4 ) k i −p ) − 2 0 # = 2 3 ) j (k ! ) 2 2 0 (k 4 2 ! 0 λ jk p 4 # 2 0 λ k a (k k ) k 0 t λ k 2 , λ jb t )Π ψ + ) 1

1 0 (k 4 3 λ k 3.1. Evaluating the 1 γ )Π 0 t k (k 3 λ k 1 λ kb  2 t k , ) )(k λ jk . ¯ + k (k
) 4 ψ 2 2 )Π · ) 4 4 0 Π k 3 1 a λ aj i 4 k k 1 k (k ψ ) ! k ψ Π 0 1 (k k 0 2 0 0 + − γ λ mj λ k a + λ bk γ 3 (k ), which sets the denominator abc t 3 − 2 (and therefore 3  k 4 3 Π 0 k )Π k k 4 ¯

2 − k k ψ  1 ¯ k ! ψ ( k ) − 1 0 + 0 − i 2 0 ψ 2 − 2 + λ k (k ) 2 λ | 3 λ k a ∂ 5 1 t 2 4 4 2 t 3 k γ (k " λ cm k 0 k k c (k k − k (k i

λ ik δ ψ + γ Π 4 k 0 + 5 + 3 + 4 3 (3) = k 3 1 k k γ )Π 0 3 δ ) d 1 1 ! ψ 4 ¯ 0 1 ψ ψ # 1 0 (k ! k γ c 2 |k 0 ) (k (k λ k a ∂ 2 0 3 (k λ k 2 t γ 4 i=1 (2π 0 (3) λ k a t k 3 2i (3) t δ γ λ jk 0 ¯ k (k Q ψ δ 3 0 ¯ 2 # 0 Π − !

ψ k # λ ij ) j λ k 2 0 ¯ ) 1 1 – 17 – ) ψ 2 Z a t " λ k 2 λ k a λ k ) 2 1 0 )Π t i t t 4 2 λ k (k 1 1 | k " (k t 0 (k k λλ λ k 0 2 ( 3 X i , , , , 3 λ jk t " i λ ij (k ) d k k ) ) ) ) i i " − − k  3 λ ij 4 4 4 4 3 i k k 3 )Π ) 3 2 ) 3 d )Π + k 3 3 k k k k 3 3 1 Π ) d 2 1 4 i=1 (2π 3 ) ) d d 1 3 2 H (k ) d | + + + + (k (k Q 2 4 (k 4 i=1 (2π 2 4 |k Pl 0 V 4 i=1 (2π 3 3 3 3 k aic λ ij λ jk 4 i=1 4 i=1 (2π (2π k λ jk Q k k k k  4 i=1 (2π Z Q Π Π ) − Q Q 4M − 0 )Π ) 4 1 − − − − Q 2 3 Z 1 2 3 k k Z 2 2 2 2 λλ k 2 4 − 0 X Z Z k | | ψ · 0 k k k k |k (k Z 2 2 5 0 0 1 ) λλ 4 X 0, as discussed below. 0 + λ ik λλ k k γ 1, using the mode functions presented in section 2 X + + + +  k λλ λλ c X X aH 1 k Π λλ ψ 1 1 1 1 X 1 a γ H + + j → 2 2 2 Pl 0 2 3 i 2 Pl 1 1 ) (k 1 1 + 1 a a a 2 k γ (k (k (k (k  a Pl 1 · i 3 ¯ i 1 ψ |k |k 1 2 2 2 2 1 Pl Pl Pl Pl aM k ( (3) (3) (3) (3) (k ¯ (k k 1 1 1  δ h δ δ δ ψ 16M 4M 2 8M 8M 8M 8M − − − × × × × × × × × × ======,1 (4) (4) (4) (4) (4) (4) (4) int ,6 int ,4 int ,7 int ,3 int ,2 int ,5 int The various terms contributing at quartic order are H H H H H H H C Details of theHaving identified quartic the loop seven quarticloop computation vertices diagram in of eq. figure (B.2),fermion expectation we values now produces proceed terms toin of the evaluate several form the of left theapproach similar vertices to to that zero. of [58], To in ensure which mathematically we sensible set results, we follow an and take the limit JCAP10(2019)018 (C.3) (C.1) (C.2) ,

 ) , 1 at lowest
h.c. h.c. (τ  3 − − r k ∗ ∗ h.c. v ) )    ∗ 1 1 − ) ∗ 1 (τ (τ ) 3 3 h.c. 1 h.c. (τ r k r k  3 , − (τ − u u r k 3 3 3  , ∗2 r k u ∗2 k k 2  ) ) h.c. u  ∂ ∂ )  1 1 3 ) ) 1 )+ − 2 2 k Pl 1 1 (τ (τ 1 p ∂ (τ . 1 1 1 ∗2 1 (τ (τ 3 ) M λ p (τ ) λ p 2 4 r 3 3 k − 1 t 3 t 2 1 V  p r k r k v 2 r k 2 2 2 (τ ) ∗ u u 2 ) (τ H u p ) 3 1 1 ) ∗ 4 r k 1 + + (τ λ p (τ · ) 4 t 1 ∗ ∗ u − 1 1 1 1 (τ − ) ) 2 λ p λ p 2 3 ) 1 1 t + 8p t 1 , (p (τ r k ∗  2 3 ) (τ (τ  u ) (τ r ) k · 3 3 1 1 2 1 ) 1 2 1 v r r k k ) λ p 1 1  1 p 1 + v v )+ (τ t 1 (p  3 3 1 3   a(τ dτ dτ ) r k k k 2 a(τ 2 1 1 (τ v ∂ ∂ 2 a(τ ) + ,λ 3 3 ) ) ) 1 1 1 2 1 r k 2 k 1 + dτ 1 ) 1 1 τ λ p a(τ τ u p 1 1 ∂ # −∞ 0 dτ · −∞ 3 (τ (τ ) ∗ ' Z 2δ 3 3 3 a(τ ) Z 1 k dτ ! τ – 18 – r r 2 k k 1 a(τ 1 −∞ ∗ ) v v (τ (k ' ,λ p ) τ Z ) (τ 3 1 1 −∞ τ r 1 2 k 3 2 + − 3 3 2 λ r ) −∞ k ) v Z 3 δ (τ k k ,λ ,λ v Z 2 1 1 1 ) 2 k 1 3 3 2  a(τ (p ) + 3 3 ), ), 3 λ λ p 2 2 ) ) λ ) ,λ a(τ  t k k 3 ; ; k 2 δ 3 1 p ,λ H ) −r 3 3 )δ 1 1 ) 3 3 ) 4 λ 1 k Pl 3 3 2 2 a(τ d d ) ) + 2 (2π (2π λ 3 a(τ ) k k " (−p p )δ 1 4 d 3 3 p M Pl 4 (2π 2 Pl 2 2 )δ a(τ (τ, τ (τ, τ d d + Z Z ) (p)χ 2 (2π (2π + (p p 1 1 a(τ M 4 1 1 1 Pl 1 p δ Z λ λ p p ij, λ (τ 2 r r at lowest order, 3M + † −r 4 Pl r Z Z 1 1 1 3 (p M F F X X + 1 χ (p λ p  δ 1 2 t δ M − r r r × × × × (p 1 X X   X (p (p)Π δ . = = 12 1 δ iλ i × × × × 4 5 λ ij E E = 0 = = = Π ) ) 2 1 3 6 (τ (τ E E E E ) ) ) ) 2 2 2 2 λ λ p p (τ (τ (τ (τ γ γ 2 2 2 2 2 2 2 2 )ˆ )ˆ λ λ λ λ p p p p (τ (τ γ γ γ γ )ˆ )ˆ )ˆ )ˆ 1 1 1 1 λ λ p p (τ (τ (τ (τ ˆ ˆ γ γ 1 1 1 1 1 1 1 1 D D However, the other three are not explicitly independent of the direction of It is convenient to note that, to quadratic order in We calculate the two point correlation function with a single loop correction; the sub- λ λ λ λ p p p p ˆ ˆ ˆ ˆ γ γ γ γ D D D D order, which can be foundlinear using term the allows for explicit several expressions derivationsfor to for spinors be the one simplified. spinors can Similarly, given show using explicit that in expressions [60]; the lack of a script indicates which ofexplicitly independent the of seven vertices of eq. (B.2) was used. Four of the results are JCAP10(2019)018 ; 1 τ 3 (C.4) (C.5) )] −k 1
= ) 1 # 1 .
0 y ) cos(2x 1 ! ) , , ,
1 ∗ )


x ) and 1 1 cos(2x h.c. − h.c. 1 (τ 1 ) τ h.c. h.c. h.c. 1 1 cos(2x − 1 − a(τ 1 )] λ p 1 x − − − t 1 ∗0 # 0 ∗ ∗ ∗ ) )] −p ) ) )

1 1 ! )+2x 1 1 1 ) 1 ∗ (τ = ) (y (y (y 3 )+2x ) [sin(2 , (τ 1 r 3 3 3 k 1 1 1 cos(2x 1 1 1 , r k r k r k 2 1 )] x λ p 1 (τ )v u u u 1 t x )] 1 1 1 dx cos(2x 1 1 1 a(τ 1 " λ p sin(2x y y y 2x (y 1 t (τ 1
∂ ∂ ∂ r ∞ x (y 3 p − sin(2x 1 ) ) ) x v r r

k ) ∂ 1 1 1 ∗
− Z v v ) 1 ∗ − ) ) 1 ∗ ) (y (y (y 1 2 1 1  ) + 1 (τ r r r − x 1 2 x 1 1 ∗0 (y u u u 0 2 1 ) (τ λ p ) r H (y 0 1 H x t 1 1 ! + + + u r 2 λ p 1 2 1 ∗ ∗ ∗ ∗ Pl ) [sin(2x i u (τ 3 1 (ϕ ) ) ) ) )t 1 ∗ ) 1 3 dτ [sin(2 x 1 1 1 )+ 1 2 1 ) 1 1 r k dx V 4 1 1 1 1 )+ (τ x 4M 2 1 (τ (y (y (y dx x 1 a(τ 1 1 )u dx ∞ 1 r r r (y 1 (τ x λ p a(τ 1 − dx x λ p r 1 v v v 1 t (y ∞ t 1 λ p τ 1 1 1 Z ∞ – 19 – u r x (τ " y y y −∞ ∞ x 3 ∗ u 2  Z

x ∂ ∂ ∂ ) t Z ) r k Z 2 ∗ ) ,λ 2 ∗ ) ) ) 1 Z ) u ) 2 2 1 2 ) 1 1 1 1 (τ 1 , ,λ 1 ) λ 1 1 ,λ (y ,λ 1 p p 1 2 p 1 3 (y (y (y 1 r λ p 2 λ 3 (y ) 2 (τ )δ · ) 3 1 t λ r r r p ) 3 1 3 1 λ + r 3 1  k ; 1 2 1 [v p h δ v v v p p 1 )δ p 3 1 λ p 3 1 + )δ 2 p 1 ) [v ( 2 1 4 t 4 4 d Pl p Pl Pl 1 2 4 2 (2π p y a(τ Pl 3 3 3 3 1 1 1 1 2 (p p + ) p 4 1 y y y y p δ Pl 4 M Pl 1 M M (p M + 2 (τ, τ 1 1 1 1 2 Z 2 2 lim −∂ →0 + δ + 2 (τ 1 1 dy M ,λ 1 2 1 (p 1 1 1 λ 2 1 p dy dy dy dy r 8π 4M λ p δ − ,λ λ (p Z t F X 1 1 48π 12π (p (p δ δ 12π Z Z r Z Z λ λ δ δ 4 4 ( × × 12π δ 4 4 4 r r r r r ≡ = H iH iH H X X X X X , 7 iH 0, to compute superhorizon quantities. We arrive at − − − − ) × × × × × E ) ; = = 0 = = = = 1 → (τ 3 2 4 5 6 1 2 2 τ E E E E E E λ p 1 ) ) ) ) ) ) γ p (τ, τ )ˆ 1 (τ (τ (τ (τ (τ (τ 1 λ 2 2 2 2 2 2 2 2 2 2 2 2 p , which makes the problematic terms subdominant. (τ λ λ λ λ λ λ p p p p p p 2 1 1 F γ γ γ γ γ γ  λ p )ˆ )ˆ )ˆ )ˆ )ˆ )ˆ ˆ γ ∼ (τ (τ (τ (τ (τ (τ D To separate the integrals, we introduce the new variables 1 1 1 1 1 1 1 1 1 1 1 1  λ λ λ λ λ λ p p p p p p · ˆ ˆ ˆ ˆ ˆ ˆ γ γ γ γ γ γ 1 D D D D D D where Following reference [58], we impose thatp the limit is approached from an orthogonal direction, we also take JCAP10(2019)018 , ,  }  ∗ (C.7) (C.6) (C.8) 2 ) 1 −I (y 2 1 3 r y 3 Λ 1 )v 1 dy +2 (y 1 r Z µI r , X ={u − s 3  )] 2 I H 1 m integrals are logarithmi- −I 2r , 1 −2ξ 3 x  . 3  Λ 2 }− cos(2x },  ∗ 1 ) ∗ 4 , ) 1 x +2 4 −I Λ 1 1  3 − (y 2 3 4 log(x) (y r ) Λ 4 r µI 1 Λ − ,  )v d x 2 2 2 − , 1 2 ) +2 I 2 3 ) )| 1 − 1 0 (y 1 H I )| + 2 r (y 1 1 2 I Pl (y µI r (ϕ [sin(2 r (y  − µI −2ξ V 1 r ). Several of the |s 2 1 <{u 3 4M   range, adiabatic subtraction agreed with the 1 1 |s x <{s I dx ) ) /f y y − 2 1 1 , ξ 0 1 y y 1 3 ∞ – 20 – were evaluated analytically and exactly. Further- log(x) 1 1 I x  −2ξ · dy 2  2 Z )+2 ) 2 log(x log(x  dy dy 2 I 2 ) ,λ sin(ϕ ,λ Z 1 ) ) 2 1 2 )| p 1 2 2 λ Z Z r 1 ,λ p λ m p p 1 + 3 1 3 1 )δ  (y λ and + ) are defined in eq.). (3.7 We have 3 1 1 )δ 3 1 p p r r r 2 + + r 3 1 3 1 3 1 ≡ 1 2 p p X X X 1 4 4 1 1 )δ v p p p p Pl Pl (y µ (p p s 4 4 2 I Pl r Pl (p δ 4 4 4 Pl Pl Pl = = = + (p (p M M d p + 2 −| δ m 1 2 2 M δ δ 2 2 1 3 1 2 M M M ,λ 2 + 2 2 I I I 1 2 2 2 ,λ )| 1 (p 3π 9π ,λ ,λ 8πM 1 (p λ 1 δ 1 1 8π λ δ δ 1 4π 3π 6π (p λ λ 4 4 δ (y ) and δ δ λ δ ) and 4 r 4 4 4 4 H H (y , H /f r H H H H (|u − − 0 s c − = = = = 0 = = = H m = 1 6 4 2 7 3 5 7  E E E E E E E cos(ϕ ) ) ) ) ) ) ) E × ) (τ (τ (τ (τ (τ (τ (τ m 2 2 2 2 2 2 2 (τ 2 2 2 2 2 2 2 2 2 λ λ λ λ λ λ λ p p p p p p p ≡ λ p γ γ γ γ γ γ γ c )ˆ )ˆ )ˆ )ˆ )ˆ )ˆ )ˆ γ )ˆ m (τ (τ (τ (τ (τ (τ (τ 1 1 1 1 1 1 1 (τ 1 1 1 1 1 1 1 The remaining integrals are UV-divergent and need to be regularized. We thus introduce In reference [44], integrals 1 1 λ λ λ λ λ λ λ p p p p p p p λ p ˆ ˆ ˆ ˆ ˆ ˆ ˆ γ γ γ γ γ γ γ ˆ γ D D D D D D D D where the functions a ultraviolet cutoff Λ.expectation (We values discuss can renormalization be further expressed below.) in terms After of some the algebra, three the integrals cally divergent and areleaves regulated the by horizon the and finitethe amount the fact of end that e-foldings of the between inflation.fluctuations when fermions even This the have when logarithmic mode a outside divergence nonzeroinflaton the is average perturbations horizon; a density analogous of that consequence behavior reference continues of was [44]. to observed source in graviton the sourced where more, it was found that in the result obtained by simply droppinglarizating off term the generated divergent pieces. byeven adiabatic Outside outside of regularization of this dominates the regime, UV theadiabatic the limit, regularization. calculated regu- and contribution Therefore, furthermore, as the in value reference of [44], the we term regularize depends by simply on dropping the the order of JCAP10(2019)018 , ,        (C.9) 1 1  (C.10) (C.11) + 1 +1 − +1 −      2 + 1 2 2 2 2  ξ ξ ξ ξ  2 2 ξ + 4ξ +4 +4 +4 +4 2 2 2 2 2 , +4 µ µ µ µ µ 2 + 4ξ  2 µ 2 , can be evaluated p p p p p 3 µ p I  2π 2π 2π 2π 2π 12ξ
p        2π 2  − ξ  2π from reference] [44 along  + 1 2  2  2 2 +4 2 8 )csch )csch )csch )csch )csch I 2 8ξ 11µ )csch µ + 4ξ − )csch + + 4ξ 2 p 2 and µ 2 4 µ 8 2π 1 µ µ p I  ) sinh(4πξ sinh(4πξ sinh(4πξ sinh(4πξ sinh(4πξ p − E      sinh(4πξ 2π γ   2  2π sinh(4πξ     2 2 ξ      2 )csch  2 2 . The third integral,  2 +4ξ ξ +4 2 −2ξ −2ξ 2 11ξ 2 + 4ξ )csch +4ξ 2 +4ξ 2 µ 2 2 µ 2 )csch + +4ξ  ) µ µ √ 2 µ 4 +4ξ +4ξ E p – 21 – + µ 1 √ √ 2 2 γ
sinh(4πξ p µ µ 8ξ + + 2ξ √ 2 2(log(2Λ) +  +  √ 2ξ √ 2ξ − ξ 2π +1     −i − sinh(4πξ i i i i 2ξ   2 µ  sinh(4πξ
+4 H i H H H H − 8ξ 2 19 16  − H + + + + 1 µ
−  −      1 2 )csch + 12 2 2 2
p   2
4µ [4(log(2Λ)+  −2ξ −2ξ 2 2 +4ξ +4ξ )
16ξ + 1 −2ξ 11 2 2 7µ E 1 3 2 µ µ  −2ξ − γ 2 +4ξ +4ξ − 2 +1 + √ √ 2 2 2 + 3i) + 1 2 µ 4 +4ξ
ξ µ µ sinh(4πξ + + 2 µ 1 +4ξ + 6iξ 2 + µ √ √ 2ξ 2ξ 2 16ξ 4 (4ξ 2 −     µ √ 1 4 2 +16  − 16 −i −i −i −i √ 2ξ 8ξ ξ 2 7µ 2 ). + 4ξ  + i −i H H H H µ − − 2 − 2 +8 .     µ 2 2 2 2 3 2 . ln(ξ [4(log(2Λ)+ µ µ µ +H +H ξ + + − − 2Λ 2 −26µ µ 2 p 2 2 ξ 2 ξ 2  3 Λ 2 4 4 1 1 ξ 4 16 µ −6µ µ ξ 2 1 − + + +6ξ 8 + + +  4π µ ξ π µ 4 µ 2 8 3 3 We reproduce here the exact analytic results for = ≈ − = ≈ − ≈ = Λ 3 2 1 I I I Finally, the third integral evaluates to divergent pieces, noting that inthese the approaches regime agree. in which adiabatic regularization is well-behaved, with their finite piece in the regime The second integral reads with the same techniques. The first integral is JCAP10(2019)018  ) 1 (D.1) (C.12) (C.13) (τ 1 s k )u 1 , (τ 1 ) and, because of r k 2 limit reads u 2 p )] ξ , 1 − + , , τ > τ ) 1 1  1 , 1 1 τ (p (τ ). 1 δ log(x) 2 , s k 2 , sin(p  log(x) log(x) 3 3 (ξ ,λ 2 )v ξ ξ 1 3 µ − 3 log(x) ξ 1 O λ 2 2 ξ ξ ) 2 3 δ (τ µ µ 2 1 2 ξ µ 1 τ µ ) µ 2 r k 2 1 2 2 2 ) ,λ µ ,λ 3 1 p 1 2 ,λ 1 ) p λ rsv 1 p λ 2 log(x)  + λ 4 )δ cos(p Pl 3 1 3 1 p 3 δ )δ 1  3 1 + 2 ξ p p 2 3 1 ) 3 1 . 1 p 1 2 p + 5 3 1 2 τ p p p (p sr 4 4 Pl Pl µ 4 1 p 1 Pl ) δ p 4 (p Pl 4 + 4 Pl 2 9π M + [p − δ 4 Pl 1 M (p + 1 2 ,λ H 1 1 δ 1 3 1 ,λ 8 2 λ (p 9π M 9π M π M  dτ 1 (p δ δ + h.c. ,λ 2 λ 1 9π M δ τ − (p 4 4 1 0 we have – 22 – 9π M δ 4 k  1, using the interaction term (B.1) and the mode δ λ 4 ∗ 4 Z 3 = H H δ 1 ) H ) → 4 2 k 2 4 2 H H 1 3 τ ,λ 4 H (τ − λ − − d 1 (2π 1 λ s k = = 0, = = = = = quartic 6 1 u n-th harmonic number. From this we find final results for )δ Z p ∗ 1 2 5 3 4 6 7 E 2 ) ) E E E E E E E p 2 4 Pl 3.1. We follow the approach, including the approximations, ) ) ) ) ) ) ) r,s (τ X (τ + 2 2 M ) (τ (τ (τ (τ (τ (τ (τ 1 λ i p 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 r k γ λ λ λ λ λ λ λ p p p p p p p u 12 )ˆ + γ γ γ γ γ γ γ (p )ˆ )ˆ )ˆ )ˆ )ˆ )ˆ )ˆ δ 2 − (τ 4 τ 1 1 ∗ (τ (τ (τ (τ (τ (τ (τ 1 denotes the λ p ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 H 2 ˆ γ λ λ λ λ λ λ λ p p p p p p p (p n 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ γ γ γ γ γ γ γ D (τ = τ H 1 D D D D D D D 1 . The parity violation is evident because this term gives a contribution s k E ip v ) e ∗ X ); then after taking the (4) ) ,3 int 2 (τ 1 vertices 2 2 2 H k dτ λ p (τ γ 1 1 τ )ˆ r k → (τ 1 Z 1 1 rsv p λ p (  The evaluation of these integrals is complicated by the fact that the integrand is oscil- ˆ γ + × × D 1 lating rapidly. Therefore, we perform a Wick rotation, noting that from which we conclude that the dominant contribution in the which we also writethat the as parity eq. violating (3.14)denoted vertex of that by the originates from main the text. portion of We interaction conclude Hamiltonian this appendix by noting to the two-point functionhelicity; of however, the the contribution graviton is which sub-dominant, is scaling as proportional to the sign of the graviton D Details of theIn cubic this loop appendix, computation wevertices shown calculate in the the contribution right from panel the of one figure loop diagram with two cubic each loop, functions calculated in section used in reference [44]. To(k start with, we drop the external momentum within the loop integral In these equations JCAP10(2019)018 , ) 2 , # x # .c. (D.4) (D.3) (D.2)  (1+ +h ∗  2 +h.c. ∗  should be   1
∗  −x ∗ 1 1 y 4 ∗ e  ) 1 2 y x  1 2 1 to find − 2 . 2 x x 1 1 y 1 x x E 1 2 y
x dx 2 ) γ
dρ x x −i 1 ) (−iy x −i ∞ 1 1  −iy r x  +3 −i s Z −i ∗ = −r  ρ (−iy −r d  r above. Physically, this )) x (−iy 2; however, we approxi- d 1 r ∗ s −r ∗ s ∗  C −r ∗  1  )d 1 1  y 1 1 )d 1 2 y π/2, and therefore we expect 1 1 +3log x y 1 2 1 2 x x y
x sinh(x )) 2 x 2 x x 1 ). → − x β (−iy −i )) ) r (−iy −i α −i 1 1  r d −i plane,  r d  (−iy −1 r  sin(α d r 2 r )+ d r d x ρ s 1 )+ (−iy + d ) ) piece, which is regulated by the finite 1 r + ∗ + 1 ∗ − cosh(x = ∗ + ∗  )s 3coth 1 1 ∗  2 1  1 (−iy x 1 1  y 1 − (x (−iy , and functions such as 1 2 y 1 1 x y 2 (−iy 1 1 −r 2 1 2 x x y 4 1 x τ r −r 2 x x x (−iy 1 x 3β )s d dx r x 1 )s −i d − – 23 – 1 (x) and then substituting −i −i −k ), x ∞  )+ r ), we can perform the integral in β −i x  1  s )+ Z =  r −r (−iy 1 −1 r −r r s d 2 (−iy 1 2 r s d s ∗ ∗ y ,λ cos(α (−iy s ,λ ∗ ∗ 1   1 ρ r λ 3   1 1 (−iy coth 3 1 3 1 λ d 3 = tan(α r 1 1 1 1 " y y p 3 p )δ ) = 2 " 1 1 2 2 1 y y )δ x x 2 1 4 4 β )d Pl Pl 2 2 2 y 2 1 x x x x 1 , and 3β 1 (1) factor. p x x 1 y 2 p x M M τ 1 O + −i −i 2 2 + 1 dy 1 −i −i (−iy β 1   dy dβ r (−iy   r r Z (p −p r s 60π (p r r s s δ Z ∞ s 540π δ s s 1 r 4   = 4   Z X r 2 × × +2( H X H × × × × +2( x − × , = 1 = τ E 1 , the integral is exponentially suppressed as ) E 1 ) y (τ 0 limit. This is dominated by the log(ρ −p 2 2 (τ λ p 2 2 = λ γ → p )ˆ γ 1 ∗ )ˆ x ρ (τ 1 1 (τ The true region of integration is shown on the left side of figure After further substituting Next we introduce polar coordinates in the λ p 1 1 ˆ λ γ p ˆ γ D D number of e-foldings between whensame the behavior mode was leaves also the observed horizon in and the the end quartic of loops inflation; in the appendix in the mate the integral by usingthat the at fixed region of integration shown on the right side. We show below where interpreted as first complex conjugating this to contribute at most a the nested integrals, both must be Wick rotated in the same direction. This gives logarithmic behavior is acontinues to consequence source of graviton the fluctuations even fact once that it the is nonzero outside fermion the horizon. energy density JCAP10(2019)018  , o 1 (D.6) (D.5) (D.8) (D.7) −irM ) 1 −2y 2 1 , . e + h.c. )) ))
−2y (−y e i i , ,
) 2,−r π/2, and so this + 2ξ + 2ξ −4πξ B 1,r e → 2,−r B −irM −irM −4iMr 1 1 2,r 2 2 1 1 (M (M y α A B 2,r ), + 1) 2 )) 1 2,r 2,r µ ), A −2βy x B B 2ξ e 2 (−x) (−x) − + 2 1 x x µ − irM 1,r y e e B ) sinh (π ) sinh (π 2,r 1 − + Γ(2irM 1,−r B (M + dy Γ(2irM B −2πξ −2πξ 1 e e 1,r 1,−r −x ) Z 1,r −x + 1) e 1r 2r A e B +2y Γ(ir (A plane. Right: the region of integration B B r Γ (2irM Γ (2irM 1,r X 2 Γ(−2irξ +1/2 + + + x B +1/2 + 2irξ β 1 − dβ i + 2irξ −Mπr −2πMr 2 ∗ −irM −irM 1 2y e e 3 2 3 2 ρ x e x −irM ∞ + +irM +irM −irM 2 2 1
1 2 1 2 2,r Z ir ir x x ) A Γ(irM irM (2x) ∗ 2,−r ) (2x) Γ(irM −2iMr 1 = = −x −x 2 1 1,r e e A – 24 – ≈ ≈ 1r 2r 1 2,r B B +i y log(ρ (−y −A x 0) 0) A −2πξ −2πξ 2 h
) 2 e e 2 ,λ µ 1 ξ 2r 1r , , λ 2,−r 3 1 + ) ) (1) uncertainty. p B )δ (2x > (2x > O 2 + 4 4 Pl ≈ A ≈ A 1−2iMr 2,r p 2 β 1,−r )) M µ + 1) A 0) 0) 2 1 2 + A 2 y M p 1 µ 1,r ∗ + 2,r ≡ (p − irM Γ(−2irM Γ(−2irM x 180π C δ 1/2−2irξ,iM (2x < (2x < −A 4 − ∗ 2 (2ξ M −1/2−2irξ,iM x x W 1,−r  H −2πξ −2πξ W 2 1 B e e −4πrξ = e 1,r β y , the integrand is exponentially suppressed as the polar angle 2 E Γ(−ir 1 A ) µ +irM +irM y 1,r Γ(−2irξ 1 1 2 2 (τ 2 2 2 2 1/2+2irξ,−iM 3C +i + 8 λ p −1/2+2irξ,−iM − − γ W nh )ˆ W = = (τ × With these approximations, we find Finally, we approximate the Whittaker functions as in reference [44]. Along the positive 1 1 1r 2r λ p A A ˆ γ D where and we note that theaxis, Whittaker we functions use are even in their second index. Along the negative Figure 2. Left: the actual region of integration in the used. At fixed axis we use (defining replacement introduces at most an JCAP10(2019)018 only, (D.9) (D.10) (D.11) (D.12) (D.13) A 1 either ,  , and where ξ  −4  4) 4) ), , ∗ − − −2 ,  ξ → ∞. We subtract 1 and )  ∗ and taking the limit 16 p integral may now be 25) . log(ρ  −8iξ −8iξ 1 + 3  µ − terms correspond to the ξ 2 2 dy log(ρ 8  ) 2 . 3 A 4ξ + 4, + 4, µ ξ πµ − )), 2 2   ,λ µ 2 + 1) 1 (4iξ (4iξ 2 λ + 2ξ part which has a divergent piece. )µ ,λ ∗ −1 −1 sinh )δ ∗ 1 + 12 log( 2 3 1 A λ B B 9 8 p (M p π/2). The integral as 1 + )δ 3 1 − + 2iξr 4 P 2 3/2 3/2 + p log(ρ  ξ ξ → . In the regime p 3−4iξ dβ 1 M ) 2 4 ξ 1−2iMr Pl ∗ 4 2 limit, is proportional to + ,λ (p (α 4πξ 4πξ 2 1 µ , this contribution to the graviton two-point 1 δ ξ ) e e λ 1 3 1 4 )πr sinh( + 6H y p )Γ(iMr (p )δ H log(ρ δ 2 4 45π M Pl piece which we renormalize away is +3 +8iξ −8iξ 2 4 p – 25 – → ∞ × 2 2 is. Regardless, the mixed term (D.11) dominates, M ,λ 17 17 H 4iξ A 2 1 + 2 2 β Γ(2iMr 3 1 λ µ + 2iξr 1 4 4 p 16 integral. We separate this into parts involving )δ µ µ (6H (0.1) (p 2 4 Pl 240π 2 = δ πMrcsch(2 p dβ ξ 4 3/2 3/2 M ξ + π π 2 ∼ O 9 AB H M 4 Γ(iMr 16 1 E i) i) − ) − = = AB (p − δ 180π E (τ = ) 2 2 4 1,r 2,r λ p C C (τ γ H +(1 +(1 + AA 2 2 limit, the purely )ˆ λ p E ξ = ) γ (τ )ˆ 1 1 (τ λ p contribution is BB 2 2 (τ ˆ γ λ p 1 1 E D B λ ) p γ )ˆ from the change to polar coordinates. This expression is reported as eq. (3.17) ˆ γ (τ ) denotes the incomplete beta function. This term has a piece which scales as D τ (τ 2 2 1 1 1 λ p p λ p γ ˆ )ˆ γ n-th harmonic number. The mixed term is (x, y D n (τ ∼ − 1 1 0. B λ p ∗ and a piece which scales approximately as Taking the large Eq.) (D.8 shows that, as claimed, at fixed ˆ γ ρ → is the D  only, and mixed. 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