Parity-violating CMB correlators with non-decaying statistical anisotropy
Nicola Bartolo, Sabino Matarrese, Marco Peloso, Maresuke Shiraishi
UUMN-TH-3435/15IPMU15-0067
Prepared for submission to JCAP
Parity-violating CMB correlators withnon-decaying statistical anisotropy
Nicola Bartolo, a,b
Sabino Matarrese, a,b,c
Marco Peloso d andMaresuke Shiraishi a,b,e a Dipartimento di Fisica e Astronomia “G. Galilei”, Universit`a degli Studi di Padova, viaMarzolo 8, I-35131, Padova, Italy b INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy c Gran Sasso Science Institute, INFN, viale F. Crispi 7, I-67100, L’Aquila, Italy d School of Physics and Astronomy, University of Minnesota, Minneapolis 55455, USA e Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),UTIAS, The University of Tokyo, Chiba, 277-8583, Japan
Abstract.
We examine the effect induced on cosmological correlators by the simultaneousbreaking of parity and of statistical isotropy. As an example of this, we compute the scalar-scalar, scalar-tensor, tensor-tensor and scalar-scalar-scalar cosmological correlators in pres-ence of the coupling L = f ( φ )( − F + γ F ˜ F ) between the inflaton φ and a vector field withvacuum expectation value A . For a suitably chosen function f , the energy in the vector field ρ A does not decay during inflation. This results in nearly scale-invariant signatures of brokenstatistical isotropy and parity. Specifically, we find that the scalar-scalar correlator of primor-dial curvature perturbations includes a quadrupolar anisotropy, P ζ ( k ) = P ( k )[1 + g ∗ (ˆ k · ˆ A ) ],and a (angle-averaged) scalar bispectrum that is a linear combination of the first 3 Legendrepolynomials, B ζ ( k , k , k ) = (cid:80) L c L P L (ˆ k · ˆ k ) P ( k ) P ( k ) + 2 perms, with c : c : c = 2 : − c (cid:54) = 0 is a consequence of parity violation, corresponding to the constant γ (cid:54) = 0).The latter is one of the main results of this paper, which provides for the first time a clearexample of an inflationary model where a non-negligible c contribution to the bispectrumis generated. The scalar-tensor and tensor-tensor correlators induce characteristic signaturesin the Cosmic Microwave Background temperature anisotropies (T) and polarization (E/Bmodes); namely, non-diagonal contributions to (cid:104) a (cid:96) m a ∗ (cid:96) m (cid:105) , with | (cid:96) − (cid:96) | = 1 in TT, TE, EEand BB, and | (cid:96) − (cid:96) | = 2 in TB and EB. The latest CMB bounds on the scalar observables( g ∗ , c , c and c ), translate into the upper limit ρ A /ρ φ (cid:46) − at γ = 0. We find that theupper limit on the vector energy density becomes much more stringent as γ grows. a r X i v : . [ a s t r o - ph . C O ] J u l ontents Pseudo-scalar fields, which can naturally emerge in global symmetry breakings, have oftenbeen employed in models of cosmological inflation (e.g., see refs. [1–11]). A typical model ofa pseudo-scalar coupled to a gauge field is L = −
12 ( ∂φ ) − V ( φ ) − F − φ f F ˜ F , (1.1)where φ is the pseudoscalar field and 1 /f expresses the strength of the axial coupling. Dueto the motion of the inflaton φ ( t ), this coupling enhances one of the two helicity states ofthe gauge field during inflation, inducing parity violation. This violation can be imprintedon the gravitational waves through the gravitational interactions of the gauge field. Mostcommon observables of such chiral tensor perturbations are the cross correlations betweentemperature anisotropies T/E-mode polarization and B-mode polarization (TB/EB) of theCosmic Microwave Background (CMB) [3, 12–15]. Since the inflaton typically speeds upduring inflation, the production of gauge quanta can potentially increase at smaller scales,giving rise to a gravity wave signal observable at terrestrial interferometers [16–18]. Thechiral nature of this signal can be probed by combining measurements from multiple inter-ferometers [19, 20]. In addition, the f ( φ ) F ˜ F interaction can also generate large primordialnon-Gaussianity [13, 21–28], and primordial black holes [29]. Finally, this coupling has alsobeen employed in models of inflationary magnetogenesis [30–38].In the framework of the pseudoscalar inflation, impacts due to statistical anisotropyhave been recently analyzed, by introducing a coherent vacuum expectation value (vev) ofthe gauge field [39–41]. In such a case, the primordial correlators are affected by the simulta-neous breaking of parity and statistical isotropy. This generates characteristic (cid:104) a (cid:96) m a ∗ (cid:96) m (cid:105) correlators of the CMB temperature anisotropies and polarization. Specifically, one findsnon-vanishing TT, TE, EE and BB correlators for | (cid:96) − (cid:96) | = 1, and non-vanishing TB– 1 –nd EB correlators for | (cid:96) − (cid:96) | = 2 [41]. Mathematically, all the CMB signals satisfying | (cid:96) − (cid:96) | = odd (even) in TT, TE, EE and BB (TE and EB) are parity-odd. These can berealized only when the primordial correlators include parity-violating information. On theother hand, the primordial correlators can generate nonzero CMB signals satisfying (cid:96) (cid:54) = (cid:96) ,only when they break rotational invariance. Hence, the above signals become distinctiveindicators of broken parity and rotational invariance in the primordial Universe. In ref. [41], we have presented a proof of the existence of these interesting signals,analyzing the simplest lagrangian (1.1). In this case, however, the vev of the vector field(which is the quantity breaking statistical isotropy) decays very rapidly during inflation, ρ A ∝ A ∝ a − (where a is the scale factor), so that one needs to assume that a vector vevis present as an ad-hoc initial condition when the CMB modes left the horizon. The rapiddecrease of ρ A then implies that only the largest CMB modes can be affected by it, so thatthe induced non-diagonal correlators are highly red-tilted (they are ∝ k − ). It was found inref. [41] that only CMB multipoles with (cid:96) (cid:46)
10 can be affected at a detectable level.As anticipated in ref. [41], a more natural initial condition, and a more interesting signal,can be obtained if the kinetic term of the vector field is modified as in the Ratra mechanism[42], so to allow for a nearly constant ρ A . Specifically, if the kinetic term is − f ( φ )4 F , and ifthe functional form of f ( φ ) is chosen such that the background evolution satisfies f ( φ ( t )) = a − ( t ), then the vector has a constant electric vev (we are using standard electromagneticnotation for simplicity, although we do not need to assume that A µ corresponds to thestandard model photon). This time dependence can be achieved through a suitable relationbetween f ( φ ) and the inflation potential [43].A model that provides a non-decaying ρ A and that breaks parity has been recentlyconsidered in ref. [36] as a model for primordial magnetogenesis [36]. It is characterized bythe lagrangian L = −
12 ( ∂φ ) − V ( φ ) + I ( φ ) (cid:18) − F + γ F ˜ F (cid:19) , (1.2)where γ is a constant. Analogously to the well studied γ = 0 case [44–54], we showin this work that, for I ( φ ) ∝ a − , the interaction between φ and A µ leaves nearly scale-invariant signatures on the primordial cosmological correlators. Differently to the γ = 0case, however, due to the I ( φ ) F ˜ F interaction, one vector helicity state is produced with agreater abundance that the other one. As mentioned above, this violation of parity can affectthe CMB correlators through the gravitational interactions of the vector field.In addition, a stronger effect arises from the direct coupling between the vector field andthe inflaton. We mentioned that the interaction in eq. (1.2) is responsible for (i) maintaininga nearly constant energy in the vector field during inflation, and (ii) enhancing one helicity ofthe vector field with respect to the other one. These effects are due to the classical evolutionof I ( φ ) and therefore to the vev of the inflaton field. However the same interaction term(1.2) also couples the vector quanta to the inflaton perturbations. As we will see, in thismodel ρ A /ρ φ (cid:28) These discussions rely on the assumption that there is no mechanism breaking parity and isotropy at latetimes. In the proposal of ref. [36], the field φ does not need to be the inflaton field. The explicit breaking of parity can be avoided if the F and the F ˜ F term are proportional to two differentfields (as for instance in supergravity) that have vevs that evolve during inflation maintaining a constant ratio[36]. – 2 –s the adiabatic perturbation of the model, with negligible error. As we show below, thecouplings δφδA and δφδA encoded in eq. (1.2) modify both the power spectrum and thebispectrum of the primordial scalar perturbations. For the power spectrum of primordialcurvature perturbations, one obtains the ACW [55] quadrupolar term P ζ ( k ) = P ( k ) (cid:104) g ∗ (ˆ k · ˆ A ) (cid:105) (1.3)with a nearly scale invariant g ∗ parameter. For the (angle-averaged) bispectrum, one obtainsthe first three terms of an expansion in Legendre polynomials [52]: B ζ ( k , k , k ) = (cid:88) L =0 c L P L (ˆ k · ˆ k ) P ( k ) P ( k ) + 2 perms . , (1.4)with c : c : c = 2 : − The three parameters { g ∗ , c , c } arise due to the breaking ofstatistical isotropy, and were already found in the − f ( φ )4 F model [51, 52]. The non-vanishingof c instead requires both breaking of statistical isotropy and parity. This is the first timethat, within these conditions, a non-negligible contribution c to the curvature bispectrum isgenerated during inflation. As we show in section 4.1, for γ = 0, the fact that such signatures have not beenobserved in the CMB data enforces the upper limit ρ A /ρ φ < ∼ − . In our work we studythe γ (cid:54) = 0 case. Our results are summarized in figure 1, where we show the upper limit on ρ A /ρ φ as a function of γ , starting from γ >
1. Strictly speaking, our analytic computation isvalid for γ (cid:29) . However, the limit is continuous in γ , and figure 1 shows that extrapolatingour lines in the γ < ∼ γ = 0. To givea measure on how strongly the upper limit decreases with γ , we note that our computationprovides the upper limit ρ A /ρ φ < ∼ × − at γ = 1, and ρ A /ρ φ < ∼ × − at γ = 2.Although having such a small ρ A is a mathematical possibility, in section 4.2 we arguethat a greater energy in the background vector field should be expected, simply from theaddition of IR modes [51]. This essentially rules out the model (1.2), under the assumptionthat it produces a constant vector energy density, that γ > O(5), and that φ is the inflaton.Our results do not immediately extend to the case in which φ is not the inflaton. However,we expect that also in that case the primordial perturbations will be significantly affected bythe effects we have studied, mostly due to the linear coupling (which exists at least due togravity) between δφ and the inflaton perturbations [25].This paper is organized as follows. In the next section, we review an inflationary modelbased on the lagrangian (1.2) and we discuss how it can lead to a constant vev of thegauge field. In section 3, we compute the scalar-scalar, scalar-tensor, tensor-tensor andscalar-scalar-scalar correlators of primordial curvature perturbations, and find the distinctobservable predictions of the lagrangian (1.2). In section 4, we estimate the observationalbounds on the energy density of the gauge field vev from the latest Planck constraints on g ∗ , c , c and c . This result is discussed in section 4 and in the concluding section. Notice that, due to a non-vanishing vev of the vector field, a statistical anisotropic bispectrum is actuallygenerated and, after an angular average (see the discussion after eq. (4.2)), it takes the form (1.4). For studiesof CMB bispectra that break statistical isotropy see refs. [56, 57]. See refs. [52, 58] for discussions on another possibility to generate non-negligible c from large-scale helicalmagnetic fields at the radiation dominated era. – 3 – A model for breaking parity and statistical isotropy
Let us consider the action [36] S = (cid:90) d x √− g (cid:34) M p R − ∂ µ φ∂ µ φ − V ( φ ) + I ( φ ) (cid:18) − F µν F µν + γ F µν F µν (cid:19)(cid:35) , (2.1)with the parameter γ being a constant. Differently from ref. [36], we identify φ with theinflaton field, and compute how the couplings in eq. (2.1) affect the primordial perturbations.For γ (cid:54) = 0, the coupling of the inflaton field to the vector explicitly breaks parity, as theproduct F is a scalar, while F ˜ F is a pseudo-scalar quantity. This coupling also affects theinflaton perturbations, since expanding φ = φ ( τ ) + δφ ( τ, x ), and denoting I ( φ ) ≡ I ( τ ),one has I ( φ ) = I + I (cid:48) φ (cid:48) δφ ≡ I ( τ ) + δI ( τ, x ) . (2.2)In principle, O( δφ ) terms in the expansion of I ( φ ) could be considered, and included in thecomputation of the bispectrum. As discussed in ref. [51] (see also ref. [59] for an explicitcheck), these higher terms give a subdominant contribution, and we can therefore disregardthem.We assume that the function I evolves in time during inflation as I ( τ ) ∝ a n ( τ ) , (2.3)and it then sets to a constant after inflation (when φ sets to a minimum). Without lossof generality, we can take this value to be 1. Strong coupling considerations put the n > ∝ I . For n >
0, this implies a very large coupling during inflation. Even if noreal quanta of the charged particles exist during inflation, loop of virtual charged particlesare out of perturbative control, which puts in question any perturbative result obtained fromthe model [60].
A non-vanishing vev of the vector field during inflation leads to anisotropic expansion. Itis well known (see for instance ref. [51]) that, for γ = 0, consistency with the CMB resultsrequires that the energy density of the vector field is much smaller than that of φ . We showbelow that the limit on the vector field energy becomes even stronger for γ (cid:54) = 0. Therefore,we can neglect the departure of the background geometry from the FLRW metric, and we use ds = a ( − dτ + dx ) in our computations [51]. As in the standard case, inflation is supportedby the inflaton potential, and the standard slow-roll condition applies (cid:15) ≡ M p (cid:16) V dVdφ (cid:17) (cid:39) We use the following notation: F µν ≡ ∂ µ A ν − ∂ ν A µ is the field strength of A µ , while ˜ F µν = η µναβ √− g F αβ isthe dual tensor, with η = 1. We also use M p = 1 / √ πG , where G is Newton’s constant. In the following,dots (primes) denote derivatives with respect to physical (conformal) time, while H ≡ ˙ aa , where a is the scalefactor. As we mentioned, the required time dependence can be obtained by suitably relating I with the inflatonpotential. See ref. [43] for details. – 4 – (cid:16) ˙ φHM p (cid:17) (cid:28)
1, with negligible corrections. At zeroth order in slow roll, the dependence ofthe scale factor on conformal time is a (cid:39) − ( Hτ ) − .We do not need to identify the vector field with the standard model photon. Wenonetheless use the “electromagnetic” convention with E = − I ( τ ) a A (cid:48) , B = I ( τ ) a ∇ × A , (2.4)where the Coulomb gauge A = ∇ · A = 0 has been assumed. In this notation, the energydensity in the gauge field acquires the familiar form ρ A = E + B . For convenience, weintroduce the canonical field V ≡ I ( τ ) A , and we expand it in a background value plusfluctuations as V = V (0) ( τ ) + δ V ( τ, x ) (consistently with spatial homogeneity, we haveimposed that the background value depends on time only; we note that with this choice theterm proportional to γ in eq. (2.1) does not contribute to the background evolution). Thegauge field vev then satisfies (cid:16) V (0) (cid:17) (cid:48)(cid:48) − I (cid:48)(cid:48) I V (0) = 0 , (2.5)leading to vanishing magnetic component, B (0) = 0, and to an electric component E (0) thatdepends on the parameter n . In particular, for n = − I (cid:48)(cid:48) I = τ ), a time-independent vevarises as E (0) ( τ ) = E vev . (2.6)This is the case that we study in this work. The fluctuations of the gauge field have two helicity states ( λ = ± δV i ( τ, x ) = (cid:90) d k (2 π ) / (cid:88) λ = ± δ ˆ V ( λ ) k ( τ ) (cid:15) ( λ ) i ( k )e i k · x , (2.7)where the quantum field δ ˆ V is decomposed as δ ˆ V ( λ ) k ( τ ) = a λ ( k ) δV λ ( τ, k ) + a † λ ( − k ) δV ∗ λ ( τ, k ) , (2.8)in terms of creation and annihilation operators that obey the algebra [ a λ ( k ) , a † λ (cid:48) ( k (cid:48) )] = δ λλ (cid:48) δ (3) ( k − k (cid:48) ). The quadratic action gives the evolution equation for the mode functions: δV (cid:48)(cid:48) λ + (cid:18) k + 2 λkγ I (cid:48) I − I (cid:48)(cid:48) I (cid:19) δV λ = 0 . (2.9)The parity violating term results in a contribution that differs in sign for the two helicities.Following ref. [36], we define the coupling strength parameter ξ ≡ − nγ , (2.10) The polarization vector (cid:15) ( λ ) i ( k ) satisfies ˆ k i (cid:15) ( λ ) i (ˆ k ) = 0, η ijk ˆ k i (cid:15) ( λ ) j (ˆ k ) = − λi(cid:15) ( λ ) k (ˆ k ), (cid:15) ( λ ) ∗ i (ˆ k ) = (cid:15) ( − λ ) i (ˆ k ) = (cid:15) ( λ ) i ( − ˆ k ) and (cid:15) ( λ ) i (ˆ k ) (cid:15) ( λ (cid:48) ) i (ˆ k ) = δ λ, − λ (cid:48) . – 5 –nd for definiteness we assume ξ >
0. As we are interested in n = −
2, this means that theparameter γ is positive. In the opposite case, one simply needs to interchange δV + ↔ δV − in the results below. Given that I ∝ a n , one finds, at zeroth order in slow roll, δV (cid:48)(cid:48) λ + (cid:18) k + 2 λk ξτ − n ( n + 1) τ (cid:19) δV λ = 0 . (2.11)Ref. [36] provided the solution to this equation with the standard adiabatic vacuuminitial condition for arbitrary n . They obtained a particularly simple expression for ξ (cid:29) n = −
2, this expression reads δV + ( τ, k ) (cid:39) − e πξ ξ / τ − √ πk / , | kτ | (cid:28) ξ (cid:28) , (2.12)while the negative helicity mode δV − ( τ, k ) is produced on a much smaller and negligibleamount [36]. Using electromagnetic convention, we can write the following power spectra inthe long wavelength regime (cid:10) δX i ( τ, k ) δY j ( τ (cid:48) , k (cid:48) ) (cid:11) ≈ δX + ( τ, k ) δY + ( τ (cid:48) , k ) (cid:15) (+) i (ˆ k ) (cid:15) (+) j (ˆ k (cid:48) ) δ (3) ( k + k (cid:48) ) , (2.13)where X, Y = E, B and δE + ( τ, k ) = − e πξ ξ / H √ πk / = 3 kτ δB + ( τ, k ) . (2.14)This expression indicates that, on superhorizon scales ( − kτ (cid:28) To compute metric perturbations, we decompose them into scalar and tensor perturbationsas standard [61], and we work in the spatially flat gauge for the scalar sector, δg scalar ij = 0. Inthis gauge, the spatial part of the metric fluctuation is given by the gravitational wave aloneas δg ij = a h ij . Moreover, since the energy density in the vector field is much smaller thanthat in the inflaton, we can identify the scalar curvature perturbation ζ ≡ − H δρ ˙ ρ (cid:39) − H ˙ φ δφ ,while the gauge field perturbations δA i are isocurvature modes that are produced as discussedin the previous subsection, and will affect ζ and h ij through their couplings to them. We have already discussed the quantization of the δA i modes. The curvature and tensorperturbations are decomposed, respectively, as ζ ( τ, x ) = (cid:90) d k (2 π ) / ζ k ( τ )e i k · x , (2.15) h ij ( τ, x ) = (cid:90) d k (2 π ) / (cid:88) λ = ± h ( λ ) k ( τ ) e ( λ ) ij ( k )e i k · x , (2.16) In the spatially flat gauge, there are also metric perturbations δg and δg i . These are non-dynamicalmodes that are integrated out, and induce gravitational couplings between the gauge quanta and ζ . As theterm proportional to γ is a topological term, no metric perturbations enter there, so these couplings originatefrom the I F term, and are given in ref. [62]. Such couplings are suppressed (technically, by an (cid:15) factor [62])with respect to the direct δφδA couplings present in eq. (2.1), and so we can simply disregard them, and set δg = δg i = 0 [41, 51, 62]. – 6 –here we have used the polarization tensor given by a product of the polarization vector as e ( λ ) ij ( k ) ≡ √ (cid:15) ( λ ) i ( k ) (cid:15) ( λ ) j ( k ). The quantized fields are expressed with the operators satisfying[ a λ ( k ) , a † λ (cid:48) ( k (cid:48) )] = δ λλ (cid:48) δ (3) δ ( k − k (cid:48) ), respectively, asˆ ζ k ( τ ) = a ( k ) ζ ( τ, k ) + a † ( − k ) ζ ∗ ( τ, k ) , (2.17)ˆ h ( λ ) k ( τ ) = a λ ( k ) h ( τ, k ) + a † λ ( − k ) h ∗ ( τ, k ) . (2.18)We treat the effect of the gauge perturbations perturbatively, and indicate the scalarand tensor correlators as (cid:68) ˆ ζ k ˆ ζ k (cid:69) = (cid:68) ˆ ζ k ˆ ζ k (cid:69) + (cid:68) ˆ ζ k ˆ ζ k (cid:69) , (cid:68) ˆ ζ k ˆ ζ k ˆ ζ k (cid:69) = (cid:68) ˆ ζ k ˆ ζ k ˆ ζ k (cid:69) + (cid:68) ˆ ζ k ˆ ζ k ˆ ζ k (cid:69) , (cid:68) ˆ h ( λ ) k ˆ h ( λ ) k (cid:69) = (cid:68) ˆ h ( λ ) k ˆ h ( λ ) k (cid:69) + (cid:68) ˆ h ( λ ) k ˆ h ( λ ) k (cid:69) , (cid:68) ˆ ζ k ˆ h ( λ ) k (cid:69) = (cid:68) ˆ ζ k ˆ h ( λ ) k (cid:69) . (2.19)The suffix zero refers to the correlators at zeroth-order in the gauge fields, namely to thestandard inflationary vacuum correlators. We disregard the zeroth-order scalar three pointfunction, as it would correspond to the non-Gaussianity from standard single-field slow-rollinflation, which is unobservable [63–66].In absence of gauge fields, we have the standard mode function solutions ζ ( τ, k ) = h ( τ, k )2 √ (cid:15) = iH (1 + ikτ )2 √ (cid:15)M p k / e − ikτ , if δA i = 0 , (2.20)leading to the standard power spectra (cid:68) ˆ ζ k ˆ ζ k (cid:69) = 2 π k P δ (3) ( k + k ) , P = H π (cid:15)M p , (2.21) (cid:68) ˆ h ( λ ) k ˆ h ( λ ) k (cid:69) = 8 π k (cid:15) P δ (3) ( k + k ) δ λ ,λ . (2.22)The suffix 1 in eq. (2.19) denotes the first non-vanishing correction from the vector fields.This requires two interaction terms, that we compute at tree level in the in-in formalism (see,e.g. ref. [67]) in the next section. In this section, we compute contributions from the gauge field to the correlators (2.19). Aswe mentioned, we treat the gauge field contributions perturbatively. The dominant couplingrelevant for the scalar modes is obtained from the direct coupling I ( − F + γ F ˜ F ). Byexpanding δ ( I ( φ )) (cid:39) − I I (cid:48) aH ζ , this gives rise to a term ∝ ζ [ A (0) + δA ] (as we discussedabove, higher order expansions of I and couplings that originate from δg and δg i can be– 7 –eglected). From this expansion we have the two terms S ζ = (cid:90) dτ d x (cid:18) − a H (cid:19) I (cid:48) I ζ (cid:104)(cid:16) E (0) i δE i − B (0) i δB i (cid:17) − γ (cid:16) E (0) i δB i + B (0) i δE i (cid:17)(cid:105) , (3.1) S ζ = (cid:90) dτ d x (cid:18) − a H (cid:19) I (cid:48) I ζ (cid:20)
12 ( δE i δE i − δB i δB i ) − γδE i δB i (cid:21) . (3.2)The I F term also leads to the dominant coupling between the gravity tensor modeand the gauge fields S h = − (cid:90) dτ d x a h ij (cid:16) E (0) i δE j + B (0) i δB j (cid:17) , (3.3)where we disregard the h δA coupling as it does not contribute at tree level to two pointcorrelators involving the graviton.In the constant vev case, i.e., I ∝ a − = ( − Hτ ) , these three terms give rise to thethree interaction hamiltonians H ζ ( τ ) = − E vev i H τ (cid:90) d p δE i ( τ, p ) ˆ ζ − p ( τ ) , (3.4) H ζ ( τ ) = − H τ (cid:90) d p d p (cid:48) (2 π ) / δE i ( τ, p ) δE i ( τ, p (cid:48) ) ˆ ζ − p − p (cid:48) ( τ ) , (3.5) H h ( τ ) = E vev i H τ (cid:90) d p δE j ( τ, p )ˆ h ij, − p ( τ ) . (3.6)In H ζ and H ζ , we have dropped the contribution of δB i with respect to δE i , as themagnetic perturbation is much smaller than the electric one at super-horizon scales, seeeq. (2.14) (the super-horizon regime dominates the time integrals of the in-in computation[51], as we discuss below). Since B (0) i = 0, and since F ˜ F ∝ E i B i , this implies that thepseudo-scalar interaction does not contribute to the leading order expression of H ζ and H ζ . For this reason, we recover the same dominant interaction hamiltonians as in ref. [51].Nonetheless, the I ( φ ) F ˜ F coupling strongly influences the phenomenological results, as itchanges the gauge field mode functions, see eq. (2.14).In the following, we perform the explicit computations of the correlators using the in-informalism. A useful intermediate result is the commutator between the zeroth-order fields.Using the zeroth-order mode functions (2.20), one finds (cid:68)(cid:104) ˆ ζ k ( τ ) , ˆ ζ k ( τ ) (cid:105)(cid:69) ≈ − iH (cid:15)M p (cid:2) τ − τ (cid:3) δ (3) ( k + k ) , (cid:68)(cid:104) ˆ h ( λ ) k ( τ ) , ˆ h ( λ ) k ( τ ) (cid:105)(cid:69) ≈ − iH M p (cid:2) τ − τ (cid:3) δ (3) ( k + k ) δ λ ,λ . (3.7) By means of the in-in formalism, the leading order correction due to the gauge field to thetwo point scalar correlation function is given by (cid:42) (cid:89) n =1 ˆ ζ k n ( τ ) (cid:43) = − (cid:90) τ dτ (cid:90) τ dτ (cid:42)(cid:34)(cid:34) (cid:89) n =1 ˆ ζ k n ( τ ) , H ζ ( τ ) (cid:35) , H ζ ( τ ) (cid:35)(cid:43) . (3.8) We note that the A (0) 2 ζ tadpole is canceled once the exact equation of motion for ζ is taken into account,and the solution (2.20) is properly modified. This effect is completely negligible, due to the fact that ρ vev E (cid:28) ρ φ . – 8 –e are interested in this result at super horizon scales, − k n τ (cid:28)
1. Modes of ζ and of δE appearing in this expression are the zeroth-order solutions, and so they are uncorrelated witheach other. So, the expectation value in eq. (3.8) splits into two separate expectation values.The dτ i time integrals are dominated by modes in the long-wavelength regimes, since thesmall-wavelength modes are highly oscillatory, giving a highly suppressed contribution to thetime integral [51]. On superhorizon scales, the electric field (contained in H ζ ( τ n )) becomesa classical (commuting) field [51]. Therefore, the expression (3.8) is evaluated to (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) = − (cid:18) E vev H (cid:19) (cid:90) τ dτ τ (cid:90) τ dτ τ (cid:90) d p (cid:90) d p ˆ E vev i ˆ E vev j (cid:104) δE i ( τ , p ) δE j ( τ , p ) (cid:105) (cid:16)(cid:68)(cid:104) ˆ ζ k ( τ ) , ˆ ζ − p ( τ ) (cid:105)(cid:69) (cid:68)(cid:104) ˆ ζ k ( τ ) , ˆ ζ − p ( τ ) (cid:105)(cid:69) + ( k ↔ k ) (cid:17) . (3.9)After computing this with eqs. (2.13) and (3.7), the time integrals are reduced to − (cid:82) τ e − k − dτ i τ i ,which is equivalent to the number of e-folds before the end of inflation at which the modeswith k leave the horizon, given by N k . With an identity: (cid:104) (cid:15) (+) i (ˆ k ) (cid:15) (+) j ( − ˆ k ) + (cid:15) (+) i (ˆ k ) (cid:15) (+) j ( − ˆ k ) (cid:105) ˆ E vev i ˆ E vev j δ (3) ( k + k )= (cid:88) s = ± (cid:15) ( s ) i (ˆ k ) (cid:15) ( − s ) j (ˆ k ) ˆ E vev i ˆ E vev j δ (3) ( k + k )= (cid:20) − (cid:16) ˆ k · ˆ E vev (cid:17) (cid:21) δ (3) ( k + k ) , (3.10)we finally obtain (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) = E π(cid:15) M p e πξ ξ N k k (cid:20) − (cid:16) ˆ k · ˆ E vev (cid:17) (cid:21) δ (3) ( k + k ) , ξ (cid:29) , (3.11)which indeed shows a non-vanishing quadrupolar asymmetry, and which is scale invariant upto the logarithmic dependence on k of N k .It is instructive to compare this result to that obtained for ξ = 2 γ = 0 [51]. We find (cid:68) ˆ ζ k ˆ ζ k (cid:69) (cid:12)(cid:12)(cid:12) ξ (cid:29) (cid:68) ˆ ζ k ˆ ζ k (cid:69) (cid:12)(cid:12)(cid:12) ξ =0 = 14 π e πξ ξ = (cid:18) δE + (cid:12)(cid:12)(cid:12) ξ (cid:29) (cid:19) (cid:18) δE + (cid:12)(cid:12)(cid:12) ξ =0 (cid:19) + (cid:18) δE − (cid:12)(cid:12)(cid:12) ξ =0 (cid:19) . (3.12)In short the difference between our result (3.11) and the result obtained for γ = ξ = 0 issimply due to the difference between the wave functions of the sourcing gauge fields (themode function in the present case is given in eq. (2.14); for ξ = 0 one has instead δE + = δE − = H √ k / [51]).By an analogous computation, the leading gauge field contributions to the scalar-tensorand to the tensor-tensor correlators are found to be (cid:68) ˆ ζ k ˆ h ( λ ) k (cid:69) = − E π(cid:15)M p e πξ ξ k − N k ˆ E vev i ˆ E vev j e ( λ ) ij (ˆ k ) δ (3) ( k + k ) δ λ , , (3.13) (cid:42) (cid:89) n =1 ˆ h ( λ n ) k n (cid:43) = E πM p e πξ ξ k − N k (cid:20) − (cid:16) ˆ k · ˆ E vev (cid:17) (cid:21) δ (3) ( k + k ) δ λ , δ λ , . (3.14)– 9 –he absence of the λ = − (cid:15) and by a factor (cid:15) with respect to the contribution to the scalar-scalar correlator. An analogous behavior was also found in the F + φF ˜ F model [41]. Again using the in-in formalism, the leading order contribution of the gauge field to the scalarbispectrum is given by : (cid:42) (cid:89) n =1 ˆ ζ k n ( τ ) (cid:43) = (cid:42) (cid:89) n =1 ˆ ζ k n ( τ ) (cid:43) (211) + (cid:42) (cid:89) n =1 ˆ ζ k n ( τ ) (cid:43) (121) + (cid:42) (cid:89) n =1 ˆ ζ k n ( τ ) (cid:43) (112) , (3.15)where we have defined (cid:42) (cid:89) n =1 ˆ ζ k n ( τ ) (cid:43) ( abc ) ≡ i (cid:90) τ dτ (cid:90) τ dτ (cid:90) τ dτ (cid:42)(cid:34)(cid:34)(cid:34) (cid:89) n =1 ˆ ζ k n ( τ ) , H ζa ( τ ) (cid:35) , H ζb ( τ ) (cid:35) , H ζc ( τ ) (cid:35)(cid:43) . (3.16)As for the power spectrum, the time integrals are dominated by modes in the super-horizon regime. Proceeding as in that computation, the first term in eq. (3.15) evaluatesto (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) (211) = − i E H (cid:90) τ dτ τ (cid:90) τ dτ τ (cid:90) τ dτ τ (cid:90) d p d p (cid:48) (2 π ) / (cid:90) d p (cid:90) d p ˆ E vev j ˆ E vev k (cid:10) δE i ( τ , p ) δE i ( τ , p (cid:48) ) δE j ( τ , p ) δE k ( τ , p ) (cid:11)(cid:68)(cid:104) ˆ ζ k ( τ ) , ˆ ζ − p − p (cid:48) ( τ ) (cid:105)(cid:69) (cid:68)(cid:104) ˆ ζ k ( τ ) , ˆ ζ − p ( τ ) (cid:105)(cid:69) (cid:68)(cid:104) ˆ ζ k ( τ ) , ˆ ζ − p ( τ ) (cid:105)(cid:69) +5 perms in k n (3.17)This expression evaluates to (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) (211) (cid:39) E H π (2 π ) / (cid:15) M p δ (3) ( k + k + k ) k k e πξ ξ (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) j (ˆ k ) (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) k (ˆ k ) ˆ E vev j ˆ E vev k (cid:90) τ Max (cid:104) − k , − k , − k (cid:105) dτ τ − τ τ (cid:90) τ − k dτ τ − τ τ (cid:90) τ − k dτ τ − τ τ +2 perms in k n , ξ (cid:29) . (3.18)Namely, the internal time variables in the integrals cover the region τ ≥ τ ≥ τ , τ ,plus lower bounds dictated by the requirement that all the modes in the interaction are inthe super-horizon regime. Starting from the other two terms in eq. (3.15), and relabeling theinternal times, we obtain an expression where the integrand is identical to the integrand of– 10 –q. (3.18), but the internal times cover the two complementary regions τ ≥ τ ≥ τ , τ , and τ ≥ τ ≥ τ , τ . This leads to (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) (cid:39) E H π (2 π ) / (cid:15) M p δ (3) ( k + k + k ) k k e πξ ξ (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) j (ˆ k ) (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) k (ˆ k ) ˆ E vev j ˆ E vev k (cid:90) τ Max (cid:104) − k , − k , − k (cid:105) dτ τ − τ τ (cid:90) τ − k dτ τ − τ τ (cid:90) τ − k dτ τ − τ τ +2 perms in k n , ξ (cid:29) . (3.19)The time integrations are dominated by the earlier possible times, giving (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) = δ (3) ( k + k + k )(2 π ) / C ξ (cid:29) k , ˆ k , ˆ E vev f ( k , k , k ; ξ )+2 perms in k n , ξ (cid:29) . (3.20)We here have defined C ξ (cid:29) k , ˆ k , ˆ E vev ≡ (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) j (ˆ k ) (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) k (ˆ k ) ˆ E vev j ˆ E vev k , (3.21) f ( k , k , k ; ξ ) ≡ E H π (cid:15) M p e πξ ξ Min[ N k , N k , N k ] N k N k k k , ξ (cid:29) , (3.22)where we recall that N k i is the number of e-fold before the end of inflation at which the modewith momentum k i left the horizon. Using the identity (cid:15) (+) ∗ i (ˆ k ) (cid:15) (+) j (ˆ k ) = (cid:104) δ ij − ˆ k i ˆ k j + i η ijk ˆ k k (cid:105) ,the angle dependence in C ξ (cid:29) k , ˆ k , ˆ E vev can be simplified as C ξ (cid:29) k , ˆ k , ˆ E vev = 14 (cid:26) − (cid:16) ˆ k · ˆ E vev (cid:17) − (cid:16) ˆ k · ˆ E vev (cid:17) + (cid:16) ˆ k · ˆ E vev (cid:17) (cid:16) ˆ k · ˆ E vev (cid:17) (cid:16) ˆ k · ˆ k (cid:17) − ˆ k · ˆ k + (cid:16) ˆ k · ˆ E vev (cid:17) (cid:16) ˆ k · ˆ E vev (cid:17) + i (cid:104) ˆ E vev · (cid:16) ˆ k − ˆ k (cid:17)(cid:105) (cid:104) ˆ E vev · (cid:16) ˆ k × ˆ k (cid:17)(cid:105)(cid:111) . (3.23)In the I ( φ ) F model (the γ = ξ = 0 case), the corresponding contribution to thebispectrum is [51] (cid:42) (cid:89) n =1 ˆ ζ k n (cid:43) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = δ (3) ( k + k + k )(2 π ) / C ξ =0ˆ k , ˆ k , ˆ E vev f ( k , k , k ; ξ = 0)+2 perm in k n , (3.24)– 11 –here C ξ =0ˆ k , ˆ k , ˆ E vev = (cid:34) (cid:88) s = ± (cid:15) ( s ) i (ˆ k ) (cid:15) ( s ) ∗ j (ˆ k ) (cid:35) (cid:34) (cid:88) s = ± (cid:15) ( s ) i (ˆ k ) (cid:15) ( s ) ∗ k (ˆ k ) (cid:35) ˆ E vev j ˆ E vev k = 1 − (cid:16) ˆ k · ˆ E vev (cid:17) − (cid:16) ˆ k · ˆ E vev (cid:17) + (cid:16) ˆ k · ˆ E vev (cid:17) (cid:16) ˆ k · ˆ E vev (cid:17) (cid:16) ˆ k · ˆ k (cid:17) , (3.25) f ( k , k , k ; ξ = 0) = f ( k , k , k ; ξ (cid:29) × δE (cid:12)(cid:12)(cid:12) ξ =0 δE + (cid:12)(cid:12)(cid:12) ξ (cid:29) , (3.26)with δE (cid:12)(cid:12)(cid:12) ξ =0 = H √ k / being the mode function of either helicity mode in the ξ = 0 case(where the two helicities are produced in equal amount). As for the power spectrum, thedifference between the ξ (cid:29) ξ = 0 result is simply due to the difference of thesourcing gauge fields. In the present case, one helicity dominates the final result, leading toviolation of parity. In this section, we compare the primordial correlators induced by the action (2.1) to thelatest CMB data, and we obtain an upper bound on the energy density in the backgroundgauge field, ρ vev E . Finally, we compare this bound with the theoretically expected value forthe vev. The primordial correlators computed in section 3 act as the initial conditions of the CMBcorrelators and can therefore be measured by CMB observations. Formally, the signatures inthe CMB scalar-scalar, scalar-tensor and tensor-tensor power spectra are essentially identicalto those obtained in the F + φF ˜ F model [41], since the primordial correlators have theidentical angular dependence. The main difference between the signatures on that modeland the signatures that we are computing here is that in the present case the gauge fieldvev is constant, so that these signatures are nearly scale invariant. On the contrary, in the F + φF ˜ F case, the vector vev (that needs to be assumed as an ad hoc initial conditionpresent when the CMB modes were generated) is rapidly decreasing, leading to observationaleffects only at the largest scales.As proven in ref. [41], the scalar-scalar correlator (3.11) can create TT, TE and EE in | (cid:96) − (cid:96) | = 0 ,
2, while the scalar-tensor (3.13) or tensor-tensor (3.14) correlator can generateTT, TE, EE, BB, TB and EB in | (cid:96) − (cid:96) | = 0 , ,
2. Specifically, TT, TE, EE and BB (TBand EB) in | (cid:96) − (cid:96) | = 1 ( | (cid:96) − (cid:96) | = 2) are distinct signatures of the anisotropic pseudoscalarinflation, since these appear only in the case that parity and rotational symmetries are brokenat the same time. However, these signal-to-noise ratios are smaller than the scalar-scalar onesbecause of the slow-roll suppression of the tensor mode. Likewise, the tensor mode also givesspecial correlations in the bispectrum due to broken parity and rotational invariance; alsosuch contributions are subdominant due to the smallness of the tensor mode. In this section,we therefore focus on the observables associated with the primordial scalar mode ζ .– 12 –or this analysis, we use the conventional g ∗ parametrization for the power spectrum[55] and the c L parametrization for the bispectrum [52], namely, (cid:42) (cid:89) n =1 ζ k n (cid:43) = δ (3) ( k + k ) P ( k ) (cid:20) g ∗ (cid:16) ˆ k · ˆ E vev (cid:17) (cid:21) , (4.1) (cid:42) (cid:89) n =1 ζ k n (cid:43) = δ (3) ( k + k + k )(2 π ) / (cid:88) L c L P ( k ) P ( k ) P L (ˆ k · ˆ k ) + 2 perms , (4.2)where P L ( x ) is the Legendre polynomial. The latter bispectrum form is obtained afteraveraging the original anisotropic bispectrum (3.20) over all directions of ˆ E vev , in the spiritof isotropic CMB measurements. This is the quantity that is immediately associated tothe angle-averaged reduced bispectrum computed from the data: the reduced bispectrum b ( k , k , k ) is obtained by averaging the bispectrum over all possible orientation of trianglesof sides of length k , k , k . The theoretical prediction for the reduced bispectrum associatedto eq. (3.20) is therefore equivalent to the theoretical prediction associated to the average ofeq. (3.20) over all possible direction of ˆ E vev [52]. Keeping into account that the CMB data force | g ∗ | (cid:28) P ( k ) (cid:39) π k P and ξ (cid:29) g ∗ (cid:39) − N π(cid:15) e πξ ξ ρ vev E ρ φ , (4.3)where N CMB is the number of e-folds before the end of inflation at which the CMB modesleave the horizon, ρ φ (cid:39) V ( φ ) (cid:39) M p H is the energy density of the inflaton, and ρ vev E ≡ E is the energy density of the gauge field.Let us now compute the average of the bispectrum (3.20) over all directions of ˆ E vev .Using (cid:90) d ˆ E vev π C ξ (cid:29) k , ˆ k , ˆ E vev = 19 P (ˆ k · ˆ k ) − P (ˆ k · ˆ k ) + 118 P (ˆ k · ˆ k ) , (4.4)and setting N k (cid:39) N k (cid:39) N k (cid:39) N CMB , we obtain ξ (cid:29) c = − N CMB π e πξ ξ g ∗ , c = − c , c = c . (4.5)We compare eqs. (4.3) and (4.5) with the results obtained for the γ = ξ = 0 case [52] ξ = 0 : g ∗ (cid:39) − N (cid:15) ρ vev E ρ φ , c = − N CMB g ∗ , c = 0 , c = c . (4.6)The Planck collaboration reported the 95% CL limits [28, 69] − . ≤ g ∗ ≤ . , − . ≤ c ≤ . , − ≤ c ≤ , − ≤ c ≤ . (4.7) In the study of CMB anisotropic bispectra [56, 57] this would correspond to single-out a monopole termin a spherical harmonic expansion of the anisotropic bispectrum [56] that does contribute to the isotropic(angle-averaged) bispectrum. This identity is easily derived using the spherical-harmonics representations of a unit vector and polar-ization vector, see [70]. These c L bounds correspond to the temperature only limits [28]: f localNL = 2 . ± . f L =1NL = − ±
43 and f L =2NL = 0 . ± . c = (6 / f localNL , c = − (12 / f L =1NL and c = − (48 / f L =2NL hold. – 13 – -30 -25 -20 -15 -10 -5
0 1 2 3 4 5 (cid:108) v e v E / (cid:108) (cid:113) | (cid:97) | g * c c c Figure 1 . Upper bounds on ρ vev E /ρ φ reported in eqs. (4.8) and (4.9), with N CMB = 60 and (cid:15) = 0 . ρ vev E /ρ φ is the most stringent of the boundsshown. Strictly speaking, the solid and dashed lines are based on an approximation that holds for | γ | (cid:29) (that is, | ξ | (cid:29) | γ | = 1, as a naive extrapolation of the linesshown in the figure is roughly consistent with the values for | γ | = 0, that are exact. From these constraints we get rough estimates of the upper bounds on ρ vev E /ρ φ using eqs. (4.3),(4.5) and (4.6): γ = 0 : ρ vev E ρ φ ≤ . × − (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from g ∗ ) , . × − (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from c ) , . × − (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from c ) , (4.8)and γ (cid:29)
12 : ρ vev E ρ φ ≤ . × − (cid:16) γ e πγ (cid:17) (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from g ∗ ) , . × − (cid:16) γ e πγ (cid:17) (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from c ) , . × − (cid:16) γ e πγ (cid:17) (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from c ) , . × − (cid:16) γ e πγ (cid:17) (cid:0) (cid:15) . (cid:1) (cid:16) N CMB (cid:17) (from c ) . (4.9)We show these bounds in figure 1, for the special choices of N CMB = 60 and (cid:15) = 0 . γ = 0 the strongest bound on ρ vev E /ρ φ is obtained from the power spectrum, at– 14 – (cid:29) the strongest bound is due to the bispectrum coefficient c . This is due the fact thatthe gauge production increases with increasing γ , and the bispectrum is affected more thanthe power spectrum by this growth (at the technical level, the source for the power spectrumis ∝ δE , while the one for the bispectrum is ∝ δE . This generates the e πξ /ξ factor in c /g ∗ in eq. (4.5)).The result we have obtained assumed ξ = 2 γ >
0, but it can be readily applied alsoto the case in which ξ = 2 γ <
0. As clear from eq. (2.11), changing sign of ξ = 2 γ simplyamounts in interchanging the role of the two helicities of the gauge field. In the computationof the inflation correlators, one simply needs to replace ξ → | ξ | and interchange (cid:15) (+) i ↔ (cid:15) ( − ) i .One can then verify that the limits (4.9) can be extended to the γ (cid:28) − region by simplyreplacing γ with | γ | . In this subsection we compare the limits on ρ vev E /ρ φ obtained in the previous subsection withwhat is naturally expected in the model (2.1).Beside producing scale invariant vector perturbations, the I ∝ a − case also sustains aconstant vector vev. This is not accidental. Due to scale invariance, super horizon modeshave the same power at any given time t . The modes also had the same power when theyleft the horizon. Equal power at horizon crossing (that happens at different time for thedifferent modes) and at the later common time t is obtained because the modes have frozenamplitude while outside the horizon. In the super-horizon regime a mode solves the sameequation of motion as a constant vev. So, if the super-horizon modes are frozen outsidethe horizon, also a completely homogeneous vev is. This explains why the I ∝ a − choiceproduces both scale invariant electric perturbations, and a constant electric vev.In presence of the vector vev, the interaction between the gauge and the inflaton per-turbations leads to anisotropic correlators of the primordial perturbations. The fact that thisanisotropy is not observed in the data results in a limit on the vev of the vector field, and onthe associated energy density ρ vev E ≡ E . The current limit is summarized in figure 1.Naively, one may simply assume that the gauge field has no homogeneous vev, in whichcase the limit computed here does not apply. While this is a mathematical possibility, suchan assumption is very unlikely in a model that is constructed to give scale invariant gaugeperturbations. Unless one makes the very ad-hoc assumption that the duration of inflation islimited to the one necessary to produce the CMB modes that we observe at the largest scales(that is, N tot = N CMB (cid:39) E vev that we have studied in this work.If we could observe many disconnected patches of the universe that have a size compa-rable with our Hubble horizon, we would observe many realizations of the IR gauge modesgenerated during the first N tot − N CMB e-folds of inflation, and we would observe a differ-ent sum E vev of these modes in each Hubble patch. Under the hypothesis of Gaussianityof the gauge perturbations (even a O (1) departure from Gaussianity would not change the This is just a statement of time translation invariance during inflation; this invariance is broken by theslow roll parameters, as the inflaton typically speeds up during inflation. This leads to departure from scaleinvariance - typical in slow roll inflation - that we are disregarding in the present discussion. – 15 –rder of magnitude estimate performed here), the values of these sum would have a Gaussiandistribution of zero mean and variance | γ | (cid:29)
12 : (cid:10) E ( x ) (cid:11) = 12 π (cid:90) IR dk k | δE + ( k ) | = 9 H π e π | γ | | γ | (cid:90) IR dkk = 9 H π e π | γ | | γ | ( N tot − N CMB ) . (4.10)We observe only one Hubble patch, and hence only one of such realizations. The variancewe have just computed gives the typical value of E obtain in any realization. We thereforeobtain | γ | (cid:29)
12 : ρ vev E ρ φ (cid:12)(cid:12)(cid:12)(cid:12) typical = 3128 π (cid:18) HM p (cid:19) e π | γ | | γ | ( N tot − N CMB ) . (4.11)The corresponding values of g ∗ and c L are | γ | (cid:29)
12 : g ∗ | typical (cid:39) − . × (cid:18) HM p (cid:19) e π | γ | | γ | . (cid:15) (cid:18) N CMB (cid:19) N tot − N CMB ,c | typical (cid:39) . × (cid:18) HM p (cid:19) e π | γ | | γ | . (cid:15) (cid:18) N CMB (cid:19) N tot − N CMB ,c | typical = − c | typical , c | typical = 12 c | typical . (4.12)Already requiring that the energy in the vector field is subdominant poses a significantlimit. For instance, for | γ | = 5 .
5, and for only 10 e-folds of inflation more than the last N CMB ones (this is a conservative assumption, as typical models of inflation give a much longerduration), one finds a subdominant vector field only if H < ∼ − M p (= 3 × GeV). Thelimit obtained in this work is much stronger. Combining the results (4.9) and (4.11), we findthe bound | γ | (cid:29)
12 : HM p < ∼ . | γ | / e π | γ | (cid:115) (cid:15) . (cid:18) N CMB (cid:19) N tot − N CMB , (4.13)which, for | γ | = 5 .
5, gives the bound HM p < ∼ − . In principle, one can assume values of HM p extremely small during inflation, although this is clearly a challenge for model building. Theabsolute minimum is obtained by imposing an instantaneous thermalization of the inflatondecay products, with the minimum reheating temperature T rh , min (cid:39) HM p > ∼ − . We thus see that values | γ | > ∼ . γ = 0 case one finds [51] γ = 0 : ρ vev E ρ φ (cid:12)(cid:12)(cid:12)(cid:12) typical = 34 π H M p ( N tot − N CMB ) , ⇒ HM p < ∼ × − (cid:115) (cid:15) . (cid:18) N CMB (cid:19) N tot − N CMB , (4.14)so that a much greater value of H during inflation can be tolerated in this case.The bounds (4.13) and (4.14) are shown in figure 2 for the specific choice of the infla-tionary parameters (cid:15) = 0 . , N CMB = 60 , N tot = 70 . Strictly speaking, the result (4.13) isvalid only at | γ | (cid:29) (that is, | ξ | (cid:29) | γ | (cid:29) | γ | (cid:46) γ = 0. – 16 – / M p | (cid:97) |10 -50 -40 -30 -20 -10
0 1 2 3 4 5 6Incompatible with BBN
Figure 2 . Upper bounds on H reported in eqs. (4.13) and (4.14), with N CMB = 60, (cid:15) = 0 .
01 and N tot − N CMB = 10. One can see that values | γ | (cid:38) . In ref. [41] we computed the non-diagonal correlators between multipoles of the CMB tem-perature anisotropy and polarization induced by the simultaneous breaking of parity androtational invariance during inflation. A very simple way to realize such a breaking is tocouple the inflaton φ with some gauge field with non-vanishing vev through a pseudo-scalarinteraction φF ˜ F . A vector field with a standard kinetic term is rapidly diluted away by theexpansion of the universe. To cope with this, in the present work we considered the recentmodification by Caprini and Sorbo [36] of the Ratra mechanism [42] for the generation ofa primordial magnetic field during inflation. This model is characterized by the interac-tion I ( φ )( − F + γ F ˜ F ), which explicitly breaks parity. For the suitable time dependence I ( φ ( t )) ∝ a ( t ), where a is the scale factor, the model produces scale invariant magneticperturbations, which is at the core of the mechanisms of ref. [42]. For I ∝ a − the modelproduces scale invariant electric perturbations. This was first used in the γ = 0 case tosustain anisotropic inflation [72], exploiting an “electromagnetic duality” in the model for I → I [73].We restricted our computations to the I ∝ a − case; however, due to the duality,we expect that our results can be readily extended to I ∝ a . We also assumed that thefield φ is the inflaton (this is not required in the magnetogenesis applications [36, 42]). Inpresence of the vector vev, the interaction between the gauge and the inflaton perturbations– 17 –eads to anisotropic correlators of the primordial perturbations. In particular, among thesecorrelators, we have shown for the first time that it is possible to generate a primordial non-Gaussian signature during inflation proportional to c in the parametrization (1.4), which isdue to broken parity and rotational invariance.The fact that the anisotropic signatures predicted by this model are not observed in thedata results in a limit on the vev of the vector field, and on the associated energy density ρ vev E ≡ E . The limit is already strong at γ = 0, where standard values of the slow rollparameter (cid:15) = 0 .
01 and of the number of observable e-folds of inflation N CMB = 60 leadto ρ vev E /ρ φ < ∼ − . Such a strong limit is due to the fact that the vector modes act asisocurvature modes that continue to source the inflaton adiabatic perturbations in the super-horizon regime, inducing a N ( N ) enhancement of the anisotropic two (three) pointcorrelators, and to the fact that the direct inflaton-vector field coupling is stronger than thegravitational one [51]. The limit becomes much more stringent with growing | γ | , see figure1, as the amount of gauge quanta produced by the moving inflaton grows exponentially with | γ | at large | γ | .The relevant question to ask is whether such stringent limits are compatible with thevev that should naturally be expected in the model. We have considered a mechanismthat, by construction, gives scale invariant perturbations for the gauge field. In typicalinflationary models, the primordial perturbations of size comparable to our horizon wereproduced N CMB (cid:39)
60 e-folds before the end of inflation. This does not mean that the scaleinvariance induced in the mechanism should stop at such wavelengths, but it is natural toassume that also modes produced earlier were scale invariant. We denote such modes asIR modes. Each of these modes is observed in our sky as classical and homogeneous vectorfield on our sky [51]. We can actually observe only the sum of such modes, and this preciselyconstitutes the homogeneous vector vev E vev from the IR modes (that eventually sum up toa vev of the vector field from the classical equations of motion) that we have studied in thiswork.Let us denote with N tot the total number of e-folds of inflation, and assume that E vev = 0at the onset of inflation (if a sizable vev is already present at the start of inflation, wesimply expect an even stronger effect, and even stronger bounds). The IR modes thatare seen as homogeneous from the point of view of the CMB modes are those producedin the first N tot − N CMB number of e-folds of inflation. The value of the same on ourHubble patch is obtained as a stochastic addition, obeying the typical random walk relation (cid:104) E (cid:105)| expected ∝ N tot − N CMB (this is completely analogous [51] to how condensates of lightscalar field develop during inflation). This was studied in ref. [51] for the γ = 0 case. Inthe present work, we extended this study to non-vanishing and large γ , where, due to thesignificant increase of the amplitude of the produced gauge quanta, the limits imposed by thiseffect become much more stringent, see figure 2. For instance, by simply allowing inflationto last 10 more e-folds than the amount necessary to produce the CMB modes (this is a veryconservative assumption), we could rule out | γ | > ∼ . I ( φ ) produces a constantelectric field. This can be achieved for I ( φ ) ∝ a n , with n = −
2. Using an “electromagneticduality” of the model, this result can be immediately translated to the I ∝ a case, althoughfor n > One can make ad-hoc assumptions to prevent this: for instance one can postulate that the duration ofinflation is only limited to N CMB e-folds, or that the scaling I ∝ a − did not hold at N > N
CMB . We do notmake such ad-hoc assumptions in discussing what should be naturally expected from this mechanism. – 18 –he discussion after eq. (2.3)). On the other hand, the n (cid:54) = − φ is not the inflaton. We hope to come back to these issues in a separatework. Acknowledgments
MS was supported in part by a Grant-in-Aid for JSPS Research under Grant Nos. 25-573and 27-10917, and in part by World Premier International Research Center Initiative (WPIInitiative), MEXT, Japan. This work was supported in part by the ASI/INAF AgreementI/072/09/0 for the Planck LFI Activity of Phase E2. The work of MP was supported in partby DOE grant de-sc0011842 at the University of Minnesota.
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