Non-Gaussianity in the Cosmic Microwave Background Anisotropies at Recombination in the Squeezed limit
aa r X i v : . [ a s t r o - ph . C O ] S e p CERN-PH-TH/2011-221
Non-Gaussianity in the Cosmic Microwave BackgroundAnisotropies at Recombination in the Squeezed limit
N. Bartolo a,b , S. Matarrese a,b and A. Riotto b,c a Dipartimento di Fisica “G. Galilei”, Universit`a di Padova, via Marzolo 8, I-131 Padova, Italy b INFN, Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy c CERN, Theory Division, CH-1211 Geneva 23, SwitzerlandE-mail: [email protected], [email protected] and [email protected]
Abstract
We estimate analytically the second-order cosmic microwave background temperature anisotropies atthe recombination epoch in the squeezed limit and we deduce the contamination to the primordial localnon-Gaussianity. We find that the level of contamination corresponds to f conNL = O (1) which is below thesensitivity of present experiments and smaller than the value O (5) recently claimed in the literature. ontents Primordial non-Gaussianity (NG) of the cosmological perturbations has become a crucial aspect of bothobservational predictions of (inflationary) early universe models and of present and future observationalprobes of the Cosmic Microwave Background (CMB) anisotropies and Large-Scale Structures (LSS) ofthe universe. The main motivation is that detecting, or simply constraining, deviations from a Gaussiandistribution of the primordial fluctuations allows to discriminate among different scenarios for the gener-ation of the primordial perturbations. A non-vanishing primordial NG encodes a wealth of informationallowing to break the degeneracy between models that, at the level of the power spectra, might resultto be undistinguishable [1]. It is impressive that a future detection of a high level of primordial NGcould rule out the standard single-field models of slow-roll inflation, which are characterized by weakgravitational interactions and thus predict tiny deviations from Gaussianity [2, 3]. Statistics like thebispectrum (three-point correlation function) of CMB anisotropies [4], and higher-order statistics, andvarious observational probes of the LSS [5], can be used to assess the level of primordial NG on variouscosmological scales.It has now been fully appreciated the importance of distinguishing different shapes of the primordialbispectrum, i.e. the dependence of the bispectrum on different triangular configurations formed by thethree wave-vectors over which, in Fourier space, the primordial fluctuations are correlated [6]. Different(classes of) inflationary models give rise to unique signals with specific triangular shapes, which thus probedifferent aspects of the physics of the primordial universe, in particular they can probe different kindsof interactions in the inflaton sector which are at the origin of the primordial non-Gaussian signal. Forexample, models in which non-linearities develop outside the horizon during inflation, or immediately afterinflation, generate a level of NG which is local, as the NG part of the primordial curvature perturbationis a local function of the Gaussian part, being generated on superhorizon scales. This is generally thecase when a light scalar field, different from the inflaton, contributes to the curvature perturbation (forexample, multi-field models of inflation [7], curvaton [8, 9] and inhomogenoeus reheating models [10, 11,12]). In momentum space, the three point function, or bispectrum, arising from the local NG is dominatedby the so-called squeezed configuration, where one of the momenta is much smaller than the other two( k ≪ k ∼ k ) and it is parametrized by the non-linearity parameter f locNL . “Equilateral” NG, peakingfor k ∼ k ∼ k is generally predicted by inflationary models where the inflaton field is characterizedby derivative interactions so that the correlation between the modes peaks when all the modes cross thehorizon during inflation at the same time (see, e.g., inflationary models with non-standard kinetic term[13], or DBI models of inflation [14]). For other interesting NG shapes/models (such as “orthogonal”
15] and “folded” (or flattened) shapes [13, 16, 17, 18, 19]) we refer the reader to the review [20]. Presentlimits on NG are summarized by the WMAP measurements − < f locNL <
74 and − < f equiNL <
266 at95% CL [21] (see [22] for CMB constraints on other types of NG). The
Planck satellite [23] is expectedto improve the sensitivity to the non-linearity parameters by an order of magnitude. In particular, forthe local case, it is expected to be sensitive to a value of the non-linearity parameter as low as f locNL = 3 at1 σ (accounting also for polarization information; for the equilateral case f equilNL = 30) [24, 25, 26, 27, 28].However there are many sources of NG in the CMB anisotropies beyond the primordial one, includingsystematic effects and astrophysical contamination. Given the deep implications that a detection ofprimordial non-Gaussianity would have, it is crucial that all possible sources of contamination for theprimordial signal are well understood and under control. In fact any non-linearities can make initiallyGaussian perturbations non-Gaussian. Such non-primordial effects can thus complicate the extractionof the primordial non-Gaussianity. One of their main effects could be that of “mimicking” a three-pointcorrelation function that is similar in shape to the primordial one, thus inducing a bias to the estimatorof primordial NG used. Therefore we have to be sure we are not ascribing a primordial origin to a signalthat is extracted from the CMB (or LSS) data using estimators of non-Gaussianity when that signalhas a different origin. Moreover we must always specify which primordial non-Gaussianity we study thecontamination from the non-primordial sources (e.g. primordial non-Gaussianity of “local”, “equilateral”or “folded” shape). In that respect the reasonable question to ask is which kind of primordial non-Gaussianity a given secondary effect can contaminate most, wether it is of the so-called local type or ofthe equilateral type.As far as CMB anisotropies are concerned, well-known examples of contamination are the NG pro-duced by (non-linear) secondary anisotropies, like, e.g., the Sunyaev-Zeldovich effect, gravitational lens-ing, Integrated-Sachs-Wolfe (or Rees-Sciama) (see, e.g. [29] for a review on CMB secondary anisotropies).Up to now it has been found that the most serious contamination to the local type of primordial NG is thethe coupling between the Integrated Sachs-Wolfe/Rees-Sciama effect and the weak gravitational lensing[30]. In the detailed analysis of [31, 32, 33] it is shown that the ISW/Rees-Sciama-lensing bispectrumproduces a bias of f contNL ≃
10 to local primordial non-Gaussianity, while the bias to equilateral primordialnon-Gaussianity is negligible (see also [34, 35]). Other secondary effects have a little impact on the (local)primordial NG with | f NL | ≤ . rder on all angular scales from the early epochs, when the cosmological perturbations were generated,to the present time, through the recombination era. These calculations set the stage for the computationof the full second-order radiation transfer function at all scales and for a a generic set of initial conditionsspecifying the level of primordial non-Gaussianity .There has been some debate about the level of contamination that the non-linear evolution at re-combination can generate. A first work, partially implementing numerically the second-order Boltzmannequations in the small scale limit, claimed a very high level of contamination to the local-type f NL [43].However it has been shown in Ref. [44] that the source of this effect, the small-scale non-linear evo-lution of the second-order gravitational potential at recombination, actually contaminates mainly theequilateral-type NG. A contamination of f contNL (equil) ≃ Planck was found [44],which is actually low w.r.t. the expected
Planck sensitivity to this kind of NG. The contamination tothe local NG from the small scale evolution of the second-order gravitational potential is also very low, f contNL (loc) ≃ . Planck . In Ref. [45] another numerical implementation ofthe second-order Boltzmann equations has been performed to compute the contamination from specificterms appearing in the second-order radiation transfer function. These are the terms that can be written(in the Poisson gauge adopted in Ref. [45]) as products of the first-order perturbations (in the form, e.g.,(Φ (1) ) , where Φ (1) is the linear gravitational potential). The contamination to local NG was found tobe negligible for an experiment like Planck being | f contNL (loc) | < f contNL (loc) = 5 up to a maximum multipole ℓ max = 2000.Such a contamination must therefore come from terms in the second-order radiaton transfer function thatare “intrinsically” of second order. The authors of Ref. [46] suggest that the main candidate is the second-order monopole perturbation (Θ (2)SW = ∆ (2)00 / (2) ), which is the usual term appearing in the CMBanisotropies due to the intrinsic photon energy density fluctuations ∆ (2)00 and the gravitational redshiftdue to the potential, both evaluated at recombination. In fact such a term on large scales reduces tothe Sachs-Wolfe effect. There has been some debate on how such a second-order contribution is able toproduce a contamination f contNL (loc) = 5 (see, e.g., [47, 48]). For example it is not clear what is the physicalmechanism generating such a squeezed signal: if it occurs due to an integration over the terms (squared infirst-order perturbations) that source the “intrinsically” second-order monopole, then, naively, one doesnot expect that a squeezed signal can be generated, because such a contribution would be associated toa causal evolution on small scales.The goal of this paper is to present a transparent computation of the bispectrum in the squeezed limitthrough a convenient coordinate rescaling [37, 49, 50]. To understand such a rescaling, it is important torecall what is generally the origin of a squeezed non-Gaussian signal: typically the local-form bispectrumis generated when short-wavelength fluctuations are modulated by long-wavelength fluctuations. Inparticular we will focus on the temperature anisotropies at recombination when the long wavelengthmode is outside the horizon, but observable at the present epoch. Thus, the effect of the long wavelengthmode imprinted at recombination can be described simply by a coordinate transformation. In this waywe can describe in a simple way the coupling of small scales to large scales that can generally produce the Of course, for specific effects on small angular scales like Sunyaev-Zeldovich, gravitational lensing, etc., fullynonlinear calculations would provide a more accurate estimate of the resulting CMB anisotropy, however, as longas the leading contribution to second-order statistics like the bispectrum is concerned, second-order perturbationtheory suffices. ocal form bispectrum. A similar cross-talk between large and small scales gives rise to the ISW-lensingcross-correlation bispectrum.The plan of the paper is the following: In Sec. II we show that a simple coordinate rescaling reproducescorrectly the second-order expressions for the temperature anisotropies in the squeezed limit. In Sec. IIIwe take advantage of this coordinate rescaling to estimate the bispectrum from the CMB anisotropiesat recombination in the squeezed limit and the corresponding contamination to the local primordialnon-Gaussianity. Sec. IV contains our conclusions. To motivate the fact that the rescaling of coordinates (that we will soon introduce) correctly capturesthe physics of the CMB anisotropies at second order in the squeezed limit, we will first show thatsuch a rescaling applied to the linear equations properly reproduces the second-order sources found inRefs. [36, 38, 39]. We first recall the Boltzmann equations at first-order in perturbation theory for theCMB anisotropies. Since this subject is rather standard, we refer the reader to standard books for moredetails [51] (see also Refs. [36, 39]). Our starting metric isd s = a ( η ) h − e d η + e − d x i , (1)where a ( η ) is the scale factor as a function of the conformal time η , and we have neglected vector andtensor perturbations. The equations of motion of the first two moments of the Boltzmann equations forthe CMB photons are ∆ (1) ′ + 43 ∂ i v (1) iγ − (1) ′ = 0 , (2) v (1) i ′ γ + 34 ∂ j Π (1) jiγ + 14 ∆ (1) ,i + Φ (1) ,i = − τ ′ (cid:16) v (1) i − v (1) γ (cid:17) , (3)where Π ij is the photon quadrupole moment and τ ′ = − n e σ T a is the differential optical depth in termsof the average number of electron number density n e and Thomson cross section σ T . Here the primesindicate differentiation with respect to η and ∂ i differentiation w.r.t. x i . The two equations above arecomplemented by the momentum continuity equation for baryons, which can be conveniently written as v (1) i = v (1) iγ + Rτ ′ h v (1) i ′ + H v (1) i + Φ (1) ,i i , (4)where we have introduced the baryon-photon ratio R ≡ ρ b / (4 ρ γ ) and we have indicated by H = a ′ /a the Hubble rate in conformal time.Eq. (4) is in a form ready for a consistent expansion in the small quantity τ − which can be performedin the tight-coupling limit. By first taking v (1) i = v (1) iγ at zero order and then using this relation in theleft-hand side of Eq. (4) one obtains v (1) i − v (1) iγ = Rτ ′ h v (1) i ′ γ + H v (1) iγ + Φ (1) ,i i . (5)Such an expression for the difference of velocities can be used in Eq. (3) to give the evolution equationfor the photon velocity in the limit of tight coupling v (1) i ′ γ + H R R v (1) iγ + 14 ∆ (1) ,i R + Φ (1) ,i = 0 . (6) otice that in Eq. (6) we are neglecting the quadrupole of the photon distribution Π (1) ij (and all thehigher moments) since it is well known that at linear order such moment(s) are suppressed in the tight-coupling limit by (successive powers of) 1 /τ ′ with respect to the first two moments, the photon energydensity and velocity. Eqs. (2) and (6) are the master equations which govern the photon-baryon fluidacoustic oscillations before the epoch of recombination when photons and baryons are tightly coupled byCompton scattering.In the tight-coupling limit the contribution to the CMB anisotropies from recombination is given byΘ (1) ( k , n , η ) = (cid:18)
14 ∆ (1)00 + Φ (1) + v (1) · n (cid:19) η = η rec . (7)We are now interested in computing the second-order CMB anisotropies in the squeezed limit, i.e. whenone of the momenta is much smaller than the other two, k ≪ k ≃ k . In particular, we will be interestedin the case in which the smallest momentum is at most equal to the wavenumber entering the horizon atequivalence, k < ∼ k eq .Instead of solving the complicated network of second-order Boltzmann equations for the CMB temper-ature anisotropies, we use the following trick. As the wavenumber k < ∼ k eq corresponds to a perturbationwhich is almost larger than the horizon at recombination and the evolution in time of the correspond-ing gravitational potential is very moderate (one can easily check, for instance, that Φ (1) k ( η ) changesits magnitude by at most 10% during the radiation epoch for k = k eq ), we can absorb the large-scaleperturbation with wavelength ∼ k − in the metric by redefining the time and the space coordinates asfollows. Let us indicate with Φ ℓ and Ψ ℓ the parts of the gravitational potentials that receive contributionsonly from the large-scale modes k < ∼ k eq (see, e.g., [49]). If the scale factor is a power law a ( η ) ∝ η α ( α = 1 and α = 2 for the period of radiation and matter domination, respectively), we can perform theredefinitions a ( η ) e ℓ d η = η α e ℓ d η = η α d η = a ( η )d η ⇒ η = e α Φ ℓ η , (8)and a ( η ) e − ℓ d x = η α e − ℓ d x = η α e − α α Φ ℓ e − ℓ d x = a ( η )d x ⇒ x = e − α α Φ ℓ e − Ψ ℓ x . (9)In particular, the combination kη = e Φ ℓ +Ψ ℓ kη , (10)where k and k are the wavenumbers in the two coordinate systems. Obviously, if one wishes to accountfor the fact that at recombination the universe is not fully matter-dominated, one should perform amore involved coordinate transformation which will eventually depend also on the parameter R . To testthe goodness of this procedure, let us consider for instance the equation of motion of the gravitationalpotential Ψ k ( η ) at second-order during the radiation epoch. From the full second-order equations (B.13)and (B.14) of Ref. [36], we see that this equation reads in the squeezed limitΨ (2) ′′ k + 4 H Ψ (2) ′ k + c s k Ψ (2) k = − c s (cid:16) Ψ (1) k + Φ (1) k (cid:17) k Ψ (1) k , (11)where in the source in the r.h.s. we have implicitly assumed an integration over the momenta k and k with the corresponding Dirac delta to ensure momentum conservation. We will adopt this convention rom now on. In fact, for a bispectrum computation, it is the kernel of the source term that matters,and from this point of view we can identify the wavenumber k as a long wavelength modulation of theperturbation in the momentum k , so that k = k + k ≃ k .It is reassuring that this equation can be simply obtained by rescaling the coordinates, following (8)and (9). Just consider the (linear) equation of motion [51, 39]Ψ (1) ′′ + 4 H Ψ (1) ′ − c s ∇ Ψ (1) = 0 , (12)whose solution is Ψ (1) k ( η ) = 3Ψ (1) k (0) ( − c s kη cos( c s kη ) + sin( c s kη ))( c s kη ) . (13)Applying (8) and (9) to Eq. (12) we obtainΨ ′′ + 4 H Ψ ′ − c s e ℓ +Ψ ℓ ) ∇ Ψ = 0 . (14)Now, expanding the perturbations into a first- and second-order parts, Ψ = Ψ (1) + Ψ (2) /
2, and identifyingin Fourier space the long wavelength mode with k , one can easily check that Eq. (14) exactly correpondsto Eq. (11). The solution of Eq. (11) isΨ (2) k ( η ) = 3Ψ (2) k (0) ( − c s kη cos( c s kη ) + sin( c s kη ))( c s kη ) + 6 3 c s kη cos( c s kη ) + ( − c s kη ) ) sin( c s kη )( c s kη ) Ψ (1) k (0)Ψ (1) k (0) . (15)Again, to have a further check of our procedure, we point out that this solution can be found by employingthe coordinate rescalings (8) and (9). Taking the expression (13) and assuming that Ψ (1) k ≃ Φ (1) k we findΨ k ( η ) = 3Ψ k (0) (cid:16) − c s k η cos( c s k η ) + sin( c s k η ) (cid:17) ( c s k η ) = 3Ψ k (0) (cid:18) − e (1) k c s k η cos (cid:18) e (1) k c s k η (cid:19) + sin (cid:18) e (1) k c s k η (cid:19)(cid:19)(cid:18) e (1) k c s k η (cid:19) . (16)Expanding at second-order one, one recovers exactly the expression (15) .We have also verified that using this procedure applied to Eqs. (2) and (6) one reobtains the cor-responding second-order Boltzmann equations already computed in Eqs. (205) and (206) of Ref. [36]expressed in terms of the bolometric temperature, i.e., in terms of the variable [∆ (2) − (∆ (1) ) ]. The bolo-metric temperature is defined as (see, e.g., [46, 45]) ( T / ¯ T ) ≡ ∆, where ∆( x , n , τ ) = R dpp f / ( R dpp ¯ f ) Notice that here we are using the short-hand notation kη = e Φ (1) k +Ψ (1) k kη which simplifies the computationgiving nonetheless the correct result. Notice that the coordinate rescaling also transforms the gravitational potentialΨ k (0). This change however is reabsorbed in the physical initial conditions. s the (normalized) brightness function, f being the photon distribution function. By Taylor expandingthis definition one finds ∆ (2) = 4Θ (2) + 16 (cid:16) Θ (1) (cid:17) = 4Θ (2) + (cid:16) ∆ (1) (cid:17) , (17)where, in the notations of Refs. [45, 36], T = T e Θ = T , ∆ TT = Θ (1) + 12 Θ (2) + 12 (cid:16) Θ (1) (cid:17) , (18)is the bolometric temperature so that one recovers the temperature variables used in [45, 36] (notice that,despite using the same symbol Θ this temperature is different from the brightness temperature defined,e.g., in Eq. (4.1) of [46]). Since the existing literature usually computed the CMB bispectrum in terms ofthe bolometric temperature (e.g., [45, 46]), our results for the CMB bispectrum will refer to this variable.Let us show in some details how the coordinate rescaling method works for Eqs. (2) and (6). Under thecoordinate rescalings (8) and (9) Eq. (2) transforms as4Θ ′ + 43 e Φ+Ψ ∂ i v iγ − ′ = 0 . (19)Expanding this equation at second-order in the perturbations (and expressing the photon velocity interms of ∆ and Ψ) one finds4Θ (2)00 ′ + 43 ∂ i v i (2) γ − ′ (2) = − (1) + Ψ (1) )(4Ψ (1) ′ − (1)00 ′ ) . (20)This equation can be confronted with Eq. (205) of Ref. [36]∆ (2) ′ + 43 ∂ i v (2) iγ − (2) ′ = S ∆ , (21)where the source term is given by S ∆ = (cid:16) ∆ (1)200 (cid:17) ′ − (1) + Ψ (1) )(4Ψ (1) ′ − ∆ (1) ′ ) − v (1) iγ (∆ (1)00 + 4Φ (1) ) ,i + 163 (Φ (1) + Ψ (1) ) ,i v i − R H R v (1)2 γ − v (1) γi ∆ (1) ,i R . (22)It is easy to check that in the squeezed limit, k ≪ k ≃ k , Eq. (21) exactly coincides to Eq. (20) whenwritten in terms of the bolometric monopole [∆ (2)00 − (∆ (1)00 ) ]. Similarly one can check that the squeezedlimit of Boltzmann equation for the second-order velocity equation of the photon-baryon system can beobtained in the same way. Let us start from the (linear) Eq. (6). By applying the coordinate rescalings(8) and (9) we obtain v ′ γ + H R R v iγ + 14 e (Φ+Ψ) ,i R + e (Φ+Ψ) Φ (1) ,i = 0 , (23)which, expanded at second-order in the perturbations, becomes v (2) i ′ γ + H R R v (2) iγ + 14 4Θ (2) ,i R + Φ (2) ,i = − R ) (Φ (1) + Ψ (1) )4Θ ,i − (1) + Ψ (1) )Φ (1) ,i . (24) his equation corresponds to the squeezed limit of the velocity continuity equation computed directly atsecond order in Ref. [36] (see Eq. (209)) v (2) i ′ γ + H R R v (2) iγ + 14 ∆ (2) ,i R + Φ (2) ,i = S iV , (25)where S iV = − R ) ∂ j Π (2) jiγ − ω ′ i − H R R ω i + 2 H R (1 + R ) ∆ (1)00 v (1) iγ + 14(1 + R ) (cid:16) ∆ (1)200 (cid:17) ,i + 83(1 + R ) v (1) iγ ∂ j v (1) jγ + 2 R R Ψ (1) ′ v (1) iγ − (1) + Ψ (1) )Φ (1) ,i − R ) (Φ (1) + Ψ (1) )∆ (1) ,i − R R ∂ i v (1)2 γ − R R ∆ (1)00 H R v (1) iγ −
14 ∆ (1) ,i R ! . (26)Here ω i is the pure second-order metric perturbation of the (0 − i ) metric tensor (which therefore doesnot appear in the first-order equation) while Π (2) ij is the second-order quadrupole moment of the photons.If one takes the source term S iV , Eq. (26), in the squeezed limit and for R = 0, one recovers exactlyEq. (24) once Eq. (25) is expressed in terms of the bolometric temperature [∆ (2)00 − (∆ (1)00 ) ]. Noticethat the second-order velocity continuity equation of the photon-baryon system in the squeezed limit isrecovered only when k Rη rec <
1, i.e. when the timescale of the modulating long wavelength mode ismuch bigger than the typical timescale of the collision term in the tight coupling limit (in particular for R ≪ R is not much smaller than unity. Since we will be concerned with a signal-to-noise ratio (
S/N ) dominated by the maximum multipole agiven experiment can reach, ℓ max ≫
1, we can use the flat-sky approximation [52] and write for thebispectrum h a ( ~ℓ ) a ( ~ℓ ) a ( ~ℓ ) i = (2 π ) δ (2) ( ~ℓ + ~ℓ + ~ℓ ) B ( ℓ , ℓ , ℓ ) . (27)Inspecting Eq. (18), the bispectrum gets two contributions. One from the intrinsically second-order termΘ (2) and one from the term (Θ (1) ) / η rec ≫ η eq in such a way that the coordinate transformations (8) and (9)can be performed in a matter-dominated period, that is we take α = 2 and x → e − ℓ / x , k → e ℓ / k , η rec → e Φ ℓ / η rec , (28) s the transformation for modes which were outside the horizon at recombination, but are subhorizon atthe time of observation. It is therefore enough to consider the rescaling acting on a ( ~ℓ ) = Z d k z π e ik z D rec Φ (1) k ′ (0) ∆ T ( ℓ, k z ) , (29)where ∆ T ( ℓ, k z ) is the radiation transfer function and D rec = ( η − η rec ) is the distance to the surface oflast scattering. We consider only the contribution at recombination and therefore∆ T ( ℓ, k z ) = 1 D S ( q ( k z ) + ℓ /D , η rec ) , S = (cid:18)
14 ∆ (1)00 + Φ (1) + v (1) · n (cid:19) η = η rec / Φ (1) k (0) . (30)Here k z is the momentum component perpendicular to the plane orthogonal to the line-of-sight and thesuperscript ′ reminds us that in the expression for k we have to set ~k k = ~ℓ/D ∗ , where ~k k is the componentperpendicular to the line-of-sight. As we have learnt in the previous section, the rescaling (28) reproducesthe second-order sources and the corresponding solutions starting from the first-order expression of thetemperature anisotropy. Therefore, we have to think of the expression (30) as a function of the rescaledcoordinates in such a way that an expansion of the long mode will give the exact second-order quantityin the squeezed limit. Notice that the rescaling (28) changes also the gravitational potentialΦ (1) k → e − ℓ Φ (1) e ℓ/ k , (31)and therefore a ( ~ℓ ) is not subject to any rescaling if one takes the Sachs-Wolfe large-scale limit in whichthe source S reduces to 1 / T ( ℓ, k z ) = aD e − / ℓ/ℓ ∗ ) . e − / | k z | /k ∗ ) . , (32) i.e. a simple exponential and a normalization coefficent a to be determined to match the amplitude ofthe angular power spectrum at the characteristic scale ℓ ≃ ℓ ∗ = k ∗ D rec . As shown in Ref. [44], the values ℓ ∗ ≃
750 and a ≃ ℓ > ℓ eq ≃
160 and boosts the angular power spectrum with respect to the Sachs-Wolfe plateau, andthe effects of Silk damping which tend to suppress the CMB anisotropies for scales ℓ > ℓ D ≃ ℓ ∗ ≃ . h a ( ~l ) a ( ~l ) i = (2 π ) δ (2) ( ~l + ~ℓ ) C ( ℓ ) , (33)with C ( ℓ ) = D (2 π ) Z d k z | ∆ T ( ℓ, k z ) | P ( k ) . (34)The exponential of the transfer function allows to cut off the integral for k ≃ k ∗ and one finds [44] C ( ℓ ) ≃ a Aπ ℓ ∗ ℓ e − ( ℓ/ℓ ∗ ) . , (35) hich holds for ℓ ≫ ℓ ∗ and we have used the amplitude of the primordial gravitational potential powerspectrum computed at first-order h Φ (1) ( k )Φ (1) ( k ) i = (2 π ) δ (3) ( k + k ) P ( k ) , P ( k ) = A/k (36)with amplitude A = 17 . × − . To compute the bispectrum, we go to the squeezed limit ℓ ≪ ℓ , ℓ (or k ≪ k , k ). In this case Φ (1) k acts as a background for the other two modes. One can therefore computethe three-point function in a two-step process: first compute the two-point function in the background ofΦ (1) k and then the result from the correlation induced by the background field. Following Ref. [37] (seealso [54]) and using the Sachs-Wolfe limit for the multipole ℓ , this procedure leads to h a ( ~ℓ ) a ( ~ℓ ) i Φ (1) k = h a ( ~ℓ ) a ( ~ℓ ) i + 5 a ( ~ℓ + ~ℓ ) C ( ℓ ) d ln (cid:2) ℓ C ( ℓ ) (cid:3) d ln ℓ . (37)The bispectrum from the intrinsically second-order term Θ (2) therefore reads B Θ (2) ( ℓ , ℓ , ℓ ) = D a ( ~ℓ ) h a ( ~ℓ ) a ( ~ℓ ) i E = (2 π ) δ (2) ( ~ℓ + ~ℓ + ~ℓ ) 5 C ( ℓ ) C ( ℓ ) d ln (cid:2) ℓ C ( ℓ ) (cid:3) d ln ℓ . (38)Notice also in all these computations one can safely neglect the rescaling of the conformal time atrecombination η rec up to terms η rec /η ≪ (1) ) /
2. In the squeezed limit it reads (see alsoRefs. [45, 36]) B ( Θ (1) ) ( ℓ , ℓ , ℓ ) = (2 π ) δ (2) ( ~ℓ + ~ℓ + ~ℓ ) 2 C ( ℓ ) C ( ℓ ) . (39)The total bispectrum therefore is B rec ( ℓ , ℓ , ℓ ) = (2 π ) δ (2) ( ~ℓ + ~ℓ + ~ℓ ) C ( ℓ ) C ( ℓ ) " (cid:2) ℓ C ( ℓ ) (cid:3) d ln ℓ . (40)Notice that if we consider the second-order CMB anisotropies on large-scales with all the modes on super-Hubble scales at recombination (i.e. with all wavenumbers ( k i η rec ≪ ∆ TT = 13 Φ rec + 118 (cid:16) Φ (1)rec (cid:17) . (41)The term (Φ (1)2rec /
18) corresponds in this limit exactly to the (Θ (1)2 /
2) term in Eq. (18). The result (41) wasfirst obtained in Ref. [55] (see also the discussion that followed in Ref. [50]). This additional contributionto primordial non-Gaussianity from the second-order gravitational potential in (Φ rec = Φ (1)rec + Φ (2)rec / (1)2rec /
18) (see Refs. [55, 49]) can contribute to a contamination of the primordial localnon-Gaussianities only on the very largest scales, 2 < ∼ ℓ < ∼ f conNL defined below). This expression is valid only when all the wavelength modes are outside the horizon at recombination, and,e.g., when written as a convolution in Fourier space, it does not hold for an external wavenumber k η rec ≪ k ′ and k ′ inside the horizon. ur goal now is to estimate the level of degradation that the NG from recombination in the squeezedlimit causes on the possible measurement of the local primordial bispectrum. A rigorous procedure is todefine the Fisher matrix (in flat-sky approximation) as F ij = Z d ℓ d ℓ d ℓ δ (2) ( ~ℓ + ~ℓ + ~ℓ ) B i ( ℓ , ℓ , ℓ ) B j ( ℓ , ℓ , ℓ )6 C ( ℓ ) C ( ℓ ) C ( ℓ ) , (42)where i (or j )= (rec , loc), and to define the signal-to-noise ratio for a component i , ( S/N ) i = 1 / q F − ii ,and the degradation parameter d i = F ii F − ii , due to the correlation bewteen the different components r ij = F − ij / q F − ii F − jj . In order to measure the contamination to the primordial bispectra one can definethat effective non-linearity parameter f conNL which minimizes the χ defined as χ = Z d ℓ d ℓ d ℓ δ (2) ( ~ℓ + ~ℓ + ~ℓ ) h f conNL B loc ( ℓ , ℓ , ℓ ; f locNL = 1) − B rec ( ℓ , ℓ , ℓ ) i C ( ℓ ) C ( ℓ ) C ( ℓ ) , (43)to find f conNL = F rec , loc F loc , loc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f locNL =1 . (44)In multipole space the bispectrum induced by a local primordial NG in the squeezed limit is given by B loc ( ℓ , ℓ , ℓ ) = − f locNL [ C ( ℓ ) C ( ℓ ) + cycl . ] . (45)Performing the integrals in the multipoles and using Eq. (35), we find f conNL ≃ −
16 + 512 " . (cid:18) ℓ max ℓ ∗ (cid:19) . − (cid:18) ℓ min ℓ max (cid:19) . (cid:18) ℓ min ℓ ∗ (cid:19) . ! ≃ . , (46)where we have taken ℓ max = 2000, ℓ ∗ = 750 and ℓ min = 1200, see Ref. [44].Let us also remark that in Refs. [44, 36] the non-linear evolution on small scales (for all the wavenum-bers ( k i η rec ≫
1) at recombination) of the second-order gravitational potential also produces a smallcontamination to the local non-Gaussianity of the order of f conNL = 0 .
3. This contribution should be addedto Eq. (46).
In this paper we have analytically estimated the level of non-Gaussianity produced by the non-linearevolution of the photon-baryon system at recombination in the squeezed limit. While a contamination O (5) was numerically obtained in [46], we do find that the total contamination to the primordial localnon-Gaussianity is smaller and not within the reach of present experiments. Our main goal was to providea clear and simple way to understand the physical origin of such a contamination. We have reached thegoal using a simple rescaling of the local coordinates of a perturbed FRW universe, which explains in avery transparent way the large-scale modulation of the perturbations which is generally at the origin of squeezed non-Gaussian signal. Let us make some brief remarks about the validity of our results. It isclear that they are valid up to terms O (Φ (1) k ∇ Φ (1) k ), which vanish in the exact squeezed limit, k → k that were not much outside theHubble radius at recombination, e.g. for k ≃ k eq . For example the initial condition for the second-ordergravitational potential would get a small correction (see Eq. (269) of [36]) from the initial condition (withvanishing primordial non-Gaussianity)Ψ (2) k (0) = 11 F ( k , k , k ) k Ψ (1) k (0)Ψ (1) k (0) , (47)where the function F ( k , k , k ) would vanish in the exact squeezed limit (see its expression in Eq. (242)of [36]. It is easy to check that, if for example, one takes k = k eq and k = 15 k eq (the latter correspondingto ℓ = 2000) one finds that the contamination to the primordial local NG induced from the correction ofthe F term in Eq. (47) is | f conNL (loc) | ≃ .
1. Indeed this must also be intended as an upper limit, since thiscorrection does not fully correlate with the local type f locNL . This example is to show that the correctionscoming from the perturbation modes k = k eq is actually subdominant. Acknowledgments
This research has been partially supported by the ASI/INAF Agreement I/072/09/0 for the Planck LFIActivity of Phase E2.
Note added
When completing this work, we have become aware of a similar work by P. Creminelli, C. Pitrou and F.Vernizzi. Our results, when overlap is possible, agree with theirs. We thank them for useful correspon-dence.
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