Cosmic microwave background bispectrum from nonlinear effectsduring recombination

S.-C. Su,1 Eugene A. Lim,1,2 and E. P. S. Shellard11Centre for Theoretical Cosmology, Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom2Theoretical Particle Physics and Cosmology Group, Physics Department, Kings College London,

Strand, London WC2R 2LS, United Kingdom(Received 4 August 2013; revised manuscript received 8 April 2014; published 1 July 2014)

We study cosmological perturbations by solving the governing Boltzmann and Einstein field equationsup to second order and calculate the corresponding CMB bispectrum during recombination. We include allthe second-order Liouville and collision terms, truncating the multipole hierarchy at l ¼ 10, consistentlyincluding all m ≠ 0 terms when calculating the bispectrum in the flat-sky limit. At this stage, we focus oncontributions at and around recombination, and we neglect second-order vector and tensor perturbations,lensing effects, and late-time nonlinear ISW effects. We find that the signal-to-noise for the bispectrum is0.69 for lmax ¼ 2000, yielding an overall signal FNL ¼ 3.19 (normalized relative to the local model).We find that the effective fNL’s of the equilateral and local type are 5.11 and 0.88, respectively. Thisrecombination signal will have to be taken into account in a quantitative analysis of the Planck data.

DOI: 10.1103/PhysRevD.90.023004 PACS numbers: 98.70.Vc, 98.80.-k

I. INTRODUCTION

Non-Gaussianities in the cosmic microwave background(CMB) provide extra clues about the Universe. With theimproving sensitivities from the current and future obser-vations [1–3], a quantitative study of non-Gaussianities isessential. It could provide crucial insights advancing ourunderstanding of the physics of the very early Universe, soits importance cannot be overstated. However, as well asprimordial effects when the initial conditions are laid downduring inflation, there can be late-time nonlinear inter-actions of the photon-baryon fluid with gravity [4]. Therehave been several past attempts at estimating the CMBbispectrum from the latter in the literature [5–8].A detailed quantitative analysis of these late-time con-

tributions is necessary, not only for debiasing potentialprimordial signals, but also as an independent test ofGeneral Relativity(GR) and cosmological perturbationtheory. Recently, Huang and Vernizzi conducted such anumerical implementation of the second-order CMB per-turbations [9], where they obtained a forecast of flocNL ≈ 0.82for the local template and the signal-to-noise S=N ¼ 0.47from the full bispectrum.1

In this paper, we present our numerical calculation ofthe bispectrum produced around recombination. More pre-cisely, we calculate all the effects due to the Comptoncollision at recombination and the early-time ISW-relatedcouplings around recombination. We obtain flocNL ¼ 0.88, inagreement with [9] and also previous work focusing on thesqueezed limit [11,12]. We emphasize that the second-orderline-of-sight (LOS) approach used here is different from the

one used in [9,10] and provides a different way to calculatethe CMB bispectrum around recombination. Our approachinterprets the quadratic second-order effects as couplingsof two well-known first-order effects in Newtonian gauge[See Eq. (18) for more details]. This allows us to separateeffects around recombination from the late-time effectsconsistently.2 Furthermore, we obtain fequNL ¼ 5.11 for theequilateral template. This value agrees very well with theprevious analytic estimate provided in [13]. The overallsignal-to-noise ðS=NÞrec ¼ 0.69 provides an effectiveFNL ¼ 3.19 [14] contribution which should be incorpo-rated in the analysis of the Planck CMB data with a forecastvariance of ΔFNL ¼ 5. First, however, we briefly describethe underlying analytic methodology before describing thenumerical pipeline that has been developed, leading to therecombination bispectrum results. A much more detaileddiscussion of these methods and results will be presented ina longer paper shortly [15].

II. SECOND-ORDER PERTURBATIONS ANDTHEIR BISPECTRA

To study non-Gaussianities generated by nonlineareffects, we solve the second-order Boltzmann equation

L½II� þ L½I;I� ¼ C½II� þ C½I;I�; ð1Þ

where L is the Liouville operator, C is the collisionoperator, the superscripts [II] and [I,I] denote linear terms

1Another recent work comes from [10].

2In principle, second-order Boltzmann equation is only enoughfor studying the effects around recombination but not at latetimes.

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of purely second-order perturbations and quadratic terms oftwo first-order perturbations, respectively. In addition, werequire knowledge of the first-order ionization fraction [16]at recombination, which encodes the effect of the perturbedlast-scattering surface (LSS), when evaluating C½I;I�.We begin with the metric expanded up to second order

gμνaðηÞ2 dx

μdxν ¼ − ð1þ 2ΦÞdη2 þ 2Bidxidη

þ ½ð1 − 2ΨÞδij þ 2X ij�dxidxj; ð2Þ

where aðηÞ is the scale factor, and Φ, Ψ, Bi’s and X ij’s areperturbations expanded in the following way

A ¼ A½I� þ 1

2A½II�; ð3Þ

with A denoting the perturbation quantities above. We willuse Newtonian gauge up to second order throughout.Moreover, we ignore the first-order and second-ordervector and tensor perturbations, i.e. Bi ¼ X ij ¼ 0. Ingeneral, the second-order vector and tensor perturbationsare non-negligible as they are sourced by the first-orderscalar perturbations nonlinearly. However, we expect theseto be subdominant.We compute the reduced bispectrum around recombi-

nation with the flat-sky and thin-shell approximation [8]

brecl1l2l3≈

r−4LSSð2πÞ2

Z∞

−∞dkz1dk

z2Pðk1ÞPðk2Þ

×Z

0

rLSS

dr1dr2dr3e−iðkz

1r1þkz

2r2þkz

3r3ÞS½I�ðk1; r1Þ

× S½I�ðk2; r2ÞS2NDðk1;k2; r3Þ þ 1 ↔ 3þ 2 ↔ 3;

ð4Þwhere LSS denotes the last-scattering surface, r≡ η0 − ηwith the present conformal time η0, PðkÞ is the primordialpower spectrum, kz is the component of k perpendicular tothe tangent plane [17]. The first-order source function S½I�

refers to SðSÞT in [18] while the second-order source functionS2ND can be expanded into its linear and quadratic partsS2ND ¼ S½II� þ S½I;I�. The linear part S½II� refers to Eq. (40)of [19] with the first-order perturbations replaced by thekernels A½II�ðk1;k2; ηÞ of the corresponding second-orderperturbations which are defined as

A½II�ðk; ηÞ≡Z

dk1dk2

ð2πÞ32 δ3ðk − k1 − k2ÞA½II�ðk1;k2; ηÞ

ð5Þ

where δ3 is the Dirac delta function in three dimension.Analogous to the well-known linear perturbation theory,S½II� contains contributions from intrinsic photon density,

Sachs-Wolfe(SW), Doppler, integrated Sachs-Wolfe(ISW)effects of purely second-order perturbations which requirethe full set of solutions from the second-order Boltzmann[20] and Einstein Field equations [21] (BEs and EFEs,respectively).In contrast, the quadratic part S½I;I� requires solutions of

first-order perturbations only and can be read in configu-ration space as3

S½I;I� ¼ 2e−τ½Sij∂jðΦþΨÞ ∂Δ∂ni − ðΦþΨÞni∂iΔ

þ 4ð−ni∂iΦþΨ0ÞΔþ ðΦ −ΨÞni∂iΦ

þ 2ΨΨ0 þ C½I;I��; ð6Þ

where Sij is the screen projector [20], 4Δ≡ δI=I is thefractional brightness, ni’s are the components of theobservational direction, C½I;I� is the quadratic collisionoperator [refers to Eq. (A.55) of [8] in Fourier space],and τ is the optical thickness.The first three terms in Eq. (6), which contain fractional

brightness Δ, are numerically problematic. It is because thecontributions of the LOS approach from these terms extendto late times at which high multipoles of the brightnessare generated through propagation and projection onto thespherical harmonics. This means that solving these terms isnumerically expensive since truncating at low multipoles isnot enough. Because we focus on the contributions at andaround recombination, we ignore the first and second termsin Eq. (6) which correspond to the lensing and time-delayeffects, respectively, and contribute at late times. The thirdterm in Eq. (6) is responsible for the first-order redshifteffects, including the SWand ISWeffects, on the first-ordertemperature anisotropies. Intuitively, it includes contribu-tions at and around recombination (i.e. couplings with SWand early-time ISW effect) as well as late-time contribu-tions (i.e. couplings with late-time ISW effect). The keyissue here is to physically distinguish between the early-time and late-time effects for those redshift-relatedcouplings, which we will pay particular attention to.Before we work on the second-order source function, it is

useful to review how we separate the early-time and late-time ISW effect at first order. The first-order fractionalbrightness at time η can be expressed in Fourier space as theLOS integral over the source ST ,

Δ½I�ðη;k; nÞ ¼ eτðηÞZ

η

0

d~ηe−ikμðη−~ηÞS½I�T ð~η;k; nÞ; ð7Þ

where μ≡ k · n=k and the first-order source function inNewtonian gauge is4

3The notation for first-order perturbations is omitted forsimplicity.

4For simplicity, we have omitted the superscript [I] on theright-hand side of the equation.

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S½I�T ¼ e−τ�Ψ0 − ikμΦþ τ0Δþ C

4

�: ð8Þ

The first two terms generate redshift effects on the back-ground CMB due to the linear perturbations of the metricwhile the last two terms contain all the effects generated atrecombination due to the Compton scattering.5 As shown in[18], we can use the integration by parts (IBPs) on thesecond term of Eq. (8) such that the source term can be

replaced by ~S½I�T

eτ ~S½I�T ≡ −δð~η − ηÞΦþ τ0ΦþΨ0 þ Φ0 þ τ0Δþ C4; ð9Þ

where δ is the Dirac delta function. The first two termsencode the SW effect while the next two terms are the ISWeffect.Integration by parts are not mere mathematical tricks.

Since we are replacing derivative terms in the integrandwith nonzero boundary terms, this means that it isimportant where we evaluate the boundary terms at.Physically, performing the IBPs “redistributes” the red-shift effects along the LOS integration. In this caseEq. (9) separates the redshift effects occurring at differenttimes physically as follows. The first two terms are theSW due to the difference of the gravitational potentialbetween recombination and time η—this is the SW effectmeasured by an observer at time η. In particular, Φ in thefirst term is a constant when η ¼ η0. The third and fourthterms (time derivatives on the potentials) include theearly-time ISW from the time-varying gravitational poten-tial due to the transition from radiation-dominated tomatter-dominated era and the late-time ISW from thetime-varying gravitational potential due to the dominationof dark energy. In this paper, we calculate the bispectrumat and around recombination. Explicitly, it means that weinclude SW and early-time ISW but exclude late-timeISW. In the following, we extend the separation to theredshift-related couplings in second order, i.e. the thirdterm of Eq. (6).The second-order fractional brightness observed at

present can be written in Fourier space as

Δ½II�ðη0;k; nÞ ¼Z

η0

0

dηe−ik·nrT kfS2NDðk1;k2; rÞg;ð10Þ

where the convolution operator T k is defined as

T kf…g≡Z

dk1dk2

ð2πÞ3=2 δðk − k1 − k2Þ…: ð11Þ

From now on, we focus on the third term of Eq. (6). We canexpress its contribution (denoted by the subscript R) to thesecond-order fractional brightness as

Δ½II�R ðη0;k; nÞ

¼ 8T k

�Zη0

0

dηe−ik·nr−τΔ1ðnÞ½Ψ02 − ik2 · nΦ2�

�;

ð12Þwhere the subscripts 1 and 2 denote the dependence on k1

and k2, respectively. As mentioned in Sec. 4.2 of [10], if weperform IBPs on the term with μ2 ≡ k2 · n=k2 in Eq. (12),we will get a time integration whose integrand contains thetime derivative of the first-order fractional brightness Δ0.We can iteratively apply IBPs on that time integration butthis approach fails to single out a unique LSS contribution.More seriously, contributions from higher multipoles growwhen we integrate over the time further after recombina-tion. That is, the approach is numerically unstable.However, there exists a unique way to separate the early-

time and late-time redshift effects which is physicallymeaningful and allows us to calculate the bispectrumaround recombination with a low-multipole truncation.The key is to realize that Eq. (12) contains a double timeintegration by substituting Δ1 (via Eq. (7) which containsan addition time integration) into the first-order fractionalbrightness. We can then exchange the inner and outer timeintegrations such that Eq. (12) is written as

Δ½II�R ¼ 8T k

�Zη0

0

d~ηe−ik1·n ~rSTð~η;k1; nÞ

×Z

η0

~ηdηe−ik2·nr½Ψ0

2ðηÞ − ik2 · nΦ2ðηÞ��: ð13Þ

Now, we perform IBPs on the integration over η andobtain

Δ½II�R ¼ 8T k

�Zη0

0

d~ηe−ik·n ~rSTð~η;k1; nÞ

×

�Φ2ð~ηÞ þ

Zη0

~ηdηeik2·nð~r−rÞ½Φ0

2ðηÞ þΨ02ðηÞ�

��:

ð14ÞWe make use of the fact that k ¼ k1 þ k2 to get theexponential with k in the first line of Eq. (14). By doing so,we do not produce the problematic term with Δ0 throughIBPs. Note that we can ignore Φ evaluated at present η0because it is a constant and does not contribute to thebispectrum.The physical meaning of Eq. (14) is clear. Analogous to

first-order redshift effects which come from the backgroundCMB signals redshifted by the first-order perturbedmetric, we have the first-order temperature anisotropies

5We ignore the reionization and other late-time scatteringsthroughout this paper.

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redshifted by the first-order SWand ISWeffects to generatesecond-order temperature anisotropies, corresponding tothe first and second term in Eq. (14), respectively. This trickallows us to distinguish between the early-time and late-time redshift effects similar to what we have done typicallyin first order.We now consistently separate the early-time and late-

time redshift effects of the first-order source function ST inEq. (14). For the first term of Eq. (14), we apply the IBPs onthe term with −ik1 · nΦ1 of the source function ST inEq. (8) and obtain

− T k

�Zη0

0

d~ηe−ik·n ~r−τik1 · nΦ1Φ2

�

¼ T k

�Zη0

0

d~ηe−ik·n ~r−τ�Φ0

1Φ2 þτ0

2Φ1Φ2

��; ð15Þ

for which we make use of the symmetry between thedummy variables k1 and k2 in the convolution and ignorethe constant boundary term evaluated at present η0. For thesecond term of Eq. (14), we exchange the inner and outertime integrations and substitute the integration over ~η withEq. (7). Putting all these together, we rewrite Eq. (12) as

Δ½II�R ¼ 8T k

�Zη0

0

dηe−ik·nr−τ�ðΔ1 þ Φ1ÞðΦ0

2 þΨ02Þ

þ τ0

2Φ1Φ2 þ

�τ0Δ1 þ

C1

4

�Φ2

��; ð16Þ

where

Δ1ðηÞ þ Φ1ðηÞ ¼ eτðηÞZ

η

0

d~ηe−ik1·nðη−~ηÞ−τð~ηÞ

×

�τ0Φ1 þ ðΦ0

1 þΨ01Þ þ τ0Δ1 þ

C1

4

�:

ð17Þ

Equation (16) contains all possible second-order couplingsinvolving first-order redshift effects: the first term includescouplings between ISW and any first-order CMB effects(e.g., Doppler-ISW, SW-ISW, and even ISW-ISW cou-plings); the second term is the SW-SW coupling; the thirdterm corresponds to couplings between CMB signalsgenerated by Compton scattering and SW effect (e.g.,Doppler-SW coupling).At second order, performing IBPs becomes tricky as

mentioned in [10]. However, the approach we takeuniquely separates SW, early-time ISW and late-timeISW effects along the LOS integration, as the couplingterms shown in Eq. (16). In contrast, [8] also performs IBPsto the third term of Eq. (6) but the integrand they obtained isdifferent from that in Eq. (16). Since different IBPs re-distribute the redshift effects differently along the time

integration, the SW and ISW effects are not manifestlyseparated in [8]. For example, any residual couplingsbetween the photon brightness and k · nΦ in the LOSintegrand do not damp out well after recombinationbecause the term k · nΦ mixes up SW and ISW effectthroughout the LOS integration. In particular, it is prob-lematic when a cutoff is used to the time integrationbecause early-time effects may be redistributed to latetimes and vice verses. In other words, integrating over theearly-time regime does not necessarily calculate the physi-cal early-time effects. We speculate that this could beone of the reasons why [8] concludes a much largersignal-to-noise.Although we perform a similar cutoff to the time

integration (explicitly from η ¼ 230 to 1050 Mpc), thisis numerically stable in Eq. (16). Indeed, the cutoff allowsus to capture the SW-related and early-time ISW-relatedcouplings but exclude the late-time ISW-related couplings.To understand how this works, we point out that only thefirst term in Eq. (16) has non-negligible contribution(ignoring reionization) outside the chosen time rangebecause of the ISW term ðΦ0 þΨ0Þ. However, ISW effectis highly suppressed after η ¼ 1050 Mpc when our uni-verse is matter-dominated and only becomes significantagain when the dark energy dominates at late times. Indeed,the early-time ISW is suppressed well before η ¼ 1050 andthus we find it safe to simply use a sharp cutoff at η ¼ 1050with negligible impact to the bispectrum. However, theextended time integration may raise some uncertainties onthe early-time ISW-related couplings due to the thin shellapproximation. We will discuss about this in details later.With Eq. (16), we can rewrite the quadratic part S½I;I� in

Eq. (6) in configuration space as

S½I;I� ¼ 2e−τ�Sij∂jðΦþΨÞ ∂Δ∂ni − ðΦþΨÞni∂iΔ

þ 4ðΦ0 þΨ0ÞðΔþ ΦÞ þ ðΦ −ΨÞni∂iΦþ 2ΨΨ0�þ 2g½ð4Δþ ~CÞΦþ 2Φ2 þ ~C½I;I��; ð18Þ

where g≡ τ0eτ and C≡ τ0 ~C. Here, we emphasize thatEq. (18) uniquely separates the SW effect, the early-timeand late-time ISW effect and allows us to have a early-timecutoff on the time integration to include the second-ordereffects generated at and around recombination.Physically, the two terms in the first line of Eq. (18) are

the lensing and time-delay effects, respectively. The IBPsperformed help to distinguish SW effect from ISW effectand clarify the physical meaning of each term. For example,we can interpret ð4Δþ ~CÞΦ and Φ2 as the photon-SW andSW-SW couplings, respectively. The term ðΦ −ΨÞni∂iΦþ2ΨΨ0 is the quadratic part of the evolution equationof photon energy p in second order, i.e. ðdp=dηÞ½I;I�.We will discuss the contributions of these effects on thebispectrum later.

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Our ultimate goal is to calculate the bispectrum asobserved today, which means that all the effects postrecombination till today are ideally included in the calcu-lation. However, since nonperturbative effects complicatethe study at low redshifts(z≲ 10), an appropriate milestonetowards this goal is the computation of the bispectrumaround recombination which is what we undertake in thispaper. This means that we cut off the LOS integration earlyafter recombination, allowing us to make a definite state-ment about the amplitude of the bispectrum at this timeclearly. We ignore contributions from lensing, time-delayand late-time ISW-related effects. The lensing and time-delay effects have been studied elsewhere [22,23]. Finally,we leave for a future publication the effects from thesecond-order vector and tensor perturbations and the late-time ISW-related effects—note that the former contributesthroughout the entire LOS integration although they shouldbe subdominant.

III. NUMERICAL IMPLEMENTATION

To calculate the bispectrum in Eq. (4), we solve the first-and second-order perturbations numerically. For bothcases, we use the Newtonian gauge as shown in [8]. Inorder to calculate the second-order perturbations, we storeall the numerically solved first-order perturbations in a gridof k and τ. Then, we feed them into the inhomogeneousparts of the second-order BEs and EFEs. There are twopoints worth to be emphasized when solving the second-order perturbations numerically. First, we solve and storethe kernels of the second-order perturbations as functionsof k1, k2, μ12 ≡ k1 · k2 and η. For conservative conver-gence, the numbers of sampling points we use are 180, 10,120 for k, μ and τ, respectively. Second, there existnumerical instabilities in Newtonian gauge in both firstand second order arising from two large terms cancelingalmost perfectly with each other and the residual is a feworders of magnitude smaller. To tackle this problem, we usetwo evolution equations which are linear combinations6 ofthe original four scalar EFEs (Eqs. (A.1)–(A.4) in [8]). Wethen use the constraint equations of the EFEs to verify thenumerically calculated second-order perturbations.The solved second-order perturbations are then fed into

the purely second-order source function S½II�. Althoughonly first-order perturbations are needed to compute thequadratic source function S½I;I�, it has to be decomposed intomultipoles S½I;I�l;m . In principle, closing the hierarchy requiresmultipoles up to infinite l, but this is numerically nottractable, and hence the usual prescription is to truncate thehierarchy once convergence is reached. The IBPs used inEq. (18) boost the convergence such that l up to 10 issufficient as shown in Fig. 1. We include all the m ≠ 0modes consistently up to l ¼ 10.

Our bispectrum at the squeezed limit is in good agree-ment with the analytical estimate in [11,12] as shown inFig. 2. From this figure, we note that the flat-sky approxi-mation performs well in the squeezed limit. The approxi-mation breaks down only when the large l≲ 200.

500 1000 1500 2000

6

4

2

2

410 2 10 6

All Q.T.

All L.T.

All

Analytic

FIG. 2. The recombination bispectrum (gray solid line) calcu-lated numerically using Eq. (4) and the analytic bispectrum (blacksolid line) are plotted in the squeezed limit. We can see that theymatch with each other very well.

600 800 1000 1200 1400 1600 1800 2000

20

15

10

5

5

up to 5

up to 8

up to 10

600 800 1000 1200 1400 1600 1800 2000

20

25

30

35

40

up to 5

up to 8

up to 10

FIG. 1. The graphs of the bispectra generated from thequadratic source function S½I;I� against l for equilateral(upper)and squeezed(lower) limit. The curves correspond to different ltruncations of the multipoles S½I;I�lm . We can see that the con-vergence occurs when l goes up to 10.

6This technique is first employed in the CMBQUICK codeintroduced by Pitrou [8].

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The breakdown occurs earlier than mentioned in [17] forthe flat-sky approximation. We believe this is because thecutoff of the LOS integration is extended to η ¼ 1050 Mpcin order to include the couplings with the early-timeISW effect. In this case, the thin-shell approximation maybreak down while the flat-sky approximation is still valid.However, these ISW-related couplings should contributemildly to the total bispectrum [see Fig. 1 in [24], and theSW-ISW and γ-ISW effects in the lower panel of Fig. 3].In the future, we will release the thin-shell and flat-skyapproximations to verify this.In addition, we remark that the numerical accuracy of

the multipoles S½I;I�l;m with m ≠ 0 cannot be checked withthe analytic solution as they contribute negligibly in thesqueezed limit. The reason is as follows. In first-order

perturbation theory, the source function S½I�T ðη;k; nÞ can be

expressed as ~S½I�T ðη; k; μÞζðkÞ, where μ≡ n · k and ζðkÞ isthe primordial perturbation. Since ~S½I�T is a function of μ, itcan be decomposed into spherical harmonics with onlym ¼ 0 modes. On the other hand, in second-order pertur-bation theory, the kernel of quadratic source function S½I;I� isa function of η and n as well as k1 and k2 which come fromthe couplings between two first-order perturbations. Thedecomposition of S½I;I� into spherical harmonics will containmodes with m ≠ 0 because k1 and k2 do not necessarily

align with each other. In other words, the couplings breakthe symmetry with m ¼ 0. However, in the squeezed limit(e.g. k1 ≪ k2;k3), the conservation law of momenta(k3 ¼ k1 þ k2) implies that k2 is almost parallel to k3.Thus, modes with m ≠ 0 are suppressed because of thesmall k1. In this case, we have only m ¼ 0 modes when k3

aligns with the z-axis of the multipole decomposition.Similarly, the second-order vector and tensor perturbationsare negligible in the squeezed limit as they are sourced asm ¼ 1 andm ¼ 2modes, respectively. Having said all this,we emphasize that m ≠ 0 modes have to be considered fornonsqueezed configurations.

IV. RESULTS AND DISCUSSION

In Fig. 3, we plot the bispectra of the terms in the sourcefunction S2ND. We can see that the main contributions comefrom the effects of the photon-SW, SW-SW and quadraticcollisions as well as the purely second-order SW, Dopplerand anisotropic stress (the term Π in [18]) effects. The sumof the photon-SW and the SW-SW effects is roughlyconstant in the plot—the total bispectrum from thesetwo effects is approximately proportional to the productof the power spectra of the long and short wavelengthmodes, i.e. ClSClL . In particular, this constant offset isalso shown in Fig. 1 of [10]. This verifies that our IBPsapproach is consistent7 with the change-of-variablesapproach first introduced by [9]. Since we decomposethe quadratic source term S½I;I� into different physicalcouplings, the physical explanation of the offset is clear:the power spectrum of the short wavelength mode comesmainly from intrinsic intensity, SW and Doppler effectswhile the power spectrum of the long wavelength mode isproportional to the square of the initial gravitationalpotential Φ (SW effect). Moreover, the offset shifts thebispectrum from the purely second-order source terms upand suppresses the correlation between the recombinationbispectrum and that of the local type.We present the full recombination bispectrum in Fig. 4,

showing isosurfaces of the bispectrum density. Althoughthe main contribution to the recombination bispectrum isconcentrated towards the edges of the tetrahedron, thebispectrum fluctuates around zero in the strongly squeezedlimit and hence its correlation with the local-type templateis somewhat suppressed. Clearly from the figure, thebispectrum does not correlate particularly well with thepopular templates—local, equilateral or orthogonal. Itpossesses its own distinct shape. In Fig. 5, we showdifferent tetrahedral cross sections through the full recom-bination bispectrum taken at different summations l1 þl2 þ l3 ¼ const. On the other hand, it contains features inthe squeezed limit and along the edges which reflect those

500 1000 1500 2000

7

6

5

4

3

2

1

10 2 10 6

ISW

SW

Doppler

0

All L.T.

500 1000 1500 2000

1

1

2

3

10 2 10 6

Collision

dpd

I,I

SW ISW

ISW

SW SW

SW

All Q.T.

FIG. 3. The bispectra of the linear terms (L.T.) and quadraticterms (Q.T.) of the source function S2ND in squeezed limit areshown in the upper and lower panel, respectively.

7The late-time ISW-related couplings are subdominant. SeeAppendix for details.

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appearing in the ISW-lensing bispectrum—the fluctuationsalong the edges of the tetrahedron. For the squeezed limitwith l1 ≪ l2, l3, this is because they both contain a termwith dðl22Cl2Þ=dl2 (see Eqs. (12) and (14) in [24]) whichcomes from modulations of large-scale fields. However,the ISW-lensing bispectrum possesses an extra term whichfluctuates according to the angle between the small l1 andthe large l2. More precisely, they have similar shapes whenl2 ¼ l3 but look very different when l1 þ l2 ¼ l3. This extrafeature weakens the correlation between the ISW-lensingand the second-order bispectra, which is only ∼30%.In Table I, we summarize the effective fNL’s, the

normalized FNL’s [25] and the signal-to-noise ratios S=Nwhich are defined as

fANL ¼ FA;rec

FA;Að19Þ

FANL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFA;A

F loc;loc

sfANL ð20Þ

ðS=NÞA ¼ ffiffiffiffiffiffiffiffiffiffiFA;A

p ð21Þ

where

FA;B ¼Xli

h2l1l2l36Cl1Cl2Cl3

bAl1l2l3bBl1l2l3

ð22Þ

hl1l2l3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2l1 þ 1Þð2l2 þ 1Þð2l3 þ 1Þ

4π

r �l1 l2 l30 0 0

�:

ð23Þ

The FNL renormalizes fNL such that equitable comparisonsamong different templates are possible. The local-type Floc

NLis small because of the fluctuating bispectrum in squeezed

FIG. 5 (color online). The cross sections of the normalizedreduced bispectrum in Fig. 4 with the conditions 1

2ðl1þ l2þ l3Þ¼

400 (upper), 1100 (middle) and 1600 (lower).

FIG. 4 (color online). The three-dimensional plot of the reducedbispectrum generated around recombination. The bispectrum isnormalized by the coefficient Dðl1; l2; l3Þ defined in [14] toremove an overall l−4 scaling. The red regions represent positivevalues, while the blue regions represent negative values of therecombination bispectrum.

TABLE I. The table of the effective fNL’s, FNL’s and S=N’s ofthe local and equilateral templates correlated to the recombinationbispectrum, as well as its total signal (autocorrelation). ForISW-lensing bispectrum, we found that its correlation with thesecond-order bispectrum is ∼30%. For ease of comparison, theFNL quantities normalize the integrated bispectrum signal forany shape relative to the fNL ¼ 1 local model. We have usedlmax ¼ 2000 throughout.

Model fNL FNL ðS=NÞEquilateral 5.11 0.66 0.028Local 0.88 0.88 0.22ISW-Lensing 7.36Autocorrelation � � � 3.19 0.69

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limit as we explained previously. The equilateral-type FequNL

is small due to the lack of support in the interior of thebispectrum. Thus, the recombination bispectrum does notcorrelate well with local and equilateral templates.These results are consistent in the squeezed limit with

[11,12]. We have also computed the equilateral typefequilNL ¼ 5.11 ðFequil

NL ¼ 0.66Þ, which is modest but consis-tent with [13]. Table II compares all the codes which arecapable for implementing the full second-order bispec-trum. While there is a relatively good agreement on thevalues of fequNL , the values of flocNL and S=N from differentcodes appear to be quite different. As we mentioned, theLOS integrand used in [8] may mix up early-time and late-time redshift couplings and fail to have convergence atlow-l truncation. This could result the larger S=N and flocNLin their calculation. Although we have taken some approx-imations, we do not think that they provide significantuncertainties. The flat-sky approximation is validated bythe matching with the analytic solution in the squeezedlimit as shown in Fig. 2. Although the thin-shell approxi-mation may add uncertainties on the bispectrum generatedby the couplings with the early-time ISW effect, thesecouplings are subdominant. The cutoff on the LOSintegration allows us to separate the early-time and late-time effects consistently and does not induce any uncer-tainties. In particular, we confirm that the m ≠ 0 modes,which are included in our calculation but not in [9,10],increase the S=N from 0.65 to 0.69. Their effect on thebispectrum is mild.Since the numerical calculation for the second-order

bispectrum is very complicated and could depend on thedetails of the implementations, a comparison betweendifferent codes is essential to resolve the discrepanciesamong different teams in the future. This will be importantfor the quantitative analysis of the Planck data where thiscontribution should be incorporated in debiasing local andequilateral signals and in determining whether there is anoverall primordial non-Gaussian signature in the data. Therecombination bispectrum will combine with the ISW-lensing bispectrum at about the 10% level and its corre-lation can affect the significance of this determination in thePlanck data (recall that ISW-lensing effect can bias the localsignal by as much as fNL ¼ 9.5 [22]). For this reason, weshall continue to incorporate more physical effects in ournumerical pipeline to improve this quantitative analysisfurther.

V. SUMMARY

In this paper, we focus on the bispectrum generatedaround recombination across the full range of multipolecombinations. We find that the effective fNL’s of theequilateral and local types are 5.11 and 0.88, respectively,while the overall signal-to-noise is ðS=NÞrec ¼ 0.69, allcalculated using lmax ¼ 2000. We note from Fig. 4 that thebispectrum possesses its own distinct features differentiat-ing it from well-known templates, such as local, equilateraland ISW-lensing. To complete the full calculation of thisbispectrum will require the inclusion of the time-delay andlensing effects [22,23], the addition of the second-ordervector and tensor perturbations, and finally the late-timeISWeffects. These will be addressed in a future publication[15]. With improving precision, this recombination bispec-trum should be included in the analysis of future CMBexperiments.

ACKNOWLEDGEMENTS

We are grateful for many informative conversations withJ. Lesgourgues, T. Tram, C. Fidler, Z. Huang and C. Pitrou(including comparisons with CMBQUICK [26]). We wouldalso like to thank J. Fergusson for many useful discussionsand help with bispectrum visualization. The numericalsimulations were implemented on the COSMOS supercom-puter, part of the DiRAC HPC Facility, which is funded bySTFC and BIS. We thank A. Kaliazin for computationalsupport and technical advice. E. A. L., S .C. S. and E. P. S. S.were supported by STFC Grant No. ST/F002998/1 and theCentre for Theoretical Cosmology. S. C. S. was supported bythe Croucher Foundation. The numerical code was devel-oped based on CAMB [27].

APPENDIX: CHANGE-OF-VARIABLEAPPROACH

In [9,10], the numerical instability of Eq. (12) wastackled by introducing a change of variables, i.e. ~Δ≡Δ − Δ2=2 with the fractional brightness Δ. Here, we willdiscuss how their approach is related to our approach basedon IBPs. Since [10] has a closer gauge choice comparedwith ours (with onlyΦ andΨ swapped), our discussion willbase on [10]. However, similar comparison works with [9].We comment that the term−ðΦþΨÞni∂iΦ is omitted in [9]while it is included in [10] and this paper. In [10], thesecond-order fractional brightness due to the redshift-related couplings is evaluated numerically as rewrittenusing our notation

Δ½II�R ðη0;k; nÞ

¼ T k

�4Δ1Δ2 − 2

Zη0

0

dηe−ik·nr−τð4τ0Δ1 þ C1ÞΔ2

�:

ðA1Þ

TABLE II. The table for comparisons among the results fromall the codes implementing the full second-order bispectrum.

This work Pitrou [8] Huang [9] Pettinari [10]

S=N 0.69 ∼1 0.47 0.47fequNL 5.11 ∼5 � � � 4.3flocNL 0.88 ∼5 0.82 0.5

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It can be shown to be mathematically equivalent to Eq. (16)but its physical interpretation is quite different. WhileEq. (16) explicitly lists out all the possible redshiftcouplings, Eq. (A1) includes extra couplings (e.g. colli-sion-collision coupling) embedded in the first term Δ2 andthen subtracts all these extra couplings in the second term.To understand this, we expand the fractional brightness Δ’sin Eq. (A1) using Eqs. (7) and (9). Equation (A1) improvesthe numerical efficiency of calculating the bispectrum.The first term contributes a shape proportional to Cl1Cl2on the bispectrum while the second term has its contribu-tion concentrated at the LSS. That means, we do not need toevaluate the first-order temperature brightness Δ½I� at latetime when the high multipoles become important andnumerically challenging.Here, we point out that the redshift couplings due to the

late-time ISW will also be included by the first term of

Eq. (A1). In other words, all the redshift-related couplingsare included. Thus, the difference between our approachand Eq. (A1) comes from the redshift couplings due to thelate-time ISW because we ignore them by a cutoff of thetime integration. Practically, the bispectrum generatedby those couplings is mild and should not contributesignificantly [24].The advantage of using Eq. (16) is that physical effects

are consistently separated along the LOS integration. Thisis useful for isolating the early-time effects. In particular,nonlinear effects become important at redshift z≲ 10. Itremains unknown how much higher-order (beyond second-order) effects contribute to the bispectrum. Indeed, [22] hasfound that the correction of lensing effect beyond secondorder contributes as large as 10% of the ISW-lensingbispectrum. An approach compatible with higher orderswill be more appropriate to study the late-time bispectrum.

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