Gravitational Lensing of the CMB: a Feynman Diagram Approach

We develop a Feynman diagram approach to calculating correlations of the Cosmic Microwave Background (CMB) in the presence of distortions. As one application, we focus on CMB distortions due to gravitational lensing by Large Scale Structure (LSS). We study the Hu-Okamoto quadratic estimator for extracting lensing from the CMB and derive the noise of the estimator up to ${\mathcal O}(\phi^4)$ in the lensing potential $\phi$. The previously noted large ${\mathcal O}(\phi^4)$ term can be significantly reduced by a reorganization of the $\phi$ expansion. Our approach makes it simple to obtain expressions for quadratic estimators based on any CMB channel. We briefly discuss other applications to cosmology of this diagrammatic approach, such as distortions of the CMB due to patchy reionization, or due to Faraday rotation from primordial axion fields.

Introduction -Primary anisotropies in the Cosmic Microwave Background were generated around 375, 000 years after the "big bang," when the universe was still in the linear regime.The CMB field can be decomposed and studied in terms of its temperature T and polarization modes E and B. Primordial scalar perturbations create only E modes of the CMB, while primordial tensor perturbations generate both parity-even E modes and parity-odd B polarization modes [1][2][3], which are important for constraining models of inflation.The magnitude of the tensor generated B-mode signal is directly related to the Hubble parameter H during inflation, and thus a detection would establish the energy scale at which inflation happened [4].
The primordial CMB generated at the surface of last scattering is statistically isotropic and Gaussian.However, during the photon's journey to us, it encounters several distorting fields, which make the CMB non-Gaussian and statistically anisotropic.Examples of such distorting fields are (a) gravitational lensing which bends the light as photons travel though the large scale structure of the universe [5][6][7][8], (b) patchy reionization which modulates the CMB intensity because of scattering when Hydrogen reionizes [9], and (c) cosmological rotation, due to parityviolating physics (e.g.due to axions), which rotates the plane of polarization of the CMB [10][11][12][13].By coupling different modes of the CMB, the distortion imprints its signature on the observed CMB by breaking statistical isotropy and introducing non-Gaussianities.All these distortions also produce B-modes that contaminate the primordial tensor B-mode signal.
One can utilize the statistical anisotropy of the observed CMB to reconstruct the distorting fields.Estimators based on the Hu-Okamoto quadratic estimator [14,15] are the most studied method for extracting these distortions.In this paper, we present a new diagrammatic way of studying distortions using such estimators and employ this method to investigate the noise properties of the estimator.We show that the previously unexplained large N (2) noise can be understood using this diagrammatic approach, and reduced by reorganizing the expansion.
Distortions in the CMB -The primordial CMB is statistically isotropic and Gaussian, and all of the information is contained in the power spectrum, , where the average is over CMB realizations and we work in the flat sky approximation.Here ℓ and k denote the Fourier modes, and the power spectrum only depends on ℓ = |ℓ|.The x, y ∈ {T, E, B} are temperature and polarization components of the CMB, which can conveniently be combined into a column vector X such that C xy are components of a 3 × 3 CMB power spectrum matrix C, Secondary distortions, such as gravitational lensing, patchy reionization and Faraday rotation, will modify the components of X [9, 11-13, 16, 17].The effect of these distortion fields on Fourier modes may generically be written as where the matrix D can mix components of the CMB.X ℓ is the primordial spectrum, and X ℓ is the lensed spectrum.We now focus on how to calculate the effect of gravitational lensing using Feynman diagrams, but we will comment on other distortions in the final discussion.
Gravitational lensing -Lensing deflects the path of CMB photons from the last scattering surface.This deflection results in a remapping of the CMB temperature/polarization pattern on the sky, n → n + d( n), and mixes the E and B polarization modes.The deflection is given by, d( n) = ∇φ( n), where the lensing potential φ( n) is obtained by integrating the gravitational potential along the line of the sight [5].There are higherorder corrections to the lensing potential [18][19][20][21][22], which FIG. 1. Feynman rules for calculating lensed CMB fields.
for simplicity we will ignore.Treating φ as a Gaussian field with power spectrum C φφ ℓ , where P gives the total momentum of all the φ fields and R (ℓ,m) encodes the mixing of E and B polarizations, Here, ϕ(ℓ, m) is the (oriented) angle between ℓ and m.The lowest order terms in Eq. ( 3) produce the familiar result [14] Feynman diagrams for lensing -Eq.( 3) yields a simple Feynman rule when calculating the average of several CMB modes x(ℓ) y(k) . . .over CMB or LSS realizations (see Fig. 1).In the calculation of such an average, each lensed field x(ℓ) is represented as a vertex with momentum ℓ flowing in.It has one straight line (the unlensed field) and arbitrary many wiggly lines (the lensing field φ) connected to it.When averaging over CMB realizations, each straight line must begin and end at a vertex.It corresponds to C xy m for the CMB components x and y, where m is the momentum flowing through the line.Similarly, each wiggly line corresponds to C φφ k when averaging over LSS realizations, where k is the wiggly line momentum.Momentum is conserved at the vertex and each unconstrained internal momentum k is integrated over with d 2 k/(2π) 2 .In addition, there is a factor corresponding to total momentum conservation (2π) 2 δ 2 (ℓ + k + . . . ) which is typically pulled out front (see e.g.Eq. ( 6)).These rules are summarized in Fig. 1 and will be illustrated with explicit examples below.
As a simple example, we calculate the lensed CMB spectra C ℓ .The diagrams contributing to C ℓ are shown in Fig. 2. Using the rules from Fig. 1 gives where both C and C are 3 × 3 matrices in {T, E, B}.Graph (a) is the unlensed spectrum C ℓ and graph (b) yields the third line.Graphs (c) and (d) give identical contributions, are multiplied by a symmetry factor of 1/2, and simplify due to R −ℓ,ℓ = 1, giving the last term on the second line.
Quadratic estimator and noise terms -Lensing breaks the statistical isotropy, correlating the CMB modes, which can be used to reconstruct the lensing field from the CMB.The superscript 0, 1 on f denotes the order in C φφ of the terms on the r.h.s.A quadratic estimator for the lensing potential can be written as and its Feynman rule is given in Fig. 1.Following [14], the normalization A L is chosen so that Eq. ( 8) yields an unbiased estimator, φxy L CMB = φ L , and the filter F xy is determined by minimizing the variance φxy Using the quadratic estimator to extract the lensing power spectrum introduces a bias φxy given by the noise terms L is expected to provide the dominant contribution to the variance.However, it has been recently noticed that the higher order noise term N (2) L can give a large contribution at small L [23,24].One of the main goals of our paper is to illustrate the power of Feynman diagrams in calculating the higher-order contributions N (1) L , which also makes it easy to track down the origin of this large contribution.
The two diagrams contributing to the lowest order noise term are shown in Fig. 3 and lead to in agreement with Ref. [14].Here C xy e ℓ(ℓ+1)σ 2 /8 ln 2 is the observed spectra, σ is the fullwidth-half-maxima of the experimental beam, and ∆ xy is experimental noise [25].We will assume fully polarized detectors for which ∆ EE = ∆ BB = √ 2∆ T T , and ∆ xx ′ = 0, x = x ′ .
Although using lensed rather than unlensed spectra in C is formally beyond the order in φ of N (0) L , it reduces the number of diagrams contributing to the higherorder noise.Specifically, corrections of the type shown in Fig. 2(b) -(d) are now already included.This approach is standard for the Gaussian noise N (0) L , but we find that also using lensed spectra in the higher-order noise terms improves their convergence.We will compare using lensed vs. unlensed spectra when we present numerical results in Fig. 8.We also use lensed spectra everywhere in the filter F of the estimator, which has been considered in Refs.[24,26].
The diagrams contributing at O(φ 2 ) are shown in Fig. 5, which we break into subgraphs involving the filter f (φ,0) defined in Eq. ( 7).This filter f (φ,0) describes the distortion of the two-point function due to lensing, as shown in Fig. 4. In these figures the "crossing out" of lines indicates that they do not produce a power spectrum in the corresponding expression.Fig. 5(a) produces the lensing spectrum C φφ L by construction.Fig. 5(b) and the corresponding cross graph can be calculated using the Feynman rules in Fig. 1, This noise contribution was first calculated by Kesden et al. [27,28].Note that corrections of the form shown in L by using lensed spectra there, and thus should not be included in N L .We will also consider using lensed spectra C ℓ and C ℓ ′ instead of C ℓ and C ℓ ′ in f (φ,0) .
There are two classes of diagrams contributing to N L .The first class of diagrams is the same form as those in Fig. 5 and Eq. ( 11), but with one of the f (φ,0) vertices replaced by the higher order f (φ,1) .The second class of diagrams is shown in Fig. 6, and involves the new filter f (φφ,0) in Fig. 7. Expressions for f (φ,1) , f (φφ,0) and N (2)  will be given in Ref. [29].We have identified the O(φ 4 ) analogue of Fig. 5(a) as the contribution that is responsible for the large size of N (2) that had been observed in Refs.[23,24].To understand this, it is useful to first discuss the contributions at order φ 2 : Fig. 5  yield C φφ and N (1) and although they are formally of the same order in the lensing expansion, C φφ is numerically larger.This is to be expected because Fig. 5(a) has less loop integrals than (b).The same is true at O(φ 4 ) and there is thus no breakdown of perturbation theory.Instead, the graph in Fig. 5(a) and corresponding higher order contributions give a convergent expansion but one that is numerically larger than the diagrams in Fig. 5(b), Fig. 6, etc.Using lensed rather than unlensed spectra in the filter f (φ,0) reduces the numerical size of this contribution to N (2) , essentially summing a class of higher order corrections.This is shown in Fig. 8, which compares lensed to unlensed spectra in the computation of N (2) and will be discussed below.
Numerical Results -In Fig. 8, we show the noise N T T L in estimating the lensing C dd L = L 2 C φφ L as a function of L for a Planck-like experiment with experimental noise ∆ T = 56 µK-arcmin and beam size σ = 7 arcmin.We consider noise calculated using two counting methods, the unlensed spectra (dotted curves) as well as lensed spectra (solid curves).Our results for the former agree with Ref. [23], showing that at small L the bias N T T (2) L is large.As we explained, this originates from higherorder corrections to Fig. 5(a), so it is not surprising that its shape is similar to C dd L .Fig. 8 clearly illustrates that using lensed spectra greatly improves the convergence of the noise terms.The main difference with Ref. [24] is that we have not only changed our estimator to use lensed spectra, but we have also reorganized our noise in terms of lensed spectra.This use of lensed spectra is responsible for the improved convergence we see, in contrast to the seemingly accidental cancellation between N T T (1) L and N T T (2) L found in Ref. [24].Discussion -We have shown how Feynman diagrams can be used to understand the CMB, illustrating their power in the context of gravitational lensing.This method allowed us to simultaneously obtain expressions for quadratic estimators based on any CMB channel and identify the origin of the (supposed) poor convergence of higher order noise terms.Additional details, as well as plots for the polarization channels, such as T T EE and EB EB, will be given in a subsequent publication [29].
Apart from lensing, there are other cosmological effects that can couple the modes of the CMB [16] such as screening from patchy reionization [9], and rotation of the plane of polarization either due to primordial magnetic fields [17,30,31] or parity-violating physics [11][12][13].The formalism presented here can be used to study these effects as well (see Eq. ( 2)).Below we discuss cosmological rotation and patchy reionization and show how Feynman rules can be derived for them.
Cosmological rotation and patchy reionization -Many theories predict parity-violating primordial fields such as axions, which have Chern-Simons couplings of the form aF µν F µν [10,32], that rotate the plane of polarization of light through an angle dα = 2d τ ȧ during propagation for a conformal time dτ .The fluctuations in the axion field a then will be imprinted in the rotation angle α of the polarization.The observed (rotated) and primordial CMB in terms of Stokes parameters are related by ( Reionization marks the time after decoupling when the vast majority of Hydrogen became ionized due to gravitational nonlinearities.When and how this process occurred is at present not well constrained.Inhomogeneous reionization produces several secondary anisotropies in the CMB.The patchy nature of reionization results in a Thomson scattering optical depth to recombination, τ (n), depending on direction n.Such optical depth fluctuations act as a modulation effect on CMB fields by suppressing the primordial anisotropies with a factor of e −τ (n) , correlating different modes by From Eqs. ( 12) and ( 14), one can obtain the corresponding Feynman rules that allow one to calculate their effect of the correlation structure of the CMB and construct the appropriate estimators and noise terms.Assuming these effects are Gaussian and statistically isotropic, the only other ingredient is C αα and C τ τ .In this case, the calculation is an expansion in α or τ instead of φ.

Fig. 2 (
b) through (d) were already part of the calculation of N (a) and (b)

FIG. 8 .
FIG. 8. Lensing signal C dd L = L 2 C φφ L and noise power spectra N T T (n) L of the quadratic estimator for C dd L .Lensed spectra are used for the solid curves, as described in the text, improving the convergence.We have assumed experimental noise ∆T T = 56 µK-arcmin and beam σ = 7 ′ .The sign of the noise N (2) L changes as a function of L, negative values are shown in red and positive values in magenta.