On the effect of non-Gaussian lensing deflections on CMB lensing measurements
Vanessa Böhm, Blake D. Sherwin, Jia Liu, J. Colin Hill, Marcel Schmittfull, Toshiya Namikawa
OOn the effect of non-Gaussian lensing deflections on CMB lensing measurements
Vanessa B¨ohm,
1, 2
Blake D. Sherwin, Jia Liu, J. Colin Hill,
5, 6
Marcel Schmittfull, and Toshiya Namikawa Berkeley Center for Cosmological Physics, University of California, Berkeley, CA 94720, USA Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 93720, USA Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Center for Computational Astrophysics, Flatiron Institute,162 5th Avenue, 10010, New York, NY, USA Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei, 10617, Taiwan (Dated: June 5, 2018)We investigate the impact of non-Gaussian lensing deflections on measurements of the CMB lens-ing power spectrum. We find that the false assumption of their Gaussianity significantly biasesthese measurements in current and future experiments at the percent level. The bias is detectedby comparing CMB lensing reconstructions from simulated CMB data lensed with Gaussian de-flection fields to reconstructions from simulations lensed with fully non-Gaussian deflection fields.The non-Gaussian deflections are produced by ray-tracing through snapshots of an N-body sim-ulation and capture both the non-Gaussianity induced by non-linear structure formation and bymultiple correlated deflections. We find that the amplitude of the measured bias is in agreementwith analytical predictions by B¨ohm et al. 2016. The bias is largest in temperature-based measure-ments and we do not find evidence for it in measurements from a combination of polarization fields(
EB, EB ). We argue that the non-Gaussian bias should be even more important for measurementsof cross-correlations of CMB lensing with low-redshift tracers of large-scale structure.
I. INTRODUCTION
Photons of the cosmic microwave background (CMB)get deflected by the the cosmic matter distribution be-tween the surface of last scattering and the observer.This effect is known as CMB lensing (see e.g., Refs. [1]and [2] for reviews). Coherent deflections distort theobserved CMB fluctuations in both temperature and po-larization in a characteristic way. The statistics of thedeflections contain a vast amount of cosmological infor-mation. They are sensitive to cosmological parametersthat determine the formation of cosmic structure, suchas a combination of σ and Ω m , the sum of neutrinomasses [3] and the presence of dark energy [4]. Theyare also a probe of the flatness of space, since curvaturechanges the relative efficiency of lensing events at dif-ferent distances. Different to other probes of large-scalestructure, CMB lensing is mostly sensitive to structuresat relatively high redshifts ( z ≈ L ≈ σ (cid:80) m ≈
30 meV if combined with suitable other probesto break degeneracies with τ and Ω m h . Common CMB lensing reconstruction uses a quadratic,weighted combination of CMB fields to recover the deflec-tion field [21, 22]. Power spectrum measurements fromthis quadratic estimator extract lensing information fromthe lensed CMB 4-point function. The 4-point estima-tor for the CMB lensing power spectrum is a biased es-timator. It is non-zero even in the absence of lensingand carries bias terms at all orders in the lensing powerspectrum [23, 24]. Other sources of systematics in CMBlensing measurements are masking, anisotropic beam ornoise properties [25, 26] and foregrounds [27–30]. Biasesto power spectrum measurements can either be estimatedand subtracted, or alleviated by suitable modifications tothe lensing estimator [31–34].Recently, Ref. [35] (hereafter BSS16) have identified anew bias to CMB lensing measurements, which arises as aconsequence of the non-Gaussian structure of the lensingdeflection field. BSS16 specifically considered the effectof a non-vanishing bispectrum of the lensing potential.In a purely analytic study, they found that the bispec-trum that arises as a consequence of non-linear structureformation can change the amplitude of the CMB lensingpower spectrum measured from CMB temperature datain current and future experiments at the percent level.Since most of these experiments rely primarily on tem-perature, a corrections to CMB lensing measurements ofthis magnitude would constitute a significant systematicand, if uncorrected, hinder the accurate estimation ofcosmological parameters. However, the theory calcula-tion in BSS16 made a number of non-trivial assumptions(see Appendix A for details) and the actual size of thebias depends on their validity. a r X i v : . [ a s t r o - ph . C O ] J un Motivated by this, we study the effect of non-Gaussianity on CMB lensing measurement in this workin a completely independent way with ray-traced lensingsimulations. Specifically, we look at the difference be-tween lensing power spectra measured with the standard4-point estimator in two different sets of simulated noisy,lensed CMB maps: one set lensed with purely Gaussiandeflection fields, the other with fully non-Gaussian de-flections obtained from ray-tracing through snapshots ofan N-body simulation. By using the same unlensed CMBand detector noise realizations for both sets, any signifi-cant difference in the measured spectra is a consequenceof the non-Gaussianity of the deflection field and canbe interpreted as a non-Gaussian bias. While the studywith simulations provides less intuition about the specificsource of a non-Gaussian bias, it is in some sense morecomplete than the theoretical analysis in BSS16, since itcaptures the full non-Gaussianity of the field, which canmanifest itself in more ways than a non-zero bispectrumand relies on fewer simplifying assumptions. We com-pare the measured non-Gaussian bias to the theoreticalprediction of BSS16. For this, we update the theoreticalprediction to also take into account the lensing bispec-trum sourced by multiple correlated deflections (so-calledpost Born corrections). The importance of the post-Bornbispectrum was recently pointed out by Ref. [36] and weuse their analytically derived expression to model it.Post-Born corrections in CMB lensing were recentlystudied in simulations in Refs. [37] and [38]. The effectof non-linear structure formation on lensing reconstruc-tions, in particular its impact on the second order lensingbias N (2) , was measured in ray-traced simulations for theinterpretation of data from the South Pole Telescope [8],but found to be irrelevant for this specific data set. Paral-lel to the work presented here, Beck et al. 2018 (in prep),have carried out a measurement of a non-Gaussian biason an independent set of ray-traced lensing simulations.This paper is organized as follows: we start with brieflyreviewing CMB lensing and CMB lensing reconstructionin Section II. In Sec. III we give a full overview of theproduction of mock CMB data maps: In subsections wedescribe the production of ray-traced lensing maps andtheir Gaussian counterparts (Sec. III A), the generationof noisy, lensed CMB simulations (Sec. III B) and thereconstruction from these mock data sets (Sec. III C).Results and their comparison to theory are presented inSec. IV. We conclude with a discussion of the resultsand a comment on the importance of the non-Gaussianbias for cross correlations with low-redshift tracers inSec. V. For details on the theoretical bias model de-rived in BSS16, we refer the reader to Appendix A andRef. [35]. II. CMB LENSING AND CMB LENSINGRECONSTRUCTION
Lensing distortions are a measure of the integratedmass distribution along the photons’ trajectories. In aflat standard cosmology and under the Born approxima-tion, the lensing convergence κ ( L ) is related to the den-sity contrast δ ( L , χ ) through the line-of-sight integration κ ( L ) = 32 Ω m H c (cid:90) χ CMB d χ W ( χ, χ CMB ) δ ( L , χ ) (1)with lensing kernel W ( χ, χ CMB ) = [1 + z ( χ )] χ ( χ CMB − χ ) χ CMB . (2)Throughout this paper, we use the flat sky approxima-tion, where L denotes the wave vector of a 2D Fouriermode on the sky.The mapping between unlensed CMB fields ( T, Q, U )and their lensed counterparts ( ˜
T , ˜ Q, ˜ U ) is determined bythe lensing deflection angle α ,˜ T ( x ) = T [ x + α ( x )] , (3)which is to good approximation curl-free and can be ex-pressed in terms of a scalar lensing potential φ ( x ) α ( x ) = ∇ φ ( x ) . (4)Similar to overdensity and gravitational potential inthree dimensions, the lensing convergence (Eq. 1) andthe lensing potential are related by the Poisson equation κ ( x ) = − ∇ φ ( x ) . (5)CMB lensing reconstruction is the recovery of the lens-ing deflection field from lensed, noisy CMB data. It iscommonly performed with an estimator that is quadraticin the lensed CMB [21, 22, 39],ˆ κ ( L ) = 12 L A XYL (cid:90) l g XY l , L ˜ X expt ( l ) ˜ Y expt ( L − l ) . (6)In Eq. 6 X and Y represent either temperature ( T ) orpolarization fields ( E/B ) and the subscript “expt” la-bels noisy, beam-deconvolved data. The weight g andthe normalization A L depend on the fiducial lensed CMBpower spectra as well as the beam and noise properties ofthe experiment (see Ref. [22] for the exact expressions ).Weight and normalization are chosen such that the esti-mator in Eq. 6 has minimum variance and is unbiased in Ref. [22] uses unlensed power spectra in the lensing weights. Re-placing them by their lensed counterparts partly removes higherorder biases from the power spectrum estimate [24, 31, 40]. the absence of any source of mode-coupling other thanlensing.A few alternatives to the quadratic estimator havebeen proposed. Some are are based on maximizing theCMB lensing posterior or sampling the joint distributionof lensing deflections and CMB [41–43]. Other estimatorsare derived from a configuration-space perspective anduse the magnification and shear of the lensed CMB fluc-tuations to estimate the lensing field [44–46]. To datethe quadratic estimator remains the most widely usedand best understood estimator for the CMB lensing de-flection field.Measurements of the CMB lensing power spectrumfrom the quadratic estimator are sensitive to the lensedCMB four-point function,ˆ C κκW X,Y Z ( L ) = 14 L A W XL A Y ZL (cid:90) l , l g W X l , L g Y Z l , L ×(cid:104) ˜ W expt ( l ) ˜ X expt ( L − l ) ˜ Y expt ( − l ) ˜ Z expt ( l − L ) (cid:105) . (7)Since the response of the CMB to lensing is non-linearin the deflection, this four-point estimator gets contribu-tions from terms at all orders in the lensing convergence.Only one of the contributing second order terms givesrise to the convergence power spectrum. The remainingterms are bias terms that need to be subtracted in orderto obtain an unbiased estimate for C κκL . They are com-monly summarized and labeled by their power in the lens-ing power spectrum: N (0) L for the bias that is sourced byGaussian CMB fluctuations (this term is present even inthe absence of lensing), N (1) L for all biases proportional to C κκL and N (2) L for biases proportional to ( C κκL ) [23, 24].The N (2) L bias can be greatly reduced by a slight modifi-cation to the lensing weights, see e.g. Refs. [24, 31, 40].Adapting this notation, the expectation value of Eq. 7,averaged over realizations of CMB and lensing deflections(and assuming that both are Gaussian fields), becomes (cid:104) ˆ C κκL (cid:105) = N (0) L + C κκL + N (1) L + O (cid:2) ( C κκL ) (cid:3) . (8)Non-linear processes, such as non-linear structure for-mation and multiple correlated deflections, introduce asmall, but detectable amount of non-Gaussianity to thelensing convergence [36, 47–49]. In the limit of small den-sity perturbations, the non-Gaussianity can be character-ized by a lensing bispectrum. A non-zero lensing bispec-trum changes the lensed temperature four-point functionand introduces an additional bias term to Eq. 8, (cid:104) ˆ C κκL (cid:105) = N (0) L + C κκL + N (1) L + N (3 / L + O (cid:2) ( C κκL ) (cid:3) . (9)This new bias was first identified in Ref. [35]. We willnow compare this theoretically derived term with mea-surements of a non-Gaussian bias in simulations. The quadratic estimator can be interpreted as a first order ap-proximation to a maximum likelihood estimator for the lensingpotential.
III. SIMULATIONS
The general work flow for isolating a non-Gaussianbias is simple: we use a set of non-Gaussian convergencemaps generated by ray-tracing through an N-body sim-ulation and a corresponding set of Gaussian convergencemaps with the same average power spectrum. The sameCMB realizations are lensed with both Gaussian and non-Gaussian convergence maps, which results in two sets oflensed CMB maps. We convolve these maps with a Gaus-sian beam before we add the same realizations of whitemeasurement noise to both sets. We beam-deconvolvethe noisy maps before we apply the standard quadraticand four-point estimators. We then compare the resultsof the reconstructions between both sets and look for sig-nificant differences. In the following sections, we providedetailed descriptions and validations for each of thesesteps. The entire procedure is also illustrated in Fig. 1.For all simulations (N-body and CMB), we use astandard ΛCDM cosmology with parameters, H =72 km / s / Mpc, Ω m = 0 . σ = 0 . w = − n s = 0 .
96 and Ω b = 0 . A. Convergence maps
We use a set of 10240 non-Gaussian convergence mapsthat was obtained from ray-tracing through snapshots ofan N-body simulation. For a detailed description of theirproduction we refer the reader to Ref. [49]. The under-lying N-body simulation is based on the public Gadget-2code [50], has a box size of 600 Mpc /h and is resolved byN = 1024 particles (corresponding to a mass resolutionof 1 . × M (cid:12) /h ). The linear matter power spectrumfor its initialization was computed with CAMB [51] andinitial conditions at z = 100 generated with N-GenIC.Snapshots were recorded between z ≈
45 and z = 0, arange which covers 99% of the growth corrected lensingkernel W ( χ, χ ∗ ) D ( z ). Convergence maps were computedwith LensTools [52] tracing 4096 light rays and calculat-ing their deflections on 3 planes per box. This proceduredoes not assume that the deflection angle is small or thatthe light rays follow unperturbed geodesics. Differentrealizations of the convergence maps were produced byrandomly rotating and shifting the potential planes [53].The resulting maps are 12 .
25 deg in size and resolved by2048 pixels measuring 0 . arcmin . We refer to thissimulation set as non-Gaussian or N-body lensing simu-lations. Their non-Gaussianity is not only a consequenceof non-linear structure formation in the N-body simula-tion, but also of the multiple deflections along the lensplanes. We do not measure or take into account the curlof the deflection field that is introduced by multiple de-flection, because we do not expect a significant bias from http://camb.info/ noisenoisenoise reconsnon-Gaussian convergence mapsGaussian convergence maps recons avg.recons autocrossautocrossT,Q,UmapsT,Q,UmapsT,Q,Umaps noisyT,E,Bmaps avg.recons dif lens and add noise noisyT,E,Bmaps reconstruct averageover sets compute average spectra FIG. 1. Schematic outline of the simulation pipeline: Squares and ellipses represent sets of 10240 simulations. We start bygenerating 3 sets of unlensed CMB and noise realizations. We then lens each CMB realization with both Gaussian and non-Gaussian convergence maps. This results in two times three sets of lensed CMB simulations, which we convolve with a Gaussianbeam. We then add the same noise realization to each corresponding set in the Gaussian and non-Gaussian branch. Afterbeam-deconvolution of the noisy maps, we run the standard quadratic estimator on each of them. We average the reconstructionresults over the three sets in each branch to beat down the reconstruction noise originating from the CMB sample variance.This leaves us with one averaged non-Gaussian and another averaged Gaussian set of reconstructed convergence maps. Wecompute the average power spectrum in each of these sets as well as the average cross correlation with the true underlyingrealizations. Any significant difference between the average power spectra in the two branches is a non-Gaussian bias. bispectra involving the curl component (see Appendix Bfor details).We further produce a second set of 10240 purely Gaus-sian convergence maps. These Gaussian simulations aregenerated by first measuring the average power spectrumof the non-Gaussian simulations and then drawing con-vergence realizations from a multivariate Gaussian withexactly this power spectrum.In Fig. 2 we compare the average power spectra ofthe Gaussian and non-Gaussian simulation set to a the-ory power spectrum computed with the anisotropy solverCLASS [54]. The missing power on the small-scale end, L > http://class-code.net/ of the non-Gaussian simulations, as expected, within thesample variance (Fig. 3, red curve). We further comparethe combined standard deviation of the average power inboth simulation sets to the size of the bias predicted byBSS16 (Fig. 3 shaded region and blue dots). The com-parison shows that the sample variance in the simulationsets is low enough to allow for a detection of a bias atthe percent level (which corresponds to the the magni-tude predicted by BSS16).To get a sense of the non-Gaussianity of the ray tracedconvergence maps, we measure their skewness, (cid:10) κ ( x ) (cid:11) after smoothing them with a Gaussian kernel on differentscales. The skewness is an integrated measure of thebispectrum . By comparing the measurement with the Note that the quadratic estimator results in an additional skew-ness in the measured maps, i.e. the measured maps have non-zeroskewness even if the underlying field is Gaussian [49]. We mea-sure the skewness in the true noiseless convergence maps and notin the reconstructions, since we are interested in quantifying thebispectrum introduced by non-linear physics. L C L C N body C Gauss C CLASS
FIG. 2. Power spectra measured from 10240 Gaussian(red) and ray-traced non-Gaussian (yellow) convergencemaps closely follow the theory curve computed with CLASS(blue). For modeling non-linear effects in the matterpower spectrum CLASS uses a version of HALOFIT [55].We use precision parameters tol perturb integration=1e-6, perturb sampling stepsize=0.01, k min tau0=0.002,k max tau0 over l max=10., halofit k per decade=3000 andl max scalars=8000 to produce the theory curve. Missingpower on small scales is owed to the finite resolution of thesimulation. ( L ) / C S i m s ( L ) ( L ) = C N body C Gauss ( L ) = bias prediction FIG. 3. For an accurate measurement of the non-linear biasin the reconstructions it is crucial that the power spectra ofthe original, non-reconstructed, Gaussian and non-Gaussiansimulations are consistent within their sample variance. Weshow that this is indeed the case by checking that their dif-ference is consistent with zero (red dots, χ /ν = 1 . ± σ, ± σ ) and compare it to the expectedsize of the non-linear bias (blue dots). theoretical prediction, (cid:10) κ ( x ) (cid:11) = (cid:90) l (cid:90) L W R ( L ) W R ( l ) W R ( | − L − l | ) × B κκκ ( L, l, | − L − l | ) (10) W R ( l ) = exp (cid:0) − l R / (cid:1) (11)we can determine the most suitable theoretical bispec- trum model for computing the bias following BSS16.We expect the bispectrum to have two contributions:one from non-linear structure formation, where the con-vergence bispectrum is an integrated measure of the bis-pectrum of large-scales structure, and a second contribu-tion from post-Born effects [36]. In the squeezed limitthese two contributions have opposite sign and partlycancel each other. We compare two different models forthe convergence bispectrum induced by non-linear struc-ture formation; one in which we model the matter bis-pectrum in tree-level perturbation theory and one inwhich we use a simulation-calibrated fit to the matterbispectrum [56] The results of the skewness measure-ment together with the different theoretical models areshown in Fig. 4. We find that the theory curve computedfrom a combination of structure formation induced andpost-born bispectra agrees well with the measurementon smoothing scales FWHM > B. CMB simulations
We produce three sets of 10240 unlensed CMB realiza-tion in temperature ( T ) and polarization ( Q, U ) basedon power spectra computed with CAMB. We use eachGaussian and non-Gaussian convergence map to lens thesame three CMB maps (one map from each set). Usingthe same lenses for three background CMBs and averag-ing over their lensing measurements reduces the Gaussianreconstruction noise and thus the noise of the bias mea-surements. The lensing algorithm is described in detailin Ref. [57] . We apply a filter that removes modes with L > In this model we replace the linear matter power spectrum byits HALOFIT counterpart This bispectrum model has 9 free parameters which are assumedto be independent from cosmology and have been measured andfixed in Ref. [56]. It also depends on cosmological parametersthrough a direct appearance of σ and indirectly by its depen-dence on the non-linear scale and the non-linear matter powerspectrum. We adapt these quantities to agree with the cosmol-ogy of the simulation. We include terms up to fifth order in this algorithm. ( x ) [ ] Gaussian simsNon-Gaussian simsgrowth (fit) + post Borngrowth (fit)growth (tree-level)
FIG. 4. The skewness measured on different scales providessome information on the bispectrum of the non-Gaussianconvergence maps. We find that we can accurately modelthe skewness by assuming that the bispectrum consists of anon-linear growth induced and post-Born induced contribu-tion. The growth-induced part is best described by using asimulation-calibrated fit to the matter bispectrum. The con-vergence maps were smoothed with a Gaussian kernel withFWHMs indicated on the x-axis and filtered to exclude modeswith
L > k min = 0 . h/ Mpc] corresponding to the box size ofthe simulation and k max = 50 [ h/ Mpc]. Outside of thesebounds we set the matter bispectrum (and matter power spec-trum in the computation of the post Born terms) to zero.From the comparison with a theory curve computed with k max = 100 [ h/ Mpc], we find that the results are not sensitiveto the k max cut off. The error bars correspond to the standarddeviation of the mean and are smaller than the marker size. l C TT l theorylensed theoryunl. simslen. sims NGlen. sims G
100 500 1000 1500 2000 2500 3000 3500 4000 l s i m s t h e o FIG. 5. The average power spectra of the lensed tempera-ture maps in Gaussian and non-Gaussian simulation branchesagree well with each other (lower panel). The agreement oflensed and unlensed realizations with the theory predictionis good except for large scales, where we find a significantdeviation at l < l C EE l theorylensed theoryunl. simslen. sims NGlen. sims G
100 500 1000 1500 2000 2500 3000 3500 4000 l s i m s t h e o FIG. 6. The average power spectra of the unlensed andlensed polarization E-modes agree with the theory predictionexcept for large scales, where we find a significant deviationat l <
After lensing, we convolve the lensed CMB maps witha Gaussian beam of width FWHM = 1 arcmin and addthe same white noise realization with a noise level of σ T = 1 µ K − arcmin in temperature and σ pol = √ σ T in polarization to each corresponding Gaussian and non-Gaussian set. The noise configurations are chosen toroughly match prospective CMB surveys.The entire procedure leaves us with 3 times 2 sets of10240 mock CMB measurements, where correspondingmaps in each of the three pairs have same CMB and noiserealizations and differ only in the underlying convergencefield. C. Lensing Reconstruction
We apply a quadratic estimator to all six sets of noisy,beam-deconvolved, lensed CMB maps in (
T T ) and ( EB )to obtain noisy estimates of the underlying convergencefields. The reconstruction pipeline is described in detailin [15].We filter scales with l <
500 from the CMB mapsprior to reconstruction since Fig. 5 indicates some incon-sistency between the power spectra of the lensed simula-tions and the theory power spectra on these scales. Wealso filter out any multipoles with l > l < .
250 500 750 1000 1250 1500 1750 2000L10 measurement from (TT, ) N (3/2)theory ( L ) C theory C Gauss C N body C N body C Gauss
FIG. 7. Average power in the cross correlation betweenconvergence maps reconstructed from the lensed temperaturemaps and the input convergence field. The measured powerfollows the theory curve (red line) for both Gaussian and non-Gaussian simulations (yellow and orange points). The differ-ence between the Gaussian and non-Gaussian reconstructionsis shown as blue points (circles if they have negative sign). Itis consistent with the theory prediction of BSS16 (light blueline). non-Gaussian branches for (
T T ) and ( EB ) reconstruc-tions. By construction, we expect all lensing biases thatare sensitive to the convergence power spectrum and thelensed or unlensed CMB power spectra (c.p. Eq. 8) tobe identical in the Gaussian and non-Gaussian simula-tions . Since we are only interested in the difference ofthe reconstructed convergence power spectra, in whichthese biases cancel out, we do not compute and removethem. Apart from the auto power spectra, we also com-pute the average power in the cross correlation betweeninput maps and reconstructed maps. This cross correla-tion is not an actual observable, but can serve as a proxyfor the non-Gaussian bias in cross correlations with othertracers of large-scale structure. Also, measurements ofthe cross power are not affected by the N (0) and N (1) bias and have lower noise. The theory prediction for thebias in cross-correlations is N crossNG ≈ / N autoNG [35]. The bispectrum of the convergence also changes the lensed CMBpower spectra but this is a sub-percent effect and not detectablein our simulations [58]
500 1000 1500 2000L0.0150.0100.0050.0000.0050.010 C r o ss B i a s / C measurement from (TT, ) theory simulations FIG. 8. We detect a non-Gaussian bias of the size pre-dicted by BSS16 (0 .
5% of the signal) in the cross correlationof temperature-based reconstructions and input maps with asignificance of 5 . σ . The p-value of the measurement for ano-bias null-hypothesis is 0.0003. The reduced χ betweenprediction and measurement is 1.02. L (TT,TT) (TT, ) FIG. 9. Covariance matrices of the power spectrum mea-surements in units of the variance. The error bars in themeasurement of the power in the cross correlation are to goodapproximation uncorrelated. The measurements of the autopower show some expected degree of correlation between thebins.
IV. RESULTS
As discussed above we measure the non-Gaussian biasfrom the difference between N autoNG ( L ) = ˆ C ˆ κ ˆ κ NG ( L ) − ˆ C ˆ κ ˆ κ G ( L ) N crossNG ( L ) = ˆ C ˆ κκ NG ( L ) − ˆ C ˆ κκ G ( L ) . (12)We start by examining this difference in the temperaturebased reconstructions.In Fig. 7 we show the measured bias from cross cor-relating reconstructions from ( T T ) with the input mapsand plot the reconstructed power spectra for compari-son. Fig. 8 shows the measured bias in units of the signal.The theory prediction from BSS16 is plotted in light bluefor comparison. The p-value of the data points assum-ing no-bias is 0 . . σ . Measuring the covariance between the data points
250 500 750 1000 1250 1500 1750 2000L10 measurement from (TT,TT) N (3/2)theory ( L ) C theory + N C Gauss C N body C N body C Gauss
FIG. 10. Measured non-Gaussian bias in the CMB lens-ing power spectrum measurement from the temperature four-point function. The reconstruction agrees well with the theoryprediction for the sum of convergence power spectrum and N (0) reconstruction noise. The measured bias is consistentwith the theory prediction of BSS16, but the null-hypothesisof no bias cannot be excluded with high statistical signifi-cance.
500 1000 1500 2000L0.020.010.000.010.02 B i a s / C measurement from (TT,TT) theory simulations FIG. 11. The non-Gaussian bias in temperature-based CMBlensing power spectrum measurements in units of the signal.The points are consistent with the theory predictions, the biasis detected with a significance of 2.84 σ . (Fig. 9, right panel), shows that the measurements indifferent bins can to good approximation be treated asuncorrelated.The bias in the auto power measured from ( T T, T T ) isdetected with a lower significance of 2 . σ (Figs. 10,11).The p-value of the measurement for a null-hypothesis ofno bias is p = 0 .
500 1000 1500 2000L0.0100.0050.0000.0050.010 B i a s / C EB based reconstruction cross auto
FIG. 12. The non-Gaussian bias to CMB lensing measure-ments from (
EB, EB ) is consistent with zero in both autoand cross correlations. L (EB,EB) (EB, ) FIG. 13. Covariance matrices of the power spectrum mea-surements from (
EB, EB ) in units of the variance. The error-bars in the measurement of the power in the cross correlationare to good approximation uncorrelated. The measurementsof the auto power show some expected degree of correlationbetween the bins.
We do not find any indication for a non-Gaussian biasin the polarization-based reconstruction (Fig. 12). Thisagrees with the intuition gained from its functional form:additional angular dependencies (as compared to the biasin temperature-based reconstruction) reduce the supportof the contributing integrals (see App. A and Ref. [35]).Again, we find correlations between the data points inthe auto power measurement (Fig. 13).
V. DISCUSSION
By comparing lensing measurements from CMB sim-ulations lensed with Gaussian and non-Gaussian conver-gence fields, we find strong indication for the existence ofa non-Gaussian bias to CMB lensing measurements fromtemperature data. The bias is at the 1% level, whichagrees with the theoretical prediction for a bispectrum-induced bias of Ref. [35] if we take into account twosources for the lensing bispectrum, non-linear structureformation and multiple correlated deflections.By measuring the bias in the cross correlations of re-
100 500 1000 1500 2000 2500 3000L86420 N ( / ) / C [ % ] redshift dependence of N (3/2) bias z < 1 z < 2 z < 3 z < z CMB
FIG. 14. The relative size of the non-Gaussian bias increasesif we only consider lenses at low redshifts z max < z CMB . Thisis a consequence of the different redshift-scalings of the com-peting terms in the lensing bispectrum from non-linear struc-ture formation and post Born effects. To illustrate this weignore any contributions to the lensing bispectrum and powerspectrum with z > z max and plot the ratio of the resultingcross bias to the power spectrum. These results suggest thatthe non-Gaussian bias could be more important for measure-ments of cross correlations of CMB lensing with low-redshifttracers. We note that the curves shown here are still pre-liminary and should be seen as a motivation to investigatethe bias on cross correlations in future work (B¨ohm et al. inprep). constructed lensing maps with the true underlying lens-ing fields, we detect the theoretically predicted bias inthe simulations at the 5 σ significance level. We detectthe non-Gaussian bias in the auto correlation with a sig-nificance of ∼ σ . The measured bias in power spectrummeasurements from a combination of E-and B-mode po-larization, ( EB, EB ), is consistent with zero. We notethat lensing B-modes at intermediate scales are more sen-sitive to smaller scales in the deflection field than lensedE-modes or temperature. A non-zero bias in
EB, EB could therefore be present in real data, if it was gener-ated by scales that are not accurately modeled in thesimulation due to its finite resolution.We point out that our results have been independentlyconfirmed by Beck et al. 2018 (in prep), who use a com-pletely different simulation set on the full sky.The good agreement between the simulations and the-ory suggests that the assumptions that entered into thetheory calculation are valid and that we can rely on it tomake predictions for different experiments. Theoreticalbias predictions for different experimental configurationare shown in Fig. 15.The non-Gaussian bias is likely to affect lensing mea-surements from CMB-experiments that are dominated bytemperature reconstruction. This includes current andupcoming experiments such as AdvACT [17] and Simons Observatory . An uncorrected non-Gaussian bias at thepercent level degrades the accuracy with which these ex-periments can measure cosmological parameters. Thenon-Gaussian bias is unlikely to affect experiments thatare polarization-dominated, such as the ground basedSPT-3G[19] and CMB-S4 experiments [20] and space-based missions like LiteBird [61] or Pico [62].It is further important to note that the smallness ofthe bias is a consequence of a somewhat coincidentalcancellation: The bias is mostly sensitive to elongatedbispectrum configurations. For these shapes, the bis-pectra from non-linear structure formation and multiplecorrelated deflections have opposite sign. The fact thatthey are in addition of similar magnitude is only true forsources at high redshifts. If we consider sources at lowredshifts or restrict contributions to the bispectra to lowredshifts, we expect this cancellation to be much less ef-ficient. The bias could therefore be more important incross-correlations of CMB lensing with low-redshift trac-ers (B¨ohm et al. in prep., Ref. [63]). We illustrate thisby plotting preliminary results for the non-Gaussian bias(corrected by a factor of 1 / z max ≤ z CMB , z source = z CMB ) in Fig. 14. These resultssuggest that the bias could be of the order of severalpercent for cross correlation measurements from temper-ature data. With these measurements getting most oftheir signal from high multipoles where the
T T estima-tor performs best, this could make this bias relevant formost future wide-field surveys. We caution at this pointthat Fig. 14 should only be seen as a motivation to inves-tigate the non-linear bias for cross correlations. By set-ting all contribution to the post-Born bispectrum abovea certain redshift to zero, it becomes negligible. For re-alistic cross correlations the expression for the post-Bornbispectrum is more complicated and its contribution tothe bias could be more important.The results shown in this work only apply to powerspectrum estimates with a quadratic estimator but sim-ilar biases could arise for alternative estimators if theyare derived under the assumption of a Gaussian deflec-tion field.Recently, Ref. [46] pointed out that a shear estima-tor [44, 45] is to good approximation robust against con-tamination from isotropic foregrounds (at the cost oflower signal to noise in the reconstruction). The factthat we find no bias in the reconstruction from
EB, EB ,for which the quadratic estimator corresponds to a shearestimator, suggests that a shear-only estimator could alsobe less sensitive to the non-linear bias (this can also beseen analytically, since the shear estimator has an ad-ditional angular dependence, which should lead to addi-tional cancellations in the bias integrals in Eq. A1) This https://simonsobservatory.org/
100 500 1000 1500 2000 2500 3000L2.01.51.00.50.0 N ( / ) / C [ % ] N (3/2) in different experiments FIG. 15. Bispectrum-induced N (3 / bias in temperature-based measurements for different experimental configurations.The lines are labeled by noise in µK − arcmin, beam FWHMin arcmin, l min and l max . The size of the bias is sensitive tothe maximal, signal-dominated CMB scale that is used in thereconstruction. possibility could be easily tested on simulations and couldbe explored in future work. We note, however, that sincethe non-linear bias can be modeled theoretically, a miti-gation at the cost of lower signal-to-noise is not crucial. ACKNOWLEDGEMENTS
We thank Dominic Beck and Giulio Fabbian for crosschecking some of the results with their simulations andAntony Lewis and Geraint Pratten for sharing their nu-merical implementations of post Born terms. VB thanksEmmanuel Schaan, Dominic Beck and Giulio Fabbianfor useful discussions. BDS was supported by an STFCErnest Rutherford Fellowship and an Isaac Newton TrustEarly Career Grant. JL is supported by an NSF As-tronomy and Astrophysics Postdoctoral Fellowship underaward AST-1602663. JCH is supported by the Friendsof the Institute for Advanced Study. MS was supportedby the Bezos fund. This work used the Extreme Sci-ence and Engineering Discovery Environment (XSEDE),which is supported by NSF grant ACI-1053575. This re-search used resources of the National Energy ResearchScientific Computing Center, a DOE Office of ScienceUser Facility supported by the Office of Science of theU.S. Department of Energy under Contract No. DE-AC02-05CH11231. Part of this work used computationalresources at the Max Planck Computing and Data Facil-ity (MPCDF).
Appendix A: Analytic prediction for abispectrum-induced CMB lensing bias
All CMB lensing analyses to date assume that thelensing convergence is a Gaussian field. However, non-linear structure formation and multiple correlatedlenses introduce a small, but detectable amount of non-Gaussianity [36, 47–49]. In the limit of small densityperturbations, the non-Gaussian structure can be char-acterized by a hierarchy of connected correlation func-tions. To lowest order, the lensing convergence acquiresa bispectrum.The lensing bispectrum introduces an additional termto the standard four-point estimator (compare Eq. 9)This new bias was first identified in Ref. [35] (BSS16). Itsname follows from the naming convention for CMB lens-ing biases, where biases are labeled by their power in thelensing power spectrum. The N / bias arises becausethe lensing bispectrum changes the lensed temperaturefour-point function.BSS16 found that the N / bias can change themeasured lensing power spectrum in temperature-basedCMB lensing analyses at the percent level. This cor-responds to a 1-2 σ effect (per L-bin) for current andupcoming CMB experiments.The estimation of the size of the bias in BSS16 is basedon the numerical evaluation of analytically derived ex-pressions. This evaluation relies on a number of assump-tions:1. The lensing bispectrum contributes to the lensedtemperature four-point function with 8 terms. Dueto the complicated structure of these terms (theyinvolve 6-dimensional coupled integrals over recon-struction weights g , the lensing bispectrum andCMB power spectra.), only two of these termswere evaluated. These two terms were chosen be-cause they factor maximally under the reconstruc-tion weights (one of them can even be split intoa product of 3 two-dimensional integrals). Theirstructure suggests that these terms are the domi-nant contributions to the bispectrum-induced bias.For temperature-only reconstruction, the two termsread N (3 / ( L ) = − A L S L (cid:90) l , l g l , L [ l · ( l − l )] × [ l · ( L − ( l − l ))] C T Tl B φ [ l − l , L − ( l − l ) , − L ] N (3 / ( L ) = 4 A L S L (cid:90) l , l g l , L ( l · l ) [ l · ( L − l )] × C T Tl B φ ( l , L − l , − L ) , (A1)with S L = (cid:90) l g l , L ( l · L ) C T Tl ≈ A − L . (A2)For polarization-based reconstruction, the struc-ture of the terms is similar, but with additionalangular dependencies (see Ref. [35] for details). Incross-correlations, all other terms vanish and thebias depends only on the two terms above.12. The evaluation in BSS16 only considered non-linearstructure formation as a source of the lensing bis-pectrum. Recently Ref. [36] pointed out that anadditional lensing bispectrum arises from multiplecorrelated lensing deflections. The effect of mul-tiple deflections is commonly ignored in the Bornapproximation. Both effects, post Born correctionsand non-linear structure formation, lead to lensingbispectra of the same order of magnitude, but forcertain triangle configurations of opposite sign.3. The modeling of the bispectrum from non-linearstructure formation in BSS16 relied on tree-levelperturbation theory, which breaks down on smallscales (and the bias was shown to be sensitive toreplacing the linear matter power spectrum by itsnon-linear (HALOFIT [55]) counterpart in the mat-ter bispectrum model).4. The theoretical modeling of the bias relied on aTaylor series expansion of the lensed CMB in thedeflection angle, and thus on the assumption ofsmall deflection angles.In this work, we use an updated analytical predic-tion for the bias, which still assumes that all but twoterms are negligible, but that takes into account the bis-pectrum from post-Born effects and uses an extended,simulation-calibrated, semi-analytic model for the matterbispectrum [56]. Updated theoretical results are shownin Fig. 15 for different experimental set-ups and togetherwith the measurement of the bias in Sec. IV. Appendix B: Non-linear bias from bispectrainvolving the curl of the lensing deflection
Allowing for multiple deflections introduces an addi-tional degree of freedom, ω , to the linear mapping be-tween the lensed and unlensed image of a source, whichdescribes a rotation of the image [64]. With this addi-tional dof, the lensing deflection angle is no longer a puregradient field, but acquires an additional curl component, α ( x ) = ∇ φ ( x ) + ∗∇ Ω( x ) , (B1)sourced by the curl potential Ω [65]. We use a ∗ to denotea rotation by 90 degrees, and, for notational simplicity,also abbreviate the combination of rotation and scalarproduct, ·∗ , in the following by ∗ .Being second order in the gravitational potential, therotation is suppressed compared to the first order con-vergence and shear distortions to the image. We thusexpect the largest bias that involves the curl poten-tial to be sourced by a “cross” bispectrum of the form B Ω ,φ,φ ( L , l , − L − l ) [36]. The curl potential can be treated in complete analogy to the scalar lensing poten-tial φ . E.g., when expressing the effect of lensing on theCMB in terms of a small perturbation to the unlensedCMB, we can write [66]˜ T = T + δ Ω T + δ φ T + δ T + δ φ T + O ( φ , Ω ) . (B2)Adapting the flat sky approximation, the first two termsare given in harmonic space by δ Ω T ( l ) = (cid:90) l (cid:48) [ l (cid:48) ∗ ( l (cid:48) − l )] T ( l (cid:48) )Ω( l − l (cid:48) ) (B3) δ φ T ( l ) = (cid:90) l (cid:48) [ l (cid:48) · ( l (cid:48) − l )] T ( l (cid:48) ) φ ( l − l (cid:48) ) . (B4)Using this perturbative framework to model the lensedtemperature 4-point function, Ref. [35] show that the twodominant terms in the N (3 / bias are sourced by con-tractions of the following expectation values over φ and T [35] N (3 / (cid:2) φ (cid:3) ← (cid:104) δ φ T δ φ T δ φ T (cid:48) T (cid:48) (cid:105) N (3 / (cid:2) φ (cid:3) ← (cid:104) δ φ T T δ φ T (cid:48) T (cid:48) (cid:105) . (B5)A bias sourced by the cross bispectrum B Ω ,κ,κ ( L , l , − L − l ) (we refer to it as ˜ N (3 / ) should therefore be dominatedby contractions of the following expectation values˜ N (3 / (cid:2) φ Ω (cid:3) ← (cid:104) δ φ T δ φ T δ Ω T (cid:48) T (cid:48) (cid:105) a + (cid:104) δ Ω T δ φ T δ φ T (cid:48) T (cid:48) (cid:105) b ˜ N (3 / (cid:2) φ Ω (cid:3) ← (cid:104) δ Ω T T δ φ T (cid:48) T (cid:48) (cid:105) . (B6)The expressions for the dominant contractions arisingfrom 1 a and 2 are identical to the auto bias (Eq. A1),but with B φ replaced by B φ Ω and S L replaced by S × L = (cid:90) l g l , L ( l ∗ L ) C T Tl = 0 . (B7)This integral vanishes because the integrand is unevenunder the angular integration. The remaining dominantcontraction from 1 b is of the form˜ N (3 / b ( L ) = − A L S L (cid:90) l , l g l , L [ l ∗ l ] × [ l · ( L − ( l − l ))] C T Tl B φ,φ, Ω [ l − l , L − ( l − l ) , − L ] . (B8)Because of the mixing of sines and cosines in the angularintegrations in Eq. B8, we expect this contribution to bestrongly suppressed compared to the corresponding termin the bias from the auto bispectrum.This short calculation suggests that biases from bis-pectra involving the curl component are likely to be neg-ligible for current and upcoming CMB experiments. By contractions we mean the terms that arise from taking the [1] A. Lewis and A. Challinor, Phys. Rep. , 1 (2006),astro-ph/0601594.[2] D. Hanson, A. Challinor, and A. Lewis, General Rela-tivity and Gravitation , 2197 (2010), arXiv:0911.0612.[3] J. Lesgourgues, L. Perotto, S. Pastor, and M. Piat,Phys. Rev. D , 045021 (2006), astro-ph/0511735.[4] B. D. Sherwin, J. Dunkley, S. Das, J. W. Ap-pel, J. R. Bond, C. S. Carvalho, M. J. Devlin,R. D¨unner, T. Essinger-Hileman, J. W. Fowler, A. Ha-jian, M. Halpern, M. Hasselfield, A. D. Hincks, R. Hlozek,J. P. Hughes, K. D. Irwin, J. Klein, A. Kosowsky, T. A.Marriage, D. Marsden, K. Moodley, F. Menanteau, M. D.Niemack, M. R. Nolta, L. A. Page, L. Parker, E. D.Reese, B. L. Schmitt, N. Sehgal, J. Sievers, D. N. Spergel,S. T. Staggs, D. S. Swetz, E. R. Switzer, R. Thornton,K. Visnjic, and E. Wollack, Physical Review Letters ,021302 (2011), arXiv:1105.0419 [astro-ph.CO].[5] K. M. Smith, O. Zahn, and O. Dor´e, Phys. Rev. D ,043510 (2007), arXiv:0705.3980.[6] C. M. Hirata, S. Ho, N. Padmanabhan, U. Seljak,and N. A. Bahcall, Phys. Rev. D , 043520 (2008),arXiv:0801.0644.[7] S. Das, B. D. Sherwin, P. Aguirre, J. W. Appel,J. R. Bond, C. S. Carvalho, M. J. Devlin, J. Dunkley,R. D¨unner, T. Essinger-Hileman, J. W. Fowler, A. Ha-jian, M. Halpern, M. Hasselfield, A. D. Hincks, R. Hlozek,K. M. Huffenberger, J. P. Hughes, K. D. Irwin, J. Klein,A. Kosowsky, R. H. Lupton, T. A. Marriage, D. Mars-den, F. Menanteau, K. Moodley, M. D. Niemack, M. R.Nolta, L. A. Page, L. Parker, E. D. Reese, B. L. Schmitt,N. Sehgal, J. Sievers, D. N. Spergel, S. T. Staggs, D. S.Swetz, E. R. Switzer, R. Thornton, K. Visnjic, andE. Wollack, Physical Review Letters , 021301 (2011),arXiv:1103.2124.[8] A. van Engelen, R. Keisler, O. Zahn, K. A. Aird, B. A.Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang,H. M. Cho, T. M. Crawford, A. T. Crites, T. de Haan,M. A. Dobbs, J. Dudley, E. M. George, N. W. Halver-son, G. P. Holder, W. L. Holzapfel, S. Hoover, Z. Hou,J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch,M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S.Meyer, M. Millea, J. J. Mohr, T. E. Montroy, T. Natoli,S. Padin, T. Plagge, C. Pryke, C. L. Reichardt, J. E.Ruhl, J. T. Sayre, K. K. Schaffer, L. Shaw, E. Shirokoff,H. G. Spieler, Z. Staniszewski, A. A. Stark, K. Story,K. Vanderlinde, J. D. Vieira, and R. Williamson, ApJ , 142 (2012), arXiv:1202.0546.[9] D. Hanson, S. Hoover, A. Crites, P. A. R. Ade, K. A.Aird, J. E. Austermann, J. A. Beall, A. N. Bender, B. A.Benson, L. E. Bleem, J. J. Bock, J. E. Carlstrom, C. L.Chang, H. C. Chiang, H.-M. Cho, A. Conley, T. M. Craw-ford, T. de Haan, M. A. Dobbs, W. Everett, J. Gallicchio, expectation value over unlensed CMB realizations. Assumingthat the unlensed CMB is Gaussian, each expectation value canbe split into a sum of three terms. We only consider the biasarising from one of these three terms, which BSS16 identifiedas the dominant one. Also, for readability, we do not write thesymmetry factors that arise from permutations of T, δT and δ T that leave the result invariant. J. Gao, E. M. George, N. W. Halverson, N. Harring-ton, J. W. Henning, G. C. Hilton, G. P. Holder, W. L.Holzapfel, J. D. Hrubes, N. Huang, J. Hubmayr, K. D.Irwin, R. Keisler, L. Knox, A. T. Lee, E. Leitch, D. Li,C. Liang, D. Luong-Van, G. Marsden, J. J. McMahon,J. Mehl, S. S. Meyer, L. Mocanu, T. E. Montroy, T. Na-toli, J. P. Nibarger, V. Novosad, S. Padin, C. Pryke,C. L. Reichardt, J. E. Ruhl, B. R. Saliwanchik, J. T.Sayre, K. K. Schaffer, B. Schulz, G. Smecher, A. A. Stark,K. T. Story, C. Tucker, K. Vanderlinde, J. D. Vieira,M. P. Viero, G. Wang, V. Yefremenko, O. Zahn, andM. Zemcov, Physical Review Letters , 141301 (2013),arXiv:1307.5830 [astro-ph.CO].[10] P. A. R. Ade, Y. Akiba, A. E. Anthony, K. Arnold, M. At-las, D. Barron, D. Boettger, J. Borrill, S. Chapman,Y. Chinone, M. Dobbs, T. Elleflot, J. Errard, G. Fab-bian, C. Feng, D. Flanigan, A. Gilbert, W. Grainger,N. W. Halverson, M. Hasegawa, K. Hattori, M. Hazumi,W. L. Holzapfel, Y. Hori, J. Howard, P. Hyland, Y. In-oue, G. C. Jaehnig, A. Jaffe, B. Keating, Z. Kermish,R. Keskitalo, T. Kisner, M. Le Jeune, A. T. Lee, E. Lin-der, E. M. Leitch, M. Lungu, F. Matsuda, T. Mat-sumura, X. Meng, N. J. Miller, H. Morii, S. Moyer-man, M. J. Myers, M. Navaroli, H. Nishino, H. Paar,J. Peloton, E. Quealy, G. Rebeiz, C. L. Reichardt, P. L.Richards, C. Ross, I. Schanning, D. E. Schenck, B. Sher-win, A. Shimizu, C. Shimmin, M. Shimon, P. Siritanasak,G. Smecher, H. Spieler, N. Stebor, B. Steinbach, R. Stom-por, A. Suzuki, S. Takakura, T. Tomaru, B. Wilson,A. Yadav, O. Zahn, and Polarbear Collaboration, Physi-cal Review Letters , 021301 (2014), arXiv:1312.6646.[11] P. A. R. Ade, Y. Akiba, A. E. Anthony, K. Arnold, M. At-las, D. Barron, D. Boettger, J. Borrill, S. Chapman,Y. Chinone, M. Dobbs, T. Elleflot, J. Errard, G. Fab-bian, C. Feng, D. Flanigan, A. Gilbert, W. Grainger,N. W. Halverson, M. Hasegawa, K. Hattori, M. Hazumi,W. L. Holzapfel, Y. Hori, J. Howard, P. Hyland, Y. In-oue, G. C. Jaehnig, A. Jaffe, B. Keating, Z. Kermish,R. Keskitalo, T. Kisner, M. Le Jeune, A. T. Lee, E. Lin-der, E. M. Leitch, M. Lungu, F. Matsuda, T. Mat-sumura, X. Meng, N. J. Miller, H. Morii, S. Moyer-man, M. J. Myers, M. Navaroli, H. Nishino, H. Paar,J. Peloton, E. Quealy, G. Rebeiz, C. L. Reichardt, P. L.Richards, C. Ross, I. Schanning, D. E. Schenck, B. Sher-win, A. Shimizu, C. Shimmin, M. Shimon, P. Siritanasak,G. Smecher, H. Spieler, N. Stebor, B. Steinbach, R. Stom-por, A. Suzuki, S. Takakura, T. Tomaru, B. Wilson,A. Yadav, O. Zahn, and Polarbear Collaboration, Physi-cal Review Letters , 021301 (2014), arXiv:1312.6646.[12] BICEP2 Collaboration, Keck Array Collaboration,P. A. R. Ade, Z. Ahmed, R. W. Aikin, K. D. Alexan-der, D. Barkats, S. J. Benton, C. A. Bischoff, J. J. Bock,R. Bowens-Rubin, J. A. Brevik, I. Buder, E. Bullock,V. Buza, J. Connors, B. P. Crill, L. Duband, C. Dvorkin,J. P. Filippini, S. Fliescher, J. Grayson, M. Halpern,S. Harrison, S. R. Hildebrandt, G. C. Hilton, H. Hui,K. D. Irwin, J. Kang, K. S. Karkare, E. Karpel, J. P.Kaufman, B. G. Keating, S. Kefeli, S. A. Kernasovskiy,J. M. Kovac, C. L. Kuo, E. M. Leitch, M. Lueker, K. G.Megerian, T. Namikawa, C. B. Netterfield, H. T. Nguyen,R. O’Brient, R. W. Ogburn, IV, A. Orlando, C. Pryke, S. Richter, R. Schwarz, C. D. Sheehy, Z. K. Staniszewski,B. Steinbach, R. V. Sudiwala, G. P. Teply, K. L. Thomp-son, J. E. Tolan, C. Tucker, A. D. Turner, A. G. Vieregg,A. C. Weber, D. V. Wiebe, J. Willmert, C. L. Wong,W. L. K. Wu, and K. W. Yoon, ApJ , 228 (2016),arXiv:1606.01968.[13] K. T. Story, D. Hanson, P. A. R. Ade, K. A. Aird,J. E. Austermann, J. A. Beall, A. N. Bender, B. A. Ben-son, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H. C.Chiang, H.-M. Cho, R. Citron, T. M. Crawford, A. T.Crites, T. de Haan, M. A. Dobbs, W. Everett, J. Gallic-chio, J. Gao, E. M. George, A. Gilbert, N. W. Halver-son, N. Harrington, J. W. Henning, G. C. Hilton, G. P.Holder, W. L. Holzapfel, S. Hoover, Z. Hou, J. D. Hrubes,N. Huang, J. Hubmayr, K. D. Irwin, R. Keisler, L. Knox,A. T. Lee, E. M. Leitch, D. Li, C. Liang, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, L. Mocanu,T. E. Montroy, T. Natoli, J. P. Nibarger, V. Novosad,S. Padin, C. Pryke, C. L. Reichardt, J. E. Ruhl, B. R.Saliwanchik, J. T. Sayre, K. K. Schaffer, G. Smecher,A. A. Stark, C. Tucker, K. Vanderlinde, J. D. Vieira,G. Wang, N. Whitehorn, V. Yefremenko, and O. Zahn,ApJ , 50 (2015), arXiv:1412.4760.[14] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Ar-naud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J.Banday, R. B. Barreiro, J. G. Bartlett, and et al., A&A , A15 (2016), arXiv:1502.01591.[15] B. D. Sherwin, A. van Engelen, N. Sehgal, M. Mad-havacheril, G. E. Addison, S. Aiola, R. Allison,N. Battaglia, D. T. Becker, J. A. Beall, J. R. Bond,E. Calabrese, R. Datta, M. J. Devlin, R. D¨unner,J. Dunkley, A. E. Fox, P. Gallardo, M. Halpern, M. Has-selfield, S. Henderson, J. C. Hill, G. C. Hilton, J. Hub-mayr, J. P. Hughes, A. D. Hincks, R. Hlozek, K. M. Huf-fenberger, B. Koopman, A. Kosowsky, T. Louis, L. Mau-rin, J. McMahon, K. Moodley, S. Naess, F. Nati, L. New-burgh, M. D. Niemack, L. A. Page, J. Sievers, D. N.Spergel, S. T. Staggs, R. J. Thornton, J. Van Lanen,E. Vavagiakis, and E. J. Wollack, Phys. Rev. D ,123529 (2017), arXiv:1611.09753.[16] G. Simard, Y. Omori, K. Aylor, E. J. Baxter, B. A.Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang,H. Cho, R. Chown, T. M. Crawford, A. T. Crites, T. deHaan, M. A. Dobbs, W. B. Everett, E. M. George,N. W. Halverson, N. L. Harrington, J. W. Henning, G. P.Holder, Z. Hou, W. L. Holzapfel, J. D. Hrubes, L. Knox,A. T. Lee, E. M. Leitch, D. Luong-Van, A. Manzotti,J. J. McMahon, S. S. Meyer, L. M. Mocanu, J. J. Mohr,T. Natoli, S. Padin, C. Pryke, C. L. Reichardt, J. E. Ruhl,J. T. Sayre, K. K. Schaffer, E. Shirokoff, Z. Staniszewski,A. A. Stark, K. T. Story, K. Vanderlinde, J. D. Vieira,R. Williamson, and W. L. K. Wu, ArXiv e-prints (2017),arXiv:1712.07541.[17] S. W. Henderson, R. Allison, J. Austermann, T. Baildon,N. Battaglia, J. A. Beall, D. Becker, F. De Bernardis,J. R. Bond, E. Calabrese, S. K. Choi, K. P. Cough-lin, K. T. Crowley, R. Datta, M. J. Devlin, S. M. Duff,J. Dunkley, R. D¨unner, A. van Engelen, P. A. Gallardo,E. Grace, M. Hasselfield, F. Hills, G. C. Hilton, A. D.Hincks, R. Hloˆzek, S. P. Ho, J. Hubmayr, K. Huffen-berger, J. P. Hughes, K. D. Irwin, B. J. Koopman, A. B.Kosowsky, D. Li, J. McMahon, C. Munson, F. Nati,L. Newburgh, M. D. Niemack, P. Niraula, L. A. Page,C. G. Pappas, M. Salatino, A. Schillaci, B. L. Schmitt, N. Sehgal, B. D. Sherwin, J. L. Sievers, S. M. Simon,D. N. Spergel, S. T. Staggs, J. R. Stevens, R. Thornton,J. Van Lanen, E. M. Vavagiakis, J. T. Ward, and E. J.Wollack, Journal of Low Temperature Physics , 772(2016), arXiv:1510.02809 [astro-ph.IM].[18] A. Suzuki, P. Ade, Y. Akiba, C. Aleman, K. Arnold,C. Baccigalupi, B. Barch, D. Barron, A. Bender,D. Boettger, J. Borrill, S. Chapman, Y. Chinone,A. Cukierman, M. Dobbs, A. Ducout, R. Dunner, T. Elle-flot, J. Errard, G. Fabbian, S. Feeney, C. Feng, T. Fujino,G. Fuller, A. Gilbert, N. Goeckner-Wald, J. Groh, T. D.Haan, G. Hall, N. Halverson, T. Hamada, M. Hasegawa,K. Hattori, M. Hazumi, C. Hill, W. Holzapfel, Y. Hori,L. Howe, Y. Inoue, F. Irie, G. Jaehnig, A. Jaffe, O. Jeong,N. Katayama, J. Kaufman, K. Kazemzadeh, B. Keating,Z. Kermish, R. Keskitalo, T. Kisner, A. Kusaka, M. L.Jeune, A. Lee, D. Leon, E. Linder, L. Lowry, F. Mat-suda, T. Matsumura, N. Miller, K. Mizukami, J. Mont-gomery, M. Navaroli, H. Nishino, J. Peloton, D. Po-letti, G. Puglisi, G. Rebeiz, C. Raum, C. Reichardt,P. Richards, C. Ross, K. Rotermund, Y. Segawa, B. Sher-win, I. Shirley, P. Siritanasak, N. Stebor, R. Stom-por, J. Suzuki, O. Tajima, S. Takada, S. Takakura,S. Takatori, A. Tikhomirov, T. Tomaru, B. Westbrook,N. Whitehorn, T. Yamashita, A. Zahn, and O. Zahn,Journal of Low Temperature Physics , 805 (2016),arXiv:1512.07299 [astro-ph.IM].[19] B. A. Benson, P. A. R. Ade, Z. Ahmed, S. W. Allen,K. Arnold, J. E. Austermann, A. N. Bender, L. E.Bleem, J. E. Carlstrom, C. L. Chang, H. M. Cho, J. F.Cliche, T. M. Crawford, A. Cukierman, T. de Haan,M. A. Dobbs, D. Dutcher, W. Everett, A. Gilbert, N. W.Halverson, D. Hanson, N. L. Harrington, K. Hattori,J. W. Henning, G. C. Hilton, G. P. Holder, W. L.Holzapfel, K. D. Irwin, R. Keisler, L. Knox, D. Kubik,C. L. Kuo, A. T. Lee, E. M. Leitch, D. Li, M. McDon-ald, S. S. Meyer, J. Montgomery, M. Myers, T. Natoli,H. Nguyen, V. Novosad, S. Padin, Z. Pan, J. Pearson,C. Reichardt, J. E. Ruhl, B. R. Saliwanchik, G. Simard,G. Smecher, J. T. Sayre, E. Shirokoff, A. A. Stark,K. Story, A. Suzuki, K. L. Thompson, C. Tucker, K. Van-derlinde, J. D. Vieira, A. Vikhlinin, G. Wang, V. Yefre-menko, and K. W. Yoon, in Millimeter, Submillimeter,and Far-Infrared Detectors and Instrumentation for As-tronomy VII , Proc. SPIE, Vol. 9153 (2014) p. 91531P,arXiv:1407.2973 [astro-ph.IM].[20] K. N. Abazajian, P. Adshead, Z. Ahmed, S. W. Allen,D. Alonso, K. S. Arnold, C. Baccigalupi, J. G. Bartlett,N. Battaglia, B. A. Benson, C. A. Bischoff, J. Borrill,V. Buza, E. Calabrese, R. Caldwell, J. E. Carlstrom,C. L. Chang, T. M. Crawford, F.-Y. Cyr-Racine, F. DeBernardis, T. de Haan, S. di Serego Alighieri, J. Dunkley,C. Dvorkin, J. Errard, G. Fabbian, S. Feeney, S. Ferraro,J. P. Filippini, R. Flauger, G. M. Fuller, V. Gluscevic,D. Green, D. Grin, E. Grohs, J. W. Henning, J. C. Hill,R. Hlozek, G. Holder, W. Holzapfel, W. Hu, K. M. Huf-fenberger, R. Keskitalo, L. Knox, A. Kosowsky, J. Ko-vac, E. D. Kovetz, C.-L. Kuo, A. Kusaka, M. Le Jeune,A. T. Lee, M. Lilley, M. Loverde, M. S. Madhavacheril,A. Mantz, D. J. E. Marsh, J. McMahon, P. D. Meerburg,J. Meyers, A. D. Miller, J. B. Munoz, H. N. Nguyen,M. D. Niemack, M. Peloso, J. Peloton, L. Pogosian,C. Pryke, M. Raveri, C. L. Reichardt, G. Rocha, A. Rotti,E. Schaan, M. M. Schmittfull, D. Scott, N. Sehgal, S. Shandera, B. D. Sherwin, T. L. Smith, L. Sorbo, G. D.Starkman, K. T. Story, A. van Engelen, J. D. Vieira,S. Watson, N. Whitehorn, and W. L. Kimmy Wu, ArXive-prints (2016), arXiv:1610.02743.[21] W. Hu, ApJ , L79 (2001), astro-ph/0105424.[22] W. Hu and T. Okamoto, ApJ , 566 (2002), astro-ph/0111606.[23] M. Kesden, A. Cooray, and M. Kamionkowski, Phys.Rev. D , 123507 (2003).[24] D. Hanson, A. Challinor, G. Efstathiou, andP. Bielewicz, Phys. Rev. D , 043005 (2011),arXiv:1008.4403 [astro-ph.CO].[25] D. Hanson, G. Rocha, and K. G´orski, MNRAS , 2169(2009), arXiv:0907.1927 [astro-ph.CO].[26] D. Hanson, A. Lewis, and A. Challinor, Phys. Rev. D , 103003 (2010), arXiv:1003.0198 [astro-ph.CO].[27] Y. Fantaye, C. Baccigalupi, S. M. Leach, and A. P. S.Yadav, J. Cosmology Astropart. Phys. , 017 (2012),arXiv:1207.0508 [astro-ph.CO].[28] A. van Engelen, S. Bhattacharya, N. Sehgal, G. P.Holder, O. Zahn, and D. Nagai, ApJ , 13 (2014),arXiv:1310.7023.[29] S. J. Osborne, D. Hanson, and O. Dor´e, J. CosmologyAstropart. Phys. , 024 (2014), arXiv:1310.7547.[30] S. Ferraro and J. C. Hill, Phys. Rev. D , 023512 (2018),arXiv:1705.06751.[31] A. Lewis, A. Challinor, and D. Hanson, J. CosmologyAstropart. Phys. , 018 (2011), arXiv:1101.2234.[32] T. Namikawa, D. Hanson, and R. Takahashi, MNRAS , 609 (2013), arXiv:1209.0091 [astro-ph.CO].[33] T. Namikawa and R. Takahashi, MNRAS , 1507(2014), arXiv:1310.2372.[34] M. S. Madhavacheril and J. C. Hill, ArXiv e-prints(2018), arXiv:1802.08230.[35] V. B¨ohm, M. Schmittfull, and B. D. Sherwin,Phys. Rev. D , 043519 (2016), arXiv:1605.01392.[36] G. Pratten and A. Lewis, J. Cosmology Astropart. Phys. , 047 (2016), arXiv:1605.05662.[37] A. Petri, Z. Haiman, and M. May, Phys. Rev. D ,123503 (2017), arXiv:1612.00852.[38] G. Fabbian, M. Calabrese, and C. Carbone, J. Cosmol-ogy Astropart. Phys. , 050 (2018), arXiv:1702.03317.[39] M. Zaldarriaga and U. Seljak, Phys. Rev. D , 123507(1999), astro-ph/9810257.[40] E. Anderes, Phys. Rev. D , 083517 (2013),arXiv:1301.2576 [astro-ph.IM].[41] C. M. Hirata and U. Seljak, Phys. Rev. D (2003),10.1103/PhysRevD.67.043001.[42] J. Carron and A. Lewis, Phys. Rev. D (2017),10.1103/PhysRevD.96.063510.[43] M. Millea, E. Anderes, and B. D. Wandelt, ArXiv e-prints (2017), arXiv:1708.06753.[44] M. Bucher, C. S. Carvalho, K. Moodley, andM. Remazeilles, Phys. Rev. D , 043016 (2012),arXiv:1004.3285 [astro-ph.CO].[45] H. Prince, K. Moodley, J. Ridl, and M. Bucher,J. Cosmology Astropart. Phys. , 034 (2018),arXiv:1709.02227.[46] E. Schaan and S. Ferraro, ArXiv e-prints (2018),arXiv:1804.06403.[47] T. Namikawa, Phys. Rev. D , 121301 (2016),arXiv:1604.08578.[48] G. Marozzi, G. Fanizza, E. Di Dio, and R. Dur-rer, J. Cosmology Astropart. Phys. , 028 (2016), arXiv:1605.08761.[49] J. Liu, J. C. Hill, B. D. Sherwin, A. Petri, V. B¨ohm,and Z. Haiman, Phys. Rev. D , 103501 (2016),arXiv:1608.03169.[50] V. Springel, MNRAS , 1105 (2005), astro-ph/0505010.[51] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. , 473 (2000), arXiv:astro-ph/9911177 [astro-ph].[52] A. Petri, Astronomy and Computing , 73 (2016),arXiv:1606.01903.[53] A. Petri, Z. Haiman, and M. May, Phys. Rev. D ,063524 (2016), arXiv:1601.06792.[54] D. Blas, J. Lesgourgues, and T. Tram, J. CosmologyAstropart. Phys. , 034 (2011), arXiv:1104.2933.[55] R. Takahashi, M. Sato, T. Nishimichi, A. Taruya, andM. Oguri, ApJ , 152 (2012), arXiv:1208.2701.[56] H. Gil-Mar´ın, C. Wagner, F. Fragkoudi, R. Jimenez, andL. Verde, J. Cosmology Astropart. Phys. , 047 (2012),arXiv:1111.4477 [astro-ph.CO].[57] T. Louis, S. Næss, S. Das, J. Dunkley, and B. Sher-win, MNRAS , 2040 (2013), arXiv:1306.6692 [astro-ph.CO].[58] A. Lewis and G. Pratten, J. Cosmology Astropart. Phys. , 003 (2016), arXiv:1608.01263.[59] M. M. Schmittfull, A. Challinor, D. Hanson,and A. Lewis, Phys. Rev. D , 063012 (2013),arXiv:1308.0286 [astro-ph.CO].[60] J. Peloton, M. Schmittfull, A. Lewis, J. Carron,and O. Zahn, Phys. Rev. D , 043508 (2017),arXiv:1611.01446.[61] A. Suzuki, P. A. R. Ade, Y. Akiba, D. Alonso, K. Arnold,J. Aumont, C. Baccigalupi, D. Barron, S. Basak,S. Beckman, J. Borrill, F. Boulanger, M. Bucher,E. Calabrese, Y. Chinone, H. Cho, A. Cukierman,D. W. Curtis, T. de Haan, M. Dobbs, A. Domin-jon, T. Dotani, L. Duband, A. Ducout, J. Dunk-ley, J. M. Duval, T. Elleflot, H. K. Eriksen, J. Er-rard, J. Fischer, T. Fujino, T. Funaki, U. Fuskeland,K. Ganga, N. Goeckner-Wald, J. Grain, N. W. Halver-son, T. Hamada, T. Hasebe, M. Hasegawa, K. Hattori,M. Hattori, L. Hayes, M. Hazumi, N. Hidehira, C. A.Hill, G. Hilton, J. Hubmayr, K. Ichiki, T. Iida, H. Imada,M. Inoue, Y. Inoue, K. D., H. Ishino, O. Jeong, H. Kanai,D. Kaneko, S. Kashima, N. Katayama, T. Kawasaki,S. A. Kernasovskiy, R. Keskitalo, A. Kibayashi, Y. Kida,K. Kimura, T. Kisner, K. Kohri, E. Komatsu, K. Ko-matsu, C. L. Kuo, N. A. Kurinsky, A. Kusaka, A. Lazar-ian, A. T. Lee, D. Li, E. Linder, B. Maffei, A. Mangilli,M. Maki, T. Matsumura, S. Matsuura, D. Meilhan,S. Mima, Y. Minami, K. Mitsuda, L. Montier, M. Na-gai, T. Nagasaki, R. Nagata, M. Nakajima, S. Naka-mura, T. Namikawa, M. Naruse, H. Nishino, T. Nitta,T. Noguchi, H. Ogawa, S. Oguri, N. Okada, A. Okamoto,T. Okamura, C. Otani, G. Patanchon, G. Pisano, G. Re-beiz, M. Remazeilles, P. L. Richards, S. Sakai, Y. Sakurai,Y. Sato, N. Sato, M. Sawada, Y. Segawa, Y. Sekimoto,U. Seljak, B. D. Sherwin, T. Shimizu, K. Shinozaki,R. Stompor, H. Sugai, H. Sugita, J. Suzuki, O. Tajima,S. Takada, R. Takaku, S. Takakura, S. Takatori, D. Tan-abe, E. Taylor, K. L. Thompson, B. Thorne, T. Tomaru,T. Tomida, N. Tomita, M. Tristram, C. Tucker, P. Turin,M. Tsujimoto, S. Uozumi, S. Utsunomiya, Y. Uzawa,F. Vansyngel, I. K. Wehus, B. Westbrook, M. Willer,N. Whitehorn, Y. Yamada, R. Yamamoto, N. Yamasaki, T. Yamashita, and M. Yoshida, ArXiv e-prints (2018),arXiv:1801.06987 [astro-ph.IM].[62] S. Hanany and Inflation Probe Mission Study Team, in
American Astronomical Society Meeting Abstracts ,American Astronomical Society Meeting Abstracts, Vol.231 (2018) p. 121.01.[63] P. M. Merkel and B. M. Sch¨afer, MNRAS , 2431 (2017).[64] A. Stebbins, ArXiv Astrophysics e-prints (1996), astro-ph/9609149.[65] C. M. Hirata and U. Seljak, Phys. Rev. D , 083002(2003), astro-ph/0306354.[66] A. Cooray, M. Kamionkowski, and R. R. Caldwell,Phys. Rev. D71