CMB Lensing

Gravitational lensing does not create a new last-scattering source. It remaps the temperature and polarization already on the sky. A spatially varying remapping couples Fourier modes that were independent before conditioning on the foreground gravitational field; that off-diagonal covariance is the signal used to reconstruct the projected matter distribution. This page derives that chain, tests it in a deterministic periodic model, and then states exactly how the ACT DR6 survey products turn it into a measurement.

Required background. Probability and Inference for Gravity supplies Probability and Statistical Inference for covariance-weighted estimators and null tests.

Helpful background. The Einstein Equation supplies Einstein Equations and the metric potentials; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects foreground physics; Relativistic Cosmology supplies the primary CMB and growth history; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for ray deflection.

Lensing is a remapping by the Weyl potential

The scalar metric used in this volume is

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2 +(1-2\Psi)d\mathbf x^2\right].$$

In the Born approximation, the lensing potential for a source at comoving distance $\chi_*$ is

$$\phi(\mathbf n)=-\int_0^{\chi_*}d\chi\, \frac{\chi_*-\chi}{\chi_*\chi} [\Phi+\Psi](\chi\mathbf n,\eta_0-\chi).$$

Thus $\boldsymbol\alpha=\nabla_{\mathbf n}\phi$ and $\kappa=-\tfrac12\nabla_{\mathbf n}^2\phi$. The sign is fixed here by the remapping convention

$$\widetilde X(\mathbf n)=X(\mathbf n+\boldsymbol\alpha), \qquad \delta X=\boldsymbol\alpha\cdot\nabla X+O(\alpha^2),$$

where $X$ may be temperature or a parallel-transported polarization field. On the sphere, $\mathbf n+\boldsymbol\alpha$ denotes displacement along the sphere’s geodesic by the tangent deflection, not addition of two unit vectors. In a small flat patch it becomes ordinary angular-coordinate addition. The formula assumes metric null propagation, the Born path, and a single source surface; post-Born corrections and a broad source kernel require a more general treatment. The geometrical remapping and its smoothing of acoustic features are reviewed in Hu and Dodelson 2002, §4.2.4, preprint printed pp. 35–36 (Open PDF) and Scott and Smoot 2026, §29.8, printed pp. 14–15 (official PDF).

Off-diagonal covariance becomes an estimator

Use the flat-sky normalization

$$X(\mathbf n)=\int\frac{d^2\boldsymbol\ell}{(2\pi)^2} X_{\boldsymbol\ell}e^{i\boldsymbol\ell\cdot\mathbf n}, \qquad \langle\phi_{\mathbf L}\phi_{\mathbf L'}\rangle =(2\pi)^2\delta_D(\mathbf L+\mathbf L')C_L^{\phi\phi}.$$

For a scalar field such as temperature, or either Stokes field in a fixed flat-sky screen basis, the first-order expansion is

$$\widetilde X_{\boldsymbol\ell}=X_{\boldsymbol\ell} -\int\frac{d^2\mathbf L}{(2\pi)^2} [\mathbf L\cdot(\boldsymbol\ell-\mathbf L)] \phi_{\mathbf L}X_{\boldsymbol\ell-\mathbf L}+O(\phi^2).$$

The E/B version additionally rotates the spin-2 basis of each Fourier mode. For an unlensed Gaussian primary field taken independent of $\phi$ in this illustrative calculation, averaging at fixed $\phi$ gives $\langle\widetilde X_{\boldsymbol\ell}\widetilde Y_{\boldsymbol\ell'}\rangle_{X,Y}=f^{XY}(\boldsymbol\ell, \boldsymbol\ell')\phi_{\mathbf L}$ for $\mathbf L=\boldsymbol\ell+\boldsymbol\ell'\ne0$. For example,

$$f^{TT}(\boldsymbol\ell,\boldsymbol\ell')= (\mathbf L\cdot\boldsymbol\ell)C_\ell^{TT} +(\mathbf L\cdot\boldsymbol\ell')C_{\ell'}^{TT}.$$

An idealized temperature quadratic estimator is therefore

$$\widehat\phi_{\mathbf L}=A_L \int\frac{d^2\boldsymbol\ell}{(2\pi)^2} \frac{f^{TT}(\boldsymbol\ell,\mathbf L-\boldsymbol\ell)} {2C_\ell^{TT,\mathrm{tot}}C_{|\mathbf L-\boldsymbol\ell|}^{TT,\mathrm{tot}}} \widetilde T_{\boldsymbol\ell} \widetilde T_{\mathbf L-\boldsymbol\ell},$$

with $A_L$ chosen for unit response. The mean field caused by masks and anisotropic noise is subtracted from the reconstructed field. Its estimated power spectrum then requires subtraction of the disconnected $N^{(0)}$ and lensing-dependent $N^{(1)}$ biases. A survey must also calibrate the response with the actual filtering and propagate the covariance. These are distinct operations, not one generic “noise subtraction.” Temperature and polarization pairs can be combined, while profile hardening reduces response to specified foreground profiles. ACT’s implementation is documented in Qu et al. 2024, §§5–6, printed pp. 13–28, especially eqs. (39), (43)–(45) (Open PDF).

Deterministic remapping benchmark

The dependency-free repository benchmark ran under Node.js 24.18.0 on Darwin/arm64 with IEEE-754 binary64 arithmetic and uses the JavaScript standard library only. On $x\in[0,2\pi)$ it samples

$$T(x)=0.8\cos3x+0.35\sin7x+0.2\cos11x$$

on $N=64,128,256,512,1024$ periodic grids and uses wrapped piecewise-linear interpolation. Its discrete transform is $a_n=N^{-1}\sum_jT_j e^{-inx_j}$, so Parseval reads $N^{-1}\sum_j|T_j|^2=\sum_n|a_n|^2$. For the constant shift $\alpha=2\pi/192$, the errors against analytic samples are:

Grid $N$	RMS error	power-ratio error	phase error (rad)	observed order
64	2.267483e-2	-4.237233e-2	1.720877e-2	—
128	5.750656e-3	-1.095396e-2	1.991897e-3	1.9793
256	1.442838e-3	-2.761677e-3	2.444318e-4	1.9948
512	3.610336e-4	-6.918787e-4	3.041464e-5	1.9987
1024	9.027866e-5	-1.730610e-4	3.797499e-6	1.9997

Linear interpolation causes the finite-grid power loss. Direct analytic sampling at $N=1024$ instead gives a power-ratio error $-2.220446\times10^{-16}$ and maximum residual from the exact phase shift $n\alpha$ of $1.304512\times10^{-15}$ rad.

The variable test injects $\alpha(x)=A\sin x$ with $A=0.02$ and fits

$$\widehat A= \frac{\sum_j[\sin x_j\,T'(x_j)][\widetilde T(x_j)-T(x_j)]} {\sum_j[\sin x_j\,T'(x_j)]^2}.$$

Across the same grids, the remap RMS falls from $1.213822\times10^{-2}$ to $7.040707\times10^{-5}$ and $\widehat A$ rises from $0.0182752509$ to $0.0199404327$. With exact sampling and $A=0.02,0.01,0.005,0.0025$, the residual from $\delta T=\alpha T'$ converges at order two, whereas $|\widehat A-A|$ converges at order three. This one-parameter fit diagnoses the small-gradient response; it is not a validation of a masked CMB quadratic estimator.

Download the MIT-licensed standard-library benchmark, then run the exact calculation with the standalone download-and-run recipe:

curl -fL https://blackhole.org/code/cosmology/cosmology-benchmarks.mjs -o cosmology-benchmarks.mjs
node cosmology-benchmarks.mjs --case cmb-lensing

Aliasing an unresolved source mode, omitting periodic wrapping, using an interpolant without a convergence test, or judging only the map mean can all make the toy test pass for the wrong reason.

ACT DR6 product binding and amplitude check

The observational example is the ACT DR6 lensing release, not the later general-purpose DR6.02 map release. The reconstruction used early DR6.01 nighttime 90 and 150 GHz maps from 2017–2021 over $9400\ \mathrm{deg}^2$, with input CMB modes $600\le\ell\le3000$. The minimum-variance baseline combines temperature and polarization with profile hardening. Its pre-unblinding bandpower range is $40\lt L\lt 763$; the post-unblinding extension reaches $L\lt 1300$. Qu et al. 2024, §§2–3, printed pp. 4–9, and §8.2, printed pp. 36–37, PDF gives the selection and amplitude test.

The public products inspected on 2026-08-19 are NASA LAMBDA’s ACT DR6 lensing-map version 1, dated 2023-11-01, and likelihood-data version 1.2, delivered 2024-02-05. The map release supplies convergence harmonics, an $N_{\mathrm{side}}=4096$ mask, multipole filter, noise curves, baseline and frequency/polarization/curl/null variants, and 400 simulation reconstructions through the full NERSC release. The likelihood supplies bandpowers, covariance, binning and fiducial spectra, response and bias correction matrices, and Cobaya code. Map calibration against Planck, beam/window transfer, polarization efficiency and angle, filtering, and the $\ell$ versus $L$ conventions remain part of the model rather than hidden preprocessing.

For a fixed fiducial bandpower vector $t$ and released data vector $d$, the Gaussian amplitude fit is

$$\widehat A=\frac{t^{\mathsf T}C^{-1}d}{t^{\mathsf T}C^{-1}t}, \qquad \sigma_A=(t^{\mathsf T}C^{-1}t)^{-1/2}.$$

For an independent bounded rerun, the archive with SHA-256 5ddc467b3c81836d0dd89349fba4f53fea799c00f4e1f43b8db093b4f31293d7 was read with act_dr6_lenslike 1.2.1 and CAMB 1.3.6. The input Planck best fit was

$$(\Omega_bh^2,\Omega_ch^2,100\theta_{\mathrm{MC}},\tau, \ln(10^{10}A_s),n_s) =(0.022377,0.12010,1.040920,0.0543,3.0447,0.96589).$$

These values are from the Planck 2018 parameter table, entry 2.5, printed p. 13 (PDF). Using the ten baseline bins, the full released covariance, Planck calibration rescaling, response and $N^{(1)}$ corrections, and the 796-simulation Hartlap factor gives

$$\widehat A_{\mathrm{lens}}=1.0122, \qquad \sigma_A=0.02282, \qquad \chi^2=13.823\quad(9\ \mathrm{dof}),$$

whose probability to exceed is $0.1288$. This reproduces, to $0.04\sigma$, the published $A_{\mathrm{lens}}=1.013\pm0.023$ and $13\%$ probability to exceed; the small offset is consistent with generating the theory curve from the rounded published cosmological parameters. The extended result is $1.016\pm0.023$. These are four-point reconstruction amplitudes, not the two-point peak-smoothing parameter also called $A_{\mathrm{lens}}$. Qu et al. 2024, §8.2, printed pp. 36–37, eqs. (46)–(48), PDF states the target and likelihood.

The simulation and null checks are indispensable. Frequency differences, temperature-only versus polarization-only reconstructions, sky and time splits, a curl estimator, and CIB-deprojected variants test distinct failure modes. Response corrections, realization-dependent $N^{(0)}$, $N^{(1)}$, mask mean fields, foreground trispectra, anisotropic noise, beam and leakage errors, and simulation mismatch must be varied separately. A stable amplitude without these tests is not evidence for an unbiased reconstruction.

Exercises

Show that an exactly sampled constant remapping on a periodic domain preserves total Fourier power and advances the phase of mode $n$ by $n\alpha$.

Solution

If $T(x)=\sum_na_ne^{inx}$, then $\widetilde T(x)=T(x+\alpha)=\sum_na_ne^{in\alpha}e^{inx}$. Therefore $\widetilde a_n=e^{in\alpha}a_n$: its phase advances by $n\alpha$ and its magnitude is unchanged. Parseval then gives $\langle|\widetilde T|^2\rangle=\sum_n|\widetilde a_n|^2 =\sum_n|a_n|^2=\langle|T|^2\rangle$. A finite-grid interpolant need not obey this identity, which is why power and phase are checked separately.

References

• ACT Collaboration, “ACT DR6 Lensing Maps,” version 1 (2023), NASA LAMBDA data release.
• ACT Collaboration, “Likelihood Data for DR6 Lensing,” version 1.2 (2024), NASA LAMBDA data release, likelihood code.
• Hu, W., and S. Dodelson, “Cosmic Microwave Background Anisotropies,” Annual Review of Astronomy and Astrophysics 40, 171–216 (2002), DOI, Open PDF.
• Planck Collaboration, “Planck 2018 Results: Cosmological Parameter Tables,” 68% table, version 2 (14 May 2019), Official PDF.
• Qu, F. J., et al. (ACT Collaboration), “The Atacama Cosmology Telescope: A Measurement of the DR6 CMB Lensing Power Spectrum and Its Implications for Structure Growth,” The Astrophysical Journal 962, 112 (2024), DOI, Open PDF.
• Scott, D., and G. F. Smoot, “Cosmic Microwave Background,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), Official PDF.