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Efficient GLS map-making and assessment of $1/f$ noise impact for future CMB satellite missions

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Marta Monelli, Guillaume Patanchon, Avinash Anand

Cosmology and Nongalactic Astrophysics · Submitted 29 Oct 2026 · 40 references

Pith's verdict: A clean, honest methods paper on chunked circulant GLS with a real fix for edge effects, but the statistical claim in Section 3.1 overstates what Eq. 3.6 actually constructs.

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arXiv ID2610.39581 (version 1)
Pith IDIZEFI6HA, Pith's permanent record of this paper and its review
CitationAvinash Anand, Guillaume Patanchon, Marta Monelli. "Efficient GLS map-making and assessment of $1/f$ noise impact for future CMB satellite missions." arXiv:2610.39581 (2026). https://pith.science/paper/IZEFI6HA
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The paper

The paper claims

STITCH preprocessing lets chunked, FFT-based CMB map-makers recover near-optimal noise suppression, with residual polarization bias below r≈2×10⁻⁷.

The reading

The paper introduces STITCH (STatistical Inpainting for Timeline CHunks), a preprocessing step that makes the fast, Fourier-based version of generalized least-squares (GLS) map-making—the optimal method for turning noisy detector timelines into sky maps—behave like the full optimal solver. Its central claim is that the boundary artifacts of chunked processing can be erased before the solve: each chunk is extended with synthetic samples drawn from the assumed stationary noise model so that the extended timeline is consistent with the periodic boundary conditions that the circulant covariance assumes. In the paper's simulations, STITCH-processed chunks recover a $BB$ noise power spectrum consistent with the reference obtained when the noise is genuinely circulant, while injecting a signal bias equivalent to $r=2.0\times10^{-7}$ at low multipoles and $r=5.2\times10^{-9}$ at high multipoles for CMB-only skies. The corrected pipeline is then used to map temporal $1/f$ noise onto angular scales for three representative satellite scanning strategies, finding that the angular knee multipole grows roughly linearly with the temporal knee frequency, that the angular slope is shallower than the temporal slope, and that scan geometry matters beyond scan speed.

What carries the argument

The load-bearing object is the conditional Gaussian extension of a chunk, constructed by maximizing $p(x|n,C)$ under the assumed circulant covariance $C$—a covariance that imposes periodic boundary conditions and is diagonalized by the discrete Fourier transform. The equation $(P_x^T C^{-1}P_x)x=-(P_x^T C^{-1}P_n)n$ carries the method: it forces the synthetic samples $x$ to be the conditional mean of the missing part of the timeline given the observed samples and the periodic noise model, so the concatenated chunk satisfies the boundary conditions the solver implicitly assumes. Because this extension is made before map-making, the circulant solver itself is unchanged and its inverse-covariance application remains two Fourier transforms plus a multiplication in Fourier space. Auxiliary pixels in the pointing matrix absorb the sky-signal contribution of the synthetic samples and are excluded from the output maps; using one auxiliary pixel per chunk is an approximation whose bias the paper finds small.

Extended reading notes

Core claim

The paper's central claim is that the boundary artifacts of chunked circulant GLS map-making are not an unavoidable cost of speed: they can be removed by statistically completing each chunk before the solve. Given a noise-only chunk $n$ of length $n_{\rm samp}$ and an assumed circulant covariance $C$ for the longer timeline, STITCH solves the conditional-maximum problem $(P_x^T C^{-1} P_x)x=-(P_x^T C^{-1} P_n)n$ for a synthetic extension $x$, with $P_n$ and $P_x$ embedding observed and synthetic samples into a timeline of length $n_{\rm tot}=n_{\rm samp}+n_{\rm ext}$. The extension samples are assigned to auxiliary pixels that are excluded from the final maps. The resulting concatenated timeline $[n,x]$ is statistically consistent with $C$, so a standard FFT-based circulant solver sees no discontinuity at the chunk boundary. The quantitative demonstration is that STITCH brings the chunked-map $BB$ noise spectrum into agreement with the reference spectrum obtained when the noise is genuinely circulant, at the cost of $E\to B$ leakage equivalent to $r=2.0\times10^{-7}$ over $2\le\ell\le29$ and $r=5.2\times10^{-9}$ over $30\le\ell\le200$ for CMB-only simulations; with a shallower noise slope $\alpha=1$ the residuals are about two orders of magnitude smaller. The same embedding-operator equation is noted to apply to gaps and flagged samples, although that extension is not tested.

Load-bearing premise

STITCH's success rests on the assumed noise model being the true one: the same power spectrum is used to simulate the noise, to synthesize the extensions, and to weight the map-making, so if $f_{\rm knee}$ or $\alpha$ is misestimated, or the real noise is non-stationary or contains gaps, the boundary residuals will reappear—an assumption the authors state explicitly but do not test.

Editorial extensions

If this is right

  • Chunked circulant GLS map-making with STITCH preprocessing can recover the same noise maps as a reference solve in which the circulant assumption holds, so future missions can keep FFT-speed processing without the usual two-orders-of-magnitude low-$\ell$ noise penalty.
  • Signal bias from the synthetic extension is small enough for B-mode science in the tested configurations: CMB-only residuals are equivalent to $r=2.0\times10^{-7}$ for $2\le\ell\le29$ and $r=5.2\times10^{-9}$ for $30\le\ell\le200$ with $\alpha=2$, and about two orders of magnitude smaller with $\alpha=1$.
  • STITCH-induced $E\to B$ leakage is subdominant to leakage from sub-pixel effects with realistic pointings: large-scale sub-pixel residuals reach about 10% of a primordial $B$-mode signal with $r=10^{-3}$ in the CMB-only case, growing by an order of magnitude when Galactic foregrounds are included.
  • For the three representative scanning strategies, the angular knee multipole $\ell_{\rm knee}$ grows approximately linearly with the temporal knee frequency $f_{\rm knee}$, so noise spectra for different $f_{\rm knee}$ approximately overlap after rescaling the multipole by $f_{\rm knee}/25\,{\rm mHz}$.
  • The fitted angular slope $\beta$ is systematically smaller than the temporal slope $\alpha$, and rescaling by scan speed alone does not align results across strategies, implying that scan geometry also determines how $1/f$ noise projects onto the sky.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As an extension, the same conditional-extension equation should be portable to any FFT-based circulant map-maker, not just the solver used in the paper; a direct port-and-compare test would establish that generality.
  • The embedding-operator form suggests a natural testable generalization: fill internal data gaps and flagged samples with the same statistical inpainting, and benchmark against a full-Toeplitz GLS reference; the paper identifies this direction but does not test it.
  • The angular noise model is empirical, fitted to 25 realizations at only two temporal slopes; a broader grid in $\alpha$ and explicit scan-geometry parameters would test whether $\ell_{\rm knee}\propto f_{\rm knee}$ and $\beta<\alpha$ hold as general mission-design trends or only for the three configurations studied.
  • Because the validation is closed-loop, a deliberately mis-specified noise model would map where STITCH's boundary suppression degrades; that robustness test is the natural next step before applying the method to real data.
Original abstract

Time-correlated $1/f$ noise is a major source of large-scale contamination in measurements of the polarized cosmic microwave background (CMB) that can limit sensitivity to primordial $B$ modes. Generalized least-squares (GLS) map-making is optimal for stationary Gaussian noise when its covariance is known, but practical implementations rely on approximations to make the computation tractable. In particular, approximating the noise covariance as circulant enables efficient map-making using fast Fourier transform, but introduces artificial edge effects when applied to finite data chunks. We introduce STITCH (STatistical Inpainting for Timeline CHunks), a preprocessing method that mitigates these effects by extending time-ordered data chunks with synthetic samples that make the extended timelines statistically consistent with the assumed circulant noise model. They can then be processed with standard circulant GLS solvers, preserving their computational efficiency while reducing residuals associated with chunk boundaries. Simulations show that STITCH substantially suppresses these edge effects, bringing the resulting noise power spectra close to those obtained when the circulant approximation holds. We demonstrate negligible impact of our procedure on the recovered signal, such as $E\to B$ leakage, even at large scales. Finally, we investigate how temporal $1/f$ noise projects onto angular scales for representative LiteBIRD-, \emph{Planck}-, and PICO-like scanning strategies. Using noise simulations spanning a range of temporal noise parameters, we fit an effective model to the resulting angular noise spectra. These results provide an empirical characterization of the temporal-to-angular noise mapping for the configurations studied and a basis for further investigations of correlated noise in future CMB polarization missions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 7,734 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.39 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.67 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Time-correlated $1/f$ noise is a major source of large-scale contamination in measurements of the polarized cosmic microwave background (CMB) that can limit sensitivity to primordial $B$ modes. Generalized least-squares (GLS) map-making is optimal for stationary Gaussian noise when its covariance is known, but practical implementations rely on approximations to make the computation tractable. In particular, approximating the noise covariance as circulant enables efficient map-making using fast Fourier transform, but introduces artificial edge effects when applied to finite data chunks. We introduce STITCH (STatistical Inpainting for Timeline CHunks), a preprocessing method that mitigates these effects by extending time-ordered data chunks with synthetic samples that make the extended timelines statistically consistent with the assumed circulant noise model. They can then be processed with standard circulant GLS solvers, preserving their computational efficiency while reducing residuals associated with chunk boundaries. Simulations show that STITCH substantially suppresses these edge effects, bringing the resulting noise power spectra close to those obtained when the circulant approximation holds. We demonstrate negligible impact of our procedure on the recovered signal, such as $E\to B$ leakage, even at large scales. Finally, we investigate how temporal $1/f$ noise projects onto angular scales for representative LiteBIRD-, \emph{Planck}-, and PICO-like scanning strategies. Using noise simulations spanning a range of temporal noise parameters, we fit an effective model to the resulting angular noise spectra. These results provide an empirical characterization of the temporal-to-angular noise mapping for the configurations studied and a basis for further investigations of correlated noise in future CMB polarization missions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Time-correlated $1/f$ noise is a major source of large-scale contamination in measurements of the polarized cosmic microwave background (CMB) that can limit sensitivity to primordial $B$ modes. background 1.00 0.06 0.03 asserted 0.98
s1 Generalized least-squares (GLS) map-making is optimal for stationary Gaussian noise when its covariance is known, but practical implementations rely on approximations to make the computation tractable. background 0.99 0.05 0.03 asserted 1.00
s2 In particular, approximating the noise covariance as circulant enables efficient map-making using fast Fourier transform, but introduces artificial edge effects when applied to finite data chunks. background 0.93 0.07 0.03 asserted 1.00
s3 We introduce STITCH (STatistical Inpainting for Timeline CHunks), a preprocessing method that mitigates these effects by extending time-ordered data chunks with synthetic samples that make the extended timelines statistically consistent with the assumed circulant noise model. contribution 0.74 0.94 0.20 asserted 1.00
s4 They can then be processed with standard circulant GLS solvers, preserving their computational efficiency while reducing residuals associated with chunk boundaries. method 0.94 0.50 0.12 asserted 0.74
s5 Simulations show that STITCH substantially suppresses these edge effects, bringing the resulting noise power spectra close to those obtained when the circulant approximation holds. result 1.00 0.95 0.24 asserted 1.00
s6 We demonstrate negligible impact of our procedure on the recovered signal, such as $E\to B$ leakage, even at large scales. result 1.00 0.94 0.18 asserted 1.00
s7 Finally, we investigate how temporal $1/f$ noise projects onto angular scales for representative LiteBIRD-, \emph{Planck}-, and PICO-like scanning strategies. contribution 0.76 0.93 0.03 asserted 0.85
s8 Using noise simulations spanning a range of temporal noise parameters, we fit an effective model to the resulting angular noise spectra. method 0.97 0.96 0.04 asserted 0.98
s9 These results provide an empirical characterization of the temporal-to-angular noise mapping for the configurations studied and a basis for further investigations of correlated noise in future CMB polarization missions. interpretation 0.81 0.90 0.23 asserted 0.91

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Noise power spectrum, P(f), as in eq. (2.3), for different choices of the model parameters. In the left panel, we vary fknee while keeping α = 1. In the right panel, we vary α at fixed fknee = 5 mHz. The white-noise power level is normalized to unity. We show the frequency range from the minimum correlation frequency considered in this work (corresponding to ∼ 23 days) to fNyquist ≡ fsamp/2. The vertical dotted line indicates the frequency corresponding to the ∼ 1.2 day chunks. {I, Q, U}). The (i nobs + t) row of A contains only three non-zero elements corresponding to the pixel pit…
Figure 2. Stokes Q maps obtained from a 1/f noise realization which is circulant over ∼ 23 day segments. The differences in the maps depend on the choice of map-making method. The reference solution (left) uses a circulant GLS over the full circulant segments. The center and right panels are instead obtained by processing the same noise realization over ∼ 1.2 day stationary, but not circulant, chunks. The large-scale features visible when using the circulant GLS (center) disappear when the TOD is preprocessed with STITCH (right). For the specifics of the simulation, see section 3.3. 3…
Figure 3. Synthetic extensions for a few differ￾ent choices of the inpainting length, next. 0 5 10 15 duration [min] −0.001 0.000 0.001 0.002 0.003 difference TOD [ µK] binning size 4 8 16
Figure 5. Impact of STITCH on the noise maps. The top row shows the I, Q, and U maps obtained by preprocessing a noise-only TOD with STITCH (fknee = 100 mHz and α = 2) and making maps with a circulant GLS. The bottom row shows the corresponding residual ∆m ≡ mb STITCH − mb ref. Note the different ranges spanned by the color bars. applying the circulant GLS (dashed teal line) suppresses these residuals, yielding a spectrum that closely matches the reference result. The good agreement between the reference and STITCH spectra also shows that neglecting information at temporal scales between ∼…
Figure 6. Top panel: angular BB power spectra obtained from the same 1/f noise realization, which is circulant over ∼ 23 day segments. The different lines were obtained with different map-making approaches. The reference solution (gray) uses a circulant GLS over the fulls circulant segments. The other two lines are instead obtained by processing the noise over ∼ 1.2 day chunks with and without STITCH preprocessing (dashed teal and orange, respectively). The breaking of the circulant assumption introduced by processing the data in chunks leads to an excess of large-scale residual noise,…
Figure 7. Impact of STITCH on the reconstructed CMB. The top row shows the I, Q, and U maps obtained by observing a CMB-only sky, preprocessing the TOD with STITCH (fknee = 100 mHz and α = 2) and making maps with a circulant GLS. The bottom row shows the corresponding residual ∆m ≡ mb STITCH − mb ref. Notice the different ranges spanned by the color bars. 101 102 ` 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6 CBB ` [µ K 2 ] input residual α = 1 residual α = 2
Figure 8. Angular BB power spectra of the residual CMB maps, ∆m ≡ mb STITCH − mb ref, for α = 2 (yellow) and α = 1 (light green). In both cases, fknee = 100 mHz. The dashed lines show the corresponding best-fit curves over the low- and high-multipole ranges. For reference, the angular power spectrum of the input CMB is shown in gray. – 12 –
Figure 9. Intuitive definition of a CMB spacecraft’s scanning parameters. The anti-Sun direction (light gray arrow) forms an angle α with the spin axis which, in turn, forms an angle β with the bore￾sight. The spacecraft precesses while spinning, with angular frequencies ωprec and ωspin, respectively. α β Tprec fspin [rpm] ωmax [ ◦/s] LiteBIRD-like 37.5 ◦ 57.5 ◦ 192.348 min 0.3 1.6 Planck-like 7.5 ◦ 85.0 ◦ 6 months 1.0 6.1 PICO-like 26.0 ◦ 69.0 ◦ 10 hours 1.0 5.6
Figure 10. Hitmaps in Galactic coordinates for the three representative scanning strategies specified in table 1. In all cases, the sky is scanned by two detector pairs (one at boresight and offset of ∼ 50 arcmin along the scanning direction) with fsamp ≃ 17 Hz, for 1 year. then fit these spectra with an effective angular noise model and use the best-fit parameters to derive empirical relations between the temporal and angular descriptions of the noise. These relations provide a practical way to translate detector-level 1/f noise properties into their expected impact on the sky and to compare…
Figure 11. Noise BB angular power spectrum for a LiteBIRD-like scanning strategy. The solid lines show the simulated NbBB ℓ averaged over 25 independent noise realizations for different choices of noise parameters. The corresponding best-fit curves given the model in eq. (4.2) are shown in dotted. By maximizing the likelihood against {p1, p2, . . .}, one can extract the best-fit parameters. In our case, however, for each combination of scanning strategy and set of parameters {σ, fknee, α}, we simulate a set of 25 independent noise angular power spectra, NbBB ℓ,i , spanned by the index i.…
Figure 12. Same average noise BB angular power spectra for a LiteBIRD-like scannning strategy as in figure 11, but plotted against the rescaled multipole ℓ/(fknee/25 mHz). This rescaling accounts for the approximately linear dependence of the angular knee scale on the temporal knee frequency, allowing the spectra with different fknee to be compared on a common effective multipole scale. higher temporal knee frequency extends the correlated-noise contribution to higher temporal frequencies, which are projected onto smaller angular scales by the scan. The fitted values in table 2 show that…
Figure 13. Impact of STITCH on the reconstructed foreground maps. The top row shows the I, Q, and U maps obtained by observing a d0s0 sky, preprocessing the TOD with STITCH (fknee = 100 mHz and α = 2) and making maps with a circulant GLS. The bottom row shows the corresponding residual ∆m ≡ mb STITCH − mb ref. Notice the different ranges spanned by the color bars. 101 102 ` 10−11 10−9 10−7 10−5 10−3 10−1 CBB ` [µ K 2 ] input residual α = 1 residual α = 2
Figure 14. BB angular power spectra of the residual foreground maps, ∆m ≡ mb STITCH − mb ref, for α = 2 (yellow) and α = 1 (light green). In both cases, fknee = 100 mHz. The dashed lines show the corresponding best-fit curves over the low- and high-multipole ranges. For reference, the angular power spectrum of the input sky is shown in gray. – 19 –
Figure 15. Angular BB power spectra quantifying the E → B leakage introduced by STITCH (solid lines) and sub-pixel effects (dashed lines). Green lines are obtained with α = 1, while yellow lines with α = 2. In all cases, we assume CMB, dust and synchrotron in input. For reference, we also plot primordial B modes with r = 10−3 (in gray). Dependence on the number of detectors We repeat the analysis with two additional detector pairs, offset by 100 arcmin along the y axis of the focal plane, approximately orthog￾onal to the scanning direction. In this case, we use a CMB-only sky with no input B…
Figure 16. Dependence on the number of detectors of the residual angular BB power spectra due to E → B leakage introduced by STITCH (solid lines) and sub-pixel effects (dashed lines). Yellow lines were obtained with two detector pairs, while orange lines with four. In all cases, we assume only CMB in input and α = 2. For reference, we also plot primordial B modes with r = 10−3 (in gray). These results indicate that sub-pixel effects can become an important source of bias for experiments targeting extremely faint primordial B modes. Beam asymmetries, which are neglected in this work, can…

Showing 15 of the paper's 16 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.39581 by the authors.

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