Simulating Partial Sky Cosmic Microwave Background Maps with 3D Fast Fourier Transforms

Emory F. Bunn; Mariona Giner Mascarell

REVIEW 4 major objections 5 minor 27 references

Simulating Partial Sky Cosmic Microwave Background Maps with 3D Fast Fourier Transforms

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A 3D power spectrum can be found that reproduces the angular CMB correlation on a partial-sky patch, so maps can be simulated in a box with fast Fourier transforms.

desk verdict Clever 3D-FFT method for partial-sky CMB simulations, but the central equivalence claim rests on adding noise to pass the KS test; worth refereeing with a request to characterize the discrepancy. read the letter →

arxiv 2507.15703 v1 pith:J34UTAXG submitted 2025-07-21 astro-ph.CO

Mariona Giner Mascarell , Emory F. Bunn This is my paper · ORCID

classification astro-ph.CO

keywords cosmicmicrowavebackgroundpartial-skysimulationsfastFouriertransformsphericalharmonicGaussianrandomfieldlinearprogrammingpowerspectrumCMB-S4

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

CMB temperature maps are usually simulated either over the full sky with spherical harmonic transforms or over small flat patches with two-dimensional FFTs. This paper attacks the growing middle ground—patches too large for flat-sky approximations but far smaller than the full sky—by simulating the field in a three-dimensional box whose power spectrum is chosen to reproduce the angular correlation function of the sphere on that patch, then sampling the box onto the sky. The paper shows, for a range of realistic smoothing scales and patch sizes, that such a 3D power spectrum can be found by linear programming, that the resulting maps pass a Kolmogorov-Smirnov test of their covariance after adding only sub-percent-level white noise, and that the FFT time factor can beat the spherical-harmonic transform for these parameters. The payoff is a practical simulation route for upcoming partial-sky experiments like CMB-S4, with a clear path to polarization maps.

What carries the argument

The central object is a solution to a linear programming problem: discretize the unknown 3D power spectrum $P_\infty(k)$ and the target correlation function, impose the inequality constraints $Ax \ge b$ with $x \ge 0$, and minimize $c^T x$ with $c_j = \sum_i A_{ij}$ so that equality is reached when a nonnegative spectrum exists. The resulting spectrum is converted to the box power spectrum via $P_{\rm box} = P_\infty (2\pi/\Delta p)^3 / (M N^2)$, and maps are generated either by a full 3D inverse FFT or, more efficiently, by a hybrid 2D FFT in the transverse directions with correlated Gaussian layers in the $z$ direction built from an eigendecomposition of the layer covariance matrix. Because a zero-mean Gaussian process is completely determined by its two-point function, matching that function is the only requirement for statistical equivalence.

What would settle it

Solve the linear program for a realistic high-resolution spectrum (e.g., a Planck-like $C_l$ with sub-degree smoothing and $\theta_{\rm max}$ around 30 degrees) and check whether the minimal feasible objective exceeds the sum of the correlation constraints; if no nonnegative solution exists, the method cannot produce exact maps in that regime. A second check is whether the noise level needed to pass the Kolmogorov-Smirnov test grows without bound as $N_{\rm side}$ increases.

Watch

Extended reading notes

Core claim

The paper establishes that for realistic angular power spectra, a nonnegative isotropic 3D power spectrum $P_\infty(k)$ exists whose flat-space correlation function equals the sphere's angular correlation function $\xi_{\rm sph}(\theta)$ with chord distance $r = 2\sin(\theta/2)$, over the range of separations inside the simulated patch. Simulating a Gaussian field with that spectrum in a rectangular box and interpolating onto spherical pixels yields maps that are statistically indistinguishable from maps made by spherical harmonic transforms, once a small amount of white noise is added to mask differences in very low-amplitude modes. For several choices of beam width and patch size the paper's FFT time factor is smaller than the $O(N_{\rm pix}^{3/2})$ scaling of the standard spherical-harmonic approach, indicating a speed advantage even before code optimization.

Load-bearing premise

The linear programming step assumes that a nonnegative isotropic 3D power spectrum exists whose correlation function exactly equals the target angular correlation on the patch; the paper checks this for its test cases but gives no proof it holds for every realistic $C_l$, beam width, and patch size.

Editorial extensions

If this is right

Partial-sky CMB simulations can be generated with FFT scaling, giving a practical middle ground between flat-sky and full-sky methods for surveys like CMB-S4.
The 2D Fourier variant with $h$ layers reduces the dominant cost to $h$ two-dimensional FFTs, which the paper's time factors show is smaller than the spherical-harmonic scaling for several tested beam and patch-size combinations.
Because the maps are Gaussian and linear, correct two-point correlations imply full statistical equivalence, so the method inherits the statistical uses of standard spherical-harmonic maps once the covariance matches.
Adding small white noise, already part of realistic simulations, masks the residual differences in very low-amplitude modes and makes the maps completely indistinguishable from the standard method.
The method generalizes in principle to spin-2 polarization maps, the natural next target for CMB experiments.

Reading between the lines

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

The feasibility of the linear program is likely tied to how well an isotropic 3D Gaussian process can mimic chord-space correlations on a curved cap; a rigorous existence criterion would turn the numerical method into a general theorem.
The box method could be adapted to other curved or flat geometries, wherever a correlation function on a subdomain can be embedded in a higher-dimensional stationary field.
The shrinking method's failure suggests that positivity of the embedded power spectrum is the real constraint; alternative basis functions or non-isotropic spectra might recover its speed without extra noise.
A compiled implementation could shift the break-even point, making the method competitive for even larger sky fractions than the table suggests.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a genuine new idea for a real gap: CMB-S4-style surveys are too big for flat-sky FFT and too small for full-sky harmonic transforms, and the paper proposes embedding the observed spherical cap in a 3D box, solving a linear program for an isotropic 3D power spectrum whose correlation function matches the angular correlation function, then FFT-sampling onto the sphere. That is a neat construction, and the two-method implementation (full 3D FFT and a hybrid 2D FFT with a Cholesky/eigendecomposition in the radial direction) is sensible. The pseudo-Cl and visual tests are encouraging, and the appendix on the shrinking method is honest about its failure.

The soft spot is exactly where the stress-test note lands. The paper's own decisive test, the Cholesky-whitened KS test, fails at zero added noise, and the authors have to add 0.36–0.7% white noise to both map sets to pass. That is not a cosmetic detail. It means the box maps' covariance is measurably different from the target xi_sph, and the discrepancy is only masked, not resolved, by adding noise. The paper says this is fine for experimental simulations because detector noise will be added anyway, but a fixed noise floor changes the map variance and can bias parameter estimates if the box method's errors are not understood. The LP step guarantees only that some P_inf exists in infinite space; it does not guarantee that the finite-box sampling and interpolation preserve the correlation, and the zero-noise KS failure is direct evidence that they don't. The paper leaves that discrepancy uncharacterized, which is the main technical weakness.

Two smaller issues: the efficiency claim is based on scaling formulas rather than measured runtimes, and no code is provided. Both are acceptable in a methods paper if labeled as preliminary, but they should be more explicit.

That said, I think the paper is worth taking seriously. The idea is novel, the presentation is clear, and the authors are honest about the limitations. The central claim is not yet established, but it is plausible and testable. The right fix is either to eliminate the zero-noise discrepancy by refining the interpolation or discretization, or to identify which modes are wrong and show that the error vanishes with resolution. I'd send it to a competent referee with that specific request. If the authors can close that gap, this could be a useful tool for Monte Carlo production.

Referee Report

4 major / 5 minor

Summary. The paper proposes a method for simulating partial-sky CMB temperature maps by constructing a three-dimensional box whose isotropic power spectrum is chosen, via linear programming, so that its two-point correlation function matches the angular correlation function ξ_sph over separations up to a chosen r_max. A 3D FFT is then used to generate a realization in the box, which is interpolated onto spherical-cap HEALPix pixels; a variant, the 2D Fourier method, performs FFTs only in the transverse directions and uses an eigendecomposition in the radial direction. The authors test the resulting maps against HEALPix simulations using visual inspection, the pseudo-Cl statistic, and a Cholesky-whitened Kolmogorov-Smirnov test, and they compare computational scaling through analytic time factors. They find that the box method passes the KS test only after adding small amounts of white noise, and they report that for some parameter choices the box time factor is smaller than the HEALPix time factor. An appendix describes a 'shrinking method' variant that is faster but requires larger noise levels and therefore is not recommended.

Significance. If the method could be made to reproduce the target angular correlation function exactly, it would fill a useful niche for experiments with sky coverage between the flat-sky and full-sky regimes. The paper is transparent about its main limitation: the zero-noise KS failure is reported explicitly, the timing comparison is honestly labeled as a scaling estimate rather than an optimized benchmark, and the appendix documents a failed variant. The mathematical construction is clearly explained and the linear-programming setup is novel in this context. However, the central claimed equivalence between box-generated and spherical-harmonic-generated maps is not established by the presented evidence, because the decisive test fails at zero noise and the noise injection used to make the test pass is not derived from the method itself. The paper would be a solid methods contribution if the residual covariance discrepancy were characterized and either removed or shown to be negligible for the intended applications.

major comments (4)

[Sections III.B and IV] The zero-noise KS failure is load-bearing for the central claim. The paper states in Section IV that 'with no noise added to the maps, the box method consistently fails the KS test,' and Table I shows that 0.36-0.7% white noise must be added before the maps pass. Since Section III.B designates the KS test as the 'final arbiter of success,' this failure is direct evidence that the box maps' covariance differs from the target ξ_sph. Adding noise to both box and HEALPix maps can make the two methods indistinguishable while leaving the box maps' covariance incorrect. The manuscript should identify the modes responsible for the discrepancy, quantify the covariance error, and demonstrate that it decreases as Δk, k_max, box size, Nside, and interpolation order are refined, or else restrict the claim to simulations that include at least a specified noise floor.
[Section III.B] The KS test as implemented is not a clean test of the method's covariance. To make the covariance matrix positive definite, the authors add 'varying amounts of noise' to the covariance matrix and a corresponding amount of white noise to the maps before applying the KS test. The noise level is tuned empirically rather than predicted from the algorithm, so the test verifies consistency only after an ad hoc correction. The paper should report the KS statistic as a function of added noise, state how the noise level is chosen, and show that the passing result is not a trivial consequence of the added noise dominating the covariance discrepancy.
[Section II.B and Appendix] The existence of a nonnegative isotropic 3D power spectrum P∞ matching ξ_sph on [0, r_max] is assumed but not proved. The linear programming problem is set up to find such a spectrum, yet the Appendix shows that positivity can fail: the shrinking method produces spectra with negative values and requires more noise. Feasibility may depend on C_l, θ_max, b, and discretization in a way that is not characterized. The paper should either prove feasibility for the class of C_l used in CMB applications, or provide numerical evidence over a wider parameter range, including cases near the boundary where the LP solution first fails.
[Sections II.D and IV] The interpolation from the box grid to the sphere is a crucial step whose error is never quantified. The two-point function of the final pixelized map is affected by the interpolation scheme, the box grid spacing Δp, the box size factors, and the HEALPix resolution Nside, but no convergence test is shown for these choices. The pseudo-Cl and KS tests include interpolation effects, but they are not isolated, and the zero-noise KS failure could originate in interpolation bias rather than in the LP solution. The authors should run the same KS test for at least one configuration while varying Nside and box size, with fixed C_l and θ_max, to show that the discrepancy converges to zero.

minor comments (5)

[Section I] The first sentence of the Introduction contains a typo: 'comic microwave background' should be 'cosmic microwave background.'
[Section IV, Table I] The noise percentage in Table I is not defined. It should state whether the noise is added relative to the map variance, the pixel variance, or the rms of the signal, and it should specify that the same noise is added to the box and HEALPix maps before the KS comparison.
[Section IV, Eq. (4.1)] The time factor comparison is explicitly order-of-magnitude, but the boldface highlighting in Table I may be read as a firm speed advantage. A sentence clarifying that the prefactor uncertainty is at least an order of magnitude would avoid overinterpretation, particularly for cases where the two time factors differ by less than a factor of two.
[Section III.A] The pseudo-Cl test is described as powerful but not a precise statistical test. It would be helpful to state explicitly whether the comparison is of ensemble-averaged pseudo-Cls over the 1000 realizations and what discrepancies would be considered significant.
[Section VI, Eq. (6.1)] The shrinking method appendix is useful, but the parameter s appears without a recommended selection criterion beyond 'small enough that a solution exists'; a brief discussion of how s was chosen for Table II would make the appendix reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 3D power spectrum is explicitly constructed from the angular correlation target, and map-level equivalence is tested against external HEALPix maps rather than assumed.

full rationale

The paper's derivation chain is self-contained and non-circular. The target correlation ξ_sph(θ) is computed directly from the input Cl via Eq. (2.1), and the 3D power spectrum P∞ is then solved for by linear programming so that its flat-space correlation function (2.4) matches ξ_sph(2 arcsin(r/2)) over the constrained interval. This is a construction, not a prediction: any solution found by the LP solver satisfies the quoted constraints by design, and the subsequent box discretization (Eq. 2.11) and simulation steps are explicit transformations of that solution. The claim that the resulting maps have the same statistical properties as HEALPix maps is not taken as a consequence of the LP construction alone; it is checked empirically against externally generated HEALPix maps via the pseudo-Cl comparison and the Cholesky-whitened Kolmogorov-Smirnov test. The whitening test remains statistically valid when the same noise variance is added to the covariance matrix and to the maps, since under the null hypothesis the whitened vector is still standard Gaussian. The added noise levels are reported transparently, not renamed as predictions, and the zero-noise KS failure is presented as a limitation requiring further testing, which is a correctness/validation concern rather than a circularity. There are no load-bearing self-citations, no imported uniqueness theorems, and no ansatz smuggled in via citation. The central derivation reduces only to the paper's own explicit equations and to standard Fourier/statistical identities, so no circular step can be exhibited.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The construction uses standard Fourier and Gaussian-process mathematics. The main load-bearing items are the assumed existence of a nonnegative isotropic P∞ and the periodic-box approximation. The noise level is tuned per test configuration, which is a pragmatic fitting step rather than part of the derivation.

free parameters (6)

Δr = b/5
Correlation function resolution chosen by hand in Section II.B; must be smaller than the smoothing scale, so it is set to one fifth of the beam width.
Δk = 2π/(3rmax)
Power spectrum discretization step chosen in Section II.B; no justification for the factor 3 other than convenience.
Δp = 0.85 Δr
Box grid spacing set in Section II.C to 'make sure we are able to capture fine details'; the 0.85 factor is an ad hoc choice.
box size factor
A 'size variable' mentioned in Section II.C for testing different box sizes; its values are not reported.
noise level for KS test = 0.36-0.7% (Table I), 1.3-5.2% (Table II)
White noise added to covariance matrix and maps so that box maps pass the KS test; tuned per configuration.
s (shrinking method smoothing) = 0.4-15 (Table II)
Constant in Eq. (6.1) chosen 'small enough that a solution exists' for the shrinking variant; ad hoc.

assumptions (5)

standard math A Gaussian random process is completely determined by its mean and two-point correlation function
Invoked in Section III.A to argue matching correlations is sufficient for correctness.
domain assumption The CMB temperature field is a realization of a homogeneous, isotropic Gaussian random process with angular power spectrum Cl
Standard cosmological assumption stated in Section I; the method inherits it.
ad hoc to paper There exists a nonnegative isotropic 3D power spectrum P∞ whose correlation matches ξsph over [0,rmax]
Assumed in Section II.B; the LP solution is found numerically for specific cases but no general feasibility proof is given.
ad hoc to paper The box correlation function equals the infinite-space correlation function for relevant separations
Section II.C uses the Fourier series relation (2.6)-(2.11) for a finite periodic box; alias effects from periodicity are not analyzed.
ad hoc to paper Interpolation from box grid to sphere pixels does not bias the two-point function
Section II.D.1 says the box map must be interpolated but does not specify the scheme or quantify its effect.

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Cite this review

Pith. "Pith review of Simulating Partial Sky Cosmic Microwave Background Maps with 3D Fast Fourier Transforms." pith.science (2026). https://pith.science/paper/J34UTAXG

@misc{pith2026250715703,
  author       = {Pith},
  title        = {Pith review of: Simulating Partial Sky Cosmic Microwave Background Maps with 3D Fast Fourier Transforms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J34UTAXG}},
  note         = {Machine review of arXiv:2507.15703}
}

read the original abstract

Simulated maps of the microwave background (CMB) radiation are generally created using one of two methods: all-sky simulations use the spherical harmonic transform, while maps covering small areas approximate the sky as flat, allowing the use of fast Fourier transforms (FFTs). Current and near-future experiments, particularly ones like CMB S4, will cover areas too large for the flat-sky approximation but significantly less than the full sky. In this regime, it can be more efficient to simulate maps in a 3-D box using FFTs, and then sample onto the observed part of the celestial sphere. We present a method for performing such simulations and show that it can be more efficient than full-sky simulations. We develop the method for scalar maps, but we expect it to be applicable to higher-spin (e.g., polarization) simulations as well.

Figures

Figures reproduced from arXiv: 2507.15703 by the authors.

**Figure 1.** FIG. 1. Flowchart for the code illustrating the chronology of the computational process. The green boxes represent the 3-D [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Two-dimensional sketch of the relative positions of the box and sphere for the 3-D Fourier method (a) and the 2-D [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. The top row shows maps made with the Box Method and the bottom row shows maps made with Healpy. For both, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

In this approach, we create a box of dimensions N × N × M , populate it with simulated values of the 3-D Fourier transform ˜T⃗k, and apply an inverse FFT to simulate a map

The 3-D Fourier Method The first method is the more straightforward algo- rithm. In this approach, we create a box of dimensions N × N × M , populate it with simulated values of the 3-D Fourier transform ˜T⃗k, and apply an inverse FFT to simulate a map. This case is especially fast when the smoothing parameter is big, leading to small box sizes. To be spe...

work page

[2] [2]

pseudo-Cls

The 2-D Fourier Method For some choices of parameters, we may be able to achieve greater efficiency by performing simulations in Fourier space in the x and y directions but in real space in the z direction. The reason for this is that the number h of layers of the box needed for the simulation in the z direction is often quite small ( h ≪ M ). Simulating ...

work page

[3] [3]

shrinking

As a result, these values should be compared in or- der of magnitude rather than precisely. Nonetheless, we highlight in bold the cases where the box time factor is smaller, suggesting that the box method may be faster. We do not present results of a direct timing comparison between the two methods because we have not attempted at this stage to optimize t...

work page

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C. L. Bennett, A. J. Banday, K. M. Gorski, G. Hinshaw, P. Jackson, P. Keegstra, A. Kogut, G. F. Smoot, D. T. Wilkinson, and E. L. Wright, Four-Year COBE DMR Cosmic Microwave Background Observations: Maps and Basic Results, Astrophys. J. Lett. 464, L1 (1996), arXiv:astro-ph/9601067 [astro-ph]

arXiv 1996

[5] [5]

Challinor, Cmb anisotropy science: a review, Pro- ceedings of the International Astronomical Union 8, 42 (2012)

A. Challinor, Cmb anisotropy science: a review, Pro- ceedings of the International Astronomical Union 8, 42 (2012)

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