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CMB and cosmic-infrared maps cross-correlate at 4.8σ, favoring a cosmological constant

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-01 12:07 UTC pith:O7OL4IR7

load-bearing objection A careful, transparent ISW/CIB measurement with a new statistic; the model-independent null is only 2.3σ, so the 4.8σ template result should be read as conditional on the mock calibration. the 3 major comments →

arxiv 2607.19648 v1 pith:O7OL4IR7 submitted 2026-07-22 astro-ph.CO

Dynamical Test of Cosmic Acceleration: k-nearest Neighbor Cross Correlation of Cosmic Microwave Background and Cosmic Infrared Background

Dongkok Kim , Donghui Jeong , Yi-Kuan Chiang , Ho Seong Hwang This is my paper
classification astro-ph.CO
keywords cosmic accelerationintegrated Sachs-Wolfe effectk-nearest neighborcross-correlationcosmic infrared backgroundcosmic microwave backgrounddark energycosmological constant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims to have detected the dynamical imprint of cosmic acceleration by cross-correlating the Planck cosmic microwave background (CMB) temperature map with a reconstructed map of the cosmic infrared background (CIB) at 100 µm. The signal is the integrated Sachs-Wolfe (ISW) and Rees-Sciama (RS) effect, which arises when CMB photons pass through gravitational potentials that change over time as dark energy drives cosmic expansion. Using a k-nearest neighbor cumulative distribution function (kNN-CDF) statistic, the authors find a positive cross-correlation with an amplitude A_kNN = 0.95 ± 0.20 (4.8σ) relative to ΛCDM mock data. This is a new, independent dynamical test that supports the standard model of a cosmological constant and, in a two-bin tomographic split, shows no evidence for evolving dark energy.

Core claim

The central claim is that the 100 µm CIB map, built from ~600 million WISE galaxies, is positively cross-correlated with the COSMIC MICROWAVE BACKGROUND temperature, and that this correlation is quantitatively consistent with the ΛCDM prediction. The null hypothesis of no correlation is rejected at p=0.02 (χ² test), and the best-fit amplitude A_kNN = 0.95 ± 0.20 gives a 4.8σ detection of the ISW/RS effect. The kNN-CDF analysis improves the detection significance by ~20% over the standard two-point angular power spectrum. The authors interpret this as dynamical evidence for the late-time evolution of gravitational potentials, consistent with a cosmological constant as the driver of cosmic acc

What carries the argument

The k-nearest neighbor cumulative distribution function (kNN-CDF) is the central statistic. For two continuous fields (CMB temperature and CIB intensity), it measures the joint probability that pixels exceed specified thresholds, normalized by the product of the individual probabilities. This statistic is sensitive to integrated higher-order clustering information, which captures the cross-correlation signal more efficiently than the two-point function. The method works by counting, for each pixel, the fraction of its k nearest neighbors in the other field above a threshold, then combining thresholds and angular scales (0.1°–0.9°) into a 45-element data vector whose covariance is estimated f

Load-bearing premise

The mock CIB map is calibrated to the real CIB only through its angular power spectrum and pixel intensity distribution, not through higher-order phase information; if the simulation's phase structure differs from reality, the template ψ_model could be biased, shifting the measured amplitude and its significance.

What would settle it

A direct comparison of the measured kNN-CDF statistic with a mock catalog that explicitly matches the CIB's two-point correlation function at all scales, while also matching higher-order (bispectrum or trispectrum) statistics, would test whether the inferred A_kNN is robust. Alternatively, repeating the analysis with a different CIB reconstruction (e.g., from Planck's 545 GHz map or from SPHEREx-like multi-band data) and a different CMB map (e.g., ACT) and checking whether the amplitude remains consistent would settle the matter.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the ISW/RS signal is real, it provides a direct dynamical measurement of the growth of gravitational potentials, consistent with a cosmological constant.
  • The improved significance over two-point analysis suggests that kNN-CDF-style statistics can extract additional non-Gaussian information from CMB–large-scale-structure cross-correlations.
  • The tomographic split into low- and high-redshift CIB bins yields amplitudes consistent with ΛCDM, placing constraints on evolving dark energy, albeit with weaker significance (3.0σ and 2.0σ).
  • The success of this approach motivates using CIB reconstructions and kNN statistics for future surveys like SPHEREx to probe dark energy dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The 4.8σ significance may be optimistic given the substantial correlations between angular bins (up to 0.93) and the reliance on mock data that are calibrated to the observed CIB only through its power spectrum and PDF; higher-order phase information is not explicitly matched.
  • The two tomographic bin amplitudes (1.41 ± 0.47 and 1.66 ± 0.81) are both above unity, hinting at a possible excess, but the large uncertainties and redshift overlap make it premature to interpret this as evidence for evolving dark energy.
  • This result could be tested by applying the same kNN-CDF cross-correlation to independent CMB maps (e.g., from ACT or SPT) and other large-scale structure tracers such as DESI galaxies or the thermal Sunyaev-Zel'dovich effect, which would provide a clean consistency check.
  • The improvement over two-point statistics suggests that kNN-CDF methods could be applied to other delicate cross-correlations, such as CMB lensing–galaxy or 21 cm–galaxy correlation, where higher-order information may help beat down noise.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper measures the cross-correlation between Planck CMB temperature and the reconstructed 100 μm CIB map of Chiang (2023) using the k-nearest neighbor cumulative distribution function (kNN-CDF). It reports a positive correlation: the null hypothesis of no correlation is rejected at p=0.02 (2.3σ) via a chi-square test, and an amplitude fit relative to ΛCDM mock data from the MXXL simulation (including ISW and RS effects) yields A_kNN=0.95±0.20 (4.8σ). The paper compares this with a two-point angular power spectrum analysis (≈4σ), performs extensive systematics tests with different CMB pipelines, dust maps, masks, and outlier cuts, and tomographically splits the CIB into two redshift bins; the results are reported as consistent with ΛCDM. The claimed improvement of kNN over two-point statistics is ~20% in detection significance.

Significance. If the claimed signal is real, the paper provides a new dynamical probe of late-time potential evolution, complementary to geometric dark-energy measurements, and demonstrates the utility of kNN-CDF for ISW-type cross-correlations. The analysis is thorough: it uses a public foreground-cleaned CIB map, constructs mocks from a large N-body simulation, and performs multiple robustness tests (CMB pipelines, masks, outlier cuts, covariance estimators). However, the model-independent null signal is only 2.3σ, and the headline 4.8σ amplitude significance is conditional on the mock template. The central risk is whether the mock CIB/ISWRS template faithfully reproduces the higher-order phase information that kNN-CDF is designed to extract; the paper's own outlier-cut sensitivity and its different amplitudes between Planck and MXXL cosmologies in the two-point analysis underline this concern. The paper is valuable and likely correct in its broad claims, but the headline significance needs stronger validation.

major comments (3)
  1. [§3.2, §4.2, Eq. (11)–(14)] The 4.8σ significance is a matched-filter amplitude A_kNN obtained by projecting the observed data vector onto a template ψ_model built from MXXL mock data. The mock CIB is calibrated to the observed CIB only through its angular power spectrum and pixel PDF (Eq. 3, Figs. 4–5), not through higher-order phase information. Since kNN-CDF is explicitly designed to capture integrated higher-order clustering, the template shape could be biased if the simulation's phase structure differs from the real CIB reconstruction (a linear combination of galaxy templates). The paper itself reports that A_kNN varies by ~10% for a 1% change in the CIB outlier cut (§4.2) and that reconstruction errors are not included in the mocks (§3.3.2). A biased template would not affect the χ²_null=2.3σ result but could over- or under-state the amplitude significance. I ask the authors to validate the template against a
  2. [§2.2, §5.1, Fig. 14] The mock ISWRS map is generated from the MXXL simulation, which assumes Ωm=0.25, σ8=0.9, differing from Planck cosmology. The two-point analysis in §5.1 shows that the best-fit amplitude changes by more than 1σ between Planck and MXXL cosmologies (A_ISW=0.97±0.25 vs 0.63±0.16). The kNN template is only from MXXL; if the true cosmology is closer to Planck, the template shape may be biased. The paper should quantify the cosmology dependence of A_kNN, for instance by constructing ψ_model from linear theory with Planck parameters or by reweighting the mock, and state the expected shift in the amplitude and significance.
  3. [§3.2, §4.1, Eq. (10)] The null-test significance is quoted as χ²_null≈66.6 for 45 degrees of freedom, p=0.02 (2.3σ). However, the covariance matrix is estimated from only 200 realizations, and the 45-element data vector is highly correlated (correlation coefficients up to 0.93 in Fig. 10). The Hartlap factor partially debiases the inverse covariance but does not by itself make the χ² statistic follow a chi-square distribution with 45 dof; the appropriate null distribution is closer to a Hotelling T². The authors should validate the p-value with an empirical null distribution (e.g., from the random realizations) or use a T² statistic. Without this, the model-independent 2.3σ rejection quoted in the abstract may not be precisely calibrated.
minor comments (5)
  1. [§3.1, Eq. (6)] The definition of ψ uses the notation P_{>T*,>I*}, P_{>T*}, P_{>I*}; please define these explicitly in the text or a table, and state that ψ is a function of θ.
  2. [Fig. 1 caption] The caption says the CIB map is 'extracted from the SFD dust map'; this could be misread as a simple subtraction. The reconstruction is based on template galaxy density fields and cross-correlations with SFD; please rephrase for clarity.
  3. [§2.2.1, Eq. (3)] The iterative procedure for finding (b_ℓ, σ_N1, σ_N2) is described qualitatively; please state the convergence criterion and the resulting uncertainties on these fitted quantities.
  4. [§5.1] The phrase 'increasing the independent data volume by (4.8/4.0)^2−1=44%' is loose; the gain in S/N does not necessarily translate directly into an independent data volume. Consider rephrasing as 'equivalent to a 44% increase in effective survey volume under inverse-variance scaling.'
  5. [Eq. (7)] There is a formatting issue: 'atop−hat' should be 'a_{ℓm}^{top-hat}' or similar.

Circularity Check

0 steps flagged

No material circularity: the ISWRS template is a forward N-body prediction, and the observed CIB–CMB cross-correlation is not an input to the mock calibration.

full rationale

The central amplitude A_kNN is measured by projecting the observed kNN-CDF data vector onto a template ψ_model built from MXXL mock CIB and ISWRS maps (Eqs. 11–14). No parameter of the mock is fitted to the observed CIB–CMB cross-correlation. The mock CIB is calibrated only to the observed CIB auto power spectrum and pixel PDF (Eq. 3, Figs. 4–5), and the mock ISWRS map is generated independently from the N-body potential evolution (Eq. 4). Equation 15 is used only as a separate linear-theory comparison, not as a calibration input. The CIB map and its bias-weighted redshift distribution come from Chiang (2023) and Chiang et al. (2025), but these are external observed data products, and the cross-correlation with Planck is not used to construct them. The paper's own caveats—mock data omit reconstruction errors (Sec. 3.3.2) and A_kNN varies by ~10% for a 1% outlier-cut change (Sec. 4.2)—are robustness limitations, not circular reductions. The model-independent χ²_nul = 2.3σ and the template-based 4.8σ are distinct measures, and the paper explicitly labels them as such. No equation in the paper defines the prediction in terms of the fitted quantity or vice versa.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

The analysis introduces no new physical entities, particles, forces, or dimensions. The free parameters are empirical calibration coefficients used to build the mock CIB map and to choose analysis thresholds. The axioms list the background assumptions about the simulation cosmology, the CIB tracer, the sufficiency of one/two-point calibration for the kNN statistic, the standard Poisson potential relation, and the statistical validity of the kNN estimator.

free parameters (6)
  • b dI/dz fitting parameters (α, z0, β; a=-0.03, b=0.1) = Not quoted in text; fitted to clustering-redshift measurements from C23/Chiang+2025
    Equation 1 is fitted to the bias-weighted redshift distribution of the 100 µm CIB, and (a,b) are described as 'arbitrarily adopted' to describe low-z emission (Section 2.1.1, Figure 1). This enters the mock CIB construction and the predicted cross-correlation.
  • Transfer function parameters (A, a, b) in bℓ = A exp(-a ℓ^b) = Not quoted; fitted to the ratio of observed to simulated CIB Cℓ, normalized at ℓ=20
    Equation 3 and the iterative procedure in Section 2.2.1 fit the transfer function to match the observed CIB auto-spectrum. This calibration shapes the mock CIB map used to build the ISW template.
  • White noise amplitudes (σ_N1, σ_N2) = (0, 0.02) MJy/sr
    Section 2.2.1: 'We find that (σ_N1, σ_N2) = (0, 0.02) MJy/sr gives the best fit to the observed Cℓ and PDF.' These noise terms are added to the mock CIB map and affect the kNN statistic.
  • Outlier cuts (CMB cut, CIB cut) = (0, 0.02)
    Section 3.3.2: the fiducial cut is chosen as the S/N-maximizing point in the mock forecast (Figure 8). It is an analysis hyperparameter that changes the measured A_kNN by up to 10% at larger sky fraction.
  • Fiducial sky fraction fsky = ≈0.49
    Section 3.3.1 and 4.2: the mask is chosen based on CIB reconstruction error thresholds, and fsky≈0.49 is adopted because σ_AkNN reaches a minimum and A_kNN variation across outlier cuts drops sharply (Figure 12).
  • Tomographic redshift boundary z_cut = ≈0.7
    Section 5.3: z_cut is chosen from the mock S/N forecast (Figure 16) and the statement that 'the reconstruction error increases with redshift.' It affects the two tomographic maps and their amplitudes.
axioms (5)
  • domain assumption MXXL simulation (H0=73, Ωm=0.25, ΩΛ=0.75, σ8=0.9, ns=1) adequately represents ΛCDM for generating both the ISWRS map and the CIB mock.
    The mock ISWRS map and mock CIB map are built from the MXXL simulation (Section 2.2). The assumed cosmology differs from Planck 2018, and Section 5.1 shows the inferred ISW amplitude depends on this choice.
  • domain assumption The C23 reconstructed 100 µm CIB map is a faithful tracer of large-scale structure, with reconstruction errors small enough after masking not to bias the cross-correlation.
    The entire analysis uses the CIB map as the matter tracer (Section 2.1.1). The paper masks high-error regions but does not model reconstruction errors in the mock; Section 3.3.1 and Appendix C discuss but do not remove this assumption.
  • domain assumption Matching the angular power spectrum and pixel PDF of the mock CIB to observation (Eq. 3) is sufficient to reproduce the higher-order phase information relevant for kNN-CDF cross-correlation.
    The mock CIB is calibrated only to one- and two-point statistics (Section 2.2.1). The kNN statistic is designed to capture integrated higher-order information; the paper validates with top-hat smoothed PDFs (Figure 5) but not with direct higher-order correlation functions.
  • standard math The Poisson equation in Fourier space, Φ(k,t) = (3/2)(H0/k)^2 Ωm,0 δ(k,t)/a, and cubic spline interpolation in scale factor provide an accurate lightcone ISWRS map.
    Section 2.2.2 uses this relation to compute the gravitational potential from MXXL density fields; the resulting ISW map is checked against linear theory (Figure 6).
  • standard math The kNN-CDF statistic ψ (Eq. 6) is an unbiased cross-correlation estimator, and the covariance estimated from 200 realizations is a valid measure of the noise.
    Section 3.1-3.2 defines ψ and the covariance. The estimator is taken from the published kNN literature; the covariance includes CMB and CIB noise but the finite number of realizations introduces a Hartlap correction (≈0.77).

pith-pipeline@v1.3.0-alltime-deepseek · 30032 in / 17253 out tokens · 214513 ms · 2026-08-01T12:07:55.301828+00:00 · methodology

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read the original abstract

The physical origin of cosmic acceleration remains one of the central questions in modern cosmology. The integrated Sachs-Wolfe (ISW) and Rees-Sciama (RS) effects, which arise from the evolution of gravitational potentials, provide a dynamical test of cosmic acceleration. We measure the cross correlation between the Planck temperature map of the cosmic microwave background (CMB) and a foreground-free map of the $100\,\mu\mathrm{m}$ cosmic infrared background (CIB) from Y.-K. Chiang (2023), using the $k$-nearest neighbor cumulative distribution function ($k$NN-CDF). As the CIB map is reconstructed from ${\sim}600$ million Wide-field Infrared Survey Explorer galaxies extending to $z\approx2.5$, the measurement is equivalent to a galaxy-CMB cross correlation. We find evidence for a positive cross correlation, rejecting the null hypothesis of no correlation at $p=0.02~(2.3\sigma)$ using the $\chi^2$ test. We further quantify its amplitude with $A_{k\mathrm{NN}}$, defined relative to an assumed cosmological model, and obtain $A_{k\mathrm{NN}}=0.95\pm0.20~(4.8\sigma)$ with respect to $\Lambda\mathrm{CDM}$ mock data including the ISW and RS effects. The $k$NN-CDF analysis improves the detection significance by ${\sim}20\%$ over the two-point correlation function. We also explore evolving dark energy by constructing two tomographic maps from the full CIB map; the results are consistent with $\Lambda$CDM. Finer tomographic reconstruction of the CIB, for example from multiband data of SPHEREx, would further tighten constraints on evolving dark energy.

Figures

Figures reproduced from arXiv: 2607.19648 by Donghui Jeong, Dongkok Kim, Ho Seong Hwang, Yi-Kuan Chiang.

Figure 1
Figure 1. Figure 1: Left: 100 µm CIB extracted from the SFD dust map in Galactic coordinate. Right: bias-weighted redshift distribution of 100 µm CIB. The black data points denote the measurements using the clustering redshift method of Y.-K. Chiang et al. (2025). The green line shows the smoothed distribution using the analytical function described in Equation 1. The purple line shows the ISW sensitivity function defined as … view at source ↗
Figure 2
Figure 2. Figure 2: Left: mock 100 µm CIB map constructed with MXXL halos. The map is smoothed using the transfer function, and white noise terms are added (see Section 2.2.1 and [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Top: angular power spectrum of 100 µm CIB from C23 (black) and MXXL simulation (blue). The fluc￾tuations are larger on a large scale in MXXL. There is also a shape difference on a small scale due to the absence of a beam function. Power spectra in the linear theory (dashed) are included to demonstrate the impact of assumed cosmol￾ogy. The power spectrum is ∼10%–20% larger with MXXL cosmology. Bottom: ratio… view at source ↗
Figure 4
Figure 4. Figure 4: Left: angular power spectrum of 100 µm CIB. Right: PDF of CIB intensity. The black lines denote the observation and blue lines denote the readjusted mock data. To match both one-point and two-point statistics, white noise terms of (σN1 , σN2 ) = (0, 0.02) MJy sr−1 are added. The ratios between mock and observational data are shown in the bottom panels. The gray shading shows the range of the standard devia… view at source ↗
Figure 5
Figure 5. Figure 5: PDF of top-hat smoothed CIB intensity. Smoothing scales of θ = 0.1, 0.5, 0.9 ◦ are shown from left to right. The black lines denote the measurements from C23 and the shading shows the 2σ range calculated with 200 realizations. The results from the mock data are shown as blue dashed lines. C23 and the mock data agree well within uncertainty, particularly at CDF < 0.99 (gray dotted vertical line). large scal… view at source ↗
Figure 6
Figure 6. Figure 6: Left: mock ISWRS map (z = 0–2.7) constructed from the density fields of MXXL simulation. Right: angular power spectrum of the mock ISWRS map (black) compared with the power spectrum in the linear theory (blue). The mock is consistent with linear theory on a large scale (ℓ ≲ 100). correlate two continuous fields by defining ψT ∗,I∗ (θ) ≡ P>T ∗,>I∗ P>T ∗ × P>I∗ − 1 , (6) where θ is the angular smoothing scal… view at source ↗
Figure 7
Figure 7. Figure 7: Left: CIB reconstruction error Ierror (MJy sr−1 ) in Galactic coordinates. The original error map from C23 is smoothed with an FWHM = 1◦ Gaussian to trace overall uncertainty structure. Middle, right: two examples of masks constructed with the smoothed error map (left panel). In addition to the generic mask, regions with Ierror larger than (0.02, 0.018)MJy sr−1 are masked. 3.3.2. Outlier Removal: Blind Tes… view at source ↗
Figure 8
Figure 8. Figure 8: Expected S/N calculated with the mock data. Each pixel denotes a different combination of outlier cuts (CMB cut, CIB cut). The S/N peaks at (0, 0.02) and de￾creases when any CMB outlier cuts or larger CIB outlier cuts are applied. The variation is dominated by the direc￾tion of the CMB cut. 4. RESULTS In this section, we present our AkNN = 0.95 ± 0.20 (4.8σ) detection of the ISWRS effect using a mask with … view at source ↗
Figure 9
Figure 9. Figure 9: Top: kNN cross-correlation statistic ψ as a function of angular scale θ. Each panel shows a different combination of thresholds (text on the upper left). Measurements from the observational data (black lines) are shown together with the distribution of random vectors (blue lines). The blue shading denotes the standard deviation of the random vectors. Bottom: deviation from the null hypothesis for a single … view at source ↗
Figure 10
Figure 10. Figure 10: Correlation matrix of kNN summary statistic ψ defined as Cij/ p CiiCjj (see Equation 9). Each 5×5 square shows the correlation between two pairs of (T ∗ , I∗ ), and the five bins within that square show the correlation between angular scales. As ψ is defined in real space, angular bins are highly correlated up to 0.93. Different combinations of (T ∗ , I∗ ) are also correlated, as a cumulative distribution… view at source ↗
Figure 11
Figure 11. Figure 11: χ 2 null as a function of fsky. χ 2 null increases up to ≈ 70 when ≈ 6% of the original map is additionally masked. The SMICA map without removal of the SZ effect shows a lower signal overall. applied. In this figure, we also present the result when the original SMICA map (no removal of the SZ effect) is used. The corresponding cross-correlation signal is always lower than the SMICA “SZ-free” map, indicat… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison with the ΛCDM mock data as a function of fsky. Different masks constructed in Section 3.3.1 are used to find a robust condition for the detection. The first three panels from the left show σAkNN , AkNN, and S/N. The distribution of the measured quantities for outlier cuts in [0, 0.05] is shown with thin gray lines (individual case), black lines (median), and gray shading (standard deviation). C… view at source ↗
Figure 13
Figure 13. Figure 13: kNN summary statistic ψ for observation (black markers) and best fit to the ΛCDM mock data (dashed lines). The blue lines and text are individual estimates for one combination of thresholds (T ∗ , I∗ ). The orange lines show the global best fit and the shading shows the 1σ range of the best-fit amplitude. Each combination of thresholds yields AkNN = 0.74–1.12 (3.0σ–4.0σ). The global fit using the full cov… view at source ↗
Figure 14
Figure 14. Figure 14: Cross angular power spectrum between CMB (ISW) and 100 µm CIB. The black data points denote the measurements from the observational data. The dashed lines are calculated using linear theory (orange: Planck, light blue: MXXL cosmology assumed). Multipoles in the range ℓ ∈ [5, 200] are averaged to five bins equally spaced in log space. The error bars show the diagonal terms of the co￾variance matrix. For bo… view at source ↗
Figure 15
Figure 15. Figure 15: kNN cross correlation between the CSFD dust map and the CMB maps, ψCSFD-CMB, as a function of angular scale θ. The text on the upper left indicates the threshold combination. The colored lines show the results obtained with the CMB maps from different pipelines. The gray shading shows the standard deviation of the random vectors, estimated from 1000 random CMB maps. Although the measured signal shows a sl… view at source ↗
Figure 16
Figure 16. Figure 16: S/N forecast of two tomographic bins as a func￾tion of redshift boundary zcut. The mock data are separated assuming a sharp boundary between two tomographic maps. The two maps reach S/N > 3 when zcut ∼ 0.8 is used. sitivity function in [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The intensity map (left) and bias-weighted redshift distribution (right) of the two tomographic maps. The maps are divided at z ∼ 0.7 with Gaussian-like tails. The green line shows the bias-weighted redshift distribution of the full CIB map and the black dashed lines show the best fit to Equation 16. tails. The best-fit amplitude (AkNN) is measured follow￾ing the same procedure presented in Sections 2 and… view at source ↗
Figure 18
Figure 18. Figure 18: Top: mass functions of MXXL lightcone halos (black marker) at four redshifts (z = 0.5, 1.0, 1.5, 2.0). We use the analytical fit (blue solid line) to the functional form of R. K. Sheth & G. Tormen (1999) (S-T in the figure). The gray dashed lines denote the mass corresponding to 50 particles. The best fit is consistent with the simulation at 1011.5 h −1 M⊙ < M200c < 1013.5 h −1 M⊙. The analytical fitting … view at source ↗
Figure 19
Figure 19. Figure 19: Top: angular power spectra of MXXL lightcone halos (gray lines) for three different mass bins (M200c ∼ 1011 , 1012, 1013 h −1 M⊙) and four epochs (z = 0.5, 1.0, 1.5, 2.0). The linear bias is measured using the best fit (dashed lines) to the power spectrum in the linear theory (black solid lines). Middle: linear halo bias as a function of mass. Bottom: ratio between measured bias and analytical fit from J.… view at source ↗
Figure 20
Figure 20. Figure 20: Top: binned error of cross angular power spec￾trum C CMB−CIB ℓ depending on the covariance estimation method. The black line shows the original method used in the main text, namely 200 realizations of CIB cross-cor￾related with random CMB. The blue line denotes the case where only random CMB is considered. The green line is the estimation using 1000 correlated random maps of CIB and (ISW+CMB). The red lin… view at source ↗
Figure 22
Figure 22. Figure 22: kNN statistic ψ measured with two tomographic maps (left: low-z, right: high-z). Black markers denote the measurements from the observational data. Solid lines are the predictions from the MXXL mock data. Dashed lines and shading show the best-fit amplitudes and corresponding 1σ uncertainty. 10 1 10 2 10 3 ` 10 10 10 8 10 6 10 4 10 2 C C M B ¡ C I B ` [ ¹ K M J y s r ¡ 1 ] A lin Planck=1.06§0.59 (1.80¾) A… view at source ↗
Figure 23
Figure 23. Figure 23: Same as [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗

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