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CMB Spectral Distortion Anisotropies from Acoustic Damping with primordial non-Gaussianity

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Pith's one-line read This paper computes CMB spectral-distortion anisotropies from acoustic damping with local-type primordial non-Gaussianity, and finds that for enhanced small-scale power the propagation of the average distortion adds a signal comparable to…

desk verdict Solid new derivation of the distortion source with full mode-coupling geometry; the enhanced small-scale power claim is an explicitly partial estimate, so read that part as illustrative. read the letter →

arxiv 2608.09457 v1 pith:R6DUHZM3 submitted 2026-08-10 astro-ph.CO

Jens Chluba , Atsuhisa Ota , Nicola Bartolo This is my paper · ORCID
classification astro-ph.CO
keywords CMBspectraldistortionsprimordialnon-Gaussianityacousticdampingdistortionanisotropiesfrequencyhierarchylocal-typesmall-scalepowerspectrumBoltzmanntransferfunctions
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

This paper aims to put predictions for CMB spectral-distortion anisotropies from acoustic damping on a common footing with the full Boltzmann transport of the distorted photon spectrum. It derives the second-order blackbody-mixing source explicitly, including velocity-dependent scattering, and computes the $\mu T$, $\mu E$, $y T$, and $y E$ cross-power spectra for local-type primordial non-Gaussianity using the frequency-hierarchy treatment. For a nearly scale-invariant primordial spectrum, the results agree broadly with earlier estimates on large angular scales but show additional small-scale damping and modified $y$-distortion correlations. For enhanced small-scale curvature power, the paper finds that anisotropies generated as the average distorted spectrum propagates through the perturbed universe become comparable to those from anisotropic dissipation, with different angular dependences and scalings ($\propto A_p$ versus $\propto A_p f_{NL}$) that in principle allow the small-scale amplitude $A_p$ and $f_{NL}$ to be constrained separately.

What carries the argument

The carrier of the calculation is the frequency-hierarchy treatment: the spectral dimension is expanded in frequency-space basis functions derived from thermalization, and the coefficients are promoted to angular multipoles, so Comptonization, double Compton and bremsstrahlung, Thomson scattering, free streaming and photon diffusion are followed in one system. On the source side, the paper builds effective anisotropic heating rates $s_{\ell 0}(\eta,k)$ from first-order photon and baryon transfer functions together with the primordial bispectrum; the mode-coupling kernel $\hat{M}_\ell$ encodes the full triangle geometry. On the transfer side, the distortion hierarchy has no restoring force: in tight coupling the monopole and dipole of a distortion do not exchange energy, so sourced modes are overdamped and then damp with a modified diffusion scale rather than oscillating. This combination is what lets the paper separate the direct anisotropic-dissipation signal from the propagation signal.

What would settle it

Evaluate the full third-order contribution to the $\mu T$ correlation in the enhanced small-scale model; if the omitted terms alter the angular pattern or the $\propto A_p$ scaling beyond the quoted uncertainty, the claimed parameter separation fails. A separate check is the large-scale $y T$ amplitude, where the new transport treatment differs visibly from older estimates and multifrequency CMB data could discriminate the two.

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Extended reading notes

Core claim

The paper's central claim is that treating spectral evolution and photon transport together changes the predicted distortion cross-spectra, and that in enhanced small-scale power scenarios it creates a separate, previously missed signal. The source is written compactly as $S^{\rm mix}=\Theta_g[\Theta_g-\Theta_{g,0}-\frac{1}{10}\Theta_{g,2}]-\frac{1}{2}(\Theta_g^2-[\Theta_g^2]_0-\frac{1}{10}[\Theta_g^2]_2)$ with $\Theta_g=\Theta+\Psi-V$, so that all blackbody-mixing terms are determined by one gauge-independent temperature variable. In Fourier space the source depends on the full triangle formed by the observable mode and the two dissipating modes, $s_{\ell 0}(\eta,k)\propto\int dk_1\,dk_2\,B(k,k_1,k_2)\, \hat{M}_\ell(\eta,k_1,k_2,\chi_1,\chi_2)$, a geometry that earlier treatments dropped by taking the dissipating modes exactly back-to-back. Distortion perturbations are overdamped rather than acoustic; their damping scale is $\sqrt{8/15}\,k_D$, which explains the faster small-scale falloff of $\mu T$ and $y T$. In the enhanced-power model, the propagation template scales as $\propto A_p$ and the anisotropic-dissipation template as $\propto A_p f_{NL}$, so the two parameters are not locked into a single product.

Load-bearing premise

The load-bearing premise is that the propagation response, computed as a partial third-order treatment, captures the leading-order effect, so that if omitted third-order terms are not small the claimed separate constraints on $A_p$ and $f_{NL}$ would be weakened.

Editorial extensions

If this is right

  • For a nearly scale-invariant primordial spectrum, the refined $\mu T$ and $\mu E$ spectra closely follow previous estimates at $\ell\lesssim 400$ but then damp faster, so small-scale anisotropy measurements need the full transport treatment.
  • The $yT$ and $yE$ spectra are more sensitive to the source evolution; tight-coupling and monopole-only approximations are adequate for $\mu$-distortion spectra but not for precision $y$-distortion predictions.
  • If the small-scale power spectrum is enhanced, the total $\mu T$ signal is a sum of an $\propto A_p$ propagation template and an $\propto A_p f_{NL}$ anisotropic-dissipation template, whose different $\ell$-dependences can break the degeneracy between $A_p$ and $f_{NL}$.
  • Distortion anisotropies are overdamped rather than acoustic, giving an extra damping factor that cannot be reproduced by attaching ordinary temperature transfer functions to a local heating modulation.

Reading between the lines

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

  • Because the propagation and anisotropic-dissipation templates also have different redshift windows, the same decomposition could be used to constrain the scale dependence of the small-scale power, for example by shifting $k_p$ in the model; this is an extension the paper leaves to future forecasts.
  • The triangle-angle dependence of $\hat{M}_\ell$ should matter more for non-local bispectrum shapes, so the full geometry may help distinguish local from equilateral or folded non-Gaussianity.
  • A natural next test is to compute the $\mu\mu$, $yy$ and $y\mu$ auto- and cross-spectra for the same models; the paper notes these are subleading, but they carry independent spectro-spatial information if measured.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a calculation of CMB spectral distortion anisotropies (μT, μE, yT, yE cross-power spectra) arising from acoustic damping in the presence of local-type primordial non-Gaussianity. The authors derive the second-order blackbody-mixing source from the Liouville and Thomson collision terms in a gauge-invariant form, project it onto angular multipoles in Fourier space, and insert the resulting effective anisotropic heating rates into the frequency-hierarchy (FH) implementation in CosmoTherm. For a nearly scale-invariant primordial spectrum they compare with earlier work, identify improved small-scale damping and y-distortion behaviour, and test tight-coupling and monopole-source approximations. For an illustrative enhanced small-scale power model they separate the signal into a propagation template, scaling as A_p, and an anisotropic-dissipation template, scaling as A_p f_NL, and argue that the different angular dependences allow A_p and f_NL to be constrained separately.

Significance. If the results hold, this is the most complete treatment to date of spectral distortion anisotropies from acoustic damping: it includes the full mode-coupling triangle geometry, frequency-dependent transport, and continuous spectral evolution through the FH method, and it is implemented in a public code (CosmoTherm). The standard-ΛCDM predictions for μT/μE/yT/yE are plausible and the comparison with previous calculations is useful. The paper's most important new phenomenological claim, however, concerns the enhanced small-scale power scenario, where the separation of A_p and f_NL rests on a partial third-order propagation calculation that the authors themselves label an estimate. Because that scenario is explicitly presented as a path to constraining A_p and f_NL separately, the incompleteness of the third-order treatment is load-bearing rather than cosmetic.

major comments (2)
  1. [§5.2 and introductory note (p.3)] The propagation response of the average distortion is computed by inserting the isotropic average distortion into the linear FH equations, which is formally an O(R^3) subset of the full third-order problem. The paper explicitly states that it 'do[es] not attempt a complete calculation at this order' and that the results 'should be regarded as an estimate.' A complete O(R^3) treatment would also include direct third-order Liouville and Thomson sources and primordial third-order correlators with different transfer functions and statistical properties. The abstract and Section 5.2 nevertheless use this estimate to claim that propagation anisotropies can be comparable to anisotropic dissipation and that A_p and f_NL can be constrained separately (Figs. 8 and 9). This claim is unsupported unless the authors either supply a quantitative bound or estimate of the omitted O(R^3) terms, or explicitly reframe the enhanced-small-scale-power results as an illustration rather than a prediction. Without such a bound, an omitted term with comparable amplitude and similar multipole dependence would mix the two templates and degrade the claimed parameter separation.
  2. [§5.2, Fig. 9] The statement that the shown Model I signals 'should be within reach of Planck, ACT and SPT' is not supported by any noise, foreground, or Fisher-forecast calculation in the manuscript. The plotted power spectra are signal-only predictions; detectability of the A_p versus f_NL separation depends on covariance, cosmic variance, foreground contamination, and the interplay with the average-distortion constraint. Since the central new phenomenological message is that the two templates can be distinguished, a quantitative forecast (or at least a clear statement that this is only an order-of-magnitude expectation) is needed before the claim can be assessed.
minor comments (4)
  1. [Fig. 4 caption] The caption shows 'R2017 x 5/6' and 'GK2012 x 1/8' but does not explain the meaning of these scaling factors; please state in the caption or text whether they are normalization corrections applied to the previous results and why they are included.
  2. [Eq. (3.4) and footnote 6] The inequality in Eq. (3.4) is labelled with a footnote saying that a factor of order unity to a few should be inserted; as written, the expression is therefore not a strict bound. Please replace the inequality by an explicit estimate with a stated uncertainty range.
  3. [§4.1] The transfer-function database uses ℓmax=10 and neglects polarization terms in the source term, with a statement that this affects heating rates around recombination by 5–10%. Since the paper emphasizes that y-distortion signals are sensitive to source details, please briefly state whether this 5–10% uncertainty propagates into the final yT/yE spectra and whether it is included in the claimed accuracy of the comparisons.
  4. [Introduction, note before §5.2] The sentence 'As a note in preparation of Sect. 5.2, where we consider an simple model...' contains a grammatical error ('an simple') and the note interrupts the introduction; consider moving this caveat to Section 5.2 itself.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the source terms and transfer functions are derived from first principles and the model parameters are inputs, not fitted outputs.

full rationale

The derivation chain is self-contained and forward. The blackbody-mixing source is obtained from the second-order Liouville and Thomson collision terms [Eqs. (2.3)�(2.8)] using first-order temperature and velocity transfer functions; it is not defined in terms of the final power spectra. The effective anisotropic heating rates [Eqs. (4.2), (4.11)] are built from the primordial bispectrum and first-order transfer functions, and the cross-power spectra are then computed by inserting these sources into the linear frequency-hierarchy system [Eqs. (5.2), (5.3)] implemented in CosmoTherm. No parameter is fitted to the target μT/E or yT/E spectra; A_s, n_s, f_NL, and the illustrative Model I parameters are inputs. The self-citations to the FH papers [24,25,27,41] and to [22,23] are references to an independently developed numerical method and to previously derived analytic damping expectations; they do not smuggle in the present results, and the numerical computation confirms rather than presupposes the cited behavior. In Sec. 5.2, the propagation template is the response of the linear transport equations to the computed average distortion; its overall ∝A_p scaling is by construction, but the angular shapes of the resulting C_l are genuine predictions. The paper explicitly flags that this propagation term is only a partial third-order estimate and that a complete O(R^3) treatment is not attempted. That is a completeness/robustness limitation that could affect the claimed A_p versus f_NL separation if omitted terms are large, but it is not circularity: the omitted terms are not equal by construction to the computed templates. Overall, no step reduces to its own input or to a fitted parameter renamed as a prediction.

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

The derivation has no fitted free parameters. The parameters listed are inputs to the illustrative Model I scenario and to the fiducial f_NL values. The key assumptions are the FH closure, the ℓ≤2 source truncation, the neglect of polarization in the source, and the partial third-order treatment of propagation effects. No new physical entities are introduced.

free parameters (5)
  • Model I amplitude A_p = 4e-7
    Chosen by hand for the illustrative enhanced small-scale power model (Sect. 5.2, Fig. 7); not fitted to data.
  • Model I pivot scale k_p = 100 Mpc^-1
    Chosen for the illustrative Model I power spectrum; sets the scale of enhanced power.
  • Model I spectral index n_p = 3
    Chosen for Model I; steeply rising small-scale power.
  • Model I cutoff k_c = 1000 Mpc^-1
    Exponential cutoff in Model I power spectrum; chosen for illustration.
  • f_NL = 1000 (standard case), 1 and 10 (Model I)
    Input parameter for local-type primordial non-Gaussianity; varied to illustrate the signals, not fitted to data.
assumptions (6)
  • domain assumption No extra source of distortions other than the mixing of blackbodies with different temperatures is present.
    Section 2 states this assumption explicitly; it excludes energy injection from other processes that would alter the distortion signals.
  • domain assumption The frequency hierarchy (FH) basis functions provide an adequate closure for the coupled spectro-spatial evolution of distortions.
    The numerical pipeline relies on the FH treatment of CosmoTherm as developed in refs. [24,25,27,41] by the same authors; the paper does not independently verify convergence of the basis truncation for these specific observables.
  • domain assumption Anisotropic distortion sources with ℓ>2 can be neglected.
    Section 2.3.3 gives source functions only up to ℓ=2 and argues from the tight-coupling suppression that higher multipoles are subdominant; this truncation is used for the spectra.
  • ad hoc to paper The propagation response of the average distortion through the perturbed medium is a partial third-order estimate that captures leading-order effects.
    The introductory note and Sect. 5.2 state that this is not a complete calculation at O(R^3) and that the results 'should be regarded as an estimate'; the separate parametrization ∝ A_p and ∝ A_p f_NL depends on this assumption.
  • domain assumption Polarization contributions to the distortion source can be neglected.
    Section 4.1 states polarization terms are neglected in the source term, affecting heating rates around recombination by 5-10%.
  • domain assumption Tight-coupling analytic transfer functions are valid for k>100 Mpc^-1.
    Section 4.1 extends the transfer-function database using tight-coupling approximations for k>100 Mpc^-1, with validation only down to z~5000-10^4; this covers the µ-era sourcing.

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Pith. "Pith review of CMB Spectral Distortion Anisotropies from Acoustic Damping with primordial non-Gaussianity." pith.science (2026). https://pith.science/paper/R6DUHZM3

@misc{pith2026260809457,
  author       = {Pith},
  title        = {Pith review of: CMB Spectral Distortion Anisotropies from Acoustic Damping with primordial non-Gaussianity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6DUHZM3}},
  note         = {Machine review of arXiv:2608.09457}
}
abstract

In this paper, we evaluate the precise distortion source and transfer functions caused by mixing of blackbodies of different temperatures using the recently developed frequency hierarchy (FH) treatment of CosmoTherm. With this we are able to evaluate the effects of primordial non-Gaussianity (PNG) on the $\mu T$, $\mu E$, $y T$ and $y E$ cross-power spectra including the coupled spectro-spatial evolution and important photon-transport effects retained by the FH treatment. For local-type PNG, we compare our results with those from previous works, illustrating new aspects that were previously not captured. We then demonstrate how the $\mu T$ and $\mu E$ signals change in the presence of enhanced curvature perturbations at small scales. For a nearly scale-invariant primordial spectrum, our results agree broadly with previous estimates on large angular scales but exhibit additional small-scale damping and modified $y$-distortion correlations arising from the scale dependence of the heating source and the distinct transport of distortion perturbations. Tight-coupling and monopole-source approximations accurately reproduce the $\mu$-distortion spectra, while the $y$-distortion signals remain more sensitive to the detailed source evolution. For enhanced small-scale power, anisotropies generated by propagation of the distorted average spectrum can become comparable to those from anisotropic dissipation. Their different angular dependences and parameter scalings principally allow the small-scale power amplitude and primordial non-Gaussianity to be constrained separately. The results presented here thus pave the path for studying PNG in new regimes using existing and upcoming high precision CMB anisotropy data to measure primordial distortion correlations.

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Reference graph

Works this paper leans on

71 extracted references · 68 canonical work pages

  1. [1]

    Akrami et al

    Y . Akrami et al. Planck 2018 results. IX. Constraints on primordial non-Gaussianity.Astron. Astrophys., 641:A9, 2020

    work page 2018

  2. [2]

    Bartolo, S

    N. Bartolo, S. Matarrese, and A. Riotto. Cosmic microwave background anisotropies at second order: I. Journal of Cosmology and Astroparticle Physics, 6:24, June 2006

    work page 2006

  3. [3]

    Cosmic Microwave Background Anisotropies up to Second Order

    N. Bartolo, S. Matarrese, and A. Riotto. Cosmic Microwave Background Anisotropies up to Second Order.astro-ph/0703496, March 2007

    work page Pith review arXiv 2007

  4. [4]

    Primordial trispectra and CMB spectral distortions.Journal of Cosmology and Astroparticle Physics, 2016(3):029, March 2016

    Nicola Bartolo, Michele Liguori, and Maresuke Shiraishi. Primordial trispectra and CMB spectral distortions.Journal of Cosmology and Astroparticle Physics, 2016(3):029, March 2016

    work page 2016

  5. [5]

    Cambridge University Press, 2022

    Daniel Baumann.Cosmology. Cambridge University Press, 2022

    2022

  6. [6]

    Biagetti, H

    M. Biagetti, H. Perrier, A. Riotto, and V . Desjacques. Testing the running of non-Gaussianity through the CMBµ-distortion and the halo bias.Physical Review D, 87(6):063521, March 2013

    work page 2013

  7. [7]

    Bianchini and G

    F. Bianchini and G. Fabbian. CMB spectral distortions revisited: A new take onµdistortions and primordial non-Gaussianities from FIRAS data.Physical Review D, 106(6):063527, September 2022

    work page 2022

  8. [8]

    Burigana, L

    C. Burigana, L. Danese, and G. de Zotti. Formation and evolution of early distortions of the microwave background spectrum - A numerical study.Astronomy&Astrophysics, 246:49–58, June 1991

    work page 1991

Show all 71 references
  1. [9]

    Giovanni Cabass, Alessandro Melchiorri, and Enrico Pajer.µdistortions or running: A guaranteed discovery from CMB spectrometry.Physical Review D, 93(8):083515, April 2016. – 29 –

    2016

  2. [10]

    B. J. Carr, K. Kohri, Y . Sendouda, and J. Yokoyama. New cosmological constraints on primordial black holes.Physical Review D, 81(10):104019–+, May 2010

    2010

  3. [11]

    J. Chluba. Green’s function of the cosmological thermalization problem.Monthly Notices of the Royal Astronomical Society, 434:352–357, September 2013

    2013

  4. [12]

    J. Chluba. Which spectral distortions doesΛCDM actually predict?Monthly Notices of the Royal Astronomical Society, 460:227–239, July 2016

    2016

  5. [13]

    Chluba, L

    J. Chluba, L. Dai, D. Grin, M. A. Amin, and M. Kamionkowski. Spectral distortions from the dissipation of tensor perturbations.Monthly Notices of the Royal Astronomical Society, 446:2871–2886, January 2015

    2015

  6. [14]

    Chluba, A

    J. Chluba, A. L. Erickcek, and I. Ben-Dayan. Probing the Inflaton: Small-scale Power Spectrum Constraints from Measurements of the Cosmic Microwave Background Energy Spectrum.The Astrophysical Journal, 758:76, October 2012

    2012

  7. [15]

    Chluba and D

    J. Chluba and D. Grin. CMB spectral distortions from small-scale isocurvature fluctuations.Monthly Notices of the Royal Astronomical Society, 434:1619–1635, September 2013

    2013

  8. [16]

    Chluba and D

    J. Chluba and D. Jeong. Teasing bits of information out of the CMB energy spectrum.Monthly Notices of the Royal Astronomical Society, 438:2065–2082, March 2014

    2014

  9. [17]

    Chluba, R

    J. Chluba, R. Khatri, and R. A. Sunyaev. CMB at 2×2 order: the dissipation of primordial acoustic waves and the observable part of the associated energy release.Monthly Notices of the Royal Astronomical Society, 425:1129–1169, September 2012

    2012

  10. [18]

    Chluba and R

    J. Chluba and R. A. Sunyaev. Superposition of blackbodies and the dipole anisotropy: A possibility to calibrate CMB experiments.Astronomy&Astrophysics, 424:389–408, September 2004

    2004

  11. [19]

    Chluba and R

    J. Chluba and R. A. Sunyaev. The evolution of CMB spectral distortions in the early Universe.Monthly Notices of the Royal Astronomical Society, 419:1294–1314, January 2012

    2012

  12. [20]

    Chluba and R

    J. Chluba and R. M. Thomas. Towards a complete treatment of the cosmological recombination problem.Monthly Notices of the Royal Astronomical Society, 412:748–764, April 2011

    2011

  13. [21]

    Chluba, G

    J. Chluba, G. M. Vasil, and L. J. Dursi. Recombinations to the Rydberg states of hydrogen and their effect during the cosmological recombination epoch.Monthly Notices of the Royal Astronomical Society, 407:599–612, September 2010

    2010

  14. [22]

    Amin, and Marc Kamionkowski

    Jens Chluba, Emanuela Dimastrogiovanni, Mustafa A. Amin, and Marc Kamionkowski. Evolution of CMB spectral distortion anisotropies and tests of primordial non-Gaussianity.Monthly Notices of the Royal Astronomical Society, 466(2):2390–2401, April 2017

    2017

  15. [23]

    Analytic approximations for the distortion transfer functions.in prep., August 2026

    Jens Chluba, Sara Evangelista, and Bryce Cyr. Analytic approximations for the distortion transfer functions.in prep., August 2026

    2026

  16. [24]

    Improved frequency hierarchy treatment for anisotropic spectral distortions.arXiv e-prints, page arXiv:2602.14963, February 2026

    Jens Chluba, Sara Evangelista, Tom Daman, and GeoffVasil. Improved frequency hierarchy treatment for anisotropic spectral distortions.arXiv e-prints, page arXiv:2602.14963, February 2026

    2026 arXiv

  17. [25]

    Spectro-spatial evolution of the CMB

    Jens Chluba, Thomas Kite, and Andrea Ravenni. Spectro-spatial evolution of the CMB. Part I. Discretisation of the thermalisation Green’s function.Journal of Cosmology and Astroparticle Physics, 2023(11):026, November 2023

    2023

  18. [26]

    The boost operator: properties, computation, and applications

    Jens Chluba and Andrea Ravenni. The boost operator: properties, computation, and applications. Monthly Notices of the Royal Astronomical Society, 548(3):stag698, May 2026

    2026

  19. [27]

    Spectro-spatial evolution of the CMB

    Jens Chluba, Andrea Ravenni, and Thomas Kite. Spectro-spatial evolution of the CMB. Part II. Generalised Boltzmann hierarchy.Journal of Cosmology and Astroparticle Physics, 2023(11):027, November 2023

    2023

  20. [28]

    Clesse, B

    S. Clesse, B. Garbrecht, and Y . Zhu. Testing inflation and curvaton scenarios with CMB distortions. Journal of Cosmology and Astroparticle Physics, 10:046, October 2014. – 30 –

    2014

  21. [29]

    R. A. Daly. Spectral distortions of the microwave background radiation resulting from the damping of pressure waves.The Astrophysical Journal, 371:14–28, April 1991

    1991

  22. [30]

    Correlating CMB spectral distortions with temperature: what do we learn on inflation?Journal of Cosmology and Astroparticle Physics, 2016(12):015, December 2016

    Emanuela Dimastrogiovanni and Razieh Emami. Correlating CMB spectral distortions with temperature: what do we learn on inflation?Journal of Cosmology and Astroparticle Physics, 2016(12):015, December 2016

    2016

  23. [31]

    Probing the scale dependence of non-Gaussianity with spectral distortions of the cosmic microwave background.Physical Review D, 91(12):123531, June 2015

    Razieh Emami, Emanuela Dimastrogiovanni, Jens Chluba, and Marc Kamionkowski. Probing the scale dependence of non-Gaussianity with spectral distortions of the cosmic microwave background.Physical Review D, 91(12):123531, June 2015

    2015

  24. [32]

    Ganc and E

    J. Ganc and E. Komatsu. Scale-dependent bias of galaxies andµ-type distortion of the cosmic microwave background spectrum from single-field inflation with a modified initial state.Physical Review D, 86(2):023518, July 2012

    2012

  25. [33]

    W. Hu, D. Scott, and J. Silk. Power spectrum constraints from spectral distortions in the cosmic microwave background.The Astrophysical Journal Letters, 430:L5–L8, July 1994

    1994

  26. [34]

    Hu and J

    W. Hu and J. Silk. Thermalization and spectral distortions of the cosmic background radiation.Physical Review D, 48:485–502, July 1993

    1993

  27. [35]

    Hu and N

    W. Hu and N. Sugiyama. Anisotropies in the cosmic microwave background: an analytic approach.The Astrophysical Journal, 444:489–506, May 1995

    1995

  28. [36]

    Hu and N

    W. Hu and N. Sugiyama. Small-Scale Cosmological Perturbations: an Analytic Approach.The Astrophysical Journal, 471:542, November 1996

    1996

  29. [37]

    Hu and M

    W. Hu and M. White. CMB anisotropies: Total angular momentum method.Physical Review D, 56:596–615, July 1997

    1997

  30. [38]

    A. F. Illarionov and R. A. Sunyaev. Comptonization, characteristic radiation spectra, and thermal balance of low-density plasma.Soviet Astronomy, 18:413–419, February 1975

    1975

  31. [39]

    Khatri and R

    R. Khatri and R. A. Sunyaev. Beyond y andµ: the shape of the CMB spectral distortions in the intermediate epoch.Journal of Cosmology and Astroparticle Physics, 9:16, September 2012

    2012

  32. [40]

    Khatri, R

    R. Khatri, R. A. Sunyaev, and J. Chluba. Mixing of blackbodies: entropy production and dissipation of sound waves in the early Universe.Astronomy&Astrophysics, 543:A136, July 2012

    2012

  33. [41]

    Spectro-spatial evolution of the CMB

    Thomas Kite, Andrea Ravenni, and Jens Chluba. Spectro-spatial evolution of the CMB. Part III. Transfer functions, power spectra and Fisher forecasts.Journal of Cosmology and Astroparticle Physics, 2023(11):028, November 2023

    2023

  34. [42]

    Komatsu, K

    E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. Seven-year Wilki...

    2011

  35. [43]

    Ma and E

    C.-P. Ma and E. Bertschinger. Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges.The Astrophysical Journal, 455:7–+, December 1995

    1995

  36. [44]

    Mehrem and A

    R. Mehrem and A. Hohenegger. A generalization for the infinite integral over three spherical Bessel functions.Journal of Physics A Mathematical General, 43(45):455204, November 2010

    2010

  37. [45]

    Analytic expressions for integrals of products of spherical bessel functions.Journal of Physics A: Mathematical and General, 24(7):1435, apr 1991

    R Mehrem, J T Londergan, and M H Macfarlane. Analytic expressions for integrals of products of spherical bessel functions.Journal of Physics A: Mathematical and General, 24(7):1435, apr 1991

    1991

  38. [46]

    Limits on primordial black holes fromµdistortions in cosmic microwave background.Physical Review D, 97(4):043525, February 2018

    Tomohiro Nakama, Bernard Carr, and Joseph Silk. Limits on primordial black holes fromµdistortions in cosmic microwave background.Physical Review D, 97(4):043525, February 2018

    2018

  39. [47]

    Nitta, E

    D. Nitta, E. Komatsu, N. Bartolo, S. Matarrese, and A. Riotto. CMB anisotropies at second order III: bispectrum from products of the first-order perturbations.Journal of Cosmology and Astroparticle Physics, 5:14, May 2009. – 31 –

    2009

  40. [48]

    Daniel Meerburg, and Subodh P

    Giorgio Orlando, P. Daniel Meerburg, and Subodh P. Patil. Primordial tensor bispectra inµ-CMB cross-correlations.Journal of Cosmology and Astroparticle Physics, 2022(2):004, February 2022

    2022

  41. [49]

    A. Ota, T. Takahashi, H. Tashiro, and M. Yamaguchi. CMBµdistortion from primordial gravitational waves.Journal of Cosmology and Astroparticle Physics, 10:29, October 2014

    2014

  42. [50]

    Cosmological constraints fromµE cross correlations.Physical Review D, 94(10):103520, November 2016

    Atsuhisa Ota. Cosmological constraints fromµE cross correlations.Physical Review D, 94(10):103520, November 2016

    2016

  43. [51]

    CMB spectral distortions as solutions to the Boltzmann equations.Journal of Cosmology and Astroparticle Physics, 2017(1):037, January 2017

    Atsuhisa Ota. CMB spectral distortions as solutions to the Boltzmann equations.Journal of Cosmology and Astroparticle Physics, 2017(1):037, January 2017

    2017

  44. [52]

    CMB spectral distortions as solutions to the Boltzmann equations.arXiv, 2017, January 2017

    Atsuhisa Ota. CMB spectral distortions as solutions to the Boltzmann equations.arXiv, 2017, January 2017

    2017

  45. [53]

    Statistical anisotropy in CMB spectral distortions.arXiv e-prints, page arXiv:1810.03928, October 2018

    Atsuhisa Ota. Statistical anisotropy in CMB spectral distortions.arXiv e-prints, page arXiv:1810.03928, October 2018

    2018 arXiv

  46. [54]

    CMB spectroscopy at third-order in cosmological perturbations

    Atsuhisa Ota and Nicola Bartolo. CMB spectroscopy at third-order in cosmological perturbations. Physical Review D, 100(4):043521, August 2019

    2019

  47. [55]

    Pajer and M

    E. Pajer and M. Zaldarriaga. New Window on Primordial Non-Gaussianity.Physical Review Letters, 109(2):021302, July 2012

    2012

  48. [56]

    Pajer and M

    E. Pajer and M. Zaldarriaga. A hydrodynamical approach to CMBµ-distortion from primordial perturbations.Journal of Cosmology and Astroparticle Physics, 2:36, February 2013

    2013

  49. [57]

    PhD thesis, University of Portsmouth, UK, January 2013

    Guido Walter Pettinari.The intrinsic bispectrum of the cosmic microwave background. PhD thesis, University of Portsmouth, UK, January 2013

    2013

  50. [58]

    C. Pitrou. The radiative transfer at second order: a full treatment of the Boltzmann equation with polarization.Classical and Quantum Gravity, 26(6):065006, March 2009

    2009

  51. [59]

    Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al. Planck 2013 Results. XXIV . Constraints on primordial non-Gaussianity.ArXiv:1303.5084, March 2013

    2013 arXiv

  52. [60]

    Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al. Planck 2015 results. XIII. Cosmological parameters.Astronomy&Astrophysics, 594:A13, September 2016

    2015

  53. [61]

    CMB spectral distortions from enhanced primordial perturbations: the role of spectator axions.Journal of Cosmology and Astroparticle Physics, 2024(8):016, August 2024

    Margherita Putti, Nicola Bartolo, Sukannya Bhattacharya, and Marco Peloso. CMB spectral distortions from enhanced primordial perturbations: the role of spectator axions.Journal of Cosmology and Astroparticle Physics, 2024(8):016, August 2024

    2024

  54. [62]

    Andrea Ravenni, Michele Liguori, Nicola Bartolo, and Maresuke Shiraishi. Primordial non-Gaussianity withµ-type and y-type spectral distortions: exploiting Cosmic Microwave Background polarization and dealing with secondary sources.Journal of Cosmology and Astroparticle Physics...

    2017

  55. [63]

    Remazeilles and J

    M. Remazeilles and J. Chluba. Extracting foreground-obscuredµ-distortion anisotropies to constrain primordial non-Gaussianity.Monthly Notices of the Royal Astronomical Society, 478(1):807–824, July 2018

    2018

  56. [64]

    Non-Gaussianity constraints with anisotropicµ distortion measurements from Planck.Monthly Notices of the Royal Astronomical Society, 515(4):5847–5868, October 2022

    Aditya Rotti, Andrea Ravenni, and Jens Chluba. Non-Gaussianity constraints with anisotropicµ distortion measurements from Planck.Monthly Notices of the Royal Astronomical Society, 515(4):5847–5868, October 2022

    2022

  57. [65]

    Angular dependence of primordial trispectra and CMB spectral distortions.Journal of Cosmology and Astroparticle Physics, 2016(10):015, October 2016

    Maresuke Shiraishi, Nicola Bartolo, and Michele Liguori. Angular dependence of primordial trispectra and CMB spectral distortions.Journal of Cosmology and Astroparticle Physics, 2016(10):015, October 2016

    2016

  58. [66]

    Stebbins

    A. Stebbins. CMB Spectral Distortions from the Scattering of Temperature Anisotropies. astro-ph/0703541, March 2007. – 32 –

    2007 arXiv

  59. [67]

    R. A. Sunyaev and Y . B. Zeldovich. Small scale entropy and adiabatic density perturbations – Antimatter in the Universe.Astrophysics and Space Science, 9:368–382, December 1970

    1970

  60. [68]

    R. A. Sunyaev and Y . B. Zeldovich. The interaction of matter and radiation in the hot model of the Universe, II.Astrophysics and Space Science, 7:20–30, April 1970

    1970

  61. [69]

    D. A. Varshalovich, A. N. Moskalev, and V . K. Khersonskii.Quantum Theory of Angular Momentum. 1988

    1988

  62. [70]

    Richard Bond, Jens Chluba, Thomas Crawford, Giulio Fabbian, Vera Gluscevic, Daniel Grin, J

    David Zegeye, Federico Bianchini, J. Richard Bond, Jens Chluba, Thomas Crawford, Giulio Fabbian, Vera Gluscevic, Daniel Grin, J. Colin Hill, P. Daniel Meerburg, Giorgio Orlando, Bruce Partridge, Christian L. Reichardt, Mathieu Remazeilles, Douglas Scott, Edward J. Wollack, and...

    2023

  63. [71]

    κm ℓ+1 ℓ+1 2ℓ+1 ˜θℓ+1,m−κ m ℓ ℓ 2ℓ+1 ˜θℓ−1,m # +δℓ0 ∂ ˜Φ ∂η−δℓ1δm0 k 3 ˜Ψ =˜sℓm +τ′

    David Zegeye, Thomas Crawford, Jens Chluba, Mathieu Remazeilles, and Keith Grainge. Square Kilometer Array as a cosmic microwave background experiment.Physical Review D, 111(6):063517, March 2025. A Useful relation and functions In this section we summarize a few useful functi...

    2025

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