Ito calculus meets the Hubble tension: Effects of small-scale electron density fluctuations on the CMB anisotropies

Geoffrey Vasil; Jens Chluba; Richard Battye

REVIEW 3 major objections 5 minor 1 cited by

Ito calculus meets the Hubble tension: Effects of small-scale electron density fluctuations on the CMB anisotropies

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Small-scale electron density fluctuations with finite time correlations broaden the CMB visibility toward higher redshifts and lower the effective Thomson scattering rate, adding damping and smearing that standard averaged recombination…

desk verdict Genuinely new formalism for clumpy-recombination CMB effects, but the spatially-uniform δ_e assumption is the load-bearing piece and it is not yet quantified; worth refereeing, needs major work. read the letter →

arxiv 2505.22242 v1 pith:PTGW5DMJ submitted 2025-05-28 astro-ph.CO astro-ph.GAhep-th

Jens Chluba , Geoffrey Vasil , Richard Battye This is my paper · ORCID

classification astro-ph.COastro-ph.GAhep-th

keywords cosmologycosmicmicrowavebackgroundelectrondensityfluctuationsItocalculusThomsonvisibilityrecombinationHubbletensionBoltzmannhierarchy

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

Small-scale, unresolved fluctuations in the free electron density during recombination leave two imprints on the CMB that standard averaged recombination treatments miss. Time correlations of the fluctuations along the line of sight broaden the Thomson visibility function toward higher redshifts, adding extra damping and smearing of anisotropies. They also reduce the effective Thomson scattering rate, which changes the photon transfer functions in ways that cannot be captured by simply rescaling the recombination history. The size of both effects is set by the combination $\zeta_e = \tau_c \sigma_e^2$, the optical depth across the coherence length times the electron-density variance. The effects are subdominant in $\Lambda$CDM but can become significant in cosmologies with early structure formation, which may matter for the Hubble tension.

What carries the argument

The central machinery is the Itô-calculus hierarchy for the moments $\kappa_p = \langle \delta_e^p X \rangle$ of the cosmological perturbation vector $X$ with respect to the electron-density fluctuation $\delta_e$. The fluctuation is driven as an Ornstein-Uhlenbeck process $d\delta_e = -\alpha\delta_e\,d\eta + \sigma\,dW$, whose exponential kernel gives a coherence time $\Delta\eta_c = 1/\alpha$. Truncating the hierarchy at second order and eliminating the fast correlation timescale converts the stochastic photon Boltzmann problem into a closed set of ordinary differential equations in which the surviving correction is proportional to $\bar\Gamma^2 \sigma_e^2 / \alpha = \bar\Gamma \zeta_e$. This produces the modified scattering rates $\bar\Gamma\{1 - f_i(\tau_c,\sigma_e)\}$ and the new line-of-sight source terms; the exponential suppression in $f_i$ keeps the correction physical when $\tau_c$ becomes large.

What would settle it

Run radiation-transfer simulations through a realistic small-scale electron-density field produced by early structure formation and compare the ensemble-averaged visibility and transfer functions with the prediction $\propto \zeta_e$; if no high-redshift broadening of the visibility and no reduction of the effective scattering rate appear, the two claimed effects are absent.

Watch

Extended reading notes

Core claim

The paper's central claim is that treating the electron density as an independent stochastic function of time with a finite correlation length changes the ensemble-averaged photon Boltzmann system. Averaging the line-of-sight visibility over correlated fluctuations yields an optical-depth correction $\Delta\tau = \int \Delta\eta_c \, \sigma_e^2 \, \bar\Gamma^2 \, d\eta'$, which broadens the visibility toward high redshift; without time correlations the correction vanishes and only the mean recombination history matters. Using Itô calculus, the authors derive a truncated hierarchy for the moments $\kappa_p = \langle \delta_e^p X\rangle$ that closes to a small set of coupled ordinary differential equations. The effective scattering rates become $\bar\Gamma\{1 - f_i(\tau_c,\sigma_e)\}$ with $f \simeq \tau_c \sigma_e^2 \exp(-\tau_c^2 \sigma_e^2)$ at leading order, so the correction saturates and cannot grow without bound. The resulting line-of-sight solution contains new source terms with different weights for the monopole, dipole and quadrupole, which is why the effect is not equivalent to changing the recombination history alone.

Load-bearing premise

Everything rests on treating the electron-density fluctuation as a time-only random field with a prescribed Ornstein-Uhlenbeck correlation and no spatial dependence on CMB scales; if the clumping is correlated with the large-scale density perturbations, mode coupling enters and the hierarchy as written needs extra terms.

Editorial extensions

If this is right

Standard Boltzmann codes that use only the mean recombination history omit an extra small-scale damping and a phase shift in the photon transfer functions that grows with $\ell$; the effect is most visible in $\ell \gtrsim 2000$ CMB spectra.
A modified average recombination history alone cannot reproduce the new corrections, because the monopole, dipole and quadrupole scattering rates are reduced by different factors $f_1,f_2,f_3$; temperature and polarization data therefore constrain the two new functions $\tau_c(z)$ and $\sigma_e(z)$ separately.
In cosmologies with early small-scale power, the corrections can reach percent level and modify the inferred cosmological parameters, including parameters relevant to the Hubble tension; the paper finds that a fixed $\zeta_e$ model can mimic part of the high-$H_0$ shift in TT but not in EE.
The damping scale $k_D^{-2}$ receives an enhanced contribution with shear-viscosity and heat-conduction terms modified by different factors, increasing small-scale damping relative to the standard tight-coupling result.

Reading between the lines

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

If the same stochastic-average logic is applied after recombination, a clumpy medium would also make Rayleigh scattering fluctuate, imprinting frequency-dependent CMB anisotropies that are not in current treatments; the paper hints at this direction but does not compute it.
The derivation drops spatial dependence of $\delta_e$ on CMB scales, so a natural test is to compute the ensemble average with $\delta_e$ correlated with the long-wavelength density field; mode-coupling terms would appear and could either enhance or cancel the claimed broadening.
The exponential suppression factor $\exp(-\tau_c^2\sigma_e^2)$ is specific to the Ornstein-Uhlenbeck kernel; measuring the actual two-point correlation of electron density from small-scale structure simulations would show whether a different kernel changes the functional form of $f_i$.

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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

Here's my take on Chluba, Vasil, and Battye. The paper does something genuinely new: it derives two CMB effects from time-correlated small-scale electron density fluctuations — an extra broadening of the Thomson visibility and a multipole-dependent reduction of the effective scattering rate — that go beyond the average recombination history. The Ito formalism is coherent, the perturbative treatment matches the moment hierarchy to about 10–20% in tested cases, and the authors are unusually explicit about what is illustrative and what is not. That is real value.

The soft spot is the load-bearing assumption of spatial uniformity of δ_e. Right after Eq. (17) they neglect spatial dependence and treat δ_e as a pure random process in time. For any physical small-scale power spectrum, the scattering term is a convolution that couples the observed mode k to high-q modes, and the statement that no mode-coupling occurs is a consequence of the uniform limit, not of the physics. The stress-test note argues that the q→0 limit is singular and that back-reaction from generated small-scale perturbations could enter at the same order as the claimed corrections. I think that concern is legitimate, though it omits the strong damping of high-q photon perturbations, so it is an open question rather than a proven fatal flaw. Either way, the paper gives no quantitative estimate of the error from this approximation, and that is the main thing a revision would have to provide.

The other weaknesses are minor in comparison. σ_e(z) and τ_c(z) are free functions with no physical model behind them; the Hubble-tension section is explicitly illustrative and does not claim a solution; and no code is shipped, so the numerics cannot be independently checked. All of these are stated in the text, which I read as honest rather than evasive.

Who is this for? CMB theory people working on recombination and small-scale clumping. I would bring it to a reading group, and I would send it to peer review rather than desk reject. But I would not want to see it accepted until the spatial-mode-coupling question is confronted quantitatively, or at least bounded in a way that does not depend on the singular uniform limit.

Referee Report

3 major / 5 minor

Summary. This paper develops a stochastic formalism for including ultra-small-scale electron density fluctuations in CMB anisotropy calculations. The free-electron field δ_e is modeled as a time-dependent Ornstein-Uhlenbeck process (Gaussian or log-normal), and Itô calculus is used to derive an ensemble-averaged Boltzmann hierarchy. The authors identify two new effects beyond changes to the average recombination history: a broadening of the Thomson visibility function from line-of-sight correlations, and a reduction of the effective Thomson scattering rate that enters the photon transfer functions differently for different multipoles. The simplified hierarchy (Eq. 85) is implemented in CosmoTherm, and illustrative transfer functions and CMB power spectra are computed for phenomenological choices of the variance σ_e and coherence optical depth τ_c. The paper closes with a discussion of the potential relevance of these effects to the Hubble tension.

Significance. If the load-bearing scale-separation assumption is valid, the paper introduces a genuinely new mechanism for modifying CMB predictions without altering the average recombination history: stochastic time-correlated scattering changes the effective damping and the photon transfer functions in a way that the standard averaged recombination treatment misses. The internal derivation is coherent, the moment hierarchy and the simplified Boltzmann hierarchy are cross-checked against each other (to roughly 10–20% in the tested cases), and the implementation in CosmoTherm makes the predictions concrete and testable. The paper is also honest about its limitations and does not circularly fit the new parameters to the target CMB spectra; σ_e and τ_c are free inputs. However, the physical interpretation of these parameters and the claimed absence of mode coupling depend on a spatial-uniformity approximation that is not quantitatively validated. This makes the central claim provisional rather than established.

major comments (3)

[3.1–3.5] The central derivation replaces the spatially fluctuating electron field δ_e(x,η) by a strictly time-dependent process δ_e(η), stated immediately after Eq. (17). For a physical field with power at wavenumber k_s ≫ k, the scattering term in Fourier space is a convolution over q, so the observed large-scale mode k couples to photon perturbations at wavenumbers k − q ≈ ±k_s. The paper's statement in Sect. 3.5 that 'no mode-coupling occurs' is therefore a direct consequence of the scale-independence of δ_e, not of the smallness of the scales. The back-reaction of the generated small-scale perturbations on the large-scale mode is of order Γ̄²σ_e²/q_s, the same parametric order as the claimed corrections in Eqs. (66) and (85), because τ_c = Γ̄ Δη_c and Δη_c ∼ 1/q_s. The limit q → 0 is singular: the generated perturbations become spatially coherent with the large-scale mode, which can overestimate the effect and artificially fix the relative weights f_1, f_2, f_3 in Eq. (83). No estimate of the error incurred by this limit is given, and the numerical validation in Sect. 6.1.1 compares two versions of the time-only model rather than a model with finite k_s. Without a quantitative estimate or a finite-k_s test, the load-bearing claim that the q → 0 limit is representative of small-scale clumping is not established.
[4.1–4.4, Eq. (83)] The simplified hierarchy is obtained by resumming a τ_c expansion whose radius of convergence is not established. The factor 1/(1 − τ_c) in Eq. (71) is discarded 'to allow a smooth transition' after being derived under τ_c < 1, so the final f_i in Eq. (83) are an ad hoc regularization of the series. The paper confirms the series only to 12th order and drops terms suppressed by 1/γ and e^{−pγ}; for τ_c ≳ 1, which occurs at high redshift in the illustrative scalings, the corrected equations are not derived from the stochastic system. The specific Hubble-tension examples use ζ_e ≈ 0.1 and τ_c ≈ 0.18, so the numerical error in those figures may be small, but the broader statement that the effects are 'significant in cosmologies with early structure formation' extrapolates beyond the validated regime. A derivation, or at least a controlled test, of the resummation for τ_c > 1 is needed before the simplified hierarchy can be used for parameter exploration in that regime.
[2.1, 2.2, and 5] The statistical model assumes δ_e is an independent Ornstein-Uhlenbeck process with an exponential correlation kernel, and higher-order correlators are omitted; the log-normal version repairs positivity but still assigns the correlation structure artificially. In a physical model generated by gravitational collapse, δ_e is correlated with the large-scale density and velocity fields, so the ensemble average over δ_e should be conditional on the long-wavelength environment. The paper acknowledges this in Sect. 2.1 but then proceeds unconditionally. The separate-universe mapping in Sect. 5 (time-independent F_b, no coupling to the perturbed radiation field) is an additional simplification. These choices are not yet tested against any simulation or analytic model of pre-recombination clumping, so the range of σ_e(η) and τ_c(η) used in Section 6 is phenomenological. This limits the strength of the Hubble-tension claims, although it does not invalidate the formalism itself.

minor comments (5)

[6.2 (Fig. 18)] Model C is selected specifically to mimic a change in H_0 from the TT spectrum, and the paper notes that the EE prediction does not match. The caption should clearly label this model as an illustration of degeneracy rather than as a viable physical solution, to avoid readers misinterpreting the figure as a proposed resolution of the Hubble tension.
[6.2] There is a typo in 'scope fo this paper' which should read 'scope of this paper'.
[2.4 and Eq. (15)] The function Li(z) is never defined in the text; please state explicitly that it is the logarithmic integral, and check the argument convention used in Eq. (15).
[Fig. 1] The caption uses the notation Γ_b without defining it; please define Γ_b ≡ Γ̄ in the caption or text.
[4.4] The text states that the line-of-sight formulation can reproduce the results of CLASS to high precision using CosmoTherm, but no comparison plot or reference to a specific figure is given. Please add a convergence test or point to where this validation is shown.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the corrected Boltzmann hierarchy is derived from explicit stochastic assumptions with free input parameters, not from the CMB spectra or from a self-citation chain.

full rationale

The paper's central result, the modified hierarchy in Eqs. (82)-(85), is obtained by two independent routes: a second-order perturbative/Langevin treatment (Sect. 3.4) and an Itô-calculus moment hierarchy (Sect. 4.1), both starting from the stated stochastic model δ_e with OU correlation (Sect. 2.2). The target CMB power spectra enter only as outputs via Eq. (48), after the ensemble averages are derived, so there is no fitted-input-called-prediction. The new parameters τ_c and σ_e^2 are free inputs with explicit physical interpretation (optical depth across the coherence length and electron-density variance); they are not tuned to reproduce the claimed damping or H0-mimicking signal. The paper even reports that Model C fails for EE, which is the opposite of a post-hoc forced agreement. The approximation of a spatially uniform δ_e(η) is stated explicitly as an assumption ('We have neglected any spatial dependence of δ_e'), not derived, and the paper flags mode-coupling as future work; an assumption being physically insecure or singular in the q→0 limit is a correctness risk, not a circular reduction. Self-citations to CosmoTherm/CosmoRec are to the numerical machinery used for illustration, and the code is benchmarked against CLASS ('we can reproduce the results of CLASS to high precision'), so the self-citations are not load-bearing. No step equates a prediction to its input by construction.

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

No new particles, forces, or dimensions are introduced. The stochastic field delta_e and the effective parameters sigma_e, tau_c, and zeta_e are statistical properties of already assumed small-scale structure, not new entities. The main ledger items are the hand-chosen parameters and the unverified stochastic modeling assumptions.

free parameters (6)

sigma_e(eta) = examples: 0.5, 2, 3 at z=1100
Variance of electron density fluctuations; chosen by hand for illustrations and normalized at z=1100; enters every correction through the functions f_i.
tau_c(eta) = examples: 0.01, 0.05, 0.1 at z=1100
Optical depth across the coherence length; chosen by hand; combined with sigma_e^2 to define zeta_e.
zeta_e = tau_c sigma_e^2 = Model C: 0.1 with sigma_e=0.75
Effective clumping parameter; Model C sets it constant to mimic an H0 shift in TT, an illustrative choice not derived from a physical model.
z_s, gamma_s = z_s=1200-1500, gamma_s=1-2
Redshift scaling parameters for tau_c in Eq. (99); hand-chosen to probe early-time effects.
z_tau, gamma_tau = z_tau=1000-2000, gamma_tau=0.25-2
Low-redshift scaling parameters in Eq. (100); illustrative choices.
sigma_b = 0.5 to 4
Log-normal baryon fluctuation variance used in the Section 5 mapping to compute average recombination histories and electron moments.

assumptions (6)

domain assumption A huge separation of scales exists between the electron-density fluctuations and the CMB modes, so delta_e has no spatial dependence on large scales.
Invoked in Sect. 2.1 and after Eq. (17); if false, convolution integrals and mode coupling appear, breaking the ensemble-average hierarchy.
ad hoc to paper delta_e follows an Ornstein-Uhlenbeck process with exponential correlation kernel and specified alpha and sigma; higher-order correlators are omitted.
Sect. 2.2 to 2.4; the quantitative size of every effect depends on this correlation model.
domain assumption The electron field is either Gaussian or log-normal, which closes the moment hierarchy at the assumed order.
Sect. 2.3 and Sect. 4.2; the Gaussian model has unphysical negative densities, and the log-normal choice is one possible positivity-preserving alternative.
ad hoc to paper Corrections can be expanded in the small parameter tau_c and the moment hierarchy truncated at second to sixth order.
Sect. 4.1 to 4.4; the simplified Boltzmann hierarchy is a truncation, and the log-normal f_i expression is an empirical resummation.
ad hoc to paper The separate-universe approach maps baryon density fluctuations to electron density fluctuations with time-independent local F_b.
Sect. 5; the authors call this extremely simplistic and expect larger variations in realistic models.
standard math Standard cosmological perturbation equations and tight-coupling approximations hold for the background.
Sect. 3.1 and Appendix B; standard GR and fluid equations from Ma and Bertschinger are assumed.

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Cite this review

Pith. "Pith review of Ito calculus meets the Hubble tension: Effects of small-scale electron density fluctuations on the CMB anisotropies." pith.science (2026). https://pith.science/paper/PTGW5DMJ

@misc{pith2026250522242,
  author       = {Pith},
  title        = {Pith review of: Ito calculus meets the Hubble tension: Effects of small-scale electron density fluctuations on the CMB anisotropies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTGW5DMJ}},
  note         = {Machine review of arXiv:2505.22242}
}

read the original abstract

In this work, we develop a novel formalism to include the effect of electron density fluctuations at ultra small scales (well below the sound horizon at last scattering) on the observed anisotropies of the Cosmic Microwave Background (CMB). We treat the electron field as an independent stochastic variable and obtain the required ensemble-averaged photon Boltzmann equations using Ito calculus. Beyond changes to the average recombination history (which can be incorporated in the standard approach) our work identifies two new effects caused by the clumpiness of the medium. The first is a correction to the Thomson visibility function caused by correlations of the electron fluctuations along the line of sight, leading to an additional broadening of the visibility towards higher redshifts which causes extra damping and smearing of the CMB anisotropies. The second effect is a reduction of the effective scattering rate in the (pre-)recombination era that affects the photon transfer functions in a non-trivial manner. These new effects are subdominant in LCDM but can be significant in cosmologies with an early onset of structure formation (e.g., due to generation of enhanced small-scale power) as suggested by a number of indicators (e.g., the abundance of high redshift galaxies observed by JWST). We discuss the relevance of these new effects to the Hubble tension, finding that corrections which cannot be captured by simple modifications to the average recombination history arise. This highlights how important an understanding of the recombination process is in cosmological inference, and that a coordinated simulation and analysis campaign is required as part of the search for the origin of the various tensions in cosmology.

Figures

Figures reproduced from arXiv: 2505.22242 by the authors.

**Figure 1.** Differential optical depth contributions. The solid black line is for Γb ≡ Γ¯, while the blue line shows Γ¯ ζe = Γ¯ 2 rs Λc for Λc = 5 × 10−3 . 200 400 600 800 1000 1200 1400 Redshift 10-5 10-4 10-3 10-2 Visibility Function Standard Recombination Λ c = 3.2 x 10-4 Λ c = 3.2 x 10-3 Λ c = 9 x 10-3 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Modifications to the Thomson visibility function for various values of Λc. Electron density variations cause a broadening of the visibility function and reduction of the maximal last-scattering probability. already capture one of the main effects of electron density perturbations along the line of sight on the CMB power spectra. To parametrize things more explicitly, let us describe the model in terms of the sound… view at source ↗

**Figure 3.** Dependence of the recombination history on Fb. The weighted free-electron fraction is shown X ∗ e = XeFb. At each redshift, the electron faction can be averaged over a distribution P(δb, σb) to obtain the average recombination history. We demand that [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 5.** Figure 5: Electron density probability distribution at various redshifts and fixed baryon density distribution. Note that here Ne/Ne,av is essentially 1+δe. 400 600 800 1000 1200 1400 1600 1800 2000 Redshift 10-3 10-2 10-1 100 Normalized second moment σb 2 = 0.25 σb 2 = 0.5 σb 2…

**Figure 4.** Figure 4: Average recombination history (upper panel) and relative difference (lower panel) for various values of σb. Baryon density perturbations lead to an acceleration of recombination around z ≃ 1100. are affected less. At z ≫ 1100, one has δe ≈ δb, which implies on average…

**Figure 7.** Figure 7: Normalized electron density moment, [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 9.** Figure 9: Model variables τc, σe and fi for the examples shown in [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 8.** Figure 8: Photon transfer functions for the temperature monopole, dipole and quadrupole for k = 0.05 Mpc−1 . The standard ΛCDM result is compared with the Gaussian and log-normal treatments for τc = 0.01 and σe = 2 at z = 1100. For the computations we used the simplified BH. exp…

**Figure 11.** Figure 11: Photon transfer function corrections for the temperature monopole and dipole at k = 0.05 Mpc−1 . We used σe = 0.5 for varying values of τc all normalized at z = 1100. In each line group, we show the results from the Gaussian (solid line) and log-normal (dashed line) s…

**Figure 10.** Figure 10: Photon transfer function corrections (with respect it the ΛCDM solution) for the temperature monopole, dipole and quadrupole at wavenumber k = 0.05 Mpc−1 . We assumed the Gaussian scenario with τc = 0.01 and σe = 0.5 at z = 1100 and compare the simplified BH approach…

**Figure 13.** Figure 13: Photon transfer function corrections for the temperature monopole and dipole at k = 0.05 Mpc−1 . We used σe = 0.5 for varying values of τc all normalized at z = 1100 with a scaling according to Eq. (99). The results were obtained with a 4th order Gaussian moment setup…

**Figure 16.** Figure 16: Photon transfer function corrections for the temperature monopole at k = 0.05 Mpc−1 . We used σe(z) = 0.5 and τc(z = 104 ) = 0.1 with a scaling according to Eq. (100a) for values of γτ and zτ as labeled. The results were obtained with a 4th order Gaussian moment setup…

**Figure 15.** Figure 15: Photon transfer function corrections for the temperature monopole, dipole and quadrupole at k = 0.05 Mpc−1 . We used σe = 0.5 normalized at z = 1100 with a scaling according to Eq. (99) and model parameters as labeled. A 4th order Gaussian moment setup was used. 6.2 …

**Figure 17.** Figure 17: CMB TT and EE power spectra, Dℓ = ℓ(ℓ + 1)Cℓ/2π. The solid black lines show the standard ΛCDM computation. The other cases where computed using the log-normal modified BH setup. For Model A, we use a setup similar to [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

**Figure 18.** Figure 18: Relative change of CMB TT and EE power spectra with respect to the ΛCDM model. The solid black lines show the difference cause by using a high value of H0. Model C was computed using the modified BH with unchanged average recombination history for a fixed value of ζe(…

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Works this paper leans on

70 extracted references · 60 canonical work pages · cited by 1 Pith paper

[1]

Abdalla E. et al. , 2022, Journal of High Energy Astrophysics, 34, 49

work page 2022
[2]

Adame A. G. et al. , 2025, , 2025, 021

work page 2025
[3]

, 2023, Astrophys

Afzal A., et al. , 2023, Astrophys. J. Lett., 951, L11

work page 2023
[4]

M., 2011, , 83, 043513

Ali-Ha \"i moud Y., Hirata C. M., 2011, , 83, 043513

work page 2011
[5]

Bailes M. et al. , 2021, Nature Reviews Physics, 3, 344

work page 2021
[6]

A., Pilaftsis A., Viatic D

Battye R. A., Pilaftsis A., Viatic D. G., 2020, , 102, 123536

work page 2020
[7]

Cambridge University Press

Baumann D., 2022, Cosmology. Cambridge University Press

work page 2022
[8]

R., Du P., Fernandez N., Weikert M

Buckley M. R., Du P., Fernandez N., Weikert M. J., 2025, arXiv e-prints, arXiv:2502.20434

arXiv 2025

Show all 70 references

[9]

Calabrese E. et al. , 2025, arXiv e-prints, arXiv:2503.14454

2025 arXiv
[10]

, 2023, , 264, 7

CCAT-Prime Collaboration et al. , 2023, , 264, 7

2023
[11]

Chen X., Kamionkowski M., 2004, , 70, 043502

2004
[12]

Chluba J., 2010, , 402, 1195

2010
[13]

Chluba J. et al. , 2021, Experimental Astronomy, 51, 1515

2021
[14]

Chluba J., Ali-Ha \"i moud Y., 2016, , 456, 3494

2016
[15]

Chluba J. et al. , 2019, , 51, 184

2019
[16]

A., 2006, , 446, 39

Chluba J., Sunyaev R. A., 2006, , 446, 39

2006
[17]

A., 2012, , 419, 1294

Chluba J., Sunyaev R. A., 2012, , 419, 1294

2012
[18]

M., 2011, , 412, 748

Chluba J., Thomas R. M., 2011, , 412, 748

2011
[19]

R., Beringue B., Meerburg P

Coulton W. R., Beringue B., Meerburg P. D., 2021, , 103, 043501

2021
[20]

Cyr B., Cotterill S., Battye R., 2025, arXiv e-prints, arXiv:2504.02076

2025 arXiv
[21]

K., Raccanelli A., Bartolo N., 2025, , 111, 063507

de Kruijf J., Vanzan E., Boddy K. K., Raccanelli A., Bartolo N., 2025, , 111, 063507

2025
[22]

Dekker A., Kravtsov A., 2025, , 111, 063516

2025
[23]

DESI Collaboration , 2025, arXiv e-prints, arXiv:2503.14738

2025 arXiv
[24]

Di Valentino E. et al. , 2021 a , Astroparticle Physics, 131, 102605

2021
[25]

Di Valentino E. et al. , 2025, arXiv e-prints, arXiv:2504.01669

2025 arXiv
[26]

Di Valentino E. et al. , 2021 b , Classical and Quantum Gravity, 38, 153001

2021
[27]

Academic Press

Dodelson S., 2003, Modern cosmology . Academic Press

2003
[28]

A., Chluba J., Rubi \ n o-Mart \' n J

Fendt W. A., Chluba J., Rubi \ n o-Mart \' n J. A., Wandelt B. D., 2009, , 181, 627

2009
[29]

Galli S., Pogosian L., Jedamzik K., Balkenhol L., 2022, Phys. Rev. D, 105, 023513

2022
[30]

Han C., Chen Z.-C., Yu H., Wu P., 2025, arXiv e-prints, arXiv:2501.09939

2025 arXiv
[31]

Hart L., Chluba J., 2020, , 493, 3255

2020
[32]

Hart L., Rotti A., Chluba J., 2020, , 497, 4535

2020
[33]

Hu W., Scott D., Sugiyama N., White M., 1995, , 52, 5498

1995
[34]

Hu W., White M., 1997, , 56, 596

1997
[35]

Jedamzik K., Abel T., 2013, JCAP, 10, 050

2013
[36]

Jedamzik K., Pogosian L., 2020, Phys. Rev. Lett., 125, 181302

2020
[37]

Kaiser N., 1983, , 202, 1169

1983
[38]

Kite T., Ravenni A., Chluba J., 2023, Journal of Cosmology and Astroparticle Physics, 2023, 028

2023
[39]

Knox L., Millea M., 2020, , 101, 043533

2020
[40]

moud Y., Sch \

Lee N., Ali-Ha \" moud Y., Sch \"o neberg N., Poulin V., 2023, , 130, 161003

2023
[41]

Lesgourgues J., 2011, ArXiv:1104.2932

2011 arXiv
[42]

Lewis A., 2013, , 2013, 053

2013
[43]

Lewis A., Challinor A., Lasenby A., 2000, , 538, 473

2000
[44]

Lewis A., Weller J., Battye R., 2006, , 373, 561

2006
[45]

Lucca M., Chluba J., Rotti A., 2024, , 530, 668

2024
[46]

P., Knox L., Chluba J., 2024 a , , 110, 083538

Lynch G. P., Knox L., Chluba J., 2024 a , , 110, 083538

2024
[47]

P., Knox L., Chluba J., 2024 b , , 110, 063518

Lynch G. P., Knox L., Chluba J., 2024 b , , 110, 063518

2024
[48]

Ma C.-P., Bertschinger E., 1995, , 455, 7

1995
[49]

H., Jedamzik K., Pogosian L., 2025 a , arXiv e-prints, arXiv:2504.15274

Mirpoorian S. H., Jedamzik K., Pogosian L., 2025 a , arXiv e-prints, arXiv:2504.15274

2025
[50]

H., Jedamzik K., Pogosian L., 2025 b , , 111, 083519

Mirpoorian S. H., Jedamzik K., Pogosian L., 2025 b , , 111, 083519

2025
[51]

Parshley S. C. et al. , 2018, ArXiv:1807.06675

2018 arXiv
[52]

, 2016, , 594, A13

Planck Collaboration et al. , 2016, , 594, A13

2016
[53]

A., Chluba J., Fendt W

Rubi \ n o-Mart \' n J. A., Chluba J., Fendt W. A., Wandelt B. D., 2010, , 403, 439

2010
[54]

F., S \'a nchez A

Sch \"o neberg N., Abell \'a n G. F., S \'a nchez A. P., Witte S. J., Poulin V., Lesgourgues J., 2022, , 984, 1

2022
[55]

Sch \"o neberg N., Vacher L., 2025, , 2025, 004

2025
[56]

Sehgal N. et al. , 2019, in Bulletin of the American Astronomical Society, Vol. 51, p. 6

2019
[57]

Sekiguchi T., Takahashi T., 2021, , 103, 083507

2021
[58]

Seljak U., Zaldarriaga M., 1996, , 469, 437

1996
[59]

Seljak U., Zaldarriaga M., 1997, Physical Review Letters, 78, 2054

1997
[60]

R., Chluba J., 2011, , 415, 1343

Shaw J. R., Chluba J., 2011, , 415, 1343

2011
[61]

R., Padmanabhan N., Finkbeiner D

Slatyer T. R., Padmanabhan N., Finkbeiner D. P., 2009, Physical Review D (Particles, Fields, Gravitation, and Cosmology), 80, 043526

2009
[62]

C., Kosowsky A., Spergel D

Thiele L., Guan Y., Hill J. C., Kosowsky A., Spergel D. N., 2021, , 104, 063535

2021
[63]

Springer New York, New York, NY

Torquato S., 2002, Random Heterogeneous Materials : Microstructure and Macroscopic Properties / by Salvatore Torquato., Interdisciplinary Applied Mathematics, 16. Springer New York, New York, NY

2002
[64]

E., Ornstein L

Uhlenbeck G. E., Ornstein L. S., 1930, Phys. Rev., 36, 823

1930
[65]

G., 2019, Nature Astronomy, 3, 891

Verde L., Treu T., Riess A. G., 2019, Nature Astronomy, 3, 891

2019
[66]

Vilenkin A., Shellard E. P. S., 2000, Cosmic Strings and Other Topological Defects

2000
[67]

Weinberg S., 1971, , 168, 175

1971
[68]

N., Ostriker J

Yu Q., Spergel D. N., Ostriker J. P., 2001, , 558, 23

2001
[69]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence a...
[70]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Abdalla E. et al. , 2022, Journal of High Energy Astrophysics, 34, 49

work page 2022

[2] [2]

Adame A. G. et al. , 2025, , 2025, 021

work page 2025

[3] [3]

, 2023, Astrophys

Afzal A., et al. , 2023, Astrophys. J. Lett., 951, L11

work page 2023

[4] [4]

M., 2011, , 83, 043513

Ali-Ha \"i moud Y., Hirata C. M., 2011, , 83, 043513

work page 2011

[5] [5]

Bailes M. et al. , 2021, Nature Reviews Physics, 3, 344

work page 2021

[6] [6]

A., Pilaftsis A., Viatic D

Battye R. A., Pilaftsis A., Viatic D. G., 2020, , 102, 123536

work page 2020

[7] [7]

Cambridge University Press

Baumann D., 2022, Cosmology. Cambridge University Press

work page 2022

[8] [8]

R., Du P., Fernandez N., Weikert M

Buckley M. R., Du P., Fernandez N., Weikert M. J., 2025, arXiv e-prints, arXiv:2502.20434

arXiv 2025