Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges

Chung-Pei Ma; Edmund Bertschinger

REVIEW

Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges

Chung-Pei Ma, Edmund Bertschinger

Astrophysics · Submitted 11 Jun 1995 · 40 references

Pith's verdict: The standard reference for linear perturbation theory in two gauges, and it holds up — the only thing worth pressing is that the 10^-3 accuracy is an estimate, not a bound.

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CitationChung-Pei Ma, Edmund Bertschinger. "Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges." arXiv:astro-ph/9506072 (1995). https://pith.science/paper/PPFIXYDP

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Editor's note

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This is the paper people cite when they need the linear equations written down correctly in both gauges, and it earns that. The real content is the parallel derivation: Einstein equations, Boltzmann hierarchies for photons and both neutrino species, fluid equations for CDM and baryons, all laid out side by side in synchronous and conformal Newtonian form, plus the explicit transformation between them (eq. 18) and the super-horizon isentropic initial conditions (eqs. 96–98). The massive neutrino treatment is the part that was genuinely missing before — full momentum dependence on a 128-point q-grid instead of the l=2 truncation everyone else used. The released code for both gauges is not a gesture; it is why the paper became infrastructure. The two-gauge agreement to 0.1% is a real cross-check on the transformation and the metric evolution, and the initial conditions are derived from the same equations rather than fitted, so there is no circularity to worry about. The COBE quadrupole appears only as a normalization for Fig. 4 and does no argumentative work. On the soft side, the stress-test note is right about the accuracy claim, though it is a reporting gap rather than an error. The 0.1% gauge agreement is not independent with respect to the things most likely to be wrong: both codes share the same Boltzmann closures (eqs. 51, 58, 65), the same q-grid, the same tight-coupling switch temperatures, and the same pre-computed recombination splines. So it bounds gauge error, not truncation error — and 0.1% is the same order as the quoted 10^-3, which leaves no visible margin. The convergence checks are described but never tabulated; a reader cannot see how Cl moves with lmax or with the switch temperature. The lmax=50 justification for massive neutrinos is stated for the non-relativistic phase, but the high multipoles are populated while the neutrinos are still relativistic, so that early phase rests entirely on the unquantified empirical check. None of this undermines the physics; it means the word 'accurate' in the abstract is doing slightly more work than the evidence shown. Who should read it: anyone building a Boltzmann code, anyone who needs the gauge transformation, and anyone working on mixed dark matter. It is a methods paper, not a controversy, so it is a reference rather than a reading-group session. Send it out; the referees will ask for the convergence table and the paper will be better for it.

Referee report

Significance

If the results hold, this is a substantial and durable contribution: a complete, self-contained reference for the linear perturbation equations in two gauges, with every species relevant to CDM, HDM, and mixed models treated at the same level of detail. Three strengths deserve explicit credit. First, the parallel presentation in two gauges with an explicit transformation (Eqs. 18, 27) is not decoration; it supplies a genuine internal cross-check, and the reported agreement of better than 0.1 percent between the two independent integrations is a meaningful validation of the metric evolution and gauge transformation. Second, the super-horizon initial conditions in Eqs. (96)–(98) are derived analytically from the same Einstein and Boltzmann equations rather than fitted to output, and the growing-mode amplitude is fixed by a single constant C, so the initial-value problem is parameter-free once the mode is chosen. Third, the numerical programs for both gauges are released, which makes the calculation reproducible and is exactly the practice this field should reward. The massive-neutrino phase-space treatment (128-point q grid, per-q Legendre hierarchy, Eq. 58) is a real advance over cr

Major objections (3)

[§8, accuracy claim] The stated accuracy, 'we estimate that our final results have a relative accuracy better than 10^-3' (§8), is load-bearing for the abstract's claim of 'the first accurate calculations' (abstract:s6), but the supporting evidence is not quantified anywhere in the paper. The convergence checks are described in one sentence ('increasing the grids of k, q, and l values as well as decreasing the integration timestep') with no numbers attached. More importantly, the 0.1 percent two-gauge agreement, which is the paper's most visible cross-check, is not independent with respect to the approximations most likely to control the error: both codes use the same truncation closures of Eqs. (51), (58), and (65), the same 128-point q grid, the same tight-coupling switch temperatures T_b = 2x10^4 K and 2x10^5 K (§5.7), and the same pre-computed recombination and baryon-temperature splines (§5.8). The gauge comparison therefore bounds gauge-transformation and metric-evolution errors, not truncation or switch errors. Note also that 0.1 percent equals 10^-3, the same order as the quoted accuracy, so the gauge difference alone consumes the entire stated error budget rather than sitting comfortably below it. I ask the authors to add a short table reporting C_l at a few multipoles (say l = 10, 100, 200, 500, 1000) for variations of l_max, q-grid size, and the two switch temperatures, so that the reader can see the margin rather than take the estimate on trust.
[§5.4, Eq. (58) and l_max = 50 for massive neutrinos] The justification given for truncating the massive-neutrino hierarchy at l_max = 50 is that 'the higher multipole moments decay rapidly once the neutrinos become nonrelativistic.' That argument applies at a > a_nr, but the high multipoles are populated while the neutrinos are still relativistic, and for the largest wavenumber in the integration (k = 10 Mpc^-1) the product k*tau reaches of order 10^3 by tau_nr. The truncation scheme of Eq. (58) is a Bessel-like extrapolation and may well handle this, but the paper does not demonstrate it. Because the massive-neutrino moments feed the shear stress that enters the Einstein equations and hence the potentials shown in Fig. 2, this is a correctness risk for the central result rather than a cosmetic one. A concrete test is available and cheap relative to the runs already performed: repeat the k = 1 and k = 10 Mpc^-1 integrations with l_max = 100 or 200 for the massive neutrinos and report the change in delta_h, sigma_h, and C_l. If the change is below 10^-3, the choice is vindicated and the sentence in §5.4 can be replaced by a quantitative statement.
[§8, Figs. 1 and 4] The paper's headline product is the angular power spectrum, yet no numerical values of C_l, and no peak positions or peak heights, are tabulated anywhere; the results exist only as figures. This makes the accuracy claim difficult to check independently and makes the paper less useful as the reference it is clearly intended to be. I recommend a small table of C_l (or of the acoustic peak positions and amplitudes) for the three models, in the same normalization as Fig. 1, so that later calculations can be compared against this work without digitizing a plot.

Minor objections (9)

[§5.5, Eqs. (59)–(60)] The polarization-dependent Thomson collision terms are attributed jointly to Bond & Efstathiou (1984, 1987) and to Kosowsky (1995), the latter listed in the references as a preprint (astro-ph/9501045). Since the polarization term is central to the paper's novelty claim, either cite a published derivation for the specific form of Eq. (60) or sketch the two-line derivation from the differential cross section d(sigma)/d(Omega) = 3 sigma_T (1 + cos^2 theta)/(16 pi) that is already quoted in the text.
[§8, truncation criteria] Two different truncation criteria are used without explanation of how they relate: l_max = 1.5 k tau_max + 10 for the temperature-fluctuation runs, and l_max = 2000 for the massless species in the potential and density runs. A sentence stating why the potential/density calculation needs the larger value (presumably because it is run to k = 10 Mpc^-1 rather than 0.5 Mpc^-1) would help the reader judge the computational cost and the adequacy of each choice.
[§5.7, switch temperatures] The statement that the switches at T_b = 2x10^4 K and 2x10^5 K 'occur early enough to preserve good accuracy in the resulting photon phase space distribution' is asserted rather than shown. This is folded into major comment 1, but even a single sentence giving the size of the change in C_l when the switch temperature is halved would settle it.
[§5.6, Eq. (68)] The baryon sound speed neglects the slow time variation of the mean molecular weight and of d ln T_b / d ln a. The text says the approximation is adequate because the baryons contribute little to the pressure of the photon-baryon fluid; a rough numerical bound on the resulting error in c_s^2 during recombination would make that statement checkable.
[§7, Eqs. (96)–(98)] The initial conditions retain only the fastest-growing physical mode and drop the D-mode, and the two residual gauge modes A and B are removed by the choice of coordinates. It would be worth stating explicitly, in one sentence, that the numerical integration starts at tau_0 = 3x10^-4 Mpc and that the neglected decaying mode contributes below the stated accuracy at that time, since the choice of starting time is what makes the truncation of the mode expansion safe.
[Table 1] The symbol glossary omits several quantities that appear repeatedly in the text and figures: k_fs and a_nr (§8, Eq. 102), F_gamma_l and G_gamma_l (§5.5), Psi_l (§5.4), and Delta_l (§6). Adding them would make the table serve its stated purpose as a guide to the reader.
[Figs. 2 and 3] Fig. 2 shows 31 curves distinguished only by the caption's 'from left to right'; labelling three or four representative wavenumbers directly on the plot would help. In Fig. 3, marking a_nr, a_eq, and a_rec on the horizontal axis of each panel would make the discussion in items (1)–(3) of §8 much easier to follow.
[§6, Eq. (91)] The large-angle result C_l ≈ (8 pi^2/9) A / [l(l+1)] for n = 1 should state explicitly that A is defined by P_psi(k) = A k^{-3} in the same convention as Eq. (90), since the numerical coefficient depends on that convention and readers will otherwise compare it to the standard Sachs-Wolfe normalization and find a discrepancy.
[Title and abstract, typesetting] The version supplied to me shows broken words and stray spacing in the title and abstract ('Co nformal', 'onu' for Omega_nu, split superscripts in the equations). This may be an artifact of the file I received, but please check the typeset version, since the abstract is the part most readers will see.

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The paper

The paper claims

The paper derives the linear theory of cosmic structure formation twice over — once in the synchronous gauge and once in the conformal Newtonian gauge — for every particle species that matters, and uses the agreement between the two to make

The reading

The paper asks a bookkeeping-shaped question with real physical stakes: if you write the linear theory of gravitational instability in two different coordinate choices — the synchronous gauge used since Lifshitz, and the conformal Newtonian gauge — do you get the same universe? Ma and Bertschinger answer by deriving side by side the complete set of Einstein, Boltzmann, and fluid equations for every species that matters in a flat cold dark matter, hot dark matter, or mixed model: CDM, baryons, photons, massless neutrinos, and massive neutrinos, with a cosmological constant allowed. The payoff is a numerical pipeline that can be run twice, once per gauge, and checked against itself; the two independent integrations agree to better than 0.1 percent. On that foundation they produce the first angular power spectra for mixed dark matter models that retain photon polarization, the full neutrino multipole hierarchy, and helium recombination. For a reader who wants to know why cosmic microwave background predictions can be trusted at the percent level, the paper's real contribution is the demonstration that a gauge choice is a computational convenience rather than a physical assumption.

What carries the argument

Three pieces carry the argument. The first is the gauge transformation of equation (18), which converts between $(h, eta)$ and $(psi, varphi)$ and is what makes a two-code cross-check possible at all. The second is the parallel set of Boltzmann hierarchies: the photon and massless-neutrino distribution functions are expanded in Legendre polynomials and reduced to coupled ordinary differential equations in equations (49), (50), (63), and (64), while the massive neutrinos keep an explicit momentum dependence and are tracked on a grid of 128 $q$-points per wavenumber with their own hierarchy in equations (56) and (57). The third is the set of truncation rules — equations (51), (58), and (65) — which close the infinite hierarchies by extrapolating the spherical-Bessel-like behavior of the high multipoles rather than simply setting them to zero, plus the tight-coupling approximation of §5.7 that replaces the stiff Thomson-scattering terms with a sound-speed equation for the combined photon-baryon fluid. Together these turn an intractable integro-differential problem into a finite system that a Runge-Kutta integrator can carry from redshift $z sim 10^9$ to today.

Extended reading notes

Core claim

The central claim is that a single, complete linear perturbation theory can be written down for all the species relevant to flat CDM, HDM, and CDM+HDM cosmologies, and that solving it in two different gauges gives the same answer. The bridge between the gauges is an explicit coordinate transformation, equation (18), which expresses the conformal Newtonian potentials $psi$ and $varphi$ in terms of the synchronous fields $h$ and $eta$ and their time derivatives. With that bridge in hand, the authors derive super-horizon isentropic initial conditions in both gauges — equations (96) and (98) — showing that the same physical mode appears as a growing density perturbation in the synchronous gauge and as a constant one in the conformal Newtonian gauge, with the difference traceable entirely to the coordinate choice. They then integrate the coupled system numerically for a standard CDM model and two mixed models with neutrino mass densities $Omega_nu = 0.2$ and $0.3$ (neutrino masses 4.7 eV and 7 eV), assuming a scale-invariant adiabatic spectrum. The two gauge integrations agree to better than 0.1 percent, and the resulting spectra are, on the authors' account, the first for mixed dark matter models that include photon polarization, higher neutrino multipole moments, and helium recombination.

Load-bearing premise

The quoted accuracy of the spectra rests on an estimate — grids, multipole ceilings, and timesteps doubled until the answer stopped moving — rather than on a proven error bound, so everything hangs on the assumption that the closure rules of equations (51), (58), and (65) and the tight-coupling switch temperatures of §5.7 converge for every wavenumber and multipole the paper uses.

Editorial extensions

If this is right

Adding a massive neutrino at $Omega_nu = 0.2$ or $0.3$ raises the anisotropy at high multipoles by roughly 10 percent at the second and third acoustic peaks, so a percent-level measurement of peak heights separates mixed dark matter from pure CDM.
Free-streaming of the massive neutrino suppresses its own perturbation growth and slows the growth of CDM in turn, with the suppression scale set by $k_{fs}(a) propto a^{1/2} m_ u$, so the shape of the matter power spectrum becomes a handle on neutrino mass.
Because the two gauges agree to 0.1 percent, the gauge-invariant content of the calculation is confirmed, and a user may choose whichever gauge simplifies an extension — for example, integrating neutrino geodesics to build N-body initial conditions.
The released code makes the entire pipeline reproducible and extendable to a cosmological constant and to additional species without rederiving the perturbation equations.

Reading between the lines

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

If two independently written Boltzmann codes ever disagree by more than about 0.1 percent on a mixed dark matter spectrum, that disagreement is more likely a truncation artifact or a coding error than new physics — the paper effectively sets a benchmark for what 'agreeing codes' should mean.
The weakest link in the numerical chain is the hierarchy closure, which rests on an asymptotic analogy to spherical Bessel functions; a genuinely independent check would come from a line-of-sight integration that never truncates the multipole expansion at all.
The sensitivity of the acoustic peak heights to $Omega_ nu$ runs in both directions: the same machinery, inverted, turns a measured peak ratio into a constraint on the sum of neutrino masses.
The gauge transformation itself is a reusable tool beyond this paper — any calculation performed in the synchronous gauge can be mapped into the conformal Newtonian gauge where the potentials have a Newtonian interpretation, which is exactly what is needed when feeding linear results into Newtonian N-body simulations.

Original abstract

This paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. It differs from others in the literature in that we give, in both gauges, a complete discussion of all particle species that are relevant to any flat cold dark matter (CDM), hot dark matter (HDM), or CDM+HDM models (including a possible cosmological constant). The particles considered include CDM, baryons, photons, massless neutrinos, and massive neutrinos (an HDM candidate), where the CDM and baryons are treated as fluids while a detailed phase-space description is given to the photons and neutrinos. Particular care is applied to the massive neutrino component, which has been either ignored or approximated crudely in previous works. Isentropic initial conditions on super-horizon scales are derived. The coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric and density perturbations are then solved numerically in both gauges for the standard CDM model and two CDM+HDM models with neutrino mass densities $\onu=0.2$ and 0.3, assuming a scale-invariant, adiabatic spectrum of primordial fluctuations. We also give the full details of the cosmic microwave background anisotropy, and present the first accurate calculations of the angular power spectra in the two CDM+HDM models including photon polarization, higher neutrino multipole moments, and helium recombination. The numerical programs for both gauges are available at http://arcturus.mit.edu/cosmics/ .

The compiled record

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

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,594 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.44 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.65 · compares with prior results 0.84 · mentions code or data availability 0.98.

The abstract, sentence by sentence

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

This paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. It differs from others in the literature in that we give, in both gauges, a complete discussion of all particle species that are relevant to any flat cold dark matter (CDM), hot dark matter (HDM), or CDM+HDM models (including a possible cosmological constant). The particles considered include CDM, baryons, photons, massless neutrinos, and massive neutrinos (an HDM candidate), where the CDM and baryons are treated as fluids while a detailed phase-space description is given to the photons and neutrinos. Particular care is applied to the massive neutrino component, which has been either ignored or approximated crudely in previous works. Isentropic initial conditions on super-horizon scales are derived. The coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric and density perturbations are then solved numerically in both gauges for the standard CDM model and two CDM+HDM models with neutrino mass densities $\onu=0.2$ and 0.3, assuming a scale-invariant, adiabatic spectrum of primordial fluctuations. We also give the full details of the cosmic microwave background anisotropy, and present the first accurate calculations of the angular power spectra in the two CDM+HDM models including photon polarization, higher neutrino multipole moments, and helium recombination. The numerical programs for both gauges are available at http://arcturus.mit.edu/cosmics/ .

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge.	contribution 0.99	0.93	0.07	asserted 1.00
s1	It differs from others in the literature in that we give, in both gauges, a complete discussion of all particle species that are relevant to any flat cold dark matter (CDM), hot dark matter (HDM), or CDM+HDM models (including a possible cosmological constant).	contribution 0.98	0.85	0.07	asserted 1.00
s2	The particles considered include CDM, baryons, photons, massless neutrinos, and massive neutrinos (an HDM candidate), where the CDM and baryons are treated as fluids while a detailed phase-space description is given to the photons and neutrinos.	contribution 0.56	0.86	0.04	asserted 1.00
s3	Particular care is applied to the massive neutrino component, which has been either ignored or approximated crudely in previous works.	gap 0.89	0.23	0.05	asserted 0.99
s4	Isentropic initial conditions on super-horizon scales are derived.	contribution 0.59	0.94	0.06	asserted 0.99
s5	The coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric and density perturbations are then solved numerically in both gauges for the standard CDM model and two CDM+HDM models with neutrino mass densities $\onu=0.2$ and 0.3, assuming a scale-invariant, adiabatic spectrum of primordial fluctuations.	method 0.94	0.97	0.09	asserted 1.00
s6	We also give the full details of the cosmic microwave background anisotropy, and present the first accurate calculations of the angular power spectra in the two CDM+HDM models including photon polarization, higher neutrino multipole moments, and helium recombination.	contribution 0.58	0.96	0.17	asserted 1.00
s7	The numerical programs for both gauges are available at http://arcturus.mit.edu/cosmics/ .	contribution 0.69	0.72	0.02	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.53 · evidence from simulation or computation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

Figures

**Figure 1.** — Top: The angular power spectrum Cl of the photon anisotropy (including polarization) in the CDM (solid), the Ων = 0.2 CDM+HDM (dashed), and the Ων = 0.3 CDM+HDM (dotted) models. Middle: The ratio of Cl for the CDM+HDM models relative to the CDM (dashed for Ων = 0.2; dotted for Ων = 0.3). Bottom: The fractional difference in Cl with and without polarization for the three models

**Figure 2.** — The scalar metric perturbations φ(k,τ ) and ψ(k,τ ) in the conformal Newtonian gauge in the Ων = 0.2 CDM+HDM model. The 31 curves from left to right correspond to 31 values of k between 10.0 Mpc−1 and 0.01 Mpc−1 . The labels τnr, τeq and τrec indicate, respectively, the time the 4.7 eV neutrinos become non-relativistic, the matter-radiation equality time, and the recombination time

**Figure 3.** — Evolution of the density fields in the synchronous ga

**Figure 4.** — The z = 0 power spectra for the pure CDM (solid), the Ων = 0.2 CDM+HDM (dashed), and the Ων = 0.3 CDM+HDM (dotted) models. In the mixed models, the upper curve is for the CDM component and the lower one is for the HDM. The COBE-compatible quadrupole moment of Qrms−PS = 17.6 µ K is used

Figures reproduced from arXiv:astro-ph/9506072 by the authors.

References

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Xiang, S., & Kiang, T. 1992, MNRAS, 259, 761 This preprint was prepared with the AAS L ATEX macros v3.0. – 40 – 10. Figure Captions Fig. 1: Top: The angular power spectrum Cl of the photon anisotropy (including polarization) in the CDM (solid), the Ω ν = 0.2 CDM+HDM (dashed), and the Ω ν = 0.3 CDM+HDM (dotted) models. Middle: The ratio of Cl for the CDM+H...

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Forward citations

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Relative velocity of dark matter and baryonic fluids and the formation of the first structures
astro-ph.CO 2010-05 conditional novelty 8.0 of 10

Supersonic relative motion between baryons and dark matter after recombination suppresses small-scale power and the abundance of the first halos, and makes their clustering scale-dependent and stochastic.
Planting neutrino SE-EdS: A < 3% accurate clustering approximation for neutrinos and other hot dark matter
astro-ph.CO 2026-11 conditional novelty 7.0 of 10

A new rational-function fit anchored to exact Einstein-de Sitter solutions gives linear neutrino density contrasts to better than 3 percent over a wide cosmological parameter range.
CAMB v2: cosmological power spectra for high-precision surveys
astro-ph.CO 2026-07 accept novelty 7.0 of 10

CAMB v2's default settings produce lensed CMB and matter power spectra converged to 10^-3 over survey-relevant scales, using a new flat-Bessel mapping for hyperspherical Bessel functions.
The Rise and Fall of Acoustic Oscillations at Cosmic Dawn
astro-ph.CO 2026-07 conditional novelty 7.0 of 10

Ignoring the percent-level phase offset between BAO and VAO features in the cosmic-dawn 21-cm power spectrum biases H(z) by ~2%; joint BAO–VAO templates are required for standard-ruler inference.

Show 55 more citing papers

Optimal and exact wide-angle power spectrum estimation
astro-ph.CO 2026-07 accept novelty 7.0 of 10

For finite-rank signals the optimal wide-angle estimator is the two-ℓ Yamamoto form, whose exact window is a finite FFT-computable sum that improves ultra-large-scale SNR by O(1).
Self-interacting neutrinos in cosmological perturbation theory -- integrating the collision kernel
astro-ph.CO 2026-06 unverdicted novelty 7.0 of 10

Derives exact rational-plus-π² closed forms for every multipole coefficient α_ℓ in the neutrino collision kernel by rewriting the kernel as angular derivatives of a Yukawa potential and reducing the integrals via a fi...
Gravitational Waves from Black Hole Reheating: The Scalar-Induced Component
hep-ph 2026-05 unverdicted novelty 7.0 of 10

Accounting for the minimal mass spread of primordial black holes from gravitational collapse suppresses the Poltergeist GW background to the level of generic scalar-induced signals and reopens ultra-light PBH parameter space.
Disentangling Similarity and Relatedness in Topic Models
cs.CL 2026-03 unverdicted novelty 7.0 of 10

Topic models lie on a spectrum from thematic-relatedness-rich to similarity-rich, and that position predicts which downstream tasks they handle well.
Isocurvature Induced Gravitational Waves at Pulsar Timing Arrays
astro-ph.CO 2025-12 unverdicted novelty 7.0 of 10

The work shows that free-streaming dark radiation isocurvature produces a qualitatively different gravitational wave spectrum than cold dark matter isocurvature and derives constraints on isocurvature power spectra ar...
Growth of Structure in Multi-species Wave Dark Matter
astro-ph.CO 2025-10 unverdicted novelty 7.0 of 10

Derives the power spectrum evolution and cross-spectra for arbitrary multi-species wave and particle dark matter, incorporating free-streaming, Jeans scales, and intrinsic fluctuations.
Accurate method for ultralight axion CMB and matter power spectra
astro-ph.CO 2024-12 conditional novelty 7.0 of 10

AxiECAMB computes CMB and matter power spectra for ultralight axions across 10^-33 to 10^-18 eV by time-averaging fast field oscillations, with metric and fluid matching that reduces errors far below cosmic variance f...
$\Lambda_{\rm s}$CDM cosmology from a type-II minimally modified gravity
astro-ph.CO 2024-02 unverdicted novelty 7.0 of 10

Λ_s VCDM is a predictive model combining Λ_s CDM with VCDM gravity via an auxiliary scalar field and sigmoid-smoothed potentials to enable stable mirror AdS-to-dS transitions with possible transient acceleration.
Invisible neutrino decay in precision cosmology
astro-ph.CO 2020-11 conditional novelty 7.0 of 10

The cosmological damping rate for invisible neutrino decay scales as (m/E)^5, not (m/E)^3, relaxing the neutrino lifetime bound by three to four orders of magnitude.
Resolving the Hubble Tension with New Early Dark Energy
astro-ph.CO 2020-06 conditional novelty 7.0 of 10

A triggered first-order phase transition in a dark sector before recombination lowers the sound horizon and yields H0 = 71.4 +/- 1.0 km/s/Mpc with the local prior, reducing the Hubble tension to 2.5 sigma without it.
New Early Dark Energy
astro-ph.CO 2019-10 conditional novelty 7.0 of 10

A first-order dark-sector phase transition before recombination supplies a short early dark energy burst that fits CMB, BAO, supernova and local H0 data better than ΛCDM, yielding H0 = 71.4 ± 1.0 km/s/Mpc.
Tetrad formalism for exact cosmological observables
astro-ph.CO 2019-08 accept novelty 7.0 of 10

The paper derives exact, coordinate-independent equations for cosmological observables on a new 'observer space-time' manifold, using tetrads to track the observer frame.
Oscillating scalar fields and the Hubble tension: a resolution with novel signatures
astro-ph.CO 2019-08 conditional novelty 7.0 of 10

A subdominant oscillating scalar field with about 10% energy density at redshift 3500 and a potential ∝ φ^{2n} (n≈3) can resolve the Hubble tension, with CMB-S4-detectable polarization signatures and self-resonance for n≈2.
Structure Formation and Exotic Compact Objects in a Dissipative Dark Sector
hep-ph 2018-12 conditional novelty 7.0 of 10

A two-particle dissipative dark sector fragments into compact objects spanning a few to millions of solar masses, with mass and compactness calculable from the dark-electron mass, dark-photon mass, and dark fine-struc...
gevolution: a cosmological N-body code based on General Relativity
astro-ph.CO 2016-04 accept novelty 7.0 of 10

gevolution solves the weak-field Einstein equations on a particle-mesh lattice, reproduces the Schwarzschild metric and Newtonian codes where they apply, and yields the gravitational slip sourced by neutrino anisotrop...
ETHOS - An Effective Theory of Structure Formation: From dark particle physics to the matter distribution of the Universe
astro-ph.CO 2015-12 accept novelty 7.0 of 10

ETHOS maps dark matter microphysics onto a small set of effective parameters controlling the linear matter power spectrum and self-interaction cross section, so many particle models can be simulated and constrained together.
Boltzmann hierarchy for interacting neutrinos I: formalism
astro-ph.CO 2014-09 conditional novelty 7.0 of 10

A first-principles derivation shows interacting neutrinos must be tracked momentum by momentum, and that the popular (c_eff^2, c_vis^2) shortcut has no correspondence to actual particle scattering.
Cosmological Non-Linearities as an Effective Fluid
astro-ph.CO 2010-04 accept novelty 7.0 of 10

Small-scale cosmological non-linearities renormalize the background only at order v squared and induce a viscous effective fluid whose parameters can be matched to simulations, with virialized scales decoupling entirely.
Large-scale instability in interacting dark energy and dark matter fluids
astro-ph 2008-04 conditional novelty 7.0 of 10

Constant-equation-of-state dark energy coupled to dark matter develops a runaway growth of the curvature perturbation on super-Hubble scales in the early radiation era, regardless of coupling strength.
Structure Formation with Generalized Dark Matter
astro-ph 1998-01 conditional novelty 7.0 of 10

A three-parameter, gauge-invariant description of dark matter stresses reproduces cold dark matter, radiation, massive neutrinos, and scalar fields as limiting cases, and lets the clustering scale be varied independen...
Probing the fate of large primordial perturbations with exoplanets
astro-ph.CO 2026-06 conditional novelty 6.5 of 10

Observed ultra-wide exoplanets already constrain the amplitude and scale of primordial power-spectrum peaks that produce ultra-compact minihalos, with misalignment signatures as a future probe.
Going off the (momentum) grid: A simple implementation of noncold relics in linear Boltzmann solvers
astro-ph.CO 2026-10 accept novelty 6.0 of 10

The authors implement and validate the GBH and DFA approximations in CLASS, matching standard massive neutrino results to sub-permille accuracy with comparable runtime.
At the EDGES of MADness: phenomenology of relic millicharged atomic dark matter
astro-ph.CO 2026-10 accept novelty 6.0 of 10

A relic ionized fraction left by dark recombination supplies a small, natural millicharged dark matter component with viable parameter space overlapping the EDGES window.
Beyond dust in local cosmography: perturbative Vlasov predictions
astro-ph.CO 2026-10 conditional novelty 6.0 of 10

Velocity dispersion from structure formation shifts local cosmographic coefficients at the sub-percent level and produces an intrinsic Hubble dipole far too small to explain the observed low-redshift dipole.
Critical magnetic Reynolds number of the weakly-collisional turbulent dynamo in supersonic plasmas
astro-ph.CO 2026-10 accept novelty 6.0 of 10

The critical magnetic Reynolds number for the weakly-collisional turbulent dynamo in supersonic plasmas (M≈2) is measured to be Rm_crit≈190-198, about 1.5 times higher than in collisional MHD.
The unified Bound Dark Energy--Bound Dark Matter model: theory and constraints from DESI DR2
astro-ph.CO 2026-10 conditional novelty 6.0 of 10

A unified model where dark energy and dark matter originate from the same confining gauge group fits cosmological data with strong Bayesian evidence against LCDM.
Big Bang Nucleosynthesis Confronts Domain Walls and First-Order Phase Transitions as the Pulsar Timing Array Explanations
hep-ph 2026-09 conditional novelty 6.0 of 10

Domain-wall and phase-transition sources of the nanohertz gravitational-wave background imprint a baryon isocurvature that shifts primordial deuterium, alleviating or overproducing D/H depending on source strength.
Less isn't more: Cosmological bounds on the neutrino masses are robust to changes in the neutrino abundance
astro-ph.CO 2026-09 accept novelty 6.0 of 10

Cosmological neutrino mass bounds are robust to changes in the neutrino abundance; reducing the neutrino abundance tightens the constraints.
Cosmological Signatures of Curvature-Coupled Dark Energy
astro-ph.CO 2026-09 conditional novelty 6.0 of 10

A curvature-coupled dark energy model with a shifted quartic coupling and inverse power-law potential predicts sizeable deviations from ΛCDM in cosmological observables and may alleviate the H0 tension.
A Multi-Axion Ladder Across Cosmic History: From Inflation, BBN, and Early Dark Energy to Late-Time Accelerated Expansion
astro-ph.CO 2026-09 conditional novelty 6.0 of 10

A multi-axion model with one BBN transient, two early dark energy components, and one late-time thawing dark energy field is evolved from z=10^11 to today, producing benchmark peak fractions of 0.99% (BBN), 7.8% and 8...
Conformally Coupled Scalar- Field Dark Energy: Cosmological Dynamics with Early- and Late- Measurements
astro-ph.CO 2026-09 conditional novelty 6.0 of 10

A conformally interacting dark-energy model is fit to CMB, BAO, and supernova data, favoring a small dark-matter-to-dark-energy energy transfer and a quintessence-like equation of state, with only marginal improvement...
Non-Minimally Coupled Warm Inflation in the Defining Frame
astro-ph.CO 2026-08 conditional novelty 6.0 of 10

In non-minimally coupled warm inflation, computing quantum and thermal effects in the defining frame suppresses the effective Einstein-frame dissipation ratio, so strong defining-frame dissipation can coexist with qua...
Emulating the nonlinear effects of modified gravity on the matter power spectrum for reconstruction
astro-ph.CO 2026-07 conditional novelty 6.0 of 10

A neural-network emulator predicts the ratio of nonlinear to linear modified-gravity matter power spectra across a 28-dimensional cosmological and MG parameter space, matching MGCAMB+ReACT to roughly 1–2%.
Phantom-divide crossing and suppressed structure growth in kinetically braided dark energy with momentum exchange
astro-ph.CO 2026-07 conditional novelty 6.0 of 10

A linearly stable kinetic-braiding dark-energy model with CDM momentum exchange realizes upward phantom-divide crossing and weak CDM gravitational clustering, altering matter and CMB spectra.
The silent dark sector: a field-theory approach to unified dark fluids with vanishing speed of sound
astro-ph.CO 2026-07 conditional novelty 6.0 of 10

All vanishing-speed-of-sound TDiff scalar theories reduce to a two-parameter mimetic action describing the dark sector as interacting dust plus vacuum; a quadratic-potential example tracks the DESI CPL background.
Defocusing dark energy: Raychaudhuri diagnostics beyond $w<-1/3$ and the phantom divide
gr-qc 2026-07 accept novelty 6.0 of 10

For sign-changing effective dark energy, smooth negative-to-positive density crossings make ρ+3p and ρ+p the correct diagnostics, give w=p/ρ a universal pole n(1+z†)/3, and force repulsion onset z_rep>z† while ρ<0.
Alleviating the Hubble Tension with Smooth Sign-Switching Dark Energy: Full CMB Constraints with DESI and PantheonPlus
astro-ph.CO 2026-07 conditional novelty 6.0 of 10

Smooth ECDM dark energy remains compatible with Planck+ACT+SPT, DESI DR2 and Pantheon+/SH0ES while alleviating the Hubble tension through a controlled late-time density transition.
A First Post-Friedmann Extension of the Schr\"odinger Approach to Cosmic Structure Formation
astro-ph.CO 2026-06 unverdicted novelty 6.0 of 10

Derives a 1PF relativistic extension of the Schrödinger approach to cold matter dynamics that requires an effective vector potential for the transverse velocity component in the cosmological frame.
Revisiting observational constraints on coupled exponential quintessence with energy and momentum transfers: degeneracy with massive neutrinos
astro-ph.CO 2026-03 conditional novelty 6.0 of 10

Allowing the neutrino mass to vary removes the 2σ evidence for momentum transfer between dark energy and dark matter in coupled quintessence, via a degeneracy with the energy-exchange coupling.
Dark sector interactions in the $w \rightarrow -1$ limit: velocity locking in pure momentum exchange models
astro-ph.CO 2025-12 conditional novelty 6.0 of 10

In momentum-exchange dark sector models, the suppression of matter clustering moves to smaller scales as w→−1 due to velocity locking, and ignoring DE perturbations erases this effect.
Cosmological perturbations in Energy-Momentum Squared Gravity
gr-qc 2025-11 conditional novelty 6.0 of 10

A covariant perturbation analysis shows EMSG changes the growth of density clumps, the decay of vorticity, and the damping of gravitational waves relative to General Relativity, giving searchable cosmological signatures.
Interpreting the Hubble tension with a cascade decaying dark matter sector
astro-ph.CO 2025-07 conditional novelty 6.0 of 10

A cascade decaying dark matter model combining early and late effects still cannot reduce the Hubble tension below about 3 sigma, and is statistically disfavored relative to Lambda CDM.
Cosmological impact of $\nu$DM interactions enhanced in narrow redshift ranges
astro-ph.CO 2025-05 conditional novelty 6.0 of 10

A redshift-limited dark matter-neutrino interaction reproduces the observed preference for non-zero coupling in CMB and cosmic shear data and evades small-scale structure bounds.
CMB Lensing Trispectrum as a Probe of Parity Violation in LSS
astro-ph.CO 2025-05 conditional novelty 6.0 of 10

The CMB lensing trispectrum is sensitive to parity violation in large-scale structure, and a parity-odd toy model predicts a detectable signal in idealized noiseless forecasts.
Joint 21-cm and CMB Forecasts for Constraining Self-Interacting Massive Neutrinos
astro-ph.CO 2025-04 conditional novelty 6.0 of 10

21-cm power spectrum forecasts show HERA and CMB-S4 combined can constrain the neutrino self-coupling G_eff to about 10 percent across strong, moderate, and mild interaction models, with futuristic lunar arrays reachi...
Clustered unified dark sector cosmology: Background evolution and linear perturbations in light of observations
astro-ph.CO 2025-02 conditional novelty 6.0 of 10

CMB secondary anisotropies constrain the clustered fraction in unified Chaplygin gas models to f greater than or similar to 0.99, leaving little room for deviations from LambdaCDM.
Origin of cosmological neutrino mass bounds: background $\textit{versus}$ perturbations
astro-ph.CO 2024-11 conditional novelty 6.0 of 10

The CMB bound on the sum of neutrino masses comes mostly from the background energy density of massive neutrinos, not from their free-streaming effects.
Atomic Dark Matter, Interacting Dark Radiation, and the Hubble Tension
hep-ph 2024-11 conditional novelty 6.0 of 10

nuADaM, a model of atomic dark matter plus self-interacting dark radiation, improves cosmological fits and raises the inferred Hubble constant to about 72.6 km/s/Mpc.
Negative neutrino masses as a mirage of dark energy
astro-ph.CO 2024-07 conditional novelty 6.0 of 10

Allowing an effective neutrino mass to take negative values exposes a 2.8-3.3σ tension with oscillation experiments under ΛCDM, which w0wa and mirage dark energy models absorb.
A Step in Understanding the Hubble Tension
astro-ph.CO 2021-10 conditional novelty 6.0 of 10

A supersymmetric dark-radiation model with an eV-scale mass threshold improves fits to CMB, BAO, and SH0ES data and reduces the Hubble tension from 4.5 sigma to 2.7 sigma.
Cosmological constraints on multi-interacting dark matter
astro-ph.CO 2020-10 conditional novelty 6.0 of 10

Simultaneous dark matter scattering off photons and dark radiation, implemented in a single Boltzmann code, lowers the S8 tension to 1.2 sigma and the H0 tension to 3.1 sigma with no degeneracy between interaction channels.
Implications of the $S_8$ tension for decaying dark matter with warm decay products
astro-ph.CO 2020-08 conditional novelty 6.0 of 10

Decaying dark matter with a 55 Gyr lifetime and 0.7 percent mass transfer into dark radiation suppresses small-scale clustering enough to reconcile Planck with weak lensing S8 measurements, though Bayesian evidence sl...
Relaxing Cosmological Neutrino Mass Bounds with Unstable Neutrinos
hep-ph 2020-07 conditional novelty 6.0 of 10

Invisible two-body neutrino decays into a light sterile state and a Goldstone boson can relax the Planck neutrino mass bound from 0.12 eV to about 1 eV, and a simple type-I seesaw extension realizes the required couplings.
Constraints on primordial gravitational waves from the Cosmic Microwave Background
astro-ph.CO 2020-04 conditional novelty 6.0 of 10

Using Planck 2018 and BICEP2/Keck data, the authors bound the primordial gravitational wave energy density to below roughly 10^-6 across all CMB-accessible frequencies, including a previously unconstrained band near t...
Gravitational Wave Production right after a Primordial Black Hole Evaporation
astro-ph.CO 2020-03 conditional novelty 6.0 of 10

Sudden reheating from evaporating primordial black holes amplifies induced gravitational waves enough that future detectors could constrain the abundance of 10^3 to 10^5 gram black holes.

See all 213 Pith citations

How later papers used it

545 later papers cite it

Evidence: substantial. 545 of the citing papers share no author with it, and 201 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 1,100 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

32 later papers by other authors are listed below: 32 used its method or tool.

Uses its method or tool 32 of 190 papers with this label

Modified the public code Class to implement the coupled Boltzmann equations of this paper.

Residual Non-Abelian Dark Matter and Dark Radiation (2016) · 1.00 · paraphrased: the licence does not allow quoting
Computed the Newtonian potential from code written in synchronous gauge using a transformation involving the synchronous gauge potentials of this paper.

The effects of the small-scale behaviour of dark matter power spectrum on CMB spectral distortion (2017) · 1.00 · paraphrased: the licence does not allow quoting
Used the closing relations defined in this paper for both the first and the second order.

The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations (2010) · 1.00 · paraphrased: the licence does not allow quoting
Followed the truncation method introduced in this paper as implemented in its code.

Dark matter self-interactions in the matter power spectrum (2022) · 1.00 · paraphrased: the licence does not allow quoting
Employed this paper's general formalism for scalar perturbations to study how the growth of perturbations is reduced by scattering.

Partially Acoustic Dark Matter, Interacting Dark Radiation, and Large Scale Structure (2016) · 1.00 · paraphrased: the licence does not allow quoting
Truncated the Boltzmann hierarchy at order l_max=5 using the approximation outlined in this paper.

Cosmological Signatures of a Mirror Twin Higgs (2018) · 1.00 · paraphrased: the licence does not allow quoting
Used the truncation prescription of this paper.

The Cosmological Evolution of Self-interacting Dark Matter (2021) · 1.00 · paraphrased: the licence does not allow quoting
Used a re-implementation of both the Newtonian and the comoving synchronous equations of this paper for its full linear theory calculations.

General relativistic 'screening' in cosmological simulations (2016) · 1.00 · paraphrased: the licence does not allow quoting
Employed this paper's general formalism for scalar perturbations in the conformal Newtonian gauge.

Reconciling Large And Small-Scale Structure In Twin Higgs Models (2016) · 1.00 · paraphrased: the licence does not allow quoting
We truncate the Boltzmann hierarchy with the traditional method in Ref. [ref] .

Cosmic microwave background spectral distortions from Rayleigh scattering at second order (2022) · Boltzmann Hierarchy equations · 1.00
Followed this paper to compute the fluid limit to the perturbed equations in k-space in the synchronous gauge for WDM.

Analytic Fluid Approximation for Warm Dark Matter (2019) · 1.00 · paraphrased: the licence does not allow quoting
Since we will keep perturbation in the these fluids to linear order, we will consider these perturbations in momentum space following the method derived in Ref. [ref] .

A novel integrated Sachs-Wolfe effect from Early Dark Energy (2023) · Linear and Nonlinear Numerical Methods · 1.00
Chose l_max=8 and obtained closed differential equations using the truncation relation of this paper.

Isocurvature Induced Gravitational Waves at Pulsar Timing Arrays (2025) · 1.00 · paraphrased: the licence does not allow quoting
Derived the initial conditions for the gravitational potentials including anisotropic stress contributions using the method described in this paper.

The Phase of the BAO on Observable Scales (2020) · 1.00 · paraphrased: the licence does not allow quoting
Effectively truncated the Boltzmann hierarchy by adopting the scheme of this paper.

Cosmological constraints on Lorentz invariance violation in the neutrino sector (2012) · 0.99 · paraphrased: the licence does not allow quoting
Followed the procedure given by this paper, in which the stress-energy tensor to linear order is given by an equation with density, pressure, anisotropic shear and fluid velocity perturbations.

Dark Matter Decaying into a Fermi Sea of Neutrinos (2010) · 0.99 · paraphrased: the licence does not allow quoting
Employed this paper's formalism to study the more subtle effects of dark gluons on the evolution of DM overdensities.

Non-Abelian dark matter and dark radiation (2015) · 0.99 · paraphrased: the licence does not allow quoting
Gave the linear perturbation theory equations in conformal Newtonian gauge for DE-DM interaction fluids according to this paper.

Observational constraints on dark matter-dark energy scattering cross section (2017) · 0.81 · paraphrased: the licence does not allow quoting
Numerically solved the coupled linearized Einstein, Boltzmann and fluid equations of this paper to describe the spatial evolution of gas around the black hole from BBN until decoupling.

Nucleosynthetic signatures of primordial origin around supermassive black holes (2021) · 0.98 · paraphrased: the licence does not allow quoting
Followed the evolution of cosmological perturbations using equations with the gravitational potential in the synchronous gauge of this paper.

Effects of electrically charged dark matter on cosmic microwave background anisotropies (2016) · 0.85 · paraphrased: the licence does not allow quoting
The conformal Newtonian gauge is parametrized as, [equation] To go from the synchronous gauge to the conformal Newtonian gauge, we use the transformation [ref] [equation] Once again, to get the leading behavior in the superhorizon limit, we approximate $η≫h$, [equation] implying [equation] Since we showed above that the sign of $σ(<0 or >0)$ depends upon the nature of DR (CDR or FDR), and we also showed $σ≈-3η$ on superhorizon scales, we see that $ϕ+ψ>0 (<0)$ for CDR (FDR).

Free-streaming and Coupled Dark Radiation Isocurvature Perturbations: Constraints and Application to the Hubble Tension (2021) · Effects of shear · 0.90
Solved numerically its equations together with the standard Boltzmann equations describing the evolution of baryons, photons and neutrinos from this paper.

The Cosmology of Atomic Dark Matter (2012) · 0.98 · paraphrased: the licence does not allow quoting
Analyzed the evolution equations of the linear perturbations in Fourier space and in the synchronous gauge of this paper.

The Early-Time Evolution of the Cosmological Perturbations in f(R) Gravity (2011) · 0.85 · paraphrased: the licence does not allow quoting
Worked in synchronous gauge following the notation and conventions of this paper.

Constraining Dark Matter-Baryon Scattering with Linear Cosmology (2013) · 0.98 · paraphrased: the licence does not allow quoting
Used equations from this paper governing the evolution of the density contrast and fluid velocity of unparticles in a spatially flat universe.

Emergent Unparticles Dark Energy can restore cosmological concordance (2023) · 0.86 · paraphrased: the licence does not allow quoting
We use the definitions for stress-energy components according to [ref] , which account for the curvature of FLRW space since the negative pressure of dark energy is a large-scale effect, and we quantize the field below with this in mind as well.

Deriving the Dark Matter-Dark Energy Interaction Term in the Continuity Equation from the Boltzmann Equation (2019) · Introduction · 0.94
Transformed an equation into k space and otherwise followed the standard procedure of linear perturbation theory of this paper to arrive at a hierarchy.

SU(2)$_{\rm CMB}$ and the cosmological model: angular power spectra (2018) · 0.97 · paraphrased: the licence does not allow quoting
Put the synchronous gauge metric into an equation of this paper to obtain four equations.

Minimal parameterizations for modified gravity (2013) · 0.88 · paraphrased: the licence does not allow quoting
Used the recursion relation between higher photon multipole moments of this paper to obtain expressions for F_gamma2 accurate to second order in tau_c.

Photons and Baryons before Atoms: Improving the Tight-Coupling Approximation (2010) · 0.98 · paraphrased: the licence does not allow quoting
Closely followed this paper to describe the perturbation equations for massive neutrinos.

Bounds on Neutrino Mass in Viscous Cosmology (2017) · 0.90 · paraphrased: the licence does not allow quoting
Gave the modified Boltzmann hierarchy for massless neutrinos following the same notation of this paper, also for the definition of the harmonic expansion coefficients.

Hints of Neutrino Dark Matter scattering in the CMB? Constraints from the Marginalized and Profile Distributions (2023) · 0.84 · paraphrased: the licence does not allow quoting
We have adopted the convention $r_≡a^4 ρ̅_/ρ_c,0$, as in Ref. [ref] , which in the 2-body decay scenario under study leads to: [equation] The first three multipoles are given by: [equation] We choose the maximum multipole $ℓ_max$ to truncate the hierarchies of equations according to the scheme proposed in Ref. [ref] for both massless and massive neutrinos, i.e.

Linear cosmological constraints on 2-body decaying dark matter scenarios and the $S_8$ tension (2021) · Linear perturbation equations · 0.93

How the literature uses it

555 arXiv papers cite this in 1,115 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 730 sentences without a confident label

Contests it 1 paper

The time dependence of the terms proportional to $c_3$ and $c_4$ would not satisfy the second order differential equation given in [ref] , which therefore does not admit, strangely enough, the standard inflationary solution for $π_ν$ given for example in [ref] .

CMB temperature anisotropy at large scales induced by a causal primordial magnetic field · Resolution for the neutrino anisotropic stress at leading order · contest 0.84

Builds on it 6 papers

The treatment of paper II generalises and extends the standard spatial Boltzmann hierarchy for early-universe perturbations [ref] to describe the full spectro-spatial evolution of the photon field.

Spectro-spatial evolution of the CMB III: transfer functions, power spectra and Fisher forecasts · Brief recap of the important equations from paper I+II · 0.91
This is a straightforward generalization, suitable to the Newtonian evolution (i.e. let $ϵ= √(q^2+a^2m^2) →am$), of the truncation scheme found in [ref] .

Neutrinos in N-body simulations · Anisotropies of Newtonian and non-relativistic NCDM components · 0.98
The first-order equation of motion ([ref]) for the perturbed phase space density $F_i$ can also be recast as an equation of motion for the phase space density contrast $Ψ_i (𝐤,𝐪_i,τ)= F_i(𝐤,𝐪_i,τ)/f̅_i(|𝐪_i|,τ)$: [equation] This is a generalisation of equation (40) of reference [ref] to include a time-dependent background distribution.

Invisible neutrino decay in precision cosmology · First-order perturbation equations · 0.89

Uses its method or tool 194 papers

Considering the drag force [ref] as the only non-gravitational force between DE and DM, the linear perturbation theory equations (in conformal newtonian gauge) for DE-DM interaction fluids are given by [ref]

Observational constraints on dark matter-dark energy scattering cross section · Dark matter-dark energy scattering models · 0.81
we have modified the equations for the time evolution of the density, velocity, shear and higher anisotropic moments of the phase space distribution in the synchronous gauge [ref] and we have included them in CLASS [ref] , using MontePython [ref] to deduce the cosmological implications of this scenario.

Cosmology of an Axion-Like Majoron · Cosmological evolution: $T<10$ eV · 1.00
To visualize DM-DR's effects on matter power spectrum, we have modified the public code Class [ref] to implement the coupled Boltzmann equations [ref] , [equation] where dot means derivative over conformal time $dτ≡dt/a$ ($a$ is the scale factor), $θ_r$ and $θ_m$ are velocity divergences of DR $A^a$ and DM $A^m$, $k$ is the co-moving wavenumber, $Ψ$ is the gravitational potential, $δ_r$ and $σ_r$ ($σ_r=0$ for $A^a$ as perfect fluid) are the density perturbation and the anisotropic stress potential of $A^a$, and $ℋ ≡ȧ/a$ is the conformal Hubble parameter.

Residual Non-Abelian Dark Matter and Dark Radiation · Numeric Results · 1.00

Compares with it 12 papers

With $π_Ai = ∂_i π_A$, the spatial parts are recast into [equation] where we write $Σ$ and $ξ_Σ$ as [equation] Then, the second-order action yields the first-order continuity and Euler equations in the presence of the stochastic noise: [equation] where a hat implies [equation] Setting the noise to zero, Eqs.[ref] and [ref] are consistent with the linearized continuity and Euler equations presented in Ref. [ref] .

Fluctuation-dissipation relation in cosmic microwave background · Equation of motion with noise · 0.95
In the pure CDM limit, where $γ_0 = 0$, we can set $f_1 = 1$ and $f_2 = γ_E - 1/2$, which satisfies Eq. ([ref]) and the initial condition of $δ_χ = -1/2 C (k τ)^2$ for $x →0$ and $θ_χ = 0$, coinciding with that for the adiabatic mode given in Eq. (96) of Ref. [ref] .

Dark matter kinetic decoupling with a light particle · Evolution of the cosmological perturbations · 0.86
The difference here with the formulas from the Ref. [ref] is that we have fixed the Newtonian gauge from the very beginning, neglected the vector and tensor perturbations, and changed the names of the Newtonian potentials, $ϕ↔ψ$, in order to be in accordance with the Ref. [ref] .

Toward a concordance teleparallel Cosmology II: Linear perturbation · Linear perturbations in $f(T)$ gravity · 0.89

Cites it as support 1 paper

Within this local framework, the scattering amplitude relies solely on Lorentz invariants and is hence equivalent to the flat-space outcome [ref] .

Boltzmann Dynamics in K-essence Cosmology: Photon Propagation in an Emergent Spacetime · Local nature of collision terms and validity of flat-space cross sections · 0.96

Cites it as background 93 papers

The formalism describing the evolution of matter and dark energy density perturbations without [ref] and with dark matter and dark energy interaction [ref] is well established.

Testing the Interaction between Dark Energy and Dark Matter with Planck Data · The phenomenological model on the interaction between dark sectors · 0.90
In linear cosmological perturbation theory, the Boltzmann equation of NCDM particles depends on the amplitude of the particle momentum and the angle between the momentum and the Fourier wavevector $k⃗$ [ref] .[The case for massless particles is simpler, since the dependence on the norm of the momentum can be factored out.] It is typically rewritten as an infinite set of coupled differential equations for the momentum-dependent multipole moments of the perturbed phase-space-density distribution function.

Rapid and accurate numerical evolution of linear cosmological perturbations with non-cold relics · Motivation – Limitations of Boltzmann hierarchy approach · 0.93
Among other features, offers a fast solver for Einstein-Boltzmann equations, a large system of ODEs which describes the evolution of linear perturbations in the Universe [ref] .

sympy2c: from symbolic expressions to fast C/C++ functions and ODE solvers in Python · Introduction · 0.91

Discussion

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