Automatic detection of fast oscillations of dark matter scalar field and updated cosmological constraints on QCDM

Amin Aboubrahim; Pran Nath

REVIEW 3 major objections 6 minor 61 references

Automatic detection of fast oscillations of dark matter scalar field and updated cosmological constraints on QCDM

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the onset of rapid oscillations in scalar-field dark matter can be detected automatically from the field dynamics, triggering a consistent oscillation-averaged treatment without a user-supplied threshold, even when…

desk verdict Real algorithmic novelty in automatic oscillation-onset detection, but the validation is thin enough that the numerical claim should be checked before trusting the derived constraints. read the letter →

arxiv 2608.13346 v1 pith:ULK4RRWL submitted 2026-08-13 astro-ph.CO

Amin Aboubrahim , Pran Nath This is my paper · ORCID

classification astro-ph.CO

keywords scalarfielddarkmatterrapidoscillationsautomaticonsetdetectionoscillationaveragingQCDMmodelmatter-darkenergyinteractioncosmologicalconstraintsBoltzmannsolver

topics Dark Matter

open problems Dark Matter Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the onset of rapid oscillations in a scalar-field dark matter component can be detected automatically from the field's own dynamics, removing the need for a user-supplied threshold that is usually tuned by trial and error. The detector watches the ratio $r = y/\theta$ of two phase-space variables; a sudden drop in this ratio marks the transition to oscillatory behavior, and the interpretation is independently confirmed by requiring the dark matter equation of state $w_\chi = -\cos\theta$ to change sign repeatedly. The paper further shows that a naive averaging of the equations fails when dark matter interacts with dark energy, leaving a spurious interaction-induced pressure, and presents a modified averaging scheme that keeps the averaged equation of state zero in the oscillating regime. Implemented in the CLASS Boltzmann solver and applied to the interacting QCDM model, the technique enables scans of the parameter space and yields updated constraints from recent BAO, supernova, and CMB data, with Bayesian evidence still favoring the standard cosmological model.

What carries the argument

The central object is the ratio $r = y/\theta$ formed from the polar-angle variables $(\Omega_\chi,\theta,y)$ introduced to recast the Klein-Gordon equation as first-order flow equations. In radiation domination the attractor $r_0 = \beta + 3\sin\theta/\theta$ drops from 5 to 2 as $\theta$ grows, so a drop in $r$ marks the onset of oscillations; the detector learns the running maximum $R_n$ and triggers on $r_n \le 0.70\,R_n$, then confirms with sign changes of $w_\chi = -\cos\theta$. The averaging is carried by the transition function $T(\theta) = \tfrac{1}{2}\bigl(1 - \tanh\bigl((\theta-\theta_*)/w\bigr)\bigr)$ that interpolates exact and averaged variables, with $\theta_*$ set automatically from the detected drop plus padding, ensuring $\langle p_\chi\rangle = 0$ in the oscillatory regime even with a dark matter-dark energy interaction.

What would settle it

Run the CLASS module with the required ratio drop varied from 0.5 to 0.9 on a fixed QCDM benchmark and rerun the MCMC fits: if the recovered $H_0$, $S_8$, or interaction-strength posteriors move by more than their quoted 68% intervals, the automatic onset is not threshold-independent. Alternatively, compare the detected $\theta_*$ with the time at which the WKB-averaged equation of state $\langle w_\chi\rangle$ first vanishes to 1% accuracy in a non-interacting quadratic model over a grid of masses; a systematic offset would indicate the detector tracks a proxy rather than the true transition.

Watch

Extended reading notes

Core claim

The central claim is that the transition to the fast-oscillation regime of a scalar-field dark matter component can be detected automatically with no user-specified onset, and that a modified averaging scheme can then be applied consistently even when the dark matter couples to a quintessence field. The detector tracks the dimensionless ratio $r = y/\theta$ in the $(\Omega_\chi,\theta,y)$ background variables; at the onset of oscillations $r$ drops from its plateau value (5 in radiation domination) to a lower attractor (2). The code flags a candidate drop when $r$ falls by at least 30% below its running maximum, then requires that the dark matter equation of state $w_\chi = -\cos\theta$ change sign at least twice as an independent confirmation before averaging begins. The averaging uses a transition function $T(\theta)$ to blend exact and averaged quantities so that the interaction term contributes no spurious pressure, preserving a zero averaged equation of state in the oscillating regime. Applied to the QCDM model, the method reproduces consistent dynamics and yields updated constraints: $H_0 = 68.37^{+0.30}_{-0.26}\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ with the full data combination, while the Bayes factor ($\ln B = -2.189$) still favors $\Lambda$CDM.

Load-bearing premise

The detector is trusted to use one fixed set of thresholds (a 30% ratio drop, minimum phase $\theta_{\min} = 6$, two sign changes, and related settings) across the whole QCDM parameter space, although the ratio-drop criterion is derived only in the non-interacting limit.

Editorial extensions

If this is right

Large parameter scans and MCMC analyses of scalar-field dark matter no longer need a hand-tuned oscillation-onset threshold; the detector finds the transition separately at each model point.
The averaging method removes the spurious interaction-induced pressure that a naive cutoff would introduce, so the dark matter equation of state correctly returns to zero in the oscillation regime.
QCDM suppresses the late-time matter power spectrum by roughly 30-35% relative to $\Lambda$CDM while leaving the CMB nearly unchanged, with the growth-rate difference accumulating below the percent level into about a 16% amplitude reduction.
Updated data combinations raise $H_0$ to $68.37 \pm 0.30\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and lower $S_8$ to $0.8119 \pm 0.0079$, but the Bayesian evidence for QCDM over $\Lambda$CDM is negative ($\ln B = -2.189$ for the full dataset).

Reading between the lines

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

The analytic derivation of the ratio drop assumes no interaction, so the robustness of the automatic trigger in strongly coupled regions is an open question; a threshold-sensitivity scan varying $f_R$ and the transition width $w$ could test whether the recovered posteriors drift outside the quoted errors.
Because the detector operates on the field variables themselves rather than a fixed $H/m_\chi$ criterion, it may transfer to other scalar dark matter potentials (quartic, axionic) with only the sign-change confirmation test needing adaptation.
The manual-mode comparison in the paper shows the automatic trigger fires earlier than the classical cutoff prescription; quantifying how much of the difference in predicted CMB and matter-power spectra comes from the choice of onset versus the improved interaction averaging would make the padding parameter practically motivated rather than heuristic.

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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

Automatic onset detection via the r=y/theta drop with a sign-change confirmation is a genuinely new algorithmic idea, and the interaction-aware averaging is a real extension beyond non-interacting axion treatments. The validation, though, is thin enough that the numerical claim should be stress-tested before the constraints are taken at face value.

What is actually new: earlier methods require a user-supplied threshold (e.g., gamma_* ~ 100 in the polar-variable formalism). Here the code learns the plateau of r, detects a 30% drop, and then confirms oscillations via sign changes in w_chi. I do not see that combination in prior work. The averaging prescription in Sec. 5, which makes the interaction contribution to vanish in the T->0 limit, is also new and looks correct in the limit shown. The paper is honest about the main model comparison: the full data give lnB = -2.189 relative to LCDM, so the data do not prefer QCDM.

The soft spots, in order of importance. First, the analytic justification of the detection criterion, Eq. (6.4), is derived only for the non-interacting case, and the paper says so explicitly. In QCDM, theta and y use V1 only, while the KG equations contain interaction sources; the interacting evolution equations for theta and y are not given. For large lambda, m_eff^2 = m_chi^2 + lambda phi^2 changes while phi rolls, and the r-drop may be shallower or occur at a different theta than the fixed thresholds assume. The validation for the interacting case is a single benchmark lambda=1e5 Mpc^-2, while the MCMC prior spans lambda in [1e-2,1e9] Mpc^-2. No sensitivity study of the hand-chosen thresholds (f_R=0.70, w=5, padding, etc.) is presented. That is not fatal, but it means the central numerical claim is under-supported. Second, the code is stated to be on Github, but there is no URL or commit hash, so the implementation cannot be checked. Minor, and addressable.

Where the paper is solid: the discussion of why naive averaging fails with interactions (Sec. 4) is clear, the perturbation construction in Sec. 5 is carefully laid out, and the MCMC setup is standard.

Who this is for: people integrating scalar-field DM into Boltzmann codes, especially for parameter scans with DM-DE interactions. They will want to read it and probably test the detector on their own models. The paper deserves a serious referee, not a desk reject; the missing pieces are addressable in revision. My recommendation is to send it to review with a request for (a) a sensitivity analysis of the thresholds across the sampled lambda range, (b) a code link or hash, and (c) at least one additional benchmark where averaged spectra are compared to exact evolution.

Referee Report

3 major / 6 minor

Summary. This paper presents an automatic method to detect the onset of fast oscillations of a scalar-field dark matter component and to switch from the exact Klein-Gordon evolution to an averaged fluid description when interactions with a dark-energy field are present. The detector monitors the ratio r=y/theta, flags a 30% drop from its running maximum, and requires a separate confirmation from sign changes of the dark-matter equation of state; the transition phase is then set to theta* = max(theta_drop + 5, 6). The averaging uses a tanh transition function T(theta) and replaces interaction terms by their oscillation averages so that <p_chi>=0 and <delta p_chi>=0 in the oscillating regime. The scheme is implemented in CLASS, demonstrated on the QCDM model at lambda=10^5 Mpc^-2, and used in MCMC analyses with Planck, ACT, SPT-3G, DESI DR2, and Pantheon+ data. The resulting constraints favor LCDM, with Bayes factors ln B between -0.569 and -2.189 for the data combinations considered.

Significance. If the automatic detector and the interacting averaging scheme are reliable, the method would be a practical improvement for scanning scalar-field dark matter models, since it removes the need to guess a transition threshold by hand. The consistency conditions in Secs. 4 and 5 are physically sensible, and the public CLASS implementation together with the MCMC pipeline are useful contributions. The main significance is methodological; the updated cosmological constraints are a secondary application. However, the current validation rests on a non-interacting analytic example and a single interacting benchmark, which is insufficient to establish the central claim that the detector and averaging work across the QCDM parameter space.

major comments (3)

[Sec. 6, Eq. (6.4)] The analytic derivation of the ratio-drop criterion is performed only for the non-interacting equations (6.1)-(6.2), and the text explicitly states that this case is used "for illustrative purposes." In QCDM, theta and y are defined using V1 only (Eqs. (3.22)-(3.24)), while the field equations (3.14)-(3.15) contain V_int,chi and V_int,phi; the evolution equations for theta and y therefore acquire interaction source terms that are not presented. The fixed detector parameters in Table 1 (f_R=0.70, theta_min=6, etc.) are then applied over the entire prior log lambda in [-2,9] of Table 2, but the only interacting validation is the single benchmark lambda=10^5 Mpc^-2 in Figs. 1-5. This is a load-bearing gap because a premature or delayed switch changes the predicted background and perturbation evolution and hence the constraints in Table 3. Please provide the interacting theta and y equations, and validate the detector on a grid of lambda (and m_chi) values by comparing the automatic theta* with the true onset from exact integration, or by demonstrating that moderate changes in theta* do not alter the predicted spectra beyond the claimed accuracy.
[Sec. 5, Eqs. (5.13)-(5.15)] The averaging prescription for perturbations is asserted rather than derived. The paper defines delta V_int and delta p-tilde_chi so that <delta p_chi> -> 0 as T -> 0, but it does not show that the resulting averaged perturbation equations correctly reproduce the exact Klein-Gordon perturbation evolution at the transition. The only numerical evidence, Figs. 3-5, is a QCDM-versus-LCDM comparison at lambda=10^5 Mpc^-2 and says nothing about the error introduced by the averaging switch. Since the stated purpose of the method is to avoid biases at the handoff (cf. refs. [29,30]), please add a direct comparison of the averaged background and perturbation outputs to the exact evolution for at least the benchmark and a few representative points across the prior, with the target accuracy quantified.
[Table 1] The automatic detector introduces seven user-specified parameters, none of which is derived or tested for sensitivity. The paper's advertised advantage over the manual gamma* is that the user need not supply a threshold, but the algorithm still depends on f_R=0.70, w=5, theta_floor=10^-12, theta_min=6, w_min=0.10, min_sign_changes=2, and theta_pad=5. For large lambda, where m_eff^2=m_chi^2+lambda phi^2 varies appreciably while phi rolls, the r=y/theta plateau drop may be shallower than 30% or occur at a different theta, and the sign-change test only confirms that w_chi oscillates, not that theta* coincides with the true onset. Please provide a sensitivity analysis of the detector parameters and report the distribution of detected theta* (or transition redshift) over the MCMC samples; otherwise the robustness of the Table 3 constraints to the detector choice is not established.

minor comments (6)

[Sec. 3, after Eq. (3.21)] "tbe perturbations" should read "the perturbations."
[Fig. 2] The caption does not state which manual threshold was used for the lower panels; please add it for reproducibility.
[Sec. 6, Eqs. (6.6)-(6.7)] Please clarify that n indexes accepted background time points and define the range of n used in the running maximum R_n.
[Table 3] The entry log(lambda/Mpc^-2) < 3.99 for CMB+DESI+PPS should state whether this is a 68% or 95% upper limit.
[Acknowledgments] "Prosum" appears to be a typo for "Program."
[References] Several DOIs, e.g., refs. [32], [45], [46], and [55], have nonstandard formats; please verify them against the publisher records.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: detection thresholds are hand-chosen but not data-fitted; imported perturbation equations from prior work are non-load-bearing for the novel detection claim.

full rationale

The automatic onset detector is derived in-paper (Sec. 6, Eqs. 6.1-6.4) from the non-interacting background equations, and the averaging prescription (Sec. 5, Eqs. 5.2-5.15) is constructed explicitly to enforce <p_chi>=0 in the oscillation regime. The fixed thresholds (Table 1) are chosen parameters, not fitted to the CMB/DESI/SN data, so the cosmological constraints in Table 3 are genuine out-of-sample predictions. The QCDM perturbation equations are imported from the authors' prior work [32]; this is a self-citation but it is not load-bearing for the central novelty (automatic detection), and the prior work is an independently published derivation rather than a fitted input. The only validation of the detector is internal (automatic vs manual modes, Fig. 2), but that is a consistency check, not a circular reduction. Hence no step in the chain equates a predicted quantity to an input by construction.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The method rests on seven hand-chosen numerical thresholds that are not derived or optimized, plus the standard WKB approximation and the authors' pre-existing QCDM model. No new physical entities are introduced.

free parameters (7)

f_R (ratio drop fraction) = 0.70
Chosen by hand in Sec. 6 (Eq. (6.9)) as the required fractional drop of r=y/theta below its running maximum; no sensitivity analysis is reported.
w (transition width) = 5.0
Width of the tanh interpolation in T(theta) (Eq. (5.1)); set as a typical value in Table 1 without derivation.
theta_floor = 1e-12
Floor on |theta| to avoid ill-conditioned division while learning the running maximum R in Sec. 6.
theta_min = 6.0
Minimum phase before the switch trigger is accepted (Eq. (6.11), Table 1).
w_min (w amplitude) = 0.10
Corridor width around zero for classifying sign changes of w_chi (Eq. (6.10), Table 1).
min_sign_changes = 2
Required number of nonzero sign changes of w_chi before oscillations are confirmed (Sec. 6).
theta_pad = 5.0
Padding added to theta_drop in Eq. (6.11) to delay averaging until a short phase interval after detection.

assumptions (4)

domain assumption WKB approximation for the rapidly oscillating chi field with slowly varying m_eff(tau)
Used in Sec. 4 to derive the averaged quantities <chi^2>, <chi'^2>, <V_int> (Eqs. (4.3)-(4.8)); the slowly-varying condition on m_eff^2=m_chi^2+lambda phi^2 is assumed during the fast-oscillation phase.
standard math The polar-variable equations (2.5)-(2.7) and the perturbed-variable relations (3.22)-(3.29) from refs. [36,37,32]
The paper imports these variables and equations without re-deriving them; they are established in the cited literature.
ad hoc to paper The QCDM action and the specific potential forms (3.3)-(3.5)
The model itself is proposed in the authors' prior work; the results are conditional on this choice of interaction lambda chi^2 phi^2 / 2.
ad hoc to paper The transition function T(theta) with tanh smoothing provides a valid interpolation between exact and averaged regimes
Defined in Eq. (5.1) without derivation; its functional form is chosen for convenience and the parameters are user-set.

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Cite this review

Pith. "Pith review of Automatic detection of fast oscillations of dark matter scalar field and updated cosmological constraints on QCDM." pith.science (2026). https://pith.science/paper/ULK4RRWL

@misc{pith2026260813346,
  author       = {Pith},
  title        = {Pith review of: Automatic detection of fast oscillations of dark matter scalar field and updated cosmological constraints on QCDM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ULK4RRWL}},
  note         = {Machine review of arXiv:2608.13346}
}

read the original abstract

It is well known that a scalar field dark matter with a quadratic potential undergoes fast oscillations when the time period represented by the mass scale in the Klein-Gordon equation becomes much smaller than that set by the Hubble parameter. This makes a solution to the equation numerically intractable. Many works in the literature have addressed the problem by either switching between solving the Klein-Gordon equation in the well-behaved regime to solving the fluid equations at the onset of oscillations, or by introducing a new set of variables that can absorb these oscillations. Despite being successful, these techniques rely on an estimate of when the oscillations start. For large scale scans of a model's parameter space, this can become cumbersome. Furthermore, the techniques have been used mainly for non-interacting dark matter models. In this work, we introduce an averaging technique with an automatic detection of the onset of oscillations, capable of capturing the non-interacting as well as the interacting dark matter scenarios. The technique, implemented in \texttt{CLASS}, is tested on the QCDM model and shows excellent detection and averaging abilities. We also update the cosmological constraints on the model using the most recent public data.

Figures

Figures reproduced from arXiv: 2608.13346 by the authors.

**Figure 1.** Top panels: plot of the energy densities of matter, DE and photons (left) and the fractional energy densities (right) versus the redshift. Bottom panels: plot of the ratio y/θ versus z (left). A sudden drop in the ratio is seen at z ∼ 107 when fast oscillations start. Plot of θ versus z (right). A sudden rise in θ is visible at around the same redshift as seen in the inset. For all panels, we take λ = 105 Mpc−2 . fi… view at source ↗

**Figure 2.** Plots of the DM and DE equations of state and the total equation of state (left) and the evolution of the DM field χ (right) as a function of z. The top panels are generated using the ‘automatic’ mode while the bottom panels use the ‘manual’ mode. For all panels, we take λ = 105 Mpc−2 . In [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Left panel: The temperature-temperature correlations in the lensed CMB power spectrum showing QCDM (solid line) and ΛCDM (dashed). Right panel: the matter power spectrum for QCDM and ΛCDM. For this benchmark, λ = 105 Mpc−2 . To verify the above interpretation, we plot in [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Plots of the Newtonian potentials, Φ (upper panel) and Ψ (lower panel) for two wavenumbers as a function of the scale factor. The fractional difference between QCDM and ΛCDM is also shown in each panel. For this benchmark, λ = 105 Mpc−2 . There is an interesting distin…

**Figure 5.** Figure 5: Plots of the density contrasts of DM in the QCDM model (solid line) and ΛCDM model (dashed line) for two values of the wavenumber. For this benchmark, λ = 105 Mpc−2 . can therefore integrate into a large difference in the final amplitude: δQCDM(a0) δΛCDM(a0) = exp Z (…

**Figure 6.** Figure 6: Triangle plot showing the 2D joint and 1D marginalized posteriors of a number cosmological parameters of QCDM for the following data set combinations: CMB (red contours), CMB+PP+DESI (blue contours) and CMB+PPS+DESI (green contours). 23 [PITH_FULL_IMAGE:figures/full_…

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Works this paper leans on

61 extracted references · 1 canonical work pages

[1]

W. Hu, R. Barkana and A. Gruzinov, Phys. Rev. Lett.85, 1158-1161 (2000) doi:10.1103/PhysRevLett.85.1158 [arXiv:astro-ph/0003365 [astro-ph]]

arXiv 2000
[2]

Ringwald, Phys

A. Ringwald, Phys. Dark Univ.1, 116-135 (2012) doi:10.1016/j.dark.2012.10.008 [arXiv:1210.5081 [hep-ph]]

arXiv 2012
[3]

L. Hui, J. P. Ostriker, S. Tremaine and E. Witten, Phys. Rev. D95, no.4, 043541 (2017) doi:10.1103/PhysRevD.95.043541 [arXiv:1610.08297 [astro-ph.CO]]

arXiv 2017
[4]

Ratra and P

B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 (1988) doi:10.1103/PhysRevD.37.3406 24

doi:10.1103/physrevd.37.3406 1988
[5]

R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett.80, 1582-1585 (1998) doi:10.1103/PhysRevLett.80.1582 [arXiv:astro-ph/9708069 [astro-ph]]

arXiv 1998
[6]

Wetterich, Nucl

C. Wetterich, Nucl. Phys. B302, 668-696 (1988) doi:10.1016/0550-3213(88)90193-9 [arXiv:1711.03844 [hep-th]]

arXiv 1988
[7]

Albrecht and C

A. Albrecht and C. Skordis, Phys. Rev. Lett.84, 2076-2079 (2000) doi:10.1103/PhysRevLett.84.2076 [arXiv:astro-ph/9908085 [astro-ph]]

arXiv 2000
[8]

Wetterich, Phys

C. Wetterich, Phys. Lett. B594, 17-22 (2004) doi:10.1016/j.physletb.2004.05.008 [arXiv:astro-ph/0403289 [astro-ph]]

arXiv 2004

Show all 61 references

[9]

Tsujikawa, Class

S. Tsujikawa, Class. Quant. Grav.30, 214003 (2013) doi:10.1088/0264- 9381/30/21/214003 [arXiv:1304.1961 [gr-qc]]

2013 arXiv
[10]

Sahni and L

V. Sahni and L. M. Wang, Phys. Rev. D62, 103517 (2000) doi:10.1103/PhysRevD.62.103517 [arXiv:astro-ph/9910097 [astro-ph]]

2000 arXiv
[11]

Matos and L

T. Matos and L. A. Urena-Lopez, Class. Quant. Grav.17, L75-L81 (2000) doi:10.1088/0264-9381/17/13/101 [arXiv:astro-ph/0004332 [astro-ph]]

2000 arXiv
[12]

Amendola and R

L. Amendola and R. Barbieri, Phys. Lett. B642, 192-196 (2006) doi:10.1016/j.physletb.2006.08.069 [arXiv:hep-ph/0509257 [hep-ph]]

2006 arXiv
[13]

Matos, A

T. Matos, A. Vazquez-Gonzalez and J. Magana, Mon. Not. Roy. Astron. Soc.393, 1359-1369 (2009) doi:10.1111/j.1365-2966.2008.13957.x [arXiv:0806.0683 [astro-ph]]

2009
[14]

J. c. Hwang and H. Noh, Phys. Lett. B680, 1-3 (2009) doi:10.1016/j.physletb.2009.08.031 [arXiv:0902.4738 [astro-ph.CO]]

2009 arXiv
[15]

D. J. E. Marsh and P. G. Ferreira, Phys. Rev. D82, 103528 (2010) doi:10.1103/PhysRevD.82.103528 [arXiv:1009.3501 [hep-ph]]

2010 arXiv
[16]

Glennon, N

N. Glennon, N. Musoke and C. Prescod-Weinstein, Phys. Rev. D107, no.6, 063520 (2023) doi:10.1103/PhysRevD.107.063520 [arXiv:2302.04302 [astro-ph.CO]]

2023 arXiv
[17]

C. G. Park, J. c. Hwang and H. Noh, Phys. Rev. D86, 083535 (2012) doi:10.1103/PhysRevD.86.083535 [arXiv:1207.3124 [astro-ph.CO]]

2012 arXiv
[18]

B. Li, T. Rindler-Daller and P. R. Shapiro, Phys. Rev. D89, no.8, 083536 (2014) doi:10.1103/PhysRevD.89.083536 [arXiv:1310.6061 [astro-ph.CO]]

2014 arXiv
[19]

L. A. Ure˜ na-L´ opez and A. X. Gonzalez-Morales, JCAP07, 048 (2016) doi:10.1088/1475- 7516/2016/07/048 [arXiv:1511.08195 [astro-ph.CO]]

2016 arXiv
[20]

D. J. E. Marsh, Phys. Rept.643, 1-79 (2016) doi:10.1016/j.physrep.2016.06.005 [arXiv:1510.07633 [astro-ph.CO]]

2016 arXiv
[21]

Magana, T

J. Magana, T. Matos, A. Suarez and F. J. Sanchez-Salcedo, JCAP10, 003 (2012) doi:10.1088/1475-7516/2012/10/003 [arXiv:1204.5255 [astro-ph.CO]]

2012 arXiv
[22]

Matos and L

T. Matos and L. A. Urena-Lopez, Phys. Rev. D63, 063506 (2001) doi:10.1103/PhysRevD.63.063506 [arXiv:astro-ph/0006024 [astro-ph]]. 25

2001 arXiv
[23]

J. A. R. Cembranos, A. L. Maroto, S. J. N´ u˜ nez Jare˜ no and H. Villarrubia-Rojo, JHEP 08, 073 (2018) doi:10.1007/JHEP08(2018)073 [arXiv:1805.08112 [astro-ph.CO]]

2018 arXiv
[24]

Foidl and T

H. Foidl and T. Rindler-Daller, Phys. Rev. D105, no.12, 123534 (2022) doi:10.1103/PhysRevD.105.123534 [arXiv:2203.09396 [astro-ph.CO]]

2022 arXiv
[25]

Aboubrahim and P

A. Aboubrahim and P. Nath, JCAP09, 076 (2024) doi:10.1088/1475-7516/2024/09/076 [arXiv:2406.19284 [astro-ph.CO]]

2024 arXiv
[26]

D. Blas, J. Lesgourgues and T. Tram, JCAP07, 034 (2011) doi:10.1088/1475- 7516/2011/07/034 [arXiv:1104.2933 [astro-ph.CO]]

2011 arXiv
[27]

Lesgourgues, [arXiv:1104.2932 [astro-ph.IM]]

J. Lesgourgues, [arXiv:1104.2932 [astro-ph.IM]]

arXiv
[28]

Hlozek, D

R. Hlozek, D. Grin, D. J. E. Marsh and P. G. Ferreira, Phys. Rev. D91, no.10, 103512 (2015) doi:10.1103/PhysRevD.91.103512 [arXiv:1410.2896 [astro-ph.CO]]

2015 arXiv
[29]

Cookmeyer, J

T. Cookmeyer, J. Cookmeyer, D. Grin and T. L. Smith, Phys. Rev. D101, no.2, 023501 (2020) doi:10.1103/PhysRevD.101.023501 [arXiv:1909.11094 [astro-ph.CO]]

2020 arXiv
[30]

Passaglia and W

S. Passaglia and W. Hu, Phys. Rev. D105, no.12, 123529 (2022) doi:10.1103/PhysRevD.105.123529 [arXiv:2201.10238 [astro-ph.CO]]

2022 arXiv
[31]

L. A. Ure˜ na-L´ opez and F. X. Linares Cede˜ no, Phys. Rev. D109, no.2, 023512 (2024) doi:10.1103/PhysRevD.109.023512 [arXiv:2307.05600 [astro-ph.CO]]

2024 arXiv
[32]

Aboubrahim, A

A. Aboubrahim, A. H. Giman and P. Nath, Phys. Rev. D114, no.2, 023501 (2026) doi:10.1103/3s8n-j6tq [arXiv:2511.22559 [astro-ph.CO]]

2026 arXiv
[33]

Aboubrahim and P

A. Aboubrahim and P. Nath, JCAP10, 081 (2025) doi:10.1088/1475-7516/2025/10/081 [arXiv:2411.11177 [astro-ph.CO]]

2025 arXiv
[34]

Aboubrahim and P

A. Aboubrahim and P. Nath, [arXiv:2601.19662 [astro-ph.CO]]
[35]

Nath and A

P. Nath and A. Aboubrahim, PoSCORFU2024, 203 (2025) doi:10.22323/1.490.0203 [arXiv:2503.09769 [astro-ph.CO]]

2025 arXiv
[36]

E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D57, 4686-4690 (1998) doi:10.1103/PhysRevD.57.4686 [arXiv:gr-qc/9711068 [gr-qc]]

1998 arXiv
[37]

Garcia-Arroyo, L

G. Garcia-Arroyo, L. A. Ure˜ na-L´ opez and J. A. V´ azquez, Phys. Rev. D110, no.2, 023529 (2024) doi:10.1103/PhysRevD.110.023529 [arXiv:2402.08815 [astro-ph.CO]]

2024 arXiv
[38]

J. A. R. Cembranos, A. L. Maroto and S. J. N´ u˜ nez Jare˜ no, JHEP03, 013 (2016) doi:10.1007/JHEP03(2016)013 [arXiv:1509.08819 [astro-ph.CO]]

2016 arXiv
[39]

Torrado and A

J. Torrado and A. Lewis, JCAP05, 057 (2021) doi:10.1088/1475-7516/2021/05/057 [arXiv:2005.05290 [astro-ph.IM]]

2021 arXiv
[40]

Lewis and S

A. Lewis and S. Bridle, Phys. Rev. D66, 103511 (2002) doi:10.1103/PhysRevD.66.103511 [arXiv:astro-ph/0205436 [astro-ph]]. 26

2002 arXiv
[41]

Lewis, Phys

A. Lewis, Phys. Rev. D87, no.10, 103529 (2013) doi:10.1103/PhysRevD.87.103529 [arXiv:1304.4473 [astro-ph.CO]]

2013 arXiv
[42]

R. M. Neal, [arXiv:math/0502099 [math.ST]]

arXiv
[43]

Gelman and D

A. Gelman and D. B. Rubin, Statist. Sci.7, 457-472 (1992) doi:10.1214/ss/1177011136

1992
[44]

Lewis, JCAP08, 025 (2025) doi:10.1088/1475-7516/2025/08/025 [arXiv:1910.13970 [astro-ph.IM]]

A. Lewis, JCAP08, 025 (2025) doi:10.1088/1475-7516/2025/08/025 [arXiv:1910.13970 [astro-ph.IM]]

2025 arXiv
[45]

Abdul Karimet al.[DESI], Phys

M. Abdul Karimet al.[DESI], Phys. Rev. D112, no.8, 083515 (2025) doi:10.1103/tr6y- kpc6 [arXiv:2503.14738 [astro-ph.CO]]

2025 arXiv
[46]

Abdul Karimet al.[DESI], Phys

M. Abdul Karimet al.[DESI], Phys. Rev. D112, no.8, 083514 (2025) doi:10.1103/2wwn-xjm5 [arXiv:2503.14739 [astro-ph.CO]]

2025 arXiv
[47]

Brout, D

D. Brout, D. Scolnic, B. Popovic, A. G. Riess, J. Zuntz, R. Kessler, A. Carr, T. M. Davis, S. Hinton and D. Jones,et al.Astrophys. J.938, no.2, 110 (2022) doi:10.3847/1538- 4357/ac8e04 [arXiv:2202.04077 [astro-ph.CO]]

2022 arXiv
[48]

Aghanimet al.[Planck], Astron

N. Aghanimet al.[Planck], Astron. Astrophys.641, A5 (2020) doi:10.1051/0004- 6361/201936386 [arXiv:1907.12875 [astro-ph.CO]]

2020 arXiv
[49]

Calabreseet al.[Atacama Cosmology Telescope], JCAP11, 063 (2025) doi:10.1088/1475-7516/2025/11/063 [arXiv:2503.14454 [astro-ph.CO]]

E. Calabreseet al.[Atacama Cosmology Telescope], JCAP11, 063 (2025) doi:10.1088/1475-7516/2025/11/063 [arXiv:2503.14454 [astro-ph.CO]]

2025 arXiv
[50]

Louiset al.[Atacama Cosmology Telescope], JCAP11, 062 (2025) doi:10.1088/1475- 7516/2025/11/062 [arXiv:2503.14452 [astro-ph.CO]]

T. Louiset al.[Atacama Cosmology Telescope], JCAP11, 062 (2025) doi:10.1088/1475- 7516/2025/11/062 [arXiv:2503.14452 [astro-ph.CO]]

2025 arXiv
[51]

Sailer, J

N. Sailer, J. Kim, S. Ferraro, M. S. Madhavacheril, M. White, I. Abril-Cabezas, J. N. Aguilar, S. Ahlen, J. R. Bond and D. Brooks,et al.JCAP06, 008 (2025) doi:10.1088/1475-7516/2025/06/008 [arXiv:2407.04607 [astro-ph.CO]]

2025 arXiv
[52]

M. S. Madhavacherilet al.[ACT], Astrophys. J.962, no.2, 113 (2024) doi:10.3847/1538- 4357/acff5f [arXiv:2304.05203 [astro-ph.CO]]

2024 arXiv
[53]

F. J. Quet al.[ACT], Astrophys. J.962, no.2, 112 (2024) doi:10.3847/1538-4357/acfe06 [arXiv:2304.05202 [astro-ph.CO]]

2024 arXiv
[54]

Carron, M

J. Carron, M. Mirmelstein and A. Lewis, JCAP09, 039 (2022) doi:10.1088/1475- 7516/2022/09/039 [arXiv:2206.07773 [astro-ph.CO]]

2022 arXiv
[55]

Camphuiset al.[SPT-3G], Phys

E. Camphuiset al.[SPT-3G], Phys. Rev. D113, no.8, 083504 (2026) doi:10.1103/7wt3- 9v2y [arXiv:2506.20707 [astro-ph.CO]]

2026 arXiv
[56]

Geet al.[SPT-3G], Phys

F. Geet al.[SPT-3G], Phys. Rev. D111, no.8, 083534 (2025) doi:10.1103/PhysRevD.111.083534 [arXiv:2411.06000 [astro-ph.CO]]

2025 arXiv
[57]

Millea and U

M. Millea and U. Seljak, Phys. Rev. D105, no.10, 103531 (2022) doi:10.1103/PhysRevD.105.103531 [arXiv:2112.09354 [astro-ph.CO]]. 27

2022 arXiv
[58]

Heavens, Y

A. Heavens, Y. Fantaye, E. Sellentin, H. Eggers, Z. Hosenie, S. Kroon and A. Mootoovaloo, Phys. Rev. Lett.119, no.10, 101301 (2017) doi:10.1103/PhysRevLett.119.101301 [arXiv:1704.03467 [astro-ph.CO]]

2017 arXiv
[59]

Heavens, Y

A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie, S. Kroon and E. Sell- entin, [arXiv:1704.03472 [stat.CO]]

arXiv
[60]

R. E. Kass and A. E. Raftery, J. Am. Statist. Assoc.90, no.430, 773-795 (1995) doi:10.1080/01621459.1995.10476572

1995
[61]

Trotta, Contemp

R. Trotta, Contemp. Phys.49, 71-104 (2008) doi:10.1080/00107510802066753 [arXiv:0803.4089 [astro-ph]]. 28

2008 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

W. Hu, R. Barkana and A. Gruzinov, Phys. Rev. Lett.85, 1158-1161 (2000) doi:10.1103/PhysRevLett.85.1158 [arXiv:astro-ph/0003365 [astro-ph]]

arXiv 2000

[2] [2]

Ringwald, Phys

A. Ringwald, Phys. Dark Univ.1, 116-135 (2012) doi:10.1016/j.dark.2012.10.008 [arXiv:1210.5081 [hep-ph]]

arXiv 2012

[3] [3]

L. Hui, J. P. Ostriker, S. Tremaine and E. Witten, Phys. Rev. D95, no.4, 043541 (2017) doi:10.1103/PhysRevD.95.043541 [arXiv:1610.08297 [astro-ph.CO]]

arXiv 2017

[4] [4]

Ratra and P

B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 (1988) doi:10.1103/PhysRevD.37.3406 24

doi:10.1103/physrevd.37.3406 1988

[5] [5]

R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett.80, 1582-1585 (1998) doi:10.1103/PhysRevLett.80.1582 [arXiv:astro-ph/9708069 [astro-ph]]

arXiv 1998

[6] [6]

Wetterich, Nucl

C. Wetterich, Nucl. Phys. B302, 668-696 (1988) doi:10.1016/0550-3213(88)90193-9 [arXiv:1711.03844 [hep-th]]

arXiv 1988

[7] [7]

Albrecht and C

A. Albrecht and C. Skordis, Phys. Rev. Lett.84, 2076-2079 (2000) doi:10.1103/PhysRevLett.84.2076 [arXiv:astro-ph/9908085 [astro-ph]]

arXiv 2000

[8] [8]

Wetterich, Phys

C. Wetterich, Phys. Lett. B594, 17-22 (2004) doi:10.1016/j.physletb.2004.05.008 [arXiv:astro-ph/0403289 [astro-ph]]

arXiv 2004