Oscillating scalar fields and the Hubble tension: a resolution with novel signatures

Mustafa A. Amin; Tristan L. Smith; Vivian Poulin

REVIEW 3 major objections 5 minor 21 cited by

Oscillating scalar fields and the Hubble tension: a resolution with novel signatures

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A scalar field that peaks at about 10% of cosmic energy near matter-radiation equality can resolve the Hubble tension, the paper argues, and predicts detectable signatures in CMB polarization and in the field's own perturbations.

desk verdict The paper that made early dark energy the default explanation to beat for the Hubble tension; rigorous and honest, with the caveat that the fit is SH0ES-dependent and quietly assumes r≲5e-3. read the letter →

arxiv 1908.06995 v1 pith:GF6SFNEB submitted 2019-08-19 astro-ph.CO gr-qchep-ph

Tristan L. Smith , Vivian Poulin , Mustafa A. Amin This is my paper · ORCID

classification astro-ph.COgr-qchep-ph MSC 83F0585A40 PACS 98.80.-k98.80.Es95.35.+d

keywords earlydarkenergyHubbletensionoscillatingscalarfieldcosmicmicrowavebackgroundpolarizationeffectivesoundspeedparametricresonanceisocurvatureperturbationsCMB-S4forecast

topics The Hubble Tension Dark Energy

open problems The Hubble Tension

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

The paper argues that a sub-dominant scalar field, frozen by Hubble friction until roughly $z_c \simeq 10^{3.5}$ and contributing about 10% of the total energy density at that redshift, can remove the $4$-$6\sigma$ disagreement between early- and late-universe measurements of the Hubble constant. Using the full linearized field equations rather than a fluid approximation, the authors show that a potential $V(\phi) \propto \phi^{2n}$ with $n \approx 3$ and a flattened large-field behavior brings Planck CMB data, BAO, supernovae, and the SH0ES distance-ladder value into agreement, shifting the inferred $H_0$ from about 67.4 to about 71.5 km/s/Mpc. The model also yields two new, testable signatures: isocurvature perturbations controlled by the tensor-to-scalar ratio, and a scale-dependent self-resonance for $n \approx 2$ that drives the scalar-field perturbations nonlinear. A sympathetic reader should care because the scenario is one of few beyond-$\Lambda$CDM proposals that improves the fit to all datasets without degrading Planck, and it predicts that next-generation CMB polarization experiments will detect the field directly.

What carries the argument

The load-bearing object is the anharmonic oscillating scalar field itself, implemented without fluid approximations through the linearized Klein-Gordon equation $\delta\varphi''_k + 2\mathcal{H}\delta\varphi'_k + [k^2+a^2V_{,\varphi\varphi}]\delta\varphi_k = -h'\varphi'/2$. The key mechanism is Hubble friction freezing the field until its effective mass $|V_{,\varphi\varphi}|^{1/2}$ drops to about $3H$, after which it oscillates; the oscillation determines both the background dilution, $w_\phi=(n-1)/(n+1)$, and, through the effective sound speed $c_s^2=[2(n-1)\varpi^2 a^2+k^2]/[2(n+1)\varpi^2 a^2+k^2]$, the perturbation evolution that Planck polarization constrains. The parameter $\Theta_i=\phi_i/f$ controls the flattening of the potential at large displacement and hence how many sub-horizon modes have $c_s^2<0.9$, which is what makes the model fit the CMB. For $n\approx2$, the oscillating background acts as a periodic pump, producing a narrow resonance band in wavenumber that stays frozen for a given co-moving mode, leading to exponential growth and nonlinearity.

What would settle it

Measure the local expansion rate with a Cepheid-independent method that is accurate to better than 1 km/s/Mpc: if the true value is near 69.8 km/s/Mpc, the SH0ES prior disappears and the EDE fraction should be consistent with zero in the combined fit. Conversely, CMB-S4 can search for the predicted scale-localized E-mode polarization deviations at multipoles $50\lesssim\ell\lesssim1000$, whose absence would rule out the scenario.

Watch

Extended reading notes

Core claim

The central claim is that an oscillating scalar field with potential $V_n(\phi)=m^2f^2[1-\cos(\phi/f)]^n$ can resolve the Hubble tension when it is frozen until $\log_{10} z_c \sim 3.5$, peaks at $f_{\rm EDE}(z_c)\sim 0.10$, and then dilutes with equation of state $w_\phi=(n-1)/(n+1)$, faster than matter. With Planck temperature and polarization, lensing, BAO, Pantheon, and SH0ES data, the field is preferred at about $3.5\sigma$, with $n=3.16^{+0.18}_{-1.16}$, $\log_{10} z_c = 3.56$, $f_{\rm EDE}=0.103$, and a large initial displacement $\Theta_i\simeq2.5$; the SH0ES $\chi^2$ improves by about 15 while the Planck fit is essentially unchanged. The same analysis shows that Planck data alone cannot distinguish the EDE from $\Lambda$CDM because of a sampling-volume degeneracy, whereas mock CMB-S4 data would detect the field at very high significance through $E$-mode polarization. The paper also identifies two signatures: isocurvature modes whose amplitude is set by the tensor-to-scalar ratio $r$, and parametric self-resonance in the $n\approx2$ case that can make field perturbations nonlinear and spatially inhomogeneous.

Load-bearing premise

The load-bearing premise is that the Hubble tension is a genuine cosmological mismatch rather than a systematic error in the Cepheid-calibrated distance ladder; the paper only argues for EDE when the SH0ES value is included as a prior.

Editorial extensions

If this is right

If the model is correct, the CMB-inferred $H_0$ shifts to about 71.5 km/s/Mpc, bringing early- and late-universe probes into agreement without degrading the Planck fit (total $\Delta\chi^2 \simeq -20$).
Planck polarization constrains $\Theta_i$ to be large (excluding $\Theta_i < 1.8$ at 95% C.L.), predicting scale-localized residuals in the $TE$ and $EE$ power spectra around multipoles $50\lesssim \ell \lesssim 1000$.
CMB-S4 would detect the EDE at roughly $10\sigma$ from $E$-mode polarization alone, independently of SH0ES, providing a decisive consistency test.
The scenario predicts a modest upward shift in $S_8$, increasing the $S_8$ tension with KiDS from about $2.3\sigma$ to $2.5\sigma$.
If $r > 5\times 10^{-3}$, isocurvature perturbations from the EDE field would be visible in the CMB and could place constraints on the scenario.

Reading between the lines

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

If the $n\approx2$ resonant growth turns nonlinear, the field's equation of state on those scales could approach $w\simeq1/3$, generating a stochastic gravitational-wave background and small-scale matter-power features that are not computed in this paper.
The Planck-only degeneracy suggests that any future CMB experiment that sees the predicted polarization pattern will effectively be measuring the scalar-field potential shape directly, not just $H_0$.
The same potential family could arise from a broad distribution of light axion-like fields, making EDE one member of a spectrum rather than a single tuned component; that connection is not developed in the paper.
A decisive independent check is a local $H_0$ measurement with accuracy below 1 km/s/Mpc: if it lands near 69.8 km/s/Mpc, the statistical case for EDE weakens substantially, which the paper itself acknowledges.

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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

You should know two things. This is the paper that made early dark energy the default explanation to beat for the Hubble tension: a scalar field at ~10% of the total energy density around zc~3500, diluting faster than matter afterward, brings Planck, BAO, SNe, and SH0ES into agreement. And it is an honest paper — nearly every limitation I would raise is disclosed in the text, often in the same breath as the claim.

What's actually new: full linearized scalar-field dynamics in place of the earlier cycle-averaged fluid treatment; the first free-n constraints, with 2

Soft spots, in proportion. The statistical case depends on SH0ES: Planck alone cannot see the EDE, and the Delta chi^2 of about 20 is mostly the SH0ES prior dropping from 16.8 to 1.7. The paper says so plainly; if distance-ladder systematics move, the 3.5 sigma moves with it. That is a conditional on the premise, not an analytic flaw. The stress-test note lands: the MCMC solves only the adiabatic sector with an implicit r<5e-3, an order of magnitude below Planck's bound, while the favored Theta_i~2.6-2.8 exponentially amplifies isocurvature. A mid-range r would likely rule out the very parameters that resolve the tension. Disclosed, but it is the strongest of the conditionals — the resolution quietly assumes something about inflation. Smaller items: n in [1,2] is excluded for computational cost, the modified code is not released, and the resonance section is linear-only and explicitly qualitative. All disclosed; all fixable.

This is a technically careful paper that deserves a serious referee. Recommend engaging with it — referee it, and treat the conditionals as the interesting content rather than grounds for rejection.

Referee Report

3 major / 5 minor

Summary. This paper studies an oscillating scalar field with potential V_n(phi)=m^2 f^2 [1-cos(phi/f)]^n as an early-dark-energy (EDE) resolution of the Hubble tension. The authors implement the exact homogeneous and linearized perturbation equations in CLASS, run MCMC fits against Planck 2015 temperature, polarization and lensing, BAO, Pantheon supernovae, and the SH0ES H0 prior, and report that an EDE with fEDE(zc)~10%, log10(zc)~3.5, and n~3 brings these data into agreement while improving the total chi^2 by about 20 relative to LCDM. They further forecast that CMB-S4 can detect the EDE cosmology from CMB data alone, and they identify two new signatures: isocurvature perturbations whose amplitude is controlled by the tensor-to-scalar ratio r, and self-resonance of scalar-field perturbations for n~2 that can become nonlinear.

Significance. If the central claim holds, the paper is a significant contribution to the Hubble-tension literature: it goes beyond fluid approximations, provides per-dataset chi^2 tables, verifies analytic initial conditions and Floquet predictions against the Boltzmann code, and produces falsifiable forecasts for CMB-S4 and for isocurvature signatures. The treatment of the perturbation dynamics and the identification of the large-Theta_i preference from Planck polarization are substantive. However, the resolution is conditional in two important ways that the paper itself discloses: the statistical preference for EDE comes from fits that include the SH0ES prior, and the analysis assumes r <~ 5e-3 so that EDE-induced isocurvature perturbations are negligible. These conditionality statements are present in the text, but they are not reflected in the abstract's unqualified wording, and the isocurvature restriction is a load-bearing assumption that is not tested against current bounds.

major comments (3)

[Sec. II B and Sec. IV A] The central MCMC claim is conditional on the assumption r <~ 5e-3, which is not sampled or constrained in the analysis. Eq. (16) and Fig. 13 show that the EDE isocurvature power is proportional to r and is exponentially amplified for the large Theta_i values preferred by the data (Table I, Theta_i ~ 2.6). With the current Planck 95% upper limit r < 0.056, an order of magnitude above the assumed threshold, most of the currently allowed r range would generate isocurvature power exceeding the ~10% cosmic-variance limit quoted in Sec. IV A, potentially ruling out the same EDE parameters that resolve the Hubble tension. Because the likelihood never includes the isocurvature sector, the reported Delta chi^2 and significance values (Tables II, V, VI) describe only the adiabatic sector. I request either that r be included in the Monte Carlo sampling with the Planck isocurvature likelihood, or that the authors derive and present the upper limit on r that the EDE+Planck fit implies, so that the final claim is not conditional on an untested inflationary parameter.
[Sec. III B, Table I, and Sec. IV B] The reported 68% interval for the potential index, n = 3.16+0.18/-1.16 from Table I, has its lower edge at the prior boundary n=2, and the region n in [1,2] is excluded from the MCMC for computational tractability. At the same time, Sec. IV B and Appendix C demonstrate that for n ~ 2 the scalar-field perturbations undergo parametric self-resonance and become nonlinear, so the linearized CLASS treatment used for the constraints is not valid over part of the posterior support. The abstract's claim that 2 <~ n <~ 3.4 is preferred at 68% confidence therefore rests on a regime where the perturbation equations break down. Please either extend the analysis to n < 2 with a controlled approximation (for example the fluid approximation of Ref. [16], as the authors themselves suggest) or provide a quantitative demonstration that the regions where resonance/nonlinearity occur are excluded by the data, before quoting a lower bound on n.
[Sec. III B and Sec. V] The statement in Sec. V that the EDE is 'indicated at ~3.5 sigma' is derived only from fits that include the SH0ES likelihood. Table II shows that the total Delta chi^2 of -20.3 is dominated by the SH0ES chi^2 dropping from 16.80 to 1.68, while the Planck chi^2 improves by only ~4. The authors explicitly do not present a Planck-only real-data MCMC because of the acknowledged sampling-volume issue (Sec. III B), and the synthetic Planck analysis in Sec. III D yields fEDE < 0.14 at 95% with Delta chi^2 = -7.8, i.e., consistent with no EDE at about 1 sigma. The statistical preference for EDE is thus a statement about the combined dataset conditional on the SH0ES prior, not about Planck data alone. The abstract and conclusions should state this conditioning more prominently, especially in the sentence 'can bring CMB, BAO, supernovae, and the SH0ES estimate of the Hubble constant into agreement.'

minor comments (5)

[Sec. II A] The heading contains a typo: 'reivew' should be 'review'.
[Sec. I] The introduction contains 'the the long-standing' and later 'the stadard six' in Sec. V; these should be corrected.
[Fig. 14] The caption of Fig. 14 states that the shape and magnitude near n~2 'should be trusted only qualitatively.' Since this figure is used to motivate the special status of n=2 and the nonlinearity criterion, please provide a numerical verification or a convergence test for the integral in Eq. (C9) near n=2.
[Sec. IV A, Eq. (16)] The expression P_phi(k)/P_zeta(k) = r (k/k0)^(-r/8-(1-n_s)) is stated without derivation or a direct reference; a one-line derivation or citation would help the reader understand the origin of the tilt terms.
[Sec. III A] The analysis uses Planck 2015 likelihoods with a footnote stating that a baseline n=3 run was checked against the new Planck 2018 release. Given that the paper was completed in 2019, consider reporting the numerical result of that check or providing a reference where it is documented; as written, the claim is not auditable.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: EDE parameters are fitted, not predicted; the new isocurvature and self-resonance signatures follow from independent physics and are checked against CLASS.

full rationale

The central EDE claim is a parameter fit, not a prediction: the MCMC samples {log10(zc), fEDE(zc), Θi, n} against CMB+BAO+SNe+SH0ES, so agreement with SH0ES is a fitted outcome. This is standard model selection, not circular reasoning, and the paper explicitly discloses that Planck alone cannot detect the EDE and that the preference is driven by the SH0ES likelihood. The predictive elements—the isocurvature amplitude set by the external tensor-to-scalar ratio r through Eq. (16), and the self-resonance growth derived from Floquet theory and verified in CLASS—do not reduce to the fitted EDE parameters. Self-citations to Refs. [12,16] are motivational and comparative; the background and linear perturbations are re-derived from the Klein-Gordon equation and implemented in a modified CLASS code, so the conclusions do not rest on an unverified self-citation chain. The disclosed limitations (implicit r≲5×10^-3, and breakdown of linear theory near n=2) are robustness caveats, not circular steps.

Assumptions & free parameters 10 free parameters · 6 assumptions · 1 invented entities

The central claim rests on ten fitted parameters: six standard LCDM parameters plus four EDE parameters (zc, fEDE, Θi, n). These are all constrained by the same datasets used to evaluate the model, so the 'resolution' is a fit rather than a free prediction. The main axioms are standard cosmology, linear perturbation theory, the assumed potential shape, and the spectator-field isocurvature relation; these are standard domain assumptions, not ad hoc inventions. The EDE field itself is an invented entity with no independent evidence yet, though the paper provides concrete observational signatures that could provide such evidence.

free parameters (10)

fEDE(zc) = 0.103 ± 0.035 (n free, best-fit 0.132); 0.107 ± 0.035 (n=3)
Peak EDE energy fraction at zc; constrained by MCMC fit to CMB+BAO+SN+SH0ES (Table I).
log10(zc) = 3.558 (+0.053/-0.110) (n free); 3.568 (+0.056/-0.140) (n=3)
Redshift of maximum EDE energy density; determined by the field mass m via the shooting method.
Θi = 2.49 (+0.52/-0.01) (n free); 2.64 (+0.36/-0.04) (n=3)
Initial field displacement in units of f; constrained mainly by Planck high-l polarization.
n = 3.16 (+0.18/-1.16), 95% C.L. n<5
Power-law index of the potential around the minimum; free parameter with flat prior 2<n<6 (Sec. III B).
ωb = 0.02261 ± 0.00024 (n free)
Baryon physical density, standard LCDM free parameter in the MCMC.
ωcdm = 0.1290 (+0.0041/-0.0045) (n free)
Cold dark matter physical density, standard LCDM free parameter.
100θs = 1.04139 (+0.00041/-0.00036) (n free)
Sound horizon angular scale, standard LCDM parameter.
10^9 As = 2.196 ± 0.055 (n free)
Primordial curvature power-spectrum amplitude, standard LCDM parameter.
ns = 0.9853 (+0.0073/-0.0079) (n free)
Primordial spectral index, standard LCDM parameter.
τreio = 0.070 ± 0.014 (n free)
Reionization optical depth, standard LCDM parameter.

assumptions (6)

standard math Friedmann-Robertson-Walker background and general relativity
Used throughout for the expansion history, Eq. (1), and Einstein equations; standard cosmology assumption.
standard math Linear scalar perturbation theory in synchronous gauge
Perturbation evolution, Eq. (9), and adiabatic initial conditions in Appendix B follow this framework.
domain assumption Scalar field is initially homogeneous and isotropic, established before the end of inflation
Stated in Sec. I: 'We assume that the field initially is (almost) perfectly homogeneous and isotropic.' Needed to set initial conditions and spectator-field isocurvature spectrum.
domain assumption Potential has the form V_n(φ)=m^2 f^2 [1-cos(φ/f)]^n with n in [2,6]
Eq. (4), motivated by axions and higher-order instanton corrections; n<2 excluded for computational tractability (Sec. III B).
domain assumption Spectator scalar field during inflation produces isocurvature perturbations with amplitude set by r
Sec. IV A, Eq. (16): P_φ/P_ζ = r (k/k0)^{-r/8-(1-n_s)}; for the main analysis the paper implicitly assumes r≲5×10^-3.
domain assumption Neutrino sector: two massless plus one massive species with Mν=0.06 eV
Sec. III A follows the Planck collaboration convention; affects the background and perturbation evolution.

invented entities (1)

Oscillating scalar field (early dark energy) with potential V_n(φ)=m^2 f^2 [1-cos(φ/f)]^n
purpose: Resolve the Hubble tension by adding ~10% extra energy density around matter-radiation equality and diluting faster than matter afterward
No direct detection exists. Falsifiable handles include CMB-S4 polarization signatures, isocurvature constraints tied to r, and self-resonance at n≈2, but none have been observed yet; the field is introduced to fit the data.

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Cite this review

Pith. "Pith review of Oscillating scalar fields and the Hubble tension: a resolution with novel signatures." pith.science (2026). https://pith.science/paper/GF6SFNEB

@misc{pith2026190806995,
  author       = {Pith},
  title        = {Pith review of: Oscillating scalar fields and the Hubble tension: a resolution with novel signatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GF6SFNEB}},
  note         = {Machine review of arXiv:1908.06995}
}

abstract

We present a detailed investigation of a sub-dominant oscillating scalar field ('early dark energy', EDE) in the context of resolving the Hubble tension. Consistent with earlier work, but without relying on fluid approximations, we find that a scalar field frozen due to Hubble friction until ${\rm log}_{10}(z_c)\sim3.5$, reaching $\rho_{\rm EDE}(z_c)/\rho_{\rm tot}\sim10$%, and diluting faster than matter afterwards can bring cosmic microwave background (CMB), baryonic acoustic oscillations, supernovae luminosity distances, and the late-time estimate of the Hubble constant from the SH0ES collaboration into agreement. A scalar field potential which scales as $V(\phi) \propto \phi^{2n}$ with $2\lesssim n\lesssim 3.4$ around the minimum is preferred at the 68% confidence level, and the {\em Planck} polarization places additional constraints on the dynamics of perturbations in the scalar field. In particular, the data prefers a potential which flattens at large field displacements. An MCMC analysis of mock data shows that the next-generation CMB observations (i.e., CMB-S4) can unambiguously detect the presence of the EDE at very high significance. This projected sensitivity to the EDE dynamics is mainly driven by improved measurements of the $E$-mode polarization. We also explore new observational signatures of EDE scalar field dynamics: (i) We find that depending on the strength of the tensor-to-scalar ratio, the presence of the EDE might imply the existence of isocurvature perturbations in the CMB. (ii) We show that a strikingly rapid, scale-dependent growth of EDE field perturbations can result from parametric resonance driven by the anharmonic oscillating field for $n\approx 2$. This instability and ensuing potentially nonlinear, spatially inhomogenoues, dynamics may provide unique signatures of this scenario.

Figures

Figures reproduced from arXiv: 1908.06995 by the authors.

**Figure 1.** Contours of constant log10fEDE(zc) (vertical/solid) and log10zc (horizontal/dashed) as a function of the axion mass, m, and decay constant, f. The red lines show the contours for n = 2 and the black for n = 3. Since H0 = 100h km/s/Mpc = 2.13h × 10−33 eV the mass parameter of the potential that helps to resolve the Hubble tension ranges between 10−28 eV . m . 10−26 eV and 0.01 . f /Mpl . 1. |Vn,φφ| so that m2n [PITH… view at source ↗

**Figure 2.** The evolution of the fraction of the total energy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Posterior distributions of the cosmological parameters reconstructed from a run to all data (including [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: 2D posterior distribution of a subset of parameters in the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Reconstructed 1D posterior of H0 and fEDE(zc). We compare the results with (blue) and without (red) high-` TT,TE,EE data, as well as keeping Θi free (full lines) and enforcing Θi = 0.1, i.e., the power-law case (dashed lines). potential, explains why that study could n…

**Figure 6.** Figure 6: Power spectrum residuals between the best-fit ΛCDM and various best-fit EDE cosmologies with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: The fraction of the total energy density in the EDE [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Effective sound speed from Eq. (12) for an EDE with n = 3, log10(zc) = 3.5 and Θi = 0.1 (top panel) or Θi = 2.8 (bottom panel). The blue shaded region show the range of k within the horizon having c 2 s < 0.9 around zc [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: The range of k within the horizon having c 2 s < 0.9 at zc as a function of Θi [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 11.** Figure 11: 1D Posterior distributions of H0 and ωcdm reconstructed from a fit to simulated Planck data (dashed lines) and CMB-S4 (full lines) in either the ΛCDM (blue) or EDE (red) cosmology. The fiducial model has {H0 = 72 km/s/Mpc, ωcdm = 0.1293}. A. Isocurvature perturbation…

**Figure 10.** Figure 10: 2D Posterior distributions of {log10(zc), fEDE(zc)} and {H0, fEDE(zc)} reconstructed from a fit to simulated Planck data and CMB-S4. The fiducial model has {H0 = 72 km/s/Mpc, fEDE(zc)= 0.115, log10(zc)= 3.53}. IV. NEW SIGNATURES AND OBSERVATIONAL CONSEQUENCES In this …

**Figure 12.** Figure 12: The evolution of the isocurvature (i.e., homo [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: The standard adiabatic (blue) and EDE-isocurvature power spectra for [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

**Figure 14.** Figure 14: The shape of the integral of the growth ratio as [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

**Figure 16.** Figure 16: Top: The dimensionless power spectrum of the field for n = 2, Θi = 2.4, zc = 104 and fEDE(zc) = 0.1 obtained using CLASS. The resonant wavenumber becomes non-linear only at late times when the fractional energy density in the field is approximately 10−3 . Bottom: The…

**Figure 15.** Figure 15: The resonant wavenumber as a function of [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]

**Figure 17.** Figure 17: Analytic and numerical evolution of several adi [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]

**Figure 18.** Figure 18: The essential features of the maximum Floquet [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]

**Figure 19.** Figure 19: The evolution of the perturbation as a function [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]

**Figure 20.** Figure 20: The Floquet chart for V (φ) = m2 f 2 [1 − cos(φ/f)]2 . The left panel shows a broader range of field values and wavenumbers, including the large field amplitude instabity band φ/f & 1. The zoom in near the origin is the band structure for φ/f 1, that is for V (φ) = (m…

**Figure 21.** Figure 21: The Floquet chart for V (φ) = m2 f 2 [1 − cos(φ/f)]n where n = 2.5. Compare with the case with n = 2 in [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]

**Figure 22.** Figure 22: 2D posterior distribution of a subset of parameters in the [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]

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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.
hi_class: Background Evolution, Initial Conditions and Approximation Schemes
astro-ph.CO 2019-09 accept novelty 6.0 of 10

hi_class now evolves the background from covariant Horndeski Lagrangians, provides consistent radiation-era initial conditions for scalar perturbations, and applies a quasi-static approximation automatically where it is safe.
The Hubble Hunter's Guide
astro-ph.CO 2019-08 conditional novelty 6.0 of 10

The least unlikely fixes to the Hubble tension raise the expansion rate in the decade before recombination, and these fixes generically leave oscillatory residuals in CMB spectra that resemble Planck's A_L and matter-...
Background-level reconstruction of scalar-field potentials from dark-energy histories and comparison with analytic potential families
astro-ph.CO 2026-03 conditional novelty 5.5 of 10

A background reconstruction maps prescribed ρ_de(z) histories to V(φ) and ranks analytic potentials by Bayesian evidence, with exponential preferred for CPL and shifted-tanh for sign-switching targets.
The $H_0$ World Cup. I. Summary of the baseline group stage results
astro-ph.CO 2026-07 conditional novelty 5.0 of 10

In a systematic head-to-head analysis, early dark energy and early modified gravity models reduce the Hubble tension to about 3σ and are favored over ΛCDM, while radiation and late-time alternatives are not.
Can the universe experience an AdS landscape since matter-radiation equality?
gr-qc 2025-06 conditional novelty 5.0 of 10

A universe with an AdS (negative cosmological constant) phase at recombination and another at low redshift is compatible with Planck, DESI, Pantheon Plus and SH0ES data, though not preferred by them.
Can the sound horizon-free measurement of $H_0$ constrain early new physics?
astro-ph.CO 2025-01 conditional novelty 5.0 of 10

Mock galaxy-survey analyses show that sound-horizon-free H0 measurements do not currently rule out BAO-compatible early dark energy models, and LambdaCDM-derived priors can bias the result.
Hubble tension: a short review of theoretical explanations
astro-ph.CO 2026-07 accept novelty 4.0 of 10

A comprehensive review finds no theoretical Hubble-tension solution yet passes all consistency tests; new early-dark-energy chains reach high H0 only when the SH0ES calibration is added.
Imprint of swampland-inspired coupled early dark energy
astro-ph.CO 2026-07 conditional novelty 4.0 of 10

A swampland-inspired DM-EDE coupling is tested against DESI DR2 BAO data, showing the EDE potential construction affects late-time dark energy constraints.
The Hubble tension: A decade review
astro-ph.CO 2026-06 unverdicted novelty 3.0 of 10

Pure early or late fixes to the Hubble tension are tightly constrained; remaining options are combined early-late interacting dark energy or new physics at the local-to-homogeneous transition.
Ultralight fuzzy dark matter review
astro-ph.CO 2025-07 unverdicted novelty 1.0 of 10

A review of fuzzy dark matter that surveys the models, wave phenomenology, numerical methods, and current observational mass constraints of ultralight dark matter.

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89 extracted references · 5 canonical work pages · cited by 21 Pith papers

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V. Poulin, K. K. Boddy, S. Bird, and M. Kamionkowski, Phys. Rev. D97, 123504 (2018), arXiv:1803.02474 [astro- ph.CO]

arXiv 2018
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Instead, the small-scale polarization measurements place a tight constraint on the initial ﬁeld displacement, Θi

Temperature-vs-polarization data Relative to several previous attempts at resolving the Hubble tension the EDE scenario presented here is not degraded when we add the small-scale Planck polariza- tion measurements. Instead, the small-scale polarization measurements place a tight constraint on the initial ﬁeld displacement, Θi. Here we explore this in deta...
[2]

First, as is demonstrated in Fig

The preference for a large initial ﬁeld displacement The initial ﬁeld value, Θi, has two main eﬀects on the EDE phenomenology. First, as is demonstrated in Fig. 2, at ﬁxed zc andfEDE(zc) the initial ﬁeld value aﬀects the asymmetry in the rise and fall of the fractional energy density contained within the EDE. In particular, smaller values of m and f requi...
[3]

Parametric resonance preliminaries Parametric resonance occurs when the eﬀective fre- quency of a harmonic oscillator varies at such a rate so as to pump energy into the oscillation. The phenomena is well-known by anyone who has been on a swing: as we pump our legs we change the moment of inertia of the pendulum and if we pump at the right rate we can in-...
[4]

S8 tension

A CLASS comparison Using our modiﬁed version of CLASS, which includes the eﬀects from self-resonance in the φ ﬁeld as well as gravitational eﬀects from other components, we can check our analytic estimates for the resonant wavenumbers as well as growth-rate of perturbations. First, we have con- ﬁrmed that for n & 2 (but not too close to n = 2), the pertur...

2019
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A detailed instability analysis of parametric resonance in power law potentials Vn∝ φ2n in an expanding uni- verse was carried out in Ref

Analytic approximations, general n. A detailed instability analysis of parametric resonance in power law potentials Vn∝ φ2n in an expanding uni- verse was carried out in Ref. [19] 9. In that work, the Floquet exponents as a function of wavenumber and am- plitude were provided for diﬀerentn. We quote the main results necessary here without re-deriving them...
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ﬂow-lines

n = 2 case and Floquet charts We have performed an analysis for the n = 2 case for two reasons. First, the growth of perturbations due to parametric resonance discussed in Sec. IV B is strongest in this case. Second, this case is particularly compelling, given that the ﬁeld evolves with a potential V = λφ4/4 around its minimum, which has been well studied...
[7]

III C and run a MCMC analysis with ﬂat priors on {ωb,ω cdm,θs,As,ns,τ reio,f EDE(zc), log10(zc), Θi} and setting n = 2

Current constraints to n = 2 We perform the same analysis as in Sec. III C and run a MCMC analysis with ﬂat priors on {ωb,ω cdm,θs,As,ns,τ reio,f EDE(zc), log10(zc), Θi} and setting n = 2. We include all previously mentioned datasets and compare the use of high- 𝓁 TT and TT,TE,EE data. Our results are reported in Table VII together with the ∆χ2 min. We sh...

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Pith tools

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

[1] [16]

Poulin, K

V. Poulin, K. K. Boddy, S. Bird, and M. Kamionkowski, Phys. Rev. D97, 123504 (2018), arXiv:1803.02474 [astro- ph.CO]

arXiv 2018

[2] [1]

Instead, the small-scale polarization measurements place a tight constraint on the initial ﬁeld displacement, Θi

Temperature-vs-polarization data Relative to several previous attempts at resolving the Hubble tension the EDE scenario presented here is not degraded when we add the small-scale Planck polariza- tion measurements. Instead, the small-scale polarization measurements place a tight constraint on the initial ﬁeld displacement, Θi. Here we explore this in deta...

[3] [2]

First, as is demonstrated in Fig

The preference for a large initial ﬁeld displacement The initial ﬁeld value, Θi, has two main eﬀects on the EDE phenomenology. First, as is demonstrated in Fig. 2, at ﬁxed zc andfEDE(zc) the initial ﬁeld value aﬀects the asymmetry in the rise and fall of the fractional energy density contained within the EDE. In particular, smaller values of m and f requi...

[4] [3]

Parametric resonance preliminaries Parametric resonance occurs when the eﬀective fre- quency of a harmonic oscillator varies at such a rate so as to pump energy into the oscillation. The phenomena is well-known by anyone who has been on a swing: as we pump our legs we change the moment of inertia of the pendulum and if we pump at the right rate we can in-...

[5] [4]

S8 tension

A CLASS comparison Using our modiﬁed version of CLASS, which includes the eﬀects from self-resonance in the φ ﬁeld as well as gravitational eﬀects from other components, we can check our analytic estimates for the resonant wavenumbers as well as growth-rate of perturbations. First, we have con- ﬁrmed that for n & 2 (but not too close to n = 2), the pertur...

2019

[6] [5]

A detailed instability analysis of parametric resonance in power law potentials Vn∝ φ2n in an expanding uni- verse was carried out in Ref

Analytic approximations, general n. A detailed instability analysis of parametric resonance in power law potentials Vn∝ φ2n in an expanding uni- verse was carried out in Ref. [19] 9. In that work, the Floquet exponents as a function of wavenumber and am- plitude were provided for diﬀerentn. We quote the main results necessary here without re-deriving them...

[7] [6]

ﬂow-lines

n = 2 case and Floquet charts We have performed an analysis for the n = 2 case for two reasons. First, the growth of perturbations due to parametric resonance discussed in Sec. IV B is strongest in this case. Second, this case is particularly compelling, given that the ﬁeld evolves with a potential V = λφ4/4 around its minimum, which has been well studied...

[8] [7]

III C and run a MCMC analysis with ﬂat priors on {ωb,ω cdm,θs,As,ns,τ reio,f EDE(zc), log10(zc), Θi} and setting n = 2

Current constraints to n = 2 We perform the same analysis as in Sec. III C and run a MCMC analysis with ﬂat priors on {ωb,ω cdm,θs,As,ns,τ reio,f EDE(zc), log10(zc), Θi} and setting n = 2. We include all previously mentioned datasets and compare the use of high- 𝓁 TT and TT,TE,EE data. Our results are reported in Table VII together with the ∆χ2 min. We sh...

work page arXiv